Page 243 - Modern Control Systems

Exercises 217

•10 -20 -50
c x = 0 0 x +
1 0
y = [0 5 50]x
-10 -20"
d. x 1 x +
0
y [0 5]x
In the following Word Match problems, match the term with the definition by writing the
correct letter in the space provided.
a. State vector The differential equation for the state vector x = Ax + Bw.
b. State of a The matrix exponential function that describes the unforced
system response of the system.
c. Time-varying The mathematical domain that incorporates the time
system response and the description of a system in terms of time, t.
d. Transition Vector containing alln state variables, x x, x 2,---, x n.
matrix
e. State A set of numbers such that the knowledge of these numbers
variables and the input function will, with the equations describing the
dynamics, provide the future state of the system.
f. State A system for which one or more parameters may vary with
differential time.
equation
g. Time domain The set of variables that describe the system.

EXERCISES

E3.1 For the circuit shown in Figure E3.1 identify a set of
state variables. A= ° '
.-1 -2_
Find the characteristic roots of the system.
Answer: - 1 , - 1
E3.4 Obtain a state variable matrix for a system with a
differential equation
2
d*y d y dy
6
j + 4 ^ 2 + i; + 8 y = 20u W-
dt* dt
FIGURE E3.1 RLC circuit.
E3.5 A system is represented by a block diagram as
E3.2 A robot-arm drive system for one joint can be repre- shown in Figure E3.5. Write the state equations in the
sented by the differential equation [8] form of Equations (3.16) and (3.17).
dv(t)
-k xv(t) - k 2y(t) + k 3i(t),
dt
where v(t) = velocity, y(t) = position, and i(t) is the U(s) Y(s)
control-motor current. Put the equations in state vari-
able form and set up the matrix form for k x = k 2 = 1.
E33 A system can be represented by the state vector dif-
ferential equation of Equation (3.16), where FIGURE E3.5 Block diagram.