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Observer-based Control for Linear Sampled-DataSystems: An Impulsive System Approach

Héctor Ríos, Laurentiu Hetel, Denis Efimov

To cite this version:Héctor Ríos, Laurentiu Hetel, Denis Efimov. Observer-based Control for Linear Sampled-Data Sys-tems: An Impulsive System Approach. Proc. 55th IEEE Conference on Decision and Control (CDC),Dec 2016, Las Vegas, United States. �hal-01371272�

Observer-based Control for Linear Sampled-DataSystems: An Impulsive System Approach

Héctor Ríos†, Laurentiu Hetel‡ and Denis Efimov§∗

Abstract—This paper deals with the sampled-data controlproblem based on state estimation for linear sampled-datasystems. An impulsive system approach is proposed based ona vector Lyapunov function method. Observer-based controldesign conditions are expressed in terms of LMIs. Someexamples illustrate the feasibility of the proposed approach.

Index Terms—Observer-based Control, Sampled-Data andImpulsive systems.

I. INTRODUCTION

IN the last decades, an enormous interest has appearedin the design of controllers and observers for continu-

ous and/or discrete dynamical systems with communicationconstraints. This interest has its motivations in systems withsampled-data control, quantization and more generally, innetworked control systems. However, all the communica-tions constraints, i.e. delays, sampling intervals, quantization,packet dropouts, and so on (for details, see [13]); implyadditional difficulties in the analysis and design comparedto the classical control systems. Regarding the observerdesign problem, one of the main issues is the scheduling:only a subset of sensors is allowed to send their datato the observer at the transmission instants. The sporadicand partial availability of system measurements requires thedevelopment of appropriate observer designs. Moreover, forcontroller design, it would be unreasonable to assume thatall states are measurable. Therefore an observer-based controlapproach is needed.

In this paper the observer-based control problem will befocused on sampled-data systems. Several methods havebeen developed to study sampled-data systems, e.g. theInput/Output stability approach [12], the discrete-time ap-proach [11], but two approaches stand out: the input delay

†Electrical & Computer Engineering, University ofCalifornia, Santa Barbara, CA 93106-9560, USA. Email:hector_riosbarajas@ece.ucsb.edu‡CRIStAL (UMR-CNRS 9189), Ecole Centrale de Lille, BP

48, Cité Scientifique, 59651 Villeneuve-d’Ascq, France. Email:laurentiu.hetel@ec-lille.fr§Non-A team @ Inria, Parc Scientifique de la Haute Borne, 40 avenue

Halley, 59650 Villeneuve d’Ascq, France; and CRIStAL (UMR-CNRS9189), Ecole Centrale de Lille, BP 48, Cité Scientifique, 59651 Villeneuve-d’Ascq, France. Emails: denis.efimov@inria.fr∗Department of Control Systems and Informatics, Saint Petersburg State

University of Information Technologies Mechanics and Optics (ITMO),Kronverkskiy av. 49, Saint Petersburg, 197101, Russia.

approach, where the system is modeled as a continuoussystem with a delay in the control input (see, e.g. [9], [8]),and the impulsive system approach, where the sampled-datasystem is treated as an impulsive system (see, e.g. [20], [5],[4], [3]).

The input delay approach has been applied in [10] todesign a sampled-data output-feedback H∞ control for linearsystems while the impulsive system approach was applied in[14] to sampled-data stabilization of linear uncertain systemsin the case of constant sampling based on piece-wise linearin time Lyapunov function. The case of variable samplingbased on a discontinuous Lyapunov function method wasintroduced by [16]. Also based on discontinuous Lyapunovfunctions, in [3] stability and stabilization conditions forperiodic and aperiodic sampled-data systems are introduced.

In the context of observer design, one approach is basedon continuous and discrete design. In [7], such an approachis used to design a discrete-continuous version of the high-gain observer for nonlinear systems. Based on a small gainapproach, in [1] an observer design is proposed for certainclasses of nonlinear systems with sampled and delayedmeasurements. Using the hybrid system approach, in [6] anobserver-protocol pair is designed to estimate the states ofa linear system under communication constraints inducedby the network. Adopting a switched observer structure, in[2] decentralized observer-based output-feedback controllersare proposed for linear systems connected via a sharedcommunication network.

