Tune the controller elements of the following control system.

The control elements are C, which is a PI controller with two free parameters, and F, a low-pass filter in the feedback path with one free parameter. For this example, load the plant G, a 9th-order model of the head-disk assembly (HDA) in a hard-disk drive.

load hinfstruct_demo G
bode(G), grid

Tune the free parameters of this control system so that the head position y tracks a step change r with a response time of about 1 ms, little or no overshoot, and no steady-state error.

First, create the tunable elements. Use tunablePID class to parameterize the PI block, and specify the filter $$F(s)$$ as a transfer function depending on a tunable real parameter $$a$$.

C0 = tunablePID('C','pi');  

a = realp('a',1); 
F0 = tf(a,[1 a]);

F0 is a genss model.

Next, express the design goals as weights on the plant model, and append them to the closed-loop system. For reasons described in detail in Fixed-Structure H-infinity Synthesis with hinfstruct, a configuration of weighting functions that achieves the design requirements for this problem is the following.

Here, LS is the desired shape of the open-loop response $$L(s)=F(s)G(s)C(s)$$.

wc = 1000;  % target crossover
s = tf('s');
LS = (1+0.001*s/wc)/(0.001+s/wc);

As discussed in Fixed-Structure H-infinity Synthesis with hinfstruct, the design requirements are satisfied if the ${{\rm H}}_{\infty }$ norm of this control structure is less than 1.

Use connect to construct a genss model representing this control structure.

% Label the block I/Os
Wn = 1/LS;  Wn.u = 'nw';  Wn.y = 'n';
We = LS;    We.u = 'e';   We.y = 'ew';
C0.u = 'e';   C0.y = 'u';
F0.u = 'yn';  F0.y = 'yf';

% Specify summing junctions
Sum1 = sumblk('e = r - yf');
Sum2 = sumblk('yn = y + n');

% Connect the blocks together
T0 = connect(G,Wn,We,C0,F0,Sum1,Sum2,{'r','nw'},{'y','ew'});

You can now use hinfstruct to find tuned values of the tunable parameters in C0 and F0 that minimize the ${{\rm H}}_{\infty }$ norm of T0. To mitigate the risk of local minima, run three optimizations, two of which are started from randomized initial values for C0 and F0:

rng('default')
opt = hinfstructOptions('Display','final','RandomStart',5);
T = hinfstruct(T0,opt);

Final: Peak gain = 3.88, Iterations = 67
Final: Peak gain = 596, Iterations = 206
       Some closed-loop poles are marginally stable (decay rate near 1e-07)
Final: Peak gain = 597, Iterations = 168
       Some closed-loop poles are marginally stable (decay rate near 1e-07)
Final: Peak gain = 3.88, Iterations = 64
Final: Peak gain = 1.56, Iterations = 96
Final: Peak gain = 1.56, Iterations = 100

The best closed-loop gain is about 1.56, so the constraint $\Vert T{\Vert }_{\infty }<1$ is nearly satisfied. The hinfstruct command returns the tuned closed-loop transfer $$T(s)$$. To validate the design, plot the tuned open-loop response L = F*G*C and compare with the target loop shape LS. To compute L, use getBlockValue to get the tuned value of $$C(s)$$ and use getValue to evaluate the filter $$F(s)$$ for the tuned value of $$a$$:

C = getBlockValue(T,'C');
F = getValue(F0,T.Blocks);  % propagate tuned parameters from T to F

L = G*C*F;
bode(LS,'r--',G*C*F,'b',{1e1,1e6}), grid, 
title('Open-loop response'), legend('Target','Actual')

The 0dB crossover frequency and overall loop shape are as expected. For further analysis of the result, see Fixed-Structure H-infinity Synthesis with hinfstruct.