Page 244 - Modern Control Systems

218 Chapter 3 State Variable Models
E3.6 A system is represented by Equation (3.16), where 0 6
A =
"0 1 1 - 5
A =
0 0
(a) Find the roots of the characteristic equation.
(a) Find the matrix ¢(0- (b) For the initial conditions (b) Find the state transition matrix ¢(0-
-
JC,(0) = -t 2(0) = l,findx(0. Answer: (a) .s = 3 , —2
Answer: (b) .Vj = 1 + t, x 2 = 1, t > 0
-2
- 3
2
E3.7 Consider the spring and mass shown in Figure 3.3 3e ' - 2e~ -6e ' + 6<T '
where M = 1 kg, k = 100 N/m, and b = 20 Ns/m. (b) ¢ ( 0 e" ' - e~ 2 3e" ' - 2e~ 2t
3
3
(a) Find the state vector differential equation, (b)
Find the roots of the characteristic equation for this
system. E3.ll Determine a state variable representation for the
system described by the transfer function
0 1 "o"
Answer: (a) x Y + 4(5 + 3)
100 -20 J _lj 7\s) =
R(s) (s + 2)(5 + 6)'
(b) s = -10, -10
E3.8 The manual, low-altitude hovering task above a E3.12 Use a state variable model to describe the circuit
moving landing deck of a small ship is very demand- of Figure E3.12. Obtain the response to an input unit
ing, particularly in adverse weather and sea condi- step when the initial current is zero and the initial
tions. The hovering condition is represented by the capacitor voltage is zero.
matrix
L = 0.2 H
0 1 0 _ T T Y Y \
• A M •-
0 0 1
0 - 6 - 3 •,;e C = 800/XF:

Find the roots of the characteristic equation.
E3.9 A multi-loop block diagram is shown in Figure
E3.9.The state variables are denoted by .Vj and .v 2. (a) FIGURE E3.12 RLC series circuit.
Determine a state variable representation of the
closed-loop system where the output is denoted by E3.13 A system is described by the two differential
y(t) and the input is /-(0- (b) Determine the character- equations
istic equation.
cfy
+ y — 2u + a%v = 0,
dt
* l 1
4— and
s
— by + Au = 0,
1 dt
' - 1 ^ + Y{s)
Ws) * 0 J J where w and y are functions of time, and u is an input
i —
' + u(t). (a) Select a set of state variables, (b) Write the
matrix differential equation and specify the elements
1 1 4 of the matrices, (c) Find the characteristic roots of the
2 s system in terms of the parameters a and b.
x 2
Answer: (c) s = -1/2 ± V l - Aab/2
FIGURE E3.9 Multi-loop feedback control system. E3.14 Develop the state-space representation of a
radioactive material of mass M to which additional
E3.10 A hovering vehicle control system is represented radioactive material is added at the rate ;(0 = Ku(t),
by two state variables, and [13] where K is a constant. Identify the state variables.