Abstract:
An apparatus and method is disclosed for automatically controlling single-input-multi-output (SIMO) systems or processes. The control output signals of a plurality of single-input-single-output (SISO) automatic controllers are combined by a combined output setter so that these SISO controllers are converted to a multi-input-single-output (MISO) automatic controller based on certain criteria; and its resulting controller output signal is able to manipulate only one actuator to control a plurality of continuous process variables or attempt to minimize a plurality of error signals between the setpoints and their corresponding process variables. Without the need of building process mathematical models, this inventive apparatus and method is useful for automatically controlling unevenly paired multivariable systems or processes where there are less system inputs than outputs including but not limited to industrial furnaces, rapid thermal processing (RTP) chambers, chemical mechanical planarization (CMP) systems, and distillation columns.

Description:
[0001]     The subject of this patent relates to automatic control of single-input-multi-output (SIMO) systems including industrial processes, equipment, facilities, buildings, homes, devices, engines, robots, vehicles, aircraft, space-vehicles, appliances and other systems, and more particularly to a method and apparatus for automatically controlling two or more continuous process variables by manipulating only one actuator.  
         [0002]     In control system applications, it is relatively easy to control single-input-single-output (SISO) systems or multi-input-multi-output (MIMO) systems that have the same amount of manipulated variables as system inputs and controlled process variables as system outputs. In practice, we often have to deal with unevenly paired multivariable systems, where the number of manipulated variables is greater or smaller than the controlled process variables. Depending on the number of system inputs and outputs, we categorize them as multi-input-single-output (MISO) systems and single-input-multi-output (SIMO) systems. These systems are much more difficult to control using conventional control methods.  
         [0003]     In U.S. patent application No. 60/474,688, we presented a method and apparatus for controlling MISO systems. In this patent, we introduce a method and apparatus for controlling SIMO systems.  
         [0004]     Single-input-multi-output (SIMO) systems are found in situations where one manipulated variable affects  2  or more process variables that need to be controlled. In these cases, there are no other manipulated variables that we can use to form several single-input-single-output (SISO) systems or one evenly paired multi-input-multi-output (MIMO) system. For instance, 
        An industrial furnace for glass melting or metal reheating typically has multiple temperature zones that need to be controlled. The number of fuel flows manipulated by control valves is typically less than the total number of temperature zones measured. In this case, we may have to design the control system to include one evenly paired MIMO system and one or more 1-input-2-output (1×2) systems.     A Rapid Thermal Processing (RTP) chamber for semiconductor wafer treatment or space material testing has many temperature zones to control. The temperature setpoints can ramp up from room temperature to a few thousand degrees Fahrenheit within 30 seconds and then ramp down. The control objective is to force each temperature point to tightly track its setpoint trajectory. However, since the number of lamps used to heat the chamber may be less than the number of temperature zones to be controlled, we are facing an unevenly paired multivariable system.     A Chemical Mechanical Planarization (CMP) system for semiconductor wafer planarization applies pressure to the polishing pad to control the thickness of the thin film deposited on the wafer. The number of pressure actuators is typically less than the number of measured thickness points. When the wafer size increases from 200 mm to 300 mm and higher, this problem becomes more severe since the measured thickness points will increase dramatically. We then have to deal with a large number of SIMO systems to control one CMP system.     A distillation column used for separating a liquid or vapor mixture into component fractions of desired purity has many process variables. A reboiler brings the liquid at the bottom to the boiling point where the component with a lower boiling point will evaporate. Plates inside the column shell enhance the separation. The reboiler steam flow is usually the appropriate manipulated variable to use to control the bottom temperature as well as the temperature of multiple plates. In this case, we have to control a 1×2 or 1×M system.        
 
         [0009]     Theoretically, a 1×M system with 1 input and M outputs, where M=2, 3, . . . is not controllable since manipulating only one input variable in its range cannot force its M output variables to move to any point within their ranges. In practice, a typical approach is to control the most important process variable and leave the other variables uncontrolled. In the distillation column example, the bottom temperature may be controlled but the plate temperatures are only monitored or loosely controlled by manipulating the reflux. In this patent, we introduce the method and apparatus to control single-input-multi-output (SIMO) systems. 
     
    
       [0010]     In the accompanying drawing:  
         [0011]      FIG. 1  is a block diagram illustrating a 2-input-1-output (2×1) Model-Free Adaptive (MFA) controller that controls a 1-input-2-output (1×2) system.  
         [0012]      FIG. 2  is a block diagram illustrating the architecture of a 2-input-1-output (2×1) Model-Free Adaptive (MFA) controller.  
         [0013]      FIG. 3  is a block diagram illustrating the simplified architecture of a 2-input-1-output (2×1) Feedback/Feedforward Model-Free Adaptive (MFA) controller.  
         [0014]      FIG. 4  is a drawing illustrating a mechanism of a Combined Output Setter that combines  2  controller outputs into one controller output.  
         [0015]      FIG. 5  is a block diagram illustrating a 3-input-1-output (3×1) Model-Free Adaptive (MFA) controller that controls a 1-input-3-output (1×3) system.  
         [0016]      FIG. 6  is a block diagram illustrating the simplified architecture of a 3-input-1-output (3×1) Model-Free Adaptive (MFA) controller.  
         [0017]      FIG. 7  is a block diagram illustrating the simplified architecture of a 3-input-1-output (3×1) Feedback/Feedforward Model-Free Adaptive (MFA) controller.  
         [0018]      FIG. 8  is a drawing illustrating a mechanism of a Combined Output Setter that combines  3  controller outputs into one controller output.  
         [0019]      FIG. 9  is a block diagram illustrating an M-input-1-output (M×1) Model-Free Adaptive (MFA) controller that controls a 1-input-M-output (1×M) system.  
         [0020]      FIG. 10  is a block diagram illustrating the simplified architecture of an M-input-1-output (M×1) Model-Free Adaptive (MFA) controller.  
         [0021]      FIG. 1I  is a block diagram illustrating the simplified architecture of an M-input-1-output (M×1) Feedback/Feedforward Model-Free Adaptive (MFA) controller.  
         [0022]      FIG. 12  is a drawing illustrating a mechanism of a Combined Output Setter that combines M controller outputs into one controller output.  
         [0023]      FIG. 13  is a block diagram illustrating a 2-input-1-output (2×1) proportional-integral-derivative (PID) controller that controls a 1-input-2-output (1×2) system.  
         [0024]      FIG. 14  is a block diagram illustrating an M-input-1-output (M×1) proportional-integral-derivative (PID) controller that controls a 1-input-M-output (1×M) system.  
         [0025]      FIG. 15  is a block diagram illustrating a multi-input-single-output (MISO) controller that controls a single-input-multi-output (SIMO) system. 
     
