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Unformatted text preview: MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Aeronautics and Astronautics
16.060: Principles of Automatic Control
Fall 2003
PROBLEM SET 5
Due: 10/23/2003
Problem 1: State Space Models for MIMO Systems
Statespace models are extremely useful for studying systems with more than one input and/or more than
one output. Problem 11.9 in Van de Vegte gives two block diagrams. The ﬁrst is for a system with two
inputs and one output, and the second is for a system with two inputs and two outputs. For each system,
determine a statespace model. Show your choice of state variables on the original block diagram.
Hint 1: Look at Example 11.3.4.
Hint 2: Deal with blocks that contain 2ndorder transfer functions (e.g. in 11.9b) by breaking them
up into two 1storder transfer functions.
Hint 3: When deriving the state equations, use the fact that the following block diagrams are equiv
alent (you should prove to yourself that they are): U X
k
s+a �
u ��+ �
x ˙ k dt
�� ��
� ��
a�
�� Problem 2: Properties of the State Transition Matrix
For the system
�
�
˙ = −1 0 �
�
x
x
1 −2
1. Find the state transition matrix �(t).
2. Show that �(0) = I .
3. Show that �−1 (t) = �(−t).
4. Show that �(t1 + t2 ) = �(t2 )�(t1 ). 1 x Problem 3: Determining Stability and Solving the State Equations
Recall from lecture:
˙
• The system � = A� + B � is stable if all the eigenvalues of A lie in the left halfplane.
x
u
x
x
• Given initial conditions � (0) and input � (t) for t > 0, the state at any time t is given by:
u
� (t) = �(t)� (0) +
x
x � t �(t − � )B� (� )d�
u (1) 0 1. For the system
�
�
˙ = −1 1 �
x
�
x
−1 −3
(a) Show whether this system is stable, marginally stable, or unstable.
(b) Find the transition matrix �(t) (you might need to consult your Laplace transform tables).
(c) Find the state response to the initial conditions x 1 (0) = 1, x2 (0) = −1.
2. Do Problem 11.25 in Van de Vegte. (Note: the “output response” means � (t), not the state response
y
x
� (t).)
Problem 4: Controllability
1. In one short paragraph, describe what it means to say that a system is uncontrollable.
2. Do Problem 11.35 in Van de Vegte. 2 ...
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 Fall '03
 willcox
 Aeronautics, Astronautics

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