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Unformatted text preview: EE221A Linear System Theory Problem Set 9 Professor C. Tomlin Department of Electrical Engineering and Computer Sciences, UC Berkeley Fall 2011 Issued 11/21; Due 12/1 Problem 1: Lyapunov Equation. (a) Consider the linear map L : R n n R n n defined by L ( P ) = A T P + PA . Show that if i + j negationslash = , i , j ( A ), the equation: A T P + PA = Q has a unique symmetric solution for given symmetric Q . (b) Show that if ( A ) C then for given Q > 0, there exists a unique positive definite P solving A T P + PA = Q (Hint: try P = integraltext e A T t Qe At dt ) Problem 2: Asymptotic and exponential stability. True or False: If a linear time-varying system is asymptotically stable, it is also exponentially stable. If true, prove, if false, give a counterexample. Problem 3: State observation problem. Consider the linear time varying system: x ( t ) = A ( t ) x ( t ) y ( t ) = C ( t ) x ( t ) This system is not necessarily observable. The initial condition at time 0 isThis system is not necessarily observable....
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This note was uploaded on 12/09/2011 for the course EE 221A taught by Professor Clairetomlin during the Fall '10 term at University of California, Berkeley.
- Fall '10
- Electrical Engineering