Education in Linear System Theory with the Geometric Approach Tools

Transcription

1 Education in Linear System Theory with the Geometric Approach Tools Giovanni Marro DEIS, University of Bologna, Italy GAS Workshop - Sept 24, 2010 Politecnico di Milano G. Marro (University of Bologna) GAS Workshop 1

2 Why Education? At the beginning the geometric control theory was taught in some university courses (Bologna, Milano and Toronto). At those times there where only two main sources, the early papers by Basile and Marro and Wonham and Morse. It is very difficult now to orientate oneselves in a huge quantity of new ideas, new terms and new discoveries born in the meantime. But, an idea for selecting good teaching material is to develop a toolbox for Matlab, that extends to the multivariable case almost all concepts studied in standard automatic control courses. It is now available and has very interesting features: it embodies the geometric approach and some basic concepts of optimal control, in a unified setting. G. Marro (University of Bologna) GAS Workshop 2

3 the beginning The textbooks in G. Marro (University of Bologna) GAS Workshop 3

4 the beginning The textbooks in (Arturo Locatelli) G. Marro (University of Bologna) GAS Workshop 4

5 the beginning The independent early papers The geometric approach was introduced independently by Basile and Marro and Wonham and Morse. The papers are: G. Basile and G. Marro, Controlled and conditioned invariant subspaces in linear system theory, J. Optim. Theory Appl., vol. 3, no. 5, pp , W. M. Wonham and A. S. Morse, Decoupling and pole assignment in linear multivariable systems: a geometric approach, SIAM J. Contr., vol. 8, no. 1, pp. 1 18, OK, the papers were independent, but their intersection was very thin. In fact, only the underlying idea was common, but the applications and algorithms were quite different. G. Marro (University of Bologna) GAS Workshop 5

6 the beginning So, at the beginning of 1970 BM WM two new names (controlled invariant and conditioned invariant); the related definitions and algorithms; duality. two new names (geometric approach and controllability subspace); one algorithm similar to that of maximal controlled invariant; a second algorithm (for the maximal controllability subspace). These differences had a noticeable influence on the subsequent researches, in particular on the development of software, since Wonham never accepted to shuffle the cards (he refused the conditioned invariant). The purpose of this talk is to show that the BM basic algorithms of controlled and conditioned invariants and duality are strictly necessary to develop a rather complete toolbox for the analysis and synthesis of multivariable control systems in the state space form. G. Marro (University of Bologna) GAS Workshop 6

7 the beginning The most recent books on geometric control theory Wonham (1974, 1979, 1985) has the great merit of having widely publicized the geometric approach. Basile and Marro (1992) emphasize duality and dual problems, that were neglected in the Wonham s book. Software for Matlab is included. Trentelman, Stoorvogel and Hautus (2001) provide a bridge towards achieving minimal H2 and H norm solutions. G. Marro (University of Bologna) GAS Workshop 7

8 the mathematical background The systems referred to (triples and quadruples) Continuous-time LTI systems: Discrete-time LTI systems: ẋ(t) = A x(t) + B u(t) y(t) = C x(t) ẋ(t) = A x(t) + B u(t) y(t) = C x(t) + D u(t) x(k + 1) = A x(k) + B u(k) y(k) = C x(k) x(k + 1) = A x(k) + B u(k) y(k) = C x(k) + D u(k) The term D u(t) or D u(k) is called feedthrough. The standard Matlab definitions of the above LTI systems are: >> sys=ss(a,b,c,0); >> sys=ss(a,b,c,0,tc); >> sys=ss(a,b,c,d); >> sys=ss(a,b,c,d,tc); where T c 0 denotes the sampling time in the discrete-time case. G. Marro (University of Bologna) GAS Workshop 8

9 the GA toolbox Dealing with subspaces in Matlab >> Y=ima(A[,1]); Y =ima >> Y=ker(A); Y =kera >> Z=sums(X,Y); Z = X + Y >> Z=ints(X,Y); Z = X Y >> Y=ortco(X); Y = X >> Y=invt(A,X); Y =A 1 X The cornerstone is ima, based on the orthogonal-triangular decomposition qr. The call Y=ima(A,1) maintains the order of the columns of A while orhonormaling their span. The algorithm for the maximum (A, B)-controlled invariant contained in C: V 1 = C V i = C A 1 (V i 1 + B) (i = 2,3,...) The algorithm for the minimum (A, C)-conditioned invariant containing B: S 1 = B S i = B + A(S i 1 C) (i = 2,3,...) (1) (2) G. Marro (University of Bologna) GAS Workshop 9

