The Matrix exponential, Dynamic Systems and Control

Transcription

1 e Matrix exponential, Dynamic Systems Control Niels Kjølsta Poulsen Department of Informatics Matematical Moelling, Builing 321 e ecnical University of Denmark DK-28 Lyngby IMM tecnical report Abstract e matrix exponential can be foun in various connections in analysis control of ynamic systems. In tis sort note we are going to list a few examples. e matrix exponential usably pops up in connection to te sampling process, watever it is in a eterministic or a stocastic setting or it is a tool for etermining a Gramian matrix. is note is intene to be use in connection to te teacing post te course in Stocastic Aaptive Control (2421) given at e Department of Informatics Matematical Moelling (IMM), e ecnical University of Denmark. is work is a result of a stuy of te literature. First version was originally written before 24. Latest revision was one in Introuction One way to give a formal efinition on te matrix exponential is troug its aylor expansion e A I + A A n! An

2 2 1 INRODUCION e numerical evaluation of te matrix exponential can in some situations be one by applying te efinition te aylor expansion. Depening on te properties of A ifferent numerical alternatives migt be use (see e.g. (Moler & Loan 1978)). e following Lemma can be foun in e.g. (Cen & Francis 1995) (page 235). Consier matrices A 11, A 12 A 22 wit aequate imensions. Let ( ) A11 A exp 12 (1) F 22 A 22 ten F 11 e A11 F 22 e A22 Proof: Let F 12 A e A11( s) A 12 e A22s s A11 A 12 A 22 Since te matrix is block upper triangular, we easily get A 2 A11 A 12 A11 A 12 A 22 A 22 A 2 11 A Consequently: A n+1 A n 11 A n 22 A n+1 11 A n+1 22 n A11 A 12 A 22 n+1 F 11 e A11 F 22 e A22 If we ifferentiate (1) we get A11 A 12 t F 22 A 22 F 22 t F 12 A 11 F 12 + A 12 F 22 Using te solution for F 22 te fact tat F 12 () we ave F 12 e A11( s) A 12 e A22s s is lemma as several applications in connection to system teory, as we will illustrate in te next sections.

3 3 2 Sampling of a eterministic system Let a eterministic (LI) system in continuous time be given by te state space escription t x Ax + Bu x() x It is well known tat te solutions to tis escription is given by x(t) e At x + t e A(t s) Bu(s) s (2) u i y i D A Plant A D Figure 1: Sample-ata control system wit te plant embee wit sensors actuators among filters converters. x e A x + x e A x + x e A x + e A( s) Bu(s) s e A( s) B s u e Aτ B τ u From a computer point of view ( using a zero orer ol sample ol network) te system (or te plant) can in iscrete time be escribe by a iscrete time moel x i+1 Φx i + Γu i were Φ e A Γ y i Cx i e A( s) B s e Aτ B τ In te latter substitution we ave use τ s. Notice te control action is assume to be constant between samples (wen we use a zero orer ol network). Also Γ A 1 (Φ I)B (Φ I)A 1 B Let ( A B exp F 22 )

4 4 2 SAMPLING OF A DEERMINISIC SYSEM k 1 k k+1 Figure 2: ZOH using te Lemma we ave F 11 e A F 12 e A( s) BIs Φ F 11 Γ F 12 e Matlab implementation of tis algoritm is liste in Appenix 9 as c2.m. 2.1 Samping of a system wit elay Consier a system wit a time elay ( < ) t x t Ax t + Bu t (k 1) k (k+1) k+ (k+1)+ Figure 3: ZOH In tis case x k+1 e A x k + or simply k+ k x k+1 e A x k + e A( ) e A(k+ s) Bu k 1 s + e As B s u k 1 + k+ k+ e A(k+ s) Bu k s e As B s u k

