Abstract:
A method and control system for an aircraft engine comprising a gas turbine driving a fan propeller with a mechanical gear-train and a dedicated pitch change mechanism for the fan propeller includes a fuel flow signal input; a pitch change mechanism signal input; a controlled plant for relating a pitch change mechanism pitch angle (BetaP) and a fuel flow (Wf) to at least two controlled outputs and a set of constraints. A decoupling control decoupling the controlled plant and/or the constraints into two separate single-input single-output (SISO) control loops for the first and second controlled outputs and a decoupling control decoupling the constraints from the decoupled controlled outputs and the constraints from one another provide gas turbine and fan propeller coordinate control while coordinately controlling constraints and outputs. A feedforward control can compensate the load change effect on engine speed and fan propeller rotor speed control.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
       [0001]    This non-provisional application claims the benefit of priority under 35 U.S.C. §119(e) to U.S. Provisional Patent Application No. 62/057595, entitled “METHOD AND SYSTEM FOR MODEL BASED CONTROL FOR VARIABLE PITCH FAN ENGINES AND TURBO-SHAFT, TURBO-PROPELLER ENGINES”, filed Sep. 30, 2014, which is herein incorporated in its entirety by reference. 
     
    
     BACKGROUND 
       [0002]    The current disclosure pertains to a control system for variable pitch fan engines and turbo-shaft, turbo-propeller engines. 
         [0003]    In some fan engines (also known as “propfan” engines), the axis of the fan propeller is parallel to or coaxial with the axis of the gas engine. Typically, in a turbo-shaft, turbo-propeller engine, the axis of one or more propellers will be perpendicular to the axis of the gas engine. In both configurations, the fan or propeller may have a fixed pitch or a variable pitch. If the pitch is variable, the engine may also have a dedicated pitch change mechanism (PCM). The propeller speed (Nr) is proportional to the gas engine power turbine shaft speed (N1) via a pure mechanical gear-train transformation, that is, Nr=Kgb*N1 where Kgb is a constant that represents the gear ratio. Controlling the fan or propeller speed, Nr, is equivalent to controlling the power turbine speed. Generally, the thrust is scheduled to be a function of propeller speed Nr, and also a function of either gas engine high pressure (HP) turbine shaft speed (N2) or engine pressure ratio (EPR). The primary challenge is to coordinate primary control of the propeller speed (Nr), the HP turbine shaft speed (N1), and any PCM pitch angle while maintaining a set of active constraints including but not limited to core pressure (Px), exhaust temperature (T), core speed rate (N2dot), and/or torque (Tq) to stay with defined limits, including but not limited to load change while rejecting external disturbances and/or internal known disturbances including but not limited to variable bleed valves and variable stator vanes. 
         [0004]    There remains a need for a systematic control method and a complete control system for a variable pitch fan engine or a turbo-shaft, turbo propeller engine that controls the fan or propeller and the engine coordinately while accommodating the active constraints. 
       BRIEF DESCRIPTION OF THE INVENTION 
       [0005]    In one aspect, an embodiment relates to a control system for an aircraft engine comprising a gas turbine driving a fan propeller with a mechanical gear-train and a dedicated pitch change mechanism for the fan propeller. The control system comprises: a fuel flow signal input; and a pitch change mechanism signal input; a controlled plant for relating a pitch change mechanism pitch angle (BetaP) from the pitch change mechanism signal input and a fuel flow (Wf) fuel flow signal input to at least two controlled outputs and at least one constraint and at least one known internal disturbance input; a decoupling control for decoupling the controlled plant into two separate single-input single-output (SISO) control loops for the first and second controlled outputs and a decoupling control for decoupling the constraints from the decoupled controlled outputs and the constraints from one another such that each of constraints is separated into a SISO control loop. The first one of the controlled outputs is either propeller speed (Nr) or power turbine shaft speed (N1) and a second one of the controlled outputs is engine core speed (N2), engine pressure ratio (EPR) or engine torque (Tq). The constraints may be core speed rate (N2dot), core pressure (Px), exhaust temperature (T) and engine torque (Tq). 
         [0006]    In another aspect, an embodiment relates to a method of controlling an aircraft engine comprising a gas turbine driving a fan propeller with mechanical gear-train, a fuel actuator, a pitch change mechanism actuator. The method comprises: receiving a fuel flow signal; receiving a pitch change mechanism signal; relating in a controlled plant a pitch change mechanism pitch angle (BetaP) from the pitch change mechanism signal and a fuel flow (Wf) fuel flow signal to at least two controlled outputs; decoupling the controlled plant into two separate single-input single-output (SISO) control loops for the first and second controlled outputs; and decoupling the constraints from the decoupled controlled outputs and the constraints from one another such that each of constraints is separated into a SISO control loop. A first one of the controlled outputs is either propeller speed (Nr) or power turbine shaft speed (N1) and a second one of the controlled outputs is engine core speed (N2), engine pressure ratio (EPR) or engine torque (Tq). The constraints may be core speed rate (N2dot), core pressure (Px), exhaust temperature (T) and engine torque (Tq). 
     
    
     
       BRIEF DESCRIPTION OF THE FIGURES 
         [0007]    The technology described herein may be best understood by reference to the following description taken in conjunction with the accompanying drawing figures in which: 
           [0008]      FIG. 1A  is a block diagram representation of a primary control architecture for an variable pitch fan engine or a turbo shaft turboprop engine for an aircraft in flight. 
           [0009]      FIG. 1B  is a block diagram representation of a primary control architecture for a variable pitch fan engine or a turbo shaft turboprop engine either in thrust reverse or for an aircraft on ground. 
           [0010]      FIG. 1C  shows block diagram representations of feed-forward control actions for a primary control architecture for a variable pitch fan engine or a turbo shaft turboprop engine. 
           [0011]      FIG. 1D  shows block diagram representations of the pitch angle servo control for a primary control architecture for a variable pitch fan engine or a turbo shaft turboprop engine. 
           [0012]      FIG. 1E  is a block diagram representation of the engine fuel flow servo control for a primary control architecture for a variable pitch fan engine or a turbo shaft turboprop engine. 
           [0013]      FIG. 2  is a block diagram representation of the constraint decoupling control architecture for a variable pitch fan engine or a turbo shaft turboprop engine. 
           [0014]      FIG. 3  is a block diagram representation of the engine-to-engine speed phase synchronization architecture for an aircraft with multiple variable pitch fan engines or turbo shaft turboprop engines. 
       
    
    
     DETAILED DESCRIPTION 
       [0015]    The current disclosure provides a control system where the fan or propeller (fan propeller) and the gas engine are treated as a single controlled plant. The control system architecture includes all outputs and constraints to be controlled, considers known disturbances rejection and is robust to drastic changes in the references. It follows that the current disclosure provides a complete, systematic but simple method and system as a holistic and separatable solution for the control of variable pitch fan engines and turbo-shaft, turbo propeller engines, whether ducted or unducted. 
         [0016]    The basic control system architecture for an embodiment of a variable pitch fan engine or turbo-shaft, turbo-propeller engine is presented in  FIGS. 1A and 1B .  FIG. 1A  presents a control system  10  for an aircraft in flight and  FIG. 1B  presents a control system architecture  12  for an aircraft on the ground or in thrust reverse. 
         [0017]    The control inputs are fuel flow (Wf)  20  from the fuel actuator (integrated into the fuel flow servo control  62 ) and PCM pitch angle (BetaP)  22  from the PCM actuator (integrated into the PCM pitch angle servo control  64 ). The other variable geometries (VG) are considered as known disturbance inputs. One of the controlled outputs,  24  may be either the propeller speed (Nr) or the power turbine shaft speed (N1) based on the relationship Nr=Kgb*N1 and denoted as Nx. A second controlled output,  26  may be any of the engine core speed (N2), engine pressure ratio (EPR) and engine torque (Tq). For clarity and brevity, the controlled outputs,  24 ,  26  presented herein for the following formulation are Nx (first controlled output  24 ) and N2 (second controlled output  26 ). Typical constraints for the control methodology may include minimum and maximum limits such as, but not limited to: minimum pressure limit (MinPx), maximum pressure limit (MaxPx), maximum temperature limit (MaxT), maximum torque limit (MaxTq), minimum speed rate limit (MinN2dot), and maximum speed rate limit (MaxN2dot). 
         [0018]    The controlled outputs  24 ,  26  N2 and Nx form the basis of feedback loops in the control system architectures  10 ,  12 . These feedback signals are combined with shaped (or filtered references) denoted as N2Ref and NxRef. The combination of the feedback signals and the shaped references N2Ref and NxRef form tracking error signals. The tracking error signals may go through reference tracking SISO controls then be combined with feedforward control actions that result from accounting for the effects of aerodynamic loading changes on the controlled outputs  24 ,  26  Nx and N2. The combination of the reference tracking SISO control outputs and the feedforward controls forms the pseudo-inputs {dot over (v)}  30 ,  32 . Application of selection logic  48  for selecting the most demanding input from the pseudo-inputs results from constraint decoupling control and a controlled output tracking control. The pseudo-inputs resulting from constraint decoupling control may replace at least one of the pseudo inputs  30 ,  32  and form the inputs for the primary decoupling control  34 . The output of the primary decoupling control forms the basis of the control input commands. The control input commands feed the fuel flow servo control  62  and the PCM pitch angle servo control  64  along with the controlled plant  28  that generates the controlled outputs  24 ,  26  and controlled constraints  50 . 
         [0019]    A controlled plant  28  comprises functional elements that describe the relationships between integrators, actuators and engine dynamics. According to the system presented herein, elements of the actuators may include first or second order transfer functions. Without loss of generality, the actuators may be considered as first order lag devices in the formulation presented herein. It follows that the control system states comprise the engine dynamic states, the actuator states and the integrator states. 
         [0020]    The controlled plant  28 , that is, the controlled portion or process of the control system  10 ,  12 , includes the states (x), the control inputs (u), the disturbance inputs (d), the outputs (y) and the constraints (y c ) according to x=[N 2  N x  Wf β p  Wf l  β pl ] T , {dot over (u)}=[{dot over (W)} f  {dot over (β)} p ] T , d=[VG 1  VG 2  VG 3  VG 4  VG 5 ] T , y=[N 2  N x ] T , and y c =[P x  T T q  Ndot] T . 
         [0021]    Assume the original controlled plant  28  is represented with the following equations: 
         [0000]        x   k+1   =f ( x   k   , {dot over (u)}   k   , d   k )   (Eq. 1a)
 