10 the GA toolbox Dualities Corollary Given a subspace B X, the orthogonal complement of an (A, B)-controlled invariant is an (A T, B )-conditioned invariant (and viceversa). As a consequence of the above corollary the following relations hold: maxv(a, B, C) = (mins(a T, B, C )) mins(a, C, B) = (maxv(a T, C, B )) When a triple Σ(A, B, C) is considered, under the settings B = imb, C = kerc and the more compact notation V (Σ) for maxv(a, B, C) and S (Σ) for mins(a, C, B), it follows that V (Σ) = (S (Σ T )) S (Σ) = (V (Σ T )) where Σ T (A T, C T, B T ) is the dual system of Σ. It will be shown that the above relations also hold for quadruples. G. Marro (University of Bologna) GAS Workshop 10

11 the GA toolbox The main properties of multivariable system Controllability Observability Internal and External Stability Right Invertibility Left Invertibility Relative Degree Invariant Zeros (Phase Minimality) Extension to Quadruples The topics distinguished in red are those for which there is a big confusion in the available literature. G. Marro (University of Bologna) GAS Workshop 11

12 the GA toolbox A quick review of the basic tools: V (Σ) and friend F V x(0) X u Σ(A, B, C, D) F x y A V V + B (A + BF) V V V ker(c + DF) The meaning of V (Σ): it denotes the maximum output-nulling subspace of Σ, i.e., the maximum subspace of the state space where it is possible to impose state trajectories that are invisible at the output, and F denotes a corresponding friend. G. Marro (University of Bologna) GAS Workshop 12

13 the GA toolbox A quick review of the basic tools: S (Σ) and friend G X u Σ(A, B, C,D) y S 0 A (S C) S G (A + GC) S S S im(b + GD) The meaning of S (Σ): referring to a discrete-time system, it is the maximal subspace of the state space reachable from the origin in at most ρ steps with trajectories that have all the states except the last one invisible at the output. G. Marro (University of Bologna) GAS Workshop 13

14 Using distributions? the GA toolbox S 0 X If the discrete-time system is the sampleddata version of a continuous-time system, it is possible to follow a (ρ 1)-step trajectory on S C, while the trajectory of the related continuous-time system is arbitrarily close to S C. Thus, by allowing distributions as inputs it is possible to maintain the state on the subspace V a = V + S C (the almost controlled invariant by Willems), invisible at the output. Unfortunately, we don t like to manage distributions as inputs, because of saturation. J. Willems, Almost invariant subspaces: an approach to high gain feedback design - Part I: Almost controlled invariant subspaces, IEEE Trans.on Autom. Cont., 26, , G. Marro (University of Bologna) GAS Workshop 14

15 The relative degree the GA toolbox But the main feature of the conditioned invarant is the extension of the concept of relative degree to the multivariable case. y 0 ρ k The relative degree in the discrete-time case. Definition In the discrete-time case the relative degree is defined as the minimum value of ρ such that, starting at the origin, from k = ρ the output is functionally controllable, while for k < ρ it can be kept to zero. Definition In the continuous-time case is similarly referred to the time derivatives of the output at the origin (i.e., to the smoothness of the output trajectories). G. Marro (University of Bologna) GAS Workshop 15

16 the GA toolbox The relative degree Theorem Refer to Σ(A, B, C) and denote with m the dimension of the subspace V + S. The relative degree is the minimum value of ρ in the condtioned invariant algorithm such that the dimension of V + S ρ is equal to m. G. Marro (University of Bologna) GAS Workshop 16

17 the GA toolbox The invariant zeros and minimality of phase Theorem Refer to Σ(A, B, C) and denote with R the maximum controllability subspace. The following equality holds: R = V S This theorem provides a more convenient computational setting for MCS, the constrained controllability subspace on V, and also provides a computational basis for the invariant zeros of Σ. G. Marro (University of Bologna) GAS Workshop 17

18 the GA toolbox The geometric definition of invariant zeros Let T = [T 1 T 2 T 3 T 4 ], with imt 1 = R = V S, imt 2 = V, im[t 1 T 3 ]=S, and T 4 such that T is nonsingular. A 11 A 12 A 13 A 14 O A 22 A 23 A 24 T 1 (A + BF)T = T 1 B = O O A 33 A 34 O O A 43 A 44 B 1 O B 3 O C T = O O C 3 C 4 The invariant zeros of the system are the eigenvalues of A 22. G. Marro (University of Bologna) GAS Workshop 18

19 A refinement the GA toolbox Since the pair (A 12,B 1 ) is controllable, it is possible to assign the eigenvalues of A 12 by a slight modification of the feedback matrix F and use a Sylvester equation to complement R with respect to V, thus obtaining a zero as the second element of the first row. A 11 O A 13 A 14 O A 22 A 23 A 24 T 1 (A + BF)T = T 1 B = O O A 33 A 34 O O A 43 A 44 B 1 O B 3 O C T = O O C 3 C 4 Then put A 22 in a block-diagonal form, with the stable eigenvalues separated from the unstable ones, by using the routine subsplit. G. Marro (University of Bologna) GAS Workshop 19