    
       [0026]     The term “mechanism” is used herein to represent hardware, software, or any combination thereof. The term “process” is used herein to represent a physical system or process with inputs and outputs that have dynamic relationships.  
       DESCRIPTION  
       [heading-0027]     A. 2-Input-1-Output Model-Free Adaptive (MFA) Controller  
         [0028]      FIG. 1  illustrates a 2-input-1-output (2×1) MFA controller that controls a 1-input-2-output (1×2) system. The control system comprises a 2-input-1-output (2×1) MFA controller  8 , a 1-input-2-output (1×2) system  10 , and signal adders  11 ,  12 ,  13 , and 
        14. The signals shown in  FIG. 1  are as follows:     r 1 (t), r 2 (t)×Setpoint 1 and Setpoint 2.     x 1 (t), x 2 (t)—System outputs of the 1×2 system.     d 1 (t), d 2 (t)—Disturbance 1 and 2 caused by noise or load changes.     y 1 (t), y 2 (t)×Measured process variables of the 1×2 system, 
            y 1 (t)=x 1 (t)+d 1 (t); and y 2 (t)=x 2 (t)+d 2 (t).    
            e 1 (t), e 2 (t)—Error between the setpoint and measured process variable, 
            e 1 (t)=r 1 (t)−y 1 (t); and e 2 (t)=r 2 (t)−y 2 (t).    
            u(t)—Output of the 2×1 MFA Controller.          
         [0038]     The control objective is for the controller to produce output u(t) to manipulate the manipulated variable so that the measured process variables y 1 (t) and y 2 (t) track the given trajectory of their setpoints r 1 (t) and r 2 (t), respectively, under variations of setpoint, disturbance, and process dynamics. In other words, the task of the MFA controller is to minimize the error e 1 (t) and e 2 (t) in an online fashion.  
         [0039]     Since there is only one manipulated variable, minimizing errors for both loops may not be possible. The control objective then can be defined as (i) to minimize the error for the more critical loop of the two, or (ii) to minimize the error for both loops with no weighting on the importance so that there may be static errors in both loops.  
         [0040]     We select the objective function for the MFA control system as  
                       E     1   ⁢   S       ⁡     (   t   )       =       ⁢       1   2     ⁢         e   1     ⁡     (   t   )       2                   =       ⁢           1   2     ⁡     [         r   1     ⁡     (   t   )       -       y   1     ⁡     (   t   )         ]       2     .                   (     1   ⁢   a     )                         E     2   ⁢   S       ⁡     (   t   )       =       ⁢       1   2     ⁢         e   2     ⁡     (   t   )       2                   =       ⁢           1   2     ⁡     [         r   2     ⁡     (   t   )       -       y   2     ⁡     (   t   )         ]       2     .                   (     1   ⁢   b     )             
 
         [0041]     The minimization of E 1s (t) and E 2s (t) is achieved by (i) the regulatory control capability of the MFA controller, whose output u(t) manipulates the manipulated variable forcing the process variables y 1 (t) and y 2 (t) to track the given trajectory of their setpoints r 1 (t) and r 2 (t), respectively; and (ii) the adjustment of the MFA controller weighting factors that allow the controller to deal with the dynamic changes, large disturbances, and other uncertainties of the control system.  
         [0042]      FIG. 2  illustrates the architecture of a 2-input-1-output Model-Free Adaptive (MFA) controller. Two multilayer neural networks  17 ,  18  are used in the design of the controller. Each neural network has one input layer  19 ,  20 , one hidden layer  21 ,  22  with N neurons, and one output layer  23 ,  24  with one neuron.  
         [0043]     Since both neural networks in each controller are identical, we will drop the subscript in the following equations to simplify. The input signal e(t) to the input layer  19 ,  20  is first converted to a normalized error signal E, with a range of −1 to 1 by using the normalization unit  25 ,  26 , where N(.) denotes a normalization function. The output of the normalization unit  25 ,  26  is then scaled by a scaling function L(.)  15 ,  16 :  
               L   ⁡     (   .   )       =         K   c       T   c       .             (   2   )             
 
 The value of E 1  at time t is computed with function L(.) and N(.):  
                 E   1     =         K   c       T   c       ⁢     N   ⁡     (     e   ⁡     (   t   )       )           ,           (   3   )             
 
 where K c &gt;0 is defined as controller gain and T c  is the user selected process time constant. K c  is used to compensate for the process steady-state gain and T c  provides information for the dynamic behavior of the process. When the error signal is scaled with these parameters, the controller&#39;s behavior can be manipulated by adjusting the parameters. 
 
