ps5 - MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 State-space 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 state-space 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 2nd-order transfer functions (e.g. in 11.9b) by breaking them up into two 1st-order 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 half-plane. 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 ...
View Full Document

Page1 / 2

ps5 - MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online