controllability_2009_11_03_01_2up

controllability_2009_11_03_01_2up - 13 - 1 Controllability...

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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 Infnite 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 ± A T 1 B A T 2 B . . . AB B ² 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 infnite time)
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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 ) = ± A T 1 B A T 2 B . . . AB B ² u (0) u (1) . . . u ( T 1) ask control questions: fnd input sequence u (0) , . . . , u ( T 1) so that x ( T ) = x des fnd all input sequences that result in x ( T ) = x des among all those, fnd the smallest, most eFcient one 13 - 4 Controllability S. Lall, Stanford 2009.11.03.01 how to control the state x ( T ) = ± A T 1 B A T 2 B . . . AB B ² 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 ±or which x ( T ) = x des , this one has the smallest norm; i.e, it minimizes T 1 X t =0 k u ( t ) k 2 = k u (0) k 2 + k u (1) k 2 + · · · + k u ( T 1) k 2 called the input energy
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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 ) = 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 2 1 0 1 . 6 0 . 8 0 1 2 1 0 . 8 1 . 6 0 . 8 0 1 1 0 0 . 8 0 . 8 x ( t ) + 0 0 0 1 0 0 u ( t ) u ( t )
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controllability_2009_11_03_01_2up - 13 - 1 Controllability...

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