Output Stabilization of Time-Varying Input Delay System using Interval Observer Technique

Output Stabilization of Time-Varying Input Delay System using Interval Observer Technique Andrey Polyakov a, Denis Efimov a, Wilfrid Perruquetti a,b and Jean-Pierre Richard a,b a - NON-A, INRIA Lille Nord-Europe b - LAGIS CNRS, Ecole Centrale de Lille (NON-A & LAGIS) Interval Predictor 1 / 22

Outline 1 Introduction (NON-A & LAGIS) Interval Predictor 2 / 22

Outline 1 Introduction 2 Problem statement (NON-A & LAGIS) Interval Predictor 2 / 22

Outline 1 Introduction 2 Problem statement 3 Interval observer and interval predictor (NON-A & LAGIS) Interval Predictor 2 / 22

Outline 1 Introduction 2 Problem statement 3 Interval observer and interval predictor 4 Control design (NON-A & LAGIS) Interval Predictor 2 / 22

Outline 1 Introduction 2 Problem statement 3 Interval observer and interval predictor 4 Control design 5 Examples (NON-A & LAGIS) Interval Predictor 2 / 22

Motivation (NON-A & LAGIS) Interval Predictor 3 / 22

Motivation I. Output-based control is usual for practice; challenging problem in many cases. (NON-A & LAGIS) Interval Predictor 3 / 22

Motivation I. Output-based control is usual for practice; challenging problem in many cases. II. Input Delay arises in models of a control system due to a physical nature of a system (transport delays, computational delay, etc); artificially, for example, in order to model a sampling effect. (NON-A & LAGIS) Interval Predictor 3 / 22

Existing results (NON-A & LAGIS) Interval Predictor 4 / 22

Existing results I. Stabilizing output-based control is designed for linear system with known and constant input delay (Olbrot 1972, Watanabe 1986); (NON-A & LAGIS) Interval Predictor 4 / 22

Existing results I. Stabilizing output-based control is designed for linear system with known and constant input delay (Olbrot 1972, Watanabe 1986); with known and time-varying input delay (Artstein 1982; Witrant, Canudas-de-Wit, Georges & Alamir 2007); with unknown and constant input delay the possible ways are: finite-time delay identification (Belkoura, Richard & Fliess 2009); adaptive control approach (Bresch-Pierti, Krstic 2009). II. For linear system with an unknown time varying input delay are designed the full state feedback control (Fridman, Seuret & Richard 2004); observer for systems with unknown time varying input and state delays (Seuret, Floquet, Richard, Spurgeon 2007). (NON-A & LAGIS) Interval Predictor 4 / 22

System description Consider the input delay control system of the form ẋ = Ax + Bu(t h(t)), y = Cx, (1) where x R n is the system state, u R m is the vector of control inputs, y R k is the measured output, A R n n, B R n m and C R k n are known matrices and input delay h(t) is assumed to be unknown but within the bounded interval: 0 h h(t) h, (2) where the minimum delay h and the maximum delay h are known. (NON-A & LAGIS) Interval Predictor 5 / 22

Basic Assumptions Assumption (1) The set Ω R n of admissible initial conditions x 0 Ω of the system (1) is assumed to be bounded and known. (NON-A & LAGIS) Interval Predictor 6 / 22

Basic Assumptions Assumption (1) The set Ω R n of admissible initial conditions x 0 Ω of the system (1) is assumed to be bounded and known. Assumption (2) The pair (A, B) is controllable and the pair (A, C ) is observable. Assumption (3) The information on the control signal u(t) on the time interval [t h, t) can be stored and used for control design. (NON-A & LAGIS) Interval Predictor 6 / 22

Problem statement Problem The main objective of this research is to design a control algorithm for exponential stabilization of the system (1), i.e. for some numbers c, r > 0 any solution of the closed-loop system (1) has to satisfy the inequality where x(0) Ω. x(t) ce rt, t > 0, (NON-A & LAGIS) Interval Predictor 7 / 22

