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This example shows how to use Robust Control Toolbox™ to design a multi-input, multi-output controller by shaping the gain of an open-loop response across frequency. This technique is applied to controlling the pitch axis of a HIMAT aircraft.

We show how to choose a suitable target loop shape and to use the `loopsyn`

function to compute a multivariable controller that optimally matches the target loop shape.

Typically, loop shaping is a tradeoff between two potentially conflicting objectives. We want to maximize the open-loop gain to get the best possible performance, but for robustness, we need to drop the gain below 0dB where the model accuracy is poor and high gain might cause instabilities. This requires a good model where performance is needed (typically at low frequencies), and sufficient roll-off at higher frequencies where the model is often poor. The frequency Wc, where the gain crosses the 0dB line, is called the crossover frequency and marks the transition between performance and robustness requirements.

**Figure 1:** Loop shaping specifications.

There are several guidelines for choosing a target loop shape `Gd`

:

**Stability Robustness**:`Gd`

should have a gain of less than 0dB at high frequencies, where the plant model is so poor that the phase error may approach 180 degrees.**Performance**:`Gd`

should have high gain where you want good control accuracy and good disturbance rejection.**Crossover and Rolloff**:`Gd`

should cross the 0dB line between these two frequency regions, and roll off with a slope of -20 to -40 dB/decade past the crossover frequency Wc.

A simple target loop shape is

where the crossover `Wc`

is the reciprocal of the rise-time of the desired step response.

For illustration purposes, let's use a six-state model of the longitudinal dynamics of the HiMAT aircraft trimmed at 25000 ft and 0.9 Mach. The aircraft dynamics are unstable, with two right-half plane phugoid modes.

**Figure 2:** NASA HiMAT aircraft

This model has two control inputs:

Elevon deflection

Canard deflection

It also has two measured outputs:

Angle of attack alpha

Pitch angle theta

The model is unreliable beyond 100 rad/sec. Fuselage bending modes and other uncertain factors induces deviations between the model and the true aircraft of as much as 20 decibels (or 1000%) beyond frequency 100 rad/sec.

ag = [ -2.2567e-02 -3.6617e+01 -1.8897e+01 -3.2090e+01 3.2509e+00 -7.6257e-01; 9.2572e-05 -1.8997e+00 9.8312e-01 -7.2562e-04 -1.7080e-01 -4.9652e-03; 1.2338e-02 1.1720e+01 -2.6316e+00 8.7582e-04 -3.1604e+01 2.2396e+01; 0 0 1.0000e+00 0 0 0; 0 0 0 0 -3.0000e+01 0; 0 0 0 0 0 -3.0000e+01]; bg = [0 0; 0 0; 0 0; 0 0; 30 0; 0 30]; cg = [0 1 0 0 0 0; 0 0 0 1 0 0]; dg = [0 0; 0 0]; G = ss(ag,bg,cg,dg); G.InputName = {'elevon','canard'}; G.OutputName = {'alpha','theta'}; clf step(G), title('Open-loop step response of HIMAT aircraft')

Our task is to control `alpha`

and `theta`

by issuing appropriate elevon and canard commands. We also want minimum spillover between channels - that is, a command in alpha should have minimum effect on theta and vice versa.

Designing a controller with `loopsyn`

involves the following steps:

Step 1: Look at the plant dynamics and responses

Step 2: Specify the desired loop shape Gd

Step 3: Use

`loopsyn`

to compute the optimal loop shaping controllerStep 4: Analyze the shaped-loop L, closed-loop T, and sensitivity S

Step 5: Verify that the closed-loop responses meet your specs.

