.ipynb .pdf

repository open issue

Contents

Homework 1: Revision

Problems

1. Find the closed-loop characteristic equation of the system illustrated in Figure 1 when:

2. $$G(s) = \frac{1}{s+1}$$ when $$H(s) = 1,\;\frac{1}{s+2}\; \mathrm{and}\; s.$$
3. $$G(s) = \frac{K(s+2)}{s\left(s^2 + s +1\right)}$$ when $$H(s) = \frac{s+1}{s+10}.$$

Figure 1: A Closed Loop System

1. A feedback control system has an open-loop transfer function

   $$G(s) = \frac{K(s + 2)}{(s + 1)(s - 5)}$$

   and unity gain feedback. Find the values of $K$ for which the system is closed-loop stable.

1. A control system has the root-locus shown in Figure 2. Find the closed-loop poles, natural frequency $\omega_n$ and gain $K$ when the damping ratio $\zeta = 0.0,\;0.1,\;0.5\;\textrm{and}\;1.0$.

   What values of gain and damping ratio satisfy the constraints $2 < \omega_n \le 10$ rads^-1.

   Is it possible to satisfy the following constraints: rise-time $T_r \le 0,4$ seconds and peak overshoot $M_p \le 0.2$ (20%) by adjusting the forward loop gain only?

   

   Figure 2: Root Locus Diagram for Question 3

1. Sketch the root-locus diagram for the system of Question 2. Find the value of the open-loop gain that yields closed-loop poles having ideal damping $\left(\zeta = 1/\sqrt{2}\right)$.

1. For the system shown in Figure 1

   $$G(s) = \frac{1}{sT + 1}$$

   and

   $$H(s) = h.$$

   Find the steady-state step error of of the closed-loop system and determine its system type number. What is the system type number when $h = 1$?

1. A general second-order closed-loop control system has the transfer function

   $$G_c(s)=\frac{b_1 s + b_0}{s^2 + a_1 s + a_0}$$

   Find suitable values of the parameters $b_1$, $b_0$, $a_1$, and $a_0$ that provide rise-time $Tr \le 0.1$ s, settling-time $T_s \le 0.5$ s, peak-overshoot $\%\mathrm{OS} \le 20\%$, zero steady-state step error and a ramp error of $0.01$.

Homework Problems Homework 2: Dominant Poles and Lead Compensation