In this paper a vector Lyapunov function-based approach[15] for stability of impulsive systems is used to designan observer-based control for linear sampled-data systems.Such an approach, proposed by [17] and [18], is based ona 2D time domain equivalence (see, e.g. [19] and [21]),and provides a stability analysis based on linear matrixinequalities (LMIs) for linear impulsive dynamical systems.It is possible to show that the sampled-data control problembased on state estimation may turn into one of findingconditions for the exponential stability of impulsive systems.Then, this vector Lyapunov function approach is applied tofind such conditions expressed in terms of LMIs, and solvethe proposed problem for linear sampled-data systems.

The outline of this work is as follows. A motivatingproblem is given in Section II. Some preliminary resultsare given in Section III. The main result is described inSection IV. Some simulation results are depicted in Section

V. Finally, some concluding remarks are discussed in SectionVI.

II. MOTIVATION

Let us consider the following sampled-data system

x(t) = Ax(t) +Bu(t), (1)y(t) = Cx(ti), ∀t ∈ [ti, ti+1), (2)u(t) = −Kx(ti), ∀t ∈ [ti, ti+1), (3)

where x ∈ Rn is the state vector, u ∈ Rm is the sampledcontrol vector, and y ∈ Rp is the sampled output vectorat each time ti for all i ∈ N, and x ∈ Rn represents anestimation of the system state x. The sampling instants tiare monotonously increasing, such that limi→∞ ti = +∞,and ti+1 − ti ∈ [Tmin, Tmax], where Tmin and Tmax arethe minimum and maximum sampling intervals, respectively;and t0 = 0. The constant matrices A, B, and C have corre-sponding dimensions while K is a design control matrix. Thecontrol u is designed by means of the following sampled-datastate observer

˙x(t) = Ax(t) +Bu(t)

+ L(y(t)− Cx(ti)), ∀t ∈ [ti, ti+1), (4)

where x ∈ Rn is the estimated state vector and L is adesign observer matrix. Define the state estimation errore(t) = x(t)−x(t). Then, the closed-loop and state estimationerror dynamics are given as follows

x(t) = Ax(t)−BKx(ti) +BKe(ti), ∀t ∈ [ti, ti+1),

e(t) = Ae(t)− LCe(ti), ∀t ∈ [ti, ti+1).

Let us define the extended state vector ξ(t) =(xT (t) eT (t) xT (ti) eT (ti))

T ∈ R4n. Thus, the abovedynamics may be written as follows

ξ(t) = Aξξ(t), ∀t ∈ R+ \ I, (5)

ξ(t) = Iξξ(t−), ∀t ∈ I, (6)

where I := {ti}i∈N is a set of impulse times, ξ(t) ∈ R4n

is the current state vector, ξ(ti) ∈ R4n represents the resetvector state, ξ(t−i ) ∈ R4n denotes the value of ξ just beforethe impulse at time ti, i.e. ξ(t−i ) = limt↑ti ξ(t), and thecorresponding matrices have the following structure

Aξ =

A 0 −BK BK0 A 0 −LC0 0 0 00 0 0 0

,

Iξ =

In 0 0 00 In 0 0In 0 0 00 In 0 0

.

Then, the sampled-data control problem based on state es-timation, i.e. find the control gain matrix K and the observergain matrix L, may turn into one of finding conditions forthe stability of the impulsive systems described by (5)-(6),under arbitrary variations of the sampling intervals.

III. STABILITY ANALYSIS FOR IMPULSIVE SYSTEMS

The entire state trajectory (ξT , τ)T can be viewed as asequence of the diagonal dynamics1 of the following 2Dsystem:

d

dt

(ξtkτ tk

)=

(Aξξ

tk

1

), ∀τ tk ∈ [0, Ti], ∀i = k ∈ N, (7)(

ξti+1

k+1

τti+1

k+1

)=

(Iξξ

ti+1

k0

), ∀τ tk = Ti, ∀i = k ∈ N, (8)

where τ ∈ R≥0 is a timer variable, ((ξtk)T , τ tk)T =(ξT (t, k), τ(t, k))T ∈ R4n+1 is the current state vec-tor, ((ξ

ti+1

k+1)T , τti+1

k+1 )T = (ξT (ti+1, k + 1), τ(ti+1, k +1))T ∈ R4n+1 represents the reset state vector, while((ξ

ti+1

k )T , τti+1

k )T = (ξT (ti+1, k), τ(ti+1, k))T ∈ R4n+1

denotes the value of (ξT , τ)T just before the jump k + 1.It is assumed that the solutions of (7)-(8) are unique for thediagonal dynamics, i.e. for all i = k.