5 5 3 Sampling of a stocastic system Consier a (LI) continuous time stocastic system t x Ax + w were te intensity of w is R. In iscrete time, i.e. for t i were is te sampling perio, te system can be escribe by x i+1 Φx i + v i were te variance of v i is Σ wic (see e.g. (Söerström 1994) p. 84) is given by or te solution to Σ e Acs R c e A c s s t Σ AΣ + ΣA + R Σ() One meto can be foun in (Söerström 1994) p Let ( ) A R exp F 21 F 22 A ten F 11 e A Φ F 22 e A Φ 1 F 12 e A( s) R s e A s s If we apply te substitution τ s we get e A( s) R s e A ( s) s e A Φ F 11 Σ F 12 F 1 22 Anoter meto is te following. Let ( A R exp F 21 F 22 A ) ten te F 12 F 11 e A Φ 1 F 22 e A Φ e A( s) Re A s s e A e As Re A s s Φ F 22 Σ F 22F 12 e Matlab implementation of tis algoritm is liste in Appenix 9 as nc2.m.

6 6 5 OBSERVABILIY 4 Controlability If we aress te question watever it is possible to rive te system t x Ax + Bu x() x from any initial state to any target state in finite time, we migt answer tat (See e.g. (Kailat 198) p. 61 or (Cen & Francis 1995) p. 236) by cecking te rank properties of te controllability Gramian matrix W c e As BB e A s s e controlability Gramian can also be foun as te solution to te following ifferential equation t W c AW c + W c A + BB W c () Define ( A BB exp F 22 A ten F 12 F 11 e A Φ 1 ) F 22 e A Φ e A( s) BB e A s s e A e As BB e A s s Φ F 22 W c F 22F 12 e Matlab implementation of tis algoritm is liste in Appenix 9 as syscwc.m. 5 Observability e ual to te controllability problem is te observability problem. observe te output from te system t x Ax y Cx If we over a finite perio of time, ten te question is watever we can etermine any initial state value. is problem is solve (See e.g. (Kailat 198) p. 615.) by cecking te rank properties of te obervervability Gramian W o e A s C Ce As s t W o A W o + W o A + C C W o ()

7 7 Define ( A C exp C F 22 A ten F 12 F 11 e A Φ e A ( s) C Ce As s e A ) F 22 e A Φ Φ F 22 W c F 22F 12 e A s C Ce As s e Matlab implementation of tis algoritm is liste in Appenix 9 as syscwo.m. 6 Sample-ata control Consier te problem of controlling a continuous time LI system x(t) Ax(t) + Bu(t) x() x() (3) t suc tat te (star continuous time) objective function J 1 2 x ( )P x( ) x (t)qx(t) + u (t)ru(t) t is minimize. e control actions are assume to be constant between samples, i.e. u(t) u i for i < t i + were is te lengt of te (constant) sampling perio. We assume for te sake of simplicity tat te orizon is a multiple of te sampling perio, i.e. N. e Bellman equation becomes in tis situation V i (x i ) min u i 1 i+ 1 i 2 x (t)qx(t) t u i Ru i + V i+1 (x i+1 ) (4) V N (x N ) 1 2 x NP x N were te Bellman function, V i (x i ), is te optimal cost to go. We will investigate te following ciate function V i (x i ) 1 2 x i S i x i wic obviously is satisfie for i N. By notation x i x(i) x N x( ). Let for sort s t i. e solution to (3) is well known is x(t) e As x i + Φ s x i + Γ s u i e A(s τ) B τ u i

