A Control Methodology for Constrained Linear Systems Based on Positive Invariance of Polyhedra

A Control Methodology for Constrained Linear Systems Based on Positive Invariance of Polyhedra Jean-Claude HENNET LAAS-CNRS Toulouse, France Co-workers: Marina VASSILAKI University of Patras, GREECE Jean-Paul BEZIAT CISI Bordeaux, FRANCE Eugênio B. CASTELAN UFSC, Florianópolis, BRAZIL. Carlos E. T. DÓREA UFBA, Bahia, BRAZIL 1

INTRODUCTION Most real systems are subject to - constraints on state and control, - disturbances and uncertainties. The positive invariance approach is able to tackle these features in a computationally efficient way. Until now, the positive invariance approach has been essentially developed in theoretical works rather than in practical applications. This conference presents some simple positive invariance concepts and methodologies useful to treat practical control problems. The basic computational ingredients of the methods are spectral assignment and Linear Programming. 2

OUTLINE OF THE PRESENTATION Part 1 : The positive invariance approach Positive invariance ; Definitions and Basic Properties Mathematical frameworks Invariance and stability Positive invariance of polyhedral sets w.r.t. linear systems : An algebraic characterization Part 2 : Linear Constrained Regulation Problems A unified model discrete-time systems / continuous-time systems state constraints / control constraints Positive invariance of polyhedral sets by state feedback Resolution of positive invariance relations Examples Part 3 : Disturbance attenuation and constrained regulation Domain of satisfactory performance Positive invariance with disturbance attenuation Regulator design with positive invariance properties The feasible domain An application in production planning 3

PART 1 THE POSITIVE INVARIANCE APPROACH Positive invariance Definitions and Basic Properties Mathematical frameworks Invariance and stability Positive invariance of polyhedral sets w.r.t. linear systems Geometric characterization Algebraic characterization 4

POSITIVE INVARIANCE Definitions and Basic Properties Definition Consider a discrete or continuous time domain, T < +, 0 2T. Let (S) be a dynamical system characterized at each time t 2T by its state vector x(t) 2X. The set X is positively invariant with respect to system (S) if and only if : x(0) 2 =) x(t) 2 8t 2T: A geometric characterization of positive invariance Let D t be the reachable set of states at time t 2T from any initial state in : D t = fx(t) jx(0) 2 g: A necessary and sufficient condition for positive invariance of the set with respect to system (S) is : D t 8t 2T: 5

POSITIVE INVARIANCE A geometric characterization Ω Ω Ω D(t) D(t ) D t 6

SOME MATHEMATICAL FRAMEWORKS Differential inclusions (some links with Viability Theory (Aubin)) _x(t) 2 F (x(t)) Differential equations _x(t) =f(x(t)) Recurrent Linear Equations - classical : x k+1 = Ax k + B 1 w k + B 2 u k - marking evolution in Petri Nets M k+1 (p) =M k (p) +Cs k - probabilistic ( Markov chains) k+1 = k P - (max,+) Algebra for Discrete Event Systems x k+1 = A x k B u k 7

POSITIVE INVARIANCE FOR DETERMINISTIC LINEAR SYSTEMS Geometric Approach à la Wonham : Invariance of subspaces E A-invariant () AE E with AE = fx 2< n jx = Ay y 2Eg. Continuous time: _x(t) =Ax(t) e At x 0 2E 8x 0 2E 8t 2< + Discrete time: x k+1 = Ax k A k x 0 2E 8x 0 2E 8k 2N: Positive invariance of closed domains (bounded or not), - continuous time: e At x 0 2 8x 0 2 8t 2< +, - discrete time: A k x 0 2 8x 0 2 8k 2N: 8

INVARIANCE AND STABILITY Direct Property Stability =) Existence of Invariant Sets If v(x) is a Lyapunov function of system (S), then =fx 2< n v(x) g with > 0 is a positively invariant set of (S). In general, such a set is closed and bounded. Example : If v(x) = x T Px is a quadratic Lyapunov function w.r.t. (S), then the ellipsoïds fx 2< n v(x) g ( > 0) are invariant sets for (S). Converse Property: Existence of closed and bounded invariant sets having the 0 state as an interior point =) Local Lyapunov Stability in a vicinity of the origin. 9

