Robust Control Toolbox™

H mixed-sensitivity synthesis method for robust control loopshaping design

Syntax

• [K,CL,GAM,INFO]=mixsyn(G,W1,W2,W3) 
  [K,CL,GAM,INFO]=mixsyn(G,W1,W2,W3,KEY1,VALUE1,KEY2,VALUE2,...) 
  

Description

[K,CL,GAM,INFO]=mixsyn(G,W1,W2,W3) computes a controller K that minimizes the H norm of the closed-loop transfer function the weighted mixed sensitivity

where S and T are called the sensitivity and complementary sensitivity, respectively and S, R and T are given by

Figure 5-13: Closed-loop transfer function for mixed sensitivity mixsyn.

The returned values of S, R, and T satisfy the following loop shaping inequalities:

where =GAM. Thus, W₁, W₃ determine the shapes of sensitivity S and complementary sensitivity T. Typically, you would choose W₁ to be small inside the desired control bandwidth to achieve good disturbance attenuation (i.e., performance), and choose W₃ to be small outside the control bandwidth, which helps to ensure good stability margin (i.e., robustness).

For dimensional compatibility, each of the three weights W₁, W₂ and W₃ must be either empty, scalar (SISO) or have respective input dimensions N_Y, N_U, and N_Y where G is N_Y-by-N_U. If one of the weights is not needed, you may simply assign an empty matrix []; e.g., P = AUGW(G,W1,[],W3) is SYS but without the second row (without the row containing W2).

Algorithm

[K,CL,GAM,INFO]=mixsyn(G,W1,W2,W3,KEY1,VALUE1,KEY2,VALUE2,...)

is equivalent to

• [K,CL,GAM,INFO]=...
      hinfsyn(augw(G,W1,W2,W3),KEY1,VALUE1,KEY2,VALUE2,...).
  

mixsyn accepts all the same key value pairs as hinfsyn.

Example

The following code illustrates the use of mixsyn for sensitivity and complementary sensitivity `loop-shaping'.

• s=zpk('s');
  G=(s-1)/(s+1)^2;
  W1=0.1*(s+100)/(100*s+1); W2=0.1;
  [K,CL,GAM]=mixsyn(G,W1,W2,[]);
  L=G*K; S=inv(1+L); T=1-S;
  sigma(S,'g',T,'r',GAM/W1,'g-.',GAM*G/ss(W2),'r-.')
  
  
  

Figure 5-14: mixsyn(G,W1,W2,[ ]) shapes sigma plots of S and T to conform to /W₁ and G/W₂, respectively.

Limitations

The transfer functions G, W₁, W₂ and W₃ must be proper, i.e., bounded as or, in the discrete-time case, as . Additionally, W₁, W₂ and W₃ should be stable. The plant G should be stabilizable and detectable; else, P will not be stabilizable by any K.

See Also
augw        Augments plant weights for control design

hinfsyn     H controller synthesis

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