Lösung des linear-quadratischen Optimierungs- problems für lineare zeitinvariante Deskriptorsysteme. mittels Riccati-Matrizengleichungen

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1 Lösung des linear-quadratischen Optimierungs- problems für lineare zeitinvariante Deskriptorsysteme mittels Riccati-Matrizengleichungen Peter C. Müller Sicherheitstechnische Regelungs- und Messtechnik Bergische Universität Wuppertal D-4097 Wuppertal

2 INRODUCION Linear-Quadratic Optimal Control Problem: State Space Model State space system: x ( t= ) Ax( t+ ) Bu( t) Performance criterion: J= x ( ) Zx( ) x Q S x + dt minimum o u S R u - Z = Z O, Q =Q O, R = R >O,Q SR S O Hamilton s equations: x A - BR S BR B x = λ Q - SR S - ( A-BR S ) λ λ =-Zx Riccati matrix equation: λ ( t) =-P( t) x ( t) ( - ) ( - - ) ( - ) -P = A-BR S P+P A-BR S -PBR B P+ Q-SR S

3 - Optimal control: P( ) = Z u t =-R B P t +S x t Properties: ( A, B) stabilizable Q-SR S, A-BR S hen: - - detectable (i) here is one and only one solution ( ) (ii) his solution is stabilizing: ( t) ( t) ( t) x =Ax + Bu () t () t - - = A-BR S -BR B P x is asymptotically stable P t 0

4 Linear-Quadratic Optimal Control Problem: Descriptor System Descriptor system: Ex ( t) =Ax ( t) +Bu( t) Performance criterion: J= x ( ) Zx( ) x Q S x + dt minimum o u S R u - Z = Z O, Q = Q O, R = R > O,Q SR S O Hamilton s function: H=λ ( Ax+Bu) - Control: ( u=r B λ Sx ) Hamilton s equations: x Q S x u S R u E x A - BR S BR B x = E λ Q - SR S - ( A-BR S ) λ E λ = Zx

5 Example of Mechanical Descriptor System Dynamics: Mq ( t) +Dq ( t) +K q ( t) =S u ( t ) +F λ ( t ) Holonomic constraints: = t + Defining O Fq() S u () t q O I O O O I O =, =, =, = x q B S E O M O A -K -D F λ S O O O F O O the standard descriptor form is obtained: Ex ( t) =Ax ( t) +Bu( t)

6 Weierstrass-Kronecker representation of a descriptor system Assumption: ( λe-a ) is a regular matrix pencil here are regular matrices: i.e. det ( λe-a ) o I V,W: O VEW =, VAW =, VB = O N O I B x x=w x Slow subsystem: x ( ) =Ax ( ) +Bu( ) A O B t t t, Fast subsystem: Nx ( ) = x ( ) +B u ( ) k- k t t t, Index k: Solution: x () k- i i i=o () () t= NB u t N O, N = O

7 Improper solution: NB O Proper solution: NB =O : x ( t) = -B u ( t) Strictly proper solution: ( t ) B = O: x = O Example: Mechanical descriptor system Schüpphaus (995): V, W A, B - F M S O I O N= O O I, B = O O O O S NB O S., = O Improper: S O ; Proper: S = O

8 RICCAI MARIX EQUAIONS Standard Approach ( ) λ t = P t Ex t : ( - ) ( - ) ( - ) -E PE= A-BR S PE+E P A-BR S -E PBR B PE+ Q-SR S - ( ), ( t) ( t) EP E=Z P =P - Optimal control: ( ) () u t =-R B P t E+S x t

9 Kurina-März Approach (005) ( ) λ t =-P t x t : ( t) ( t ) EP =P E - ( ) - - = + - -EP A-BRS P+P A-BRS -PBRBP Q-SRS Optimal control: ( ) EP =Z - u t =-R B P t +S x t

10 SOLUIONS OF HE RICCAI EQUAIONS General Properties (i) If ( ) P is a solution of the standard approach then t ( t) ( t) P =P E is a solution of the Kurina-März approach (ii) If X is a basis for the null space of E, EX =O, then there are necessary conditions for the existence of solutions P, P : P : ( - ), ( - ) Q-SR S X=O ZX=O, E P A-BR S X=O P : EPX=O, ZX=O

