Stabilization for switched linear systems: Hybrid observer-based method

This paper investigates interval observer-based controller design for switched linear systems involving additional disturbance and measurement noise, whose modes need not to be cooperative. First, by giving the upper bound and the lower bound of the disturbance and the measurement noise, we construct hybrid interval observer for the considered switched linear system by means of a switched coordinate transformation, which can transform the observer error systems into cooperative ones. The interval observer can supply certain state information at any instant. Compared with the interval observer design for switched systems with arbitrary switching sequences or dwell time switching based on common Lyapunov function, the difﬁculty consists in characterizing the jump of the multiple Lyapunov functions. Then, by using the multiple Lyapunov functions method and average dwell time scheme, some sufﬁcient conditions are derived and applied to build the interval observer-based state feedback controller. Finally, we provide an example to illustrate the validity of the derived results.


INTRODUCTION
Switched systems exist widely in many practical systems [1][2][3], such as circuit systems [4], traffic control systems [5], etc. Stability analysis is a fundamental problem of switched systems. When all the subsystems are stable, the switching signal may not guarantee the stability of the whole switched system, that is, the overall stability is not equal to the simple superposition of each single model stability. Research on stability of switched systems has been a hot topic and many achievements have been obtained [6][7][8][9][10][11][12][13][14][15][16]. For example, the global finite-time stabilization of a class of switched nonlinear systems with the powers of positive odd rational numbers was studied in [14]. Work in [15] provided multiple Lyapunov function-based small-gain theorems for switched interconnected nonlinear systems. Long and Zhao [16] discussed H ∞ control of switched nonlinear systems in p-normal form by using multiple Lyapunov functions. Note that all the above-mentioned results focused on the stability properties and control synthesis of the switched system with synchronous switching signals. However, in practice, there is often a lag between the system and its associated controller, so the asynchronous switching signal is inevitable. Under asynchronous switching, the robust observer design problems, the output-feedback control, the adaptive output feedback fuzzy stabilization for switched systems were discussed in [17][18][19], respectively.
State estimation is an important method for state feedback of dynamic systems with unmeasurable states. Interval observers, as a new method of state estimation, mainly deal with the state estimation of dynamic systems with large uncertainties. Since the interval observer method was first proposed in reference [20], it has attracted a lot of attention and has been applied in vehicle positioning [21], fault detection [22], etc. Unlike the traditional observers, the interval observer realizes the state estimation by providing the upper and the lower bounds of the state of the observed system at any time and brings part of the uncertain information into the process of the observer architecture as part of the design, which makes the observer have great tolerance for uncertainties [23][24][25]. Due to the dynamic complexity of switched systems, the interval observer-based controller design for switched systems is much more complicated than usual single model systems. Therefore, there are few literatures about the design of interval observers for switched systems [26][27][28][29][30][31][32]. By monotony method, Huang et al. [26] discussed the problem of interval observer design for discrete-time switched systems. For a class of switched systems with additional disturbance, Wang et al. [28] studied the design of hybrid interval observers for switched linear time-invariant systems. In [29], based on the condition that A i -L i C i are both Metzler and Hurwitz, an interval observer was designed for linear switched systems. But in many practical cases, A i -L i C i do not always meet the above conditions. Motivated by the above-mentioned concerns, the present work studies the design of hybrid interval observer-based controller for continuous-time switched linear systems with disturbance and measurement noise under asynchronous switching, whose modes need not be cooperative. First, by giving the upper bound and the lower bound of the disturbance and the measurement noise, we construct hybrid interval observer for the considered switched linear system by means of a coordinate transformation, which can transform the observer error systems into cooperative ones. Second, we study the asynchronous control of switched systems based on the improved interval observer by allowing the switching delay to be time-varying. Third, we derive the sufficient condition to guarantee the stability of the switched system when the switching signals satisfying an average dwell time scheme, and further establish the interval observerbased state-feedback controller gains. And last, we provide an example to illustrate the validity of the derived results.
The contribution compared with the literature can be summarised as the following aspects: (1) Unlike the literatures [26,27], which focused on synchronous and asynchronous interval observers for switched discrete-time systems and did not deal with the controller design, we study the asynchronous control based on interval observers for switched continuous-time systems, that is, the asynchronous phenomenon occurs between the controller and the subsystems. (2) In [28], a switched time-varying coordinate transformation method was used for switched time-invariant systems, and the derived sufficient conditions for the construction of interval observers are related to the switching times t i , which makes it difficult to check these conditions. Compared to the system considered in [28], the system considered in this paper is more extensive. Besides, the interval observer constructed in this paper is time-invariant, and the sufficient conditions for its existence are time-independent and easier to be checked. (3) The interval observers encountered in [29,30] were designed for cooperative error systems. The interval observers available in the paper are made for the systems, which may not be cooperative. (4) To avoid the switched system being transformed into a hybrid system, Ethabet et al. [31] designed the interval observers in the original basis 'x' and only required that the upper and the lower bounds for the initial states make the errors E + q (0) and E − q (0) are nonnegative. But in fact, regardless of the use of the original basis 'x', the coordinate has been used to ensure the cooperativity, thus it has to make sure that the errors E + q and E − q are non-negative at each switching time, which would certainly require the interval observer system be a hybrid system. (5) In [32], the hybrid framer of the switched system was designed to satisfy the upper and the lower bounds, i.e. x ≤ x ≤x, but the framer was not necessarily asymptotically stable. Besides, the conditions for the construction were derived based on common Lyapunov function. This paper adopts multiple Lyapunov functions method for the asymptotical stability of the hybrid framer, so the sufficient conditions given in the paper are less conservative.
The structure of the paper is as follows: Section 2 describes preliminaries and problem formulation. In Section 3, the main results are given. Simulation results are given to illustrate the effectiveness of the proposed methods in Section 4. Section 5 concludes the paper.
Notations. ℝ n denotes the n dimensional Euclidean space. For a matrix P, P > 0 (P < 0) means that P is positive definite (negative definite); max (P ) and min (P ) denote the maximum and minimum eigenvalues of the matrix P, respectively. I and 0 denote the identity matrix and zero matrix with appropriate dimensions, respectively. E denotes the vector whose elements are all ones. For any two vectors x 1 , x 2 ∈ ℝ n , the relations 21 x 22 ] T , then x 11 ≤ x 21 and x 12 ≤ x 22 . P T and P −1 denote the transpose and the inverse of a square matrix P. A matrix M ∈ R n×n is said to be Metzler if each off-diagonal entry of it is non-negative. Euclidean norm for a vector u ∈ ℝ n u will be denoted as |u| e . For a measurable and locally essentially bounded function u : ℝ + → ℝ n u , the symbol ||u[t 1 , t 2 ]|| = esssu p t 1 ≤t ≤t 2 {|u(s)| e }, if t 2 = +∞, then we will simply write ||u|| ∞ . M = max{A, B} is the matrix where each entry is m i, j = max{a i, j , b i, j }. Let us define A + = max{A, 0}, A − = A + − A; thus, the elementwise absolute value will be denoted as |A| = , k ∈ ℕ. The notation will be simplified, e.g. by omitting the arguments of the functions, whenever no confusion can arise from the context.

