Output regulation for a class of positive switched systems

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Output regulation for a class of positive switched systems Peng Wang a,b,∗, Dan Ma a,b, Jun Zhao a,b a State

Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang,110819, China b College of Information Science and Engineering, Northeastern University, Shenyang,110819, China Received 5 October 2017; received in revised form 19 December 2018; accepted 20 December 2018 Available online xxx

Abstract In this paper, a complete procedure for the study of the output regulation problem is established for a class of positive switched systems utilizing a multiple linear copositive Lyapunov functions scheme. The feature of the developed approach is that each subsystem is not required to has a solution to the problem. Moreover, two types of controllers and switching laws are devised. The first one depends on the state together with the external input and the other depends only the error. The conditions ensuring the solvability of the problem for positive switched systems are presented in the form of linear matrix equations plus linear inequalities under some mild constraints. Two examples are finally given to show the performance of the proposed control strategy. © 2019 Published by Elsevier Ltd on behalf of The Franklin Institute.

1. Introduction Switched systems, which comprise continuous dynamics coupled with discrete logic rules determining the active subsystems (modes) at switching instants, are a class of hybrid dynamical systems [1–4]. In practice, many power systems, mechanical systems, chemical processes and thermal models can be modeled as switched systems. Over the past few years,

∗ Corresponding author at: State Key Laboratory of Synthetical Automation for Process Industries, College of Information Science and Engineering, Northeastern University, Shenyang, China. E-mail addresses: [email protected] (P. Wang), [email protected] (J. Zhao).

https://doi.org/10.1016/j.jfranklin.2018.12.029 0016-0032/© 2019 Published by Elsevier Ltd on behalf of The Franklin Institute.

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many basic problems concerning switched systems have been extensively investigated. For example, stability and stabilization [5,6], dissipativity theory synthesis [7–10], filter design [11–14], tracking control problem [15–17]. Besides the above problems, some researchers have focused on the output regulation problem (ORP) for dynamic systems. The problem is an important control problem in the study of switched systems. The earlier study of this problem can be traced back to the work in [21,22]. The key goal of the problem has three aspects: achieving the tracking error to the origin asymptotically and rejecting a family of external disturbances, and meanwhile guaranteeing the internal stability. Moreover, this problem stems from various applications such as disturbance rejection for flight vehicles, coordination and manipulation of robots, and vibration suppression of high speed trains [23], and several others. On the other hand, many physical systems involve variables that are always confined to the positive orthant. The interest is that positive systems capture the positivity characteristic of this kind of systems [24]. Further, positive switched systems (PSSs) can be viewed as a special class of switched systems that satisfies the positivity constraint. Examples of PSSs can be encountered in numerous engineering fields, such as queueing models and HIV mitigation therapy [25]. Because of significant applications of PSSs, in recent decades, this class of system has attracted the attention of many researchers, e.g [26–30]. Despite the study of on PSSs can be found in vast literature, the ORP of PSSs is still a challenging problem. The ORP of PSSs is closely related to both the switching law and the positivity of systems. Recently, the ORP has been considered for switched linear and nonlinear systems using different switching laws. A zero-error manifold switching policy is employed to investigate the ORP in which the exosystems switch synchronously with the plant [36]. However, the resulting condition is difficult to check. In [37] it is shown that the ORP of switched bimodal linear systems has been addressed by the use of a parameter-dependent switching strategy. Besides, the solution of the optimal ORP on the basis of randomly switching was derived in [38]. Under average dwell time constraints, the ORP and the almost ORP were studied of switched impulsive systems [20] and switched non-linear systems [39], respectively. Unluckily, the above approaches demand that the problem for all subsystems is solvable. Thus, they cannot be applied to the case in which the problem for subsystems is unsolvable. In this paper, we take into account that each subsystem is not required to have a solution to the problem. Thus, the problem for PSSs will become more and more difficult. Also no related results have been reported for PSSs by now, which partially motivates the study. In addition, most previous results and the well-developed methods on ORP for general switched systems are also not directly applicable to PSSs because they take no account of the positivity constraint. For instance, when addressing stability problem with the requirement of output regulation, we know that quadratic Lyapunov functions are commonly and great choices for generally switched systems [20], but in general they lead to conservative results, and specialized tools, based on linear copositive Lyapunov functions (LCLFs) [31], turn out to be more appropriate for PSSs [29,30]. Compared with the linear matrix inequality (LMI) method for generally linear systems, [34] proposed linear programming (LP) approach to treat stability issue of positive systems. Based on LP approach, [17] applied a multiple LCLFs technique to study L1 output tracking control for PSSs. Adopting similar approach, the ORP for non-switched positive systems with constant exogenous signals was addressed in [33]. It is worth mentioning that the LP approach is more simple under the LCLFs framework than the LMI approach and possesses a low computational complexity [34]. Please cite this article as: P. Wang, D. Ma and J. Zhao, Output regulation for a class of positive switched systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2018.12.029

