Abstract:
Modem rotary machine production requires built-in fault detection and diagnoses. The occurrence of faults, e.g. increased friction or loose bonds has to be detected as early as possible. Theses faults generate a nonlinear behavior. Therefore, a method for fault detection and diagnosis of a rotary machine is presented. Based on a rotor system model for the faulty and un-faulty case, subspace-based identification methods are used to compute singular values that are used as features for fault detection. The method is tested on an industrial rotor balancing machine.

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
       [0001]    Applicants claim priority under 35 U.S.C. §119 of European Application No. 07017478.4 filed on Sep. 6, 2007. 
       BACKGROUND OF THE INVENTION 
       [0002]    The invention relates to a method for fault detection and diagnosis of a rotary machine, in particular a balancing machine, wherein a rotor having an imbalance is rotated and excites a vibration in the rotary machine due to the imbalance-caused force, and wherein the rotational speed of the rotor and the vibrations are measured in order to obtain input data quantitative for the rotational speed and the vibrations. 
       SUMMARY OF THE INVENTION 
       [0003]    A method for fault detection and diagnosis of a rotary machine, in particular a balancing machine is provided. A rotor having an imbalance is rotated and excites a vibration in the rotary machine due to the imbalance-caused force. The rotational speed of the rotor and the vibrations are measured in order to obtain input data quantitative for the rotational speed and the vibrations. 
         [0004]    In accordance with the method, it is assumed that a process of dynamic behavior of the rotary machine can be modeled by a linear system in the un-faulty state. An over-determined set of linear equations is formed, which contains input and output data of the process and unknown states of the assumed linear system. The number of states needed to accurately model the process is extracted by using mathematical operations such as orthogonal or oblique projections to form a matrix of which the rank equals the assumed linear system. Singular values are computed by using Singular Value Decomposition for obtaining an approximate indication for the order of the assumed linear state. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0005]      FIG. 1  shows a rotary system model for normal condition; 
           [0006]      FIG. 2  shows the fault state of sliding friction in sensor connection; 
           [0007]      FIG. 3  shows the fault state of sliding friction between bearing support socket and ground; and 
           [0008]      FIG. 4  shows singular values for normal and faulty states. 
       
    
    
     DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS 
     1 Introduction 
       [0009]    Fault detection and diagnosis is increasingly important for modern rotary machines. Currently, mostly limit checking and periodic maintenance cycles are used to detect faults. Sometimes signal-based fault detection is applied. However, these methods mainly rely on the knowledge of experts. This situation can be improved by model-based fault detection. As more information (proper excitation, process model, several measurements) is used, more accurate fault detection can be performed. Standard rotary systems behave basically linear. In case of specific faults, e.g. sliding friction, loose bonds and motion blocks, linear relations no longer hold. An indication for the nonlinearity can be used to detect these faults. Taking into account the noisy environment the method presented will use subspace approaches to estimate singular values. These singular values can be used to detect these faults. 
       2 Modeling 
       [0010]    In order to design a model-based fault detection and diagnosis system, the dynamic behavior of the rotor system needs to be modeled. In a first step, a general model with two degrees of freedom for stiff rotors is given. For lower rotary speeds a simplified model can be applied. 
         [0011]    It is assumed that the rotor is not fully balanced, so that an imbalance force F u  and torque M u  exist. The rotor is situated on two independent bearing supports. Their movement speeds in horizontal plane are denoted by {dot over (x)} 1 ,{dot over (x)} 2 . Plunger coil sensors are used to measure these speeds, resulting in measurement values {dot over (s)} 1 ,{dot over (s)} 2  (see also  FIG. 1 ). 
       2.1 Model for Stiff Rotors 
       [0012]    Rotor and bearings are assumed to be stiff, the ground connection of the two bearing supports is modeled by two spring-damper systems. It is assumed furthermore that the sensor force feedback on the rotor movement can be neglected. 
         [0000]      m r &gt;&gt;m s1  m r &gt;&gt;m s2    
         [0000]      c 1 &gt;&gt;c s1  c 2 &gt;&gt;c s2    
         [0013]    As long as the system stays in normal condition, it can be described by a linear state space system. 
         [0000]        {dot over (x)}   m ( t )= A   m   x   m ( t )+ B   m   u   m ( t )  (1) 
         [0000]        y   m ( t )= C   m   x   m ( t ) 
         [0014]    Applying Newton&#39;s law of motion for a rotary mass it follows 
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         [0015]    However, this detailed model is not needed for the regarded rotary system. As it is operated in sub-critical region, certain simplifications can be made. 
       2.2 Reduced Model 
       [0016]    Under the assumption that the machine is driven with sub-critical rotary speed, i.e. (ω r &lt;&lt;ω crit ) where ω r  is the actual rotary speed and ω crit  is the lower of the two critical speeds according to system (1) it can be assumed that 
         [0000]      m 1 {umlaut over (x)} 1 &lt;&lt;d 1 {dot over (x)} 1 &lt;&lt;c 1 x 1    
         [0000]      m 2 {umlaut over (x)} 2 &lt;&lt;d 2 {dot over (x)} 2 &lt;&lt;c 2 x 2    
         [0017]    With these simplifications the model reduces to a model of order two: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
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         [0018]    The system is observable and controllable, the poles are on the stability limit. Inputs and outputs are exchanged in order to match the state space structure. Discretization with small sampling time T 0  leads to 
         [0000]        {dot over (x)}   d ( k+ 1)= A   d   x   d ( k )+ B   d   u   d ( k )  (16) 
         [0000]        y   d ( k )= C   d   x   d ( k )  (17) 
         [0000]    
       
         
           
             
               
                 
                   
                     
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         [0019]    This model will be used as basis for fault detection. 
       2.3 Nonlinear Fault States 
       [0020]    The presented method is used to detect two fault states where sliding friction occurs. 
       2.3.1 Sliding Friction in Sensor Connection 
       [0021]    If the sensor is not connected properly to the left moving rotor support, the force is propagated via sliding friction. The propagated force F Rs =f({dot over (x)} 1 −{dot over (s)} 1 ) is modeled as Coulomb dry friction. The sensor dynamics are described by the frequency response G s (w). 
         [0000]    
       
         
           
             
               
                 
                   
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         [0022]    The sensor force feedback on the rotor movement is neglected. Thus, the relation is nonlinear in the outputs equation only. 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
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                                 ) 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   26 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       u 
                       p 
                     
                      
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       ( 
                       
                         
                           
                             F 
                             u 
                           
                            
                           
                             ( 
                             t 
                             ) 
                           
                         
                          
                         
                             
                         
                          
                         
                           
                             M 
                             u 
                           
                            
                           
                             ( 
                             t 
                             ) 
                           
                         
                       
                       ) 
                     
                     T 
                   
                 
               
               
                 
                   ( 
                   27 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       x 
                       p 
                     
                      
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       ( 
                       
                         
                           
                             
                               x 
                               . 
                             
