Page 240 - Modern Control Systems

Chapter 3 State Variable Models

CHECK

In this section, we provide three sets of problems to test your knowledge:True or False, Multiple
Choice, and Word Match. To obtain direct feedback, check your answers with the answer key
provided at the conclusion of the end-of-chapter problems.
In the following True or False and Multiple Choice problems, circle the correct answer,
1. The state variables of a system comprise a set of variables that
describe the future response of the system, when given the present
state, all future excitation inputs, and the mathematical model
describing the dynamics. True or False
2. The matrix exponential function describes the unforced response of
the system and is called the state transition matrix. True or False
3. The outputs of a linear system can be related to the state variables
and the input signals by the state differential equation. True or False
4. A time-invariant control system is a system for which one or more
of the parameters of the system may vary as a function of time. True or False
5. A state variable representation of a system can always be written
in diagonal form. True or False
6. Consider a system with the mathematical model given by the differential equation:
3
2
d y d y .Jy
5-4+ 10-—r + 5— + 2 v = u
1
dt 3 dt dt
A state variable representation of the system is:
"-2 -1 -0.4" "l
a. x = 1 0 0 X + 0 u
0 1 0 _0_
y = 0 0 0.2]x
"-5 - 1 -0.7~ " - 1 ~
b. x = 1 0 0 X + 0 u
0 1 0 0_
y = [0 0 0.2]x
"-2 - f "l
c. x = x + u
1 0_ 0
y = [i 0]x
" - 2 - 1 -0.4" ~r
d. x = 1 0 0 X + 0 u
0 1 0 _0_
y = [1 0 0.2]x
For Problems 7 and 8, consider the system represented by
X = A3i + B u,
where

A = and B =
7. The associated state-transition matrix is:
a. $(f,0) = [5f]
"1 5r"
b. ¢(¢,0) =
10 1