Regional Input-to-State Stabilization of Discrete-Time Nonlinear - PowerPoint PPT Presentation

Regional Input-to-State Stabilization of Discrete-Time Nonlinear Quadratic Systems Carlos E. de Souza Department of Systems and Control National Laboratory for Scienti fi c Computing (LNCC/MCTI) Petr´ opolis, RJ, Brazil Seminar presented at Universit´ e Catholique de Louvain (UCL) September 25, 2014 Regional Input-to-State Stabilization of Discrete-Time Nonlinear Quadratic Systems – p. 1

Focus of this Seminar Deals with the problem of regional stabilization for, possibly open-loop unstable, discrete-time nonlinear quadratic systems subject to persistent magnitude bounded disturbances. The class of quadratic systems is an extension of bilinear systems, where quadratic terms in the state variables are also considered. Nonlinear quadratic systems can precisely represent a large class of systems, such as, distillation columns, induction motors, heating and air conditioning systems, and Lokta-Volterra type predator-prey systems. Regional Input-to-State Stabilization of Discrete-Time Nonlinear Quadratic Systems – p. 2

Quadratic systems also fi nd application as a quadratic approxi- mation around an operation point (via a Taylor series expansion) of a general nonlinear system. We address the design of a static nonlinear state feedback controller achieving local input-to-state stability with a guaranteed bounding set for the system state trajectories under nonzero initial conditions and persistent magnitude bounded disturbances. The control gain can be quadratic in the state variables (which are potentially less conservative than linear controllers), and the design is in terms of numerically tractable linear matrix inequality (LMI) problems (state-dependent LMIs). Regional Input-to-State Stabilization of Discrete-Time Nonlinear Quadratic Systems – p. 3

Motivation for Input-to-State Stability (ISS) ISS (Sontag, TAC 1989) is a stability notion that characterizes the input/output (i/o) and internal behavior of nonlinear systems subject to a disturbance input. The notion of ISS encompasses the paradigms of i/o and state-space stability. ISS describes the dependence of the size of the system state trajectory on the magnitude of the initial state and disturbance input. Regional Input-to-State Stabilization of Discrete-Time Nonlinear Quadratic Systems – p. 4

Linear Matrix Inequalities (LMIs) • Standard form: F ( x ) = F 0 + x 1 F 1 + · · · + x p F p > 0 – x = [ x 1 · · · x p ] ′ ∈ R p is the variable to be determined; – F i ’s are given symmetric matrices; – The inequality means that the matrix F ( x ) is positive de fi nite. • Matrices as variables: e.g., consider the matrix inequality: A ′ P + PA < 0 . P = P ′ is the variable. A is a given matrix; The previous inequality can be written as an LMI in the entries of P . • LMIs can be numerically ef fi ciently solved via polynomial-time algorithms. Regional Input-to-State Stabilization of Discrete-Time Nonlinear Quadratic Systems – p. 5

OUTLINE Problem Formulation Features of the Control Method Controller Design An Example Concluding Remarks Regional Input-to-State Stabilization of Discrete-Time Nonlinear Quadratic Systems – p. 6

PROBLEM FORMULATION Consider the discrete-time nonlinear quadratic system S : x + = A ( x ) x ( k ) + B ( x ) u ( k ) + E ( x ) d ( k ) , k ∈ Z + , x + := x ( k +1) , x (0) = x 0 , x ∈ X ⊂ R n • u ( k ) ∈ R n u : control input; d ( k ) ∈ D ⊂ R n w : disturbance input; • X : polytopic region of the state-space containing the origin x =0 ; • D : set of admissible disturbances D = { s ∈ R n w : ∥ s ∥ ≤ δ } δ : a given positive scalar de fi ning the “size" of D . Regional Input-to-State Stabilization of Discrete-Time Nonlinear Quadratic Systems – p. 7

