controllability_2009_11_03_01

# controllability_2009_11_03_01 - 13 1 Controllability S Lall...

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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 Lall...

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