Incremental passivity-based output regulation for switched nonlinear systems via average dwell-time method

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Incremental passivity-based output regulation for switched nonlinear systems via average dwell-time method Hongbo Pang, Jun Zhao PII: DOI: Reference:

S0016-0032(19)30103-6 https://doi.org/10.1016/j.jfranklin.2018.09.039 FI 3794

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Journal of the Franklin Institute

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3 November 2017 15 September 2018 24 September 2018

Please cite this article as: Hongbo Pang, Jun Zhao, Incremental passivity-based output regulation for switched nonlinear systems via average dwell-time method, Journal of the Franklin Institute (2019), doi: https://doi.org/10.1016/j.jfranklin.2018.09.039

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Hongbo Panga,b,∗, Jun Zhaob a College

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Incremental passivity-based output regulation for switched nonlinear systems via average dwell-time methodI

of Science,Liaoning University of Technology,Jinzhou 121001,P.R.ofChina of Information Science and Engineering, Northeastern University, State Key Laboratory of Synthetical Automation of Process Industries, Shenyang 110819, China

b College

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Abstract

This paper investigates the output regulation problem for a class of switched nonlinear systems with at least a feedback incrementally passive subsystem via average dwell time method. First, the output regulation problem for switched nonlinear system via full information feedback is solved. The stabilizing con-

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trollers consist of the state feedback controllers and linear output feedback controllers. In some particular cases, it is unnecessary to verify that all the solutions of the switched nonlinear system converge to the bounded steady-state solution,

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while we only have to verify the regulated outputs converge to zero directly. Second, a dynamic error-feedback stabilizer for each subsystem and a switched internal model whose subsystems all are incrementally passive are designed to

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solve the output regulation problem for the switched nonlinear system under a composite switching signal with average dwell times. The stabilizer and the

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internal model are interconnected in a more simple way and allowed to switch asynchronously. Finally, two examples are provided to show the effectiveness of the obtained results.

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Keywords: Switched nonlinear systems; Incremental passivity; I This work was supported by the National Natural Science Foundation of China under Grants 61703190,61773098, and 111 Project, Grant/Award Number: B16009. ∗ Corresponding author Email address: [email protected] (Hongbo Pang) URL: (Hongbo Pang)

Preprint submitted to Journal of LATEX Templates

April 13, 2019

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Switched internal model; Output regulation; Average dwell time.

1. Introduction

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. Output regulation is one of the most important problems in control theory and engineering. In general, the output regulation includes tracking exosystemgenerated reference signals asymptotically and rejecting the effect of exosystem5

generated disturbances. The output regulation problem is more challengeable than the stabilization problem. This problem is also a very interesting problem and has been widely applicated in many mechanical systems, and other practical

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fields [1,2]. More results on this topic can be found [3,4].

. Recently, switched systems have received more and more attention due to 10

the widespread applications in control community [5-9]. A switched system is a special hybrid dynamical system which is composed of a finite number of subsystems and a switching rule that governs the switching among them [10].

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Naturally, it is also of interest to consider the output regulation problem for switched nonlinear systems. It is more difficult to solve the output regulation problem because of the interactions of continuous dynamics and discrete dy-

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namics. Several methods which have been developed to study stability are still useful for the output regulation problem of switched nonlinear systems, such as

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multiple Lyapunov functions method [11-13] and the average dwell time method [13-15] and so on. In [11-13], the full information feedback and error feedback output regulation problems for a class of switched nonlinear systems were solved

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using multiple Lyapunov functions method and average dwell time method, respectively. However, it is better to solve the output regulation problem using

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the internal model principle as non-switched systems. In [14,15], a switched internal model was designed to solve the globally decentralized output regulation

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problem for a class of large-scale switched noninear systems by using average dwell time method. On the other hand, passivity concept was firstly proposed by Willems [16]. Since a storage function of a passive system can be chosen as a Lyapunov func2

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tion,passivity theory was often used for solving a variety of nonlinear control 30

problems [17-20], such as the nonlinear output regulation problem [19, 20]. The concept of conventional passivity was extended to incremental passivity which

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can describe a more extensive class of physical systems in [21]. This concept was originally proposed from an operator point of view [22, 23]. Incremental

passivity can provide an effective tool for constructing incremental Lyapunov 35

functions for incremental stability analysis [24] and convergence analysis [25,

26]. The invariance of incremental passivity under the feedback interconnection implies that an incrementally passive controller can be designed to drive the

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trajectories to converge to a steady-state solution. Therefore, the incremental passivity theory is useful for the output regulation problem of nonlinear sys40

tems [21, 27-29]. Moreover, incremental balanced truncation provided a model reduction technique that guarantees the preservation of relevant stability property and incremental passivity property [30,31].

With the growing research attention on switched nonlinear systems, the stud-

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ies on passivity of switched nonlinear systems have naturally emerged. Passivity, feedback passification and passivity-based stabilization problems of switched

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nonlinear systems were studied in [32-36]. As non-switched systems, passivity has been applied to solve the output regulation problem [37, 38]. The incremental passivity property was still important for switched nonlinear systems

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[39-43]. The incremental passivity for switched nonlinear systems was defined using weak-storage functions and multiple supply rates in [39]. But the adjacent

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storage functions were required to be connected at each switching time. [4042] proposed a more general incremental passivity concept. This incremental passivity property allowed the adjacent storage functions increase at the switch-

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ing points and was applied to solve the output regulation problem even if each

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subsystem was not incrementally passive. To the best of our knowledge, there have been no results on the output regulation problem for switched nonlinear systems with at least a feedback incrementally passive subsystem. It is assumed that some active subsystems are not feedback incrementally passive. However, even if each subsystem was incrementally passive, the output regulation problem 3

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may be not solvable. Moreover, the difficulty in solving the output regulation problem for switched nonlinear systems is to design an internal model. Therefore, it is interesting to study this problem. How to solve this output regulation

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problem? It is well known that average dwell time method is useful for studying stability of a switched system which are composed of both stable and unstable 65

subsystems [44]. [35, 36] have adopted this idea to solve the stablization and H∞ control problems.

