82 CHAPTER 2. MATHEMATICAL MODELS IN CONTROL
- A system is described by the following differential equation
d^2 x
dt^2
+2
dx
dt
+3x=1
with the initial conditionsx(0) = 1andx ̇(0) =− 1. Show a block diagram
of the system, giving its transfer function and all pertinent inputs and
outputs.
- Develop the state model (state and output equations) for the system be-
low.
R(s)−→
5 s+1
s^4 +2s^3 +s^2 +5s+10
−→C(s)
- A system is defined as
d^2 x
dt^2
+12
dx
dt
+30x=f(x)
Linearize the system for the following functionsf(x).
(a)f(x)=sinxforx=0 (b)f(x)=sinxforx=π
(c)f(x)=e−xforx≈ 0
- Afirst-order system is modeled by the state and output equations as
dx(t)
dt
= − 3 x(t)+4u(t)
y(t)=x(t)
(a) Find the Laplace transform of the set of equations and obtain the
transfer function.
(b) If the inputu(t)is a unit step function, withx(0) = 0,findy(t),
t> 0.
(c) If the inputu(t)is a unit step function, withx(0) =− 1 ,findy(t),
t> 0.
(d) Obtain aMatlabsolution to verify your results.
- Given the state equations
∑
x ̇ 1 (t)
x ̇ 2 (t)
∏
=
∑
02
− 1 − 3
∏∑
x 1 (t)
x 2 (t)
∏
+
∑
0
1
∏
u(t)
y(t)=
£
10
§
∑
x 1 (t)
x 2 (t)
∏
(a) Find the Laplace transform of the set of equations and obtain the
transfer function.