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Motivated by the above discussion, we are interested in disentangling the ORP of PSSs by dual design of the switching law and a set of controllers for subsystems. Specifically, the problem of each subsystem is not required to be solvable. Two cases will be considered: The first one is that both the exogenous input ω and the system state x are available. We devise jointly the full information feedback controller and the state-input dependent switching rule to solve the problem. The second case is that only the output tracking error can be available from measurements. The same property is also assured via the joint design of the dynamic error feedback controller and an error-dependent switching law. Compared with the existing works, the main contributions of this paper are summarized as three aspects. (i) Whenever the external input can be measured, a novel state-input dependent switching rule is proposed. This method is useful and is less conservative. Thus, the individual coordinates transformations is adopt to change the subsystems, which means that each group of regulator equations has its own solution for each positive subsystem. (ii) Most of existing results take no account of limiting the sign of tracking error, while this paper considers the positivity of tracking error. There are two-tier meanings: for one thing, by means of the positivity of tracking error, the overshoot phenomenon is avoided in the course of tracking; for another, we design an error based switching law dependent on the partition of the nonnegative state space instead of the whole state space. (iii) Although the ORP of each positive subsystem is unsolvable, the problem of PSSs can become feasible by dual design of feedback controllers and suitable switching laws, which greatly enlarge the degree of freedom in the design. Moreover, all conditions can be easily checked by employing LP tools under some mild constraints. The body of the paper is outlined as follows. Section 2 characterizes the ORP. Section 3 establishes the solvability conditions of the problem by dual design of controllers and switching laws. Two illustrative examples and simulations are given in Section 4. Finally, the conclusions are offered in Section 5. n Notation. Rn means the n-dimensional Euclidean space, and the set R+ is denoted by n n R+ = {x ∈ R : x  0}. aij denotes the entry in row i and column j of a matrix A, aij > 0 ( ≥ 0) for all i, j stand for A0 (0) and A is called Metzler matrix if aij ≥ 0 for all i = j. AT denotes the transpose of matrix A. Denote vec operation which is comprised of the column vector of a matrix gained by means of stacking the columns of the matrix one above the other. let  be the Kronecker product, for any matrices P, Q, R with compatible dimensions, we have that vec (PQR) = (RT  P )vec (Q). I is an identity matrix.

2. Problem formulation and preliminaries Consider the switched system described by x˙(t ) = Aσ (t ) x(t ) + Bσ (t ) u(t ) + Pσ (t ) ω(t ) e(t ) = Cσ (t ) x(t ) − Qσ (t ) ω(t ),

(1)

where x(t) ∈ Rn and u(t) ∈ Rl refer to the system state and the control input, respectively. ω(t ) ∈ Rr denotes the exogenous signal including the reference inputs to be tracked or/and the disturbances to be rejected. The second equation defines the error e(t) ∈ Rp between the actual plant output Cσ x and a reference signal Qσ w that the plant output is required to track. In order to guarantee the positivity of the actual output and the reference signal, we require that Cσ 0 and Qσ 0 are constant matrices with proper dimensions. Moreover, the ω(t) is Please cite this article as: P. Wang, D. Ma and J. Zhao, Output regulation for a class of positive switched systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2018.12.029

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are generated by ω˙ (t ) = Sω(t ),

(2)

where S is a Metzler matrix. The switching signal σ (t ) : [0, ∞ ) → Iˆm = {1, . . . , m} is a piecewise constant function of time or state. In accordance with the switching signal σ (t), the switching sequence can be expressed by the following form:  = {x0 ; (i0 , t0 ), (i1 , t1 ), . . . , (ik , tk ), . . . , |ik ∈ Iˆm , k ∈ N }, where x0 , t0 , N are the initial state, the initial time and the set of nonnegative integers, respectively. It is shown that the il th subsystem is activated when t ∈ [tl , tl+1 ). m is the number of positive subsystems of the PSSs. Assume that Ai , Bi , Pi are constant matrices with proper dimensions and finite many switchings occur during any finite time. Some necessary assumption, lemmas and definition for the switched system are given for subsequent developments. Assumption 1. The eigenvalues of the Metzler matrix S are in the closed right half-plane. Lemma 1 [29]. The system x˙(t ) = Aσ (t ) x(t ) is positive for any switched signal σ (t) if and only if each Ai is Metzler matrix, ∀i ∈ Iˆm . Lemma 2 [29]. Let Ai be Metzler matrix, then the following conditions are equivalent: (a) Ai are Hurwitz, ∀i ∈ Iˆm . (b) There exists vectors ν i 0 in Rn such that ATi νi ≺ 0 hold, ∀i ∈ Iˆm . Lemma 3 [32]. Ai are Metzler matrices if and only if there exists positive constants ζ i such that Ai + ζi I  0, ∀i ∈ Iˆm . Definition 1 [24]. The system (1) is said to be a PSS if for any switched signal σ (t), any non-negative initial condition and non-negative inputs, the state trajectory x(t)0 for all t ≥ 0. Remark 1. According to [24], a PSS satisfying the condition in Definition 1 is usually called internally PSS, which is explicit distinction from externally PSSs. In this paper, namely, we only cope with the internally PSSs. Lemma 4 [24]. The system (1) is positive under any switching signal σ (t) if and only if Ai are Metzler matrices, Bi 0, Pi 0, ∀i ∈ Iˆm . Now, the ORP of the systems (1) and (2) is to be given below. This problem of the PSSs means that we need to devise suitable switching rules and feedback control strategies, which achieve the following objectives: reaching asymptotic tracking, accomplishing disturbance rejection, making the system positive, stabilizing the overall closed-loop systems. According to the available information, two different regulators are in a position to handle the above problem that contains the full information feedback and the error feedback. ORP using full information feedback and switching. The full information controllers can be devised if both x(t) and ω(t) are available for feedback. Given the systems (1) and (2), co-design the full information feedback controller u(t ) = Kσ (t ) x(t ) + Lσ (t ) ω(t )

