Determine the range of values for the gain K for which the closed-loop system shown in Fig. 13.4 is stable.

Question

Determine the range of values for the gain K for which the closed-loop system shown in
Fig. 13.4 is stable.

Accepted Answer

First, find the system transfer function:

[latex]T_{CL} \left(s ight) =\frac{\frac{K\left(s+40 ight) }{s\left(s+10 ight) } }{1+\frac{K\left(s+40 ight)}{s\left(s+10 ight)\left(s+20 ight)} } .[/latex]

Hence the system characteristic equation is

[latex]s\left(s+10 ight)\left(s+20 ight)+K\left(s+40 ight)=0,[/latex]

which, after multiplying, becomes

[latex]s^{3} +30s^{2} +\left(200+K ight) s+40K=0[/latex].

The necessary conditions for stability are

[latex]200+K\gt 0, 40K\gt 0.[/latex]

Both inequalities are satisfied for K > 0, which satisfies the necessary conditions for system stability. But this is not enough. The Hurwitz necessary and sufficient conditions for stability of this third-order system are [latex]D_{1} \gt 0[/latex] and [latex]D_{2} \gt 0[/latex]. The first inequality is satisfied because [latex]D_{1} =a_{2} =30[/latex]. The second Hurwitz determinant [latex]D_{2} [/latex] is

[latex]D_{2} =\begin{vmatrix} a_{2} & a_{0} \ a_{3} & a_{1} \end{vmatrix} =\begin{vmatrix} 30 & 40K \ 1 & \left(200+K ight) \end{vmatrix}=\left(6000-10K ight) .[/latex]

Combining the necessary and sufficient set of conditions yields the range of values of K for which the system is stable:

[latex]0\lt K\lt 600.[/latex]

Question 13.2: Determine the range of values for the gain K for which the c...

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