         [0000]        y   k   =h ( x   k   , {dot over (u)}   k   , d   k )   (Eq. 1b)
 
         [0000]        y   c,k   =h   c ( x   k   , {dot over (u)}   k   , d   k )   (Eq. 1c)
 
         [0022]    At sample k, the system states, x k , the inputs {dot over (u)} k−1 , the disturbances d k  and the outputs y k   − =h(x k , {dot over (u)} k−1 , d k )are known. Thus, the deviation variables are expressed about the current operating condition. 
         [0023]    Define the deviation variables from the conditions: 
         [0000]        {tilde over (x)}   j   =x   j   −x   k   , {dot over (ũ)}   j   ={dot over (u)}   j   −{dot over (u)}   k−1   , {tilde over (d)}   j   =d   j   −d   k   , {tilde over (y)}   j   =y   j   −y   k   − , and  {tilde over (y)}   c,j   =y   c,j   −y   c,k   −   
         [0024]    The local linearized model of the system, in terms of deviation variables, may be derived as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
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         [0000]    where F k is the free response over one sample and 
         [0000]        F   k   =f ( x   k   , {dot over (u)}   k−1   , d   k )− x   k   =x   k+1   −   −x   k  
 
         [0025]    For the states associated with engine dynamics and actuator dynamics where the state update is not explicitly dependent on the input, that is, there exist: 
         [0000]        x   k+1   =f ( x   k   , {dot over (u)}   k   , d   k )= x   k+1   −   =f ( x   k   , {dot over (u)}   k−1   , d   k ) 
         [0000]      and 
         [0000]        x   k   =f ( x   k−1   , {dot over (u)}   k−1   , d   k−1 )= x   k   −   =f ( x   k−1   , {dot over (u)}   k−2   , d   k−1 ). 
         [0000]      Therefore, 
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         [0000]        F   a,k   =x   a,k+1   −   −x   a,k   =x   a,k+1   −x   a,k   ≈x   a,k   −x   a,k−1 . 
         [0000]    For integrators, F i,k =x i,k+1 −x i,k =T s ·{dot over (u)} R,k−1 , where T s  is the sampling time. For the controlled outputs and constraints, 
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         [0026]    Using the inputs of the integrators as the control inputs for the controlled plant, D u =0 and D cu =0 due to the increased relative degrees introduced by actuators and integrators. The state space perturbation model for the control system is: 
         [0000]        {tilde over (x)} ( k+ 1)= A{tilde over (x)} ( k )+ B{dot over (ũ)} ( k )+ B   d   {tilde over (d)} ( k )+ F   k    (Eq. 2a)
 
         [0000]        {tilde over (y)} ( k )= C{tilde over (x)} ( k )+ F   d   {tilde over (d)} ( k )   (Eq. 2b)
 
         [0000]        {tilde over (y)} ( k )= C   c   {tilde over (x)} ( k )+ D   cd   {tilde over (d)} ( k )   (Eq. 2c)
 
         [0027]    Note that by definition, {tilde over (x)}(k)=0, {tilde over (d)}(k)=0 and {tilde over (y)} c (k)=0. Let d(k)−d(k−1)={circumflex over (d)}(k) and allow the following approximations: 
         [0000]      {tilde over (d)}(k+1)≈{circumflex over (d)}(k)
 
         [0000]      {tilde over (d)}(k+2)≈2{circumflex over (d)}(k)
 
         [0000]      {tilde over (d)}(k+3)≈3{circumflex over (d)}(k)
 
         [0028]    Turning now to the control method and system, consider the primary decoupling control  34 . Recall that the controlled outputs, y  24 ,  26  for the propeller speed and engine speed control include variables N2 (second controlled output  26 ) and Nx (first controlled output  24 ) each with relative degree 3. Therefore, 
         [0000]    
       
         
           
             
               
                 
                   
                       
                   
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                                     d 
                                     ~ 
                                   
                                   k 
                                 
                               
                             
                           
                         
                       
                     
                      
                     
                       
 
                     
                      
                     
                       
                         
                           
                             
                                 
                             
                              
                             
                               
                                 
                                   y 
                                   ~ 
                                 
                                 
                                   i 
                                   , 
                                   
                                     k 
                                     + 
                                     3 
                                   
                                 
                               
                               = 
                                 
                                
                               
                                 
                                   
                                     C 
                                     i 
                                   
                                    
                                   
                                     A 
                                     3 
                                   
                                    
                                   
                                     
                                       x 
                                       ~ 
                                     
                                     k 
                                   
                                 
                                 + 
                                 
                                   
                                     C 
                                     i 
                                   
                                    
                                   
                                     A 
                                     2 
                                   
                                    
                                   B 
                                    
                                   
                                     
                                       
                                         u 
                                         . 
                                       
                                       ~ 
                                     
                                     k 
                                   
                                 
                                 + 
                                 
                                   
                                     C 
                                     i 
                                   
                                    
                                   
                                     A 
                                     2 
                                   
                                    
                                   
                                     B 
                                     d 
                                   
                                    
                                   
                                     
                                       d 
                                       ~ 
                                     
                                     k 
                                   
                                 
                                 + 
                                 
                                   
                                     C 
                                     i 
                                   
                                    
                                   
                                     AB 
                                     d 
                                   
                                    
                                   
                                     
                                       d 
                                       ~ 
                                     
                                     
                                       k 
                                       + 
                                       1 
                                     
                                   
                                 
                                 + 
                               
                             
                           
                         
                       
                       
                         
                           
                               
                              
                             
                               
                                 
                                   C 
                                   i 
                                 
                                  
                                 
                                   B 
                                   d 
                                 
                                  
                                 
                                   
                                     d 
                                     ~ 
                                   
                                   
                                     k 
                                     + 
                                     2 
                                   
                                 
                               
                               + 
                               
                                 
                                   D 
                                   di 
                                 
                                  
                                 
                                   
                                     d 
                                     ~ 
                                   
                                   
                                     k 
                                     + 
                                     3 
                                   
                                 
                               
                               + 
                               
                                 
                                   
                                     C 
                                     i 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         A 
                                         2 
                                       
                                       + 
                                       A 
                                       + 
                                       I 
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   F 
                                   k 
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             = 
                               
                              
                             
                               
                                 
                                   E 
                                   i 
                                 
                                  
                                 
                                   
                                     
                                       u 
                                       . 
                                     
                                     ~ 
                                   
                                   k 
                                 
                               
                               + 
                               
                                 
                                   ( 
                                   
                                     
                                       
                                         C 
                                         i 
                                       
                                        
                                       
                                         AB 
                                         d 
                                       
                                     
                                     + 
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       
                                         C 
                                         i 
                                       
                                        
                                       
                                         B 
                                         d 
                                       
                                     
                                     + 
                                     
                                       3 
                                        
                                       
                                           
                                       
                                        
                                       
                                         D 
                                         di 
                                       
                                     
                                   
                                   ) 
                                 
                                  
                                 
                                   
                                     d 
                                     ~ 
                                   
                                   k 
                                 
                               
                               + 
                               
                                 
                                   
                                     C 
                                     i 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         A 
                                         2 
                                       
                                       + 
                                       A 
                                       + 
                                       I 
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   F 
                                   k 
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             = 
                               
                              
                             
                               
                                 
                                   K 
                                   px 
                                 
                                  
                                 
                                   F 
                                   k 
                                 
                               
                               + 
                               
                                 
                                   E 
                                   i 
                                 
                                  
                                 
                                   
                                     
                                       u 
                                       . 
                                     
                                     ~ 
                                   
                                   k 
                                 
                               
                               + 
                               
                                 
                                   K 
                                   pd 
                                 
                                  
                                 
                                   
                                     d 
                                     ^ 
                                   
                                   k 
                                 
                               
                             
                           
                         
                       
                     
                      
                     
                       
 
                     
                      
                     
                         
                     
                      
                     
                       
                         This 
                          
                         
                             
                         
                          
                         is 
                       
                       , 
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     3 
                   
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         y 
                         ~ 
                       
                       
                         i 
                         , 
                         
                           k 
                           + 
                           3 
                         
                       
                     
                     = 
                     
                       
                         
                           K 
                           px 
                         
                          
                         
                           F 
                           k 
                         
                       
                       + 
                       
                         E 
                          
                         
                           
                             
                               u 
                               . 
                             
                             ~ 
                           
                           k 
                         
                       
                       + 
                       
                         
                           K 
                           pd 
                         
                          
                         
                           
                             d 
                             ^ 
                           
                           k 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     4 
                   
                   ) 
                 
               
             
           
         
       
     
         [0029]    The input-output mapping or decoupling matrix between the control inputs and controlled outputs is: 
         [0000]    
       
         
           
             E 
             = 
             
               
                 [ 
                 
                   
                     
                       
                         
                           C 
                           1 
                         
                          
                         
                           A 
                           2 
                         
                          
                         B 
                       
                     
                   
                   
                     
                       
                         
                           C 
                           2 
                         
                          
                         
                           A 
                           2 
                         
                          
                         B 
                       
                     
                   
                 
                 ] 
               
               = 
               
                 [ 
                 
                   
                     
                       
                         e 
                         11 
                       
                     
                     
                       
                         e 
                         12 
                       
                     
                   
                   
                     
                       
                         e 
                         21 
                       
                     
                     
                       
                         e 
                         22 
                       
                     
                   
                 
                 ] 
               
             
           
         
       
     
         [0000]    By introducing the pseudo-input {dot over (v)}  30 ,  32  and letting {tilde over (y)} i,k+3 ={dot over (v)} i , the input-output relation is to be shaped into the desired dynamics in the continuous-time domain as: 
         [0000]    
       
         
           
             
               
                 
                   y 
                   ~ 
                 
                 i 
               
               
                 
                   v 
                   . 
                 
                 i 
               
             
             = 
             
               1 
               
                 s 
                 · 
                 
                   ( 
                   
                     
                       
                         T 
                         
                           
                             pd 
                              
                             
                                 
                             
                              
                             1 
                           
                           , 
                           i 
                         
                       
                        
                       s 
                     
                     + 
                     1 
                   
                   ) 
                 
                 · 
                 
                   ( 
                   
                     
                       
                         T 
                         
                           
                             pd 
                              
                             
                                 
                             
                              
                             2 
                           
                           , 
                           i 
                         
                       
                        
                       s 
                     
                     + 
                     1 
                   
                   ) 
                 
               
             
           
         
       
     
         [0000]    Equivalently, in the discrete-time domain with sampling time, T s : 
         [0000]    
       
         
           
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         
                           
                             
                               y 
                               ~ 
                             
                             i 
                           
                            
                           
                             ( 
                             z 
                             ) 
                           
                         
                         
                           
                             
                               v 
                               . 
                             