Motivated by the above discussion, we will study the output regulation for switched nonlinear systems via average dwell time method. Compared with the

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existing papers, there are three distinct features. First, compared with the full information regulators designed in [11-13], the stabilizing controllers consist of the state feedback controllers which rendered at least a subsystem of controlled plant incrementally passive and linear output feedback controllers which stabilized the resulting closed-loop system. In some particular cases, this paper dose not need to verify that all the solutions of the switched nonlinear system converge to the bounded steady-state solution, while we only have to verify the

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regulated outputs converge to zero directly. Second, the error feedback reg-

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ulators and a composite switching law with average dwell time are designed to solve the error feedback regulation problem. The error feedback regulators comprise the stabilizing controllers and a switched internal model whose subsystems are incrementally passive. Compared with [14,15], the stabilizer and the

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internal model are interconnected in a more simple way and allowed to switch

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asynchronously. Moreover, the structure of the stabilizers is independent on the specific structure of the internal model. Thirdly, compared with [40-42], some

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subsystems are allowed to be non-incrementally passive, when they are active.

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2. Preliminaries 2.1 Review of incremental passivity . In this section, we will recall the concept of incremental passivity for nonlinear systems [21,27].

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Definition 1 [21,27] Consider a system x˙ = f (x, u, ω) , e = h (x, ω)

with state x(t) ∈ Rn , the input u ∈ Rm , and the output ∈ Rm , a bounded

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(1)

exogenous signal ω (t) ∈ Rs . System (1) is said to be incrementally passive if

there exists a C 1 function V (x, x′ ) : R2n → R+ , called a storage fuction, such that for any two inputs u, u′ , and any two solutions of system (1) x (t),x′ (t), corresponding to these inputs, the respective outputs e = h (x, ω) and e′ =

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h (x′ , ω), the inequality

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∂V ∂V T V˙ = f (x, u, ω) + f (x′ , u′ , ω) ≤ (e − e′ ) (u − u′ ) ∂x ∂x′

(2)

holds. If, in addition, there exists a constant λp > 0 such that

∂V ∂V T V˙ = f (x, u, ω) + f (x′ , u′ , ω) ≤ −λp V + (e − e′ ) (u − u′ ) ∂x ∂x′

(3)

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holds, then system (1) is said to be exponentially incrementally passive. If there exists a constant λn > 0 such that

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∂V ∂V T V˙ = f (x, u, ω) + ′ f (x′ , u′ , ω) ≤ λn V + (e − e′ ) (u − u′ ) ∂x ∂x holds, then system (1) is said to be non-incrementally passive. Remark 1 Notice that if f (0, 0, ω) = 0, h (0, ω) = 0(i.e. u ≡ 0, x ≡ 0 is

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the solution of system (1) with the zero output) then an incrementally passive system is also passive in the conventional sense with the storage function

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V˜ (x) = V (x, 0) . The notion of exponentially incremental passivity introduced in Definition 1 is more general than the notion of exponential passivity defined in [18].

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2.2 Problem formulation . A switched nonlinear system under consideration is described by ( ) x˙ = fσ(t) x, uσ(t) , ω , e = hσ(t) (x, ω) ,

5

(4)

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where x(t) ∈ Rn is the state, e ∈ Rm is the regulated output/regulated error, σ (t) : [0, ∞) → I = {1, 2, · · · M } is a piecewise constant function of time, called 110

a switching signal, which will be determined later [10]. Thus, (fσ(t) , hσ(t) ) :

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[0, ∞) → {(f1 , h1 ) , · · · , (fi , hi ) , · · · , (fM , hM ) ; i ∈ I} , i.e. fσ(t) , hσ(t) are also piecewise functions. Here, for the i-th subsystem,fi , hi , i ∈ I are smooth in

x and continuous in ω. ui ∈ Rm is the input vector of the -th subsystem.

Corresponding to the switching signal, the switching sequence is described as 115

Σ = {x0 ; (i0 , t0 ) , (i1 , t1 ) , . . . , (ik , tk ) , . . . |ik ∈ I, k ∈ N } with the initial time t0 ,

the initial state x0 , the switching time tk and the set of nonnegative integers

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N . When t ∈ [tk , tk+1 ), σ(t) = ik , i.e., the ik -th subsystem is active. The

exogenous signal ω (t) including exogenous commands, exogenous disturbances is generated by the exosystem

ω˙ = s (ω) , 120

ω (t0 ) ∈ W,

(5)

where W ⊂ Rs is a given compact and positively invariant set of initial condi-

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tions and s is a smooth function.

The output regulation problems for system (4) can be formulated as follows:

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Problem 1. [42] (Output regulation via full information feedback): Design a set of feedback controllers of the form

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(6)

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ui = αi (x, e, ω) = ηi (x, ω) + φi (e, ω) , i ∈ I,

where αi , ηi and φi are smooth mappings and a switching signal σ(t) such that

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for all initial conditions ω (t0 ) ∈ W and x0 ∈ Rn , the solutions (x (t) , ω (t)) of

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the system

( ) x˙ = fσ(t) x, ω, ασ(t) (x, e, ω) ,

(7)

ω˙ = s (ω)

are bounded for t ≥ t0 and lim e (t) = 0. t→∞

Remark 2 If the states x and ω are available for measurements, Problem 1

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should be considered. Compared with classical full information feedback controller, ηi (x, ω) , i ∈ I in the controller (6) play an role in rendering at least a 6

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subsystem of system (4) incrementally passive. φi (e, ω) , i ∈ I in (6) consist of steady control and output feedback stabilizer. For system (4) with at least an incrementally passive subsystem, we also should consider problem 1. Even if the states x are not required to be available for measurements and ω are available

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for measurements, the output regulation problem can be solved by the design of

the full information feedback controllers the output regulation problem can be solved by the design of the full information feedback controllers ui = φi (e, ω).