(3)

and a switching law σ (t) such that Please cite this article as: P. Wang, D. Ma and J. Zhao, Output regulation for a class of positive switched systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2018.12.029

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(i) The system (1) with the expected controller (3) is positive and asymptotically stable under the designed switching law σ (t) when ω(t ) = 0. (ii) For each nonnegative (x(0), ω(0)), the solution (x(t), ω(t)) of x˙(t ) = (Aσ (t ) + Bσ (t ) Kσ (t ) )x(t ) + (Pσ (t ) + Bσ (t ) Lσ (t ) )ω(t ) ω˙ (t ) = Sω(t )

(4)

satisfies lim e(t ) = lim (Cσ (t ) x(t ) − Qσ (t ) ω(t )) = 0.

t→∞

t→∞

(5)

ORP using error feedback and switching. Depending only on the measured output error e(t), we can devise the following dynamic error feedback controller ξ˙(t ) = Fσ (t ) ξ (t ) + Gσ (t ) e(t ), u(t ) = Hσ (t ) ξ (t ),

(6)

where the gain matrices Fi , Gi , Hi , i ∈ Iˆm are to be determined. Given the systems (1) and (2), it consists of designing dynamic error controller (6) and a switching law σ (t) such that (i) The system (1) with the desired controller (6) is positive and asymptotically stable under the designed switching law σ (t) when ω(t ) = 0. (ii) For each nonnegative (x(0), ξ (0), ω(0)), the solution (x(t), ξ (t), ω(t)) of x˙(t ) = Aσ (t ) x(t ) + Bσ (t ) Hσ (t ) ξ (t ) + Pσ (t ) ω(t ) ξ˙(t ) = Fσ (t ) ξ (t ) + Gσ (t )Cσ (t ) x(t ) − Gσ (t ) Qσ (t ) ω(t )

(7)

ω˙ (t ) = Sω(t ) satisfies lim e(t ) = lim (Cσ (t ) x(t ) − Qσ (t ) ω(t )) = 0

t→∞

t→∞

(8)

Our key goal is to determine controller gains and switching rules jointly such that the ORP can be addressed for PSSs. 3. Main result Building on the available information, two different methods are possible for solving the ORP for PSSs: the full information controller together with a state-input dependent switching law and the error feedback controller as well as an error-dependent switching law - both to be discussed below. Now, we start with the full information case. 3.1. Full information feedback regulator In this subsection, the full information controller and a novel state-input dependent switching law can be devised jointly if the external input ω(t) and the state x(t) are available for both feedback and switching. Please cite this article as: P. Wang, D. Ma and J. Zhao, Output regulation for a class of positive switched systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2018.12.029

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Theorem 1. Consider the ORP of the systems (1) and (2). If (i) there exist matrices  i , Ti , ∀ i ∈ Iˆm fulfilling the linear matrix equations: Ai Ti + Bi i + Pi = Ti S,

(9)

Ci Ti = Qi .

(10)

(ii) for given scalars π ij and vectors μi , there exist constants β i and a series of vectors ν i , λi for all i, j ∈ Iˆm such that the following conditions νi  0, μi  0, βi > 0, πi j ≤ 0,

(11)

vec (Ai )(μTi BiT νi ) + (I  Bi μi )vec (λTi ) + βi vec (I )  0,

(12)

ATi νi

+ λi +

m 

πi j (νi − ν j ) ≺ 0,

(13)

j=1 m 

πi j (T j − Ti )T ν j 0,

(14)

j=1

then, under the switching law T σ (t ) = arg min{x˜cl,i (t )νi }

(15)

i∈Iˆm

and the full information feedback controller in the form of Eq. (3)   μi λT μi λT u(t ) = T Ti x(t ) + i − T Ti Ti ω(t ), μi Bi νi μi Bi νi the ORP is solved for the systems (1) and (2). Proof. Consider the different coordinates transformations x˜cl,i (t ) = x(t ) − Ti ω(t ) and let Li = i − Ki Ti . Applying the regulator Eqs. (9) and (10), substituting the controller (3) into the system (1) and using the different coordinates transformations, one can show that the closedloop system of the new variable x˜cl,σ (t ) (t ) can be represented in the form x˙˜cl,σ (t ) (t ) = A˜ σ (t ) x˜cl,σ (t ) (t ),

(16)

e(t ) = Cσ (t ) x˜cl,σ (t ) (t ),

where A˜ i = Ai + Bi Ki , i ∈ Iˆm . Therefore, the ORP of the systems (1) and (2) can be transformed into the stabilization problem of the closed-loop system (16). First of all, we prove the positivity of the closed-loop system (16). Since μi 0 with μi ∈ Rr , Bi 0 with Bi ∈ Rn × r and ν i 0 with ν i ∈ Rn , we can easily obtain μTi BiT νi > 0. On the basis of Eq. (12), we can finally show that Ai +

1 μTi BiT νi

Bi μi λTi +

βi I T μi BiT νi

 0.