Remark 1. In this approach the model (5)-(6) correspondsonly to the sequence of diagonal dynamics of the 2D model(7)-(8). This model transformation is used to obtain sufficientconditions for stability, based on vector Lyapunov functions.An equivalence between models is not required since neces-sary and sufficient conditions are not provided.

In the present section some definitions and results forthe stability of impulsive systems, in the framework of 2Dsystems, are introduced (see [18]).

Let |q| denote the Euclidean norm of a vector q. Thefollowing stability definition is introduced:

Definition 1. [18]. A 2D system described by (7)-(8), is saidto be exponentially diagonal ξtk-stable (EDξtk-S) if there existpositive constants κ1, κ2, κ3, and c such that 0 < κ1 < 1and

|ξti+1

k+1 |2 ≤ cκk+1

1 |ξ00 |2, ∀τ tk = Ti, (9)

|ξtk|2 ≤ κ2|ξtik |2, ∀τ tk ∈ [0, Ti], (10)

|τ tk| ≤ κ3, (11)

for all i = k ∈ N.

Note that condition (11) holds by definition, i.e. |τ tk| ≤ κ3,with κ3 = Tmax. Denote ztk := ((ξtk)T , τ tk)T . In order to givethe stability conditions a vector Lyapunov approach is used2,i.e.

V (ztk, zti+1

k+1) =

(V1(ztk)

V2(zti+1

k+1)

),

where V1(·) > 0, V2(·) > 0, for all t ≥ 0, and V1(0) = 0,V2(0) = 0. Now, let us introduce the following definition.

Definition 2. The divergence operator of a function Valong the trajectories of system (7)-(8) is defined for all

1The diagonal dynamics make reference only to those dynamics given by(7)-(8) corresponding to i = k, for all i, k ∈ N and for all t ∈ R≥0.

2The use of vector Lyapunov functions offers a more flexible frameworksince each component of the vector Lyapunov function can satisfy less rigidrequirements as compared to a single Lyapunov function (see e.g. [15]).

t ∈ [ti, ti+1) as follows

divV (ztk, zti+1

k+1) =dV1(ztk)

dt+ V2(z

ti+1

k+1)− V2(zti+1

k ). (12)

Note that V1 is differentiable with respect to continuoustime t while the difference in V2 is calculated in discretetime k. Thus, the following theorem is introduced.

Theorem 1. [18]. Assume that there exist positive constantsc1, c2, c3, c4 and c5 such that the vector Lyapunov functionV (ztk, z

ti+1

k+1) and its divergence along the trajectories of thesystem (7)-(8) satisfy, for all τ tk ∈ [0, Ti], i = k ∈ N, thefollowing inequalities:

c1|ξtk|2 ≤ V1(ztk) ≤ c2|ξtk|2, (13)

c3|ξpk|2 ≤ V2(zpk) ≤ c4|ξpk|

2, ∀p = ti, ti+1 (14)

divV ≤ −c5(|ξtk|2 + |ξti+1

k |2), (15)

c2 (c4 − c5) ≤ c1c5 ∨ Ti ≤c2c5α, (16)

c2c5γ ≤ Ti, (17)

where γ = − ln[

c3c5+c3

]and α = − ln

[1− c1c5

c2(c4−c5)

]for all

c2 (c4 − c5) > c1c5. Then, the 2D system (7)-(8) is EDξtk-Sfor any sequence {Ti}i∈N such that Ti ∈ [ c2c5 γ,

c2c5α].