• A ( x ) , B ( x ) , E ( x ) : af fi ne matrix functions of x = [ x 1 · · · x n ] ′ : n n n � � � A ( x )= A 0 + B ( x )= B 0 + E ( x )= E 0 + x i A i , x i B i , x i E i i =1 i =1 i =1 A i , B i , E i , i = 0 , 1 , . . . , n : given constant real matrices. Admissible Stabilizing Controllers We consider static nonlinear state feedback controllers K : R n �→ R n u u ( x ) = K ( x ) x, K ( x ) = K 1 ( x ) + K 2 ( x ) K 1 ( x ) = K 0 + � n K 2 ( x ): quadratic in x i =1 x i K i , Regional Input-to-State Stabilization of Discrete-Time Nonlinear Quadratic Systems – p. 8

Regional Stabilization Problem For a given state-space region X , and sets D and R 0 of admissible initial states satisfying R 0 ⊂ X (given, or to be found), determine a controller K such that: the closed-loop system is locally input-to-state stable; and the state trajectory x ( k ) driven by any x 0 ∈ R 0 and d ( k ) ∈ D stays con fi ned to some state-space region R containing the equilibrium point x = 0 and satisfying R 0 ⊆ R ⊂ X . – The set R is referred to as trajectories bounding set . – In the absence of d , R is a stability region for the closed-loop system, i.e., a set of initial states such that x ( k ) → 0 as k → ∞ . Regional Input-to-State Stabilization of Discrete-Time Nonlinear Quadratic Systems – p. 9

Input-to-State Stability Consider the nonlinear system S 0 x + = f ( x, ν ) , x ∈ R n , ν ∈ R n ν , x (0) = x 0 , f (0 , 0) = 0 De fi nition . System S 0 is input-to-state stable if there exist positive scalars ρ 1 and ρ 2 , a KL -function α 1 ( · , · ) , and a K -function α 2 ( · ) s.t. ∥ x ( k, x 0 , ν ) ∥ ≤ α 1 ( ∥ x 0 ∥ , k ) + α 2 ( ∥ ν ∥ ∞ ) , ∀ k ∈ Z + , ∀ x 0 : ∥ x 0 ∥ ≤ ρ 1 , ν : ∥ ν ∥ ∞ ≤ ρ 2

Input-to-State Stability Consider the nonlinear system S 0 x + = f ( x, ν ) , x ∈ R n , ν ∈ R n ν , x (0) = x 0 , f (0 , 0) = 0 De fi nition . System S 0 is input-to-state stable if there exist positive scalars ρ 1 and ρ 2 , a KL -function α 1 ( · , · ) , and a K -function α 2 ( · ) s.t. ∥ x ( k, x 0 , ν ) ∥ ≤ α 1 ( ∥ x 0 ∥ , k ) + α 2 ( ∥ ν ∥ ∞ ) , ∀ k ∈ Z + , ∀ x 0 : ∥ x 0 ∥ ≤ ρ 1 , ν : ∥ ν ∥ ∞ ≤ ρ 2 ∥ ν ∥ ∞ := sup k ∈ Z + ∥ ν ( k ) ∥ ; K -function: positive de fi nite and strictly increasing; KL -function β ( s, t ) : (i) β ( · , t ) is a K -function for each t ≥ 0 ; (ii) decreases to 0 as t → ∞ for each s ≥ 0 .

Input-to-State Stability Consider the nonlinear system S 0 x + = f ( x, ν ) , x ∈ R n , ν ∈ R n ν , x (0) = x 0 , f (0 , 0) = 0 De fi nition . System S 0 is input-to-state stable if there exist positive scalars ρ 1 and ρ 2 , a KL -function α 1 ( · , · ) , and a K -function α 2 ( · ) s.t. ∥ x ( k, x 0 , ν ) ∥ ≤ α 1 ( ∥ x 0 ∥ , k ) + α 2 ( ∥ ν ∥ ∞ ) , ∀ k ∈ Z + , ∀ x 0 : ∥ x 0 ∥ ≤ ρ 1 , ν : ∥ ν ∥ ∞ ≤ ρ 2 • The equilibrium x = 0 is locally asymptotically stable. • x ( t ) is bounded and x ( k ) → 0 if ν ( k ) → 0 as k → ∞ . • ISS generalizes the bound ∥ x ( k ) ∥ ≤ c 1 ∥ x 0 ∥ α k + c 2 ∥ ν ∥ ∞ , which holds for asymptotically stable linear systems x + = Ax + B ν . Regional Input-to-State Stabilization of Discrete-Time Nonlinear Quadratic Systems – p. 10

