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hw7_2009_11_19_01_solutions

hw7_2009_11_19_01_solutions - EE263 S Lall 2009.11.19.01...

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EE263 S. Lall 2009.11.19.01 edited Homework 7 Solutions Due Thursday 12/3 1. Observability. 0640 Consider the system x ( t + 1) = 1 0 . 5 0 0 . 25 0 0 . 5 0 . 25 0 . 125 0 . 125 x ( t ) + 0 1 1 u ( t ) y ( t ) = bracketleftbig 1 0 . 5 0 bracketrightbig x ( t ) (a) Suppose we observe the outputs y (0) = 1 y (1) = 2 y (2) = 0 The control signals at time t = 0 , 1 were u (0) = 2 u (1) = 1 What was the initial state x (0)? (b) Suppose the input u ( t ) = 0 for all t . Find the unit magnitude initial state x (0) which maximizes the energy in the output summationdisplay t =0 bardbl y ( t ) bardbl 2 (c) Again with input u ( t ) = 0 for all t , is there any non-zero initial state which will give an output y with zero energy? Solution. (a) We have y (0) y (1) y (2) = P u (0) u (1) u (2) + Jx 0 where P = 0 0 0 CB 0 0 CAB CB 0 J = C CA CA 2 The matrix J is full rank, since the system is observable, so we find the unique x 0 satisfying this equation is x 0 = 188 374 94 (b) Let V be the solution to the Lyapunov equation V A T V A = C T C Then the energy in the output is summationdisplay t =0 bardbl y ( t ) bardbl 2 2 = x T 0 V x 0 1

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EE263 S. Lall 2009.11.19.01 edited The unit magnitude initial state that maximizes x T 0 V x 0 is the eigenvector of V with the largest eigenvalue, which is x 0 0 . 85 0 . 48 0 . 20 (c) No, since the system is observable. One way to see this is because all of the eigenvalues of V are strictly positive. 2. Output feedback for maximum damping. 53020 Consider the discrete-time linear dynamical system x ( t + 1) = Ax ( t ) + Bu ( t ) , y ( t ) = Cx ( t ) , with A R n × n , B R n × m , C R p × n . In output feedback control we use an input which is a linear function of the output, that is, u ( t ) = Ky ( t ) , where K R m × p is the feedback gain matrix . The resulting state trajectory is identical to that of an autonomous system, x ( t + 1) = ¯ Ax ( t ) . (a) Write ¯ A explicitly in terms of A , B , C , and K . (b) Consider the single-input, single-output system with A = 0 . 5 1 . 0 0 . 1 0 . 1 0 . 5 0 . 1 0 . 2 0 . 0 0 . 9 , B = 1 0 0 , C = bracketleftbig 0 1 0 bracketrightbig . In this case, the feedback gain matrix K is a scalar (which we call simply the feedback gain .) The question is: find the feedback gain K opt such that the feedback system is maximally damped. By maximally damped, we mean that the state goes to zero with the fastest asymptotic decay rate (measured for an initial state x (0) with non-zero coefficient in the slowest mode.) Hint: You are only required to give your answer K opt up to a precision of ± 0 . 01, and you can assume that K opt [ 2 , 2]. Solution. From the equations given in the problem, x ( t + 1) = Ax ( t ) + Bu ( t ) = Ax ( t ) + BKy ( t ) = Ax ( t ) + BKCx ( t ) = ( A + BKC ) x ( t ) Therefore, ˜ A = A + BKC . Since x ( t ) = ( ˜ A ) t x (0), this is a discrete-time autonomous system. Thus, the system is stable iff all the eigenvalues of ˜ A are less than 1 in magnitude, that is, vextendsingle vextendsingle vextendsingle λ parenleftBig ˜ A parenrightBigvextendsingle vextendsingle vextendsingle < 1. Since the largest eigenvalue in magnitude corresponds to the slowest mode, it determines asymptotic decay rate. Therefore, this feedback system is maximally
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