                             1 
                           
                            
                           
                             ( 
                             t 
                             ) 
                           
                         
                          
                         
                             
                         
                          
                         
                           
                             x 
                             1 
                           
                            
                           
                             ( 
                             t 
                             ) 
                           
                         
                          
                         
                             
                         
                          
                         
                           
                             
                               x 
                               . 
                             
                             2 
                           
                            
                           
                             ( 
                             t 
                             ) 
                           
                         
                          
                         
                             
                         
                          
                         
                           
                             x 
                             2 
                           
                            
                           
                             ( 
                             t 
                             ) 
                           
                         
                       
                       ) 
                     
                     T 
                   
                 
               
               
                 
                   ( 
                   28 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       y 
                       p 
                     
                      
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       ( 
                       
                         
                           
                             
                               s 
                               . 
                             
                             1 
                           
                            
                           
                             ( 
                             t 
                             ) 
                           
                         
                          
                         
                             
                         
                          
                         
                           
                             
                               s 
                               . 
                             
                             2 
                           
                            
                           
                             ( 
                             t 
                             ) 
                           
                         
                       
                       ) 
                     
                     T 
                   
                 
               
               
                 
                   ( 
                   29 
                   ) 
                 
               
             
             
               
                 
                   
                     A 
                     p 
                   
                   = 
                   
                     A 
                     m 
                   
                 
               
               
                 
                   ( 
                   30 
                   ) 
                 
               
             
             
               
                 
                   
                     B 
                     p 
                   
                   = 
                   
                     B 
                     m 
                   
                 
               
               
                 
                   ( 
                   31 
                   ) 
                 
               
             
           
         
       
     
       2.3.2 Sliding Friction in Ground Connection 
       [0023]    If the left rotor support is not properly connected to the ground, the bearing support socket (mass m g ) may move on the ground. FIG. ( 3 ) shows the dynamic behavior of this fault state. It is assumed that dry Coulomb friction persists between rotor support socket and the ground. 
         [0024]    The dynamics may be described by a linear system with nonlinear feedback according to FIG. ( 3 ). 
         [0000]        {dot over (x)}   1 ( t )= A   1   x   1 ( t )+ B   1   u   1 ( t ) 
         [0000]        y   1 ( t )= C   1   x   1 ( t )  (32) 
         [0000]    with 
         [0000]    
       
         
           
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         u 
                         1 
                       
                        
                       
                         ( 
                         t 
                         ) 
                       
                     
                     = 
                     
                       
                         ( 
                         
                           
                             
                               F 
                               u 
                             
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
                            
                           
                               
                           
                            
                           
                             
                               M 
                               u 
                             
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
                            
                           
                               
                           
                            
                           
                             
                               F 
                               
                                 R 
                                  
                                 
                                     
                                 
                                  
                                 1 
                               
                             
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
                         
                         ) 
                       
                       T 
                     
                   
                 
               
               
                 
                   ( 
                   33 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         x 
                         1 
                       
                        
                       
                         ( 
                         t 
                         ) 
                       
                     
                     = 
                     
                       
                         ( 
                         
                           
                             
                               
                                 x 
                                 . 
                               
                               1 
                             
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
                            
                           
                               
                           
                            
                           
                             
                               x 
                               1 
                             
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
                            
                           
                               
                           
                            
                           
                             
                               
                                 x 
                                 . 
                               
                               2 
                             
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
                            
                           
                               
                           
                            
                           
                             
                               x 
                               2 
                             
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
                            
                           
                               
                           
                            
                           
                             
                               
                                 x 
                                 . 
                               
                               g 
                             
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
                            
                           
                               
                           
                            
                           
                             
                               x 
                               g 
                             
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
                         
                         ) 
                       
                       T 
                     
                   
                 
               
               
                 
                   ( 
                   34 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         y 
                         1 
                       
                        
                       
                         ( 
                         t 
                         ) 
                       
                     
                     = 
                     
                       
                         ( 
                         
                           
                             
                               
                                 s 
                                 . 
                               
                               1 
                             
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
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                                 s 
                                 . 
                               
                               2 
                             
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
                         
                         ) 
                       
                       T 
                     
                   
                 
               
               
                 
                   ( 
                   35 
                   ) 
                 
               
             
             
               
                 
                   
                     A 
                     1 
                   
                   = 
                   
                     ( 
                     
                       
                         
                           
                               
                           
                         
                         
                           
                               
                           
                         
                         
                           
                               
                           
                         
                         
                           
                               
                           
                         
                         
                           
                             
                               ( 
                               
                                 
                                   
                                     m 
                                     r 
                                   
                                    
                                   
                                     d 
                                     1 
                                   
                                    
                                   
                                     l 
                                     1 
                                     2 
                                   
                                 
                                 + 
                                 
                                   
                                     ⊖ 
                                     r 
                                   
                                    
                                   
                                     d 
                                     1 
                                   
                                 
                               
                               ) 
                             
                             
                               
                                 m 
                                 r 
                               
                                
                               
                                 ⊖ 
                                 r 
                               
                             
                           
                         
                         
                           
                             
                               ( 
                               
                                 
                                   
                                     m 
                                     r 
                                   
                                    
                                   
                                     c 
                                     1 
                                   
                                    
                                   
                                     l 
                                     1 
                                     2 
                                   
                                 
                                 + 
                                 
                                   
                                     ⊖ 
                                     r 
                                   
                                    
                                   
                                     c 
                                     1 
                                   
                                 
                               
                               ) 
                             
                             
                               
                                 m 
                                 r 
                               
                                
                               
                                 ⊖ 
                                 r 
                               
                             
                           
                         
                       
                       
                         
                           
                               
                           
                         
                         
                           
                             A 
                             m 
                           
                         
                         
                           
                               
                           
                         
                         
                           
                               
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                               
                           
                         
                         
                           
                               
                           
                         
                         
                           
                               
                           
                         
                         
                           
                               
                           
                         
                         
                           
                             
                               ( 
                               
                                 
                                   
                                     - 
                                     
                                       l 
                                       2 
                                     
                                   
                                    
                                   
                                     m 
                                     r 
                                   
                                    
                                   
                                     l 
                                     1 
                                   
                                    
                                   