In practice, the whole states are often unavailable for measurements and the

control approach via state feedback would not be able to be implemented. Thus, Problem 2 should be considered.

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Problem 2. [40] (Output regulation via error feedback): Design a set of dynamic error-feedback controllers for the subsystems of the form ξ˙ = ηi (ξ, e) ,

ui = αi (ξ, e) , i ∈ I,

(8)

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where ξ ∈ Rq is the measured state, ηi (ξ, e) and αi (ξ, e) are smooth functions and a switching signal σ(t), such that for all initial conditions ω (t0 ) ∈

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W ,(x (t0 ) , ξ (t0 )) ∈ Rn+q , all solutions of the closed-loop system (4), (5) and

(8) are bounded for t ≥ t0 and lim e (t) = 0. t→∞

This paper will solve the output regulation problems for system (4) using

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the incremental passivity property of subsystems. However, it is unnecessary to require each susbsystem to be incrementally passive. At least a subsystem is assumed to be feedback incrementally passive. For convenience, the subsystem

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of system (4) is feedback incrementally passive for i ∈ IP ⊆ I and non-feedback incrementally passive for i ∈ IP ⊆ I.

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The objective is to derive a switching law that incorporates an average dwell

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time approach to solve the output regulation problems for system (4). To this end, we first recall the definition of average dwell time.

Definition 2 [44 ]For a switching signal σ(t) and any t > τ > 0, if Nσ (τ, t) ≤

N0 + t−τ τa holds for N0 ≥ 0,τa > 0, where Nσ (τ, t) denotes the switching numbers of σ(t)over the interval (τ, t), then τa and N0 are called the average dwell time 7

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and the chatter bound, respectively. Also, the activation time ratio of incrementally passive subsystems and the definition of class K functions are introduced as follows.

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Definition 3 [35] For any 0 ≤ T1 < T2 , let Tp[T1 ,T2 ] denote the total activation time of the incrementally passive subsystems during [T1 , T2 ]. Then rp[T1 ,T2 ] = 165

Tp[T1 ,T2 ] T2 −T1

is called passivity rate of system (4). Obviously, 0 < rp[T1 ,T2 ] ≤ 1.

Definition 4 [45] A function k : R+ → R+ is called a class K∞ function

if it is continuous, strictly increasing, K (0) = 0 and K(r) → ∞ as r → ∞.

A continuous function β : R+ × R+ → R+ is class KL functions and for all 170

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fixed s, the map r → β (r, s) is class K∞ function and for all fixed r, the map s → β (r, s)strictly decreasing and β (r, s) → 0, when s → ∞.

Next, we will give a notion of convergent switched systems. Definition 5 A switched system

x˙ = Fσ(t) (x, ω (t)) ,

(9)

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where ω (t) ∈ Rs is a piecewise continuous and bounded external signal, σ (t) denotes a switching signal for t ≥ t0 , Fi , i ∈ I are continuous in x and ω, is called globally uniformly convergent system if there exists an unique globally

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bounded solution xω (t) on R, and a function β ∈ KL such that for all initial

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conditions

∥x (t) − xω (t)∥ ≤ β (∥x (t0 ) − xω (t0 )∥ , t − t0 )

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holds. The solution xω (t) is called a steady-state solution. Remark 3 Convergence property of a switched system means that all the

trajectories associated with the same switching signal converge to the same bounded reference trajectory independently of their initial conditions. When

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(9) has only one subsystem, Definition 5 is reduced to the notion of convergent systems in [26,46].

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3. Output regulation via full information In this section, we will solve the output regulation problem for system (4)

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To this end, we need the following assumptions.

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based on the full information feedback via average dwell time method.

Assumption 1 [40] For any solution of the exosystem starting from ω (t0 ) ∈ W ,there exist x ¯ω (t) and u ¯iω that are bounded on R+ , satisfying x ¯˙ω (t) = fi (¯ xω (t) , u ¯iω (t) , ω (t)) , ∀t ≥ t0 , 0 = hi (¯ xω (t) , ω (t)) , ∀i ∈ I.

Remark 4 Assumption 1 is the necessary solvability condition of the output

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(10)

regulation problem for switched systems and has been adopted for non-switched systems [21]. (10) implies that the regulator equation of each subsystem of system (4) is solvable. The first equation in (10) implies all subsystems of system (4) have a common solution x ¯ω (t) corresponding to the solution of the exosystem starting from ω (t0 ) ∈ W . The second equation in (10) implies the

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output of subsystem of system (4) is zero at the common solution x ¯ω (t),which implies that the output of of system (4) is zero at x ¯ω (t) . This tell us that x ¯ω (t)

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is a steady-state solution of system (4) corresponding to the inputs u ¯iω (t) , i ∈ I. A common submanifold, where the output is zero, is generated. Similar to [11], the regulator equations for switched nonlinear systems can also be formulated:

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∂π(ω) ∂ω s (ω)

= fi (π (ω) , ci (ω) , ω) ,

(11)

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0 = hi (π (ω) , ω) ,

where π (ω) and ci (ω) defined on a set W are differentiable. The existence of π (ω) and ci (ω) is a necessary condition for the solvability of the output

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regulation problem as non-switched case. If π (ω) and ci (ω) satisfy (11) then Assumption 1 holds with x ¯ω (t) = π (ω) and u ¯iω (t) = ci (ω).The first equation in

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(11) expresses the fact that there exists a common submanifold in the state space of subsystems of the composite system (4) and (5). Namely, the graph of the common mapping x ¯ω (t) = π (ω), which is rendered locally invariant by u ¯iω (t) = ci (ω). The second equation in (11) means the output of system (4) is zero at 9

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each point of this common submanifold. However, the solvability of the regulator 210

equations implies the existence of common solution x ¯ω (t) = π (ω).The number of the regulator equations may be more than degrees of freedom (independent

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variables). In this case, the solvability of the regulator equations is an open problem which is the common coordinate transformation problem [47].

