Stability Analysis Using a Bode Plot Consider the feedback control system in Figure 10.65 , where [latex]KG(s) = K\frac{100}{s(s+5)(s+20)}[/latex] and [latex]K[/latex] is assumed to be [latex]1[/latex]. Plot the Bode magnitude and phase curves using MATLAB. Give comments on the stability of the closed-loop system.

Note that the Bode plot is drawn based on the loop gain [latex]L(s) = K\frac{100}{s(s+5)(s+20)}[/latex] The MATLAB command used to sketch the Bode plot is bode. Assuming [latex]K = 1[/latex], the following is the MATLAB session:

Question 10.19: Stability Analysis Using a Bode Plot Consider the feedback c...

Modeling and Analysis of Dynamic Systems

Stability Analysis Using a Bode Plot

Consider the feedback control system in Figure 10.65, where

$KG(s) = K\frac{100}{s(s+5)(s+20)}$

and $K$ is assumed to be $1$ . Plot the Bode magnitude and phase curves using MATLAB. Give comments on the stability of the closed-loop system.

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>> num = [100];
>> den = conv(conv([1 0],[1 5]),[1 20]);
>> sysL = tf(num,den);
>> bode(sysL);

As observed in Figure 10.75, the phase curve crosses $-180°$ at frequency $10 \text{rad/s}$ , and the corresponding magnitude is approximately $−28 \text{dB}$ . This implies that a gain of $28 \text{dB}$ can be added before the magnitude plot reaches $0 \text{dB}$ at that frequency. Thus, the $\text{GM}$ is $28 \text{dB}$ . The magnitude curve crosses $0 \text{dB}$ at frequency $1 \text{rad/s}$ , and the corresponding phase is $-104°$ . This implies that a phase of $76°$ can be subtracted before the phase plot reaches $-180°$ . Thus, the $\text{PM}$ is $76°$ . According to Equations 10.61 and 10.62, the closedloop system with the proportional controller $K = 1$ is stable.

$\begin{cases}\text{GM}>0 \text{db} \text{stable}\\\text{GM}=0 \text{db} \text{marginally stable} \\ \text{GM}<0 \text{db} \text{unstable} \end{cases}$ (10.61)

$\begin{cases}\text{PM}>0° \text{stable}\\\text{PM}=0° \text{marginally stable} \\ \text{PM}<0° \text{unstable} \end{cases}$ (10.62)

More information on stability can be extracted from the $\text{GM}$ . Specifically, the $\text{GM}$ when $K = 1$ gives the stability range of $K$ for which the proportional feedback control system is stable. In our case, $\text{GM} = 28 \text{dB}$ or $25$ , and this indicates that the closed-loop system is stable for $0 < K < 25$ . We can also use the MATLAB command margin as follows:

>> [gm,pm,wcg,wcp] = margin(sysL)

which returns $\text{GM}, \text{PM}$ , and the associated frequencies as defined in Figure 10.74. In our case, the $\text{GM}$ returned by the command margin is $25$ . Note that the stability range of $K$ can be determined in this way only for systems that change from being stable to unstable as $K$ increases.