INTERCONNECTION AND DECOMPOSITION PROBLEMS WITH nd BEHAVIORS
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1 INTERCONNECTION AND DECOMPOSITION PROBLEMS WITH nd BEHAVIORS Diego Napp, Marius van der Put and H.L. Trentelman University of Groningen, The Netherlands INTERCONNECTIONANDDECOMPOSITION PROBLEMS WITH nd BEHAVIORS p.1/13
2 OUTLINE Introduction : Motivation and aim. Modules and behaviors. Regular interconnection, controllable and autonomous Translating system problems to algebraic problems. A useful theorem. Controllable-autonomous decomposition with finite dimensional intersection. Almost regular interconnection. INTERCONNECTIONANDDECOMPOSITION PROBLEMS WITH nd BEHAVIORS p.2/13
3 INTRODUCTION The behavioral approach is an elegant framework for the analysis and design of multidimensional systems. Regular Interconnection problem: Given a plant behavior B and a control objective B d find a controller behavior B c such that the interconnection of B and B c gives B d. If the restrictions of B c are independent of the restrictions of the plant then the interconnection is regular (as in a feedback loop). INTERCONNECTIONANDDECOMPOSITION PROBLEMS WITH nd BEHAVIORS p.3/13
4 INTRODUCTION The behavioral approach is an elegant framework for the analysis and design of multidimensional systems. Regular Interconnection problem: Given a plant behavior B and a control objective B d find a controller behavior B c such that the interconnection of B and B c gives B d. If the restrictions of B c are independent of the restrictions of the plant then the interconnection is regular (as in a feedback loop). Autonomous-controllable Decomposition problem: When is it possible to write a system as the direct sum of its controllable part and an autonomous part? INTERCONNECTIONANDDECOMPOSITION PROBLEMS WITH nd BEHAVIORS p.3/13
5 INTRODUCTION 1D case: Regular Interconnection problem studied by J.C. Willems. Auton-control decomposition problem studied by Kalman. nd case: Regular Intercon. problem studied by P. Rocha, E.Zerz, J. Wood... Auton-control decomposition problem studied by E.Zerz, Valcher... INTERCONNECTIONANDDECOMPOSITION PROBLEMS WITH nd BEHAVIORS p.4/13
6 INTRODUCTION 1D case: Regular Interconnection problem studied by J.C. Willems. Auton-control decomposition problem studied by Kalman. nd case: Regular Intercon. problem studied by P. Rocha, E.Zerz, J. Wood... Auton-control decomposition problem studied by E.Zerz, Valcher... Very strong conditions needed for solving the problems in nd systems! When does there exists B c such that the regular interconnection of B and B c gives almost B d? When can we decompose the system into the almost direct sum of the controllable part and the autonomous part? What does almost mean? INTERCONNECTIONANDDECOMPOSITION PROBLEMS WITH nd BEHAVIORS p.4/13
7 MULTIDIMENSIONAL SYSTEMS The system Σ = (A, q, B) B = Solution set of a system of linear constant-coefficient PDE s. A = C (R n, R) or D (R n, R) (for discrete case A = R Nn ) INTERCONNECTIONANDDECOMPOSITION PROBLEMS WITH nd BEHAVIORS p.5/13
8 MULTIDIMENSIONAL SYSTEMS The system Σ = (A, q, B) B = Solution set of a system of linear constant-coefficient PDE s. A = C (R n, R) or D (R n, R) (for discrete case A = R Nn ) D := R[ξ 1,..., ξ n ] the polynomial ring. We can find R D q such that R( x 1,..., B = {w A q the equation holds} = ker(r) x n )w = 0 INTERCONNECTIONANDDECOMPOSITION PROBLEMS WITH nd BEHAVIORS p.5/13
9 MULTIDIMENSIONAL SYSTEMS The system Σ = (A, q, B) B = Solution set of a system of linear constant-coefficient PDE s. A = C (R n, R) or D (R n, R) (for discrete case A = R Nn ) D := R[ξ 1,..., ξ n ] the polynomial ring. We can find R D q such that R( x 1,..., x n )w = 0 B = {w A q the equation holds} = ker(r) For discrete case take the shift operator σ i instead of x i. INTERCONNECTIONANDDECOMPOSITION PROBLEMS WITH nd BEHAVIORS p.5/13
10 MODULES AND BEHAVIORS B := < R > = module of D q generated by the rows of R Oberst showed that B 1:1 B nd linear systems Modules over D INTERCONNECTIONANDDECOMPOSITION PROBLEMS WITH nd BEHAVIORS p.6/13
11 MODULES AND BEHAVIORS B := < R > = module of D q generated by the rows of R Oberst showed that B 1:1 B nd linear systems Modules over D B 1 B 2 = B 1 B 2 (B 1 + B 2 ) = B 1 B 2 (B 1 B 2 ) = B 1 + B 2 B has finite dimension over R B D q has finite codimension ( i.e. D q /B has finite dimension ). INTERCONNECTIONANDDECOMPOSITION PROBLEMS WITH nd BEHAVIORS p.6/13
12 INTERCONNECTION Interconnection of systems is the basis for control in the behavioral framework and are descrived by interconnections of the corresponding behaviors. Thus B 1 =ker(r 1 ), B 2 =ker(r 2 ) B 1 B 2 =ker ( R 1 R 2 ) INTERCONNECTIONANDDECOMPOSITION PROBLEMS WITH nd BEHAVIORS p.7/13
13 INTERCONNECTION Interconnection of systems is the basis for control in the behavioral framework and are descrived by interconnections of the corresponding behaviors. Thus B 1 =ker(r 1 ), B 2 =ker(r 2 ) B 1 B 2 =ker The interconnection is called regular if rank(r 1 )+rank(r 2 ) =rank ( R 1 R 2 ) ( R 1 R 2 ) or if B 1 B 2 = 0 Expresses the idea of restricting what is not yet restricted. INTERCONNECTIONANDDECOMPOSITION PROBLEMS WITH nd BEHAVIORS p.7/13