8 8 6 SAMPLED-DAA CONROL were Φ s e As Γ s If we furtermore introuces te integrals e A(s τ) B τ e Aτ B τ Q 1 1 Φ s QΦ s s Q 12 1 Φ s QΓ s s Q 2 1 Γ s QΓ s s ten te inner part of te minimization in (4) can be written as I 1 x 2 i u Q 1 Q 12 xi i + 1 Q 12 Q 2 u i 2 u i Ru i + 1 x 2 i u Φ S i+1φ Φ S i+1γ i Γ S i+1φ Γ S i+1γ or as I 1 2 x i u i at means tat te control is given by: Q 1 + Φ S i+1φ Q 12 + Φ S i+1γ Q 12 + Γ S i+1φ R + Q 2 + Γ S i+1γ xi u i xi u i u i R + Q 2 + Γ S i+1 Γ 1 Q 12 + Γ S i+1 Φ xi were S i is given by te recursion S i Q 1 +Φ S i+1 Φ Q 12 + Φ S i+1 Γ R + Q2 + Γ S i+1 Γ 1 Q 12 + Γ S i+1 Φ S N P is ensures tat te ciate function satisfy te Bellman equation. Notice, te solution to tis problem coincie wit te solution to a iscrete time problem, just wit transforme weigt matrices. We will now use te matrix exponential for etermine te tese weigt matrices. Let Q1 Q Σ 12 Q 12 Q 2 For etermining te matrices, efine te square matrix A B A en by te Lemma e As e As s ea(s t) t I e As s eat t I Φs Γ s I If we furtermore efine te matrix Q c Q

9 REFERENCES 9 it is straigt forwar to ceck tat Σ e A s Q c e As s Φ s I Γ s Q Φs Γ s I s If we compute te matrix: F 22 ten finally F 12 F 11 e A ( A Q exp c A F 22 e A e A ( s) Q c e As s e A ) e A s Q c e As s F 22 Σ F 22F 12 Φ Γ I e Matlab implementation of tis algoritm is liste in Appenix 9 as conc2.m. References Cen,. & Francis, B. (1995). Optimal Sample-Data Control Systems, Communications Control Engineering Series, Springer-Verlag New York Inc. Kailat,. (198). Linear Systems, Prentice Hall. Moler, C. & Loan, C. V. (1978). Nineteen ubious ways to compute te exponential of a matrix, SIAM Review, Vol. 2, No. 4 (Oct., 1978), pp (4): Söerström,. (1994). Discrete-ime Stocastic Systems: Estimation Control, Prentice Hall.

10 1 7 SAMPLED-DAA CONROL 7 Sample-ata control e problem in section 6 can be treate in a more general framework (see e.g. (Cen & Francis 1995) page 238). Here te problem will be solve in te same setting as in section 6. Consier te problem of controlling a continuous time LI system x(t) t Ax(t) + Bu(t) x() x() (5) y(t) Cx(t) + Du(t) suc tat te objective function J 1 2 x ( )P x( ) y(t) 2 t (6) is minimize. Notice te traitional weigts is embee in te output matrices, C D. e alternative formulation J 1 2 x ( )P x( ) + 1 x (t) u (t) Q S x(t) 2 S t R u(t) is easily obtaine from te metos escribe in 8. e control actions are, as in section 6, assume to be constant between samples, i.e. u(t) u i for i < t i + It is quite easy to ceck tat te cost function in (6) is equivalent wit te iscrete time problem of controlling te system x i+1 Φx i + Γu i x x suc tat te cost function J 1 2 x NP x N N 1 i x i u i Q 11 Q 12 Q 12 Q 22 xi u i is minimize. Here: Let Σ Q 22 Q 12 ( ) Q11 Q 12 Q 21 Q 22 Q 11 D + C e A t C D + C t e A s C Ce As s t e As B s D + C C Q c D e As B s t t e As B s t C D Q S S R

11 11 Define te square matrix A A B en by te Lemma e At e At ea(t s) s e At t eas s I I It is straigt forwar to ceck tat Σ e A s C C D e As s e A s Q c e As s D Compute te matrix ( A Q exp F 22 A en Σ F 22F 12 F 22 Φ Γ I ) e Matlab implementation of tis algoritm is liste in Appenix 9 as smplq.m. 8 Manipulation of cost functions Connection between output point of view cost functions. Consier z Cx + Du ( C D ) ( ) x u en J z 2 ( x t u ) C were D C D ( x u ( ) C Q C C D D C D D ) ( x t u ) ( x Q u On te oter, given Q we can easily fin C D. Assuming Q to be positive efinite we can perform a colesky factorization, i.e. fin H suc tat en (using a matlab notation) C H(:, 1 : n) Q H H D H(:, n + 1 : en) If Q is not positive efinite (but still symmetric) we ave to user SVD instea. en Q USU were U is an upper triangular matrix S is a iagonal matrix. In tis case H SU )

