ADVANCED CONTROL SYSTEM (EC-503 (C)) - Important Questions

Derive the pulse-transfer function of a continuous-time plant when it is sampled by a Zero-Order Hold (ZOH) with sampling period $T_s$. Explain the mapping from the continuous-time domain to the discrete-time domain and show how poles of the continuous plant relate to poles of the pulse-transfer function. In your derivation show expressions for $\Phi= e^{A T_s}$ and $\Gamma=\int_{0}^{T_s} e^{A\tau}B\,d\tau$ for a state-space plant $\dot{x}(t)=Ax(t)+Bu(t)$.

Use the Jury stability test to determine whether the following discrete-time polynomial is stable (all roots inside the unit circle). Carry out the complete Jury table and conclude about stability for

$$P(z)=z^{3}-0.5\,z^{2}+0.2\,z-0.1.$$

Explain the phase-plane method. For the nonlinear system

$$\dot{x}_{1}=x_{2},\qquad \dot{x}_{2}=-x_{1}+x_{2}\left(1 - x_{1}^{2} - x_{2}^{2}\right),$$

find and classify all equilibrium points. Sketch typical trajectories in the phase plane and comment on stability of each equilibrium.

State Lyapunov's direct method for stability of autonomous systems. Using the Lyapunov candidate $V(x)=x^{T}Px$ with $P=P^{T}>0$, show how to test stability for a linear system $\dot{x}=Ax$. Apply this procedure to determine stability of the system with

$$A=\left[\begin{matrix}-1 & 0\\[4pt]0 & -2\end{matrix}\right].$$

Design a compensator (specify lead or lag) using the root-locus method to meet the following requirements for a given plant: damping ratio $\zeta=0.5$ and 2% settling time $T_{s}\leq 2\ \text{s}$. Describe the design steps, sketch the root-locus with compensator, and verify that the closed-loop system meets the specifications.

Explain commonly used PID tuning rules such as Ziegler–Nichols (both reaction curve and ultimate gain methods) and Cohen–Coon. For the plant

$$G(s)=\frac{1}{s\left(s+2\right)},$$

obtain PID parameters using Ziegler–Nichols and Cohen–Coon. Implement both controllers in MATLAB Simulink, report step-response metrics (rise time, settling time, overshoot, steady-state error), and compare the results discussing advantages and disadvantages of each tuning rule.

Derive the discrete-time state-space model obtained by exact sampling (ZOH) for the continuous-time system $\dot{x}(t)=Ax(t)+Bu(t)$. Show that the sampled model is

$$x[k+1]=\Phi\,x[k]+\Gamma\,u[k],$$

and explicitly give $$\Phi= e^{A T_s},\qquad \Gamma=\int_{0}^{T_s} e^{A\tau}B\,d\tau.$$

Compute $\Phi$ and $\Gamma$ for

$$A=\left[\begin{matrix}0 & 1\\[4pt]-2 & -3\end{matrix}\right],\qquad B=\left[\begin{matrix}0\\[4pt]1\end{matrix}\right],$$

with sampling period $T_s=0.1\ \text{s}$ (you may leave matrix exponentials in closed form).

Explain the bilinear (Tustin) transform and derive the substitution for $s$ in terms of $z$. Using the Tustin transform with prewarping, convert the continuous-time lead compensator

$$G_c(s)=K\frac{1+\tau s}{1+\alpha\tau s}\qquad\left(0<\alpha<1\right)$$

to a discrete-time compensator $G_c(z)$ for sampling period $T_s$ and show how frequency prewarping is applied to preserve a specified analog frequency $\omega_{p}$.

Given the open-loop pulse-transfer function

$$G(z)=\frac{0.5\,z}{z^{2}-1.2\,z+0.4},$$

use a suitable digital controller to place the dominant closed-loop poles at $0.5\pm j\,0.1$. Show controller design steps, compute the resulting closed-loop transfer function, and verify pole locations.