(17)

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By Lemma 1, Ai + μT B1T νi Bi μi λTi are Metzler matrices. Namely, Ai + Bi Ki are Metzler matrices i i and Ci 0 for i ∈ Iˆm , then one can verify that the closed-loop system (16) is a positive system. Choose the multiple LCLFs candidate for the closed-loop system (16) as T Vσ (t ) (x˜cl,σ (t ) ) = x˜cl,σ (t ) νσ (t ) .

(18)

Computing the time derivative of Vi (x˜cl,i ) along with the trajectory of the ith subsystem of the closed-loop system (16), we have T V˙i (x˜cl,i (t )) = x˜cl,i (t )(Ai + Bi Ki )T νi .

(19)

Performing the different coordinates transformations, we have x˜cl,i (t ) = x(t ) − Ti ω(t ) and x˜cl, j (t ) = x(t ) − T j ω(t ). In addition, Vi (x˜cl,i ) − V j (x˜cl, j ) = x˜cl,i (t )νi − x˜cl, j (t )ν j T = x˜cl,i (t )(νi − ν j ) + ωT (t )(T j − Ti )T ν j .

(20)

On account of the switching law (15) and the scalars π ij ≤ 0, whenever the ith subsystem is active, one derives ⎛ ⎞ ⎛ ⎞ m m   T x˜cl,i (t )⎝ πi j (νi − ν j )⎠ + ωT (t )⎝ πi j (T j − Ti )T ν j ⎠ ≥ 0, j=1

j=1

which infers that when the ith subsystem is activated, we get ⎛ ⎞ m  T T V˙i (x˜cl,i (t )) ≤ x˜cl,i (t )(Ai + Bi Ki )T νi + x˜cl,i (t )⎝ πi j (νi − ν j )⎠ ⎛ +ωT (t )⎝

m 

⎞

j=1

πi j (T j − Ti )T ν j ⎠

j=1

⎡ ⎤ m  T

T (Ai + Bi Ki )T νi + πi j (νi − ν j ) ⎥ x˜ (t ) ⎢ j=1 ⎢ ⎥ = cl,i m  ⎣ ⎦ ω(t ) T πi j (T j − Ti ) ν j j=1

≤ 0. Then, we can demonstrate that the closed-loop system (16) is asymptotically stable, i.e. limt→∞ x˜cl,σ (t ) (t ) = 0. Meanwhile, we obtain lim e(t ) = lim (Cσ (t ) x(t ) − Qσ (t ) ω(t )) = lim Cσ (t ) x˜cl,σ (t ) (t ) = 0.

t→∞

t→∞

t→∞

(21)

Therefore, it can be verified that the ORP of the systems (1) and (2) can be achieved. Remark 2. Due to the terms KiT BiT νi (i ∈ Iˆm ) of variables Ki and ν i being mutually coupling, the solvability conditions of the ORP attained by Theorem 1 may not be computationally feasible. Consequently, we introduce variables λi = KiT BiT νi in Eq. (13) to convert nonlinear inequalities into linear inequalities. In other words, the solution for the ORP is obtained by exploiting LP tools when the scalars π ij and the vectors μi are fixed. Moreover, using the Please cite this article as: P. Wang, D. Ma and J. Zhao, Output regulation for a class of positive switched systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2018.12.029

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vectorization operator vec(•), one can rewrite the inequality (12) as vec (μTi BiT νi Ai + Bi μi λTi + βi I ) = vec (μTi BiT νi Ai + Bi μi λTi I + βi I ) = vec (μTi BiT νi Ai ) + vec (Bi μi λTi I ) + vec (βi I ) = vec (Ai )(μTi BiT νi ) + (I  Bi μi )vec (λTi ) + βi vec (I )  0.

(22)

Obviously, Eq. (22) is a standard linear inequality and the conversion form of Eq. (22) is different from [32]. Remark 3. Concerning this result, it is important to remark that it is difficult to use individual coordinates transformations to change the subsystems, since we will have a switched systems with subsystems expressed in different coordinates. This difficulty has been overcome in Theorem 1 by design of a novel state-input dependent switching signal. Different from the typically common solution method [10,18,19] and impulsive dynamic method [20], our method is novel and is less conservative. 3.2. Output regulation via error feedback Theorem 1 gives a result of the ORP utilizing the full information state feedback. However, in practice, system states are usually unavailable. In this subsection, we consider that only the output error e(t) is measurable. The next theorem provides solvability conditions to the ORP of the systems (1) and (2) by means of the dynamic error feedback and presents a error-dependent switching law assuring that the ORP is solved. In order to devise the errordependent switching law, we need to assume that C = Ci , Q = Qi (∀ i ∈ Iˆm ) in the following theorem. Theorem 2. Consider the ORP of the systems (1) and (2) utilizing the dynamic error feedback. If (i) there exist matrices T, , Hi , Fi , (∀ i ∈ Iˆm ) meeting the linear matrix equations: Ai T + Bi Hi  + Pi = T S,

(23a)

S = Fi ,

(23b)

CT = Q.