                             i 
                           
                            
                           
                             ( 
                             z 
                             ) 
                           
                         
                       
                       = 
                       
                         
                           ( 
                           
                             
                               1 
                               - 
                               
                                 λ 
                                 
                                   
                                     pd 
                                      
                                     
                                         
                                     
                                      
                                     1 
                                   
                                   , 
                                   i 
                                 
                               
                             
                             
                               z 
                               - 
                               
                                 λ 
                                 
                                   
                                     pd 
                                      
                                     
                                         
                                     
                                      
                                     1 
                                   
                                   , 
                                   i 
                                 
                               
                             
                           
                           ) 
                         
                          
                         
                           ( 
                           
                             
                               1 
                               - 
                               
                                 λ 
                                 
                                   
                                     pd 
                                      
                                     
                                         
                                     
                                      
                                     2 
                                   
                                   , 
                                   i 
                                 
                               
                             
                             
                               z 
                               - 
                               
                                 λ 
                                 
                                   
                                     pd 
                                      
                                     
                                         
                                     
                                      
                                     2 
                                   
                                   , 
                                   i 
                                 
                               
                             
                           
                           ) 
                         
                          
                         
                           ( 
                           
                             
                               T 
                               s 
                             
                             
                               z 
                               - 
                               1 
                             
                           
                           ) 
                         
                       
                     
                     , 
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     5 
                   
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         λ 
                         
                           
                             pd 
                              
                             
                                 
                             
                              
                             1 
                           
                           , 
                           i 
                         
                       
                       = 
                       
                          
                         
                           - 
                           
                             Ts 
                             
                               T 
                               
                                 
                                   pd 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                                 , 
                                 i 
                               
                             
                           
                         
                       
                     
                     , 
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         λ 
                         
                           
                             pd 
                              
                             
                                 
                             
                              
                             2 
                           
                           , 
                           i 
                         
                       
                       = 
                       
                          
                         
                           - 
                           
                             Ts 
                             
                               T 
                               
                                 
                                   pd 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                 
                                 , 
                                 i 
                               
                             
                           
                         
                       
                     
                     , 
                     and 
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       
                         y 
                         ~ 
                       
                       
                         i 
                         , 
                         
                           k 
                           + 
                           3 
                         
                       
                     
                     - 
                     
                       
                         ( 
                         
                           1 
                           + 
                           
                             λ 
                             
                               
                                 pd 
                                  
                                 
                                     
                                 
                                  
                                 1 
                               
                               , 
                               i 
                             
                           
                           + 
                           
                             λ 
                             
                               
                                 pd 
                                  
                                 
                                     
                                 
                                  
                                 2 
                               
                               , 
                               i 
                             
                           
                         
                         ) 
                       
                        
                       
                         
                           y 
                           ~ 
                         
                         
                           i 
                           , 
                           
                             k 
                             + 
                             2 
                           
                         
                       
                     
                     + 
                     
                       
                         ( 
                         
                           
                             λ 
                             
                               
                                 pd 
                                  
                                 
                                     
                                 
                                  
                                 1 
                               
                               , 
                               i 
                             
                           
                           + 
                           
                             λ 
                             
                               
                                 pd 
                                  
                                 
                                     
                                 
                                  
                                 2 
                               
                               , 
                               i 
                             
                           
                           + 
                           
                             λ 
                             
                               
                                 pd 
                                  
                                 
                                     
                                 
                                  
                                 1 
                               
                               , 
                               i 
                             
                           
                           + 
                           
                             λ 
                             
                               
                                 pd 
                                  
                                 
                                     
                                 
                                  
                                 2 
                               
                               , 
                               i 
                             
                           
                         
                         ) 
                       
                        
                       
                         
                           y 
                           ~ 
                         
                         
                           i 
                           , 
                           
                             k 
                             + 
                             1 
                           
                         
                       
                     
                     - 
                     
                       
                         λ 
                         
                           
                             pd 
                              
                             
                                 
                             
                              
                             1 
                           
                           , 
                           i 
                         
                       
                        
                       
                         λ 
                         
                           
                             pd 
                              
                             
                                 
                             
                              
                             2 
                           
                           , 
                           i 
                         
                       
                        
                       
                         
                           y 
                           ~ 
                         
                         
                           i 
                           , 
                           k 
                         
                       
                     
                   
                   = 
                   
                     
                       
                         
                           v 
                           ~ 
                         
                         
                           i 
                           , 
                           k 
                         
                       
                        
                       
                         ( 
                         
                           1 
                           - 
                           
                             λ 
                             
                               
                                 pd 
                                  
                                 
                                     
                                 
                                  
                                 1 
                               
                               , 
                               i 
                             
                           
                         
                         ) 
                       
                     
                      
                     
                       ( 
                       
                         λ 
                         
                           
                             pd 
                              
                             
                                 
                             
                              
                             2 
                           
                           , 
                           i 
                         
                       
                       ) 
                     
                      
                     
                       T 
                       s 
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     5 
                   
                   ) 
                 
               
             
           
         
       
     
         [0000]    Substituting for {tilde over (y)} i,k+2 , {tilde over (y)} i,k+1  and {tilde over (y)} i,k  in (Eq. 5), yields: 
         [0000]        {tilde over (y)}   i,k+3   =K   pdv,i   {dot over (v)}   i,k   +K   pdx,i   F   k   +K   pdd,i   {circumflex over (d)}   k    (Eq. 6)
 
         [0000]      That is, 
         [0000]        {tilde over (y)}   i,k+3   =K   pdv   {dot over (v)}   k   +K   pdx   F   k   +K   pdd   {circumflex over (d)}   k    (Eq. 7)
 
         [0030]    Comparing (Eq. 7) to (Eq. 4) yields: 
         [0000]      E{dot over (ũ)} k +K pdv {dot over (v)} k +(K pdx −K px )F k +(K pdd −K pd ){circumflex over (d)} k    (Eq. 8)
 
         [0000]    Therefore, the primary decoupling control  34  is: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             u 
                             
                               . 
                               ∼ 
                             
                           
                           k 
                         
                         = 
                           
                          
                         
                           
                             
                               E 
                               
                                 - 
                                 1 
                               
                             
                              
                             
                               K 
                               pdv 
                             
                              
                             
                               
                                 v 
                                 . 
                               
                               k 
                             
                           
                           + 
                           
                             
                               
                                 E 
                                 
                                   - 
                                   1 
                                 
                               
                                
                               
                                 ( 
                                 
                                   
                                     K 
                                     pdx 
                                   
                                   - 
                                   
                                     K 
                                     px 
                                   
                                 
                                 ) 
                               
                             
                              
                             
                               F 
                               k 
                             
                           
                           + 
                           
                             
                               
                                 E 
                                 
                                   - 
                                   1 
                                 
                               
                                
                               
                                 ( 
                                 
                                   
                                     K 
                                     pdd 
                                   
                                   - 
                                   
                                     K 
                                     pd 
                                   
                                 
                                 ) 
                               
                             
                              
                             
                               
                                 d 
                                 ^ 
                               
                               k 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               K 
                               V 
                             
                              
                             
                               
                                 v 
                                 . 
                               
                               k 
                             
                           
                           + 
                           
                             
                               K 
                               X 
                             
                              
                             
                               F 
                               k 
                             
                           
                           + 
                           
                             
                               K 
                               D 
                             
                              
                             
                               
                                 d 
                                 ^ 
                               
                               k 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               K 
                               V 
                             
                              
                             
                               
                                 v 
                                 . 
                               
                               k 
                             
                           
                           + 
                           
                             
                               K 
                               X 
                             
                              
                             
                               
                                 x 
                                 ^ 
                               
                               k 
                             
                           
                           + 
                           
                             
                               K 
                               D 
                             
                              
                             
                               
                                 d 
                                 ^ 
                               
                               k 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     9 
                   
                   ) 
                 
               
             
           
         
       
     
         [0000]    Consequently, the decoupled controlled plant  28  is: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             x 
                             ~ 
                           
                            
                           
                             ( 
                             
                               k 
                               + 
                               1 
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             A 
                              
                             
                               
                                 x 
                                 ~ 
                               
                                
                               
                                 ( 
                                 k 
                                 ) 
                               
                             
                           
                           + 
                           
                             
                               BK 
                               v 
                             
                              
                             
                               
                                 v 
                                 . 
                               
                                
                               
                                 ( 
                                 k 
                                 ) 
                               
                             
                           
                           + 
                           
                             
                               BK 
                               D 
                             
                              
                             
                               
                                 d 
                                 ^ 
                               
                                
                               
                                 ( 
                                 k 
                                 ) 
                               
                             
                           
                           + 
                           
                             
                               BK 
                               X 
                             
                              
                             
                               F 
                               X 
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             
                               B 
                               d 
                             
                              
                             
                               
                                 d 
                                 ~ 
                               
                                
                               
                                 ( 
                                 k 
                                 ) 
                               
                             
                           
                           + 
                           
                             F 
                             k 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             A 
                              
                             
                               
                                 x 
                                 ~ 
                               
                                
                               
                                 ( 
                                 k 
                                 ) 
                               
                             
                           
                           + 
                           
                             
                               B 
                               N 
                             
                              
                             
                               
                                 v 
                                 . 
                               
                                
                               
                                 ( 
                                 k 
                                 ) 
                               
                             
                           
                           + 
                           
                             
                               B 
                               dN 
                             
                              
                             
                               
                                 d 
                                 ^ 
                               
                                
                               
                                 ( 
                                 k 
                                 ) 
                               
                             
                           
                           + 
                           
                             
                               B 
                               d 
                             
                              
                             
                               
                                 d 
                                 ~ 
                               
                                
                               
                                 ( 
                                 k 
                                 ) 
                               
                             
                           
                           + 
                           
                             
                               K 
                               bf 
                             
                              
                             
                               F 
                               k 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     
                       Eq 
                       . 
                       