Next, to obtain sufficient conditions of solvability of the output regulation 215

problem for system (4), we make the following assumptions.

Assumption 2 There exist C 1 feedback controllers ui = ηi (x, ω) + vi with ηi (¯ xω , ω) = 0, µ ≥ 1, positive constants λp and λn , nonnegative smooth func-

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tions Vi (x, x′ ) and K∞ functions α1 and α2 , such that

1.α1 (∥x − x′ ∥) ≤ Vi (x, x′ ) ≤ α2 (∥x − x′ ∥) Vi ≤ µVj

(12)

2. For i ∈ IP ⊆ I and any solution ω (t) of the exosystem (5) starting from 220

ω (t0 ) ∈ W , the i-th subsystem of close-loop system

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( ) x˙ = fσ(t) x, ησ(t) (x, ω) + vσ(t) , ω ,

(13)

e = hσ(t) (x, ω)

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with the input vi and the output e is exponentially incrementally passive i.e. T V˙ i ≤ −λp Vi + (vi − vi′ ) (e − e′ ) .

For i ∈ In ⊆ I, the i-th subsystem of close-loop system (13) is non-

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3.

(14)

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incrementally passive, i.e. T V˙ i ≤ λn Vi + (vi − vi′ ) (e − e′ ) .

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Remark 5 For the i-th subsystem of system (13), (14) and (15) represent

the excess of incremental passivity and the shortage” of incremental passivity,

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respectively. The term λp Vi and λn Vi measure the amount of the excess of

incremental energy and the shortage of incremental energy, respectively. Exponentially incremental passivity condition is stronger than incremental passivity condition, but this avoids the use of exponential small-time norm-observation

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assumption [35]. 10

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Assumption 3 [35]There exists constant T0 ≥ 0 and passivity rate γp ∈ (0, 1] such that for all 0 ≤ T1 < T2 , the activation time of feedback incrementally passive subsystems during [T1 , T2 ] satisfies TP [T1 ,T2 ] ≥ γp (T2 − T1 ) − T0 . 235

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Remark 6 The constant T0 can be interpreted as an initial offset on the

activation time of the non-incrementally passive subsystems. This allows the switched system start with a non-incrementally passive subsystem. When T0 =

0, we have to start with an incrementally passive subsystem because of γp > 0. The main result of this section is obtained as follows.

Theorem 1 Suppose that Assumptions 1-3 hold. Then, (i) the output regula-

tion problem is solved by the controllers vi = u ¯iω − Ki e under the average dwell

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time τa satisfying

λ∗ = λp γp − λn (1 − γp ) −

ln µ > 0, τa

(16)

where Ki are positive definite matrixes. (ii) If λ∗ ≥ 0 then the output regulation problem is solvable for system (4) with the common output e = h (x, ω).

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Proof. Multiplying both sides by eλt , (14) and (15) become

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where

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d ( λt ) T e Vi ≤ eλt (vi − vi′ ) (e − e′ ) , dt

(17)

  λ , when i ∈ I . p P λ=  −λ when i ∈ I . n

n

Integrating (17) over [s, t] for ∀t > s ≥ t0 gives

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Vi (x (t) , x′ (t)) ≤ e−λ(t−s) Vi (x (s) , x′ (s)) +

∫

t

e−λ(t−τ ) Γi (τ ) dτ,

(18)

s

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where Γi (τ ) = (vi (τ ) − vi′ (τ )) (e (τ ) − e′ (τ )) . According to Assumption 1,¯ xω (t) corresponding to the inputs vi = u ¯iω (t) is

a common bounded solution of closed-loop system (13) and e = hi (¯ xω (t) , ω (t)) =

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0, i ∈ I.

Substituting νi′ = u ¯iω , x = x ¯ω , e′ = 0,vi = u ¯iω − Ki e into (18) gives −λ(t−s)

Vi (x (t) , x ¯ω (t)) ≤ e

Vi (x (s) , x ¯ω (s)) − 11

∫

t s

e−λ(t−τ ) eT (τ ) Ki e (τ ) dτ, (19)

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  λ , when i ∈ I . p P where λ =  −λ , when i ∈ I . n n For any given t > t0 , there exists a positve integer k such that t ∈ [tk , tk+1 ).

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Thus,Nσ(t) (t0 , t) = k. By (19), we get  ∫ tp −λ(t −τ ) T  p  ¯ω (tp )) ≤ e−λ(tp −tp−1 ) Vip (x (tp−1 ) , x ¯ω (tp−1 )) − tp−1 e e (τ )Ki e (τ ) dτ,  Vip (x (tp ) , x  p = 1, · · · , k − 1;   ∫t   V (x (t) , x ¯ω (t)) ≤ e−λ(t−tk ) Vik (x (tk ) , x ¯ω (tk )) − tk e−λ(t−τ ) eT (τ )Ki e (τ ) dτ, ik

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  λ , when i ∈ I . p k P where λ =  −λ , when i ∈ I . n k n Since Vi ≤ µVj holds for µ ≥ 1, we obtain that

Vik (x (t) , x ¯ω (t)) ≤ µe−λ(t−tk ) Vik−1 (x (tk ) , x ¯ω (tk )) −

∫t

tk

e−λ(t−τ ) eT (τ )Ki e (τ ) dτ

≤ µe−λ(t−tk ) (e−λ(tk −tk−1 ) Vik−1 (x (tk−1 ) , x ¯ω (tk−1 )) − ∫ t e−λ(t−τ ) eT (τ )Ki e (τ ) dτ ) − tk e−λ(t−τ ) eT (τ )Ki e (τ ) dτ tk−1

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∫ tk

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≤ µNσ (t0 ,t) e−λp Tp[t0 ,t] +λn Tn[t0 ,t] Vi0 (x(t0 ), x ¯ω (t0 )) ∫ t N (τ,t) −λ T +λ T T p n p[t0 ,t] n[t0 ,t] e e (τ ) Ki e (τ ) dτ − t0 µ σ

(21)

According to the definition of the average dwell time and Assumption 3, we have

t−t0 τa ) ln µ

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(N0 +

− λp Tp[t0 ,t] + λn Tn[t0 ,t]

≤ N0 ln µ − (λp γp − λn (1 − γp ) −

Let λ∗ = λp γp − λn (1 − γp ) −

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ln µ τa .

ln µ τa )(t

− t0 ) + (λp + λn ) T0 .