 (ii) for given scalars π ij , there exist a set of vectors gi ∈ Rn , q = q1 α is ∈ Rn , s = 1, 2, . . . , n, ∀i ∈ Iˆm , constants εi such that

(23c) q2

···

qn

T

∈ Rn ,

gi  0, q  0, αis  0, εi > 0, πi j ≤ 0,

(24a)

− Fi − εi I 0,

(24b)

− Bi Hi 0,

(24c)

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− CT

n 

αis 0,

9

(24d)

s=1

ATi C T gi + C T

n 

αis +

m 

s=1

πi j C T (gi − g j ) ≺ 0,

(24e)

j=1

HiT BiT C T gi + FiT q ≺ 0,

(24f)

then, under the switching law σ (t ) = arg min{eT (t )gi }

(25)

i∈Iˆm

and the error feedback controller in the form of Eq. (6), the ORP is achieved for the systems (1) and (2). Moreover, the associated gain matrices of the controller is given by  T  T Gi = ai1 ai2 · · · ain = q1−1 αi1 q2−1 αi2 · · · qn−1 αin . Proof. Taking the coordinates transformation x¯(t ) = x(t ) − T ω(t ), ξ¯(t ) = ξ (t ) − ω(t ) and using Eqs. (6) and (23a), one has ˙¯ ) = Aσ (t ) (x¯(t ) + T ω(t )) + Bσ (t ) Hσ (t )(ξ¯(t ) + ω(t )) + (Pσ (t ) − T S)ω(t ) x(t = Aσ (t ) x¯(t ) + Bσ (t ) Hσ (t ) ξ¯(t ).

(26)

Similar to the process above, and from Eq. (23b) we have ξ¯(t ) = Fσ (t ) (ξ¯(t ) + ω(t )) + Gσ (t )C x¯(t ) − Sω(t ) = Fσ (t ) ξ¯(t ) + Gσ (t )C x¯(t ).

(27)

Moreover, from Eq. (23c), we get e(t ) = Cx(t ) − Qω(t ) = C(x¯(t ) − T ω(t )) − Qω(t ) = C¯ x¯(t ).

(28)

In view of Eqs. (26)–(28), the closed-system (7) can be expressed as  x(t ˆ˙ ) = A¯ σ (t ) xˆ(t ), e(t ) = C¯ xˆ(t ),

  Bi Hi , C¯ = C 0 . where xˆ(t ) = col (x¯(t ), ξ¯(t )), A¯ i = GACi F i

(29)

i

Next, on account of Eq. (24b) and Lemma 3, we have that Fi are Metzler matrices, i ∈ Iˆm . According to Eq. (24c) and (24d), it is obviously that Bi Hi 0, C0 and Gi C0, i ∈ Iˆm . Thus, the enlarged system (29) is positive. In fact, it can be shown that the ORP of the systems (1) and (2) is in a position to be converted into a stabilization problem for the enlarged system (29). Please cite this article as: P. Wang, D. Ma and J. Zhao, Output regulation for a class of positive switched systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2018.12.029

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Construct the following multiple LCLFs T

C gi . Vσ (t ) (xˆ ) = xˆT νσ (t ) and νσ (t ) = q

(30)

According to the special structure of ν i , i ∈ Iˆm , we have xˆT (t )νi = x¯T (t )C T gi + ξ¯T (t )q = eT (t )gi + ξ¯T (t )q.

(31)

From Eq. (31), one attains min{xˆT (t )νi } = min{eT (t )gi } + ξ¯T (t )q. i∈Iˆm

i∈Iˆm

(32)

In term of Eqs. (25) and (32), we know that σ (t ) = arg min{eT (t )gi } = arg min{xˆT (t )νi }. i∈Iˆm

i∈Iˆm

(33)

In what follows, by virtue of Eq. (33) and the scalars π ij ≤ 0, when the ith subsystem is active, it follows that ⎛ ⎞ m  xˆT (t )⎝ πi j (νi − ν j )⎠ ≥ 0. j=1

Differentiating Vi (xˆ ) along with the trajectory of the ith subsystem of the enlarged system (29) gives

T

T T

x¯(t ) Bi Hi Ai C gi ˙ Vi (xˆ(t )) = ¯ Gi C Fi q ξ (t )

T

T T

x¯(t ) Bi Hi Ai C gi ≤ ¯ Gi C Fi q ξ (t ) ⎛ ⎞

T 

m x¯(t ) ⎝ − g g j ⎠ + ¯ πi j C T i q−q ξ (t ) j=1

≤ 0.

(34)

Furthermore, using the technique of matrix decomposition, we obtain GTi q =

n 

ais qs =

s=1

n 

αis .

(35a)

s=1

Substituting Eq. (35a) into Eq. (24e) yields ATi C T gi + C T GTi q +

m 

C T πi j (gi − g j ) ≺ 0.

