Interval observers design for continuous-time

CONFIDENTIAL. Limited circulation. For review only.

Interval observers design for continuous-time switched systems Haifa Ethabet ∗ Tarek Raissi ∗∗ Messaoud Amairi ∗ Mohamed Aoun ∗ Research Laboratory Modeling, Analysis and Control of Systems (MACS) LR16ES22, National Engineering School of Gabes (ENIG), University of Gabes 6029 Gabes, Tunisia (e-mail: [email protected], [email protected], [email protected]) ∗∗ Conservatoire National des Arts et Metiers (CNAM), Cedric - lab 292, Rue St-Martin, 75141 Paris Cedex 03, France (e-mail: [email protected]) ∗

Abstract: This paper deals with interval observers design for linear switched systems. The considered systems are subject to disturbances which are assumed to be unknown but bounded. First, observer gains are computed to ensure the stability of the estimation error. Then, under some changes of coordinates an interval observer is designed. The proposed methodology is illustrated through a numerical example. Keywords: Interval observer, stability, continuous-time switched systems, hybrid systems, cooperativity 1. INTRODUCTION

tic assumptions. This approach assumes that the measurement and process errors are unknown but bounded.

Switched systems are one of the most important classes of Hybrid Dynamical Systems (HDS). They consist of a family of continuous or discrete-time subsystems and a switching rule governing the switching among them. At each time only one subsystem is active. Several works treated switched systems such as (Djemai and Defoort (2015); Liberzon (2012); Branicky (1998)). Among these studies, control and state estimation were crucial and fundamental problems.

In the last decade, several works have investigated estimation problems in an unknown but bounded error (UBBE) framework for different classes of continuous systems such as Linear Time-Invariant (LTI), Linear TimeVarying (LTV) and Linear Parameter-Varying (LPV) systems (see Ra¨ıssi et al. (2012); Mazenc and Bernard (2011); Videau et al.; Ra¨ıssi et al. (2010); Mazenc and Bernard (2010); Gouz´e et al. (2000); Lamouchi et al. (2016); Yousfi et al. (2016b,a) and the references therein).

In order to study the stability of switched systems, specific results have been developed. For example, in (Hu et al. (2002)) a common Lyapunov function yields sufficient conditions for the global asymptotic stability. However, it may not always be possible to get this common function. Therefore, multiple Lyapunov functions were proposed for instance in (Liberzon and Morse (1999)). Approaches based on average dwell time have also been studied to ensure the stability of switched systems (see Hetel (2007); Serres et al. (2011) and the references therein).

The main limitation of the interval observer theory is the cooperativity feature. Considering that most of systems are not cooperative, it has been shown in the literature that any linear invariant-time system could be changed into a cooperative form. The technique proposed in (Mazenc and Bernard (2011)) considers that it is possible to design an interval observer via a time-varying change of coordinates based on the Jordan representation. In (Ra¨ıssi et al. (2012)) an invariant change of coordinates is used to ensure the cooperativity property.

As far as the stability problem is widely concerned, it is worth pointing out that the state is not always directly measured but may be estimated from the input and the output of the process. State estimation for switched systems remains a challenging problem by reason of the combined discrete and continuous states (see Hocine et al. (2005); Tian et al. (2009); Birouche et al. (2006); Arichi et al. (2015)). In practice, measurements are usually subject to noises. To compute robust estimates, interval observers assume that noises and disturbances are bounded without any stochas-

The extension of interval observers to switched systems has not been fully considered in the literature. To the best of the authors knowledge, only a preliminary work has been developed in (He and Xie (2016, 2015)) with the strong assumption that there exist gains Lq such that (Aq − Lq Cq ), q ∈ 1, N , are Metzler (the off-diagonal elements are nonnegative). Unfortunately, this assumption is rarely verified. In this paper, the main contribution is to design an interval observer for switched linear systems subject to distur-

Preprint submitted to 20th IFAC World Congress. Received November 7, 2016.