(22)

Substituting (22) into (21) gives

∗

Vik (x (t) , x ¯ω (t)) ≤ eN0 ln µ−λ (t−t0 )+(λp +λn )T0 Vi0 (x (t0 ) , x ¯ω (t0 )) ∫ t N ln µ−λ∗ (t−τ )+(λ +λ )T T p n 0 − t0 e 0 e (τ )Ki e (τ ) dτ.

(23)

When λ∗ > 0, it is easy to deduce from (11) that α1 (∥x (t) − x ¯ω (t)∥) ≤ eN0 ln µ−λ

∗

(t−t0 )+(λp +λn )T0

α2 (∥x (t0 ) − x ¯ω (t0 )∥) ,

( ) ∗ ¯ω (t0 )∥) . (24) ∥x (t) − x ¯ω (t)∥ ≤ α1−1 eN0 ln µ−λ (t−t0 )+(λp +λn )T0 α2 (∥x (t0 ) − x 12

(20)

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When t → ∞, lim x (t) = x ¯ω (t).Therefore, the state of the closed-loop t→∞

system (7) is bounded and lim e (t) = 0.

t→∞ ∫t When λ∗ = 0, e = h (x, ω), since (23) and (24) hold, t0 eT (τ ) Ki e (τ )dτ < ∞

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holds and x(t) is bounded. According to Barbalat Lemma, lim e (t) = 0.

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t→∞

Remark 7.Compared with [11, 12], the full information feedback regulators in Theorem 1 comprise two components: the steady-state control and the linear output feedback stabilizing controllers. If λ∗ = 0, e = h (x, ω),then this 270

paper dose not need to verify that all the solutions of system (4) converge to

outputs converge to zero directly.

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the bounded steady-state solution, while we only have to verify the regulated

Remark 8.It should be noted that passivity-based output regualtion of switched stochastic delay systems was studied via average dwell time method in [37]. The 275

passivity property of each subsystem was required in [11,12, 37]. However, the focus of our work is on the case that some subsystems are non-incrementally passive. The output regulation problems of some subsystems may be not solvable.

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From Theorem 1, the passivity rate needs to be larger than constant

λn λn +λp .

When λn > λp , the total of activation time of incrementally passive subsystems is longer than the total of activation time of non-incrementally passive subsystems.

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280

Remark 9. In (16),λp γp denotes the average rate of exponential decay of

PT

the incremental storage functions, while λn (1 − γp ) measures their exponential growth due to the non-incrementally passive subsystems, and

measures their

exponential growth due to the switches. Thus, (16) can be interpreted that the

CE

285

ln µ τa

storage functions of the switched system are decreasing on average.

AC

4. Output regulation via error feedback In this section, the error feedback regulators are designed to solve the error

feedback regulation problem for system (4) under a composite switching signal

290

with average dwell times. We first define a switched internal model of system (4).

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Definition 6. Consider a switched system ( ) τ˙ = φστ (t) τ, e˜στ (t) , vτ = γ (τ ) ,

(25)

CR IP T

where στ (t) : [0, ∞) → Iτ = {1, 2, · · · Mτ } is the switching signal of system

(25) and dim e˜στ (t) = dim hσ(t) , dim γ = dim uσ(t) , and ϕi , γ are continuous 295

functions, for any solution ω (t) of the exosystem starting from ω (t0 ) ∈ W , there exists τ¯ω (t) that is bounded on R+ satisfies

τ¯˙ω (t) = φi (¯ τω , 0) = ϕ (¯ τω ) , u ¯iω (t) = γ (¯ τω (t)) , i ∈ I, ∀t ≥ t0 .

(26)

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Thus, system (25) is said to have the internal model property.

Corresponding to the switching signal, we have the switching sequence Στ = {τ0 ; (iτ 0 , t0 ) , (iτ 1 , tτ 1 ) , . . . , (iτ l , tτ l ) , . . . |iτ l ∈ Iτ , l ∈ N } .

(27)

Remark 10.For each subsystem of system (4), each subsystem of (25) is an 300

internal model in the usual sense given in [21]. On the hand, system (25)

M

generates a set of steady-state controllers. Therefore, system (25) is called a switched internal model of system (4).

model system. 305

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Now, we make the main standing assumptions on system (4) and the internal

Assumption 4.There exist the dynamic error feedback controllers

PT

ξ˙ = ηsi (ξ, e) ,

(28)

ui = θsi (ξ, e) + vi

with continuous functions ηsi (ξ, e) and θsi (ξ, e) satisfying ηsi (0, 0) = 0, θsi (0, 0) =

CE

0 ,i ∈ I, constant µs ≥ 1, positive numbers λsp , λsn and nonnegative smooth

functions Vsi (X, X ′ ) and class K∞ functions αs1 (·),αs2 (·) such that the follow-

AC

ing conditions hold. 1.αs1 (∥X − X ′ ∥) ≤ Vsi (X, X ′ ) ≤ αs2 (∥X − X ′ ∥) , Vi (X, X ′ ) ≤ µs Vj (X, X ′ ) , X = (x, ξ) .

310

2. For i ∈ IP ⊆ I, the i-th subsystem of close-loop system (4) and (28) with

the input vi and output e is exponentially incrementally passive, i.e. T V˙ si ≤ −λsp Vsi + (vi − vi′ ) (e − e′ )

14

(29)

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3. When i ∈ In ⊆ I,, the i-th subsystem of close-loop system (4) and (28) is non-incrementally passive, i.e. T V˙ si ≤ λsn Vsi + (vi − vi′ ) (e − e′ ) .