A. Exponential Diagonal ξtk-Stability: Quadratic LyapunovFunctions

Consider that V1 and V2 take the following quadraticstructure

V (ztk, zti+1

k+1) =

((ξtk)TP1(τ tk)ξtk

(ξti+1

k+1)TP2(τti+1

k+1 )ξti+1

k+1

), (18)

where P1 is continuously differentiable with respect to t,symmetric, bounded, and positive definite matrix for all τ tk ∈[0, Ti], i = k ∈ N, while P2 is a symmetric and positivedefinite matrix, i.e.

0 < c1I ≤ P1(τ tk) ≤ c2I, ∀τ tk ∈ [0, Ti], (19)0 < c3I ≤ P2(τpk ) ≤ c4I, p = ti, ti+1. (20)

Thus, based on the previous choice for V1 and V2, ifTheorem 1 is applied to the ideal and uncertain impulsivesystem (5)-(6), then the following result is obtained.

Corollary 1. Consider the vector Lyapunov functionV (ztk, z

ti+1

k+1) in (18). Assume that there exist matricesP1(τ tk) = PT1 (τ tk) > 0, continuously differentiable on t andbounded for all τ tk ∈ [0, Ti], i = k ∈ N, P2(0) = PT2 (0) > 0and P2(Ti) = PT2 (Ti) > 0 satisfying (19)-(20), and aconstant c5 > 0, such that the following matrix inequality

P1(τ tk)Aξ +ATξ P1(τ tk)

+dP1(t)dt

+ c5I4n0

0ITξ P2(0)Iξ

−P2(Ti) + c5I4n

≤ 0, (21)

holds for all τ tk ∈ [0, Ti], for all i = k ∈ N, and constraints

(16)-(17) are satisfied with c1, c2, c3, c4 and c5. Then thesystem (7)-(8), with f = 0, is EDξtk-S for any sequence{Ti}i∈N such that Ti ∈ [ c2c5 γ,

c2c5α].

Now the goal is to apply the conditions for exponentialdiagonal ξtk-stability of the impulsive systems (7)-(8) bymeans of the statements given by Corollary 1; and solvethe sampled-data control problem based on state estimationfor system (1)-(3) in a constructive way.

IV. OBSERVER-BASED CONTROL DESIGN

In this section a particular choice for P1 and P2 is pro-posed. Then, by means of the statements given by Corollary1, the control gain matrix K and the observer gain matrix Lwill be found to provide a stabilization of the state dynamicsx as well as an estimation x, i.e. stabilization of the extendedstate ξ in (5)-(6). Thus, the following proposition gives asolution to the sampled-data control problem based on stateestimation.

Theorem 2. Consider that P1 and P2 have the followingstructure for all τ tk ∈ [0, Ti], i = k ∈ N

P1(τ tk) =τ tkP11 + (Ti − τ tk)P12

Ti, P2(τ tk) = P21 + τ tkP22,

where P1l = diag(P−(1)1l , P

(2)1l , P

−(3)1l , P

−(4)1l ), for l = 1, 2,

with P(q)1l = P

(q)T

1l > 0, for l = 1, 2, q = 1, 4, P−(q)1l =

(P(q)1l )−1, and P2l = PT2l > 0, for l = 1, 2. Then, the system

(5)-(6) is EDξtk-S if there exist matrices P (q)1l , for j, l = 1, 2,

q = 1, 4, and P2l = PT2l > 0, for l = 1, 2; such that theLMIs (22)-(23) with3

φ11(Θ) = AP(1)11 + P

(1)11 A

T + P(1)11 /Θ− 2P

(1)11 + ΘP

(1)12 ,

φ12(Θ) = P(2)11 A+ATP

(2)11 + (P

(2)11 − P

(2)12 )/Θ +Q12,

φ13(Θ) = −2P(1)11 + ΘP

(3)12

, φ14(Θ) = −2P(1)11 + ΘP

(4)12 ,

φ15(Θ) = ITξ P21Iξ − P21 −ΘP22 +Q15,

φ21(Θ) = AP(1)12 + P

(1)12 A

T − P (1)12 /Θ,

φ22(Θ) = P(2)12 A+ATP

(2)12 + (P

(2)11 − P

(2)12 )/Θ +Q22,

φ23(Θ) = −2P(1)12 + ΘP

(3)12

, φ24(Θ) = −2P(1)12 + ΘP

(4)12 ,

φ25(Θ) = ITξ P21Iξ − P21 −ΘP22 +Q25,

hold for the finite set Θ ∈ {Tmin, Tmax}, Qj = QjT =

diag(Q−1j1 , Qj2, Q

−1j3 , Q

−1j4 , Qj5) > 0 for j = 1, 2, fixed

Λj = ΛjT > 0 for j = 1, 2, and where

YK1 = KP(1)11 , YK2 = KP

(1)12 , (24)

