Problem 3 : Controllability Gramian (a) Consider the following state equation: -0.5 0 x = -...
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Problem 3 : Controllability Gramian (a) Consider the following state equation: -0.5 0 x = - [ 85 9] = + [8]. x U 0 x(0) = [0:1] (i) (2 points) Show that there exists an input u(t) that drives x(0) to 0 in finite time. (ii) (3 points) Compute (by hand) the particular input u(t) that achieves this in 3 seconds. (iii) (3 points) Verify that the input achieves this by plotting y [1_0] x= x(t) and y = [0_1] x = x (t) versus t using the MATLAB command 1sim (sys,u,t, x0), where sys = ss (A,B,C,D) is an object representing the state-space model. (iv) (2 points) If possible, compute the controller canonical form (CCF) of the above state equation. If you cannot, why not? (b) (5 points) Given a controllable state equation, prove that the following input steers the state from x(to) = xo to x(t) = x for arbitrary x and x: u(t) = BT eAT (to-t) -t)W(to, t)(e(to-t) x xo), where W(to, tf) is the controllability gramian. (Hint: See proof of Theorem 3.2.) Problem 3 : Controllability Gramian (a) Consider the following state equation: -0.5 0 x = - [ 85 9] = + [8]. x U 0 x(0) = [0:1] (i) (2 points) Show that there exists an input u(t) that drives x(0) to 0 in finite time. (ii) (3 points) Compute (by hand) the particular input u(t) that achieves this in 3 seconds. (iii) (3 points) Verify that the input achieves this by plotting y [1_0] x= x(t) and y = [0_1] x = x (t) versus t using the MATLAB command 1sim (sys,u,t, x0), where sys = ss (A,B,C,D) is an object representing the state-space model. (iv) (2 points) If possible, compute the controller canonical form (CCF) of the above state equation. If you cannot, why not? (b) (5 points) Given a controllable state equation, prove that the following input steers the state from x(to) = xo to x(t) = x for arbitrary x and x: u(t) = BT eAT (to-t) -t)W(to, t)(e(to-t) x xo), where W(to, tf) is the controllability gramian. (Hint: See proof of Theorem 3.2.)
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Solutions Step 1 The system is controllable if for any initial state xo and any final state xf there exists an input ut that will drive the system fro... View the full answer
Related Book For
Income Tax Fundamentals 2013
ISBN: 9781285586618
31st Edition
Authors: Gerald E. Whittenburg, Martha Altus Buller, Steven L Gill
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