PRELIMINARIES AND PROBLEM FORMULATION
Consider the following switched linear systeṁ where x(t ) ∈ ℝ n is the system state; x(t 0 ) = x 0 is the initial state vector and assumed to be bounded by the known bound: |x 0 | ≤ E, where > 0 is a scalar constant; u(t ) ∈ ℝ q is the input; y(t ) ∈ ℝ p is the measurable output vector.
(t ) : ℝ + → M = {1, 2, … , m} is a piecewise constant function of time t and called switching signal, m is the number of subsystems. Corresponding to (t ), it has the switching sequence {( 0 , t 0 ), … , ( k , t k ), … , | k ∈ M, k ∈ ℕ}, which means that the k th subsystem is active when t ∈ [t k , t k+1 ), k ∈ ℕ. For any i ∈ M , A i , B i , and C i are known real constant matrices of appropriate dimensions, w i (t ) and v i (t ) are the disturbance and the measurement noise, respectively. We assume that the state of the system does not jump at the switching instants and that only finitely many switches can occur in any finite interval. In addition, for any i ∈ M , we assume that the pair (A i , C i ) are observable.
The description of the main result requires the following definitions and lemmas.
with x ∈ ℝ n , w i ∈ ℝ l and with f i , i ∈ M of class C 1 . The disturbance w i are Lipschitz continuous and such that there exist two known bounds w u (t ), w l (t ) ∈ ℝ l , Lipschitz continuous, and such that, for all t ≥ 0 Moreover, the initial condition x(0) = x 0 is assumed to be bounded by two known bounds: Then, the dynamical systeṁ , w l (t )) ∈ ℝ 2l , the switched signal , and bounds for the solution with i (i ∈ M ), H u , H l , G Lipschitz continuous of appropriate dimension, is called an interval observer (resp. an exponentially stable interval observer) for (2) if (i) for all Lipschitz continuous functionw(t ), all the solutions of (5) are defined over [0, +∞); (ii) for any vectors x 0 , x 0 , andx 0 in ℝ n satisfying (4), the solutions of (2), (5) with respectively x 0 , Z 0 = G (t 0 ,x 0 , x 0 ) as an initial condition at t = t 0 , denoted respectively x(t ), Z (t ), are defined for all t ≥ t 0 and satisfy, for all t ≥ t 0 , the inequalities (5) is globally uniformly asymptotically stable (resp. globally uniformly exponentially stable) under the switching signal whenw is identically equal to zero.
holds for a > 0 and N 0 ≥ 0, then a is called average dwell time and N 0 is called a chattering bound. Denoted by S ave [ a , N 0 ] the class of switching signals with average dwell time a and chattering bound N 0 . Lemma 1. [28] Consider the systeṁ with x ∈ ℝ n , w ∈ ℝ n , where  is Metzler and w ≥ 0 is a Lipschitz continuous function, then