         [0046]     The E 1  signal then goes iteratively through a series of delay units  27 ,  28 , where z −1  denotes the unit delay operator. A set of normalized and scaled error signals E 2  to E N  is then generated. In this way, a continuous signal e(t) is converted to a series of discrete signals, which are used as the inputs to the neural network. These delayed error signals E i , i=1,2, . . . N, are then conveyed to the hidden layer through the neural network connections. This is equivalent to adding a feedback structure to the neural network. Then the regular static multilayer neural network becomes a dynamic neural network.  
         [0047]     A Model-Free Adaptive controller uses a dynamic block such as a dynamic neural network. A dynamic block is just another name for a dynamic system, whose inputs and outputs have dynamic relationships.  
         [0048]     Each input signal can be conveyed separately to each of the neurons in the hidden layer  21 ,  22  via a path weighted by an individual weighting factor w ij , where i=1,2, . . . N, and j=1,2, . . . N. The inputs to each of the neurons in the hidden layer are summed by adder  29 ,  30  to produce signal p j . Then the signal p j  is filtered by an activation function  31 ,  32  to produce q i , where j denotes the jth neuron in the hidden layer.  
         [0049]     A piecewise continuous linear function ƒ(x) mapping real numbers to [0,1] is used as the activation function in the neural network as defined by  
                 f   ⁡     (   x   )       =   0     ,           ⁢       if   ⁢           ⁢   x     &lt;     -     b   a                 (     4   ⁢   a     )                   f   ⁡     (   x   )       =     ax   +   b       ,           ⁢       if   -     b   a       ≤   x   ≤     b   a               (     4   ⁢   b     )                   f   ⁡     (   x   )       =   1     ,           ⁢       if   ⁢           ⁢   x     &gt;     b   a               (     4   ⁢   c     )             
 
 where a is an arbitrary constant and b=½. 
 
         [0051]     Each output signal from the hidden layer is conveyed to the single neuron in the output layer  23 ,  24  via a path weighted by an individual weighting factor h j , where j=1,2, . . . N. These signals are summed in adder  33 ,  34  to produce signal z(.), and then filtered by activation function  35 ,  36  to produce the output o(.) of the neural network  17 ,  18  with a range of 0 to 1.  
         [0052]     A de-normalization function  37 ,  38  defined by 
 
 D ( x )=100 x,   (5) 
 
 maps the o(.) signal back into the real space to produce the controller signal v(t). 
 
         [0054]     The algorithm governing the input-output of the controller is seen by the following difference equations:  
                   p   j     ⁡     (   n   )       =       ∑     i   =   1     N     ⁢           ⁢         w   ij     ⁡     (   n   )       ⁢       E   i     ⁡     (   n   )             ,           (   6   )                     q   j     ⁡     (   n   )       =     f   ⁡     (       p   j     ⁡     (   n   )       )         ,           (   7   )                         o   ⁡     (   n   )       =       ⁢     f   ⁡     (       ∑     j   =   1     N     ⁢           ⁢         h   j     ⁡     (   n   )       ⁢       q   j     ⁡     (   n   )           )         ,                 =       ⁢       a   ⁢       ∑     j   =   1     N     ⁢           ⁢         h   j     ⁡     (   n   )       ⁢       q   j     ⁡     (   n   )             +   b       ,                 (   8   )             
 
 where the variable of function f(.) is in the range specified in Equation (4b), and o(n) is bounded by the limits specified in Equations (4a) and (4c). The controller signal v(t) becomes  
                     v   ⁡     (   t   )       =       ⁢           K   c     ⁡     (   .   )       ⁢     e   ⁡     (   t   )         +     D   ⁡     (     o   ⁡     (   t   )       )                       =       ⁢           K   c     ⁡     (   .   )       ⁢     e   ⁡     (   t   )         +     100   [       a   ⁢       ∑     j   =   1     N     ⁢           ⁢         h   j     ⁡     (   n   )       ⁢       q   j     ⁡     (   n   )             +   b     ]         ,                 (   9   )             
 
 where n denotes the nth iteration; o(t) is the continuous function of o(n); D(.) is the de-normalization function; and K c (.)&gt;0, the controller gain  41 ,  42 , is a parameter used to adjust the magnitude of the controller. This is the same parameter as in the scaling function L(.)  15 ,  16  and is useful to fine tune the controller performance or keep the system stable. 
 
         [0057]     An online learning algorithm as described in U.S. Pat. No. 6,556,980 B1 is an example of one algorithm that can be used to continuously update the values of the weighting factors of the MFA controller as follows: 
 
Δ w   ij ( n )= a   2   ηe ( n ) E   i ( n ) h   j ( n ),  (10) 
 
Δ h   j ( n )= aηe ( n ) q   j ( n ).  (11) 
 
         [0058]     The equations (1) through (11) work for both process direct-acting or reverse acting types. Direct-acting means that an increase in the process input will cause its output to increase, and vice versa. Reverse-acting means that an increase in the process input will cause its output to decrease, and vice versa. To keep the above equations working for both direct and reverse acting cases, e(t) is calculated differently based on the acting type of the process as follows: 
 
 e ( t )= r ( t )− y ( t ), if direct acting  (12a) 
 
 e ( t )=−[ r ( t )− y ( t )]. if reverse acting  (12b) 
 
         [0059]     This is a general treatment for the process acting types. It applies to all Model-Free Adaptive controllers to be introduced below.  
         [0060]     We can consider that there are two single-input-single-output (SISO) MFA controllers  44  and  46  in this design with input signals e 1 (t) and e 2 (t), and output signals v 1 (t) and v 2 (t). Then the Combined Output Setter  48  can be used to combine the signals v 1 (t) and v 2 (t) to produce the controller output u(t).  
         [0061]      FIG. 3  is a block diagram illustrating the simplified architecture of a 2-input-1-output (2×1) Feedback/Feedforward Model-Free Adaptive (MFA) controller. The 2×1 Feedback/Feedforward MFA controller  58  comprises two SISO MFA controllers  50 ,  51 , two Feedforward MFA controllers  52 ,  53 , two signal adders  54 ,  55 , and one Combined Output Setter  56 .  
         [0062]     The compensation-type Feedforward MFA controller described in the U.S. Pat. No. 6,556,980 B1 is an example of how a Feedforward MFA controller is designed. Due to the adaptive capability of the feedback MFA controller, we can design a Feedforward MFA controller with a first-order dynamic block as follows:  
                   G   fc     ⁡     (   S   )       =           Y   f     ⁡     (   S   )         D   ⁡     (   S   )         =         K   sf     ⁢     K   cf             T   cf     ⁢   S     +   1           ,           (   13   )             
 
 where D(S) and Y f (S) are the Laplace transform of signals d(t) and y f (t), the input and output signals of the Feedforward controller, respectively, G fs (S) is the Laplace transfer function of the Feedforward controller, K cf  is the feedforward gain, T cf  is the feedforward time constant, and K sf  is the feedforward sign factor. We can select the constants K cf , T cf , and K sf  based on the basic understanding of the process. The system can also be fine tuned by adjusting these constants. 
 