In this example, the aircraft model `G`

has two unstable right-half plane phugoid-mode poles. It has one zero, which is in the left-half plane:

plant_poles = pole(G)

`plant_poles = `*6×1 complex*
-5.6757 + 0.0000i
0.6898 + 0.2488i
0.6898 - 0.2488i
-0.2578 + 0.0000i
-30.0000 + 0.0000i
-30.0000 + 0.0000i

plant_zeros = tzero(G)

plant_zeros = -0.0210

We'll use `sigma`

to plot the min and max I/O gain as a function of frequency:

clf, sigma(G,'g',{.1,100}); title('Singular value plot for aircraft model G(s)');

**Figure 4:** Singular value plot for aircraft model G(s).

For this design we use the target loop shape Gd(s)=8/s corresponding to a rise time of about 1/8=0.125 sec.

s = zpk('s'); % Laplace variable s Gd = 8/s; sigma(Gd,{.1 100}) grid title('Target loop shape Gd(s).') % create textarrow pointing to crossover frequency Wc hold on; plot([8,35],[0,21],'k.-'); plot(8,0,'kd'); plot([.1,100],[0 0],'k'); text(3,23,'Crossover Frequency \omega_c = 8'); hold off;

**Figure 5:** Target loop shape Gd(s).

Now, we're ready to design an H-infinity controller `K`

, such that the gains of the open-loop response G(s)*K(s) match the target-loop shape Gd as well as possible (while stabilizing the aircraft dynamics).

[K,CL,GAM] = loopsyn(G,Gd); GAM

GAM = 1.6445

The value GAM=1.6445 indicates that the target-loop shape was met within +/-4.3dB (using 20*log10(GAM)=4.32). Compare the singular values of the open-loop L=G*K with the target-loop shape `Gd`

:

L = G*K; % form the compensated loop L sigma(Gd,'b',L,'r--',{.1,100}); grid legend('Gd (target loop shape)','L (actual loop shape)');

**Figure 6:** Singular values of L compared to Gd

Next we'll compare the gains of the open-loop transfer L, closed-loop transfer T, and sensitivity function S.

T = feedback(L,eye(2)); T.InputName = {'alpha command','theta command'}; S = eye(2)-T; % SIGMA frequency response plots sigma(inv(S),'m',T,'g',L,'r--',Gd,'b',Gd/GAM,'b:',... Gd*GAM,'b:',{.1,100}) legend('1/\sigma(S) performance',... '\sigma(T) robustness',... '\sigma(L) open loop',... '\sigma(Gd) target loop shape',... '\sigma(Gd) \pm GAM(dB)'); % Make lines wider and fonts larger % set(findobj(gca,'Type','line','-not','Color','b'),'LineWidth',2); h = findobj(gca,'Type','line','-not','Color','b'); set(h,'LineWidth',2);

**Figure 7:** Sigma frequency response plots.

Finally, in this step we plot the step responses of the closed-loop system T.

```
step(T,8)
title('Responses to step commands for alpha and theta');
```

**Figure 8:** Responses to step commands for alpha and theta.

Our design looks good. The alpha and theta controls are fairly decoupled, overshoot is less the 15 percent, and peak time is only 0.5 sec.

We just designed a 2-input, 2-output controller `K`

with satisfactory performance. However this controller has fairly high order:

size(K)

State-space model with 2 outputs, 2 inputs, and 16 states.

We can now use model reduction algorithms to try to simplify this controller while retaining its performance characteristics. First we compute the Hankel singular values of `K`

to understand how many controller states effectively contribute to the control law:

hsv = hankelsv(K); semilogy(hsv,'*--') grid title('Hankel singular values of K') xlabel('Order')

**Figure 9:** Hankel singular values of K.

The Hankel singular values (HSV) measure the relative energy of each state in a balanced realization of `K`

. Notice that the HSV for the 10th state is four orders of magnitude smaller than for the 9th state, so try computing a 9th-order approximation of K

Kr = reduce(K,9); order(Kr)

ans = 9

The simplified controller `Kr`

has order 9 compared to 16 for the initial controller `K`

. Compare the approximation error `K-Kr`

across frequency with the gains of `K`

.

sigma(K,'b',K-Kr,'r-.') legend('K','error K-Kr')

This plot suggests that the approximation error is small in relative terms, so go ahead and compare the closed-loop responses with the original and simplified controllers `K`

and `Kr`

Tr = feedback(G*Kr,eye(2)); step(T,'b',Tr,'r-.',8) title('Responses to step commands for alpha and theta'); legend('K','Kr')

**Figure 11:** Responses to step commands for alpha and theta.

The two responses are indistinguishable so the simplified 9th-order controller `Kr`

is a good substitute for `K`

for implementation purposes.

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