                                     d 
                                     1 
                                   
                                 
                                 + 
                                 
                                   
                                     ⊖ 
                                     r 
                                   
                                    
                                   
                                     d 
                                     1 
                                   
                                 
                               
                               ) 
                             
                             
                               
                                 m 
                                 r 
                               
                                
                               
                                 ⊖ 
                                 r 
                               
                             
                           
                         
                         
                           
                             
                               ( 
                               
                                 
                                   
                                     - 
                                     
                                       l 
                                       2 
                                     
                                   
                                    
                                   
                                     m 
                                     r 
                                   
                                    
                                   
                                     l 
                                     1 
                                   
                                    
                                   
                                     c 
                                     1 
                                   
                                 
                                 + 
                                 
                                   
                                     ⊖ 
                                     r 
                                   
                                    
                                   
                                     c 
                                     1 
                                   
                                 
                               
                               ) 
                             
                             
                               
                                 m 
                                 r 
                               
                                
                               
                                 ⊖ 
                                 r 
                               
                             
                           
                         
                       
                       
                         
                           
                               
                           
                         
                         
                           
                               
                           
                         
                         
                           
                               
                           
                         
                         
                           
                               
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             
                               c 
                               1 
                             
                             
                               m 
                               g 
                             
                           
                         
                         
                           
                             
                               d 
                               1 
                             
                             
                               m 
                               g 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           
                             
                               - 
                               
                                 c 
                                 1 
                               
                             
                             
                               m 
                               g 
                             
                           
                         
                         
                           
                             
                               - 
                               
                                 d 
                                 1 
                               
                             
                             
                               m 
                               g 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                     
                      
                     
                         
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   36 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       B 
                       1 
                     
                     = 
                     
                       ( 
                       
                         
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             0 
                           
                         
                         
                           
                             
                               B 
                               m 
                             
                           
                           
                             
                                 
                             
                           
                           
                             0 
                           
                         
                         
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             0 
                           
                         
                         
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             
                               1 
                               / 
                               
                                 m 
                                 g 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   37 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       C 
                       1 
                     
                     = 
                     
                       ( 
                       
                         
                           C 
                           m 
                         
                          
                         
                           
                             
                               0 
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               0 
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   38 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       F 
                       Ri 
                     
                     = 
                     
                       
                         F 
                         d 
                       
                        
                       
                         sign 
                          
                         
                           ( 
                           
                             
                               
                                 x 
                                 . 
                               
                               g 
                             
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   39 
                   ) 
                 
               
             
           
         
       
     
       2.3.3 Summary 
       [0025]    The two described fault states introduce nonlinear behavior into the state space relation, either in the output equation or in the system equation. An approximation of this behavior with the reduced model according to section 2.2 turns out to be inaccurate. 
       2.4 Imbalance Force and Torque 
       [0026]    It is assumed that the rotor is not fully balanced. The remaining imbalances cause an imbalance force F u  and torque M u . It is assumed that the rotor speed ω r  and rotor roll angle φ r  are known and the imbalance amplitudes A u1 , A u2  and angles φ u1 , φ u2  are measured or known. The imbalance force and torque can be modeled as 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       F 
                       u 
                     
                      
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       A 
                       
                         u 
                          
                         
                             
                         
                          
                         1 
                       
                     
                      
                     
                       
                         ω 
                         r 
                         2 
                       
                        
                       
                         ( 
                         t 
                         ) 
                       
                     
                      
                     
                       cos 
                        
                       
                         ( 
                         
                           
                             
                               ϕ 
                               r 
                             
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
                           + 
                           
                             ϕ 
                             
                               u 
                                
                               
                                   
                               
                                
                               1 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   40 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       + 
                       
                         A 
                         
                           u 
                            
                           
                               
                           
                            
                           2 
                         
                       
                     
                      
                     
                       
                         ω 
                         r 
                         2 
                       
                        
                       
                         ( 
                         t 
                         ) 
                       
                     
                      
                     
                       cos 
                        
                       
                         ( 
                         
                           
                             
                               ϕ 
                               r 
                             
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
                           + 
                           
                             ϕ 
                             
                               u 
                                
                               
                                   
                               
                                
                               2 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   41 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       M 
                       u 
                     
                      
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       l 
                       1 
                     
                      
                     
                       A 
                       
                         u 
                          
                         
                             
                         
                          
                         1 
                       
                     
                      
                     
                       
                         ω 
                         r 
                         2 
                       
                        
                       
                         ( 
                         t 
                         ) 
                       
                     
                      
                     
                       cos 
                        
                       
                         ( 
                         
                           
                             
                               ϕ 
                               r 
                             
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
                           + 
                           
                             ϕ 
                             
                               u 
                                
                               
                                   
                               
                                
                               1 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   42 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       - 
                       
                         l 
                         2 
                       
                     
                      
                     
                       A 
                       
                         u 
                          
                         
                             
                         
                          
                         2 
                       
                     
                      
                     
                       
                         ω 
                         r 
                         2 
                       
                        
                       
                         ( 
                         t 
                         ) 
                       
                     
                      
                     
                       cos 
                        
                       
                         ( 
                         
                           
                             
                               ϕ 
                               r 
                             
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
                           + 
                           
                             ϕ 
                             
                               u 
                                
                               
                                   
                               
                                
                               2 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   43 
                   ) 
                 
               
             
           
         
       
     
       3 Estimation of the Degree of Linearity 
       [0027]    As an approximate indication for the degree of linearity, common subspace-based methods are well-suited. The mathematical approach that is used can be described as follows:
       It is assumed that the process can be modeled by a linear state space system in the un-faulty state.   Given the input and output data of the process, an over-determined set of linear equations is constructed. The set of equations contains input data, output data and unknown states of the state space system.   The number of states needed to accurately model the process is extracted by mathematic operations such as orthogonal or oblique projections. The number of observable and controllable states equals the system order.   The problem of determining what number of states is needed to model the system is transformed to a matrix rank determination. This determination is performed approximately by computing the singular values via Singular Value Decomposition (SVD).       
 