         [0064]     The control signals u 1 (t) and u 2 (t) are calculated based on the following formulas: 
 
 u   1 ( t )= v   1 ( t )+ v   f1 ( t ),  (14a) 
 
 u   2 ( t )= v   2 ( t )+ v   f2 ( t ),  (14b) 
 
 where v 1 (t) and V 2 (t) are the feedback MFA controller outputs and V f1 (t) and v f2 (t) are the Feedforward MFA controller outputs. There may be cases where there is only one active feedforward controller. If FFC1 is inactive, v f1 (t)=0 and then u 1 (t)=v 1 (t). If FFC2 is inactive, v f2 (t)=0 and then u 2 (t)=v 2 (t). 
 
         [0066]      FIG. 4  is a drawing illustrating a mechanism of a Combined Output Setter that can combine  2  controller outputs u 1 (t) and u 2 (t) into one controller output u(t). By moving the knob R  60 , we can adjust the amount of control signals u 1 (t) and u 2 (t) to be contributed to the actual controller output u(t), which can be calculated based on the following formula: 
   u ( t )= Ru   1 ( t )+(1 −R ) u   2 ( t ),  (15)  
 where 0≦u(t)≦100; 0≦R≦1; and R is a constant. 
 
         [0068]     Here we introduce an alternative mechanism for the Combined Output Setter based on the controller gains. The formula to combine multiple controller outputs into one output using controller gains follows:  
                 u   ⁡     (   t   )       =           K   c1         K   c1     +     K   c2         ⁢       u   1     ⁡     (   t   )         +         K   c2         K   c1     +     K   c2         ⁢       u   2     ⁡     (   t   )             ,           (   16   )             
 
 where K c1 , and K c2  are MFA controller gains for SISO MFA 1 and SISO MFA 2, respectively. This allows the 2×1 MFA controller to dynamically tighten the more important loop of the two. We can easily set a higher controller gain for the more important loop to minimize its error as the highest priority and allow the other loop to be relatively in loose control. On the other hand, we can also easily set both controller gains at an equal value so that both loops are treated with equal importance. 
 
         [0070]     Both mechanisms represented in Equations (15) and (16) can be used as the Combined Output Setter  48  and  56  in  FIG. 2  and  FIG. 3 , respectively.  
         [0071]     To expand the design, we can rescale the control output signal u(t) from its 0% to 100% range to an engineering value range by using a linear function. In addition, control limits and constraints can be applied to these signals for safety or other reasons to limit the control actions. These design concepts can be readily applied to all the controllers presented in this patent.  
         [heading-0072]     B. 3-Input-1-Output Model-Free Adaptive Controller  
         [0073]      FIG. 5  illustrates a 3-input-1-output (3×1) MFA controller that controls a 1-input-3-output (1×3) system. The control system comprises a 3-input-1-output (3×1) MFA controller  62 , a 1-input-3-output (1×3) system  64 , and signal adders  65 ,  66 ,  67 ,  68 ,  69 , and  70 . The signals shown in  FIG. 5  are as follows: 
        r 1 (t), r 2 (t), r 3 (t)—Setpoint 1, 2 and 3.     x 1 (t), x 2 (t), x 3 (t)—System outputs of the 1×3 system.     d 1 (t), d 2 (t), d 3 (t)—Disturbance 1, 2, and 3 caused by noise or load changes.     y 1 (t), y 2 (t), y 3 (t)—Measured process variables of the 1×3 system, 
            y 1 (t)=x 1 (t)+d 1 (t); y 2 (t)=x 2 (t)+d 2 (t); and y 3 (t)=x 3 (t)+d 3 (t).    
            e 1 (t), e 2 (t), e 3 (t)—Error between the setpoint and measured process variable, 
            e 1 (t)=r 1 (t)−y 1 (t); e 2 (t)=r 2 (t)−y 2 (t); and e 3 (t)=r 3 (t)−y 3 (t).    
            u(t)—Output of the 3×1 MFA Controller.          
         [0082]     The control objective is for the controller to produce output u(t) to manipulate the manipulated variable so that the measured process variables y 1 (t), y 2 (t), and y 3 (t) track the given trajectory of their setpoints r 1 (t), r 2 (t), and r 3 (t), respectively, under variations of setpoint, disturbance, and process dynamics. In other words, the task of the MFA controller is to minimize the error e 1 (t), e 2 (t), and e 3 (t) in an online fashion.  
         [0083]     Since there is only one manipulated variable, minimizing errors for all three loops may not be possible. The control objective then can be defined as (i) to minimize the error for the most critical loop, or (ii) minimize the error for all 3 loops with no weighting on the importance so that there may be static errors in all loops.  
         [0084]      FIG. 6  illustrates the simplified architecture of a 3-input-1-output Model-Free Adaptive (MFA) controller  78 . It includes 3 SISO MFA controllers  72 ,  73 , and  74 , and a Combined Output Setter  76 . Each of the SISO MFA controllers can be the same as described in the 2×1 MFA controller case. The controller output signals v 1 (t), v 2 (t), and v 3 (t) are used as inputs to the Combined Output Setter  76  to produce the controller output u(t).  
         [0085]      FIG. 7  is a block diagram illustrating the simplified architecture of a 3-input-1-output (3×1) Feedback/Feedforward Model-Free Adaptive (MFA) controller. The 3×1 Feedback/Feedforward MFA controller  89  comprises three SISO MFA controllers  79 ,  80 ,  81 , three Feedforward MFA controllers  82 ,  83 ,  84 , three signal adders  85 ,  86 ,  87 , and one Combined Output Setter  88 . The design of the Feedforward MFA controllers can be the same as described in the 2×1 Feedback/Feedforward MFA controller case.  
         [0086]     The control signals u 1 (t), u 2 (t), and u 3 (t) are calculated based on the following formulas: 
 
 u   1 ( t )= v   1 ( t )+ v   f1 ( t ),  (17a) 
 
 u   2 ( t )=v 2 ( t )+ v   f2 ( t ),  (17b) 
 
 u   3 ( t )= v   3 ( t )+ V   f3 ( t )  (17c) 
 
 where v 1 (t), v 2 (t), and v 3 (t) are the feedback MFA controller outputs and v f1 (t), v f2 (t), and v f3 (t) are the Feedforward MFA controller outputs. If a Feedforward MFA controller is not active, its v fj (t)=0, then u j (t)=v j (t), j=1, 2, 3. 
 