CONFIDENTIAL. Limited circulation. For review only.

bances. The measurement noise and the state disturbance are unknown but bounded with known bounds. State estimation conditions are given in terms of Linear Matrix Inequalities (LMIs). The proposed methodology allows one to overcome the strong limitations in (He and Xie (2016, 2015)). It will be shown that the constructive methodology can be applied for a large class of linear switched systems. This paper is organized as follows. Some preliminaries are described in Section 2. Interval state estimation for switched linear systems is stated in Section 3. A numerical example is presented in Section 4 to illustrate the efficiency of the proposed method. Section 5 concludes the paper. 2. PRELIMINARIES The sets of real and integer numbers are denoted by R and N respectively. The elementwise absolute value of a vector x ∈ Rn will be denoted by |x|. Denote the sequence of integers 1, ..., N by 1, N . Ep denotes a (p × 1) vector whose elements are equal to 1. I is the identity matrix of proper dimension. For a matrix P = P T , the relation P ≺ 0 (P ≻ 0) means that the matrix P ∈ Rn is negative (positive) definite. Denote by x and x ¯ the lower and upper bounds of the state x such that x ≤ x ≤ x ¯. The relation ≤ should be interpreted elementwise for vectors and matrices. For a matrix A ∈ Rm×n , define A+ = max {0, A} and A− = A+ − A. Lemma 1. (Chebotarev et al. (2015)) Let x ∈ Rn be a ¯ and A ∈ Rm×n be a constant vector satisfying x ≤ x ≤ x matrix, then (1) A+ x − A− x¯ ≤ Ax ≤ A+ x¯ − A− x  A matrix A = {aij } ∈ Rn×n is called Metzler if all its offdiagonal elements are nonnegative, i.e. aij ≥ 0, ∀i = 6 j. Lemma 2. (Gouz´e et al. (2000)) Consider the system described by:  x˙ (t) = Ax + u x (0) = x0 where A is a Metzler matrix and u ≥ 0. If x0 ≥ 0, then x (t) ≥ 0, ∀ t ≥ 0. (2)  Lemma 3. (Jiang et al. (2002)) Let δ be a scalar satisfying δ > 0 and P ∈ Rn be a symmetric positive definite matrix, then 1 2xT y ≤ xT P x + δy T P −1 y x, y ∈ Rn (3) δ  For the sake of clarity, let us introduce some definitions of interval observers. Consider the following system:  x˙ = Ax + φ (t) y = Cx where φ is a continuous function of time.

(4)

Assume that there exist two known functions φ¯ and φ : R → Rn Lipschitz continuous such that φ (t) ≤ φ (t) ≤ φ (t) for all t ≥ 0. Theorem 1. (Gouz´e et al. (2000)) If there exists a gain K such that (A − KC) is Metzler and if x0 ≤ x0 ≤ x ¯0 , then the system  x ¯˙ = A¯ x + φ¯ + K (y − C x ¯) (5) x˙ = Ax + φ + K (y − Cx) is a framer for the system (4) such that ¯ (t) , x (t) ≤ x (t) ≤ x

∀t ≥ 0

Theorem 2. (Gouz´e et al. (2000)) The system (5) is called an interval observer for system (4) if the lower (x − x) and upper (¯ x − x) estimation errors are asymptotically stable. 3. INTERVAL STATE ESTIMATION FOR SWITCHED LINEAR SYSTEMS In this section an interval observer is designed for a linear switched system that would satisfy both the stability and the cooperativity conditions. Consider a Switched Linear System (SLS) described by:  x˙ (t) = Aq x (t) + Bq u (t) + w (t) , q ∈ 1, N , N ∈ N (6) ym (t) = Cq x (t) + v (t) where x ∈ Rn , u ∈ Rm , ym ∈ Rp , w ∈ Rn , v ∈ Rp are respectively the state vector, the input, the output, the disturbance and the measurement noise. Aq , Bq and Cq are constant matrices of proper dimensions. q is the index of the active subsystem and N is the number of linear subsystems. The measurement noise and the state disturbance are assumed to be unknown but bounded with a priori known bounds such that |w (t)| ≤ w, ¯ |v (t)| ≤ V¯ Ep ,

∀t ≥ 0 ∀t ≥ 0

(7) (8)

where w ¯ ∈ Rn and V¯ is a scalar. The aim is to derive two trajectories x (t) and x ¯ (t) where ¯ (t) , ∀t ≥ 0, despite the disturbances, x (t) ≤ x (t) ≤ x starting from the initial condition x0 which is assumed to be bounded by two known bounds: x0 ≤ x0 ≤ x ¯0