(36a)

j=1

At last, Eq. (34) implies the enlarged system (29) is asymptotically stable, i.e. limt→∞ xˆ(t ) = 0. Furthermore, we finally obtain lim e(t ) = lim C¯ xˆ(t ) = 0.

t→∞

t→∞

(37)

The proof is completed. Please cite this article as: P. Wang, D. Ma and J. Zhao, Output regulation for a class of positive switched systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2018.12.029

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Remark 4. It is observed that the sufficient condition of Theorem 2 is presented in the form of a set of linear inequalities if the scalars π ij and the vectors μi are fixed. Compared with the existing results [10,20] that ones use the linear matrix inequalities technique or dilated linear matrix inequalities to addresses the ORP, while the obtained linear inequalities can be easily checked by means of LP tools. 4. Illustrative examples In this section, we provide two examples to display the effectiveness of the main results. The first example is for the full information feedback controller and switching case, while the second one is for the error feedback controller and switching case. Example 1. Consider the switched linear model of a turbofan engine established in [40]. We use the proposed control scheme of the output regulation to the speed control of a turbofan engine model during the acceleration procedure. The switched linear model is obtained by linearising the non-linear model of the turbofan engine. This model was used to address the stabilisation problem of PSSs in [41] and the dynamical behavior of this model with respect to exogenous input signal is described as follows:





˙ f (t ) N N f (t ) ω1 (t ) = + , A B W + P σ (t ) σ (t ) f σ (t ) ˙ c (t ) Nc (t ) ω2 (t ) N (38)



N f (t ) ω (t ) − Qσ (t ) 1 , Ps30 = Cσ (t ) Nc (t ) ω2 (t ) where the state of the linearized system is x = [N f , Nc ]T and the control input is u = W f . Nc and Nf denote the core speed and fan speed, respectively. Wf is the fuel flow and Ps30 represents the static pressure.  means the increment. And the reference output is generated by



ω˙1 (t ) ω (t ) =S 1 , (39) ω˙2 (t ) ω2 (t ) The exogenous signal ω1 (t) represents the reference fan speed increment and ω2 (t) refers to the reference core speed increment. The subsystem matrices are given by



1.4467 −3.8557 230.6739 , B1 = , A1 = 0.4690 −4.7081 653.5547



1.4001 −3.7401 231.5508 , B2 = , A2 = 0.4752 −4.5586 657.3084





0 2.6371 0.5093 4.2645 1.2444 0.6 , P2 = , S= , P1 = 6.6449 11.8977 0.2143 2.8311 0 0.5     C1 = 1.2 1.1 , C2 = 1.12 1.32 ,     Q1 = 0.4 0.39 , Q2 = 0.43 0.44 . According to the above matrices, the solution of the regulator Eqs. (9) and (10) can be readily obtained as

  1 1.5 1 = 0.005 0.001 , T1 = , 2 2.5 Please cite this article as: P. Wang, D. Ma and J. Zhao, Output regulation for a class of positive switched systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2018.12.029

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4.5 ω1

4

ω

2

3.5 3

ω (t)

2.5 2 1.5 1 0.5 0 0

2

4

6

8

10

t(sec) Fig. 1. The exosystem ω(t).

 2 = 0.01

 0.01 , T2 =



2 1.5

1.5 . 2

By implementing LP with μ1 = μ2 = 1, π21 = −4, and π12 = −3, one can find that the conditions in Theorem 1 are feasible with the following solution ν1 = [0.0895 0.0492]T , ν2 = [0.0966 0.0418]T which further derives the controller gain matrices as     K1 = 0.0038 0.0014 , K2 = 0.0049 0.0013 . Since all conditions of Theorem 1 have been satisfied, it is verified that the ORP for the PSS Eq. (1) has been solved under both the designed controller (16) and the switching signal Eq. (15). The simulation results are shown in Figs. 1–4. From Fig. 1, we can see that the exosystem (39) is unstable. It is observed that the desired switching signal is presented in Fig. 2. Fig. 3 describes the trajectory of the closed-loop system state x˜cl,σ (t ), whose initial value is x˜cl,σ (0) = [6.5 2.3]T . Finally, Fig. 4 shows that the output error of PSS (1) truly decays to zero asymptotically. In practice, complete access to the state is unpractical from the view of measurements, and the control approach via state feedback would not be implemented [35]. Therefore, we devise an error-dependent switching strategy to solve the OPR in Theorem 2. Example 2. A numerical example will be provided to favor Theorem 2.



1.5 −5 1.5 , B1 = , A1 = 0.5 −3.8 0.5 Please cite this article as: P. Wang, D. Ma and J. Zhao, Output regulation for a class of positive switched systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2018.12.029

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Fig. 2. The switching signal σ .

Fig. 3. The closed-loop system state x˜cl,σ (t ).

Fig. 4. The error output e(t). Please cite this article as: P. Wang, D. Ma and J. Zhao, Output regulation for a class of positive switched systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2018.12.029

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Fig. 5. The switching signal σ (t).

Fig. 6. The output error of the enlarged systems.





−4 1.4 0.4 , B2 = , A2 = 0.5 −5 2



1.0 5.0250 1.7500 3.725 , P2 = , P1 = 7.5500 1.25 9.95 13.25



0 0 0.35 3.2 4.8 , C= , Q= S= 0 0.5 0 3.5 7

3.2 . 8.75

By resorting to the above parameters and solving regulator Eq. (23a)–(23c), we can obtain the following matrices in Theorem 2.



