Problems on Closed-Loop Analysis and Design

Abstract

Select positive values of parameters $\beta$, $\tau _1$, and $\tau _2$ [s] in the following transfer functions: $L_1(s) = K \frac{(\beta s + 1)}{s(\tau _1 s + 1) (\tau _2 s +1)} \quad L_2(s) = K \frac{(\beta s + 1)}{s^3}.$ Analyze the root locus when the gain K varies between 0 and $\infty$. Indicate the values of the gain for which the closed-loop system is unstable. Confirm the expected results from theory using the interactive tool root_locus (card 7.2). For $L_2(s)$, propose a controller that stabilizes the closed-loop system. Analyze the stability as a function of K making use of the Nyquist stability criterion. Confirm the results using the interactive tool Nyquist_criterion (card 7.3). Using the interactive tool stability_margins (card 7.4) and for a value of $K=1$, determine the PM and the GM of the feedback systems with loop transfer functions $L_1(s)$ and $L_2(s)$. Indicate the value of K for which these systems would become unstable and compare it with the results of exercises 1 and 2. For $L_2(s)$, propose a controller designed in the frequency domain that stabilizes the closed-loop system. For a value of K providing a stable closed loop, analyze how much time delay can be added to $L_1(s)$ until the closed loop becomes unstable. Supplementary material: Closed-loop stability using the Routh–Hurwitz criterion.