Robust Control Toolbox™

Scale state/uncertainty while preserving uncertain input/output map of uncertain system

Syntax

• usysout = ssbal(usys) 
  usysout = ssbal(usys,Wc) 
  usysout = ssbal(usys,Wc,FSflag) 
  usysout = ssbal(usys,Wc,FSflag,BLTflag) 
  

Description

usysout = ssbal(usys) yields a system whose input/output and uncertain properties are the same as usys, a uss object. The numerical conditioning of usysout is usually better than that of usys, improving the accuracy of additional computations performed with usysout. usysout is a uss object. The balancing algorithm uses mussv to balance the constant uncertain state-space matrices in discrete time. If usys is a continuous-time uncertain system, the uncertain state-space is mapped by using a bilinear transformation into discrete time for balancing.

usysout = ssbal(usys,wc) defines the critical frequency wc for the bilinear prewarp transformation from continuous time to discrete time. The default value of wc is 1 when the nominal uncertain system is stable and 1.25*mxeig when it is unstable. mxeig corresponds to the value of the real, most positive pole of usys.

usysout = ssbal(usys,wc,FSflag) sets the scaling flag FSflag to handle repeated uncertain parameters. Setting FSflag=1 uses full matrix scalings to balance the repeated uncertain parameter blocks. FSflag=0, the default, uses a single, positive scalar to balance the repeated uncertain parameter blocks.

usysout = ssbal(usys,wc,FSflag,BLTflag) sets the bilinear transformation flag, BLTflag. By default, BLTflag=1 and transforms the continuous-time system usys to a discrete-time system for balancing. BLTflag=0 results in balancing the continuous-time state-space data from usys. Note that if usys is a discrete-time system, no bilinear transformation is performed.

ssbal does not work on an array of uncertain systems. An error message is generated to alert you to this.

Example

Consider a two-input, two-output, two-state uncertain system with two real parameter uncertainties, p1 and p2.

• p2=ureal('p2',-17,'Range',[-19 -11]); 
  p1=ureal('p1',3.2,'Percentage',0.43); 
  A = [-12 p1;.001 p2]; 
  B = [120 -809;503 24]; 
  C = [.034 .0076; .00019 2]; 
  usys = ss(A,B,C,zeros(2,2)) 
  USS: 2 States, 2 Outputs, 2 Inputs, Continuous System 
    p1: real, nominal = 3.2, variability = [-0.43  0.43]%, 1 
  occurrence 
    p2: real, nominal = -17, range = [-19  -11], 1 occurrence          
  usys.NominalValue 
  a = 
            x1     x2 
     x1    -12    3.2 
     x2  0.001    -17 
  
  b = 
           u1    u2 
     x1   120  -809 
     x2   503    24 
  
  c = 
              x1       x2 
     y1    0.034   0.0076 
     y2  0.00019        2 
  
  d = 
         u1  u2 
     y1   0   0 
     y2   0   0 
  
  Continuous-time model. 
  ssbal is used to balance the uncertain system usys.
  
  usysout = ssbal(usys) 
  USS: 2 States, 2 Outputs, 2 Inputs, Continuous System 
    p1: real, nominal = 3.2, variability = [-0.43  0.43]%, 1 
  occurrence 
    p2: real, nominal = -17, range = [-19  -11], 1 occurrence  
          
  usysout.NominalValue 
  a = 
               x1        x2 
     x1       -12    0.3302 
     x2  0.009692       -17 
  
  b = 
             u1      u2 
     x1  0.7802   -5.26 
     x2    31.7   1.512 
  
  c = 
              x1       x2 
     y1    5.229   0.1206 
     y2  0.02922    31.74 
  
  d = 
         u1  u2 
     y1   0   0 
     y2   0   0 
  
  Continuous-time model. 
  

See Also
canon       Forms canonical state-space realizations

c2d         Converts continuous-time models to discrete-time

d2c         Converts discrete-time models to continuous-time

mussv       Sets bounds on the Structure Singular Value (µ)

mussvextract  Extracts compressed data returned from mussv

ss2ss       Changes state coordinates for state-space models

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