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hw8_2011_05_26_01_solutions

# hw8_2011_05_26_01_solutions - EE263 S Lall 2011.05.26.01...

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EE263 S. Lall 2011.05.26.01 Homework 8 Solutions Due Thursday 6/2 at 5 PM. No extensions permitted. 1. Scalar time-varying linear dynamical system. 50110 Show that the solution of ˙ x ( t ) = a ( t ) x ( t ), where x ( t ) R , is given by x ( t ) = exp parenleftbiggintegraldisplay t 0 a ( τ ) parenrightbigg x (0) . (You can just differentiate this expression, and show that it satisfies ˙ x ( t ) = a ( t ) x ( t ).) Find a specific example showing that the analogous formula does not hold when x ( t ) R n , with n > 1. Solution. Differentiating the given expression, we obtain ˙ x ( t ) = parenleftbigg d dt integraldisplay t 0 a ( τ ) parenrightbigg exp parenleftbiggintegraldisplay t 0 a ( τ ) parenrightbigg x (0) = a ( t ) exp parenleftbiggintegraldisplay t 0 a ( τ ) parenrightbigg x (0) = a ( t ) x ( t ) . For the second part, we look for a counterexample with x ( t ) R 2 . We let A ( t ) = braceleftbigg A 1 0 t < 1 A 2 t 1 , where A 1 = bracketleftbigg 0 1 0 0 bracketrightbigg , A 2 = bracketleftbigg 0 0 1 0 bracketrightbigg . Then we have x (2) = (exp A 2 )(exp A 1 ) x (0) = bracketleftbigg 1 1 0 1 bracketrightbigg bracketleftbigg 1 0 1 1 bracketrightbigg x (0) = bracketleftbigg 1 1 1 2 bracketrightbigg x (0) . The formula above gives x (2) = exp( A 1 + A 2 ) x (0) = bracketleftbigg 0 1 1 0 bracketrightbigg x (0) = bracketleftbigg 1 . 5431 1 . 1752 1 . 1752 1 . 5431 bracketrightbigg x (0) . Choosing almost any x (0) ( e.g. , x (0) = e 1 ) will give us a contradiction. 2. Toeplitz Matrices 0710 (a) Write a routine sys_toeplitz.m which computes the Toeplitz matrix corresponding to a discrete-time linear dynamical system. In particular, recall that, if x ( t + 1) = Ax ( t ) + Bu ( t ) y ( t ) = Cx ( t ) + Du ( t ) 1

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EE263 S. Lall 2011.05.26.01 and the initial condition x (0) = 0, then y (0) y (1) y (2) . . . y ( N ) = T u (0) u (1) u (2) . . . u ( N ) where T is the Toeplitz matrix T = D CB D CAB CB D . . . . . . CA N - 1 B CA N - 2 B . . . CB D We will use this matrix to find minimum-norm controllers for linear systems. Since we often are only concerned about the trajectory at particular times, it is useful to construct only those rows and columns of T necessary. The function should take arguments of the form T=sys_toeplitz(A,B,C,D,out_times,in_times); where out_times and in_times are row vectors. For example, if out_times=[1,2,4]; in_times=[0:10] then sys_toeplitz should return just the 3 × 11 block matrix T so that y (1) y (2) y (4) = T u (0) u (1) . . . u (10) (b) Suppose we have a hovercraft moving in two-dimensions, with unit mass, acted upon by input forces u 1 ( t ) and u 2 ( t ) through the center of mass in the x and y directions. Let u ( t ) = bracketleftbigg u 1 ( t ) u 2 ( t ) bracketrightbigg and let the position of the vehicle at time t be q ( t ) R 2 . Construct a linear dynamical system model in continuous time, and its discretization. Use sampling period h = 1.
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hw8_2011_05_26_01_solutions - EE263 S Lall 2011.05.26.01...

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