(9)

To design an interval observer for (6), a necessary condition is given in the following assumption. Assumption 1. There exist gains Lq such that the matrices (Aq − Lq Cq ) are Metzler for q ∈ 1, N . The matrices Lq (q ∈ 1, N ) denote the observer gains associated with each subsystem q. A candidate interval observer structure for the estimation of x¯, x is described by:  x ¯˙ = (Aq − Lq Cq ) x ¯ + Bq u + w ¯ + Lq ym + |Lq | V¯ Ep ¯ + Lq ym − |Lq | V¯ Ep x˙ = (Aq − Lq Cq ) x + Bq u − w (10) Similarly to T heorem 1, the following theorem gives the conditions for achieving partially the desired design goal.

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CONFIDENTIAL. Limited circulation. For review only.

Theorem 3. Consider the system described by (6). Let Assumption 1 be satisfied. For any initial condition x0 ≤ x0 ≤ x¯0 , if there exist observer gains Lq such that (Aq − Lq Cq ) Metzler , ∀q ∈ 1, N (11) then the system (10) is a framer for the system (6) with ¯ (t) , ∀t ≥ 0 (12) x (t) ≤ x (t) ≤ x  Proof. Let e¯ (t) = x ¯ − x be the upper observation error and e (t) = x − x be the lower observation error. Let us show that e¯ and e are positive. From (6) and (10) the dynamics of the interval estimation errors are given by: ¯q e¯˙ (t) = x ¯˙ − x˙ = (Aq − Lq Cq ) e¯ (t) + Γ

(13)

e˙ (t) = x˙ − x˙ = (Aq − Lq Cq ) e (t) + Γq

(14)

and

where

2 , Wq = SLq δq then the framer (10) is an asymptotically stable interval observer for (6). αq =

 Proof. As mentioned in T heorem 2, the global asymptotic stability of the interval observer is ensured by applying a common Lyapunov function to the estimation errors. Consider the following Lyapunov function: T

V (¯ e) = e¯(t) S¯ e (t) where S is a symmetric positive definite matrix. The derivative of V is given by:

T V˙ (¯ e) = e¯˙ S¯ e + e¯T S e¯˙ h i T eT Sw = e¯T (Aq − Lq Cq ) S + S(Aq − Lq Cq ) e¯ − 2¯

+2¯ eT SLq v + 2¯ eT S w ¯ + 2¯ eT S |Lq | V¯ Ep

where

¯q = w Γ ¯ − w + Lq v + |Lq | V¯ Ep (15) ¯ Γq = w + w ¯ − Lq v + |Lq | V Ep (16) As |v (t)| ≤ V¯ Ep , ∀t ≥ 0 and by construction (w ¯ − w) ¯ q and Γq are positive for and (w + w) ¯ are positive then Γ all q ∈ 1, N . In addition, since Lq are computed to verify (11) and x0 and x ¯0 are chosen such that  e¯ (0) = x ¯0 − x0 ≥ 0 e (0) = x0 − x0 ≥ 0 then, according to Lemma 2, e¯ (t) and e (t) are positive ∀t ≥ 0. Thus, x (t) ≤ x (t) ≤ x ¯ (t) , ∀t ≥ 0 (17)  Remark 1. The framer (10) is initialized with the initial conditions x0 and x ¯0 for the active subsystem (q = 1). At the switching time instant, the output of the previous active subsystem (q = i) is used to initialize (10) with the subsystem (q = i + 1). In addition, T heorem 3 ensures only the inclusion relation x (t) ≤ x (t) ≤ x ¯ (t). However, the errors (x − x) and (¯ x − x) are not guaranteed to be bounded. For the stability analysis of (13) and (14), let us introduce the following lemma. Lemma 4. (Liberzon and Morse (1999)) Consider the following switched linear system: (18) x˙ (t) = Aq x (t) , q ∈ 1, N The switched system (18) is globally asymptotically stable if there exists a matrix S = S T > 0 such that
(19) V˙ (x) = xT Aq T S + SAq x < 0, q ∈ 1, N where V (x) is the common Lyapunov function given by: V (x) = xT Sx (20)  Theorem 4. Let Assumption 1 be satisfied. Given scalars δq > 0, if there exists a symmetric positive definite matrix S ∈ Rn for all q ∈ 1, N such that (21) Aq T S + SAq − Cq T Wq T − Wq Cq + αq S ≺ 0

