Unformatted text preview: Linear System Theory (4541.512) – Spring 2011 Prof. Songhwai Oh Homework 4. (Due: 6/1/2011)
1. Let be the impulse response matrix of a linear system. Show that the map
(1) is bounded if and only if, for all and , the map
(2) is bounded, where is an element of at -th row and -th column. 2. Consider a linear time-varying system
represented as Show that . A solution to the Lyapunov equation can be satisﬁes 3. Consider a LTI system , where (a) Find the transfer function.
(b) Is the system asymptotically stable? (Explain)
(c) Is the system BIBO stable? (Explain)
(d) Is the system controllable? Is the system observable? (Explain)
4. Consider a LTI system
new input. and a state feedback control , where is a (a) Find the transfer function of the closed-loop system, i.e., the transfer function of the closed-loop
system with input and output .
(b) Show that if the original system is controllable then the closed-loop system is also controllable.
5. Consider the controllability and observability grammians
over the time period
, where and of a LTI system (a) Find the controllability and observability grammians under similarity transformations. Note that
the transformed system is
(b) Prove that the eigenvalues of the product are constants under similarity transformations. 1 ...
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- Spring '10
- impulse response matrix, Prof. Songhwai Oh