Lecture 2: Discrete-time Linear Quadratic Optimal Control
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1 ME 33, U Berkeley, Spring 04 Xu hen Lecture : Discrete-time Linear Quadratic Optimal ontrol Big picture Example onvergence of finite-time LQ solutions Big picture previously: dynamic programming and finite-horizon discrete-time LQ this lecture: infinite-horizon discrete-time LQ and its properties Lecture : Discrete-time Linear Quadratic Optimal ontrol ME33 -
2 Review: solution of the general discrete-time LQ problem system dynamics: performance index: x (k + ) = A(k)x (k) + B (k)u (k), x (k 0 ) = x o () J = x T (N)Sx (N) + N { x T (k)q (k)x (k) + u T (k)r (k)u (k) k=k 0 Q (k) = T (k) (k) 0, S = S T 0, R (k) = R T (k) 0 optimal J o = x T o P (0)x o achieved by the state-feedback control law: u o (k) = R (k) + B T (k)p (k + )B (k) B T (k)p (k + )A(k)x (k) Riccati equation: P (k) = A T (k)p (k + )A(k) + Q (k) A T (k)p (k + )B (k) R (k) + B T (k)p (k + )B (k) B T (k)p (k + )A(k) with the boundary condition P (N) = S. Lecture : Discrete-time Linear Quadratic Optimal ontrol ME33 - Example: double integrator plant dynamics: x (k + ) x (k + ) performance index: where = y (k) = T 0 x (k) x (k) 0 x (k) x (k) J N (x 0 ) = x T (N)Sx (N) + Q = T + T u (k), T = N{ x T (k)qx (k) + Ru (k), R > 0 next: examine the convergence of P (k) with different P (N) = S ( 0) P (k) = A T P (k + )A+Q A T P (k + )B R + B T P (k + )B B T P (k + )A Lecture : Discrete-time Linear Quadratic Optimal ontrol ME33-3
3 Example: double integrator N = 0, P (N) = P P P P P time index P (k) = A T P (k + )A + Q A T P (k + )B R + B T P (k + )B B T P (k + )A Lecture : Discrete-time Linear Quadratic Optimal ontrol ME33-4 Example: double integrator N = 30, P (N) = P P P P P time index P (k) = A T P (k + )A + Q A T P (k + )B R + B T P (k + )B B T P (k + )A Lecture : Discrete-time Linear Quadratic Optimal ontrol ME33-5
4 Example: double integrator N = 30, P (N) = P P P P P time index P (k) = A T P (k + )A + Q A T P (k + )B R + B T P (k + )B B T P (k + )A Lecture : Discrete-time Linear Quadratic Optimal ontrol ME33-6 Example: double integrator observations: P (k) is indeed always symmetric regardless of the boundary condition P (N)( 0), the solution of the Riccati equation converges to the same steady state P s the control law u o (k) = R + B T P (k + )B B T P (k + )Ax (k) thus converges (backwards) to u o (k) = R + B T P s B B T P s Ax (k) {{ Lecture : Discrete-time Linear Quadratic Optimal ontrol ME33-7
5 From finite-horizon to infinite-horizon LQ in the case of N, it turns out that (A, B) is controllable or stabilizable guaranteed convergence of P (k) to a bounded P s intuition: if (A, B) is unstabilizable, then there are unstable uncontrollable modes that may cause lim J N (x 0 ) = lim N N { x T (N)Sx (N) + N{ x T (k)qx (k) + u T (k)r (k)u (k) = yielding J o N (x 0) = x T o P (0)x 0 = lim N P (0) = Lecture : Discrete-time Linear Quadratic Optimal ontrol ME33-8 Infinite-horizon discrete-time LQ for LTI systems system dynamics: x (k + ) = Ax (k) + Bu (k), x (k 0 ) = x o () performance index: J = { x T (k)qx (k) + u T (k)ru (k), Q 0, R 0 k=k 0 optimal state-feedback control law: u o (k) = R + B T P s B B T P s Ax (k) {{ Algebraic Riccati equation: P s = A T P s A + Q A T P s B R + B T P s B B T P s A Lecture : Discrete-time Linear Quadratic Optimal ontrol ME33-9