20 the GA toolbox The second cornerstone of the GA toolbox This is done by using the command: >> [Xs,Xu,X0]=subsplit(A[,1]); that provides the A-invariant subspaces of the stable, unstable and boundary modes of a real, square matrix A. The two-arguments call refers to the discrete-time case. Referring to the previous change of basis, it allows to define V g, the maximum controlled invariant internally stabilizable (i.e., incorporating all and only the stable invariant zeros). This is simply obtained with the call: >> [V,F]=vstarg(sys); It also allows design of suitable filters that perform cancellation of all stable and unstable zeros. G. Marro and E. Zattoni, Unknown-state, unknown-input recontruction in discrete-time non-minimum phase systems: geometric methods, Automatica, 46, , G. Marro (University of Bologna) GAS Workshop 20

21 the GA toolbox Extending to systems with feedthrough - An early result B.D.O. Anderson, Output nulling invariant and controllability subspaces, Proceedings of the 6th IFAC Congress, August G. Marro (University of Bologna) GAS Workshop 21

22 the GA toolbox Extending to systems with feedthrough - Trick # 1 D Σ u Σ(A,B, C) + + y Σ(L,I q,i q) z For the quadruple Σ(A,B,C,D) the maximum output-nulling subspace is the maximum (A, B)-controlled invariant V such that for a suitable F (A + BF) V V, V ker(c + DF) Let us refer to he extended system shown in the figure, represented by: x A O B ˆx = A = B = C = z C L D O I q where L denotes an arbitrary diagonal matrix. A basis matrix V of V and friend F are easily deduced from the correspoding V and F of Σ, that have the following structures: V V =, F O = F O G. Marro (University of Bologna) GAS Workshop 22

23 the GA toolbox Extending to systems with feedthrough - Trick # 2 Σ D v Σ(L,I p,i p) u Σ(A,B, C) + + y For the quadruple Σ(A,B,C,D) the minimum input-containing subspace S is the minimum (A, C)-conditioned invariant S such that for a suitable G (A + GC) S S, S (B + GD) Let us refer to he extended system shown in the figure, represented by: x A B O ˆx = A = B = C = u C D O L Ip where L denotes an arbitrary diagonal matrix. A basis matrix S of S and friend G are easily deduced from the correspoding S and G of Σ, that have the following structures: S O S =, G O I G = p O G. Marro (University of Bologna) GAS Workshop 23

24 the GA toolbox Anyway, duality is still valid! V ( Σ(A,B,C,D) ) = (S ( Σ(A T,C T,B T,D T ) )) S ( Σ(A,B,C,D) ) = (V ( Σ(A T,C T,B T,D T ) )) Remark The system on the right is the dual system of Σ, referred to as Σ T. Hence equivalent but more compact notations are V (Σ) = ( S (Σ T ) ) S (Σ) = ( V (Σ T ) ) G. Marro (University of Bologna) GAS Workshop 24

25 the GA toolbox Some properties of systems expressed in geometric terms >> [V,F]=vstar(sys); u Σ x y >> [S,G]=sstar(sys); >> r=reldeg(sys); >> z=gazero(sys); >> [V,F]=vstarg(sys); Left invertibility V S = {0} Right invertibility V + S = X Relative degree minimal ρ such that V + S ρ = X Minimality of phase Z(Σ) C g where C g is the set of stable modes in the complex plane. G. Marro (University of Bologna) GAS Workshop 25

26 The most important solved problems The measurable disturbance decoupling problem h Σ c u Σ y Equations ẋ(t) = Ax(t) + B u(t) + H h(t) y(t) = C x(t) Let B = imb, H = imh, C = kerc. The measurable disturbance decoupling problem It will be shown that the solution in geometric terms depends on A, B, H, C. The solution of a synthesis problem with geometric techniques as a rule includes A n & s structural condition: H V ( Σ(A,B,C) ) ; A suff. stabilizability condition: Σ to be stable and minimum-phase. Within the GA Toolbox the problem is solved with the command: >> sysc=hud(sys,h). G. Marro (University of Bologna) GAS Workshop 26

27 The most important solved problems The exact feedforward model following h Σ c u Σ Σ m y + η =0 y m _ Σ Exact model following as a measurable disturbance decoupling problem with stability (Σ m absorbs the relative degree). The problem is solvable if the relative degree of Σ m is greater or equal to that of Σ, and both Σ and Σ m are stable and minimum-phase. The poles of Σ m appear as zeros of Σ; If Σ m consists of p independent SISO systems, we achieve exact row-by-row decoupling at no cost. G. Marro (University of Bologna) GAS Workshop 27

