Closed-Loop Impulse Control of Oscillating Systems

Transcription

1 Closed-Loop Impulse Control of Oscillating Systems A. N. Daryin and A. B. Kurzhanski Moscow State (Lomonosov) University Faculty of Computational Mathematics and Cybernetics Periodic Control Systems, 2007

2 Outline 1 Problem 2 Dynamic Programming Approach 3 Numerical Algorithm 4 Ellipsoidal Approximation 5 Asymptotic Solution ( t ) 6 Unilateral Impulses 7 Double Constraint Approach 8 Generalized Impulse Control Problem

3 Oscillating System C 1 L 1 k 1 m 1 k 2 w 1 C 2 L 2 m 2 w 2 m N 1 w N 1 k N m N w N V CN L N F

4 Oscillating System m 1 ẅ 1 = k 2 (w 2 w 1 ) k 1 w 1 m i ẅ i = k i+1 (w i+1 w i ) k i (w i w i 1 ) m ν ẅ ν = k ν+1 (w ν+1 w ν ) k ν (w ν w ν 1 ) + u(t) m N ẅ N = k N (w N w N 1 ) w i = w i (t) displacements from the equilibrium m i masses of the loads k i stiffness coefficients u(t) = du dt impulse control (U BV )

5 N ρ(ξ)w tt (t,ξ) = [Y (ξ)w ξ (t,ξ)] ξ, t > t 0, 0 < ξ < L w(t,0) = 0, w ξ (t,l) = u(t)/y (L), t t 0 w(t 0,ξ) = w 0 (ξ), w t (t 0,ξ) = ẇ 0 (ξ), 0 ξ L w(t,ξ) displacement from the equilibrium u(t) = du dt impulse control ρ(ξ) mass density Y (ξ) Young modulus

6 Oscillating System Normalized matrix form: dx(t) = Ax(t)dt + BdU(t) ( ) w 1 (t) w(t) x(t) = w(t) = ẇ(t). w N (t) This system is completely controllable.

7 Impulse Control Problem Problem (1) Minimize J(U( )) = Var [t 0,t 1 ] U( ) + ϕ(x(t 1 + 0)) over U( ) BV [t 0,t 1 ] where x(t) is the trajectory generated by control input u(t) = du dt starting from x(t 0 0) = x 0. u(t) = 2N i=1 h i δ(t τ i ) Important particular case: ϕ(x) = I ( x {0}) completely stop oscillations on fixed time interval [t 0,t 1 ].

8 The Value Function Definition The minimum of J(U( )) with fixed initial position x(t 0 0) = x 0 is called the value function: V (t 0,x 0 ) = V (t 0,x 0 ;t 1,ϕ( )). V (t 0,x 0 ) = inf x 1 R n { p,x1 e (t 1 t 0 )A x 0 ϕ(x 1 ) + sup p R B n T e (t 1 )A T p C[t0,t 1 ] The value function is convex and its conjugate equals ( ) V (t 0,p) = ϕ (e (t 0 t 1 )A T p) + I e (t 0 t 1 )A T p B [t0,t 1 ] B where p [t0,t 1 ] = T e (t 1 )A T p. C[t0,t 1 ] }.

9 Dynamic Programming Equation The value function V (t,x;t 1,ϕ( )) satisfies the Principle of Optimality V (t 0,x 0 ;t 1,ϕ( )) = V (t 0,x 0 ;τ,v(τ, ;t 1,ϕ( ))), τ [t 0,t 1 ] The value function it is the solution to the Hamilton Jacobi Bellman quasi-variational inequality: min {H 1 (t,x,v t,v x ), H 2 (t,x,v t,v x )} = 0, V (t 1,x) = V (t 1,x;t 1,ϕ( )). H 1 = V t + V x,ax, H 2 = min V x,bu + 1 = B T V x + 1. u S 1

10 The Control Structure H 1 (t,x) = 0 (t,x) H 2 (t,x) = 0 wait du(t) = 0 choose jump direction d = B T V x choose jump amplitude minα 0 : H 1 (t,x + αd) = 0 jump U(τ) = α d χ(τ t)

11 Numerical Algorithm The value function is V (t 0,x 0 ) = max p R n B T e (t 1 t)at p 1 t [t 0,t 1 ] p,x 0. Replace B T e (t 1 t)a T p 1 by a finite number of linear inequalities, and [t 0,t 1 ] with a finite number of time instants: ˆV (t 0,x 0 ) = which is a LP problem. max D p R n E q i,b T e (t 1 t)at p 1,i=1,M t=θ 1,θ 2,...,θ K p,x 0

12 Numerical Algorithm Finding control for given (t,x) is a LP ranging problem. The error estimate is V (t,x) ˆV (t,x) V (t,x) [ 1 + O ( K 1)],

13 Ellipsoidal Approximation X ν [t] backward reach set under condition Var U ν V (t,x) = min {ν x X ν [t]} We look for an approximation of X ν [t]. Ellipsoids: E (q,q) = { x x q,q 1 (x q) 1 } ρ(l E (q,q)) = l,q + l,ql 1 2 (see Kurzhanski and Vályi, 1997)

14 Ellipsoidal Approximation Ellipsoidal approximation is derived through comparison principle for Hamilton Jacobi equations (Kurzhanski, 2006): X ν [t] = E (0,(ν k(t))z(t)) {Ż = AZ + ZA T η(t)bb T k = 1 4 η(t) { Z(t 1 ) = 0 k(t 1 ) = 0 Here η(t) 0 is a parameter function X ν [t] = cl ν( ) X ν [t]