T

1 1.5 1.2 0 0 , H1 = , T = , 1 = 2 = 2 2.5 1.3 0 0

T

0.5 0.4 −1.4 1.1 −1.6 , H2 = . F1 = , F2 = 0.46 −1.5 1.6 0.56 −1.3 Please cite this article as: P. Wang, D. Ma and J. Zhao, Output regulation for a class of positive switched systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2018.12.029

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2.5

2

State

1.5

1

0.5

0 0

5

10 t(sec)

15

20

Fig. 7. The state xˆ(t ) response of the enlarged systems.

Let π12 = −0.29, π21 = −0.15. Then, the LP problem Eq. (24a)–(24f) of Theorem 2 is feasible. A feasible solution of the gain matrices is G1 =

0.0812 0.0748

0.1924 0.1772

T

, G2 =

0.1737 0.1600

0.0010 0.0009

T .

The effectiveness of the proposed control strategy is shown in Figs. 5–7. Let the initial condition xˆ(0) = [2.5 0.8 0.6 1.8]T to be given. According to Eq. (25), the error-dependent switching signal is presented in Fig. 5. Fig. 6 describes the output tracking error of the augmented positive systems. In addition, Fig. 7 shows that the state trajectories of the augmented systems decay to zero asymptotically. 5. Conclusions In this paper, we have investigated the ORP for a class of PSSs by exploiting a multiple LCLFs method. Two kind of switching laws, including the state-input-dependent switching law and the error-dependent switching law, have been designed. In particular, even though the ORP for each positive subsystem cannot be disentangled, the problem for the PSSs has been solved using the joint designed controllers and switching laws. The solvability conditions for the problem have also been established. All conditions are given by linear matrix equations and linear inequalities, which can be easily computed by means of standard linprog toolbox. The effectiveness of the objective results has been exhibited through two examples. In our work, the target plant is a linear dynamic system. However, many physical systems possess the property of networks, such as compartmental systems, multiagent systems [42], and several others. One of the future research topics is to extend the main results to networked control systems utilizing the sampled-data control strategy [43]. Please cite this article as: P. Wang, D. Ma and J. Zhao, Output regulation for a class of positive switched systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2018.12.029

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Acknowledgment This work was supported by the National Natural Science Foundation of China under Grants 61773098, 61603079 and the 111 Project (B16009). References [1] D. Liberzon, A.S. Morse, Basic problems in stability and design of switched systems, IEEE Trans. Autom. Control 19 (5) (1999) 59–70. [2] D. Liberzon, Switching in Systems and Control, Boston, 2003. [3] M.S. Branicky, Multiple Lyapunov functions and other analysis tools for switched and hybrid systems, IEEE Trans. Autom. Control 43 (1998) 475–482. [4] H. Lin, P.J. Antsaklis, Stability and stabilizability of switched linear systems: a survey of recent results, IEEE Trans. Autom. Control 54 (2) (2009) 308–322. [5] Y.E. Wang, X.M. Sun, F. Mazenc, Stability of switched nonlinear systems with delay and disturbance, Automaitca 69 (2016) 78–86. [6] X.J. Su, L.G. Wu, P. Shi, C.L.P. Chen, Model approximation for fuzzy switched systems with stochastic perturbation, IEEE Trans. Fuzzy Syst. 23 (5) (2015) 1458–1473. [7] H. Yang, V. Cocquempot, B. Jiang, Fault tolerance analysis for switched systems via global passivity, IEEE Trans. Circuits Syst. II, Exp. Briefs 55 (12) (2008) 1279–1283. [8] J. Zhao, D.J. Hill, Dissipativity theory for switched systems, IEEE Trans. Autom. Control 53 (4) (2008) 941–953. [9] P. Wang, J. Zhao, Dissipativity of positive switched systems using multiple linear supply rates, Nonlinear Anal. Hybrid Syst. 32 (2019) 37–53. [10] J. Li, J. Zhao, Incremental passivity and incremental passivity-based output regulation for switched discrete-time systems, IEEE Trans. Cybern. 47 (5) (2017) 1122–1132. [11] L.X. Zhang, P. Shi, E.K. Boukas, C.H. Wang, Robust l2 − l∞ filtering for switched linear discrete time-delay systems with polytopic uncertainties, IET Control Theory Appl. 1 (3) (2007) 722–730. [12] X.J. Su, P. Shi, L.G. Wu, Y.-D. Song, Fault detection filtering for nonlinear switched stochastic systems, IEEE Trans. Autom. Control 61 (5) (2016) 1310–1315. [13] D. Wang, Z. Wang, G.Y. Li, W. Wang, Fault detection for switched positive systems under successive packet dropouts with application to the leslie matrix model, Int. J. Robust Nonlinear Control 26 (13) (2016) 2807–2823. [14] D. Zhang, P. Shi, W.A. Zhang, L. Yu, Energy-efficient distributed filtering in sensor networks: a unified switched system approach, IEEE Trans. Cybern. 47 (7) (2017) 1618–1629. [15] B. Niu, Y.J. Liu, G.D. Zong, Z.Y. Han, J. Fu, Command filter-based adaptive neural tracking controller design for uncertain switched nonlinear output-constrained systems, IEEE Trans. Cybern. 47 (10) (2017) 3160–3171. [16] L.J. Long, T. Si, Small-gain technique-based adaptive NN control for switched pure-feedback nonlinear systems, IEEE Trans. Cybern. 99 (2018) 1–12. [17] J. Wang, J. Zhao, Output tracking control with l1 -gain performance for positive switched systems, J. Frankl. Inst. 354 (10) (2017) 3907–3918. [18] X.X. Dong, J. Zhao, Output regulation for a class of switched nonlinear systems: an average dwell-time method, Int. J. Robust Nonlinear Control 23 (4) (2013) 439–449. [19] L.L. Li, C.L. Jin, X. Ge, Output regulation control for switched stochastic delay systems with dissipative property under error-dependent switching, Int. J. Syst. Sci. 49 (2) (2018) 383–391. [20] C.Z. Yuan, F. Wu, Almost output regulation of switched linear dynamics with switched exosignals, Int. J. Robust Nonlinear Control 27 (16) (2017) 3197–3217. [21] B.A. Francis, W.M. Wonham, The internal model principle of control theory, Automatica 12 (5) (1976) 457–465. [22] E.J. Davison, The robust control of a servomechanism problem for linear time-invariant multivariable systems, IEEE Trans. Autom. Control 21 (1) (1976) 25–34. [23] J. Huang, Nonlinear Output Regulation: Theory and Applications, USA: SIAM, Philadelphia, PA, 2004. [24] L. Farina, S. Rinaldi, Positive Linear Systems: Theory and Applications, John Wiley and Sons, New York, 2000. [25] E. Hernandez-Vargas, R. Middleton, P. Colaneri, F. Blanchini, Discrete-time control for switched positive systems with application to mitigating viral escape, Int. J. Robust Nonlinear Control 21 (10) (2011) 1093–1111. [26] J. Lian, J. Liu, New results on stability of switched positive systems: an average dwell-time approach, IET Control Theory Appl. 7 (13) (2013) 1651–1658. Please cite this article as: P. Wang, D. Ma and J. Zhao, Output regulation for a class of positive switched systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2018.12.029