         [0032]    The computed singular values give an approximate indication for the order of the presumed state space system. If the given process strongly obeys the reduced model equations according to section (2.2), the method indicates a process of order two. If nonlinear behavior resides and the linear model does not fit, the indication becomes indistinct. 
       3.1 Linearity Indication 
       [0033]    This subsection briefly describes the computation of the features for linearity indication. The used algorithm is partially known as MOESP (Multivariable Output-Error State Space). 
       3.1.1 State Space System 
       [0034]    The input/output relation is assumed to match a linear state space relation according to equation (44). N samples of inputs and outputs are available. 
         [0000]        x ( k+ 1)= Ax ( k )+ Bu ( k )+ Bn ( k )  (44) 
         [0000]        y ( k )= Cx ( k )+ Cm ( k )  (45) 
         [0035]    The system is observable and controllable of order n. m(k) and n(k) represent white noise sequences. Matrices A, B, C, D and the states x(k) can be transformed by a regular transformation to 
         [0000]      Ā=T −1 AT 
         [0000]        B =T −1 B 
         [0000]        C =CT 
         [0000]          x   ( k )= T   −1   x ( k )  (46) 
       3.1.2 Alignment in Block Hankel Matrices 
       [0036]    The measured data is aligned in block Hankel Matrices 
         [0000]    
       
         
           
             
               
                 
                   
                     U 
                     p 
                   
                   = 
                   
                     ( 
                     
                       
                         
                           
                             u 
                              
                             
                               ( 
                               0 
                               ) 
                             
                           
                         
                         
                           
                             u 
                              
                             
                               ( 
                               1 
                               ) 
                             
                           
                         
                         
                           … 
                         
                         
                           
                             u 
                              
                             
                               ( 
                               
                                 j 
                                 - 
                                  
                               
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             u 
                              
                             
                               ( 
                               1 
                               ) 
                             
                           
                         
                         
                           
                             u 
                              
                             
                               ( 
                               2 
                               ) 
                             
                           
                         
                         
                           … 
                         
                         
                           
                             u 
                              
                             
                               ( 
                               j 
                               ) 
                             
                           
                         
                       
                       
                         
                           ⋮ 
                         
                         
                           ⋮ 
                         
                         
                           ⋱ 
                         
                         
                           ⋮ 
                         
                       
                       
                         
                           
                             u 
                              
                             
                               ( 
                               
                                  
                                 - 
                                 1 
                               
                               ) 
                             
                           
                         
                         
                           
                             u 
                              
                             
                               ( 
                                
                               ) 
                             
                           
                         
                         
                           … 
                         
                         
                           
                             u 
                              
                             
                               ( 
                               
                                  
                                 + 
                                 j 
                                 - 
                                 2 
                               
                               ) 
                             
                           
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   47 
                   ) 
                 
               
             
             
               
                 
                   
                     U 
                     
                       f 
                        
                       
                           
                       
                     
                   
                   = 
                   
                     ( 
                     
                       
                         
                           
                             u 
                              
                             
                               ( 
                                
                               ) 
                             
                           
                         
                         
                           
                             u 
                              
                             
                               ( 
                               
                                  
                                 + 
                                 1 
                               
                               ) 
                             
                           
                         
                         
                           … 
                         
                         
                           
                             u 
                              
                             
                               ( 
                               
                                  
                                 + 
                                 j 
                                 - 
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                             u 
                              
                             
                               ( 
                               
                                  
                                 + 
                                 1 
                               
                               ) 
                             
                           
                         
                         
                           
                             u 
                              
                             
                               ( 
                               
                                  
                                 + 
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                               ) 
                             
                           
                         
                         
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                             u 
                              
                             
                               ( 
                               
                                  
                                 + 
                                 j 
                               
                               ) 
                             
                           
                         
                       
                       
                         
                           ⋮ 
                         
                         
                           ⋮ 
                         
                         
                           ⋱ 
                         
                         
                           ⋮ 
                         
                       
                       
                         
                           
                             u 
                              
                             
                               ( 
                               
                                 
                                   2 
                                    
                                   
                                       
                                   
                                    
                                    
                                 
                                 - 
                                 1 
                               
                               ) 
                             
                           
                         
                         
                           
                             u 
                              
                             
                               ( 
                               
                                 2 
                                  
                                 
                                     
                                 
                                  
                                  
                               
                               ) 
                             
                           
                         
                         
                           … 
                         
                         
                           
                             u 
                              
                             
                               ( 
                               
                                 
                                   2 
                                    
                                   
                                       
                                   
                                    
                                    
                                 
                                 + 
                                 j 
                                 - 
                                 2 
                               
                               ) 
                             
                           
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   48 
                   ) 
                 
               
             
             
               
                 
                   
                     Y 
                     p 
                   
                   = 
                   
                     ( 
                     
                       
                         
                           
                             y 
                              
                             
                               ( 
                               0 
                               ) 
                             
                           
                         
                         
                           
                             y 
                              
                             
                               ( 
                               1 
                               ) 
                             
                           
                         
                         
                           … 
                         
                         
                           
                             y 
                              
                             
                               ( 
                               
                                 j 
                                 - 
                                  
                               
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             y 
                              
                             
                               ( 
                               1 
                               ) 
                             
                           
                         
                         
                           
                             y 
                              
                             
                               ( 
                               2 
                               ) 
                             
                           
                         
                         
                           … 
                         
                         
                           
                             y 
                              
                             
                               ( 
                               j 
                               ) 
                             
                           
                         
                       
                       
                         
                           ⋮ 
                         
                         
                           ⋮ 
                         
                         
                           ⋱ 
                         
                         
                           ⋮ 
                         
                       
                       
                         
                           
                             y 
                              
                             
                               ( 
                               
                                  
                                 - 
                                 1 
                               
                               ) 
                             
                           
                         
                         
                           
                             y 
                              
                             
                               ( 
                                
                               ) 
                             
                           
                         
                         
                           … 
                         
                         
                           
                             y 
                              
                             
                               ( 
                               
                                  
                                 + 
                                 j 
                                 - 
                                 2 
                               
                               ) 
                             
                           
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   49 
                   ) 
                 
               
             
             
               
                 
                   
                     Y 
                     f 
                   
                   = 
                   
                     ( 
                     
                       
                         
                           
                             y 
                              
                             
                               ( 
                                
                               ) 
                             
                           
                         
                         
                           
                             y 
                              
                             
                               ( 
                               
                                  
                                 + 
                                 1 
                               
                               ) 
                             
                           
                         
                         
                           … 
                         
                         
                           
                             y 
                              
                             
                               ( 
                               
                                  
                                 + 
                                 j 
                                 - 
                                 1 
                               
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             y 
                              
                             
                               ( 
                               
                                  
                                 + 
                                 1 
                               
                               ) 
                             
                           
                         
                         
                           
                             y 
                              
                             
                               ( 
                               
                                  
                                 + 
                                 2 
                               
                               ) 
                             
                           
                         
                         
                           … 
                         
                         
                           
                             y 
                              
                             
                               ( 
                               
                                  
                                 + 
                                 j 
                               
                               ) 
                             