14 INTERCONNECTION Interconnection of systems is the basis for control in the behavioral framework and are descrived by interconnections of the corresponding behaviors. Thus B 1 =ker(r 1 ), B 2 =ker(r 2 ) B 1 B 2 =ker The interconnection is called regular if rank(r 1 )+rank(r 2 ) =rank ( R 1 R 2 ) ( R 1 R 2 ) or if B 1 B 2 = 0 Expresses the idea of restricting what is not yet restricted. Given B, if there exists B c s.t. B d = B B c, then B d is said to be implementable from B. And it is called regular implementable if the interconnection is regular. INTERCONNECTIONANDDECOMPOSITION PROBLEMS WITH nd BEHAVIORS p.7/13
15 CONTROLLABILITY A B is said to be controllable if for all w 1,w 2 B and all sets U 1,U 2 R n with disjoint closure, there exist a w B such that w U1 = w 1 U1 and w U2 = w 2 U2. Let N be a D-module. Then n N is called a torsion element if there exists a nonzero d D such that dn = 0. B is controllable iff D q /B is torsion free INTERCONNECTIONANDDECOMPOSITION PROBLEMS WITH nd BEHAVIORS p.8/13
16 CONTROLLABILITY A B is said to be controllable if for all w 1,w 2 B and all sets U 1,U 2 R n with disjoint closure, there exist a w B such that w U1 = w 1 U1 and w U2 = w 2 U2. Let N be a D-module. Then n N is called a torsion element if there exists a nonzero d D such that dn = 0. B is controllable iff D q /B is torsion free A B A is said to be strongly controllable if it is a direct summand of A. A free module is a module having a free basis A B A is said to be strongly controllable iff D q /B is free INTERCONNECTIONANDDECOMPOSITION PROBLEMS WITH nd BEHAVIORS p.8/13
17 CONTROLLABILITY A B is said to be controllable if for all w 1,w 2 B and all sets U 1,U 2 R n with disjoint closure, there exist a w B such that w U1 = w 1 U1 and w U2 = w 2 U2. Let N be a D-module. Then n N is called a torsion element if there exists a nonzero d D such that dn = 0. B is controllable iff D q /B is torsion free A B A is said to be strongly controllable if it is a direct summand of A. A free module is a module having a free basis A B A is said to be strongly controllable iff D q /B is free B is autonomous if D q /B is a torsion module INTERCONNECTIONANDDECOMPOSITION PROBLEMS WITH nd BEHAVIORS p.8/13
18 "Almost" regular implementability problem Almost regular implementability problem: Given B and a desired B d B when does there exists B c s.t. B d B B c finite codimension with regular interconnection? Equivalent to the algebraic problem: Given are N 1 N D q when does there exists N 2 N D q s.t. N 1 + N 2 N finite codimension and N 1 N 2 = 0? Note that N 1 = B, N = B d, N 2 = B c INTERCONNECTIONANDDECOMPOSITION PROBLEMS WITH nd BEHAVIORS p.9/13
19 "Almost" direct summand controllable-autonomous decomposition Almost direct summand control-aut decomposition problem: Given B, is it possible to decompose it as B = B contr +B aut and B contr B aut finite dimension Equivalent to the algebraic problem: Given a D-module M. Let M t be its torsion part, does there exists a D-module A such that M t + A M with finite codimension and M t A = 0 INTERCONNECTIONANDDECOMPOSITION PROBLEMS WITH nd BEHAVIORS p.10/13
20 A useful theorem From now on take D = R[ξ 1, ξ 2 ]. Theorem: Let M be a finitely generated D-module with no torsion. Then there exists a free D-module M + such that M M + and M + /M has finite dimension. INTERCONNECTIONANDDECOMPOSITION PROBLEMS WITH nd BEHAVIORS p.11/13
21 A useful theorem From now on take D = R[ξ 1, ξ 2 ]. Theorem: Let M be a finitely generated D-module with no torsion. Then there exists a free D-module M + such that M M + and M + /M has finite dimension. Corollary: Given a controllable 2D behavior B A q there exists a strongly controllable sub-behavior B B INTERCONNECTIONANDDECOMPOSITION PROBLEMS WITH nd BEHAVIORS p.11/13
22 Solution to the "almost" regular implementability problem Given B and a desired B d B there exists B c s.t. B d B B c finite codimension with regular interconnection Given are N 1 N D q there exists N 2 N D q s.t. N 1 + N 2 N finite codimension and N 1 N 2 = 0 Given are N 1 N D q, take N 2 N + 2 and N N + finite codimension then N + 1 of N + finite codimension is direct summand Remember that N 1 = B, N = B d Direct summands can be computed!! INTERCONNECTIONANDDECOMPOSITION PROBLEMS WITH nd BEHAVIORS p.12/13
23 Solution to "almost" direct summand control-aut decomposition Given B, is it possible to decompose it as B = B contr +B aut and B contr B aut finite dimension Given a D-module M. Let M t be its torsion part, there exists a D-module A such that M t + A M with finite codimension and M t A = 0 Solution: Yes, it is possible of n=2. Example: Take n = 3 i.e. D = R[ξ 1, ξ 2, ξ 3 ] then ) B = ker ( x 1 x 2 2 x 2 1 of its controllable part and an autonomous part. is not an almost direct summand INTERCONNECTIONANDDECOMPOSITION PROBLEMS WITH nd BEHAVIORS p.13/13
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