JID: FI

ARTICLE IN PRESS

[m1+;April 22, 2019;7:0]

P. Wang, D. Ma and J. Zhao / Journal of the Franklin Institute xxx (xxxx) xxx

17

[27] Y.H. Tong, C.H. Wang, L.X. Zhang, Stabilisation of discrete-time switched positive linear systems via timeand state-dependent switching laws, IET Control Theory Appl. 6 (11) (2012) 1603–1609. [28] X.W. Liu, C.Y. Dang, Stability analysis of positive switched linear systems with delays, IEEE Trans. Autom. Control 56 (7) (2011) 1684–1690. [29] O. Mason, R. Shorten, On linear copositive Lyapunov functions and the stability of switched positive linear systems, IEEE Trans. Autom. Control 52 (7) (2007) 1346–1349. [30] E. Fornasini, M.E. Valcher, Linear copositive Lyapunov functions for continuous-time positive switched systems, IEEE Trans. Autom. Control 55 (8) (2010) 1933–1937. [31] X.D. Zhao, L.X. Zhang, P. Shi, M. Liu, Stability of switched positive linear systems with average dwell time switching, Automatica 48 (6) (2012) 1132–1137. [32] J.F. Zhang, Z.Z. Han, F.B. Zhu, l1 -gain analysis and control synthesis of positive switched systems, Int. J. Syst. Sci. 46 (12) (2015) 2111–2121. [33] B. Roszak, E.J. Davison, The multivariable servomechanism problem for positive LTI systems, IEEE Trans. Autom. Control 55 (9) (2010) 2204–2209. [34] M. A. Rami, F. Tadeo, Controller synthesis for positive linear systems with bounded controls, IEEE Trans. Circuits Syst. II, Exp. Briefs 54 (2) (2007) 151–155. [35] Z. Shu, J. Lam, H.J. Gao, B.Z. Du, L.G. Wu, Positive observers and dynamic output-feedback controllers for interval positive linear systems, IEEE Trans. Circuits Syst. I, Reg. Pap. 55 (10) (2008) 3209–3222. [36] V. Gazi, Output regulation of a class of linear systems with switched exosystems, Int. J. Control 80 (10) (2007) 1665–1675. [37] Z.Z. Wu, F.B. Amara, Regulator synthesis for bimodal linear systems, IEEE Trans. Autom. Control 56 (2) (2011) 390–394. [38] W.J. Lee, P.P. Khargonekar, Optimal output regulation for discrete-time switched and Markovian jump linear systems, SIAM J. Control Optim. 47 (1) (2008) 40–72. [39] X.X. Dong, J. Zhao, Solvability of the output regulation problem for switched non-linear systems, IET Control Theory Appl. 6 (8) (2012) 1130–1136. [40] H. Richter, Advanced Control of Turbofan Engines, Springer, New York, 2012. [41] J. Wang, J. Zhao, Stabilisation of switched positive systems with actuator saturation, IET Control Theory Appl. 10 (6) (2016) 717–723. [42] D. Wang, N. Zhang, J.L. Wang, W. Wang, Cooperative containment control of multiagent systems based on follower observers with time delay, IEEE Trans. Syst. Man Cybern. Syst. 47 (1) (2017) 13–23. [43] J. Lian, C. Li, B. Xia, Sampled-data control of switched linear systems with application to an f-18 aircraft, IEEE Trans. Ind. Electron. 64 (2) (2017) 1332–1340.

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