                           
                         
                       
                       
                         
                           ⋮ 
                         
                         
                           ⋮ 
                         
                         
                           ⋱ 
                         
                         
                           ⋮ 
                         
                       
                       
                         
                           
                             y 
                              
                             
                               ( 
                               
                                 
                                   2 
                                    
                                   
                                       
                                   
                                    
                                    
                                 
                                 - 
                                 1 
                               
                               ) 
                             
                           
                         
                         
                           
                             y 
                              
                             
                               ( 
                               
                                 2 
                                  
                                 
                                     
                                 
                                  
                                  
                               
                               ) 
                             
                           
                         
                         
                           … 
                         
                         
                           
                             y 
                              
                             
                               ( 
                               
                                 
                                   2 
                                    
                                   
                                       
                                   
                                    
                                    
                                 
                                 + 
                                 j 
                                 - 
                                 2 
                               
                               ) 
                             
                           
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   50 
                   ) 
                 
               
             
           
         
       
     
         [0000]    with 
         [0000]        u ( k )=( {dot over (s)}   i ( k ) {dot over (s)}   2 ( k )) T    
         [0000]        y ( k )=( F   u ( k ) M   u ( k )) T    
         [0037]    N number of measurements 2i maximum order that can be indicated. User-chosen. 
         [0038]    j=N−2i+1 if all measurements are used. 
         [0039]    The matrices contain all available data and therefore all available information. A set of linear equations is formed which contains these Hankel Matrices and the state Vectors x(k). To explain the procedure, the noise influence is set to zero at this stage. 
         [0000]        Y   p =Γ i   X   p   +H   i   U   p    
         [0000]        Y   f =Γ i   X   f   +H   i   U   f   (51) 
         [0000]        X   f   =A   i   X   p +Δ i   U   p    
         [0040]    To develop this set of equations, following matrices are used: 
         [0000]    
       
         
           
             
               
                 
                   
                     Γ 
                     i 
                   
                   = 
                   
                     
                       ( 
                       
                         
                           C 
                           T 
                         
                          
                         
                             
                         
                          
                         
                           
                             ( 
                             CA 
                             ) 
                           
                           T 
                         
                          
                         
                             
                         
                          
                         
                           
                             ( 
                             
                               CA 
                               2 
                             
                             ) 
                           
                           T 
                         
                          
                         
                             
                         
                          
                         … 
                          
                         
                             
                         
                          
                         
                           
                             ( 
                             
                               CA 
                               
                                 i 
                                 - 
                                 1 
                               
                             
                             ) 
                           
                           T 
                         
                       
                       ) 
                     
                     T 
                   
                 
               
               
                 
                   ( 
                   52 
                   ) 
                 
               
             
             
               
                 
                   
                     Δ 
                     i 
                   
                    
                   
                     ( 
                     
                       
                         A 
                         
                           i 
                           - 
                           1 
                         
                       
                        
                       B 
                        
                       
                           
                       
                        
                       
                         A 
                         
                           i 
                           - 
                           2 
                         
                       
                        
                       B 
                        
                       
                           
                       
                        
                       … 
                        
                       
                           
                       
                        
                       AB 
                        
                       
                           
                       
                        
                       B 
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   53 
                   ) 
                 
               
             
             
               
                 
                   
                     H 
                     i 
                   
                   = 
                   
                     ( 
                     
                       
                         
                           D 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           … 
                         
                         
                           0 
                         
                       
                       
                         
                           CB 
                         
                         
                           D 
                         
                         
                           0 
                         
                         
                           … 
                         
                         
                           0 
                         
                       
                       
                         
                           CAB 
                         
                         
                           CB 
                         
                         
                           D 
                         
                         
                           … 
                         
                         
                           0 
                         
                       
                       
                         
                           ⋮ 
                         
                         
                           ⋮ 
                         
                         
                           ⋮ 
                         
                         
                           … 
                         
                         
                           … 
                         
                       
                       
                         
                           
                             
                               CA 
                               
                                 i 
                                 - 
                                 2 
                               
                             
                              
                             B 
                           
                         
                         
                           
                             
                               CA 
                               
                                 i 
                                 - 
                                 3 
                               
                             
                              
                             B 
                           
                         
                         
                           
                             
                               CA 
                               
                                 i 
                                 - 
                                 4 
                               
                             
                              
                             B 
                           
                         
                         
                           … 
                         
                         
                           D 
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   54 
                   ) 
                 
               
             
           
         
       
     
         [0041]    The state matrices X p  and X f  are defined analogously to the input/output Hankel Matrices: 
         [0000]        X   p =( x (0) x (1) . . .  x ( j− 1))  (55) 
         [0000]        X   f =( x ( i ) x ( i+ 1) . . .  x ( i+j− 1))  (56) 
         [0042]    The set of equations (51) can easily be verified by direct insertion. By removing the unknown states from this set of equations the solution for Y f  yields: 
         [0000]    
       
         
           
             
               
                 
                   
                     Y 
                     f 
                   
                   = 
                   
                     
                       ( 
                       
                         
                           H 
                           f 
                         
                          
                         
                             
                         
                          
                         
                           
                             Γ 
                             i 
                           
                           ( 
                           
                             
                               Δ 
                               i 
                             
                             - 
                             
                               
                                 A 
                                 i 
                               
                                
                               
                                 Γ 
                                 i 
                                 † 
                               
                                
                               
                                 H 
                                 i 
                               
                             
                           
                           ) 
                         
                          
                         
                           Γ 
                           i 
                         
                          
                         
                           A 
                           i 
                         
                          
                         
                           Γ 
                           i 
                           † 
                         
                       
                       ) 
                     
                      
                     
                       ( 
                       
                         
                           
                             
                               U 
                               f 
                             
                           
                         
                         
                           
                             
                               U 
                               p 
                             
                           
                         
                         
                           
                             
                               Y 
                               p 
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   57 
                   ) 
                 
               
             
           
         
       
     
         [0043]    The notation † stands for the Moore-Penrose-Pseudoinverse. Equation (57) yields direct information on the linearity indication features. If the model is purely linear, the row space of Y f  can be fully described by the row spaces 
         [0000]    
       
         
           
             
               ( 
               
                 
                   
                     
                       U 
                       f 
                     
                   
                 
                 
                   
                     
                       U 
                       p 
                     
                   
                 
                 
                   
                     
                       Y 
                       p 
                     
                   
                 
               
               ) 
             
             . 
           
