The infinite horizon positive LQ-problem for linear continuous time - PowerPoint PPT Presentation

The infinite horizon positive LQ-problem for linear continuous time systems Charlotte Beauthier 1 Joint work with M. Laabissi 2 and J. J. Winkin 1 1 University of Namur (FUNDP) – Belgium 2 University Chouaib Doukkali – Morocco CESAME Seminar UCL, Louvain-La-Neuve May 23, 2006 Ch. Beauthier The positive LQ-problem for linear continuous time systems

Motivations Problem Statement Stable Case Unstable Case Conclusion and future work Outline Motivations 1 Problem Statement 2 Preliminary Concepts Problem Stable Case 3 Main Results Compartmental systems Design Methodology for Q and R Unstable Case 4 Conclusion and future work 5 Ch. Beauthier The positive LQ-problem for linear continuous time systems

Motivations Problem Statement Stable Case Unstable Case Conclusion and future work Outline Motivations 1 Problem Statement 2 Stable Case 3 Unstable Case 4 Conclusion and future work 5 Ch. Beauthier The positive LQ-problem for linear continuous time systems

Motivations Problem Statement Stable Case Unstable Case Conclusion and future work Motivations ◮ The class of positive linear time-invariant systems → Dynamical models where all the variables should remain nonnegative. → Large class of applications : see [Farina, Rinaldi, 2000] Economics models Chemical processes Compartmental systems Biological systems L. Farina, S. Rinaldi Positive Linear Systems : Theory and applications Wiley, New York, 2000. Ch. Beauthier The positive LQ-problem for linear continuous time systems

Motivations Problem Statement Stable Case Unstable Case Conclusion and future work Motivations ◮ Feedback control law for a positive system → A IM : Keeping positivity for the closed-loop system. � Stabilizability and holdability problem. Ch. Beauthier The positive LQ-problem for linear continuous time systems

Motivations Problem Statement Stable Case Unstable Case Conclusion and future work Motivations A. Berman, M. Neumann, R. J. Stern Nonnegative matrices in dynamic systems John Wiley and Sons, 1989. → Existence of state feedback. S. Boyd, L. El Ghaoui, E. Feron, V. Balakrishnan Linear matrix inequalities in system and control theory Society for industrial and applied mathematicals, Philadelphia, 1994. → Compute of state feedback � LMI. Ch. Beauthier The positive LQ-problem for linear continuous time systems

Motivations Problem Statement Stable Case Unstable Case Conclusion and future work Motivations ◮ Feedback of LQ type → P OSITIVE LQ- PROBLEM : Given a positive system, ? Conditions ? such that the resulting LQ-optimal closed-loop system is positive ? Design methodology for choosing the weighting matrices in the quadratic cost ? Ch. Beauthier The positive LQ-problem for linear continuous time systems

Motivations Problem Statement Stable Case Unstable Case Conclusion and future work Motivations Positive optimal control ◮ W. P . M. H. Heemels, S. J. L. Van Eijndhoven, A. A. Stoorvogel Linear quadratic regulator problem with positive controls Int. J. Control, Vol. 70, No. 4, 551-578, 1998. Ch. Beauthier The positive LQ-problem for linear continuous time systems

Motivations Problem Statement Stable Case Unstable Case Conclusion and future work Motivations Iterative scheme for the solution of the Riccati equation ◮ and unsigned matrices. Chun-hua Guo and A. J. Laub On a Newton-like Method for Solving Algebraic Riccati Equations SIAM J. Matrix Anal. Appl., Vol. 21, No. 2, 2000. Chun-hua Guo and A. J. Laub On the iterative solution of a class of nonsymmetric algebraic Riccati equations SIAM J. Matrix Anal. Appl., Vol. 22, No. 2, 2000. Ch. Beauthier The positive LQ-problem for linear continuous time systems

Motivations Problem Statement Stable Case Unstable Case Conclusion and future work Motivations Nash Riccati equation and positive games theory. ◮ G. Jank, D. Kremer Open loop Nash games and positive systems - Solvability conditions for non symmetric Riccati equations Lehrstuhl II für Mathematik, RWTH Aachen, Germany, May 2004. Ch. Beauthier The positive LQ-problem for linear continuous time systems

Motivations Problem Statement Preliminary Concepts Stable Case Problem Unstable Case Conclusion and future work Outline Motivations 1 Problem Statement 2 Preliminary Concepts Problem 3 Stable Case Unstable Case 4 Conclusion and future work 5 Ch. Beauthier The positive LQ-problem for linear continuous time systems

Motivations Problem Statement Preliminary Concepts Stable Case Problem Unstable Case Conclusion and future work Preliminary Concepts - Positive Systems Consider the following linear time-invariant system description, denoted by R = [ A , B ] : � ˙ x ( t ) = A x ( t ) + B u ( t ) , x ( 0 ) = x 0 ◮ Definitions of positive system • A linear system R = [ A , B ] is said to be positive if ∀ x 0 ≥ 0 , ∀ u ( t ) ≥ 0 , ∀ t ≥ 0 : x ( t ) ≥ 0 • A linear system ˙ x = A x ( t ) is said to be positive if ∀ x 0 ≥ 0 , ∀ t ≥ 0 : x ( t ) ≥ 0 . Ch. Beauthier The positive LQ-problem for linear continuous time systems

