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Unformatted text preview: EE363 Prof. S. Boyd EE363 homework 1 1. LQR for a triple accumulator. We consider the system x t +1 = Ax t + Bu t , y t = Cx t , with A = 1 0 0 1 1 0 0 1 1 , B = 1 , C = bracketleftBig 0 0 1 bracketrightBig . This system has transfer function H ( z ) = ( z- 1)- 3 , and is called a triple accumulator, since it consists of a cascade of three accumulators. (An accumulator is the discrete- time analog of an integrator: its output is the running sum of its input.) We’ll use the LQR cost function J = N- 1 summationdisplay t =0 u 2 t + N summationdisplay t =0 y 2 t , with N = 50. (a) Find P t (numerically), and verify that the Riccati recursion converges to a steady- state value in fewer than about 10 steps. Find the optimal time-varying state feedback gain K t , and plot its components ( K t ) 11 , ( K t ) 12 , and ( K t ) 13 , versus t . (b) Find the initial condition x , with norm not exceeding one, that maximizes the optimal value of J . Plot the optimal....
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- Winter '09
- Determinant, Characteristic polynomial, Triangular matrix, semidefinite quadratic form