PARTICULAR SETS POSITIVELY INVARIANT FOR LINEAR SYSTEMS : stability domains Polyhedral Lyapunov functions v(x) =max i f j(qx) ij g () i Q 2< gn g n, rank(q) =n and a positive vecteur in < g. The set S(Q ) is a positively invariant polytope defined by: S(Q ) =fx 2< n : ; Qx g Compact polyhedral sets containing the zero state v(x) =max i maxf ;(Qx) i (p 1 ) i (Qx) i (p 2 ) i g Under rank(q) =n and vectors p 1 p 2 strictly positive, the set S(Q p 1 p 2 )=fx 2< n : ;p 1 Qx p 2 g is not empty, compact (closed and bounded), contains the zero state and positively invariant. 10

GENERAL POLYHEDRAL SETS FOR LINEAR SYSTEMS Unbounded polyhedra: R(G ) =fx 2< n : Gx g This polyhedron is unbounded if Ker G 6= f0g. Property : A necessary condition for positive invariance of R(G ) with respect to linear system (S), is invariance of the subspace Ker G for system (S). Polyhedral cones: Representation 1: image of the positive orthant K = fx 2< n jx = By y 2< m +g K = B< m +: Representation 2: polyhedron K = fx 2< n jgx 0g: A particular cone : < n + is a positively invariant set for any system x k+1 = Ax k with matrix A non-negative (componentwise). 11

AN ALGEBRAIC CHARACTERIZATION OF INCLUSION OF POLYEDRA Extended Farkas Lemma Consider two polyhedra in < n, denoted R(L!) and R(G ). A necessary and sufficient condition for : R(L!) R(G ) is the existence of a non-negative matrix, U, such that: U:L = G U:! Remark: The inclusion R(L!) R(G ) is equivalent to: Lx! =) Gx : The row-vectors of matrix U can be interpreted as dual vectors. 12

POSITIVE INVARIANCE OF POLYEDRA Discrete-time linear system (S D ) : x k+1 = Ax k. Theorem NSC for positive invariance of R(G ) w.r.t. (S D ): 9H 0 (componentwise non-negative ) such that: HG = GA H : Remark : If rank(g) =n and >0, H ; I should be a -M-matrix and (S D ) is stable. Continuous-time linear systems (S C ) : _x(t) =Ax(t). Theorem NSC for positive invariance of R(G ) w.r.t. (S C ): 9H essentially non-negative (H ij 0 8 (i j 6= i)) such that: HG = GA H 0: Remark : If rank(g) = n and > 0, H should be a -Mmatrice and (S) is stable. 13

POSITIVE INVARIANCE OF SYMMETRICAL POLYEDRA Symmetrical polyhedra: S(Q ) :=fx 2< n jqxj g Q 2< qn Theorem NSC for positive invariance of S(Q ) w.r.t. (S D ): 9H 2< qq such that: HQ = QA jhj : Remark : If rank(q) = n and > 0, positive invariance of S(Q ) implies stability of (S D ). Theorem NSC for positive invariance of S(Q ) w.r.t. (S C ): 9H 2< qq such that: HQ = QA ^H 0: with ( ^H ii = H ii ^H ij = jh ij j Remark : If rank(q) = n and > 0, positive invariance of S(Q ) implies stability of (S D ). 14

PART 2 LINEAR CONSTRAINED REGULATION PROBLEMS A unified model discrete-time systems / continuous-time systems state constraints / control constraints Positive invariance of polyhedral sets by state feedback Resolution of positive invariance relations by Linear Programming by eigenstructure assignment Examples 15

A UNIFIED MODEL OF CONSTRAINED LINEAR SYSTEMS p[x t ] = Ax t + Bu t B 2< nm m n: (S) Continuous-time case : p is the derivative operator _x t = Ax t + Bu t Discrete-time case : p is the advance operator x t+1 = Ax t + Bu t Case of linear constraints on the state vector: ; Qx t with rank Q = r n i > 0 i =1 :: r: Constraints generate a polyhedral domain S(Q ) in the state space: S(Q ) =fx 2< n ; Qx g 16