         [0088]      FIG. 8  is a drawing illustrating a mechanism of a Combined Output Setter that can combine  3  controller outputs u 1 (t), u 2 (t), and u 3 (t) into one controller output u(t). By moving the knobs R 1  and R 2 , we can adjust the amount of control signals u 1 (t), u 2 (t), and u 3 (t) to be contributed to the actual controller output u(t), which can be calculated based on the following formula: 
   u ( t )= R   1   u   1 ( t )+ R   2   u   2 ( t )+(1 −R   1   −R   2 ) u   3 ( t ),  (18)  
 where 0≦u(t)≦100; 0≦R 1 &lt;1; 0≦R 2 &lt;1; 0≦R 1 +R 2 ≦1; and R 1  and R 2  are constants. 
 
         [0090]     Similar to the 2×1 case, the controller gain weighted Combined Output Setter algorithm is given in the following formula:  
                 u   ⁡     (   t   )       =           K   c1       K   sum       ⁢       u   1     ⁡     (   t   )         +         K   c2       K   sum       ⁢       u   2     ⁡     (   t   )         +         K   c3       K   sum       ⁢       u   3     ⁡     (   t   )             ,           (     19   ⁢   a     )                   K   sum     =       K   c1     +     K   c2     +     K   c3         ,           (     19   ⁢   b     )             
 
 where K c1 , K c2  and K c3  are MFA controller gains for SISO MFA 1, 2, and 3, respectively. This allows the 3×1 MFA controller to dynamically tighten the most important loop of the three. We can easily set a higher controller gain for the most important loop to minimize its error and allow the other loops to be relatively in loose control. We can also set these 3 controller gains at an equal value so that all loops are treated with equal importance. Both mechanisms represented in Equations (18) and (19) can be used as the Combined Output Setter  76  and  88  in  FIG. 6  and  FIG. 7 , respectively. 
 
 C. Single-Input-Multi-Output Model-Free Adaptive Controller 
 
         [0093]      FIG. 9  illustrates an M-input-1-output (M×1) MFA controller that controls a 1-input-M-output (1×M) system. The control system comprises an M-input-1-output (M×1) MFA controller  94 , a 1-input-M-output (1×M) system  96 , and signal adders  98 ,  99 ,  100 ,  102 ,  103 , and  104 . The signals shown in  FIG. 9  are as follows: 
        r 1 (t), r 2 (t), . . . , r M (t)—Setpoint 1, 2, . . . , M.     x 1 (t), x 2 (t), . . . , x M (t)—System outputs of the 1×M system.     d 1 (t), d 2 (t), . . . , d M (t)—Disturbance 1, 2, . . . , M caused by noise or load changes.     y 1 (t), y 2 (t), . . . , y M (t)—Measured process variables of the 1×M system, 
            y 1 (t)=x 1 (t)+d 1 (t); y 2 (t)=x 2 (t)+d 2 (t); . . . ; and y M (t)=x M (t)+d M (t).    
            e 1 (t), e 2 (t), . . . , e M (t)—Error between the setpoint and measured process variable, 
            e 1 (t)=r 1 (t)−y 1 (t); e 2 (t)=r 2 (t)−y 2 (t); . . . ; and e M (t)=r M (t)−y M (t).    
            u(t)—Output of the M×1 MFA Controller.          
         [0102]     The control objective is for the controller to produce output u(t) to manipulate the manipulated variable so that the measured process variables y 1 (t), y 2 (t), . . . , y M (t) track the given trajectory of their setpoints r 1 (t), r 2 (t), . . . , r M (t), respectively, under variations of setpoint, disturbance, and process dynamics. In other words, the task of the MFA controller is to minimize the error e 1 (t), e 2 (t), . . . , e M (t) in an online fashion.  
         [0103]     Since there is only one manipulated variable, minimizing errors for all M loops may not be possible. The control objective then can be defined as (i) to minimize the error for the most critical loop, or (ii) to minimize the error for all M loops with no weighting on the importance so that there may be static errors in all loops.  
         [0104]      FIG. 10  illustrates the simplified architecture of an M-input-1-output Model-Free Adaptive (MFA) controller  114 . It includes M SISO MFA controllers  106 ,  107 ,  108  and a Combined Output Setter  112 . Each of the SISO MFA controllers can be the same as described in the 2×1 MFA controller case. The controller output signals v 1 (t), v 2 (t), . . . , v M (t) are used as inputs to the Combined Output Setter  112  to produce the controller output u(t).  
         [0105]      FIG. 11  is a block diagram illustrating the simplified architecture of an M-input-1-output (M×1) Feedback/Feedforward Model-Free Adaptive (MFA) controller. The M×1 Feedback/Feedforward MFA controller  128  comprises M SISO MFA controllers  116 ,  117 ,  118 , M Feedforward MFA controllers  119 ,  120 ,  121 , M signal adders  122 ,  123 ,  124 , and one Combined Output Setter  126 . The design of the Feedforward MFA controllers can be the same as described in the 2×1 Feedback/Feedforward MFA controller case.  
         [0106]     The control signals u 1 (t), u 2 (t), . . . , u M (t) are calculated based on the following formulas: 
 
 u   1 ( t )= v   1 ( t )+ v   f1 ( t ),  (20a) 
 
 u   2 ( t )= v   2 ( t )+ v   2 ( t )  (20b) 
 
. . . 
 
 u   M ( t )= v   M ( t )+ v   fM ( t ),  (20c) 
 
 where v 1 (t), v 2 (t), . . . , v M (t) are the feedback MFA controller outputs and v f1 (t), v f2 (t), . . . , v fM (t) are the feedforward MFA controller outputs. If a Feedforward MFA controller is not active, its v fi (t)=0, then u j (t)=v j (t), j=1, 2, . . . , M. 
 
