H∞ mixed-sensitivity synthesis method for robust control loopshaping design
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
Closed-loop transfer function Ty1u1 for
The returned values of S, R, and T satisfy the following loop shaping inequalities:
where γ =
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 NY, NU,
and NY where G is NY-by-NU.
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
This example shows 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-.') legend('S','T','GAM/W1','GAM*G/ss(W2)','Location','SouthWest')
mixsyn command shapes the singular values of
T to conform to
The transfer functions G, W1, W2 and W3 must
be proper, i.e., bounded as s → ∞
or, in the discrete-time case, as z → ∞.
Additionally, W1, W2 and W3 should
be stable. The plant G should be stabilizable and
P will not be stabilizable by