controllability_2009_11_03_01

controllability_2009_11_03_01 - 13 - 1 Controllability S....

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Unformatted text preview: 13 - 1 Controllability S. Lall, Stanford 2009.11.03.01 13. Controllability Controlling the state Example: mass-spring system The reachable set Cayley-Hamilton Controllability Example: controlling identical masses Energy required to reach desired state Controllability ellipsoid Example: energy used against time Infinite time problems Lyapunov equations Example: solving Lyapunov equations Solutions of Lyapunov equations 13 - 2 Controllability S. Lall, Stanford 2009.11.03.01 The Key Points of This Section we can compute minimum energy inputs so that x ( T ) = x des we can measure controllability by looking at the SVD of the matrix bracketleftbig A T 1 B A T 2 B ... AB B bracketrightbig for large T the singular values and left singular vectors give us the controllability ellipsoid , which tell us strong and weak directions in the state-space to compute this as T , we solve the Lyapunov equation W AWA T = BB T the eigenvectors of W are the axis directions and lengths of the controllability ellipsoid (over infinite time) 13 - 3 Controllability S. Lall, Stanford 2009.11.03.01 Controlling the State discrete-time LDS, x ( t ) R n and u ( t ) R m x ( t + 1) = Ax ( t ) + Bu ( t ) x (0) = 0 look at state at time T x ( T ) = bracketleftbig A T 1 B A T 2 B ... AB B bracketrightbig u (0) u (1) . . . u ( T 1) ask control questions: find input sequence u (0) ,...,u ( T 1) so that x ( T ) = x des find all input sequences that result in x ( T ) = x des among all those, find the smallest, most efficient one 13 - 4 Controllability S. Lall, Stanford 2009.11.03.01 how to control the state x ( T ) = bracketleftbig A T 1 B A T 2 B ... AB B bracketrightbig u (0) . . . u ( T 1) = H T u (0) . . . u ( T 1) minimum norm solution is u (0) . . . u ( T 1) = H T x des among all input sequences for which x ( T ) = x des , this one has the smallest norm; i.e, it minimizes T 1 summationdisplay t =0 bardbl u ( t ) bardbl 2 = bardbl u (0) bardbl 2 + bardbl u (1) bardbl 2 + + bardbl u ( T 1) bardbl 2 called the input energy 13 - 5 Controllability S. Lall, Stanford 2009.11.03.01 example: mass-spring system m 1 m 2 m 3 k 1 k 2 k 3 b 1 b 2 b 3 masses m i = 1 , spring constants k = 1 , damping constants b = 0 . 8 x ( t ) = 1 1 1 2 1 1 . 6 . 8 1 2 1 . 8 1 . 6 . 8 1...
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controllability_2009_11_03_01 - 13 - 1 Controllability S....

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