A UNIFIED MODEL OF CONSTRAINED LINEAR SYSTEMS Case of linear constraints on the control vector: ; Mu t with rank M = c m i > 0 i=1 :: c: Constraints generate the polyhedral domain S(M ) in the control space: S(M ) =fu 2< c ; Mu g Under a state-feedback regulation law: u t = Fx t the linear control constraints define a polyhedron in the state space, S(MF ). S(MF )=fx 2< n ; MFx g 17

POSITIVE INVARIANCE BY STATE FEEDBACK Positive Invariance relations A necessary and sufficient condition for S(Q ) to be a positively invariant set of system (S) is the existence of a matrix H 2< rr and of a scalar such that: HQ = Q(A + BF) ~H with ~H = jhj 0 < 1 in the discrete-time case, ~H = ^H 0 in the continuous-time case. A structural interpretation y t = Qx t can be interpreted as an output vector. The first relation imposes (A+BF)-invariance of Ker Q. 18

POSITIVE INVARIANCE BY LINEAR PROGRAMMING Basic LP formulation: Minimize subject to HG ; GBF = GA H H ij 0 8(i j) in the discrete-time case H ij 0 8(i j 6= i) in the continuous-time case 0 in the discrete-time case Result: R(G ) is (A + BF)-invariant if: 1 in the discrete-time case, 0 in the continuous-time case. Remark Many additional constraints can be added to this problem, to take into account: - control constraints - parametric uncertainties - regional pole placement. 19

POSITIVE INVARIANCE BY EIGENSTRUCTURE ASSIGNMENT This scheme applies to systems with linear constraints on the state vector or on the control vector or on the output vector. In this scheme, the domain of constraints S(Q ) is made positively invariant by state feedback. The construction applies only if rank(q) =r m: The problem is decomposed into two stages: (A+BF)-invariance of Ker Q. This is equivalent to locating n;r closed-loop generalized eigenvectors in a lkerq. Resolution of positive invariance conditions in < n =Ker Q 20

SPECTRAL SUFFICIENT CONDITIONS A spectral condition - discrete time case If matrix H satisfying : HQ = QA has the real Jordan form, and its eigenvalues, i + j i verify: then, 9 >0 such that: j i j + j i j1 jhj : And polyhedron S(Q ) is positively invariant. 1 I 0 1 R Spectral domain 21

SPECTRAL SUFFICIENT CONDITIONS A spectral condition - continuous time case If matrix H satisfying : HQ = QA has the real Jordan form, and its eigenvalues, i + j i verify: then, 9 >0 such that: i ;j i j ^H 0: And polyhedron S(Q ) is positively invariant. I 0 R Spectral domain 22

POSITIVE INVARIANCE BY EIGENSTRUCTURE ASSIGNMENT Consider the system matrix (Rosenbrock 1970): P () = " # I ; A ;B Q 0 rm (A+BF)-invariance of Ker Q is possible if and only if the equation " # 0 = 0 P ( i ) " vi w i # has at least n ; r solutions ( i v i ), with vectors v i independent. Structural condition If r m and r < n, condition rank(qb) = r is sufficient for (A,B)-invariance of Ker Q. Remark: If any invariant zero is unstable, (A+BF)-invariance of Ker Q and closed-loop stability will not be simultaneously obtained. 23

EIGENSTRUCTURE ASSIGNMENT IN Ker Q If rank(qb) =r m, the following technique can be applied: 1. Select n ; r stable closed-loop poles: - The p invariant zeros of (A B Q) have to be selected as closed-loop poles. They must be stable. - The n ; p ; r remaining closed-loop poles are selected in the stable region as desired. - J 1 is the Jordan form of the restriction of (A+BF) to Ker Q. 2. Eigenvectors spanning Ker Q - Define V 1 =(v 1 ::: v n;r ) such that: GV 1 =0 (A + BF)V 1 = V 1 J 1 - For any i, solve P ( i ) " vi w i # = " 0 0 # 24