6 Infinite-horizon discrete-time LQ for LTI systems conditions for a meaningful solution: (A, B) controllable/stabilizable and (A, ) observable/detectable (Q = T ) guaranteed closed-loop asymptotic stability for x (k + ) = (A B )x (k) A cl x (k) (A,B) controllable/stabilizable P s and bounded : already shown observability P s 0 : with u o (k) = x (k) and Q = T x T o P s x o = { x T (k)qx (k) + u T (k)ru (k) = { xo T ( ) T A k cl = R / where W cl is the observability gramian for { x T (k) T R / R / T A k cl x o R / x (k) = x T o W cl x o x (k + ) = (A B )x (k) = A cl x (k) (3) ỹ (k) = R / x (k) Lecture : Discrete-time Linear Quadratic Optimal ontrol ME33-0 Infinite-horizon discrete-time LQ for LTI systems observability P s 0 (continued): observability is invariant under static output feedback control x (k + ) = (A B )x (k) = A cl x (k) ỹ (k) = R / x (k) is observable if the open-loop system x (k + ) = Ax (k) + Bu (k) ỹ (k) = R / x (k) is observable (which holds as (A,) is observable). Hence P s = W cl is positive definite under observability. Analogous analysis can be applied to the detectability case. Lecture : Discrete-time Linear Quadratic Optimal ontrol ME33 -
7 Infinite-horizon discrete-time LQ for LTI systems closed-loop stability of x (k + ) = (A B )x (k) = A cl x (k) ỹ (k) = R / x (k) comes from a transformation from Riccati equation to Lyapunov equation: P s = A T P s A + Q A T P s B R + B T P s B B T P s A = A T P s A + Q A T P s B R + B T P s B {{ K T s R + B T P s B R + B T P s B B T P s A {{ = (A B ) T P s (A B ) + A T P s B K T s B T P s B + T K T s = (A B ) T P s (A B ) + T + K T s R ( ) R + B T P s B (4) (A B ) T P s (A B ) P s = R / T R / 0 (5) observability of (4) plus P s 0 closed-loop stability from (5) Lecture : Discrete-time Linear Quadratic Optimal ontrol ME33 - Remark Theorem (An extension of Lyapunov theory based on observability). if we find from a Lyapunov equation A T PA P = Q where P 0, Q = T 0, and (A,) is observable, then the system x (k + ) = Ax (k) is asymptotically stable. Proof: Since P 0 and Q 0, the system is stable in the sense of Lyapunov. All eigenvalues of A are hence on or inside the unit circle. Pick any eigenvalue-eigenvector pair (λ,v) where λ is on the unit circle. Then v = v Qv = v ( A T PA P ) ( ) v = λ v Pv = 0, which implies v = 0. We thus have Av = λv v = 0 Av = λv = 0. A n v = 0 A. A n v = 0 A. A n has to be full-column rank Lecture : Discrete-time Linear Quadratic Optimal ontrol ME33-3
8 Infinite-horizon discrete-time LQ for LTI systems an example of closed-loop stability requirement: consider x (k + ) = x (k) + u (k), Q = 0, R = the state constraint Q is zero, the optimal control is u o (k) = 0 the closed-loop system is thus unstable on the other hand, (A,) is clearly unobservable the Riccati equation however still converges, as (A, B) is controllable Lecture : Discrete-time Linear Quadratic Optimal ontrol ME33-4 Infinite-horizon discrete-time LQ for LTI systems Excellent closed-loop properties guaranteed closed-loop stability (just shown moments ago) good margins for single-input systems (see ME3 class reader): ( ) Phase margin > sin R R + B T P s B Gain margin > R R+B T P s B guaranteed stability for % loop gain change (see ME3 class reader): 00 + R R+B T P s B < loop gain change < 00 R R+B T P s B Lecture : Discrete-time Linear Quadratic Optimal ontrol ME33-5
9 Summary Review: general disccrete-time LQ problem Example: double integrator 3 onvergence of the Riccati equation solution 4 Infinite-horizon discrete-time LQ and its properties Lecture : Discrete-time Linear Quadratic Optimal ontrol ME33-6
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