controllability_2009_11_03_01_2up

# controllability_2009_11_03_01_2up - 13 1 Controllability S...

This preview shows pages 1–4. Sign up to view the full content.

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)

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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 )

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 08/23/2010 for the course EE 263 taught by Professor Boyd,s during the Fall '08 term at Stanford.

### Page1 / 12

controllability_2009_11_03_01_2up - 13 1 Controllability S...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online