Key Lemma (Jiang and Wang, Automatica, 2005) Consider the nonlinear system x + = f ( x, ν ) , f (0 , 0) = 0 , x ∈ R n , ν ∈ U ⊂ R n ν Let V : Λ �→ R + be continuous and b 1 , . . . , b 4 positive scalars s.t. b 1 ∥ x ∥ 2 ≤ V ( x ) ≤ b 2 ∥ x ∥ 2 , ∀ x ∈ Λ ⊂ R n ; ∆ V ( x ):= V ( x + ) − V ( x ) ≤ − b 3 ∥ x ∥ 2 + b 4 ∥ ν ∥ 2 , ∀ x ∈ Λ , ν ∈ U Then, the system is locally input-to-state stable. ( V ( x ) : ISS-Lyapunov function ) Regional Input-to-State Stabilization of Discrete-Time Nonlinear Quadratic Systems – p. 11

FEATURES OF THE CONTROL METHOD Scaled System Representation x + = A ( x ) x ( k ) + B ( x ) u ( k ) + δ E ( x ) w ( k ) , � � s ∈ R n w : ∥ s ∥ ≤ 1 w ( k ) ∈ W , ∀ k ∈ Z + , W = Decomposition of B ( x ) � � � � B 0 I n B ( x ) = Ψ ( x ) ′ Ψ ( x )= , , Π ( x ) B � � ′ , � � ′ Π ( x ) := x ⊗ I n = x 1 I n · · · x n I n B = 1 · · · B ′ B ′ n Regional Input-to-State Stabilization of Discrete-Time Nonlinear Quadratic Systems – p. 12

Polytopic Region X X is assumed to be a symmetric polytope with respect to the state-space origin. • Intersection of half-spaces representation X = { x ∈ R n : | c ′ i x | ≤ 1 , i = 1 , . . . , r } , c i ∈ R n , i = 1 , . . . , r : de fi ne the 2 r faces of X • Convex hull representation X = Co { v 1 , v 2 , . . . , v κ } v i ∈ R n , i = 1 , . . . , κ : vertices of X Regional Input-to-State Stabilization of Discrete-Time Nonlinear Quadratic Systems – p. 13

ISS-Lyapunov Function Candidate P = P ′ > 0 V ( x ) = x ′ Px, Input-to-State Stability Condition In order to have the condition ∆ V ( x ) ≤ − b 3 ∥ x ∥ 2 + b 4 ∥ w ∥ 2 of the Key Lemma satis fi ed, we impose the following condition: � � ∥ w ∥ 2 − V ( x ) , ∀ x ∈ X , w ∈ W ∆ V ( x ) ≤ β β : positive scalar to be found. Regional Input-to-State Stabilization of Discrete-Time Nonlinear Quadratic Systems – p. 14

Set R 0 of Admissible Initial States � � x ∈ R n : x ′ P 0 x ≤ 1 R 0 := P 0 = P ′ 0 > 0 such that R 0 ⊂ X . Trajectories Bounding Set R De fi ned by a normalized level set of the Lyapunov function V ( x ) : R := { x ∈ R n : x ′ P x ≤ 1 } and such that R 0 ⊆ R ⊂ X . Regional Input-to-State Stabilization of Discrete-Time Nonlinear Quadratic Systems – p. 15