(22)

(23)

From Lemma 3, it is clear that V˙ (¯ e) ≤ e¯T B1 e¯ + C1

(24)

where B1 = Aq T S + SAq − Cq T Wq T − Wq Cq +

2 S δq

(25)

and   C1 = wT [−δq S] w + w ¯ T [δq S] w ¯ + v T δq LTq SLq v   (26) +Ep T δq V¯ LTq SLq V¯ Ep

Similarly, the derivative of the common Lyapunov function for the lower estimation error yields: V˙ (e) = e˙ T (t) Se (t) + eT (t) S e˙ (t) h i T = eT (Aq − Lq Cq ) S + S(Aq − Lq Cq ) e ¯ − 2eT SLq v +2eT Sw + 2eT S w +2eT S |Lq | V¯ Ep

≤ eT B 1 e + C2

(27)

where   C2 = wT [δq S] w + w ¯T [δq S] w ¯ − v T δq LTq SLq v i h T (28) +Ep T δq V¯ |Lq | S |Lq | V¯ Ep

From (21), it is assumed that B1 ≺ 0. In addition, the noises and disturbances are bounded it follows that C1 is bounded. Therefore the error e¯ is bounded. The same arguments show that the error e is also bounded.

 Remark 2. In some cases, the existence of a common Lyapunov function is not always guaranteed. To ensure the stability of the interval observer (10), multiple Lyapunov functions could be calculated instead of the common Lyapunov function. Nevertheless, this methodology is not investigated in this paper.

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CONFIDENTIAL. Limited circulation. For review only.

The methodology described above, although simple, is not always constructive. Indeed, it is not always possible to find gains Lq such that Assumption 1 is satisfied. Hence, the main idea consists in finding a change of coordinates that transforms the observation errors into cooperative forms. The changes of coordinates proposed for instance in (Mazenc and Bernard (2011); Ra¨ıssi et al. (2012)) for continuous systems can be used to transform the matrices (Aq − Lq Cq ) into a Metzler form. Therefore, let us assume that there exists a non singular transformation matrix P such that, with the new coordinates z = P x, the system (6) is transformed into the form  z˙ = P Aq P −1 z + P Bq u + P w (29) y = Cq P −1 z + v A Luenberger based candidate observer for the system (29) can be written in the new coordinates z as:  z¯˙ = P (Aq − Lq Cq ) P −1 z¯ + P Bq u + P + w ¯ − P −w    ¯  +P Lq ym + |P Lq | V Ep (30)  −1 + −  = P (A − L C ) P z + P B u + P w − P w ¯ z ˙  q q q q  +P Lq ym − |P Lq | V¯ Ep

V (ez ) = ez T M ez The derivatives of V (¯ ez ) and V (ez ) are given by: T V˙ (¯ ez ) = e¯˙ z M e¯z + e¯Tz M e¯˙ z

≤ e¯Tz B2 e¯z + C3

≤ ez T B2 ez + C4

From systems (29) and (30), the dynamics of the interval estimation errors are given by:

(41)

where

and

∀q ∈ 1, N

Let e¯z (t) = z¯ − z be the upper observation error and ez (t) = z − z be the lower one.

(40)

V˙ (ez ) = e˙ Tz M ez + ez T M e˙ z

where

z¯ (0) = P + x ¯0 − P − x0 (31) + z (0) = P x0 − P − x ¯0 (32) P is the solution of the Sylvester equation given by P Aq − Rq P = Qq Cq , Qq = P Lq (33) and Rq = P (Aq − Lq Cq ) P −1

