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This example shows how to design a robust controller for a two-loop autopilot that controls the pitch rate and vertical acceleration of an airframe. The controller is robust against gain and phase variations in the multichannel feedback loop.

The airframe dynamics and the autopilot are modeled in Simulink. See Tuning of a Two-Loop Autopilot for more information about this model and for additional design and tuning options.

```
open_system('rct_airframe2')
```

As in the example Tuning of a Two-Loop Autopilot, trim the airframe for and . The trim condition corresponds to zero normal acceleration and pitching moment ( and steady). Use `findop`

to compute the corresponding operating condition. Then, linearize the airframe model at this trim condition.

opspec = operspec('rct_airframe2'); % Specify trim condition % Xe,Ze: known, not steady opspec.States(1).Known = [1;1]; opspec.States(1).SteadyState = [0;0]; % u,w: known, w steady opspec.States(3).Known = [1 1]; opspec.States(3).SteadyState = [0 1]; % theta: known, not steady opspec.States(2).Known = 1; opspec.States(2).SteadyState = 0; % q: unknown, steady opspec.States(4).Known = 0; opspec.States(4).SteadyState = 1; % controller states unknown, not steady opspec.States(5).SteadyState = [0;0]; op = findop('rct_airframe2',opspec); G = linearize('rct_airframe2','rct_airframe2/Airframe Model',op); G.InputName = 'delta'; G.OutputName = {'az','q'};

Operating point search report: --------------------------------- opreport = Operating point search report for the Model rct_airframe2. (Time-Varying Components Evaluated at time t=0) Operating point specifications were successfully met. States: ---------- <strong>Min</strong> <strong>x</strong> <strong>Max</strong> <strong>dxMin</strong> <strong>dx</strong> <strong>dxMax</strong> <strong>_____</strong> <strong>__________</strong> <strong>_____</strong> <strong>_____</strong> <strong>___________</strong> <strong>_____</strong> (1.) rct_airframe2/Airframe Model/Aerodynamics & Equations of Motion/ Equations of Motion (Body Axes)/Position 0 0 0 -Inf 984 Inf -3048 -3048 -3048 -Inf 0 Inf (2.) rct_airframe2/Airframe Model/Aerodynamics & Equations of Motion/ Equations of Motion (Body Axes)/Theta 0 0 0 -Inf -0.0097235 Inf (3.) rct_airframe2/Airframe Model/Aerodynamics & Equations of Motion/ Equations of Motion (Body Axes)/U,w 984 984 984 -Inf 22.69 Inf 0 0 0 0 2.4587e-11 0 (4.) rct_airframe2/Airframe Model/Aerodynamics & Equations of Motion/ Equations of Motion (Body Axes)/q -Inf -0.0097235 Inf 0 -1.7215e-16 0 (5.) rct_airframe2/MIMO Controller -Inf 0.00065361 Inf -Inf -0.0089973 Inf -Inf 3.7592e-20 Inf -Inf 0.030259 Inf Inputs: ---------- <strong>Min</strong> <strong>u</strong> <strong>Max</strong> <strong>____</strong> <strong>__________</strong> <strong>___</strong> (1.) rct_airframe2/delta trim -Inf 0.00043574 Inf Outputs: None ----------

Design an H-infinity controller that responds to a step change in vertical acceleration under 1 second. Use a mixed-sensitivity formulation with a lowpass weight `wS`

that penalizes low-frequency tracking error and a highpass weight `wT`

that enforces adequate roll-off. Build an augmented plant with `azref,delta`

as inputs and the filtered `wS*e,wT*az,e,q`

as outputs. (For information about the mixed-sensitivity formulation, see Mixed-Sensitivity Loop Shaping.)

sb = sumblk('e = azref-az'); wS = makeweight(1e2,5,0.1); wS.u = 'e'; wS.y = 'we'; wT = makeweight(0.1,50,1e2); wT.u = 'az'; wT.y = 'waz'; Paug = connect(G,wS,wT,sb,{'azref','delta'},{'we','waz','e','q'});

Compute the optimal H-infinity controller using `hinfsyn`

. In this configuration there are two measurements `e,q`

and one control `delta`

.

[Knom,~,gam] = hinfsyn(Paug,2,1); gam

gam = 0.5928

Verify the open-loop gain against `wS,wT`

.

f = figure(); sigma(Knom*G,wS,'r--',1/wT,'g--'), grid legend('open-loop gain','> wS at low freq','< 1/wT at high freq')

Plot the closed-loop response.