         
       
     
         [0044]    The row space of a matrix is the space spanned by its row vectors. If a matrix J is of full rank, its row space equals the row space of J z  if J=J s J z . 
       3.1.3 Feature Extraction 
       [0045]    For order extraction many different methods are known. The most common are N4SID, MOESP and CVA. As the underlying system is on the stability limit, the algorithm with the most direct order computation, MOESP, is used. Tests with real data as described in the following have approved this choice. MOESP uses a direct RQ decomposition of aligned Block Hankel Matrices: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ( 
                       
                         
                           
                             
                               U 
                               f 
                             
                           
                         
                         
                           
                             
                               U 
                               p 
                             
                           
                         
                         
                           
                             
                               Y 
                               p 
                             
                           
                         
                       
                       ) 
                     
                     = 
                     
                       
                         ( 
                         
                           
                             
                               
                                 R 
                                 
                                   U 
                                   f 
                                 
                               
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               
                                 R 
                                 
                                   U 
                                   p 
                                 
                               
                             
                             
                               0 
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               0 
                             
                             
                               
                                 R 
                                 
                                   Y 
                                   p 
                                 
                               
                             
                             
                               0 
                             
                           
                           
                             
                               
                                 
                                   H 
                                   f 
                                 
                                  
                                 
                                   R 
                                   
                                     U 
                                     f 
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     Γ 
                                     i 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         Δ 
                                         i 
                                       
                                       - 
                                       
                                         
                                           A 
                                           i 
                                         
                                          
                                         
                                           Γ 
                                           i 
                                           † 
                                         
                                          
                                         
                                           H 
                                           i 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   R 
                                   
                                     U 
                                     p 
                                   
                                 
                               
                             
                             
                               
                                 
                                   Γ 
                                   i 
                                 
                                  
                                 
                                   A 
                                   i 
                                 
                                  
                                 
                                   Γ 
                                   i 
                                   † 
                                 
                                  
                                 
                                   R 
                                   
                                     Y 
                                     p 
                                   
                                 
                               
                             
                             
                               0 
                             
                           
                         
                         ) 
                       
                        
                       
                         ( 
                         
                           
                             
                               
                                 Q 
                                 
                                   U 
                                   r 
                                 
                               
                             
                           
                           
                             
                               
                                 Q 
                                 
                                   U 
                                   p 
                                 
                               
                             
                           
                           
                             
                               
                                 Q 
                                 
                                   Y 
                                   p 
                                 
                               
                             
                           
                           
                             
                               
                                 Q 
                                 
                                   Y 
                                   f 
                                 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                       
                   
                    
                   with 
                    
                   
                       
                   
                    
                   
                     
 
                   
                    
                   
                       
                   
                    
                   
                     
                       
                         U 
                         f 
                       
                       = 
                       
                         
                           R 
                           
                             U 
                             f 
                           
                         
                          
                         
                           Q 
                           
                             U 
                             f 
                           
                         
                       
                     
                     , 
                     
                       
                         U 
                         p 
                       
                       = 
                       
                         
                           R 
                           
                             U 
                             p 
                           
                         
                          
                         
                           Q 
                           
                             U 
                             p 
                           
                         
                       
                     
                     , 
                     
                       
                         Y 
                         p 
                       
                       = 
                       
                         
                           R 
                           
                             Y 
                             p 
                           
                         
                          
                         
                           Q 
                           
                             Y 
                             p 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   58 
                   ) 
                 
               
             
           
         
       
     
         [0046]    From the first part, a matrix β r , with rank=system order is extracted. 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           β 
                           r 
                         
                         = 
                         
                           ( 
                           
                             
                               
                                 Γ 
                                 i 
                               
                                
                               
                                 ( 
                                 
                                   
                                     Δ 
                                     i 
                                   
                                   - 
                                   
                                     
                                       A 
                                       i 
                                     
                                      
                                     
                                       Γ 
                                       i 
                                       t 
                                     
                                      
                                     
                                       H 
                                       i 
                                     
                                   
                                 
                                 ) 
                               
                             
                              
                             
                               R 
                               
                                 U 
                                 p 
                               
                             
                              
                             
                                 
                             
                              
                             
                               Γ 
                               i 
                             
                              
                             
                               A 
                               i 
                             
                              
                             
                               Γ 
                               i 
                               † 
                             
                              
                             
                               R 
                               
                                 Y 
                                 p 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             ( 
                             
                               
                                 Γ 
                                 i 
                               
                                
                               
                                   
                               
                                
                               
                                 Γ 
                                 i 
                               
                             
                             ) 
                           
                            
                           
                             ( 
                             
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         Δ 
                                         i 
                                       
                                       - 
                                       
                                         
                                           A 
                                           i 
                                         
                                          
                                         
                                           Γ 
                                           i 
                                           t 
                                         
                                          
                                         
                                           H 
                                           i 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   
                                     
                                       A 
                                       i 
                                     
                                      
                                     
                                       Γ 
                                       i 
                                       t 
                                     
                                      
                                     
                                       R 
                                       
                                         Y 
                                         p 
                                       
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   
                     
                       
                         ( 
                         59 
                         ) 
                       
                     
                   
                   
                     
                       
                         ( 
                         60 
                         ) 
                       
                     
                   
                 
               
             
           
         
       
     
         [0047]    The rank of β r  equals the number of linear independent vectors in its column space, which means that 
         [0000]      rank(β r )=rank(Γ i )= n   (61) 
         [0048]    The column space of a matrix is the space spanned by its column vectors. If a matrix J is of full rank, its column space equals the column space of J z  if J=J s J z . 
         [0049]    A proper method can extract the ‘true’ order of the underlying system. In the case of a faulty, nonlinear behavior, the linearity indication differs from the described model of order two. To extract the matrix rank in an approximate way, Singular Value Decomposition (SVD) is used. The SVD of β r  yields 3 matrices U 1 , S 1 , V 1   
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           β 
                           r 
                         
                         = 
                         
                           
                             U 
                             1 
                           
                            
                           
                             S 
                             1 
                           
                            
                           
                             V 
                             1 
                             T 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               U 
                               1 
                             
                              
                             
                               ( 
                               
                                 
                                   
                                     
                                       σ 
                                       1 
                                     
                                   
                                   
                                     0 
                                   
                                   
                                     … 
                                   
                                   
                                     0 
                                   
                                 
                                 
                                   
                                     0 
                                   
                                   
                                     
                                       σ 
                                       2 
                                     
                                   
                                   
                                     … 
                                   
                                   
                                     0 
                                   
                                 
                                 
                                   
                                     ⋮ 
                                   
                                   
                                     ⋮ 
                                   
                                   
                                     ⋱ 
                                   
                                   
                                     ⋮ 
                                   
                                 
                                 