Motivations Problem Statement Preliminary Concepts Stable Case Problem Unstable Case Conclusion and future work Preliminary Concepts - Positive Systems ◮ A well-known characterization of positive systems T HM : A linear system R = [ A , B ] is positive ⇔ A is a Metzler matrix and B ≥ 0 . where • A is a Metzler matrix ⇔ ∀ i � = j : a ij ≥ 0 . • B is a nonnegative matrix , denoted by B ≥ 0, ⇔ ∀ i , j : b ij ≥ 0 . Ch. Beauthier The positive LQ-problem for linear continuous time systems

Motivations Problem Statement Preliminary Concepts Stable Case Problem Unstable Case Conclusion and future work Preliminary Concepts - M-Matrix • A is a Z-matrix if ∀ i � = j : a ij ≤ 0 , → − A is a Metzler matrix. → A is a Z-matrix ⇔ ∃ s ∈ IR , B ≥ 0 such that A = s I − B . See e.g. [Bapat, Raghavan, 1997]. R. B. Bapat, T. E. S. Raghavan Nonnegative matrices and applications Encyclopedia of mathematics and its applications 64, Cambridge university press, 1997. Ch. Beauthier The positive LQ-problem for linear continuous time systems

Motivations Problem Statement Preliminary Concepts Stable Case Problem Unstable Case Conclusion and future work Preliminary Concepts - M-Matrix • A is a M-matrix if ∃ s ≥ ρ ( B ) , B ≥ 0 such that A = s I − B . • A is a nonsingular M-matrix if ∃ s > ρ ( B ) , B ≥ 0 such that A = s I − B . Ch. Beauthier The positive LQ-problem for linear continuous time systems

Motivations Problem Statement Preliminary Concepts Stable Case Problem Unstable Case Conclusion and future work Preliminary Concepts - M-Matrix Theorem Let A a Z-matrix, then the following assertions are equivalent : A is nonsingular and A − 1 ≥ 0 . 1 There exists a vector x ≥ 0 such that A x >> 0 . 2 There exists a vector x >> 0 such that A x >> 0 . 3 All eigenvalues of A have positive real parts. 4 A is a nonsingular M-matrix. 5 Ch. Beauthier The positive LQ-problem for linear continuous time systems

Motivations Problem Statement Preliminary Concepts Stable Case Problem Unstable Case Conclusion and future work Preliminary Concepts - Kronecker product The Kronecker product of two matrices A = ( a ij ) ∈ IR m × n and B = ( b ij ) ∈ IR p × q is the matrix A ⊗ B given by :   a 11 B · · · a 1 n B . .  ∈ IR mp × nq . . A ⊗ B :=   . · · · .  a m 1 B · · · a mn B For each matrix A = ( a ij ) ∈ IR m × n , we associate the vector vect ( A ) ∈ IR mn defined by : vect ( A ) := [ a 11 , . . . , a m 1 , . . . , a 1 n , . . . , a mn ] T . Ch. Beauthier The positive LQ-problem for linear continuous time systems

Motivations Problem Statement Preliminary Concepts Stable Case Problem Unstable Case Conclusion and future work Preliminary Concepts - Kronecker product → Rewriting of some equations e.g. the Sylvester equation : A X + X B = C ⇔ [ I ⊗ A + B T ⊗ I ] vect ( X ) = vect ( C ) . → If A and B are M-matrices Then [ I ⊗ A + B T ⊗ I ] is also a M-matrix. � σ ([ I ⊗ A + B T ⊗ I ]) = σ ( A ) + σ ( B ) P . Lancaster and M. Tismenetsky The theory of matrices 2nd ed., Academic Press, Orlando, Floride, 1985. Ch. Beauthier The positive LQ-problem for linear continuous time systems

Motivations Problem Statement Preliminary Concepts Stable Case Problem Unstable Case Conclusion and future work Problem Statement Given a linear time-invariant system [ A , B ] . Minimize � ∞ [ x ( t ) T Q x ( t ) + u ( t ) T R u ( t )] dt , J ( x 0 , u ) = 0 with R ∈ IR m × m positive definite (pd) symmetric matrix Q ∈ R n × n positive semidefinite (psd) symmetric matrix Ch. Beauthier The positive LQ-problem for linear continuous time systems

Motivations Problem Statement Preliminary Concepts Stable Case Problem Unstable Case Conclusion and future work Well-known results If  ( A , B ) is (exponentially) stabilizable  and ( Q , A ) is (exponentially) detectable  Then the algebraic Riccati equation (ARE) A T X + X A − X B R − 1 B T X = − Q has a unique stabilizing psd solution X . Moreover : → Optimal control : state feedback type : u opt ( t ) = K opt x ( t ) = − R − 1 B T Xx ( t ) → Optimal cost : J ( x 0 , u opt ) = x T 0 X x 0 . Ch. Beauthier The positive LQ-problem for linear continuous time systems