Lemma 3. [31]
Let > 0 be a scalar and S ∈ ℝ n×n be a symmetric positive definite matrix, then

MAIN RESULT
In the section, we will give the design of the interval observerbased controller for the system (1) under asynchronous switching.

Hybrid interval observer
We suppose that the function w i , i ∈ M is external disturbance with known bounds.

Assumption 1.
The disturbance and the measurement noise are assumed to be unknown but bounded with priori known bounds such that −w ≤ w i (t ) ≤w, | i (t )| ≤̄E are verified for ∀t ∈ ℝ + and ∀i ∈ M, wherew ∈ ℝ n and̄is a positive scalar.
then an asymptotically interval observer for the subsystem i of (1) is given by:ẋ Proof. The proof of the lemma is similar to the proof of Theorem 5 in [31].
Remark 1. For the single subsystem, we adopt the interval observer framework in [31]. However, for the interval observer of switched systems, we wish to point out that: To avoid the switched system being transformed into a hybrid system, Ethabet et al. [31] designed the interval observers in the original basis 'x' and only required that the upper and the lower bounds for the initial states make the errors E + q (0) and E − q (0) are non-negative. But in fact, regardless of the use of the original basis 'x', the coordinate has been used to ensure the cooperativity, thus it has to make sure that the errors E + q and E − q are non-negative at each switching time, which would certainly require the interval observer system to be a hybrid system. For the switched system (1), we construct the following hybrid dynamical systemṡ where Remark 2. In [28], under average dwell time scheme, a switched time-varying interval observer was established for switched linear time-invariant systems. However, the derived sufficient conditions for the construction of interval observer are related to the switching times t i , which makes it difficult to check these conditions. Compared to the system considered in [28], the system considered in this paper is more extensive. Besides, the interval observer constructed in this paper is time-invariant, and the sufficient conditions for its existence are time-independent, so they are easier to be checked.
Remark 3. In [32], the hybrid framer of the switched system was designed to satisfy the upper and the lower bounds, i.e.
x ≤ x ≤x, but the framer was not necessarily asymptotically stable. Besides, the conditions for the construction were derived based on common Lyapunov function. In the paper, multiple Lyapunov functions method is adopted for the asymptotical stability of the hybrid framer, so the sufficient conditions given in the paper are less conservative.
Remark 4. In [27], the interval observer design problem for discrete-time switched systems with asynchronous switching law was investigated. It was assumed that the switching signal of the interval observer is asynchronous with the one of the subsystems. Unlike the literature, we study the asynchronous control based on interval observer for switched continuous-time systems, that is, the asynchronous phenomenon occurs between the controller and the subsystems.