CR IP T

(30)

Assumption 5.Each subsystem of system (25) is exponentially incrementally 315

passive. Namely, there exist positive definite smooth functions Vτ i (τ, τ ′ ) , i ∈ Iτ , class K∞ functions ατ 1 (·) , ατ 2 (·) , µτ ≥ 1, positive constant λτ p such that the following inequalities hold.

(31)

T V˙ τ i (τ, τ ′ ) ≤ −λτ Vτ i (τ, τ ′ ) + (˜ ei − e˜′i ) (vτ − vτ′ )

(32)

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ατ 1 (∥τ − τ ′ ∥) ≤ Vτ i (τ, τ ′ ) ≤ ατ 2 (∥τ − τ ′ ∥) , Vi (τ, τ ′ ) ≤ µτ Vj (τ, τ ′ )

Remark 11. Since the internal system (25) depends on the structure of the 320

exosystem (5) as well as on the properties of system (4), it is hard to verify the existence of an incrementally passive internal model. However, it can be

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designed to be incrementally passive structurely.

Theorem 2.Suppose that Assumptions 1, 3, 4 and 5 hold. Then, the output

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regulation problem is solved by the controllers ui = θsi (ξ, e) + vτ − Ki e,

325

PT

τ˙ = φστ (τ, −e) , vτ = γ (τ ) , ξ˙ = ηsi (ξ, e) ,

where Ki are positive definite matrixes and θsi are continuous functions, under

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a class of switching law σ ′ = (σ, στ ) with the average dwell time τa >

ln µs λsp γp − λsn (1 − γp )

and ττ a >

ln µτ λτ

(33)

AC

satisfying

(λτ − λsp ) Tp + (λτ + λsn ) Tn = Nστ (t) (t1 , t2 ) ln µτ − Nσ(t) (t1 , t2 ) ln µs , t1 ≥ t2 . (34) Proof. The close-loop system (4) and (28) is interconnected with the switched

internal model system (25) through vσ = vτ + vsσ , e˜σ′ = −e + vIM σ′ where 330

dim e˜σ′ = dim vIM σ′ , dim γσ′ = dim vsσ . Thus, the switching signal of the 15

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interconnection system is σ ′ (t) = (σ (t) , στ (t)) : [0, ∞) → I ′ = I × Iτ =

{(k, i) |k ∈ I; i ∈ Iτ } . Corresponding to the switching signal σ ′ (t), we obtain the switching sequence

335

(35)

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{ } Σ′ = (x0 , ξ0 , τ0 ) ; (i′0 , t0 ) , (i′1 , t′1 ) , . . . , (i′l , t′l ) , . . . i′j ∈ I ′ , j ∈ N ,

( ) where i′j = σ ′ t′j = (σ (tk ) , στ (tτ l )) = (ik , iτ l ). The storage functions of the

interconnection system are chosen as S(k,l) = Vsk (X, X ′ ) + Vτ l (τ, τ ′ ).

Similar to the proof of Theorem 1, it follows from (29), (30) and (32) that for ∀t > s ≥ t0 ,

Vτ i (τ (t) , τ ′ (t)) ≤ e−λτ (t−s) Vsi (τ (s) , τ ′ (s)) +

t

e−λs (t−τ ) Γs (τ ) dτ, (36)

∫

s

t

e−λτ (t−τ ) Γτ (τ ) dτ ,

(37)

s

  λ , when i ∈ I ( ) sp P T ′ where λs = , Γs (τ ) = (e (τ ) − e′ (τ )) vσ(τ ) (τ ) − vσ(τ (τ ) . )  −λ , when i ∈ I sn n ) [ For any given t > t0 ,,t ∈ t′j , t′j+1 by (36) and (37), we get  ∫ tp −λ (t −τ )   e s p Γs (τ ) dτ, V (X(tp ), X ′ (tp )) ≤ e−λs (tp −tp−1 ) Vsip−1 (X(tp−1 ), X ′ (tp−1 )) + tp−1   sip p = 1, · · · , k − 1;     V (X(t), X ′ (t)) ≤ e−λs (t−tk ) V (X(t ), X ′ (t )) + ∫ t e−λs (t−τ ) Γ (τ ) dτ, sik

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340

∫

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Vsi (X (t) , X ′ (t)) ≤ e−λs (t−s) Vsi (X (s) , X ′ (s)) +

sik

k

k

tk

s

PT

and  ∫t p −λτ (tτ p −tτ (p−1) ) ′  e−λτ (tτ p −τ ) Γτ (τ ) dτ,  Vτ iτ p (τ (tτ (p−1) ), τ ′ (tτ (p−1) )) + tττ(p−1)  Vτ iτ p (τ (tp ), τ (tp )) ≤ e 

CE

p = 1, 2, · · · , k − 1;   ∫t   V ′ −λτ (t−tτ l ) Vτ iτ l (τ (tτ l ), τ ′ (tτ l )) + tτ l e−λτ (t−τ ) Γτ (τ ) dτ τ iτ l (τ (t), τ (t)) ≤ e

Let µ = max {µs , µτ }. Since Vsi ≤ µs Vsj , Vτ i ≤ µτ Vτ j holds for µs ≥ 1, µτ ≥

AC

1, we obtain that ∫t Vsik (X(t), X ′ (t)) ≤ µs e−λs (t−tτ k ) Vsik−1 (X(tk ), X ′ (tk )) − tk e−λs (t−τ ) Γs (τ ) dτ ( ) ( ) ∫t k e−λs (t−τ ) Γs (τ ) dτ ≤ µe−λτ (t−tτ k ) e−λs (tτ k −tτ (k−1) ) Vsiτ k τ (tτ (k−1) ), τ ′ (tτ (k−1) ) + tττ(k−1) ∫t (38) + tτ k e−λs (t−τ ) Γs (τ ) dτ −λ