         [0108]      FIG. 12  is a drawing illustrating a mechanism of a Combined Output Setter that can combine M controller outputs u 1 (t), u 2 (t), . . . , u M (t) into one controller output u 1 (t). By moving the knobs R 1 , R 2 , . . . , R M-1 , we can adjust the amount of control signals u 1 (t), u 2 (t), . . . , u M (t) to be contributed to the actual controller output u(t), which can be calculated based on the following formula: 
   u ( t )= R   1   u   1 ( t )+ R   2   u   2 ( t )+ . . . +(1 −R   1   − . . . −R   M-1 ) u   M ( t )  (21)  
 where M=3, 4, 5, . . . ; 0≦u(t)≦100; 0≦R 1 ≦1; 0≦R 2 ≦1; . . . 0≦R M-1 ≦1; 0≦R 1 +R 2 + . . . +R M-1 ≦1; and R 1 , R 2 , R M-1  are constants. 
 
         [0110]     Similarly, a controller gain weighted Combined Output Setter algorithm is given in the following formula:  
                 u   ⁡     (   t   )       =           K   c1       K   sum       ⁢       u   1     ⁡     (   t   )         +         K   c2       K   sum       ⁢       u   2     ⁡     (   t   )         +   …   +         K   cM       K   sum       ⁢       u   M     ⁡     (   t   )             ,           (     22   ⁢   a     )                   K   sum     =       K   c1     +     K   c2     +   …   +     K   cM         ,           (     22   ⁢   b     )             
 
 where K c1 , K c2 , . . . , K cM  are MFA controller gains for SISO MFA 1, 2, and M, respectively. This allows the M×1 MFA controller to dynamically tighten the most important loop. We can easily set a higher controller gain for the most important loop to minimize its error as the highest priority and allow the other loops to be relatively in loose control. We can also set these M controller gains at an equal value so that all loops are treated with equal importance. 
 
         [0112]     Both mechanisms represented in Equations (21) and (22) can be used as the Combined Output Setter  112  and  126  in  FIG. 10  and  FIG. 11 , respectively.  
         [0113]     To expand the design, we can rescale the control output signal u(t) from its 0% to 100% range to an engineering value range by using a linear function. In addition, control limits and constraints can be applied to these signals for safety or other reasons to limit the control actions.  
         [heading-0114]     D. 2-Input-1-Output PID Controller  
         [0115]      FIG. 13  illustrates a 2-input-1-output (2×1) PID controller that controls a 1-input-2-output (1×2) system. The control system comprises a 2-input-1-output (2×1) PID controller  134 , a 1-input-2-output (1×2) system  136 , and signal adders  137 ,  138 ,  139 , and  140 . Within the 2×1 PID controller  134 , there are two SISO PID controllers  142 ,  143  and one Combined Output Setter  144 . The signals shown in  FIG. 13  are as follows: 
        r 1 (t), r 2 (t)—Setpoint 1 and Setpoint 2.     x 1 (t), x 2 (t)—System outputs of the 1×2 system.     d 1 (t), d 2 (t)—Disturbance 1 and 2 that are caused by noise or load changes.     y 1 (t), y 2 (t)—Measured process variables of the 1×2 system, 
            y 1 (t)=x 1 (t)+d 1 (t); and y 2 (t)=x 2 (t)+d 2 (t).    
            e 1 (t), e 2 (t)—Error between the setpoint and measured process variable, 
            e 1 (t)=r 1 (t)−y 1 (t); and e 2 (t)=r 2 (t)−y 2 (t).    
            u(t)—Output of the 2×1 PID Controller.     u 1 (t), u 2 (t)—Output of SISO PID 1 and PID 2.          
         [0125]     The control objective is for the controller to produce output u(t) to manipulate the manipulated variable so that the measured process variables y 1 (t) and y 2 (t) track the given trajectory of their setpoints r 1 (t) and r 2 (t), respectively. In other words, the task of the 2×1 PID controller is to minimize the error e 1 (t) and e 2 (t) in an online fashion.  
         [0126]     Since there is only one manipulated variable, minimizing errors for both loops may not be possible. The control objective then can be defined as (i) to minimize the error for the more critical loop of the two, or (ii) to minimize the error for both loops with no weighting on the importance so that there may be static errors in both loops.  
         [0127]     The standard PID algorithm has the following form:  
                   u   j     ⁡     (   t   )       =       K   p     ⁢     {       e   ⁡     (   t   )       +       1     T   i       ⁢     ∫       e   ⁡     (   t   )       ⁢     ⅆ   t           +       T   d     ⁢       ⅆ     e   ⁡     (   t   )           ⅆ   t           }         ,           (   23   )             
 
 where K p  is the Proportional Gain, T i  is the Integral Time in second/repeat, T d  is the Derivative Time in repeat/second, and u j (t) is the output of the jth PID, j=1, 2. 
 
         [0129]     The Combined Output Setter illustrated in  FIG. 4  can be used to combine the two SISO PID controller outputs u 1 (t) and u 2 (t) into one controller output u(t). By moving the knob R, we can adjust the amount of control signals u 1 (t) and u 2 (t) to be contributed to the actual controller output u(t), which can be calculated based on the following formula: 
 
 u ( t )= Ru   1 ( t )+(1 −R ) u   2 ( t ),  (24) 
 
 where 0≦u(t)≦100; 0≦R≦1; and R is a constant. 
 
         [0131]     A controller gain weighted Combined Output Setter algorithm is given in the following formula:  
                 u   ⁡     (   t   )       =           K     p   ⁢           ⁢   1           K     p   ⁢           ⁢   1       +     K     p   ⁢           ⁢   2           ⁢       u   1     ⁡     (   t   )         +         K     p   ⁢           ⁢   2           K     p   ⁢           ⁢   1       +     K     p   ⁢           ⁢   2           ⁢       u   2     ⁡     (   t   )             ,           (   25   )             
 
 where K p1  and K p2  are proportional gains for PID 1 and PID 2, respectively. This allows the 2×1 PID controller to dynamically tighten the more important loop of the two. We can easily set a higher proportional gain for the more important loop to minimize its error as the highest priority and allow the other loop to be relatively in loose control. We can also easily set both controller gains at an equal value so that both loops are treated with equal importance. 
 