Interval observer-based control
Suppose that the systems (17) where The control signal going into the plant is of the form whereK i and K i are the controller gains of the subsystem i, d (t ) is the uncertain switching delay, satisfying 0 ≤ d (t ) ≤ d .
Here we assume that the maximal switching delay d is known a priori without loss of generality, and d ≤ t i+1 − t i , i ∈ ℕ.
In the paper, we set 1 (t ) = (t ), 2 (t ) = 1 (t − d (t )) and adopt the merging switching signal ′ = ( 1 (t ), 2 (t )) to study the asynchronous switching signals in a unified framework. The merging action means that the set of switching times of σ′ is the union of the sets of switching times of 1 and of 2 .
We introduce two lemmas.

Under Assumption 1, for the gain matrices L i , i ∈ M given by Lemma 4 and known positive constants , , and
where then with the interval observer-based controller (27), whenw and̄are identically equal to zero, the switched system (1) is globally uniformly exponentially stable for any switching signal with average dwell time satisfying Proof. The proof of this result takes the following two steps.
Step 1. We use the mathematical induction method to prove that x ≤ x ≤x for all t ≥ 0, wherex and x are computed by (21) and (22).
Since F 0 is assumed to be Metzler. Consider (17), (19), and (21)-(24), by Lemma 4, it holds that x ≤ x ≤x for all t ∈ 0 . Now, suppose that for all t ∈ i−1 , the inequality holds, which implies is true for all t ∈ i−1 by Lemma 2 and (21) and (22). We will show that for any t ∈ i , (35) is true. In fact, since F i is assumed to be Metzler, consider (17)- (22), by Lemma 4, it holds that x ≤ x ≤x for all t ∈ i .

Theorem 2. Consider matrices
j , then with the interval observer-based controller (27), whenw and̄are identically equal to zero, the switched system (1) is globally uniformly exponentially stable for any switching signal with average dwell time satisfying Moreover, the interval observer-based state-feedback controller gains are given byK j = H 1 jP −1 Proof. LetK jP j = H 1 j , K jF j = H 2 j , multiplying both sides of (30) by

NUMERICAL EXAMPLE
In this section, we consider regulator systems used in semiconductor technology [34]. In the system shown in Figure 1, Sw1 is a bipolar transistor and Sw2 is a diode. The regulator has two switching modes: mode 1, Sw1 is closed and Sw2 is off; mode 2, Sw1 is off and Sw2 is closed. Selecting the state variable T , control input u(t ) = V s , where x 1 is the inductance current I L , x 2 is the capacitor voltage V c and V s is the power supply voltage. Under the different modes ( (t ) = 1, 2), FIGURE 1 A circuit diagram of the buck/boost regulator [34] the system matrices of the buck/boost regulator are given by (t ) = 1 : Here, we choose the system parameters L = 1H , C = 1F , R 1 = 0.5Ω, R s == 2Ω, and suppose the other system matrices are , v 1 (t ) = v 2 (t ) = sin(t ) (2 + t ) 2 withv =  For the parameters = 0.1, = 0.11, = 15, solving the LMIs in Lemma 4, we obtain the interval observer-based statefeedback controller gains

CONCLUSION
This paper has investigated asynchronous interval observerbased controller for switched systems with additional disturbance and measurement noise. By giving the upper bound and the lower bound of the disturbance and the measurement noise, we have constructed hybrid interval observer for the considered switched linear system by means of a coordinate transformation, which can transform the observer error systems into cooperative ones. Then, based on multiple Lyapunov functions method and average dwell time scheme, some sufficient conditions have been derived and applied to build the interval observer-based state feedback controller. The design of the event-triggered interval observer-based controller is of great significance which deserves further study.

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