15 Ellipsoidal Approximation x x 9

16 Asymptotic Solution ( t ) ḧ i = ω 2 i h i + b i u, i = 1,N. h 1, dh 1 /dt Time t h 2, dh 2 /dt Time t h 3, dh 3 /dt Time t

17 Asymptotic Solution ( t ) cl t t 1 X 1 [t] = C = N j=1 X 1 [t] { backward reach set under } condition Var U 1 C j = (h,ḣ) ω 2 j hj 2 + ḣ2 j bj 2 ellipsoids V C j V = max ω2 j h2 j + ḣ2 j t j=1,n bj 2 Control strategy: Optimal : jump if B T V x = 1 h j = 0 for all maximizers j in (*). Useless after first jump. ε-optimal: jump if B T V x 1 ε: Var U( ) V 1 ε (*)

18 Unilateral Impulses Additional constraint: du 0 (du 0). General case: KdU 0 (K matrix) or du K (K cone). The minimum number of impulses is the same 2N. Numerical procedures apply with minor modifications. Asymptotic solution does not change. The problem may be not solvable on small time intervals.

19 Unilateral Impulses Minimal Control Norm Unilateral Impulses Bilateral Impulses t

20 Impulse vs Bang-Bang Controls 25 Bang Bang Control 20 Minimal Control Norm Impulse Control t

21 Double Constraint Approach Problem (2) Minimize J(u) = t1 t 0 u(t) dt + ϕ(x(t 1 )) over controls u(t) satisfying u(t) µ, where x(t) is the trajectory generated by control u starting from x(t 0 ) = x 0. Here controls are bounded functions. Optimal controls only take values µ, 0, µ. V µ (t,x) is the value function for Problem 2. 0 V µ (t,x) V (t,x) = O(µ 1 ) for each (t,x)

22 Double Constraint Approach 5 mu = u = 0 Not Solvable 1 u = µ x u = µ Not Solvable u = x 1

23 Double Constraint Approach u = 0 u = µ 10 5 x u = µ u = x 1

24 Generalized Impulse Control Problem Problem (3) Minimize J(u) = ρ [u] + ϕ(x(t 1 + 0)) over distributions u D 1 [α,β], (α,β) [t 0,t 1 ] where x(t) is the trajectory generated by control u starting from x(t 0 0) = x 0. Here ρ [u] is the conjugate norm to the norm ρ on C 1 [α,β]: ρ[ψ] = max ψ(t) 2 + ψ (t) 2. t [α,β] u(t) = 2N i=1 h (0) i δ(t τ i ) + h (1) i δ (t τ i ).

25 Reduction to Impulse Control Problem u D 1 : u = du 0 dt + d2 U 1 dt 2 U 0,U 1 BV Problem 3 reduces to a particular case of Problem 1 for the system ẋ = Ax + Bu, B = ( B AB ) and the control u = du dt, U(t) = ( ) U0 (t). U 1 (t) Error bound for numerical algorithm: V (t,x) ˆV(t,x) V (t,x) [ 1 + O ( K 1 + M 2)],

26 Examples Chain of 5 springs String (10 elements)

27 References Bellman, R. (1957). Dynamic Programming. Princeton Univ. Press. Bensoussan, A. and J.-L. Lions (1982). Contrôle impulsionnel et inéquations quasi variationnelles. Paris. Crandall, M. G. and P.-L. Lions (1983). Viscosity solutions of Hamilton Jacobi equations. Trans. Amer. Math. Soc. 277, Daryin, A. N., A. B. Kurzhanski and A. V. Seleznev (2005). A dynamic programming approach to the impulse control synthesis problem. In: Proc. Joint 44th IEEE CDC-ECC IEEE. Seville. Demyanov, V. F. (1974). Minimax: Directional Derivates. Nauka. Moscow. Dykhta, V. A. and O. N. Sumsonuk (2003). Optimal impulsive control with applications. Fizmatlit. Moscow. Gusev, M. I. (1975). On optimal control of generalized processes under non-convex state constraints. In: Differential Games and Control Problems. UNC AN SSSR. Sverdlovsk. Kalman, R. E. (1960). On the general theory of control systems. In: Proc. 1st IFAC Congress. Vol. 1. IFAC. Butterworths. London. Krasovski, N. N. (1957). On a problem of optimal regulation. Prikl. Math. & Mech. 21(5),

28 References Krasovski, N. N. (1968). The Theory of Motion Control. Nauka. Moscow. Kurzhanski, A. B. (1975). Optimal systems with impulse controls. In: Differential Games and Control Problems. UNC AN SSSR. Sverdlovsk. Kurzhanski, A. B. and I. Vályi (1997). Ellipsoidal Calculus for Estimation and Control. SCFA. Birkhäuser. Boston. Kurzhanski, A. B. and Yu. S. Osipov (1969). On controlling linear systems through generalized controls. Differenc. Uravn. 5(8), Miller, B. M. and E. Ya. Rubinovich (2003). Impulsive Control in Continuous and Discrete-Continuous Systems. Kluwer. N.Y. Motta, M. and F. Rampazzo (1995). Space-time trajectories of nonlinear systems driven by ordinary and impulsive controls. Differential and Integral Equations 8, Neustadt, L. W. (1964). Optimization, a moment problem and nonlinear programming. SIAM J. Control 2(1), Rockafellar, R. T. (1970). Convex Analysis. Vol. 28 of Princeton Mathematics Series. Princeton University Press.

29 Continuous and Smooth Controls It is required to use continuous or smooth controls. Control force is produced by an integrator t τν τ2 F(t) = u(τ 1 )dτ 1 dτ ν t } 0 t 0 {{ t 0 } ν times u(t) is the new control variable. Hard bound (geometrical constraint) on control: u(t) P = [ µ,µ] Examples: Continuous Control Smooth Control