12 12 9 CODES 9 Coes e coes liste below is te Matlab implementations of te algoritms escribe in te previous capters. e listning occur in teir strippe version, i.e. witout any significant input cecking or options. function A,B c2(a,b,) Usage A,B c2(a,b,) Fin te iscret time moel x(t+1)a*x(t)+b*u(t) wen te continuous time moel ot(x)a*x+b*u is sample wit as sampling perio te input is constant between samples. n,msize(b); Fexpm(A B*; zeros(m,n+m)); AF(1:n,1:n); BF(1:n,n+1:en); function (S,A)nc2(A,R,) Usage: (S,A)nc2(A,R,) Fin te iscret time moel x(t+1)a*x(t)+v(t) v(t) ~ N(,S) wen te continuous time moel ot(x)a*x+w is sample wit as sampling perio. e continuous time wite noise is assume to ave te intensity R. nlengt(a); Fexpm(-A R; zeros(n,n) A *); F12F(1:n,n+1:en); F22F(n+1:en,n+1:en); AF22 ; SA*F12;

13 13 function Wc,Asyscwc(A,B,) Usage: Wc,Asyscwc(A,B,) Fin te controlability Gramian Wc int_^ exp(as)bb exp(a s)s for te continuous time system ot(x)a*x+b*u If tis system is sample wit as sampling perio te input is constant between samples A contains te iscrete time system matrix. nlengt(a); Fexpm(-A B*B ; zeros(n,n) A *); F12F(1:n,n+1:en); F22F(n+1:en,n+1:en); AF22 ; WcA*F12; function Wo,Asyscwo(C,A,) Usage: Wo,Asyscwo(C,A,) Fin te observability Gramian Wo int_^ exp(a s)c Cexp(As)s for te continuous time system ot(x)a*x ycx If tis system is sample wit as sampling perio te input is constant between samples A contains te iscrete time system matrix. nlengt(a); Fexpm(-A C *C; zeros(n,n) A*); F12F(1:n,n+1:en); F22F(n+1:en,n+1:en); AF22; WcA *F12;

14 14 9 CODES function Q1,Q2,Q12,A,Bconc2(A,B,Q,R,) Usage: Q1,Q2,Q12,A,Bconc2(A,B,Q,R,) ransform te Continuous time control problem ot(x)ax+bu Jint x Qx + u Ru s into te iscrete time LQ control problem x(i+1)a*x+b*u J sum (x u ) Q1 Q12; Q12 Q2 (x;u) n,msize(b); QcQ zeros(n,m); zeros(m,n) R*; AcA B; zeros(n,n+m); Fexpm(-A Qc; zeros(n.n) A*); F22F(n+m+1:en,n+m+1:en); F12F(1:n+m,n+m+1:en); QF22 *F12/; Q1Q(1:n,1:n); Q2Q(n+1:en,n+1:en); Q12Q(1:n,n+1:en); AF22(1:n,1:n); BF22(1:n,n+1:en); function A,B,C,Dsmplq(A,B,C,D,) Usage A,B,C,Dsmplq(A,B,C,D,) syssmplq(sysc,) ransform te H2 continuous time problem ot(x)a*x+b*u yc*x+d*u Jint y *W*y s into a iscrete time H2 problem x(t+1)a*x+b*u yc*x+d*u

15 15 Jsum y *W*y if nargin5, typ1; elseif nargin2, typ2, A,B,C,Dssata(sysc); else isp( Wrong argument list ); return 1 en n,msize(b); QcC ;D *C D; AcA B; zeros(n,n+m); Fexpm(-A Qc; zeros(n.n) A*); F22F(n+m+1:en,n+m+1:en); F12F(1:n+m,n+m+1:en); QF22 *F12; AF22(1:n,1:n); BF22(1:n,n+1:en); U,Ssv(Q); Hsqrt(S)*U ; CH(:,1:n); DH(:,n+1:en); if typ2, Ass(A,B,C,D,), en