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Unformatted text preview: UNIVERSITY OF TORONTO FACULTY OF APPLIED SCIENCE & ENGINEERING ECE 311S DYNAMIC SYSTEMS & CONTROL FINAL EXAM, April 14, 2011 SOLUTION 1. Consider the system with input u ( t ) and output y ( t ) modelled by the differential equa tion 4 y + y = u. (a) [2 marks] Take Laplace transforms with zero initial conditions and in that way find the transfer function from u ( t ) to y ( t ). (b) [3 marks] Is the system BIBO stable? If yes, give a bound on  y ( t )  in terms of a bound on  u ( t )  . If not, give a bounded u ( t ) for which y ( t ) is not bounded. Soln (a) G ( s ) = 1 4 s 2 + 1 (b) It is not BIBO stable since there are poles on the imaginary axissee Theorem 3.5.1. If the input is a sinusoid of the resonant frequency, the output is unbounded. That is, take U ( s ) = G ( s ). 1 2. (a) [2 marks] Consider the state model x = Ax + Bu y = Cx. If x (0) = 0, then u ( t ) and y ( t ) are related by a convolution equation y ( t ) = g ( t ) * u ( t ). Write the expression for g ( t ) in terms of A,B,C . (b) [5 marks] Consider the system with input u ( t ) and output y ( t ) modelled by the differential equation 4 y + y = 2 u + u. Derive a state model. For full marks you must show what the components of x are. Soln (a) From equation (3.3) in the course notes and since D = 0 g ( t ) = C e At B. This is the inverse Laplace transform of C ( sI A ) 1 B . (b) The transfer function is G ( s ) = 2 s + 1 4 s 2 + 1 . Follow the method in the final example in Section 2.6. 2 3. (a) [2 marks] Suppose A is a 3 3 matrix for which det( sI A ) = s 3 + 2 s 2 + s + 3 ....
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This note was uploaded on 01/30/2012 for the course ECE ECE311 taught by Professor Francis during the Spring '11 term at University of Toronto Toronto.
 Spring '11
 Francis

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