EIGENSTRUCTURE ASSIGNMENT IN A COMPLEMENTARY SUBSPACE OF Ker Q 1. Selection of r appropriate closed-loop poles: The eigenvalues 0 i of (A + BF)j(<n =Ker Q) are selected so as to satisfy: j i j + j i j < 1 in the discrete-time case i < ;j i j in the continuous-time case J 2 : real Jordan form of (A + BF)j(< n =Ker Q). 2. Eigenvectors spanning R Vectors v 0 i and w0 i P ( 0 i ) " v 0 i w 0 i # = are computed to satisfy: " 0 e i #, with e i = 2 6 4 0 : 1 : 0 3 7 5 i: 25

CONSTRUCTION OF THE GAIN MATRIX Let V =[V 1 j V 2 ] be the matrix of the desired real generalized eigenvectors, and W =[W 1 j W 2 ] the associated input directions. The selected real Jordan form of (A + BF) is: J = " # J1 0 0 J 2 The feedback gain matrix providing the desired eigenstructure assignment is: F = WV ;1 By construction, it satisfies for some positive vector : J 2 Q = Q(A + BF) ~J 2 26

EXAMPLE 1 : State Constraints Consider the following data: A = 2 6 4 B = 0:4832 0:8807 ;0:7741 ;0:6135 0:6538 ;0:9626 ;0:2749 0:4899 0:9933 2 6 4 ;0:8360 0:4237 0:7469 ;0:2613 ;0:0378 0:2403 The open-loop system has unstable eigenvalues: (A) = 2 6 4 3 7 5 0:4481 + j0:9683 0:4481 ; j0:9683 1:2341 The state constraints are defined by ; Qx 3 7 5 3 7 5 with: Q = " ;0:6538 0:7741 ;0:9933 0:4899 ;0:9626 ;0:8360 # = System (A B Q) has an stable invariant zero at 0:6647. " 2:5 2:5 # 27

Positive invariance of S(Q ) and global asymptotic stability are obtained when selecting: 8 >< >: 1 = 0:6647 ( stable zero) 0 1 = 0:1+j0:6 0 2 = 0:1 ; j0:6 6 4 2 0-2 -4-6 -6-4 -2 0 2 4 6 Invariance of the domain of constraints 28

5 4 3 2 1 0-1 -2 0 5 10 15 20 25 30 Convergence in < 3 =Ker Q 0-1 -2-3 -4-5 -6 0 5 10 15 20 25 30 Convergence in Ker Q 29

EXAMPLE 2 : Control Constraints Third order system: A = 2 4 0:125 ;1:375 0:375 ;2:500 ;0:500 2:500 0:625 1:125 ;0:125 B = 2 4 5:00 ;1:00 1:00 2:00 1:00 0:00 The open-loop system has two unstable eigenvalues, (;3:0 2:0), and one stable, 0:5. 1:00 ;1:00 ;1 2 Su k 1 2 with S = : 2:00 1:00 3 5 3 5 In this example, r = m =2. The stable pole, 1 =0:5 is left unchanged. We select 2 =0:4 ; 0:4j 3 =0:4+0:4j, and S;1 as the matrix of real input vectors associated with these last two eigenvalues. 30

EXAMPLE 2 (contd.) The matrix of closed-loop generalized real eigenvectors becomes: 2 3 0:707 0:825 ;3:050 V = 4 0:0 1:061 0:494 5 0:707 1:528 ;3:102 under the feedback gain matrix: 0:470 0:625 ;0:470 F = 1:696 0:495 ;1:696 to obtain: J = 2 4 0:5 0 0 0 0:4 ;0:4 0 ;0:4 0:4 3 5 : A better eigenstructure assignment is obtained by simply inverting the order of 2 and 3. Under the new feedback gain matrix, 0:511 0:733 ;0:511 F = 0:658 ;0:160 ;0:658 the size of the invariant domain is increased by more than 40%. 31