                           
                       
                        
                       10 
                     
                      
                     a 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       y 
                       ~ 
                     
                      
                     
                       ( 
                       k 
                       ) 
                     
                   
                   = 
                   
                     C 
                      
                     
                       
                         x 
                         ~ 
                       
                        
                       
                         ( 
                         k 
                         ) 
                       
                     
                      
                     
                       D 
                       d 
                     
                      
                     
                       
                         d 
                         ~ 
                       
                        
                       
                         ( 
                         k 
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     
                       Eq 
                       . 
                       
                           
                       
                        
                       10 
                     
                      
                     b 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         y 
                         ~ 
                       
                       c 
                     
                      
                     
                       ( 
                       k 
                       ) 
                     
                   
                   = 
                   
                     
                       C 
                       c 
                     
                      
                     
                       
                         x 
                         ~ 
                       
                        
                       
                         ( 
                         k 
                         ) 
                       
                     
                      
                     
                       D 
                       cd 
                     
                      
                     
                       
                         d 
                         ~ 
                       
                        
                       
                         ( 
                         k 
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     
                       Eq 
                       . 
                       
                           
                       
                        
                       10 
                     
                      
                     c 
                   
                   ) 
                 
               
             
           
         
       
     
         [0031]    When the PCM pitch angle  22  needs to follow a scheduled pitch angle reference  36 , for example, in a thrust reverse operation (as shown in  FIG. 1B ), the primary decoupling control  34  performs an online reconfiguration to reduce to a lower order system. From the original 2×2 primary decoupling control  34 , E{dot over (ũ)} k =K pdv {dot over (v)} k +(K pdx −K px )F k +(K pdd −K pd ){circumflex over (d)} k =M p,k , or, more compactly, 
         [0000]    
       
         
           
             
               
                 
                   [ 
                   
                     
                       
                         
                           e 
                           11 
                         
                       
                       
                         
                           e 
                           12 
                         
                       
                     
                     
                       
                         
                           e 
                           21 
                         
                       
                       
                         
                           e 
                           22 
                         
                       
                     
                   
                   ] 
                 
                  
                 
                   [ 
                   
                     
                       
                         
                           
                             u 
                             
                               . 
                               ∼ 
                             
                           
                           
                             1 
                             , 
                             k 
                           
                         
                       
                     
                     
                       
                         
                           
                             u 
                             
                               . 
                               ∼ 
                             
                           
                           
                             2 
                             , 
                             k 
                           
                         
                       
                     
                   
                   ] 
                 
               
               = 
               
                 [ 
                 
                   
                     
                       
                         
                           M 
                           
                             p 
                             , 
                             k 
                           
                         
                          
                         
                           ( 
                           1 
                           ) 
                         
                       
                     
                   
                   
                     
                       
                         
                           M 
                           
                             p 
                             , 
                             k 
                           
                         
                          
                         
                           ( 
                           2 
                           ) 
                         
                       
                     
                   
                 
                 ] 
               
             
             , 
           
         
       
     
         [0000]    the system separates {dot over (u)} 2  term, alternatively known as u2dot  38 , from the controlled plant  28  and treats u2dot  38  as a known disturbance input. Consequently, the system transforms the original 2×2 control into a 1×1 control for N2 (second controlled output  26 ) and Nx (first controlled output  24 ), respectively, 
         [0000]    
       
         
           
             
               i 
               . 
               e 
               . 
               
                 
                   [ 
                   
                     
                       
                         
                           e 
                           11 
                         
                       
                     
                     
                       
                         
                           e 
                           21 
                         
                       
                     
                   
                   ] 
                 
                  
                 
                   [ 
                   
                     
                       u 
                       
                         . 
                         ∼ 
                       
                     
                     
                       1 
                       , 
                       k 
                     
                   
                   ] 
                 
               
             
             = 
             
               
                 [ 
                 
                   
                     
                       
                         
                           M 
                           
                             p 
                             , 
                             k 
                           
                         
                          
                         
                           ( 
                           1 
                           ) 
                         
                       
                     
                   
                   
                     
                       
                         
                           M 
                           
                             p 
                             , 
                             k 
                           
                         
                          
                         
                           ( 
                           2 
                           ) 
                         
                       
                     
                   
                 
                 ] 
               
               - 
               
                 
                   [ 
                   
                     
                       
                         
                           e 
                           12 
                         
                       
                     
                     
                       
                         
                           e 
                           22 
                         
                       
                     
                   
                   ] 
                 
                  
                 
                   [ 
                   
                     
                       u 
                       
                         . 
                         ∼ 
                       
                     
                     
                       2 
                       , 
                       k 
                     
                   
                   ] 
                 
               
             
           
         
       
     
         [0000]    where the PCM pitch angle (BetaP)  22  determines {dot over (ũ)} 2,k  (i.e. {dot over (ũ)} 2,k  follows the reference schedule provided by BetaP). 
         [0032]    In this way, BetaP  22  does not serve as a speed tracking control input. Instead, the control system treats BetaP as a known disturbance for speed tracking Therefore, the system includes only one control input, fuel flow (Wf)  20  from the fuel actuator, to serve the two controlled outputs  24 ,  26 . 
         [0033]    Consequently, the control system  12  may then instantiate a compromise for tracking the two controlled speed outputs  24 ,  26  (i.e. N2 and Nx) by one control input (Wf). The control system  12  may select the most demanding {dot over (ũ)} 1,k  amongst {dot over (ũ)} 1,k =(1/e 11 )[M p,k (1)−e 12 *{dot over (ũ)} 2,k ] and {dot over (ũ)} 1,k =(1/e 21 )[M p,k (2)−e 22 *{dot over (ũ)} 2,k ] such that the most demanding speed tracking is satisfied and the other speed tracking is naturally fallout, and the {dot over (ũ)} 1,k  applies to two speed trackings proportionally corresponding to their dynamic relation inherited from the decoupling matrix E. The control system  12  may also apply a weighting factor to each of the two speed tracking outputs where the weighting factor is based on a relative priority of the two controlled outputs  24 ,  26  for the given operating condition. Alternatively, the control system  12  may control the higher priority speed tracking output with more effort whereby the lower priority output naturally falls out. Weighting factors add the flexibility to adjust the natural ratio, inherited from decoupling matrix E, of {dot over (ũ)} 1,k  proportionally acting on the two speed trackings for compensating potential model error in decoupling matrix E and addressing controlled outputs priority concern. Now the most demanding {dot over (ũ)} 1,k  is selected amongst the weighted {dot over (ũ)} 1,k =(1/e 11 )[M pW,k (1)−e 12 *{dot over (ũ)} 2,k ] and {dot over (ũ)} 1,k =(1/e 21 )[M pW,k (2)−e 22 *{dot over (ũ)} 2,k ]. 
         [0034]    Because the primary decoupling control  34  has decoupled the control plant  28  into two separate single-input single-output (SISO) control loops for N2 and NX and shaped their desired dynamics, respectively, the control system may include additional SISO lead/lag to enhance the N2 and NX tracking  40 ,  42  respectively, which may be expressed for v1dot  30  and v2dot  32  as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       v 
                       . 
                     
                     i 
                   
                   = 
                   
                     
                       
                         
                           
                             k 
                             pci 
                           
                            
                           
                             ( 
                             
                               1 
                               + 
                               
                                 
                                   τ 
                                   pci 
                                 
                                  
                                 s 
                               
                             
                             ) 
                           
                         
                         
                           1 
                           + 
                           
                             
                               T 
                               pci 
                             
                              
                             s 
                           
                         
                       
                        
                       
                         ( 
                         
                           
                             y 
                             Ri 
                           
                           - 
                           
                             y 
                             i 
                           
                         
                         ) 
                       
                     
                     = 
                     
                       
                         
                           
                             k 
                             pci 
                           
                            
                           
                             ( 
                             
                               1 
                               + 
                               
                                 
                                   τ 
                                   pci 
                                 
                                  
                                 s 
                               
                             
                             ) 
                           
                         
                         
                           1 
                           + 
                           
                             
                               T 
                               pci 
                             
                              
                             s 
                           
                         
                       
                        
                       
                         
                           y 
                           ^ 
                         
                         i 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     11 
                   
                   ) 
                 
               
             
           
         
       
     
         [0035]    After decoupling the plant  28  and shaping its desired dynamics, the control system  10 ,  12  determines the controlled output tracking  40 ,  42  by the performance of the following SISO control relationship: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         y 
                         ~ 
                       
                       i 
                     
                     
                       
                         y 
                         ^ 
                       
                       i 
                     
                   
                   = 
                   
                     
                       1 
                       
                         s 
                         · 
                         
                           ( 
                           
                             
                               
                                 T 
                                 
                                   
                                     pd 
                                      
                                     
                                         
                                     
                                      
                                     1 
                                   
                                   , 
                                   i 
                                 
                               
                                
                               s 
                             
                             + 
                             1 
                           
                           ) 
                         
                         · 
                         
                           ( 
                           
                             
                               
                                 T 
                                 
                                   
                                     pd 
                                      
                                     
                                         
                                     
                                      
                                     2 
                                   
                                   , 
                                   i 
                                 
                               
                                
                               s 
                             
                             + 
                             1 
                           
                           ) 
                         
                       
                     
                     · 
                     
                       
                         1 
                         + 
                         
                           
                             T 
                             pci 
                           
                            
                           s 
                         
                       
                       
                         
                           k 
                           pci 
                         
                          
                         
                           ( 
                           
                             1 
                             + 
                             
                               
                                 τ 
                                 pci 
                               
                                
                               s 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     12 
                   
                   ) 
                 
               
             
           
         
       
     
         [0036]    The control system  10 ,  12  determines the first operand of the transfer function 
         [0000]    
       
         
           
             ( 
             
               i 
               . 
               e 
               . 
               