YL1 = P(2)11 L, YL2 = P

(2)12 L; (25)

and constraints (16)-(17) also hold with c1 = λmin(P11),c2 = λmax(P12), c3 = λmin(P21), c4 = λmax(P21 + ΘP22)and c5 = min(λmin(Q1), λmin(Q2)).

3The LMI variables are P (q)1l for l = 1, 2, q = 1, 4; P2l for l = 1, 2;

and YKj ,YLj for j = 1, 2. The matrices Qjl for j = 1, 2, l = 1, 5, canbe declared as variables or fixed values; while matrices Λj for j = 1, 2,are fixed.

φ11(Θ) 0 −BYK1BYK1

0 P(1)11 0 0 0 0 0 0

? φ12(Θ) 0 0 0 0 0 0 0 0 YL1C 0

? ? φ13(Θ) 0 0 0 P(1)11 P

(1)11 0 0 0 0

? ? ? φ14(Θ) 0 0 0 0 P(1)11 P

(1)11 0 P

(1)11

? ? ? ? φ15(Θ) 0 0 0 0 0 0 0? ? ? ? ? −Q11 0 0 0 0 0 0

? ? ? ? ? ? −ΘP(3)11 0 0 0 0 0

? ? ? ? ? ? ? −Q13 0 0 0 0

? ? ? ? ? ? ? ? −ΘP(4)11 0 0 0

? ? ? ? ? ? ? ? ? −Q14 0 0

? ? ? ? ? ? ? ? ? ? −Λ−11 0

? ? ? ? ? ? ? ? ? ? ? −Λ1

≤ 0, (22)

φ21(Θ) 0 −BYK2BYK2

0 P(1)12 P

(1)12

0 0 0 0 0 0

? φ22(Θ) 0 0 0 0 0 0 0 0 0 YL2C 0

? ? φ23(Θ) 0 0 0 0 P(1)12 P

(1)12 0 0 0 0

? ? ? φ24(Θ) 0 0 0 0 0 P(1)12 P

(1)12 0 P

(1)12

? ? ? ? φ25(Θ) 0 0 0 0 0 0 0 0

? ? ? ? ? −ΘP(1)11 0 0 0 0 0 0 0

? ? ? ? ? ? −Q21 0 0 0 0 0 0

? ? ? ? ? ? ? −ΘP(3)11 0 0 0 0 0

? ? ? ? ? ? ? ? −Q23 0 0 0 0

? ? ? ? ? ? ? ? ? −ΘP(4)11 0 0 0

? ? ? ? ? ? ? ? ? ? −Q24 0 0

? ? ? ? ? ? ? ? ? ? ? −Λ−12 0

? ? ? ? ? ? ? ? ? ? ? ? −Λ2

≤ 0. (23)

Υ1(Θ) =

P−(1)11 A + AT P

−(1)11 +

(P−(1)11 −P−(1)

12 )

Θ0 −P−(1)

11 BK P−(1)11 BK 0

?P

(2)11 A + AT P

(2)11 +

(P(2)11 −P (2)

12 )

Θ0 −P (2)

11 LC 0

? ?(P

−(3)11 −P−(3)

12 )

Θ0 0

? ? ?(P

−(4)11 −P−(4)

12 )

Θ0

? ? ? ? φ15(Θ)

≤ −Q1,

(26)

Υ2(Θ) =

P−(1)12 A + AT P

−(1)12 +

(P−(1)11 −P−(1)

12 )

Θ0 −P−(1)

12 BK P−(1)12 BK 0

?P

(2)12 A + AT P

(2)12 +

(P(2)11 −P (2)

12 )

Θ0 −P (2)

12 LC 0

? ?(P

−(3)11 −P−(3)

12 )

Θ0 0

? ? ?(P

−(4)11 −P−(4)

12 )

Θ0

? ? ? ? φ25(Θ)

≤ −Q2.