1 x2 0 x1-1 -1 0 1 (1.a) 1 x2 0 x1-1 -1 0 1 (1.b) The invariant domains in projection 32

PART 3 DISTURBANCE ATTENUATION AND CONSTRAINED REGULATION Domain of satisfactory performance Positive invariance with disturbance attenuation Regulator design with positive invariance properties The feasible domain An application in production planning The dynamical model A closed-loop production policy 33

DOMAIN OF SATISFACTORY PERFORMANCE Discrete-time linear system: x k+1 = Ax k + B 2 u k + B 1 w k w k 2< q is the disturbance input vector. It is random and takes its value in a closed and bounded polyhedral set in < q : w k 2 R[L ] =fw 2< q jlw g Constraints on the state vector: S x x k x 8k 2N Constraints on the control input vector: S u u k u 8k 2N Combined performance requirements: Z s x k + Z u u k : 8k 2N: Target state: (x u ). The problem is formulated as a regulation problem. Under a stabilizing linear state feedback, u k = Fx k, all the constraints and requirements define a polyhedral performance domain: with 2< q, 0. R[Q ] =fx 2< n j Qx :g 34

POSITIVE INVARIANCE WITH DISTURBANCE ATTENUATION Autonomous linear system (S): with w k 2 R[L ]. x k+1 = A 0 x k + B 1 w k Polyhedral set in the state space < n : R[G ] =fx 2< n j Gx :g Positive invariance of R[G ]: x 0 2 R[G ] =) x n 2 R[G ] 8n 2N 8fw k g w k 2 R[L ]: A geometric characterization: P(R[G ]) R[G ] with P(R[G ]) = fy = A 0 x+b 1 w j ( x 2 R[G ] w 2 R[L ]: 35

POSITIVE INVARIANCE ; A geometric characterization R(G,η) P(R(G,η)) P(R[G ]) R[G ] 36

POSITIVE INVARIANCE WITH DISTURBANCE ATTENUATION Positive invariance theorem: A NSC for positive invariance of R[G ] w.r.t. (S) for any disturbance vector w k in R[L ], is the existence of two nonnegative matrices H and M such that : HG = GA 0 ML = GB 1 H + M : This theorem is obtained by application of Farkas Lemma to the inclusion condition P(R[G ]) R[G ]. 37

POSITIVE INVARIANCE OF SYMMETRICAL POLYHEDRA Positive invariance theorem: A NSC for positive invariance of S(; )=fx 2< n j; ;x g: w.r.t. (S) for any disturbance vector w k in S( )=fw 2< q j; w g is the existence of two matrices H and M such that: H; = ;A 0 M = ;B 1 jhj + jmj : 38

REGULATOR DESIGN Spectral assignment methodology: Selection of the closed-loop spectrum : (A + B 2 F ) Closed-loop real Jordan form : J Matrix of generalized real eigenvectors: V such that : JV ;1 = V ;1 (A + BF): V =[V 1 j V 2 ] and W =[W 1 j W 2 ] with for i real in (A + B 2 F ), [ i I ; A ; B] " vi w i # =0 and F = WV ;1 : 39

A SPECTRAL SUFFICIENT CONDITION FOR POSITIVE INVARIANCE WITHOUT DISTURBANCES If matrix J satisfying : J; =;A 0 has the real Jordan form, and its eigenvalues, i + j i verify: then, 9 >0 such that: j i j + j i j1 jjj : And polyhedron S(; ) is positively invariant (in the undisturbed case). 1 I 0 1 R Spectral domain j i j + j i j1 40

SUFFICIENT CONDITIONS FOR POSITIVE INVARIANCE UNDER BOUNDED DISTURBANCES 1. Positive invariance with contractivity. Set ;=V ;1 and A 0 = A + B 2 F. If we impose the tighter conditions : with 0 i 1, then, row by row : j i j + j i j < 1 ; i ; i A 0 = J i ; jj i j (1 ; i ): 2. Disturbance attenuation Condition P(S(; )) S(; ) is obtained through the algebraic conditions : M = ;B 1 jmj diag( l ) 41