                 1 
                 
                   s 
                   · 
                   
                     ( 
                     
                       
                         
                           T 
                           
                             
                               pd 
                                
                               
                                   
                               
                                
                               1 
                             
                             , 
                             i 
                           
                         
                          
                         s 
                       
                       + 
                       1 
                     
                     ) 
                   
                   · 
                   
                     ( 
                     
                       
                         
                           T 
                           
                             
                               pd 
                                
                               
                                   
                               
                                
                               2 
                             
                             , 
                             i 
                           
                         
                          
                         s 
                       
                       + 
                       1 
                     
                     ) 
                   
                 
               
             
             ) 
           
         
       
     
         [0000]    by the desired decoupled dynamics selection. The control system  10 ,  12  selects the second operand of the transfer function based on the desired output tracking.
 
Given N 2R −N 2  and N xR −N x , the primary control system includes known inputs {dot over (v)} 1   N     2    and {dot over (v)} 2   N     x   .
 
         [0037]    Turning now to the constraints decoupling control  44  as presented in  FIG. 2 , the control system includes the assumption that all constraints have relative degree 2, without loss of generality, based on the decoupled control plant  46 . The input-output mapping between the control pseudo-inputs (e.g. v1dot  30 , v2dot  32 ) and all of the constraint outputs  50  may be expressed as: 
         [0000]    
       
         
           
             
               
                 
                   y 
                   ~ 
                 
                 ci 
               
                
               
                 ( 
                 k 
                 ) 
               
             
             = 
             
               
                 
                   C 
                   ci 
                 
                  
                 
                   
                     x 
                     ~ 
                   
                    
                   
                     ( 
                     k 
                     ) 
                   
                 
               
               + 
               
                 
                   D 
                   cdi 
                 
                  
                 
                   
                     d 
                     ~ 
                   
                    
                   
                     ( 
                     k 
                     ) 
                   
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     
                       y 
                       ~ 
                     
                     
                       ci 
                       , 
                       
                         k 
                         + 
                         1 
                       
                     
                   
                   = 
                     
                    
                   
                     
                       
                         C 
                         ci 
                       
                        
                       A 
                        
                       
                         
                           x 
                           ~ 
                         
                         k 
                       
                     
                     + 
                     
                       
                         C 
                         ci 
                       
                        
                       
                         B 
                         N 
                       
                        
                       
                         
                           v 
                           . 
                         
                         k 
                       
                     
                     + 
                     
                       
                         C 
                         ci 
                       
                        
                       
                         ( 
                         
                           
                             
                               B 
                               dN 
                             
                              
                             
                               
                                 d 
                                 ^ 
                               
                               k 
                             
                           
                           + 
                           
                             
                               B 
                               d 
                             
                              
                             
                               
                                 d 
                                 ~ 
                               
                               k 
                             
                           
                         
                         ) 
                       
                     
                     + 
                   
                 
               
             
             
               
                 
                     
                    
                   
                     
                       
                         C 
                         ci 
                       
                        
                       
                         K 
                         bf 
                       
                        
                       
                         F 
                         k 
                       
                     
                     + 
                     
                       
                         D 
                         cdi 
                       
                        
                       
                         
                           d 
                           ^ 
                         
                         
                           k 
                           + 
                           1 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       
                         C 
                         ci 
                       
                        
                       
                         K 
                         bf 
                       
                        
                       
                         F 
                         k 
                       
                     
                     + 
                     
                       
                         ( 
                         
                           
                             
                               C 
                               ci 
                             
                              
                             
                               B 
                               dN 
                             
                           
                           + 
                           
                             D 
                             cdi 
                           
                         
                         ) 
                       
                        
                       
                         
                           d 
                           ^ 
                         
                         k 
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     
                       y 
                       ~ 
                     
                     
                       ci 
                       , 
                       
                         k 
                         + 
                         2 
                       
                     
                   
                   = 
                     
                    
                   
                     
                       
                         C 
                         ci 
                       
                        
                       
                         A 
                         2 
                       
                        
                       
                         
                           x 
                           ~ 
                         
                         k 
                       
                     
                     + 
                     
                       
                         C 
                         ci 
                       
                        
                       
                         AB 
                         N 
                       
                        
                       
                         
                           v 
                           . 
                         
                         k 
                       
                     
                     + 
                     
                       
                         C 
                         ci 
                       
                        
                       
                         AB 
                         dN 
                       
                        
                       
                         
                           d 
                           ^ 
                         
                         k 
                       
                     
                     + 
                     
                       
                         C 
                         ci 
                       
                        
                       
                         AB 
                         d 
                       
                        
                       
                         
                           d 
                           ^ 
                         
                         k 
                       
                     
                     + 
                   
                 
               
             
             
               
                 
                     
                    
                   
                     
                       
                         C 
                         ci 
                       
                        
                       
                         ( 
                         
                           
                             
                               B 
                               dN 
                             
                              
                             
                               
                                 d 
                                 ^ 
                               
                               
                                 k 
                                 + 
                                 1 
                               
                             
                           
                           + 
                           
                             
                               B 
                               d 
                             
                              
                             
                               
                                 d 
                                 ^ 
                               
                               
                                 k 
                                 + 
                                 1 
                               
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         D 
                         cdi 
                       
                        
                       
                         
                           d 
                           ^ 
                         
                         
                           k 
                           + 
                           2 
                         
                       
                     
                     + 
                     
                       
                         
                           C 
                           ci 
                         
                          
                         
                           ( 
                           
                             A 
                             + 
                             I 
                           
                           ) 
                         
                       
                        
                       
                         K 
                         bf 
                       
                        
                       
                         F 
                         k 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       
                         
                           C 
                           ci 
                         
                          
                         
                           ( 
                           
                             A 
                             + 
                             I 
                           
                           ) 
                         
                       
                        
                       
                         K 
                         bf 
                       
                        
                       
                         F 
                         k 
                       
                     
                     + 
                     
                       
                         E 
                         ci 
                       
                        
                       
                         
                           v 
                           . 
                         
                         k 
                       
                     
                     + 
                     
                       ( 
                       
                         
                           
                             C 
                             ci 
                           
                            
                           
                             B 
                             dN 
                           
                         
                         + 
                         
                           
                             C 
                             ci 
                           
                            
                           
                             B 
                             d 
                           
                         
                         + 
                       
                     
                   
                 
               
             
             
               
                 
                   
                       
                      
                     
                       2 
                        
                       
                           
                       
                        
                       
                         D 
                         cdi 
                       
                     
                     ) 
                   
                    
                   
                     
                       d 
                       ^ 
                     
                     k 
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       
                         
                           C 
                           ci 
                         
                          
                         
                           ( 
                           
                             A 
                             + 
                             I 
                           
                           ) 
                         
                       
                        
                       
                         K 
                         bf 
                       
                        
                       
                         F 
                         k 
                       
                     
                     + 
                     
                       
                         E 
                         ci 
                       
                        
                       
                         
                           v 
                           . 
                         
                         k 
                       
                     
                     + 
                     
                       
                         ( 
                         
                           
                             
                               C 
                               ci 
                             
                              
                             
                               B 
                               dN 
                             
                           
                           + 
                           
                             
                               C 
                               ci 
                             
                              
                             
                               B 
                               d 
                             
                           
                           + 
                           
                             2 
                              
                             
                               D 
                               cdi 
                             
                           
                         
                         ) 
                       
                        
                       
                         
                           d 
                           ^ 
                         
                         k 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       
                         K 
                         cpxi 
                       
                        
                       
                         F 
                         k 
                       
                     
                     + 
                     
                       
                         E 
                         ci 
                       
                        
                       
                         
                           v 
                           . 
                         
                         k 
                       
                     
                     + 
                     
                       
                         K 
                         cpdi 
                       
                        
                       
                         
                           d 
                           ^ 
                         
                         k 
                       
                     
                   
                 
               
             
           
         
       
     
         [0038]    The input-output mapping matrix between the control pseudo-inputs (e.g. v1dot  30 , v2dot  32 ) and constraint outputs  50  may be expressed as: 
         [0000]    
       
         
           
             
               E 
               c 
             
             = 
             
               
                 [ 
                 
                   
                     
                       
                         
                           C 
                           
                             C 
                              
                             
                                 
                             
                              
                             1 
                           
                         
                          
                         
                           AB 
                           N 
                         
                       
                     
                   
                   
                     
                       
                         
                           C 
                           
                             C 
                              
                             
                                 
                             
                              
                             2 
                           
                         
                          
                         
                           AB 
                           N 
                         
                       
                     
                   
                   
                     
                       
                         
                           C 
                           
                             C 
                              
                             
                                 
                             
                              
                             3 
                           
                         
                          
                         
                           AB 
                           N 
                         
                       
                     
                   
                   
                     
                       
                         
                           C 
                           
                             C 
                              
                             
                                 
                             
                              
                             4 
                           
                         
                          
                         
                           AB 
                           N 
                         
                       
                     
                   
                 
                 ] 
               
               = 
               
                 [ 
                 
                   
                     
                       
                         e 
                         
                           c 
                            
                           
                               
                           
                            
                           11 
                         
                       
                     
                     
                       
                         e 
                         
                           c 
                            
                           
                               
                           
                            
                           12 
                         
                       
                     
                   
                   
                     
                       
                         e 
                         
                           c 
                            
                           
                               
                           
                            
                           21 
                         
                       
                     
                     
                       
                         e 
                         
                           c 
                            
                           
                               
                           
                            
                           22 
                         
                       
                     
                   
                   
                     
                       
                         e 
                         
                           c 
                            
                           
                               
                           
                            
                           31 
                         
                       
                     
                     
                       
                         e 
                         
                           c 
                            
                           
                               
                           
                            
                           32 
                         
                       
                     
                   
                   
                     
                       
                         e 
                         
                           c 
                            
                           
                               
                           
                            
                           41 
                         
                       
                     
                     
                       
                         e 
                         
                           c 
                            
                           
                               
                           
                            
                           42 
                         
                       
                     
                   
                 
                 ] 
               
             
           
         
       
     
       Consequently, 
       [0039]        {tilde over (y)}   c,k+2   =K   cpx   F   k   +E   c   {dot over (v)}   k   +K   cpd   {circumflex over (d)}   k    (Eq. 13)
 
         [0040]    The control system includes defining the constraint tracking error  52  as: ŷ ci (k+j)=y cri (k+j)−y ci (k), j=0, 1, 2. Note that the constraint reference  54  may be a maximum limit, a minimum limit or both. 
         [0041]    The desired constraint tracking response for the control system may be expressed as: 
         [0000]      ( ŷ   ci ( k+ 2)− {tilde over (y)}   ci ( k+ 2))+ k   ci,1 ( ŷ   ci ( k+ 1)− {tilde over (y)}   ci ( k+ 1))+ k   ci,0 ( ŷ   ci ( k )− {tilde over (y)}   ci ( k ))=0   (Eq. 14)
 
         [0042]    The poles for the tracking dynamics may be set as λ cd1  and λ cd2  within the unit circle. Consequently, 
         [0000]        k   ci,1 =−(λ cd1 +λ cd2 ),  k   ci,0 =λ cd1 ·λ cd2    (Eq. 15)
 
         [0000]    Substituting {tilde over (y)} ci,k+1  into (Eq. 12) yields: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       y 
                       ~ 
                     
                     
                       ci 
                       , 
                       
                         k 
                         + 
                         1 
                       
                     
                   
                   = 
                     
                    
                   
                     
                       
                         C 
                         ci 
                       
                        
                       A 
                        
                       
                         
                           x 
                           ~ 
                         
                         k 
                       
                     
                     + 
                     
                       
                         C 
                         ci 
                       
                        
                       
                         B 
                         N 
                       
                        
                       
                         
                           v 
                           . 
                         
