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Improving Stability While Preserving Open-Loop Characteristics

This example shows how to use Robust Control Toolbox™ function ncfsyn to improve the stability robustness of a closed-loop system while approximately maintaining the high-gain and low-gain characteristics of the open-loop response.

Plant Model

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

$$\frac{16}{s^2+0.16 s+16}$$

A Bode plot clearly shows the resonant peak.

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

Closed-Loop Objectives and Initial Controller Design

The closed-loop system should

  • Be insensitive to noise, including 60dB/decade attenuation beyond 20 rad/sec

  • Have integral action, and a sensitivity bandwidth of at least 0.5 rad/sec

  • Have gain-crossover frequencies no larger than 7

We can translate these requirements into a desired shape for the open-loop gain and seek a controller that enforces this shape. For example, a controller Kprop 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

The desired high-frequency rolloff, integral action, and location of the gain-crossover frequency are all evident from the graph. Unfortunately, this controller does not stabilize the closed-loop system, as evidenced by the Stable field from the allmargin calculation.

allmargin(P*Kprop)
ans = 

     GainMargin: 0.1096
    GMFrequency: 4.1967
    PhaseMargin: -20.4811
    PMFrequency: 5.6155
    DelayMargin: 1.0552
    DMFrequency: 5.6155
         Stable: 0

Closed-loop Stability and Margin Improvement Using ncfsyn

The command ncfsyn can be used to obtain closed-loop stability and improved stability margins without significant degradation in the large (>>1) and small (<<1) gain regions of the open-loop response. A value of Gamma (the 3rd output argument) less than 3 indicates success (modest gain degradation along with acceptable robustness margins). Note that ncfsyn assumes positive feedback so you must flip the sign of Kprop. The new controller, K, stabilizes the nominal dynamics, and appears to have decent robustness margins, at least when measured in the classical sense.

[negK,~,Gamma] = ncfsyn(P,-Kprop);
K = -negK;   % flip sign back
Gamma
allmargin(P*K)
Gamma =

    1.9883


ans = 

     GainMargin: [6.3299 11.1423]
    GMFrequency: [1.6110 15.1667]
    PhaseMargin: [80.0276 -99.6641 63.7989]
    PMFrequency: [0.4472 3.1460 5.2319]
    DelayMargin: [3.1236 1.4443 0.2128]
    DMFrequency: [0.4472 3.1460 5.2319]
         Stable: 1

As expected, the closed-loop is stable, and with Gamma less than 2, the gain and phase margins are adequate by rule-of-thumb standards.

Bode Magnitude of Controllers

Gamma gives an indication of the gain degradation (in going from Kprop to K) we should expect, both the in the large and small loop-gain regions. With Gamma approximately 2, we expect only about 6dB (20*log10(Gamma)) gain reduction in the large loop-gain regions, and no more than 6dB gain increase in the small loop-gain regions. Bode magnitude plots confirm this.

subplot(1,2,1)
bodemag(Kprop,'r',K,'g'); grid
legend('Proposed Controller, Kprop','Redesigned Controller, K')
title('Controller Gains')
subplot(1,2,2)
bodemag(P*Kprop,'r',P*K,'g'); grid
legend('P*Kprop','P*K')
title('Open-Loop Gains')

Figure 1: Controller and Open-Loop gains

Closed-Loop Response to Impulse at the Plant Input

An impulsive disturbance at the plant input is damped out effectively in a few seconds. For comparison, the uncompensated plant response is also shown.

TF = 4;
subplot(1,2,1)
impulse(feedback(P,K),'b',P,'r',TF);
legend('Closed-Loop','Open-Loop')
title('Plant response, y, to impulse at plant input')
subplot(1,2,2);
impulse(-feedback(K*P,1),'b',TF)
legend('Closed-Loop Control action, u')

Figure 2: Controller and Open-Loop gains

Sensitivity and Complementary Sensitivity Functions

Finally, 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

Conclusions

In this example, the function ncfsyn is used to adjust a proposed controller in order to achieve closed-loop stability, while attempting to maintain the general large and small loop-gain characteristics of the open-loop function PK.

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