Interval observation (Gouze, Rapaport & Hadj-Sadok 2000) x(t) = Ã x(t) + f (t), y = C x where x R n, y R k, Ã R n n, C R k n, and f : R R n : f (t) f (t) f (t), where f (t) and f (t) are known. (NON-A & LAGIS) Interval Predictor 8 / 22

Interval observation (Gouze, Rapaport & Hadj-Sadok 2000) x(t) = Ã x(t) + f (t), y = C x where x R n, y R k, Ã R n n, C R k n, and f : R R n : where f (t) and f (t) are known. f (t) f (t) f (t), ẋ(t) = Ãx(t) + f (t) + L( Cx(t) y(t)), ẋ(t) = Ãx(t) + f (t) + L( Cx(t) y(t)), x(0) x(0) x(0) and Ã + L C Hurwitz and Metzler matrix. (NON-A & LAGIS) Interval Predictor 8 / 22

Illustration of the interval observation (NON-A & LAGIS) Interval Predictor 9 / 22

Interval observer for Time-varying Input Delay System Lemma Under Assumptions 2-3 there always exist matrices L R n k and S R n n, det(s) = 0 such that S 1 (A + LC )S Hurwitz and Metzler, (4) and the interval observer of the form ẋ(t) = Ãx(t) + min s [t h,t h] Bu(s) + L( Cx(t) y(t)), ẋ(t) = Ãx(t) + max s [t h,t h] Bu(s) + L( Cx(t) y(t)), x(0) x(0) x(0), Ã = S 1 AS, B = S 1 B, L = S 1 L, C = CS, x = S 1 x, (5) guarantees x(t) x(t) x(t) t > 0, (6) and x(t) x(t), x(t) x(t) if u(t) 0 for t +. (NON-A & LAGIS) Interval Predictor 10 / 22

Some remarks (NON-A & LAGIS) Interval Predictor 11 / 22

Some remarks I. Let Hurwitz and Metzler matrix R be given and we need to find S, L: S 1 (A + LC )S = R. Denote X = S 1 and Y = S 1 L. Then XA + YC = RX. (7) If the matrix R has disjoint spectrum and the pair (A, C ) is observable then the equation (7) has a solution. II. The condition x(0) x(0) x(0) may be guaranteed, since the set of admissible initial conditions Ω is assumed to be known and bounded. For example, if Ω = {x R n : x T Px < 1}, P 0, then x T S T PS x < 1 and x i (0) = x i (0) = 1/λ min (S T PS), i = 1, 2,..., n. (NON-A & LAGIS) Interval Predictor 11 / 22

Interval predictor let us introduce the following predictor variables: 0 z(t) = eãh x(t) + e Aθ min Bu(s)d θ, (8) h s [t+θ h+h,t+θ] 0 z(t) = eãh x(t) + e Aθ max Bu(s)d θ, (9) h s [t+θ h+h,t+θ] which are correctly defined due to Assumption 3. (NON-A & LAGIS) Interval Predictor 12 / 22

Control design Let us define the control in the form u(t) = Kz(t), z(t) = z(t) + z(t), (10) 2 where K R m n. (NON-A & LAGIS) Interval Predictor 13 / 22

Control design Let us define the control in the form u(t) = Kz(t), z(t) = z(t) + z(t), (10) 2 where K R m n. Remark Let us mention that the control function can be selected in a more general form u(t) = K z(t) + Kz(t), where K, K R m n. (NON-A & LAGIS) Interval Predictor 13 / 22

Main Result Theorem If for some given α, β, γ R + the matrices X, Z, R i, S i R n n, i = 1, 2,..., 2n and the matrix Y R m n satisfy the following LMI system ( ) We W ez 0, X 0, Z 0, R i 0, S i 0, (11) W T ez W z then the system (1) together with the control (10) for K = YX 1 is exponentially stable with the convergence rate : r min{α, β, γ}. (NON-A & LAGIS) Interval Predictor 14 / 22