T

+λ

T

≤ µNσ(t) (t0 ,t) e sp p[t0 ,t] sn n[t0,t ] Vsi0 (X(t0 ), X ′ (t0 )) ∫t + t0 µNσ(t) (τ,t) e−λsp Tp⌊τ,t⌋ +λsn Tn[τ,t] Γs (τ ) dτ 16

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345

Similarly, we have ′

Vτ iτ l (τ (t), τ (t)) ≤ µ

Nστ (t0 ,t) −λτ (t−t0 )

e

′

Vτ iτ 0 (τ (t0 ), τ (t0 )) +

∫

t

µNστ (τ,t) e−λτ (t−τ ) Γτ (τ ) dτ (39)

t0

CR IP T

′ Since (34) holds, substituting vσ(t) = vτ + vsσ(t) , e˜σ′ (t) = −e + vIM σ′ (t) , vσ(t) = ′ ′ vτ′ + vsσ(t) , e˜′σ′ (t) = −e′ + vIM σ ′ (t) into (38) and (39) gives

Vsik (X(t), X ′ (t)) + Vτ iτ l (τ (t), τ ′ (t))

−λ T +λ T ≤ µNσ(t) (t0 ,t) e sp p[t0 ,t] sn n[t0,t ] (Vsi0 (X(t0 ), X ′ (t0 )) + Vτ iτ 0 (τ (t0 ), τ ′ (t0 ))) (40) ∫t + t0 µNσ(t) (τ,t) e−λsp Tp⌊τ,t⌋ +λsn Tn[τ,t] Γ (τ ) dτ,

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( )T ( )T ′ ′ where Γ (τ ) = vsσ(t) − vsσ(t) (e − e′ ) + vIM σ′ (t) − vIM (vτ − vτ′ ) . σ ′ (t)

The feedback controllers are designed as vIM i = 0 and vsi = −Ki e. Ac-

350

cording to Assumption 1 and (26), (¯ xω (t) , 0, τ¯ω (t)) is a bounded solution of the interconnection switched system with vIM i = 0 and vsi = −Ki e.The rest of proof is similar to Theorem 1.

Remark 12. When e = h (x, ω), the output regulation problem is solvable

355

ln µs λp γp −λn (1−γp ) ,and

ττ a ≥

M

under a class of switching law σ ′ = (σ, στ ) with the average dwell times τa ≥ ln µτ λτ .

ED

Remark 13. Compared with [12, 13], a switched internal model is designed to generate steady-state control. The stabilizing controllers consist of the dynamic error-feedback controllers which rendered at least a subsystem of controlled

360

PT

plant incrementally passive and linear output feedback controllers. Compared with [14, 15], the switching signal of system (25) is different from the switching signal of system (4). Thus, the internal model is allowed to switch asynchronous-

CE

ly with system (4). This gives more design freedom of the switching law.

AC

Remark 14.If στ = σ then the internal model system is designed as τ˙ = ( ) φσ(t) τ, e˜σ(t) , vτ = γσ(t) (τ ) satisfying τ¯˙ω (t) = ϕ (¯ τω ) , u ¯iω (t) = γi (¯ τω (t)) , i ∈

365

I, ∀t ≥ t0 , λτ ≥ λsp .

Namely, each system has an incrementally passive internal model. For i ∈ Ip

let λτ = λsp . For i ∈ In , let λτ = −λnp .(40) can be written as follows −λ

T

+λ

T

Vτ iτ l (τ (t), τ ′ (t)) ≤ µNσ (t0 ,t) e sp p[t0 ,t] sn n[t0,t ] Vτ iτ 0 (τ (t0 ), τ ′ (t0 )) ∫t + t0 µNστ (t) (τ,t) e−λsp Tp⌊τ,t⌋ +λsn Tn[τ,t] Γτ (τ ) dτ. 17

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Similar to the proof of Theorem 2, the output regulation problem is solved by the controllers ui = θsi (ξ, e) + vτ − Ki e, τ˙ = φi (τ, −e) , vτ = γi (τ ) under the average dwell time τa >

ln µs λsp γp −λsn (1−γp ) .

CR IP T

370

5. Examples

In this section, two examples will be presented to demonstrate the validity of our developed theoretical results.



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375

Example 1 Consider system (4) described by  ( ) −x1 x21 + 4 + 12 x2 + 3 + u11 + ω 6   1  2 x1 − x2 + 1 + 13 u21 + 12 ω 2 − x4 f1 (x, u1 , ω) =    2x4 − x3  −x3 − 6x4 + 12 x2 + 7ω 2 x1 + x2 + 19 + 0.5u12 − 22ω 2

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   −2x1 + x2 + 18 + u22 − 17ω 2 − x4 f2 (x, u2 , ω) =    2x4 − x3  −x3 − 8x4 + x2 + 8ω 2 

e=

e1





=

x1 − ω 2

2



       

       

(41)

 = Cx − Dω 2

CE

PT

x2 − 2ω ) ) ( ( T with u1 = u11 , u21 , u1 = u12 , u22 , x = (x1 , x2 , x3 , x4 ) , C =  the exosystem ω˙ = 0,  where  1 0 0 0 1  ,D =  . 0 1 0 0 2 e2

1    0 T and V2 (x, x′ ) = 12 (x − x′ ) P2 (x − x′ ), where P1 =    0  0

AC 380

We choose the storage functions of subsystems as V1 (x, x′ ) =

18

0

0

3

0

0

1

0

0

1 2

T

(x− x′ ) P1 (x − x′ ) 0   0   and P2 =  0   2

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

2 0

0

0



CR IP T

     0 1 0 0   .    0 0 1 0    0 0 0 2 Differentiating Vi gives

T T V˙ 1 ≤ −1.5V1 + (u1 − u ˆ1 ) (e − eˆ) , V˙ 2 ≤ 3V2 + (u2 − u ˆ2 ) (e − eˆ) .