CL = feedback(G*Knom,diag([1 -1])); step(CL(:,1)), grid

Compute the disk margins at the plant input and outputs. That the `az`

loop uses negative feedback while the `q`

loop uses positive feedback, so change the sign of the loop gain of the `q`

loop before using `diskmargin`

.

loopsgn = diag([1 -1]);

Examine the margins at the plant input.

DM = diskmargin(Knom*loopsgn*G)

DM = struct with fields: GainMargin: [0.3918 2.5524] PhaseMargin: [-47.2105 47.2105] DiskMargin: 0.8740 LowerBound: 0.8740 UpperBound: 0.8740 Frequency: 28.8842 WorstPerturbation: [1x1 ss]

For the margins at plant outputs, use the multiloop margin to account for simultaneous, independent variations in both output channels.

[~,MM] = diskmargin(loopsgn*G*Knom)

MM = struct with fields: GainMargin: [0.4994 2.0025] PhaseMargin: [-36.9262 36.9262] DiskMargin: 0.6678 LowerBound: 0.6678 UpperBound: 0.6691 Frequency: 23.6845 WorstPerturbation: [2x2 ss]

Finally, compute the margins against simultaneous variations at the plant input and outputs.

MMIO = diskmargin(loopsgn*G,Knom)

MMIO = struct with fields: GainMargin: [0.6866 1.4565] PhaseMargin: [-21.0565 21.0565] DiskMargin: 0.3717 LowerBound: 0.3717 UpperBound: 0.3725 Frequency: 24.8030 WorstPerturbation: [1x1 struct]

The disk-based gain and phase margins exceed 2 (6dB) and 35 degrees at the plant input and the plant outputs. Moreover, `MMIO`

indicates that this feedback loop can withstand gain variations by a factor 1.45 or phase variations of 21 degrees affecting all plant inputs and outputs at once.

Next, investigate the impact of gain and phase uncertainty on performance. Use the `umargin`

control design block to model a gain change factor of 1.4 (3dB) and phase change of 20 degrees in each feedback channel. Use `getDGM`

to fit an uncertainty disk to these amounts of gain and phase change.

GM = 1.4; PM = 20; DGM = getDGM(GM,PM,'balanced'); ue = umargin('e',DGM); uq = umargin('q',DGM); Gunc = blkdiag(ue,uq)*G; Gunc.OutputName = {'az','q'};

Rebuild the closed-loop model and sample the gain and phase uncertainty to gauge the impact on the step response.

CLunc = feedback(Gunc*Knom,loopsgn); step(CLunc(:,1),3) grid

The plot shows significant variability in performance, with large overshoot and steady-state error for some combinations of gain and phase variations.

Redo the controller synthesis to account for gain and phase variations using `musyn`

. This synthesis optimizes performance uniformly for the specified range of gain and phase uncertainty.

Punc = connect(Gunc,wS,wT,sb,{'azref','delta'},{'we','waz','e','q'}); [Kr,gam] = musyn(Punc,2,1); gam

D-K ITERATION SUMMARY: ----------------------------------------------------------------- Robust performance Fit order ----------------------------------------------------------------- Iter K Step Peak MU D Fit D 1 51.06 26.33 26.62 4 2 7.029 5.24 5.304 8 3 1.681 1.156 1.157 4 4 0.9705 0.9702 0.9822 10 5 0.962 0.9619 0.9622 10 6 0.9625 0.9623 0.9629 10 Best achieved robust performance: 0.962 gam = 0.9619

Compare the sampled step responses for the nominal and robust controllers. The robust design reduces both overshoot and steady-state errors and gives more consistent performance across the modeled range of gain and phase uncertainty.

```
CLr = feedback(Gunc*Kr,loopsgn);
rng(0) % for reproducibility
step(CLunc(:,1),3)
hold
rng(0)
step(CLr(:,1),3)
grid
```

Current plot held

The robust controller achieves this performance by increasing the (nominal) disk margins at the plant output and, to a lesser extent, the I/O disk margin. For instance, compare the disk-based margins at the plant outputs for the nominal and robust designs. Use `diskmarginplot`

to see the variations of the margins with frequency.

Lnom = loopsgn*G*Knom; Lrob = loopsgn*G*Kr; clf diskmarginplot(Lnom,Lrob) title('Disk margins at plant outputs') grid legend('Nominal','Robust')

Examine the margins against variations simultaneous variations at the plant inputs and outputs with the robust controller.