         [0133]     Both mechanisms represented in Equations (24) and (25) can be used as the Combined Output Setter  144  in  FIG. 13 .  
         [0134]     To expand the design, we can rescale the control output signal u(t) from its 0% to 100% range to an engineering value range by using a linear function. In addition, control limits and constraints can be applied to these signals for safety or other reasons to limit the control actions.  
         [0135]     Since PID is not an adaptive controller, proper manual tuning of its parameters K p , T i , and T d  is required. When process dynamics change, frequent manual tuning of the parameters may be required. Model-Free Adaptive (MFA) controllers will outperform the PIDs because of their adaptive capability.  
         [heading-0136]     E. Multi-Input-Single-Output PID Controller  
         [0137]      FIG. 14  illustrates an M-input-1-output (M×1) PID controller that controls a 1-input-M-output (1×M) system. The control system comprises an M-input-1-output (M×1) PID controller  146 , a 1-input-M-output (1×M) system  148 , and signal adders  149 ,  150 ,  151 ,  152 ,  153 , and  154 . Within the M×1 PID controller  146 , there are M SISO PID controllers  155 ,  156 ,  157  and one Combined Output Setter  158 . The signals shown in  FIG. 14  are as follows: 
        r 1 (t), r 2 (t), . . . , r M (t)—Setpoint 1, 2, . . . , M.     x 1 (t), x 2 (t), . . . , X M (t)—System outputs of the 1×M system.     d 1 (t), d 2 (t), . . . , d M (t)—Disturbance 1, 2, . . . , M caused by noise or load changes.     y 1 (t), y 2 (t), . . . , y M (t)—Measured process variables of the 1×M system, 
            y 1 (t)=x 1 (t)+d 1 (t); y 2 (t)=x 2 (t)+d 2 (t); . . . ; and y M (t)=x M (t)+d M (t).    
            e 1 (t), e 2 (t), . . . , e M (t)—Error between the setpoint and measured process variable, 
            e 1 (t)=r 1 (t)−y 1 (t); e 2 (t)=r 2 (t)−y 2 (t); . . . ; and e M (t)=r M (t)−y M (t).    
            u(t)—Output of the M×1 PID Controller.     u 1 (t), u 2 (t), . . . , u M (t)—Output of SISO PID 1, 2, . . . , M, respectively.          
         [0147]     The control objective is for the controller to produce output u(t) to manipulate the manipulated variable so that the measured process variables y 1 (t), y 2 (t), . . . , y M (t) track the given trajectory of their setpoints r 1 (t), r 2 (t), . . . , r M (t), respectively. In other words, the task of the PID controller is to minimize the error e 1 (t), e 2 (t), . . . , e M (t) in an online fashion.  
         [0148]     Since there is only one manipulated variable, minimizing errors for all M loops may not be possible. The control objective then can be defined as (i) to minimize the error for the most critical loop, or (ii) to minimize the error for all M loops with no weighting on the importance so that there may be static errors in all loops.  
         [0149]     The standard PID algorithm has the following form:  
                   u   j     ⁡     (   t   )       =       K   p     ⁢     {       e   ⁡     (   t   )       +       1     T   i       ⁢     ∫       e   ⁡     (   t   )       ⁢     ⅆ   t           +       T   d     ⁢       ⅆ     e   ⁡     (   t   )           ⅆ   t           }         ,           (   26   )             
 
 where K p  is the Proportional Gain, T i  is the Integral Time in second/repeat, T d  is the Derivative Time in repeat/second, and u j (t) is the output of the jth PID, j=1,2, . . . M. 
 
         [0151]     Similarly, we can expand the 2×1 case to the M×1 case. The Combined Output Setter illustrated in  FIG. 12  can be used to combine M PID controller outputs u 1 (t), u 2 (t), . . . , u M (t) into one controller output u(t). By moving the knobs R 1 , R 2 , . . . , R M-1 , we can adjust the amount of control signals u 1 (t), u 2 (t), . . . , u M (t) to be contributed to the actual controller output u(t), which can be calculated based on the following formula: 
 
 u ( t )  R   1   u   1 ( t )+ R   2   u   2 ( t )++(1−R 1   − . . . −R   M-1 ) u   M ( t ),  (27) 
 
 where M=3, 4, 5, . . . , 0≦u(t)≦100; 0≦R 1 ≦1; 0≦R 2 ≦1; . . . ; 0≦R M-1 ≦1; 0≦R 1 +R 2 + . . . +R M-1 ≦1; and R 1 , R 2  . . . , R M-1 , are constants. 
 
         [0153]     Similarly, a controller gain weighted Combined Output Setter algorithm is given in the following formula:  
                 u   ⁡     (   t   )       =           K     p   ⁢           ⁢   1         K   sum       ⁢       u   1     ⁡     (   t   )         +         K     p   ⁢           ⁢   2         K   sum       ⁢       u   2     ⁡     (   t   )         +   …   +         K   pM       K   sum       ⁢       u   M     ⁡     (   t   )             ,           (     28   ⁢   a     )                   K   sum     =       K     p   ⁢           ⁢   1       +     K     p   ⁢           ⁢   2       +   …   +     K   pM         ,           (     28   ⁢   b     )             
 
 where K p1 , K p2 , . . . , K pM  are proportional gains for PID 1, 2, . . . , M, respectively. This allows the M×1 PID controller to dynamically tighten the most important loop. We can easily set a higher proportional gain for the most important loop to minimize its error as the highest priority and allow the other loops to be relatively in loose control. We can also set these M proportional gains at an equal value so that all loops are treated with equal importance. Both mechanisms represented in Equations (27) and (28) can be used as the Combined Output Setter  158  in  FIG. 14 . 
 