                         k 
                       
                     
                     + 
                     
                       
                         C 
                         ci 
                       
                        
                       
                         ( 
                         
                           
                             
                               B 
                               dN 
                             
                              
                             
                               
                                 d 
                                 ^ 
                               
                               k 
                             
                           
                           + 
                           
                             
                               B 
                               d 
                             
                              
                             
                               
                                 d 
                                 ~ 
                               
                               k 
                             
                           
                         
                         ) 
                       
                     
                     + 
                   
                 
               
             
             
               
                 
                     
                    
                   
                     
                       
                         C 
                         ci 
                       
                        
                       
                         K 
                         bf 
                       
                        
                       
                         F 
                         k 
                       
                     
                     + 
                     
                       
                         D 
                         cdi 
                       
                        
                       
                         
                           d 
                           ^ 
                         
                         
                           k 
                           + 
                           1 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       
                         C 
                         ci 
                       
                        
                       
                         K 
                         bf 
                       
                        
                       
                         F 
                         k 
                       
                     
                     + 
                     
                       
                         ( 
                         
                           
                             
                               C 
                               ci 
                             
                              
                             
                               B 
                               dN 
                             
                           
                           + 
                           
                             D 
                             cdi 
                           
                         
                         ) 
                       
                        
                       
                         
                           d 
                           ^ 
                         
                         k 
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     
                       
                         y 
                         ~ 
                       
                       ci 
                     
                      
                     
                       ( 
                       
                         k 
                         + 
                         2 
                       
                       ) 
                     
                   
                   = 
                     
                    
                   
                     
                       [ 
                       
                         
                           
                             
                               y 
                               ^ 
                             
                             ci 
                           
                            
                           
                             ( 
                             
                               k 
                               + 
                               2 
                             
                             ) 
                           
                         
                         + 
                         
                           
                             k 
                             
                               ci 
                               , 
                               1 
                             
                           
                            
                           
                             
                               
                                 y 
                                 ^ 
                               
                               ci 
                             
                              
                             
                               ( 
                               
                                 k 
                                 + 
                                 1 
                               
                               ) 
                             
                           
                         
                         + 
                         
                           
                             k 
                             
                               ci 
                               , 
                               0 
                             
                           
                            
                           
                             
                               
                                 y 
                                 ^ 
                               
                               ci 
                             
                              
                             
                               ( 
                               k 
                               ) 
                             
                           
                         
                       
                       ] 
                     
                     - 
                   
                 
               
             
             
               
                 
                     
                    
                   
                     [ 
                     
                       
                         
                           k 
                           
                             ci 
                             , 
                             1 
                           
                         
                          
                         
                           
                             
                               y 
                               ^ 
                             
                             ci 
                           
                            
                           
                             ( 
                             
                               k 
                               + 
                               1 
                             
                             ) 
                           
                         
                       
                       + 
                       
                         
                           k 
                           
                             ci 
                             , 
                             0 
                           
                         
                          
                         
                           
                             
                               y 
                               ^ 
                             
                             ci 
                           
                            
                           
                             ( 
                             k 
                             ) 
                           
                         
                       
                     
                     ] 
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       [ 
                       
                         
                           
                             
                               y 
                               ^ 
                             
                             ci 
                           
                            
                           
                             ( 
                             
                               k 
                               + 
                               2 
                             
                             ) 
                           
                         
                         + 
                         
                           
                             k 
                             
                               ci 
                               , 
                               1 
                             
                           
                            
                           
                             
                               
                                 y 
                                 ^ 
                               
                               ci 
                             
                              
                             
                               ( 
                               
                                 k 
                                 + 
                                 1 
                               
                               ) 
                             
                           
                         
                         + 
                         
                           
                             k 
                             
                               ci 
                               , 
                               0 
                             
                           
                            
                           
                             
                               
                                 y 
                                 ^ 
                               
                               ci 
                             
                              
                             
                               ( 
                               k 
                               ) 
                             
                           
                         
                       
                       ] 
                     
                     - 
                   
                 
               
             
             
               
                 
                     
                    
                   
                     [ 
                     
                       
                         k 
                         
                           ci 
                           , 
                           1 
                         
                       
                        
                       
                         
                           
                             y 
                             ^ 
                           
                           ci 
                         
                          
                         
                           ( 
                           
                             k 
                             + 
                             1 
                           
                           ) 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       [ 
                       
                         
                           
                             
                               y 
                               ^ 
                             
                             ci 
                           
                            
                           
                             ( 
                             
                               k 
                               + 
                               2 
                             
                             ) 
                           
                         
                         + 
                         
                           
                             k 
                             
                               ci 
                               , 
                               1 
                             
                           
                            
                           
                             
                               
                                 y 
                                 ^ 
                               
                               ci 
                             
                              
                             
                               ( 
                               
                                 k 
                                 + 
                                 1 
                               
                               ) 
                             
                           
                         
                         + 
                         
                           
                             k 
                             
                               ci 
                               , 
                               0 
                             
                           
                            
                           
                             
                               
                                 y 
                                 ^ 
                               
                               ci 
                             
                              
                             
                               ( 
                               k 
                               ) 
                             
                           
                         
                       
                       ] 
                     
                     - 
                   
                 
               
             
             
               
                 
                     
                    
                   
                     
                       
                         k 
                         
                           ci 
                           , 
                           1 
                         
                       
                        
                       
                         C 
                         ci 
                       
                        
                       
                         K 
                         bf 
                       
                        
                       
                         F 
                         k 
                       
                     
                     - 
                     
                       
                         
                           k 
                           
                             ci 
                             , 
                             1 
                           
                         
                          
                         
                           ( 
                           
                             
                               
                                 C 
                                 ci 
                               
                                
                               
                                 B 
                                 dN 
                               
                             
                             + 
                             
                               D 
                               cdi 
                             
                           
                           ) 
                         
                       
                        
                       
                         
                           d 
                           ^ 
                         
                         k 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       [ 
                       
                         
                           
                             
                               y 
                               ^ 
                             
                             ci 
                           
                            
                           
                             ( 
                             
                               k 
                               + 
                               2 
                             
                             ) 
                           
                         
                         + 
                         
                           
                             k 
                             
                               ci 
                               , 
                               1 
                             
                           
                            
                           
                             
                               
                                 y 
                                 ^ 
                               
                               ci 
                             
                              
                             
                               ( 
                               
                                 k 
                                 + 
                                 1 
                               
                               ) 
                             
                           
                         
                         + 
                         
                           
                             k 
                             
                               ci 
                               , 
                               0 
                             
                           
                            
                           
                             
                               
                                 y 
                                 ^ 
                               
                               ci 
                             
                              
                             
                               ( 
                               k 
                               ) 
                             
                           
                         
                       
                       ] 
                     
                     - 
                   
                 
               
             
             
               
                 
                     
                    
                   
                     
                       
                         K 
                         cdxi 
                       
                        
                       
                         F 
                         k 
                       
                     
                     - 
                     
                       
                         K 
                         cddi 
                       
                        
                       
                         
                           d 
                           ^ 
                         
                         k 
                       
                     
                   
                 
               
             
           
         
       
     
         [0043]    If the reference is constant, ŷ ci (k+2)=ŷ ci (k+1)=ŷ_ci(k) and: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
                               y 
                               ~ 
                             
                             ci 
                           
                            
                           
                             ( 
                             
                               k 
                               + 
                               2 
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     y 
                                     ^ 
                                   
                                   ci 
                                 
                                  
                                 
                                   ( 
                                   
                                     k 
                                     + 
                                     2 
                                   
                                   ) 
                                 
                               
                               + 
                               
                                 
                                   k 
                                   
                                     ci 
                                     , 
                                     1 
                                   
                                 
                                  
                                 
                                   
                                     
                                       y 
                                       ^ 
                                     
                                     ci 
                                   
                                    
                                   
                                     ( 
                                     
                                       k 
                                       + 
                                       1 
                                     
                                     ) 
                                   
                                 
                               
                               + 
                               
                                 
                                   k 
                                   
                                     ci 
                                     , 
                                     0 
                                   
                                 
                                  
                                 
                                   
                                     
                                       y 
                                       ^ 
                                     
                                     ci 
                                   
                                    
                                   
                                     ( 
                                     k 
                                     ) 
                                   
                                 
                               
                             
                             ] 
                           
                           - 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             
                               K 
                               cdxi 
                             
                              
                             
                               F 
                               k 
                             
                           
                           - 
                           
                             
                               K 
                               cddi 
                             
                              
                             
                               
                                 d 
                                 ^ 
                               
                               k 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               ( 
                               
                                 1 
                                 + 
                                 
                                   k 
                                   
                                     ci 
                                     , 
                                     1 
                                   
                                 
                                 + 
                                 
                                   k 
                                   
                                     ci 
                                     , 
                                     0 
                                   
                                 
                               
                               ) 
                             
                              
                             
                               
                                 y 
                                 ^ 
                               
                               
                                 ci 
                                 , 
                                 k 
                               
                             
                           
                           - 
                           
                             
                               K 
                               cdxi 
                             
                              
                             
                               F 
                               k 
                             
                           
                           - 
                           
                             
                               K 
                               cddi 
                             
                              
                             
                               
                                 d 
                                 ^ 
                               
                               k 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               k 
                               ci 
                             
                              
                             
                               
                                 y 
                                 ^ 
                               
                               
                                 ci 
                                 , 
                                 k 
                               
                             
                           
                           - 
                           
                             
                               K 
                               cdxi 
                             
                              
                             
                               F 
                               k 
                             
                           
                           - 
                           
                             
                               K 
                               cddi 
                             
                              
                             
                               
                                 d 
                                 ^ 
                               
                               k 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     16 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     That 
                      
                     
                         
                     
                      
                     is 
                   
                   , 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       y 
                       ~ 
                     
                     
                       c 
                       , 
                       
                         k 
                         + 
                         2 
                       
                     
                   
                   = 
                   
                     
                       
                         K 
                         c 
                       
                        
                       
                         
                           y 
                           ^ 
                         
                         
                           c 
                           , 
                           k 
                         
                       
                     
                     - 
                     
                       
                         K 
                         cdx 
                       
                        
                       
                         F 
                         k 
                       
                     
                     - 
                     
                       
                         K 
                         cdd 
                       
                        
                       
                         
                           
                             d 
                             ^ 
                           
                           k 
                         
                         . 
                       