(39)

and

i h i h T T ¯ − w T δq P − M P − w C3 = w ¯ T δq P + M P + w   −wT δq P T M P w i h T +Ep T δq V¯ |P Lq | M |P Lq | V¯ Ep   (42) +v T δq LTq P T M P Lq v h h i i T T C4 = −wT δq P + M P + w + w ¯ T δq P − M P − w ¯   +wT δq P T M P w i h T +Ep T δq V¯ |P Lq | M |P Lq | V¯ Ep   (43) −v T δq LTq P T M P Lq v

i h  T T B2 = P (Aq − Lq Cq ) P −1 M + M P (Aq − Lq Cq ) P −1 +

1 M δq T

= P −1 Aq T P T M + M P Aq P −1 +

1 M δq

T

e¯˙ z (t) = z¯˙ − z˙ = P (Aq − Lq Cq ) P −1 e¯
  + P +w ¯ − P − w − P w + |P Lq | V¯ Ep ¯q +P Lq v = R¯ e+Υ (34) and e˙ z (t) = z˙ − z˙ = P (Aq − Lq Cq ) P −1 e 
 + P w − P +w − P −w ¯ + |P Lq | V¯ Ep

(35) −P Lq v = Rq e + Υq where 
 ¯ q = P +w Υ ¯ − P − w − P w + |P Lq | V¯ Ep + P Lq v (36)
  ¯ + |P Lq | V¯ Ep − P Lq v (37) Υq = P w − P + w − P − w It has been shown in several works that the existence of the solution of the Sylvester equation P is not restrictive. In (Mazenc and Bernard (2011)), it has been shown that the Jordan canonical form can also be used and P can be constant or time-varying. In addition, a constant change of coordinates has been proposed in (Ra¨ıssi et al. (2012)). Similarly to the proof of T heorem 4, the asymptotic stability of the observer (30) is ensured by applying a common Lyapunov function to the observation errors as follows: V (¯ ez ) = e¯Tz M e¯z (38)

(44) − P −1 Cq T Lq T P T M − M P Lq Cq P −1 ≺ 0 The existence of a common transformation matrix P such that P (Aq − Lq Cq ) P −1 for all q ∈ 1, N are Metzler seems to be difficult. The LMI presented in (44) is a nonlinear inequality. Therefore, the stability of the observer (30) can not be easily ensured. However, it is rare, even impossible, to determine a non singular transformation matrix P to transform the system (6) into a cooperative form such that P (Aq − Lq Cq ) P −1 (q ∈ 1, N ) are Metzler. As an issue to this problem, a second method is proposed. The main idea consists in redesigning two conventional observers in the original base “x”. Then, stability conditions will be given in terms of LMI by applying a common Lyapunov function to the estimation errors. Consider the SLS (6) and two point observers described by  +
− ˙ = (Aq − Lq Cq ) xˆ+ + Bq u + P −1 P + w  ¯ + P w ¯ x ˆ  q q q    +Lq ym + |Lq | V¯ Ep
−   ¯ − Pq− w ¯ x ˆ˙ = (Aq − Lq Cq ) x ˆ− + Bq u + Pq−1 −Pq+ w    +Lq ym − |Lq | V¯ Ep (45)

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with Pq , q ∈ 1, N , are chosen as in the following theorem. Theorem 5. If the initial condition x0 verifies x0 (t) ≤ x0 (t) ≤ x¯0 (t) and if there exist non singular transformation matrices Pq such that Pq (Aq − Lq Cq ) Pq−1 are Metzler, then the system (45) is a framer with x (t) ≤ x (t) ≤ x ¯ (t) (46) where   x = Qq + Pq x ˆ− − Qq − Pq xˆ+

As Qq = Pq −1 , then x≤x≤x ¯ where

(53)

  x = Qq + Pq x ˆ+ ˆ− − Qq − Pq x

ˆ− ˆ+ − Qq − Pq x x¯ = Qq + Pq x For the stability analysis, Let us now show that Eq+ and Eq− are asymptotically stable or simply show that (ˆ x+ − x) − and (x − xˆ ) are asymptotically stable. 

(47)

Let e+ = (ˆ x+ − x) and e− = (x − xˆ− ) be the observation errors.

and Qq = Pq −1 ; Qq + = max {0, Qq } ; Qq − = Qq + − Qq In addition, if there exists a symmetric definite positive matrix M such that (48) Aq T M + M Aq − Cq T Wq T − Wq Cq + σq M ≺ 0 where 3 σq = and Wq = M Lq δq then (45) is asymptotically stable.