THE CONSTRAINED CONTROL SCHEME Construction of a domain such that : 1. The zero state lies in the interior of, 2. is D;invariant with respect to the controlled system, for any disturbance vector w k in R[L ]. 3. R[Q ] x 2 R(Q,ρ) Ω x 1 An admissible domain 42

THE DOMAIN OF ADMISSIBLE INITIAL STATES The size of the invariant domain is maximized through maximization of the components of. The scheme is completed by solving the following Linear Program: subject to : Maximize C = nx i=1 c i i with all c i > 0 jqv j ; i A 0 = J i ; jj i j (1 ; i ) M = ;B 1 jmj diag( l ) 43

APPLICATION TO PRODUCTION PLANNING θ 5 1 5 1 4 θ 4 π 51 π 53 π 43 π 52 3 1 θ 3 π 32 π 31 θ 2 1 1 2 1 θ 1 A Petri net representation of a product structure = 2 6 4 0 0 0 0 0 0 0 0 0 0 31 32 0 0 0 0 0 43 0 0 51 52 53 0 0 3 7 5 : 44

THE PRODUCTION MODEL Stock equation for product i : 8 >< >: y ik = y i k;1 + u i k;i ; N X j=1 ij v jk ; d ik : s ik = y ik + s i if y ik ;s i s ik = 0 if y ik ;s i with a backorder of ; y ik ; s i : Stock equation in vector form : (1 ; q ;1 )y k =[diag(q ; i ) ; ]v k ; d k Decomposition of the demand vector d k = d + w k with E(d k )=d E(w k )=0. Boundedness assumption : ;w w k w with w d: 45

THE PRODUCTION MODEL Steady-state nominal control Property : v 0. v =(I ; ) ;1 d Change of variable u k = v k ; v to obtain : (1 ; q ;1 )y k =[;+T 1 q ;1 + T q ; ]u k ; w k State vector : State equation : x k =[y T k;1 ut k;1 u T k; ]T x k+1 = Ax k + B 1 w k + B 2 u k with A = 2 6 4 I T 1 T O O O I O O........ O O I O 3 7 5 B 1 = 2 6 4 I Ọ. O 3 7 5 B 2 = 2 6 4 T 0 I Ọ. O 3 7 5 : 46

AN INVARIANT CONTROLLER v k = v +[F G 1 G ]x k D-Invariance of a domain S(; ) by choosing : is obtained F = ;(I ; ) ;1 G j = ;(I ; ) ;1 E j for j =1 ::: with, E j == diag(e j i ) with ( e j i =1if j i e j i =0if j> i : S(; ) defines a set of admissible initial states under: S(; ) R[Q ]: 47

THE ADMISSIBLE SET OF INITIAL STATES The positively invariant domain for the closed-loop system, S(; ) is defined by: ;= 2 6 4 I E 1 E O ;(I ; ) O O O ;(I ; )........ O O O ;(I ; ) 3 7 5 =[:::] T with 0: D;invariance of S(; ) is obtained if i max(w i w i ) 8i =1 ::: N: Inclusion S(; ) R[Q ] with maximization of the components i is obtained by Linear Programming. 48

CONCLUSIONS The proposed methodology has been mainly based on spectral assignment. This approach also offers degrees of freedom in the choice of the eigenstructure. They can be used to improve: the robustness of the control scheme minimization of condition number k(v )=kv k 2 kv ;1 k 2 the size of the set of admissible initial states S(; ). Positively invariant controls generally provide local solutions valid only if x 0 2 S(; ): 49

CONCLUSIONS (contd.) A dual-mode control scheme can be constructed if x 0 =2 S(; ): The state is first attracted to S(; ) (in openloop). Then the closed-loop scheme can be applied. The concepts of (A,B)-invariance and D-(A,B)- invariance have been studied to generalize the positive invariance approach to non-linear control laws and resolution of optimization problems such as: maximization of the set of admissible initial states optimal attenuation of bounded persistent disturbances (`1 problem). 50