                                   
                                     0 
                                   
                                   
                                     0 
                                   
                                   
                                     … 
                                   
                                   
                                     
                                       σ 
                                       i 
                                     
                                   
                                 
                               
                               ) 
                             
                           
                            
                           
                             V 
                             1 
                             T 
                           
                         
                       
                     
                   
                 
               
               
                 
                   
                     
                       
                         ( 
                         62 
                         ) 
                       
                     
                   
                   
                     
                       
                         ( 
                         63 
                         ) 
                       
                     
                   
                 
               
             
           
         
       
     
         [0050]    U, and V 1  are orthogonal matrices. S 1  is a diagonal matrix which contains the singular values σ i . In case of a fault-free, not noisy system, n singular values are nonzero while all other singular values are zero. 
         [0000]      σ 1 &gt; . . . &gt;σ n &gt;0  (64) 
         [0000]      σ n+1 = . . . =σ 2i =0  (65) 
       3.2 Noise Influence on Feature Extraction 
       [0051]    Under the influence of noise (process noise as well as measurement noise), the SVD no longer yields clear order decisions. To represent noise influences, it will be assumed that the measurements y(k) are contaminated by white noise n(k)=(n 1 (k)n 2 (k)) T . The excitations u(k) contain noise m(k)=(m 1 (k)m 2 (k)) T . With 
         [0000]    
       
         
           
             
               N 
               p 
             
             = 
             
               ( 
               
                 
                   
                     
                       n 
                        
                       
                         ( 
                         0 
                         ) 
                       
                     
                   
                   
                     
                       n 
                        
                       
                         ( 
                         1 
                         ) 
                       
                     
                   
                   
                     … 
                   
                   
                     
                       n 
                        
                       
                         ( 
                         
                           j 
                           - 
                            
                         
                         ) 
                       
                     
                   
                 
                 
                   
                     
                       n 
                        
                       
                         ( 
                         1 
                         ) 
                       
                     
                   
                   
                     
                       n 
                        
                       
                         ( 
                         2 
                         ) 
                       
                     
                   
                   
                     … 
                   
                   
                     
                       n 
                        
                       
                         ( 
                         j 
                         ) 
                       
                     
                   
                 
                 
                   
                     ⋮ 
                   
                   
                     ⋮ 
                   
                   
                     ⋱ 
                   
                   
                     ⋮ 
                   
                 
                 
                   
                     
                       n 
                        
                       
                         ( 
                         
                            
                           - 
                           1 
                         
                         ) 
                       
                     
                   
                   
                     
                       n 
                        
                       
                         ( 
                          
                         ) 
                       
                     
                   
                   
                     … 
                   
                   
                     
                       n 
                        
                       
                         ( 
                         
                            
                           + 
                           j 
                           - 
                           2 
                         
                         ) 
                       
                     
                   
                 
               
               ) 
             
           
         
       
       
         
           
             
               N 
               f 
             
             = 
             
               ( 
               
                 
                   
                     
                       n 
                        
                       
                         ( 
                          
                         ) 
                       
                     
                   
                   
                     
                       n 
                        
                       
                         ( 
                         
                            
                           + 
                           1 
                         
                         ) 
                       
                     
                   
                   
                     … 
                   
                   
                     
                       n 
                        
                       
                         ( 
                         
                            
                           + 
                           j 
                           - 
                           1 
                         
                         ) 
                       
                     
                   
                 
                 
                   
                     
                       n 
                        
                       
                         ( 
                         
                            
                           + 
                           1 
                         
                         ) 
                       
                     
                   
                   
                     
                       n 
                        
                       
                         ( 
                         
                            
                           + 
                           2 
                         
                         ) 
                       
                     
                   
                   
                     … 
                   
                   
                     
                       n 
                        
                       
                         ( 
                         
                            
                           + 
                           j 
                         
                         ) 
                       
                     
                   
                 
                 
                   
                     ⋮ 
                   
                   
                     ⋮ 
                   
                   
                     ⋱ 
                   
                   
                     ⋮ 
                   
                 
                 
                   
                     
                       n 
                        
                       
                         ( 
                         
                           
                             2 
                              
                             
                                 
                             
                              
                              
                           
                           - 
                           1 
                         
                         ) 
                       
                     
                   
                   
                     
                       n 
                        
                       
                         ( 
                         
                           2 
                            
                           
                               
                           
                            
                            
                         
                         ) 
                       
                     
                   
                   
                     … 
                   
                   
                     
                       n 
                        
                       
                         ( 
                         
                           
                             2 
                              
                             
                                 
                             
                              
                              
                           
                           + 
                           j 
                           - 
                           2 
                         
                         ) 
                       
                     
                   
                 
               
               ) 
             
           
         
       
     
         [0000]    and subsequent formation of M p  and M f , equation (51) is enhanced to 
         [0000]        Y   p =Γ i   X   p   +H   i   U   p   +H   i   N   p   +M   p    
         [0000]        Y   f =Γ i   X   f   +H   i   U   f   +H   i   N   f   +M   f   (66) 
         [0000]        X   f   =A   i   X   p +Δ i   U   p +Δ i   N   p    
         [0052]    The solution for Y f  then yields 
         [0000]    
       
         
           
             
               
                 
                   
                     Y 
                     f 
                   
                   = 
                     
                    
                   
                     
                       ( 
                       
                         
                           H 
                           f 
                         
                          
                         
                             
                         
                          
                         
                           
                             Γ 
                             i 
                           
                           ( 
                           
                             
                               Δ 
                               i 
                             
                             - 
                             
                               
                                 A 
                                 
                                     
                                 
                                 i 
                               
                                
                               
                                 Γ 
                                 i 
                                 † 
                               
                                
                               
                                 H 
                                 i 
                               
                             
                           
                           ) 
                         
                          
                         
                             
                         
                          
                         
                           Γ 
                           i 
                         
                          
                         
                           A 
                           i 
                         
                          
                         
                           Γ 
                           i 
                           † 
                         
                       
                       ) 
                     
                      
                     
                       ( 
                       
                         
                           
                             
                               U 
                               f 
                             
                           
                         
                         
                           
                             
                               U 
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                   67 
                   ) 
                 
               
             
             
               
                 
                   
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                   ( 
                   68 
                   ) 
                 
               
             
           
         
       
     
         [0053]    and the SVD is performed on 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       β 
                       i 
                     
                     = 
                       
                      
                     
                       
                         β 
                         r 
                       
                       + 
                       
                         
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                                 i 
                               
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                                         H 
                                         i 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
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                                 i 
                               
