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exam_solution - UNIVERSITY OF TORONTO FACULTY OF APPLIED...

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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 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. Sol’n (a) G ( s ) = 1 4 s 2 + 1 (b) It is not BIBO stable since there are poles on the imaginary axis—see 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
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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 y + y = - 2 ˙ u + u. Derive a state model. For full marks you must show what the components of x are. Sol’n (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
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3. (a) [2 marks] Suppose A is a 3 × 3 matrix for which det( sI - A ) = s 3 + 2 s 2 + s + 3 . Does e At converge to 0 as t tends to ? Justify your answer.
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