Consider a process that can be described by the transfer function matrix
$G_{p}(s) = \begin{bmatrix} \frac{2}{10s+1} & \frac{1.5}{s+1} \\ \frac{1.5}{s+1} & \frac{2}{10s+1} \end{bmatrix}$
Assume that two proportional feedback controllers are to be used so that $G_{c1}= K_{c1} and G_{c2}=K_{c2}$ . Determine the values of $K_{c1} and K_{c2}$ that result in closed-loop stability for both the 1-1/2-2 and 1-2/2-1 configurations.

Question

Consider a process that can be described by the transfer function matrix

[latex]G_{p}(s) = \begin{bmatrix} \frac{2}{10s+1} & \frac{1.5}{s+1} \ \frac{1.5}{s+1} & \frac{2}{10s+1} \end{bmatrix}[/latex]

Assume that two proportional feedback controllers are to be used so that [latex]G_{c1}= K_{c1} and G_{c2}=K_{c2}[/latex]. Determine the values of [latex]K_{c1} and K_{c2}[/latex] that result in closed-loop stability for both the 1-1/2-2 and 1-2/2-1 configurations.

Accepted Answer

The characteristic equation for the closed-loop system (1-1/2-2 pairing) is obtained by substitution into Eq. 18-20 and collecting powers of s as follows (which can be obtained
using a symbolic math toolbox):

[latex](1+G_{c1}G_{p11})(1+G_{c2}G_{p22})-G_{c1}G_{c2}G_{p12}G_{p21}=0 (18.20)[/latex]

[latex]a_{4}s^{4} +a_{3}s^{3} +a_{2}s^{2} +a_{1}s+ a_{0}=0 (18.23)[/latex]

where

[latex] a_{4}=100[/latex]

[latex]a_{3}=20K_{c1} +20K_{c2}+220 [/latex]

[latex]a_{2}=42K_{c1}+42K_{c2}-221K_{c1}K_{c2}+141 [/latex]

[latex]a_{1}=24K_{c1}+24K_{c2}-37K_{c1}K_{c2}+22 [/latex]

[latex]a_{0}=2K_{c1}+2K_{c2}+1.75K_{c1}K_{c2}+1 [/latex]

Note that the characteristic equation in Eq. 18-23 is fourthorder, even though each individual transfer function in [latex]G_{p}(s)[/latex] is first order.

The controller gains that result in a stable closed-loop system can be determined by applying direct substitution to the stability criterion (Chapter 11) for specified values of [latex]K_{c1} and K_{c2}[/latex]. The resulting stability regions are shown in Fig. 18.6. If either [latex]K_{c1} or K_{c2}[/latex] is close to zero, the other controller gain can be an arbitrarily large, positive value and still have a stable closed-loop system. This result is a consequence of having process transfer functions that are first order without time delay, which is an idealistic case. MIMO
control systems normally have an upper bound for stability for both controller gains for all values of [latex]K_{c1}[/latex].

A similar stability analysis can be performed for the 1-2/2-1 control configuration (see Fig. 18.7). A comparison of Figs. 18.6 and 18.7 indicates that the 1-2/2-1 control scheme results in a larger stability region because a wider range of controller gains can be used. For example, suppose that [latex]K_{c1} = 2[/latex]. Then Fig. 18.6 indicates that the 1-1/2-2 configuration will be stable if [latex]−0.8 < K_{c2} < 0.5[/latex]. By contrast, Fig. 18.7 shows that the corresponding stability limits for the 1-2/2-1 configuration are [latex]−0.3 < K_{c2} < 2.0[/latex].

This example illustrates that closed-loop stability depends on the control configuration as well as the numerical values of the controller settings. If PI control had been considered instead of proportional-only control, the stability analysis would have been much more complicated due to the larger number of controller settings and the higher-order characteristic equation.

Question 18.2: Consider a process that can be described by the transfer fun...

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