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                           ) 
                         
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                             ( 
                             
                               
                                 
                                   
                                     N 
                                     f 
                                   
                                 
                               
                               
                                 
                                   
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                                     N 
                                     p 
                                   
                                 
                               
                               
                                 
                                   
                                     M 
                                     p 
                                   
                                 
                               
                             
                             ) 
                           
                           / 
                           
                             U 
                             f 
                           
                         
                       
                     
                   
                 
               
               
                 
                   
                     = 
                       
                      
                     
                       
                         β 
                         r 
                       
                       + 
                       
                         E 
                         n 
                       
                     
                   
                 
               
             
              
             
               ( 
               
                 
                   
                     
                       U 
                       p 
                     
                   
                 
                 
                   
                     
                       Y 
                       p 
                     
                   
                 
               
               ) 
             
           
         
       
     
         [0000]    where 
         [0000]      β i =2 i× 2 i  matrix with rank that has to be estimated 
         [0000]      β r =2 i× 2 i  matrix with rank  n  (n=system order) 
         [0000]        E   N =2 i× 2 i  Matrix with full rank (noise representation) 
         [0054]    The notation 
         [0000]    
       
         
           
             / 
             
               
                 U 
                 f 
               
                
               
                 ( 
                 
                   
                     
                       
                         U 
                         p 
                       
                     
                   
                   
                     
                       
                         Y 
                         p 
                       
                     
                   
                 
                 ) 
               
             
           
         
       
     
         [0000]    stands for the orthogonal projection onto the row space of 
         [0000]    
       
         
           
             
               ( 
               
                 
                   
                     
                       U 
                       f 
                     
                   
                 
                 
                   
                     
                       U 
                       p 
                     
                   
                 
                 
                   
                     
                       Y 
                       p 
                     
                   
                 
               
               ) 
             
             . 
           
         
       
     
         [0000]    where only the part lying in the row space of 
         [0000]    
       
         
           
               
             
               ( 
               
                 
                   
                     
                       U 
                       p 
                     
                   
                 
                 
                   
                     
                       Y 
                       p 
                     
                   
                 
               
               ) 
             
           
         
       
     
         [0000]    is considered. In literature, this projection is referred to as ‘Oblique Projection’. 
         [0055]    It is shown in [VOdM96] that E n →0 for N→∞. 
         [0056]    An infinite number of measurements is not achievable. However, if the Signal-to-Noise-Ratio (SNR) and the measurement number N is sufficiently high, the system impact on the Singular Values is larger than the noise influence (see [Lju99]) and E n &lt;&lt;β r  holds and therefore rank (β i )=rank (β r ). Normally, the influence of E n  is not fully negligible and the Singular Values computation yields 
         [0000]      σ 1 &gt; . . . &gt;σ n &gt;&gt;σ n+1 , . . . , σ 2i   (69) 
         [0057]    The singular values representing the system structure dominate, all successive singular values represent the noise influence and are considerably lower. 
       4 Results 
       [0058]    As test rig, a industrial production machine for rotors with mass m r ≈25 kg is used. It is equipped with 2 standard plunger coil sensors. The fundamentation can be adjusted by common clamps. The machine is driven in sub-critical rotary speed. The environment consists of normal industrial surrounding, e.g. other machines, noise etc. 
         [0059]    Three different runs are analyzed:
       Un-faulty standard run   Run with sliding friction in sensor connection   Run with sliding friction in ground connection       
 
         [0063]    The features described in the preceding section are computed for the 3 described states. All tests are done on the same machine with equal bearings and rotor. For each run, a timespan of 5 seconds is examined. The singular values are computed according to the preceding chapter. FIG. ( 4 ) gives an overview over the different features. The values are given in logarithmic scale, the standard deviation over all regarded runs is indicated. 
         [0064]    The first two Singular Values, which represent the linear behavior, remain nearly equal. The third and fourth singular value refer to model inaccuracies and noise. In case of nonlinear behavior, their values are considerably higher than in the purely linear, un-faulty case. Principally, these values can be used as a feature for the appearance of nonlinear faults in linear systems. 
         [0065]    The invention described above presents a subspace-based method to detect the occurrence of nonlinear fault states in linear systems. A rotor system was used as example. The fault-free state as well as two different fault cases have been modeled and tested on an industrial rotor balancing machine. It has been shown that the computed singular values are highly sensible to nonlinear fault states and are well-suited as features for the occurrence of the considered friction faults. 
         [0000]    
       
         
               
             
               
               
               
             
           
               
                   
               
               
                 Symbols and Abbreviations 
               
             
          
           
               
                 Symbol 
                 Unit 
                 Description 
               
               
                   
               
               
                 F u   
                 [N] 
                 imbalance-caused force 
               
               
                 M u   
                 [Nm] 
                 imbalance-caused torque 
               
               
                 {dot over (x)} 1   
                 [m/sec] 
                 movement speed of left rotor support 
               
               
                 {dot over (x)} 2   
                 [m/sec] 
                 movement speed of right rotor support 
               
               
                 {dot over (s)} 1 , {dot over (s)} 2   
                 [m/sec] 
                 measurements by speed sensors 
               
               
                 c 1 , c 2   
                 [N/m] 
                 spring constants 
               
               
                 d 1 , d 2   
                 [N/m] 
                 damper constants 
               
               
                 m r   
                 [kg] 
                 rotor mass 
               
               
                 Θ r   
                 [kgm 2 ] 
                 rotor moment of inertia 
               
               
                 l 1 , l 2   
                 [m] 
                 distance to the center of gravity 
               
               
                 A u1 , A u2   
                 [l] 
                 imbalance amplitude 
               
               
                 φ u1 , φ u2   
                 [l] 
                 imbalance amplitude 
               
               
                 ω r   
                 [rad/sec] 
                 rotation speed 
               
               
                 φ r   
                 [rad] 
                 rotor roll angle 
               
               
                 n(k), m(k) 
                   
                 white noise sequences 
               
               
                   
               
             
          
         
       
     
         [0066]      FIG. 1 : Rotary system model for normal condition. The rotor (mass m r ) movement in horizontal plane can be modeled with linear spring-damper systems (left drawing, birdview). The sensor plunger coils (masses m s1 ,m s2 ) are connected directly to the bearings. As example, the connection of plunger coil  1  is given in the right drawing (side view). 
         [0067]      FIG. 2 : Fault state of sliding friction in sensor connection 
         [0068]      FIG. 3 : Fault state of sliding friction between bearing support socket and ground. 
         [0069]      FIG. 4 : Singular values for normal and faulty states