W z = W e = Π 1 B 1 Y... B 2n Y Y T B 1 T e β h S 1... 0............ Y T B 2n T...... e β h S 2n, Π 2 X Ã T + Y T B T B 1 Y... B 2n Y ÃX + BY Π 3 B 1 Y... B 2n Y Y T B 1 T Y T B 1 T e α h R 1... 0............... Y T B 2n T Y T B 2n T 0... e α h R 2n ( Z C W ez = T L T eãt h [ ] ) I n I n 0, 0 0 Π 1 = (Ã + L C )Z + Z (Ã + L C ) T + βz, Π 2 = ÃX + BY + X Ã T + Y T B T + αx, Π 3 = 1 4 2n i=1(r i + e γh S i ) 2 hx, h := h h, where B i R n m, i = 1, 2,..., n is such that i-th row of B i coincides with i-th row of B but all other rows of B i are zero; B n+i = B i, i = 1, 2..., n. For sufficiently small h the LMI is feasible. (NON-A & LAGIS) Interval Predictor 15 / 22

Double integrator A = ( 0 1 0 0 ) ( 0, B = 1 ), C = ( 1 0 ) (NON-A & LAGIS) Interval Predictor 16 / 22

Double integrator A = ( 0 1 0 0 ) ( 0, B = 1 ), C = ( 1 0 ) Solving the Silvester s equation (7) for ( ) 3.0000 2.3200 R = 0.2700 0.4100 we derive and Ã= S = ( 0.4346 2.5663-0.0736-0.4346 ( 3.6234 0.2599 1.5558 9.1860 ) ( 0.0079, B= -0.1102 ) ( 0.9479, L = 0.0948 ) ), C= ( -3.6234-0.2599 ). (NON-A & LAGIS) Interval Predictor 16 / 22

Figure: Controlled double integrator: Interval Predictor-based Feedback. (NON-A & LAGIS) Interval Predictor 17 / 22

Figure: Controlled double integrator: Interval Predictor-based Feedback. Figure: Controlled double integrator: Results of Choi & Lim 2010. (NON-A & LAGIS) Interval Predictor 17 / 22

Linear oscillator A = ( 0 1 1 0 ) ( 0, B = 1 ), C = ( 1 0 ). Solving the Silvester s equation (7) for ( ) 44.5200 46.0040 R = 4.4520 14.8400 leads to ( S = 10 3 0.0592 0.0958 0.4526 1.5394 ( 14.6857 49.7339 Ã = 4.3566 14.6857 ) ( 0.9994, L = 0.0016 ), B = 10 3 ( 2.0026 1.2384 C = ( 59.2393 95.7922 ). Finally, using Sedumi-1.3 for MATLAB we solve LMI system (11) for α = β = γ = 0.2, h = 1, h = 2 and obtain K = ( 137.1771 469.0443 ). (NON-A & LAGIS) Interval Predictor 18 / 22 ). ),

Figure: Linear oscillator h(t) = 1 + 0.5(1 sign(cos(0.5t))), x(0) = (0, 1) T and v(t) = 0. (NON-A & LAGIS) Interval Predictor 19 / 22

Unstable system A = ( 0 1 0.1 0.2 ) ( 0, B = 1 ), C = ( 1 0 ), h = 1, and h = 2. h(t) = 1.5 + 0.5 sin(t), x(0) = (1, 0) T and v(t) = 0. (NON-A & LAGIS) Interval Predictor 20 / 22

Summary (NON-A & LAGIS) Interval Predictor 21 / 22

Summary The interval observer and interval predictor for linear systems with a unknown time-varying input delay are introduced. (NON-A & LAGIS) Interval Predictor 21 / 22

Summary The interval observer and interval predictor for linear systems with a unknown time-varying input delay are introduced. The interval predictor-based output control algorithm is presented. Control problems with another delays (output or/and state) and system uncertainties can be also tackled by this technique. (NON-A & LAGIS) Interval Predictor 21 / 22

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