                     
                   
                 
               
               
                 
                     
                 
               
             
           
         
       
     
         [0044]    However, if the constraint reference  54  is time varying, ŷ ci (k+2)≠ŷ ci (k+1)≠ŷ_ci(k). The control system includes the assumption that the references changes linearly within 3 samples by defining ŷ Rci,k =y Rci (k)−y Rci (k−1), y Rci (k+1)=y Rci (k)+αŷ Rci,k , and y Rci (k+2)=y Rci (k)+2αŷ Rci,k . Consequently, 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
                               y 
                               ~ 
                             
                             ci 
                           
                            
                           
                             ( 
                             
                               k 
                               + 
                               2 
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     y 
                                     ^ 
                                   
                                   ci 
                                 
                                  
                                 
                                   ( 
                                   
                                     k 
                                     + 
                                     2 
                                   
                                   ) 
                                 
                               
                               + 
                               
                                 
                                   k 
                                   
                                     ci 
                                     , 
                                     1 
                                   
                                 
                                  
                                 
                                   
                                     
                                       y 
                                       ^ 
                                     
                                     ci 
                                   
                                    
                                   
                                     ( 
                                     
                                       k 
                                       + 
                                       1 
                                     
                                     ) 
                                   
                                 
                               
                               + 
                               
                                 
                                   k 
                                   
                                     ci 
                                     , 
                                     0 
                                   
                                 
                                  
                                 
                                   
                                     
                                       y 
                                       ^ 
                                     
                                     ci 
                                   
                                    
                                   
                                     ( 
                                     k 
                                     ) 
                                   
                                 
                               
                             
                             ] 
                           
                           - 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             
                               K 
                               cdxi 
                             
                              
                             
                               F 
                               k 
                             
                           
                           - 
                           
                             
                               K 
                               cddi 
                             
                              
                             
                               
                                 d 
                                 ^ 
                               
                               k 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               ( 
                               
                                 
                                   2 
                                    
                                   α 
                                    
                                   
                                     
                                       y 
                                       ^ 
                                     
                                     
                                       Rci 
                                       , 
                                       k 
                                     
                                   
                                 
                                 + 
                                 
                                   
                                     k 
                                     
                                       ci 
                                       , 
                                       1 
                                     
                                   
                                    
                                   α 
                                    
                                   
                                     
                                       y 
                                       ^ 
                                     
                                     
                                       Rci 
                                       , 
                                       k 
                                     
                                   
                                 
                                 + 
                                 
                                   k 
                                   
                                     ci 
                                     , 
                                     0 
                                   
                                 
                               
                               ) 
                             
                              
                             
                               
                                 y 
                                 ^ 
                               
                               
                                 ci 
                                 , 
                                 k 
                               
                             
                           
                           - 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             
                               K 
                               cdxi 
                             
                              
                             
                               F 
                               k 
                             
                           
                           - 
                           
                             
                               K 
                               cddi 
                             
                              
                             
                               
                                 d 
                                 ^ 
                               
                               k 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               [ 
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         2 
                                          
                                         α 
                                       
                                       + 
                                       
                                         
                                           k 
                                           
                                             ci 
                                             , 
                                             1 
                                           
                                         
                                          
                                         α 
                                       
                                     
                                     ) 
                                   
                                    
                                   
                                     
                                       y 
                                       ^ 
                                     
                                     
                                       Rci 
                                       , 
                                       k 
                                     
                                   
                                 
                                 + 
                                 
                                   k 
                                   ci 
                                 
                               
                               ] 
                             
                              
                             
                               
                                 y 
                                 ^ 
                               
                               
                                 ci 
                                 , 
                                 k 
                               
                             
                           
                           - 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             
                               K 
                               cdxi 
                             
                              
                             
                               F 
                               k 
                             
                           
                           - 
                           
                             
                               K 
                               cddi 
                             
                              
                             
                               
                                 d 
                                 ^ 
                               
                               k 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     17 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     That 
                      
                     
                         
                     
                      
                     is 
                   
                   , 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       y 
                       ~ 
                     
                     
                       c 
                       , 
                       
                         k 
                         + 
                         2 
                       
                     
                   
                   = 
                   
                     
                       
                         K 
                         c 
                         * 
                       
                        
                       
                         
                           y 
                           ^ 
                         
                         
                           c 
                           , 
                           k 
                         
                       
                     
                     - 
                     
                       
                         K 
                         cdx 
                       
                        
                       
                         F 
                         k 
                       
                     
                     - 
                     
                       
                         K 
                         cdd 
                       
                        
                       
                         
                           
                             d 
                             ^ 
                           
                           k 
                         
                         . 
                       
                     
                   
                 
               
               
                 
                     
                 
               
             
           
         
       
     
         [0045]    In preferred embodiments of the control system, the control priority of Nx,  24  is greater than the control priority of N2  26 , that is, N2  26  is the tradeoff target. Therefore, the control system may compare all the constraints and N2. The control system may then select the variable with the largest demand for {dot over (v)} 1 , alternatively known as v1dot  30 , as the new controlled output. 
         [0046]    The constraint decoupling control  44  presented herein is applicable at least for variable pitch fan engines and turbo-shaft, turbo propeller engines. Other applications more generally include, but are limited to, open-rotor turbine engines. 
         [0047]    Note that when Nx  24  is controlled by v2dot  32  alone, the ith constraint input-output mapping may be expressed as ŷ k =e ci,1 {dot over (v)} 1 (k)+e ci,2 {dot over (v)} 2 (k). The term {dot over (v)} 2 (k) (e.g. v2dot  32 ) determined by tracking Nx  24  is a known input for any of the constraints and noted as {dot over (v)} 2,k   N     x   . 
         [0048]    The control system includes the constraint decoupling control  44  derived by comparing (Eq. 13) and (Eq. 17): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             c 
                           
                            
                           
                             
                               v 
                               . 
                             
                             k 
                           
                         
                         = 
                           
                          
                         
                           
                             
                               K 
                               c 
                               * 
                             
                              
                             
                               
                                 y 
                                 ^ 
                               
                               
                                 c 
                                 , 
                                 k 
                               
                             
                           
                           - 
                           
                             
                               ( 
                               
                                 
                                   K 
                                   cdx 
                                 
                                 + 
                                 
                                   K 
                                   cpx 
                                 
                               
                               ) 
                             
                              
                             
                               F 
                               k 
                             
                           
                           - 
                           
                             
                               ( 
                               
                                 
                                   K 
                                   cdd 
                                 
                                 + 
                                 
                                   K 
                                   cpd 
                                 
                               
                               ) 
                             
                              
                             
                               
                                 d 
                                 ^ 
                               
                               k 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               K 
                               c 
                               * 
                             
                              
                             
                               
                                 y 
                                 ^ 
                               
                               
                                 c 
                                 , 
                                 k 
                               
                             
                           
                           - 
                           
                             
                               K 
                               cx 
                             
                              
                             
                               F 
                               k 
                             
                           
                           - 
                           
                             
                               K 
                               cd 
                             
                              
                             
                               
                                 d 
                                 ^ 
                               
                               k 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               
                                 K 
                                 c 
                                 * 
                               
                                
                               
                                 
                                   y 
                                   ^ 
                                 
                                 
                                   c 
                                   , 
                                   k 
                                 
                               
                             
                             - 
                             
                               
                                 K 
                                 cx 
                               
                                
                               
                                 
                                   x 
                                   ^ 
                                 
                                 k 
                               
                             
                             - 
                             
                               
                                 K 
                                 cd 
                               
                                
                               
                                 
                                   d 
                                   ^ 
                                 
                                 k 
                               
                             
                           
                           = 
                           
                             M 
                             
                               cp 
                               , 
                               k 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     18 
                   
                   ) 
                 
               
             
             
               
                 
                   Alternatively 
                   , 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       [ 
                       
                         
                           
                             
                               e 
                               
                                 c 
                                  
                                 
                                     
                                 
                                  
                                 11 
                               
                             
                           
                           
                             
                               e 
                               
                                 c 
                                  
                                 
                                     
                                 
                                  
                                 12 
                               
                             
                           
                         
                         
                           
                             
                               e 
                               
                                 c 
                                  
                                 
                                     
                                 
                                  
                                 21 
                               
                             
                           
                           
                             
                               e 
                               
                                 c 
                                  
                                 
                                     
                                 
                                  
                                 22 
                               
                             
                           
                         
                         
                           
                             
                               e 
                               
                                 c 
                                  
                                 
                                     
                                 
                                  
                                 31 
                               
                             
                           
                           
                             
                               e 
                               
                                 c 
                                  
                                 
                                     
                                 
                                  
                                 32 
                               
                             
                           
                         
                         
                           
                             
                               e 
                               
                                 c 
                                  
                                 
                                     
                                 
                                  
                                 41 
                               
                             
                           
                           
                             
                               e 
                               
                                 c 
                                  
                                 
                                     
                                 
                                  
                                 42 
                               
                             
                           
                         
                       
                       ] 
                     
                      
                     
                       [ 
                       
                         
                           
                             
                               
                                 v 
                                 . 
                               
