UNIVERSITY OF TORONTO FACULTY OF APPLIED SCIENCE AND ENGINEERING

DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING

ECE356S – System and Signal Analysis II

Midterm Test – March 1999

(10 pts)

1. Suppose the transition matrix e^{At} is given by

e^{At} = [ \begin{matrix}\frac{1}{2}(3e^{−t} − e^{−3t}) & \frac{1}{2}(e^{−t} − e^{−3t}) \\ −\frac{3}{2}(e^{−t} − e^{−3t}) & \frac{1}{2}(3e^{−3t} − e^{−t}) \end{matrix} ]

Determine the matrix A.

(20 pts)

2. Consider the following feedback system

\frac{K}{s + K}  \frac{1}{s(s + 4)}

where K > 0 is an adjustable parameter.

(i) Sketch the root locus of the closed loop system as a function of the parameter K, and determine the range of values of K for closed loop stability.

(ii) The relative degree of the loop transfer function is 3. Do the results of (i) contradict the usual root locus behaviour for relative degree 3 systems? Explain your answer.

(20 pts)

3. Consider the following feedback system

-  K  \frac{s + 3}{(s + 2)^{2}(s − 1)}

The gain K is > 0.

(i) Putting

P (s) = \frac{s + 3}{(s + 2)^{2}(s − 1)}

sketch the Nyquist plot of P (s). (ii) Using the results of (i), determine the range of values of the gain K for which the closed

loop system is stable.