Robust Control Toolbox™

Fit frequency response magnitude data with stable, minimum-phase state-space model

Syntax

• B = fitmagfrd(A,N) 
  B = fitmagfrd(A,N,RD) 
  B = fitmagfrd(A,N,RD,WT) 
  B = fitmagfrd(A,N,RD,WT,C) 
  

Description

B = fitmagfrd(A,N) is a stable, minimum-phase ss object, with state-dimension N, whose frequency response magnitude closely matches the magnitude data in A. A is a 1-by-1 frd object, and N is a nonnegative integer.

B = fitmagfrd(A,N,RD) forces the relative degree of B to be RD. RD must be a nonnegative integer whose default value is 0. You can specify the default value for RD by setting RD to an empty matrix.

B = fitmagfrd(A,N,RD,WT) uses the magnitude of WT to weight the optimization fit criteria. WT can be a double, ss or frd. If WT is a scalar, then it is used to weight all entries of the error criteria (A-B). If WT is a vector, it must be the same size as A, and each individual entry of WT acts as a weighting function on the corresponding entry of (A-B). The default value for WT is 1, and you can specify it by setting WT to an empty matrix.

B = fitmagfrd(A,N,RD,WT,C) enforces additional magnitude constraints on B, relative to A as:

• if C(w) == -1, then enforce abs(B(w)) <= abs(A(w)
• if C(w) == 1, then enforce abs(B(w)) >= abs(A(w))
• if C(w) == 0, then no additional constraint

where w denotes the frequency. C should be a scalar, double, or frd, with C.Frequency equal to A.Frequency.

Examples

You can use the fitmagfrd command to fit frequency response magnitude data. Create frequency response data from a fifth-order system.

• sys = tf([1 2 2],[1 2.5 1.5])*tf(1,[1 0.1]); 
  sys = sys*tf([1 3.75 3.5],[1 2.5 13]); 
  omeg = logspace(-1,1); 
  sysg = abs(frd(sys,omeg)); 
  bodemag(sysg,'r'); 
  
  
  
  

Fit the magnitude data with a minimum-phase, stable third-order system:

• ord = 3; 
  b1 = fitmagfrd(sysg,ord); 
  b1g = frd(b1,omeg); 
  bodemag(sysg,'r',b1g,'k:');
  
  
  
  

Fit the magnitude data with a third order system constrained to lie below and above the given data.

• b2 = fitmagfrd(sysg,ord,[],[],-1); 
  b2g = frd(b2,omeg); 
  b3 = fitmagfrd(sysg,ord,[],[],1); 
  b3g = frd(b3,omeg); 
  bodemag(sysg,'r',b1g,'k:',b2g,'b-.',b3g,'m--') 
  
  
  
  
  
  

Algorithm

fitmagfrd uses a version of log-Chebychev magnitude design, solving

•    min f     subject to (at every frequency point in A):  
             |d|^2 /(1+ f/WT) < |n|^2/A^2 < |d|^2*(1 + f/WT) 
  

plus additional constraints imposed with C. n, d denote the numerator and denominator, respectively, and B = n/d. n and d have orders (N-RD) and N, respectively. The problem is solved using bisection on f and linear programming for fixed f. An alternate approximate method, which cannot enforce the constraints defined by C, is B = fitfrd(genphase(A),N,RD,WT).

Limitations

This input frd object must be either a scalar 1-by-1 object or, a row, or column vector.

References

Oppenheim, A.V., and R.W. Schaffer, Digital Signal Processing, Prentice Hall, New Jersey, 1975, p. 513.

See Also
fitfrd      Fits frequency response data with state-space model

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