EECE 340 Homework, Spring 2008

1 American University of Beirut Department of Electrical and Computer Engineering EECE 440 Signals and Systems Homework 2: Due July 18, 2006 Part I Problem 1 Consider a unity feedback linear time-invariant control system whose input is r(t) and output is c(t). The forward transfer function of this system is given by: 4) 2)(S S(S )1 20(S G(S) + + + = Represent this system in a state variable form by determining the state equation and the output equation. Problem 2 The open-loop transfer function of a unity feedback control system is given by: 1 aS 4S 4S S 2 )1 (S A(S) 2 3 4 + + + + + = Determine the stability of the system (state the reason) for the following two cases when a=0 and a=8. Problem 3 The dynamic equations of a linear time-invariant system whose output is y(t) and input r(t) are given by: a. Determine the stability of the above system. b. Determine the gains (f1, f2) of the input [ ] ⎥⎦ ⎤ ⎢⎣ ⎡ = )t( X )t( f 2 X f1 )t(r 2 1 in order to force the closed-loop system poles in the desired locations in the s-plane (-5 and –3). [ ] ⎥⎦ ⎤ ⎢⎣ ⎡ = ⎥⎦ ⎤ ⎢⎣ + ⎡ ⎥⎦ ⎤ ⎢⎣ ⎡ ⎥⎦ ⎤ ⎢⎣ = ⎡− ⎥⎦ ⎤ ⎢⎣ ⎡ ′ ′ )t( X )t( 1 1 X t(r ), y )t( 1 1 )t( X )t( X 0 2 2 1 )t( X )t( X 2 1 2 1 2 1 2 Problem 4 A temperature control system is shown below. Sensor 1 measures the temperature, and this sensor is slow compared to the rate at which the temperature can change. Sensor 2 is called the rate sensor. PI Controller Gp(S) + K (1+1/S) 2/(S+2) - - Sensor 2 S Sensor 1 10/(S+10) 1. Find the values of K that will result in the system being oscillatory. 2. With K equal to the value found in (1), determine the period of oscillation. 3. Determine the range of K for stability. 4. Determine the steady-state error for a constant input provided that the system is stable. Problem 5 Consider a linear control system whose system transfer function is given by: 1 s K R )s( C )s( + = a. For K=10, find the unit step response. b. Find the time constant. The time constant T is defined as the time T that satisfies the following equation: )] (c o) .0 368 (c[ ) (c (c T) ∞ − ∞ = − . c. Can you have another definition or meaning of the time constant? d. For K=10, close the loop with a constant feedback gain H. Design H so that the unit- step response reaches (and never decreases below) the 63% of its final value within the first 0.5 sec. 3 Problem 6 The open-loop transfer function of a unity feedback control system is given by: 10) 1)(s s( z) z s( 10 G )s( + + + = Determine z so that the step response percentage overshoot is 0.1.% Problem 7 The open-loop transfer function of a unity feedback control system is given by: s 2s s 10 3s )s( G 2 3 + + + = Determine the steady-state value (final value) of the output signal for a unit-step input. Problem 8 The closed-loop transfer function of a control system is given by: 3 6s 7s 5s 5s s s 1 s )s( R )s( C 2 3 4 5 6 + + + + + + + = 1. Construct the RH table of the above system using the ∈-method. 2. Choose ∈ > 0 and determine the number of poles of the closed-loop system in each half of the complex plane. 3. Choose ∈ < 0 and determine the number of poles of the closed-loop system in each half of the complex plane. 4. Comment on the results obtained in parts 2 and 3. 5. Using another procedure, determine the number of poles of the system in each half of the complex plane. 4 Part 2: Matlab Project 1. Stability using MATLAB. You are requested to determine a procedure to check on the stability of a LTI continuous using MATLAB. Give procedure and a minimum of two examples. 2. State Variable Analysis using MATLAB. Determine a MATLAB procedure for representing a system defined by its transfer function in a State-variable representation. Give a minimum of two examples.