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This example shows how to use `ncfsyn`

to shape the open-loop response while enforcing stability and maximizing robustness. `ncfsyn`

measures robustness in terms of the normalized coprime stability margin computed by `ncfmargin`

.

The plant model is a lightly damped, second-order system.

$$P(s)=\frac{16}{{s}^{2}+0.16s+16}$$.

A Bode plot shows the resonant peak.

P = tf(16,[1 0.16 16]); bode(P)

The design objectives for the closed-loop are the following.

Insensitivity to noise, including 60dB/decade attenuation beyond 20 rad/sec

Integral action and a bandwidth of at least 0.5 rad/s

Gain crossover frequencies no larger than 7 rad/s

In loop-shaping control design, you translate these requirements into a desired shape for the open-loop gain and seek a compensator that enforces this shape. For example, a compensator consisting of a PI term in series with a high-frequency lag component achieves the desired loop shape.

K_PI = pid(1,0.8); K_rolloff = tf(1,[1/20 1]); Kprop = K_PI*K_rolloff; bodemag(P*Kprop); grid

Unfortunately, the compensator `Kprop`

does not stabilize the closed-loop system. Examining the closed-loop dynamics shows poles in the right half-plane.

pole(feedback(P*Kprop,1))

`ans = `*4×1 complex*
-20.6975 + 0.0000i
0.4702 + 5.5210i
0.4702 - 5.5210i
-0.4029 + 0.0000i

`ncfsyn`

You can use `ncfsyn`

to enforce stability and adequate stability margins without significantly altering the loop shape. Use the initial design `Kprop`

as loop-shaping pre-filter. `ncfsyn`

assumes a positive feedback control system (see `ncfsyn`

), so flip the sign of `Kprop`

and of the returned controller.

```
[K,~,gamma] = ncfsyn(P,-Kprop);
K = -K; % flip sign back
gamma
```

gamma = 1.9903

A value of the performance `gamma`

less than 3 indicates success (modest gain degradation along with acceptable robustness margins). The new compensator `K`

stabilizes the plant and has good stability margins.

allmargin(P*K)

`ans = `*struct with fields:*
GainMargin: [6.2984 10.9082]
GMFrequency: [1.6108 15.0285]
PhaseMargin: [79.9812 -99.6214 63.7590]
PMFrequency: [0.4467 3.1469 5.2304]
DelayMargin: [3.1253 1.4441 0.2128]
DMFrequency: [0.4467 3.1469 5.2304]
Stable: 1

With `gamma`

approximately 2, the expect at most `20*log10(gamma)`

= 6dB gain reduction in the high-gain region and at most 6dB gain increase in the low-gain region. The Bode magnitude plot confirms this. Note that `ncfsyn`

modifies the loop shape mostly around the gain crossover to achieve stability and robustness.

subplot(1,2,1) bodemag(Kprop,'r',K,'g',{1e-2,1e4}); grid legend('Initial design','NCFSYN design') title('Controller Gains') subplot(1,2,2) bodemag(P*Kprop,'r',P*K,'g',{1e-3,1e2}); grid legend('Initial design','NCFSYN design') title('Open-Loop Gains')

**Figure 1:** Compensator and open-loop gains.

With the `ncfsyn`

compensator, an impulse disturbance at the plant input is damped out in a few seconds. Compare this response to the uncompensated plant response.

subplot(1,2,1) impulse(feedback(P,K),'b',P,'r',5); legend('Closed loop','Open loop')

subplot(1,2,2); impulse(-feedback(K*P,1),'b',5) title('Control action')

**Figure 2:** Response to impulse at plant input.

The closed-loop sensitivity and complementary sensitivity functions show the desired sensitivity reduction and high-frequency noise attenuation expressed in the closed-loop performance objectives.

S = feedback(1,P*K); T = 1-S; clf bodemag(S,T,{1e-2,1e2}), grid legend('S','T')

In this example, you used the function `ncfsyn`

to adjust a hand-shaped compensator to achieve closed-loop stability while approximately preserving the desired loop shape.