                               
                                 1 
                                 , 
                                 k 
                               
                             
                           
                         
                         
                           
                             
                               
                                 v 
                                 . 
                               
                               
                                 2 
                                 , 
                                 k 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             
                               M 
                               
                                 cp 
                                 , 
                                 k 
                               
                             
                              
                             
                               ( 
                               1 
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             
                               M 
                               
                                 cp 
                                 , 
                                 k 
                               
                             
                              
                             
                               ( 
                               2 
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             
                               M 
                               
                                 cp 
                                 , 
                                 k 
                               
                             
                              
                             
                               ( 
                               3 
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             
                               M 
                               
                                 cp 
                                 , 
                                 k 
                               
                             
                              
                             
                               ( 
                               4 
                               ) 
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 and 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       [ 
                       
                         
                           
                             
                               e 
                               
                                 c 
                                  
                                 
                                     
                                 
                                  
                                 11 
                               
                             
                           
                         
                         
                           
                             
                               e 
                               
                                 c 
                                  
                                 
                                     
                                 
                                  
                                 21 
                               
                             
                           
                         
                         
                           
                             
                               e 
                               
                                 c 
                                  
                                 
                                     
                                 
                                  
                                 31 
                               
                             
                           
                         
                         
                           
                             
                               e 
                               
                                 c 
                                  
                                 
                                     
                                 
                                  
                                 41 
                               
                             
                           
                         
                       
                       ] 
                     
                      
                     
                       [ 
                       
                         
                           v 
                           . 
                         
                         
                           1 
                           , 
                           k 
                         
                       
                       ] 
                     
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             
                               
                                 M 
                                 
                                   cp 
                                   , 
                                   k 
                                 
                               
                                
                               
                                 ( 
                                 1 
                                 ) 
                               
                             
                           
                         
                         
                           
                             
                               
                                 M 
                                 
                                   cp 
                                   , 
                                   k 
                                 
                               
                                
                               
                                 ( 
                                 2 
                                 ) 
                               
                             
                           
                         
                         
                           
                             
                               
                                 M 
                                 
                                   cp 
                                   , 
                                   k 
                                 
                               
                                
                               
                                 ( 
                                 3 
                                 ) 
                               
                             
                           
                         
                         
                           
                             
                               
                                 M 
                                 
                                   cp 
                                   , 
                                   k 
                                 
                               
                                
                               
                                 ( 
                                 4 
                                 ) 
                               
                             
                           
                         
                       
                       ] 
                     
                     - 
                     
                       
                         [ 
                         
                           
                             
                               
                                 e 
                                 
                                   c 
                                    
                                   
                                       
                                   
                                    
                                   12 
                                 
                               
                             
                           
                           
                             
                               
                                 e 
                                 
                                   c 
                                    
                                   
                                       
                                   
                                    
                                   22 
                                 
                               
                             
                           
                           
                             
                               
                                 e 
                                 
                                   c 
                                    
                                   
                                       
                                   
                                    
                                   32 
                                 
                               
                             
                           
                           
                             
                               
                                 e 
                                 
                                   c 
                                    
                                   
                                       
                                   
                                    
                                   42 
                                 
                               
                             
                           
                         
                         ] 
                       
                        
                       
                         [ 
                         
                           
                             v 
                             . 
                           
                           
                             2 
                             , 
                             k 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                     
                 
               
             
           
         
       
     
         [0049]    For each constraint tracking, the control system calculates the demanded {dot over (v)} ci,1 (k) as: 
         [0000]        {dot over (v)}   ci,1 ( k )=( e   ci1 ) −1   [M   cp,k ( i )− e   ci2   {dot over (v)}   2,k   N     x   ]  (Eq. 19)
 
         [0050]    Given the constraints, expressed as minimum and maximum limits  56 , minimum pressure limit (MinPx), maximum pressure limit (MaxPx), maximum temperature limit (MaxT), maximum torque limit (MaxTq), minimum speed rate limit (MinN2dot), and maximum speed rate limit (MaxN2dot), the control system includes the following relationships: 
         [0000]      Max Px  requests  {dot over (v)}   c1,1   + =( e   c11 ) −1   [M   cp,k   + (1)− e   c12   {dot over (v)}   2,k   Nx ];
 
         [0000]      Min Px  requests  {dot over (v)}   c1,1   − =( e   c11 ) −1   [M   cp,k   − (1)− e   c12   {dot over (v)}   2,k   Nx ];
 
         [0000]      Max T  requests  {dot over (v)}   c2,1   + =( e   c21 ) −1   [M   cp,k   + (2)− e   c22   {dot over (v)}   2,k   Nx ];
 
         [0000]      Max Tq  requests  {dot over (v)}   c3,1   + =( e   c31 ) −1   [M   cp,k   + (3)− e   c32   {dot over (v)}   2,k   Nx ];
 
         [0000]      Max N 2dot requests  {dot over (v)}   c4,1   + =( e   c41 ) −1   [M   cp,k   + (4)− e   c42   {dot over (v)}   2,k   Nx ]; and
 
         [0000]      Min N 2dot requests  {dot over (v)}   c4,1   − =( e   c41 ) −1   [M   cp,k   −1 (4)− e   c42   {dot over (v)}   2,k   Nx ].
 
         [0000]    Each constraint tracking is embedded in the constraint decoupling control  44  described above. 
         [0051]    The control system&#39;s selection logic  48  determines the most demanding output among all the control constraints outputs  50  and N2  26 . Then the control system selects the pseudo-input {dot over (v)} (e.g. v1dot  30 , v2dot  32 ) requested by the most demanding output (e.g. any of the control constraints  50  or N2 and assigns that pseudo-input to be v1dot. 
         [0052]    The control reference shaping  58  is for both primary control references (i.e. primary control reference shaping  58 A) and constraint control references (i.e. constraint control reference shaping  58 B). In general, the control reference shaping  58  fits a reference into one or more desired trajectories. Control reference shaping  58  performs functions related to reference rate scheduling and reference coordination, filtering, and reference prediction action. 
         [0053]    As contemplated herein for use with the control system, primary control reference shaping  58 A functions for reference rate scheduling and reference coordination for N2 and Nx references based on thrust forward operation or thrust reverse operation, and the transition associated between the two operations. For constraint references, constraint control reference shaping  58 B provides reference prediction based on the known information of the constant or varying constraint limits. For example, MinPx, MaxPx, MaxT, and MaxTq may be constant limits, MaxN2dot and MinN2dot may be varying limits, and MaxN2dot and MinN2dot may be continuous variables. Therefore, it is contemplated that the control system includes the prediction of the N2dot limits to be linear extensions of the current N2dot limit rate over the samples within the relative degree of N2dot. 
         [0054]    In general, if control reference shaping robustly and accurately manages the reference change rates, all the references are properly coordinated, and the feedback control has proper bandwidth, then a feedforward control action is not necessary. However, when the external disturbances are significant, the control system may include a feedforward control action as shown in  FIG. 1C  to enhance the control system bandwidth while rejecting external disturbances. 
         [0055]    In the presently disclosed control system, the feedforward control action operates to effectively consider the impact of changes to the aerodynamic load  60  on the control system  10 ,  12 . Significant changes to the aerodynamic load  60  may result in changes to the aircraft&#39;s speed and propeller blade pitch angle. The changes to the aerodynamic load  60  may be known in advance as scheduled, or the control system may estimate the load change by, for example, a torque sensor measurement with consideration of the aerodynamic delay and mechanical transport delay. The control system may map the load change to a corresponding reaction change in speed and blade pitch angle and incorporate the adjustments in an issued request for actuator control commands. In this way, the control system responds to the load changes more responsively while tracking the speed references. The integration of control elements for both forward path compensation and feedback path compensation as well as feedforward control action for the PCM pitch angle servo control  64  is shown in  FIG. 1D . The integration of control elements for both forward path compensation and feedback path compensation for the fuel flow servo control  62  is shown in  FIG. 1E . 
         [0056]    Referring now to  FIG. 3 , for an aircraft with multiple engines  102 ,  104  where each engine operates with its own control system  110 ,  112 , the control systems  110 ,  112  and a speed phase synchronization regulator  114  perform the engine-to-engine speed phase synchronization (i.e. the relative rotational position of the propellers) by a series of control steps. Each engine control system  110 ,  112  measures the propeller speed phase  116 ,  118  and broadcasts a filtered phase signal  120 ,  122  to one or more of the other engine control systems. Each engine control system includes a speed phase synchronization regulator. One engine  102  is specified to be the master and the second engine  104  is specified as the slave. Each engine control system receives the phase signals from other engines and the master engine ID via data bus. Each control system of slave engines calculates the difference of the speed phase signals  124  by comparing the speed phase signal from the master engines  102  with the speed phase signal from the slave engine itself  104  and then compares the difference with a phase difference reference  126  (in general it is zero) to generate a phase difference error  128 . The phase difference error  128  is sent to the speed phase synchronization regulator  114  and the speed phase synchronization regulator  114  calculates a speed reference bias  130 . The control system adds the speed reference bias  130  to the slave engine propeller speed reference such that the slave engine speed phase is synchronized to the master engine speed phase. 
         [0057]    The control system described above treats the fan or propeller and the gas engine as one controlled plant and includes all the outputs and constraints to be controlled, considers known disturbances rejection, and covers the operational conditions where large changes occur in the reference schedules. Consequently, the technical benefits include a complete, systematic but simple method and system as a holistic and separatable solution for the control of variable pitch fan engines, and turbo-shaft turbo-propeller engines, including ducted and unducted variable pitch fan engines, and turbo-shaft turbo-propeller engines. 
         [0058]    It is to be understood the control system architectures disclosed herein may be provided in any manner known to those of ordinary skill, including software solutions, hardware or firmware solutions, and combinations of such. Such solutions would incorporate the use of appropriate processors, memory (and software embodying any algorithms described herein may be resident in any type of non-transitory memory), circuitry and other components as is known to those of ordinary skill. 
         [0059]    This written description uses examples to disclose the invention, including the best mode, and also to enable any person skilled in the art to practice the invention, including making and using any devices or systems and performing any incorporated methods. The patentable scope of the invention is defined by the claims, and may include other examples that occur to those skilled in the art. Such other examples are intended to be within the scope of the claims if they have structural elements that do not differ from the literal language of the claims, or if they include equivalent structural elements with insubstantial differences from the literal languages of the claims.