This example shows how to use Robust Control Toolbox™ to tune the standard configuration of a longitudinal autopilot. We thank Professor D. Alazard from Institut Superieur de l'Aeronautique et de l'Espace for providing the aircraft model and Professor Pierre Apkarian from ONERA for developing the example.

The longitudinal autopilot for a supersonic passenger jet flying at Mach 0.7 and 5000 ft is depicted in Figure 1. The autopilot main purpose is to follow vertical acceleration commands issued by the pilot. The feedback structure consists of an inner loop controlling the pitch rate and an outer loop controlling the vertical acceleration . The autopilot also includes a feedforward component and a reference model that specifies the desired response to a step command . Finally, the second-order roll-off filter

is used to attenuate noise and limit the control bandwidth as a safeguard against unmodeled dynamics. The tunable components are highlighted in orange.

**Figure 1: Longitudinal Autopilot Configuration.**

The aircraft model is a 5-state model, the state variables being the aerodynamic speed (m/s), the climb angle (rad), the angle of attack (rad), the pitch rate (rad/s), and the altitude (m). The elevator deflection (rad) is used to control the vertical load factor . The open-loop dynamics include the oscillation with frequency and damping ratio = 1.7 (rad/s) and = 0.33, the phugoid mode = 0.64 (rad/s) and = 0.06, and the slow altitude mode = -0.0026.

load ConcordeData G bode(G,{1e-3,1e2}), grid title('Aircraft Model')

Note the zero at the origin in . Because of this zero, we cannot achieve zero steady-state error and must instead focus on the transient response to acceleration commands. Note that acceleration commands are transient in nature so steady-state behavior is not a concern. This zero at the origin also precludes pure integral action so we use a pseudo-integrator with = 0.001.

When the control system is modeled in Simulink, you can use the `slTuner`

interface to quickly set up the tuning task. Open the Simulink model of the autopilot.

```
open_system('rct_concorde')
```

Configure the `slTuner`

interface by listing the tuned blocks in the Simulink model (highlighted in orange). This automatically picks all Linear Analysis points in the model as points of interest for analysis and tuning.

ST0 = slTuner('rct_concorde',{'Ki','Kp','Kq','Kf','RollOff'});

This also parameterizes each tuned block and initializes the block parameters based on their values in the Simulink model. Note that the four gains `Ki,Kp,Kq,Kf`

are initialized to zero in this example. By default the roll-off filter
is parameterized as a generic second-order transfer function. To parameterize it as

create real parameters
, build the transfer function shown above, and associate it with the `RollOff`

block.

wn = realp('wn', 3); % natural frequency zeta = realp('zeta',0.8); % damping Fro = tf(wn^2,[1 2*zeta*wn wn^2]); % parametric transfer function setBlockParam(ST0,'RollOff',Fro) % use Fro to parameterize "RollOff" block

The autopilot must be tuned to satisfy three main design requirements:

1. **Setpoint tracking**: The response
to the command
should closely match the response of the reference model:

This reference model specifies a well-damped response with a 2 second settling time.

2. **High-frequency roll-off**: The closed-loop response from the noise signals to
should roll off past 8 rad/s with a slope of at least -40 dB/decade.

3. **Stability margins**: The stability margins at the plant input
should be at least 7 dB and 45 degrees.

For setpoint tracking, we require that the gain of the closed-loop transfer from the command to the tracking error be small in the frequency band [0.05,5] rad/s (recall that we cannot drive the steady-state error to zero because of the plant zero at s=0). Using a few frequency points, sketch the maximum tracking error as a function of frequency and use it to limit the gain from to .

Freqs = [0.005 0.05 5 50]; Gains = [5 0.05 0.05 5]; Req1 = TuningGoal.Gain('Nzc','e',frd(Gains,Freqs)); Req1.Name = 'Maximum tracking error';

The `TuningGoal.Gain`

constructor automatically turns the maximum error sketch into a smooth weighting function. Use `viewSpec`

to graphically verify the desired error profile.

viewSpec(Req1)

Repeat the same process to limit the high-frequency gain from the noise inputs to and enforce a -40 dB/decade slope in the frequency band from 8 to 800 rad/s

Freqs = [0.8 8 800]; Gains = [10 1 1e-4]; Req2 = TuningGoal.Gain('n','delta_m',frd(Gains,Freqs)); Req2.Name = 'Roll-off requirement'; viewSpec(Req2)

Finally, register the plant input
as a site for open-loop analysis and use `TuningGoal.Margins`

to capture the stability margin requirement.

addPoint(ST0,'delta_m') Req3 = TuningGoal.Margins('delta_m',7,45);

We are now ready to tune the autopilot parameters with `systune`

. This command takes the untuned configuration `ST0`

and the three design requirements and returns the tuned version `ST`

of `ST0`

. All requirements are satisfied when the final value is less than one.

[ST,fSoft] = systune(ST0,[Req1 Req2 Req3]);

Final: Soft = 0.965, Hard = -Inf, Iterations = 80

Use `showTunable`

to see the tuned block values.

showTunable(ST)

Block 1: rct_concorde/Ki = d = u1 y1 -0.02968 Name: Ki Static gain. ----------------------------------- Block 2: rct_concorde/Kp = d = u1 y1 -0.009867 Name: Kp Static gain. ----------------------------------- Block 3: rct_concorde/Kq = d = u1 y1 -0.2843 Name: Kq Static gain. ----------------------------------- Block 4: rct_concorde/Kf = d = u1 y1 -0.02231 Name: Kf Static gain. ----------------------------------- wn = 4.8 ----------------------------------- zeta = 0.509

To get the tuned value of
, use `getBlockValue`

to evaluate `Fro`

for the tuned parameter values in `ST`

:

```
Fro = getBlockValue(ST,'RollOff');
tf(Fro)
```

ans = 23.07 -------------------- s^2 + 4.89 s + 23.07 Continuous-time transfer function.

Finally, use `viewSpec`

to graphically verify that all requirements are satisfied.

```
figure('Position',[100,100,550,710])
viewSpec([Req1 Req2 Req3],ST)
```

We now verify that the tuned autopilot satisfies the design requirements. First compare the step response of
with the step response of the reference model
. Again use `getIOTransfer`

to compute the tuned closed-loop transfer from `Nzc`

to `Nz`

:

Gref = tf(1.7^2,[1 2*0.7*1.7 1.7^2]); % reference model T = getIOTransfer(ST,'Nzc','Nz'); % transfer Nzc -> Nz figure, step(T,'b',Gref,'b--',6), grid, ylabel('N_z'), legend('Actual response','Reference model')

Also plot the deflection and the respective contributions of the feedforward and feedback paths:

T = getIOTransfer(ST,'Nzc','delta_m'); % transfer Nzc -> delta_m Kf = getBlockValue(ST,'Kf'); % tuned value of Kf Tff = Fro*Kf; % feedforward contribution to delta_m step(T,'b',Tff,'g--',T-Tff,'r-.',6), grid ylabel('\delta_m'), legend('Total','Feedforward','Feedback')

Finally, check the roll-off and stability margin requirements by computing the open-loop response at .

OL = getLoopTransfer(ST,'delta_m',-1); % negative-feedback loop transfer margin(OL); grid; xlim([1e-3,1e2]);

The Bode plot confirms a roll-off of -40 dB/decade past 8 rad/s and indicates gain and phase margins in excess of 10 dB and 70 degrees.

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