MMIO = diskmargin(loopsgn*G,Kr)

MMIO = struct with fields: GainMargin: [0.6492 1.5404] PhaseMargin: [-24.0166 24.0166] DiskMargin: 0.4254 LowerBound: 0.4254 UpperBound: 0.4263 Frequency: 19.6040 WorstPerturbation: [1x1 struct]

Recall that with the nominal controller, the feedback loop could withstand gain variations of a factor of 1.45 or phase variations of 21 degrees affecting all plant inputs and outputs at once. With the robust controller, those margins increase to about 1.54 and 24 degrees.

View the range of simultaneous gain and phase variations that the robust design can sustain at all plant inputs and outputs.

diskmarginplot(MMIO.GainMargin)

`diskmargin`

returns the smallest change in gain and phase that destabilizes the feedback loop. It can be insightful to inject this perturbation in the nonlinear simulation to further analyze the implications for the real system. For example, compute the destabilizing perturbation at the plant outputs for the nominal controller.

```
[~,MM] = diskmargin(Lnom);
WP = MM.WorstPerturbation;
bode(WP)
title('Smallest destabilizing perturbation')
```

The worst perturbation is a diagonal, dynamic perturbation that multiplies the open-loop response at the plant outputs. With this perturbation, the closed-loop system becomes unstable with an undamped pole at the frequency `w0 = MM.Frequency`

.

damp(feedback(WP*G*Knom,loopsgn))

Pole Damping Frequency Time Constant (rad/seconds) (seconds) -1.88e-03 1.00e+00 1.88e-03 5.33e+02 -2.55e-02 1.00e+00 2.55e-02 3.92e+01 -3.20e-02 1.00e+00 3.20e-02 3.12e+01 -5.16e-02 1.00e+00 5.16e-02 1.94e+01 -1.25e-01 1.00e+00 1.25e-01 7.98e+00 -3.85e+00 + 5.04e+00i 6.07e-01 6.34e+00 2.60e-01 -3.85e+00 - 5.04e+00i 6.07e-01 6.34e+00 2.60e-01 -8.38e+00 + 1.17e+01i 5.81e-01 1.44e+01 1.19e-01 -8.38e+00 - 1.17e+01i 5.81e-01 1.44e+01 1.19e-01 -3.73e-14 + 2.37e+01i 1.58e-15 2.37e+01 2.68e+13 -3.73e-14 - 2.37e+01i 1.58e-15 2.37e+01 2.68e+13 -2.95e+01 1.00e+00 2.95e+01 3.39e-02 -3.33e+01 1.00e+00 3.33e+01 3.00e-02 -6.04e+01 + 2.28e+01i 9.36e-01 6.46e+01 1.66e-02 -6.04e+01 - 2.28e+01i 9.36e-01 6.46e+01 1.66e-02 -3.86e+01 + 5.36e+01i 5.85e-01 6.61e+01 2.59e-02 -3.86e+01 - 5.36e+01i 5.85e-01 6.61e+01 2.59e-02 -1.10e+03 1.00e+00 1.10e+03 9.05e-04

w0 = MM.Frequency

w0 = 23.6845

Note that the gain and phase variations induced by this perturbation lie on the boundary of the range of safe gain/phase variations computed by `diskmargin`

. To confirm this result, compute the frequency response of the worst perturbation at the frequency `w0`

, convert it to a gain and phase variation, and visualize it along with the range of safe gain and phase variations represented by the multiloop disk margin.

DELTA = freqresp(WP,w0); clf diskmarginplot(MM.GainMargin) title('Range of stable gain and phase variations') hold on plot(20*log10(abs(DELTA(1,1))),abs( angle(DELTA(1,1))*180/pi),'ro') plot(20*log10(abs(DELTA(2,2))),abs( angle(DELTA(2,2))*180/pi),'ro')

To simulate the effect of this worst perturbation on the full nonlinear model in Simulink, insert it as a block before the controller block, as done in the modified model `rct_airframeWP`

.

```
close(f)
open_system('rct_airframeWP')
```

Here the MIMO Controller block is set to the nominal controller `Knom`

. To simulate the nonlinear response with this controller, compute the trim deflection and `q`

initial value from the operating condition `op`

.

delta_trim = op.Inputs.u + [1.5 0]*op.States(5).x; q_ini = op.States(4).x;

By commenting the WorstPerturbation block in and out, you can simulate the response with or without this perturbation. The results are shown below and confirm the destabilizing effect of the gain and phase variation computed by `diskmargin`

.

**Figure 1: Nominal response.**

**Figure 2: Response with destabilizing gain/phase perturbation.**

`umargin`

| `diskmargin`

| `diskmarginplot`

| `musyn`