         [0155]     Since PID is a general-purpose controller, the 2×1 and M×1 PID controllers presented in this patent apply to all alternative forms of PID algorithms. They may be P only, PI, PD, or PID controllers, in analog or digital formulas, with various definitions of variables, parameters and units, etc. This Mxl PID controller with the Combined Output Setter will be more powerful than a single-input-single-output PID controller when controlling a 1×M system. However, since it is not an adaptive controller, it may not be able to handle large dynamic changes in the systems. Proper manual tuning of PID parameters is always required. The M×1 Model-Free Adaptive (MFA) controller presented in this patent is a more preferred solution for controlling a 1×M system.  
         [heading-0156]     F. Multi-Input-Single-Output Controller  
         [0157]      FIG. 15  illustrates a multi-input-single-output (MISO) controller that controls a single-input-multi-output (SIMO) system. The control system comprises a multi-input-single-output (MISO) controller  160 , a single-input-multi-output (SIMO) system  162 , and signal adders  164 ,  165 ,  166 ,  168 ,  169 , and  170 . Within the MISO controller  160 , there are M single-input-single-output (SISO) controllers  172 ,  173 ,  174  and one Combined Output Setter  176 . The signals shown in  FIG. 15  are as follows: 
        r 1 (t), r 2 (t), . . . , r M (t)—Setpoint 1, 2, . . . , M.     x 1 (t), x 2 (t), . . . , x M (t)—System outputs of the SIMO system.     d 1 (t), d 2 (t), . . . , d M (t)—Disturbance 1, 2, . . . , M caused by noise or load changes.     y 1 (t), y 2 (t), . . . , y M (t)—Measured process variables of the SIMO system, 
            y 1 (t)=x 1 (t)+d 1 (t); y 2 (t)=x 2 (t)+d 2 (t); . . . ; and y M (t)=x M (t)+d M (t).    
            e 1 (t), e 2 (t), . . . , e M (t)—Error between the setpoint and measured process variable, 
            e 1 (t)=r 1 (t)−y 1 (t); e 2 (t)=r 2 (t)—y 2 (t); . . . ; and e M (t)=r M (t)−y M (t).    
            u(t)—Output of the MISO Controller.     u 1 (t), u 2 (t), . . . , u M (t)—Output of SISO controller C 1 , C 2 , . . . , C M .          
         [0167]     The control objective is for the controller to produce output u(t) to manipulate the manipulated variable so that the measured process variables y 1 (t), y 2 (t), . . . , y M (t) track the given trajectory of their setpoints r 1 (t), r 2 (t), . . . , r M (t), respectively. In other words, the task of the MISO controller is to minimize errors e 1 (t), e 2 (t), . . . , e M (t) in an online fashion.  
         [0168]     Since there is only one manipulated variable, minimizing errors for all M loops may not be—possible. The control objective then can be defined as (i) to minimize the error for the most critical loop, or (ii) to minimize the error for all M loops with no weighting on the importance so that there may be static errors in all loops.  
         [0169]     The MISO controller comprises M single-input-single-output (SISO) controllers  172 ,  173 ,  174 . Without losing generality, we assume the control outputs for the SISO controllers C 1 , C 2 , . . . , C M  are calculated based on the following formulas, respectively: 
 
 u   1 ( t )= f ( e   1 ( t ), t, P 11 , P 12 , . . . P 1l )  (29a) 
 
 u   2  ( t )= f ( e   2 ( t ), t, P 21 , P 22 , . . . , P 2l ,  (29b) 
 
. . . 
 
 u   M ( t )= f ( e   M ( t ), i, P M1 , P M2 , . . . , P Mt ),  (29c) 
 
 where t is time, P 11 , P 12 , . . . , P 1l  are tuning parameters for controller C 1 , P 21 , P 22 , . . . , P 2l  are tuning parameters for controller C 2 , . . . , and P M1 , P M2 , . . . , P Ml  are tuning parameters for controller C M . 
 
         [0171]     The Combined Output Setter illustrated in  FIG. 12  can be used to combine M SISO controller outputs u 1 (t), u 2 (t), . . . , u M (t) into one controller output u(t). By moving the knobs R 1 , R 2 , . . . , R M-1 , we can adjust the amount of control signals u 1 (t), u 2 (t), . . . , u M (t) to be contributed to the actual controller output u(t), which can be calculated based on the following formula: 
 
 u ( t )= R   1   u   1 ( t )+ R   2   u   2 ( t )+ . . . +(1 −R   1   − . . . R   M-1 ) u   M ( t )  (30) 
 
 where M=3, 4, 5 . . . , 0≦u(t)≦100; 0≦R 1 ≦1; 0≦R 2 ≦1; . . . ; 0≦R M-1 ≦1; 0≦R 1 +R 2 + . . . +R M-1 &lt;1; and R 1 ., R 2  . . . , R M-1  are constants. 
 
         [0173]     Similarly, a controller gain weighted Combined Output Setter algorithm is given in the following formula, assuming the controller gain is the first parameter P 1 :  
                 u   ⁡     (   t   )       =           P   11       K   sum       ⁢       u   1     ⁡     (   t   )         +         P   21       K   sum       ⁢       u   2     ⁡     (   t   )         +   …   +         P   M1       K   sum       ⁢       u   M     ⁡     (   t   )             ,           (     31   ⁢   a     )                   K   sum     =       P   11     +     P   21     +   …   +     P   M1         ,           (     31   ⁢   b     )             
 
 where P 11 , P 21 , . . . , P Ml  are gains for controller C 1 , C 2 , . . . , C M , respectively. This allows the MISO controller to dynamically tighten the most important loop. We can easily set a higher controller gain for the most important loop to minimize its error as the highest priority and allow the other loops to be relatively in loose control. We can also set these M controller gains at an equal value so that all loops are treated with equal importance. Both mechanisms represented in Equations (30) and (31) can be used as the Combined Output Setter  176  in  FIG. 15 . 
 
         [0175]     This is a general case example of converting M single-input-single-output (SISO) controllers including but not limited to Model-Free Adaptive (MFA) controllers, or proportional-integral-derivative (PID) controllers, or any other form of SISO controllers to a multi-input-single-output (MISO) controller to control a single-input-multi-output (SIMO) system.