28 The most important solved problems From exact feedforward to exact feedback model following r + h u Σ _ c Σ y + η = 0 r + h u _ Σ c Σ y + η = 0 Σ m y m _ Σ m y m _ Exact feedback model following. A structurally equivalent connection. h Σ m Σ c r u Σ y m + Σ + m y Another structurally equivalent connection. + η = 0 _ r + h u Σ _ c Σ m Σ y m y + η = 0 Exact feedback model following, possibly with row-by-row decoupling and multiple internal models. _ G. Marro (University of Bologna) GAS Workshop 28

29 Optimal control Geometric approach and optimal control h u F Σ The Kalman regulator problem x y Consider the LQR-H 2 problem ẋ(t) = Ax(t) + B u(t) + H h(t) y(t) = C x(t) + D u(t) with the cost J = 0 y T (t)y(t) dt The solution of the corresponding Riccati equation in the Matlab environment: >> Q = C *C; R = D *D; S = D *C; >> [X,L,K] = care(a,b,q,r,s); F=-K; It is requested that R is nonsingular (the relative degree of the quadruple (A,B,C,D) is zero). G. Marro (University of Bologna) GAS Workshop 29

30 Optimal control Geometric approach and optimal control Consider the continuous-time Hamiltonian system: [ ẋ(t) ṗ(t) ] [ ][ A 0 x(t) = Q A T p(t) 0 = [ S T B T ] [ x(t) p(t) ] [ B + S ] + [ R ] u(t) ] u(t) The Hamiltonian system is a quadruple (Â, B,Ĉ, D) whose output must be maintained at zero for the given initial state x 0. Hence the LQR problem is easily interpreted as a standard disturbance localization problem of the geometric approach for the above Hamiltonian system. The corresponding solution with the Geometric Approach Toolbox is: >> [V,F,X]=vstargh2(A,B,C,D); The regular, singular and cheap cases are handled without distinction. G. Marro (University of Bologna) GAS Workshop 30

31 Optimal control H 2 -optimal model following u Σ, Σ 1 y y 1 ẋ(t) = Ax(t) + B u(t) y(t) = C x(t) + D u(t) The Stoorvogel problem y 1 (t) = C 1 x(t) + D 1 u(t) System Σ 1 has the same matrices A and B as Σ, but different output matrices C 1 and D 1. Its feature is that exact decoupling problem with respect the output C 1, D 1 reflects in H 2 optimal decoupling problem with respect the output C, D. >> sys=ss(a,b,c,d); >> sys1=stoor(sys); A. Stoorvogel, The singular H 2 control problem, Automatica, vol. 28, pp , G. Marro (University of Bologna) GAS Workshop 31

32 Optimal control H 2 -optimal model following u y Σ, Σ 1 y 1 The Stoorvogel problem ẋ(t) = Ax(t) + B u(t) y(t) = C x(t) + D u(t) y 1 (t) = C 1 x(t) + D 1 u(t) Theorem 1 the exact disturbance decoupling problem with state feedback for any input disturbance matrix H such that imh V 1, where V 1 is referred to Σ 1, corresponds to a minimal H 2 -norm solution for Σ; 2 the number of outputs of Σ 1 is equal to that of Σ; 3 Σ 1 is minimum-phase; 4 Σ 1 has the same global relative degree as Σ; 5 the steady-state gain of Σ 1 is equal to that of Σ. G. Marro (University of Bologna) GAS Workshop 32

33 Optimal control The proposed multivariable linear regulator regulator d r + h + u _ Σ c _ Σ u s y 1 Σ o model (possibly q SISO) + e =0 _ y m Σ m controlled system y (reduced-order) observer The H 2 norm of the overall system between input r and the tracking error y y m is minimal for this system Σ and this model Σ m. In the GA Toolbox environment this regulator is designed with the call: >> modfol (Σ and Σ m are entered in interactive mode). G. Marro (University of Bologna) GAS Workshop 33

34 The proposed multivariable linear regulator G. Marro, F. Morbidi and D. Prattichizzo, H2-optimal model following: A geometric approach, Proceedings of the 3rd IFAC Symposium on System Structure and Control., Foz de Iguaçu, Brazil, October 17-19, Conclusion The geometric approach has been developed for about forty years gradually, as new linear and nonlinear problems arose and were analyzed. It provides a significant insight into systems and control, based on very few elementary tools, on which the overall theory is based. Although the geometric tools are very simple and supported by exhaustive computational machinery, it is rather difficult to get a complete panorama of them, since their presentation in the literature by several authors is not uniform in style and has very often been covered in unnecessarily heavy mathematics. G. Marro (University of Bologna) GAS Workshop 34

35 The Geometric Approach Toolbox G. Marro (University of Bologna) GAS Workshop 35

36 The Geometric Approach Toolbox (cont d) The GA Toolbox is freely downloadable from: G. Marro (University of Bologna) GAS Workshop 36

37 Sorry, but... G. Marro (University of Bologna) GAS Workshop 37