(27)

Proof: After some algebraic manipulations on matrixinequality (21) given by Corollary 1, it is possible to obtaininequalities (26) and (27), that should be hold for the finiteset Θ ∈ {Tmin, Tmax}.

Then, applying some quadratic non-singular transforma-tions, Schur’s complement and Λ-inequality to (26) and (27),respectively; one can obtain the LMIs given by (22) and (23).The complete proof is omitted due to lack of space.

Remark 2. Theorem 2 provides a particular way to solve theproposed problem, i.e. find the control gain matrix K andthe observer gain matrix L such that the system (5)-(6) isstable. Then, one can obtain K from (24), and L from (25),using any of the two equalities, respectively.

Note that a different selection for P1 and P2, even for Lya-punov functions with non-quadratic structure, may decreasethe conservatism. More complex tools like sum-of-squares[3], looped-functional approach [5], or convex characteriza-tions [4], may be applied to improve the application of thismethod.

V. SIMULATION RESULTS

Let us consider system (1)-(2) with

A =

(0 10 0

), B =

(01

)C =

(1 0

).

This example represents a double-integrator that has awide range of applications. Theorem 2 is applied togetherwith a bisection-like approach using SeDuMi solver amongYALMIP in MATLAB to find a solution for the LMIs (22)-(23) and the corresponding control and observer gains.

Based on Theorem 2, it is possible to show that theimpulsive system (5)-(6) is EDS for all 2.40 > Ti > 0 witha set of feasible control and observer gains. The followingfeasible results are obtained for different fixed values of Ti:

L(Ti=0.5) =

(4.06971.8351

), K(Ti=0.5) =

(0.0441 0.4030

),

L(Ti=1.0) =

(11.50315.4886

), K(Ti=1.0) =

(0.0522 0.4188

),

L(Ti=2.0) =

(9.40474.7571

), K(Ti=2.0) =

(0.0708 0.5789

).

The trajectories of the system, for different values of Ti, are

020

4060

80100

−10

12

34

−1

−0.5

0

0.5

1

Time k

Ti = 0.5

x1(t) vs x1(t)

x2(t)vsx2(t)

010

2030

4050

−20

24

6−1

−0.5

0

0.5

1

Time k

Ti = 1

x1(t) vs x1(t)

x2(t)vsx2(t)

05

1015

2025

−4−2

02

46

−2

−1

0

1

2

Time k

Ti = 2

x1(t) vs x1(t)

x2(t)vsx2(t)

05

1015

20

−40−20

020

40−20

−10

0

10

Time k

Ti = 3

x1(t) vs x1(t)

x2(t)vsx2(t)

x(t)x(t)

x(t)x(t)

x(t)x(t)

x(t)x(t)

Figure 1. Trajectories of the sampled-data system, their estimations, and the state estimation error for different values of Ti.

depicted in Fig. 1.

VI. CONCLUSIONS

In this paper a vector Lyapunov function-based approachfor stability of impulsive systems is used to design anobserver-based control for linear sampled-data systems. Thisapproach is based on a 2D time domain equivalence andprovides a stability analysis based on LMIs. Since thesampled-data control problem based on state estimation maybe turned into one of finding conditions for the exponentialstability of impulsive systems, the vector Lyapunov functionapproach is applied to find such conditions expressed interms of LMIs, and solve the proposed problem for linearsampled-data systems. Some numerical examples illustratethe feasibility of the proposed approach. The analysis ofuncertain sampled-data linear/nonlinear systems is in thescope of the future research.

ACKNOWLEDGMENT

This work was supported in part by Conseil RegionNord-Pas de Calais (ESTIREZ), the Government of RussianFederation (Grant 074-U01), the Ministry of Education andScience of Russian Federation (Project 14.Z50.31.0031), andANR ROCC-SYS (ANR-14-CE27-0008).

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