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Unformatted text preview: EE263 S. Lall 2009.11.19.01 Homework 7 Due Thursday 12/3 1. Observability. 0640 Consider the system x ( t + 1) = 1 . 5 − . 25 . 5 . 25 . 125 − . 125 x ( t ) + 1 1 u ( t ) y ( t ) = bracketleftbig 1 . 5 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? 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 = . 5 1 . . 1 − . 1 . 5 − . 1 . 2 . . 9 , B = 1 , C = bracketleftbig 1 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 ± . 01, and you can assume that K opt ∈ [ − 2 , 2]. 1 EE263 S. Lall 2009.11.19.01 3. A method for rapidly driving the state to zero. 53055 We consider the discrete-time linear dynamical system x ( t + 1) = Ax ( t ) + Bu ( t ) , where A ∈ R n × n and B ∈ R n × k , k < n , is full rank. The goal is to choose an input u that causes x ( t ) to converge to zero as t → ∞ . An engineer proposes the following simple method: at time t , choose u ( t ) that minimizes bardbl x ( t + 1) bardbl . The engineer argues that this scheme will work well, since the norm of the state is made as small as possible at every step. In this problem you will analyze this scheme. (a) Find an explicit expression for the proposed input u ( t ) in terms of x ( t ), A , and B . (b) Now consider the linear dynamical system x ( t + 1) = Ax ( t ) + Bu ( t ) with u ( t ) given by the proposed scheme ( i.e. , as found in (3a)). Show that x satisfies an autonomous linear dynamical system equation x ( t + 1) = Fx ( t ). Express the matrix F explicitly in terms of A and B ....
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This note was uploaded on 08/23/2010 for the course EE 263 taught by Professor Boyd,s during the Fall '08 term at Stanford.
- Fall '08