Consider the following Lyapunov function:
T V e+ = e+ M e+ (54) where M is a symmetric positive definite matrix. As in proof of T heorem 4, the derivative of the Lyapunov function can be given as follows:



ˆ− ˆ+ − Qq − Pq x x ¯ = Qq + Pq x

 Proof. Consider the errors Eq+ = Pq xˆ+ − Pq x and Eq− = Pq x − Pq x ˆ− . Let us show that x¯ − x ≥ 0 and x − x ≥ 0 where x¯ and x are computed by (47).


T T V˙ e+ = e˙ + M e+ + e+ M e˙ + h i T T = e+ (Aq − Lq Cq ) M + M (Aq − Lq Cq ) e+
T T ¯ − 2e+ M w ¯ + Pq− w +2e+ M Pq −1 Pq+ w T T +2e+ M Lq v + 2e+ M |Lq | V¯ Ep

According to Lemma 3, we have
T V˙ e+ ≤ e+ B3 e+ + C5 where

From (6) and (45) the dynamics of the errors Eq+ and Eq− are given by:

T

= Aq T M + M Aq − Cq T Wq T +

= Rq Eq+ + γq+

(49)

where
  ¯ − Pq w +Pq Lq v+Pq |Lq | V¯ Ep (50) ¯ + Pq− w γq+ = Pq+ w Similarly to Eq+ , the dynamics of Eq− are given by: ˆ˙ − E˙ q− = Pq x˙ − Pq x = Rq Eq− + γq−

(51)

where 
 γq− = Pq w − −Pq+ w ¯ − Pq− w ¯ + Pq |Lq | V¯ Ep − Pq Lq v (52) According to Lemma 1 we have −Pq+ w ¯ − Pq− w ¯ ≤ Pq w ≤ Pq+ w ¯ + Pq− w ¯ Since Pq (Aq − Lq Cq ) Pq−1 are assumed to be Metzler, and by construction γq+ and γq− are positive for all t ≥ 0. Then, if x¯0 and x0 are chosen such that Eq+ (0) and Eq− (0) are positive, then the errors Eq+ (t) and Eq− (t) stay positive ∀t ≥ 0 such that Pq x ˆ− ≤ Pq x ≤ Pq xˆ+

3 M δq 3 − Wq Cq + M (56) δq

B3 = (Aq − Lq Cq ) M + M (Aq − Lq Cq ) +

+ E˙ q+ = Pq xˆ˙ − Pq x˙

= Pq (Aq − Lq Cq ) x ˆ − Pq (Aq − Lq Cq ) x
  + − ¯ − Pq w + Pq Lq v + Pq |Lq | V¯ Ep ¯ + Pq w + Pq w
ˆ+ − Pq x + γq+ = Pq (Aq − Lq Cq ) Pq−1 Pq x

(55)

and i h   T T ¯ + v T δq L q T M L q v C5 = w ¯T δq Pq+ Pq−1 M Pq−1 Pq+ w i h T T ¯ − wT [δq M ] w +w ¯ T δq Pq− Pq−1 M Pq−1 Pq− w i h T (57) +Ep T δq V¯ |Lq | M |Lq | V¯ Ep

The noise v and disturbance w are bounded, it follows that C5 is bounded. Therefore, if B3 ≺ 0, the observation error e+ is bounded. The same arguments allow one to show that the observation error e− is also bounded. In addition, since Pq and Qq are bounded for all (q ∈ 1, N ) then Eq+ and Eq− are bounded.  The cooperativity property has motivated the need for state transformation. However, computing a transformation matrix Pq for each subsystems, such that zq = Pq x, transforms the switched system (6) into a hybrid one. Besides, the stability analysis in the base “z” seems to be difficult. Hence, a non singular common transformation matrix should be used to change (6) into a cooperative form. Nonetheless, this matrix P is rarely, even impossible, found. These limitations prove the interest of the methodology proposed above. Thus, the key point of the proposed

Preprint submitted to 20th IFAC World Congress. Received November 7, 2016.

CONFIDENTIAL. Limited circulation. For review only.

