EE 6300 Exam#1 Oct. 23, 2014

75 minutes

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1. The block diagram of a control system is given below.

s + 2

s + 1

1

s

1

s

u y

Obtain a state-space representation of the system without any block-diagram reduction. (25pts)

2. A control system is described by

x(t) =

[

−2 −8−2 −2

]

x(t) +

[

21

]

u(t),

andy(t) =

[

1 0]

x(t),

where u, x, and y are the input, the state, and the output variables, respectively.

(a) Determine x(t) for t ≥ 0, when x(0) =[

1 −1]T

, and u(t) = 0 for t ≥ 0. Show all your workclearly. (10pts)

(b) Determine x(t) for t ≥ 0, when x(0) = 0, and u(t) = 1 for t ≥ 0. Show your work clearly.(10pts)

(c) Determine x(t) for t ≥ 0, when x(0) =[

1 −1]T

, and u(t) = 1 for t ≥ 0. Show your workclearly. (10pts)

3. A time-varying linear control system is described by

x(t) =

[

−t 10 −t

]

x(t) +

[

01

]

u(t),

where u and x are the input and the state variables, respectively. Determine the state transitionmatrix Φ(t, t0). Show all your work clearly. (25pts)

4. A control system is described in state-space representation, such that

x(t) = Ax(t) + Bu(t),

y(t) = Cx(t) + Du(t),

where u, x, and y are the input, the state, and the output variables, respectively. For the followingA, B, C, and D matrices, determine whether the system is asymptotically stable, marginally stable,or unstable in the state sense; and whether it is bounded-input-bounded-output stable or not. Justifyyour answer.

1

(a)

A =

0 −1 1 0 01 0 0 1 00 0 0 −1 00 0 1 0 00 0 0 0 0

, B =

11000

, C =[

0 0 1 0 1]

, and D = 0.

(05pts)

(b)

A =

0 −1 1 0 01 0 0 1 00 0 0 −1 00 0 1 0 00 0 0 0 −1

, B =

11001

, C =[

0 0 1 0 1]

, and D = 0.

(05pts)

(c)

A =

0 −1 0 0 01 0 0 0 00 0 0 −1 00 0 1 0 00 0 0 0 0

, B =

11001

, C =[

0 0 1 0 1]

, and D = 0.

(05pts)

(d)

A =

0 −1 0 0 01 0 0 0 00 0 0 −1 00 0 1 0 00 0 0 0 1

, B =

11001

, C =[

0 0 1 0 1]

, and D = 0.

(05pts)

2