11 Standard Riccati Equation for Proper Systems Proper system: NB =O Representation of standard Riccati equation for LI descriptor systems in Weierstrass-Kronecker form: P - = P = V PV P P P -P = A P +P A P B +P B R B P +B P +Q -, - - -P N=A P N+P - P B +P B R B P +B P N+Q, -N P N = P N+N P -N P B +P B R B P +B P N+Q. - Requirements: Solution: Q B =O, Q B =O. NPB=O, PB=O. - - u ( t) =-R - B P ( t) x ( t) -P =A P +P A P BR B P +Q,

12 KM-Riccati Equation for Proper Systems Proper system: NB =O Representation of KM-Riccati equation for LI descriptor systems in Weierstrass-Kronecker form: - V PW P P P = = P P Symmetry conditions:, = P=P P PN, NP=PN. Riccati equations: - -P =A P +P A P B + P B R B P + B P + Q -, -P N=P +A P N- P B P B R B P N B P Q , -P N=P +P - N P B P B R B P N B P Q

13 Requirements: PB=O NPB O., = - Auxiliary matrix: P = B P B : P +P -P R P +B Q B = O P =R-R R+B Q B, orthogonal matrix / ( ) / Solution: + ( ) + + ( ) - - -P= A B R+BQ B BQ P P A B R+BQ B BQ - - -PB R+BQ B BP+ Q - Q B R+BQ B BQ. () ( ) () - ( ) () u =- R+B Q B B P B Q x t t - t. Résumé: he Kurina-März approach is more general than the standard approach because the requirements QB=O, QB = O are not needed. If these requirements hold then both approaches agree.

14 Improper LI descriptor systems Counter example: Inserting solution o NB O x =x +b u, o=x +b u, b ( ) o J= xqx+ ru dt minimum. x= -bu- bu, x= -bu into the performance criterion J= ( r+ bqb) u + ( q b b +q b ) uu+q b u dt minimum o the solution is obtained by classical calculus of variations: ü - u=o, = ω ω r+ q b bqb u o ωt u () ( + o o ω ) ( o ω ) u t = u e + u e u =- o b x, u =- 0 o b b x b x ωt

15 Closed-loop control system: ( b ω ) b b b b + b x= x - x, x= x- x. Eigenvalues: λ = + ω (unstable), In contrast, Kurina-März approach leads to following results: Symmetry condition: Riccati equations: P o p = p b r p. O=- p +q, O= p + p - r b p bp b p + +q, p = p - r b p b p + +q.

16 Optimal control =- r + ( + ) Closed-loop control system implies u b p x bp b p x q x = - ( p +q ) x. CONRADICION he direct solution implies independent x ( t ), x( ) requires interdependent states.. t while the KM-Riccati approach Résumé: he Riccati approaches fail for improper LI descriptor systems.

17 CONCLUSION Proper LI descriptor systems ( NB = O ) : Kurina-März approach can be successfully applied to solve the linearquadratic optimal control problem Standard approach can be applied only for QB= O, QB= O. hese requirements have not to be assumed for the Kurina-März approach. herefore, the Kurina-März approach is superior to the standard approach. Improper LI descriptor systems ( ) NB O : Both Riccati approaches fail.

18 For improper systems a completely different approach has to be applied: Fast subsystem: x k- i () i () t = NB u () t i=o k- i k = NBζ i+() t N B v () t Extension: i=o ( ζ ( t) = u ( t),ζ ( t) = u ( t),...,ζ ( t) = u k- ) k- ( t) ( k- ( t) = ) ( t) v u Integrator chain: ( t) = ( t ), ( t) = ( t) ( t) = ( t ), ( t) = ( t) ζ ζ ζ ζ,..,ζ ζ ζ v 3 k- k- k- Optimization of the extended slow subsystem with respect to leads to a dynamic feedback (instead of a static feedback) control. x, ζ,..,ζ,v k-