4. NUMERICAL EXAMPLE Consider the linear switched system subject to disturbances described by:  x˙ (t) = Aq x (t) + Bq u (t) + w (t) , ∀q ∈ 1, 3 (58) y (t) = Cq x (t) + v (t) where     −0.5 2 −1.5 0.262 , A2 = A1 = 0 −1 0 −1   −0.6 1.5 A3 = 0 −1       1 1 0 , B3 = , B2 = B1 = 1 0 1 1 0 1 1 1 C1 = [ ] , C2 = [ ] , C3 = [ 1.5 ] w (t) and v (t) are uniformly distributed bounded signals T such that −w ¯ ≤ w (t) ≤ w ¯ with w ¯ = [0.03 0.03] and ¯ ¯ ¯ −V Ep ≤ v (t) ≤ V Ep with V = 0.3.



   1.0000 1.5000 1.0000 1.5000 , Q2 = −0.2593 1.3599 −0.9853 0.5876   1.0000 1.0000 Q3 = −0.3299 1.1899

The results of simulation of the obtained observer are depicted in Fig. 1(a) for both coordinates where solid lines present the state and dashed lines present the estimated bounds. The switching between the three subsystems is governed by the switching signal plotted in Fig. 1(b). The initial state x0 is assumed to be bounded such that x0 ≤ x0 ≤ x ¯0 where T x¯0 = [ 1.5 1.5 ] T x0 = [ −1.5 −1.5 ] 2

upper bound xmax lower bound xmin

1

state x

x1

In the next section, the effectiveness and the performance of the proposed method is illustrated via a numerical example.

Q1 =

0

-1 -2 -3 0

5

10

15

20

0

5

10

15

20

Now, using the Matlab LMI toolbox, one can solve the LMI defined by (48). One feasible solution is given by:       0.5734 0.9397 −0.6555 , L3 = , L2 = L1 = −0.2512 −0.2497 −0.1011   81.6804 21.7133 , δq = 84.5569, ∀q ∈ 1, 3 M= 21.7133 55.5904

30

35

40

30

35

40

1

0

-1

-2

To verify the cooperativity property, a transformation of coordinates must be used such that P (Aq − Lq Cq ) P −1 are Metzler. However, it is not possible to compute this common transformation matrix P . Hence, non singular transformation matrices Pq are calculated. Consequently, a conventional observer (45) is constructed for the system (58).

25

2

x2

technique consists in computing non singular matrices Pq and ensuring the estimation without any change of coordinates. Note that a similar approach has been investigated for discrete systems (non switched systems) in (Mazenc et al. (2014)).

T ime (s) 25

(a) State and estimated bounds 4 switching signal

3.5 3 2.5 2 1.5 1 0.5 0 0

5

10

T ime (s) (b) Switching signal 15

20

25

30

35

40

Fig. 1. Interval state estimation for the switched system with disturbances.

The results show that, despite the disturbances, the state Note that Pq are computed to verify that Pq (Aq − Lq Cq ) Pq −1 is always inside the upper and the lower trajectories. are Metzler for all q ∈ 1, 3 and given by: The interval observer has exhibited approved stability     0.7776 −0.8577 0.2845 −0.7262 properties. The relation , P2 = P1 = 0.1483 0.5718 0.4770 0.4841 ¯ (t) , ∀t ≥ t0 x (t) ≤ x (t) ≤ x   0.7829 −0.6580 is always verified. P3 = 0.2171 0.6580 As shown in Fig. 1(a), the interval is quite large at All conditions of Theorem 5 are satisfied; it follows that the beginning, although its width decreases despite the uncertainties on the measurements. Finally, the interval the system (45) is asymptotically stable verifying observer remains stable despite the switching instants. x (t) ≤ x (t) ≤ x¯ (t) for all t ≥ 0, ∀q ∈ 1, 3. with and

x = Qq + Pq xˆ− − Qq − Pq x ˆ+ ˆ− x ¯ = Qq + Pq xˆ+ − Qq − Pq x

where Qq = Pq−1 are given by:

5. CONCLUSION This paper investigates state estimation for switched linear systems subject to disturbances. An interval observer is designed under some transformations where two conventional observers are reformulated in the base “x”. The stability and the cooperativity conditions are represented in terms

Preprint submitted to 20th IFAC World Congress. Received November 7, 2016.

CONFIDENTIAL. Limited circulation. For review only.

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Preprint submitted to 20th IFAC World Congress. Received November 7, 2016.