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385

( )T (a) When ω is available for measurements, x ¯ω (t) = ω 2 , 2ω 2 , 2ω 2 , ω 2 is steady( ) 1 solution of the regulator equations (10) with u ¯11ω = u ¯21ω = 3 ω 2 − 1 ,¯ u2ω = ) ( ( ) 2 2 2 ¯2ω = 18 −1 + ω . According to Theorem 1, the output regu38 −1 + ω , u lation problem is solved by ui = u ¯iω − e under a class of switching signal with

the average dwell time τa ≥

ln µ λp γp −λn (1−γp ) , λp

= 1.5, λ n = 3.

Now, let µ = 3.5 and the passivity rate γp = 0.8. Then, the average dwell time is chosen as τa = 2 > 1.83. The simulation results are depicted in Figs 1-4 390

for the initial states x (0) = (68.1, 80.7, 9, 7) , ω (0) = 0.8. The state response

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of closed-loop system (41) with ui = u ¯iω − e is shown in Fig. 1, which tell us that the state of closed-loop system is bounded. Moreover, the exogenous signal

ED

ω (t) presented in Fig. 2 is also bounded. From Fig. 4, the regulated output converges to zero under the switching signal as shown in Fig. 3 . Therefore, 395

the output regulation problem is solvable under a class of switching signal with

PT

average dwell time τa = 2. Thus, the simulation results well illustrate the theory presented.

CE

(b) When ω is unavailable for measurements, a switched internal model with

AC

400

each subsytem being incrementally  0  3  Subsystem 1: τ˙ =  3 0  0 0  0  38  Subsystem 2: τ˙ =  0 18  0 0

passive is designed as follows.     . 3 0 0   τ.  e˜, v1 (τ ) =   0 3 0 

    e˜, v2 (τ ) =  

38 0

0

0 18

0

The common storage function are chosen as Wk (τ1 , τ 2 ) = 19

1 2



 τ.

T

(τ1 − τ2 ) (τ1 − τ2 ) , k =

ACCEPTED MANUSCRIPT

90

x1 x2

80

x x

state response x

60 50 40 30 20

0

0

2

4

6

4

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10

−10

3

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70

8

10

12

14

16

18

20

time(sec)

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Figure 1: State response of the switched system.

1.6 1.4 1.2

PT

exogenous signal ω

ω

ED

1.8

1

0.8

CE

0.6 0.4

AC

0.2 0

−0.2

0

2

4

6

8

10

12

14

time (sec)

Figure 2: The exogenous signal ω (t).

20

16

18

20

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80

e

1

e

70

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2

60

regulated output

50

40

30

20

0

−10

0

2

4

6

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10

8

10

12

14

16

18

20

time(sec)

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Figure 3: The regulated output of the switched system.

σ

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2.2

2

1.6

1.4

CE

switching signal σ

PT

1.8

1.2

AC

1

0.8

0

2

4

6

8

10

12

14

16

time(sec)

Figure 4: The switching signal of the switched system.

21

18

20

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1, 2.

is a steady-solution of closed-loop system   3 0  u1 = v1 (τ ) − e, τ˙ = −  0 3  0 0  38  u2 = v2 (τ ) − 2e, τ˙ = −  0  0

0 18



  38 0   e, v2 (τ ) =   0 18

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

(41) with the controllers:    3 0 0   e, v1 (τ ) =   0 3 0

CR IP T

405

We consider the case that the switched model and the switched system switch ( ) ( )T synchronously. Thus,τω = ω 2 − 1, ω 2 − 1, ω 2 − 1 , x ¯ω (t) = ω 2 , 2ω 2 , 2ω 2 , ω 2

0



 τ,

0 0



 τ.

Therefore, the output regulation problem is solvable under the average dwell τa ≥

ln µ λp γp −λn (1−γp ) , λp

= 1.5, λ n = 3.

Let µ = 3,the passivity rate γp = 0.8 and the average dwell time τa = 2 >

410

1.83. The simulation results are depicted in Figs 5-8. It can be seen from Figs.

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5-7 that all the solutions of closedloop system starting from the initial states x (0) = (49.1, 39.7, 48, 43) , τ (0) = (67, 39, 29), ω (0) = 0.8 are bounded and the

415

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output converges to zero. The switching signal is given by Fig. 8. Therefore, the output regulation problem is solvable under average dwell time τa = 2. Thus, the simulation results well illustrate the theory presented.

PT

Example 2 Consider a switched stirred tank reactor (CT SR) system, which was previously studied in [48, 49].

CE

x˙ 1 =

AC

x˙ 2 =

Fσ(t) V Fσ(t) V

( ) x1,in,σ(t) − x1 + Kσ(t) φσ(t) (x1 , x2 ) + d1σ(t) , ( ) x2,in,σ(t) − x2 − ∆Hσ(t) (x1 , x2 ) φσ(t) (x1 , x2 )

(42)

+γσ(t) (x2c − x2 ) + γσ(t) uσ(t) + d2σ(t) ,

where σ (t) : R+ → I = {1, 2};x1 ∈ R and x1,in,k ∈ R are the chemical species

420

concentration in the reactor and in the input flow, respectively. And x2,in,k , x2 ∈ R are the reactor and the input flow temperatures, respectively. For each k, φk (x1 , x2 ) is the reaction rate, Kk denotes the stoichiometric coefficient,

Fk V

is

the ratio between the feed stream Kowrate and the volume of the reactor called 22

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50

80

x x

state response x

30

x

1

τ

τ3

3

40

4

20 10 0

20

0

−20

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−10

−40

−20 −30

2

τ

60

2

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40

1

state response τ

x

0

10

20

30

time (sec)

−60

40

0

10

20

time (sec)

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Figure 5: State response of the switched system.

ω

ED

1.8 1.6 1.4

PT

1.2 1

0.8

CE

0.6 0.4

AC

0.2 0

−0.2

0

5

10

15

20

25

30

time (sec)

Figure 6: The exogenous signal ω (t).

23

35

40

30

40

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50

e1 e

40

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2

regulated output

30

20

10

0

−20

−30

0

5

10

15

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−10

20

25

30

35

40

time(sec)

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Figure 7: The regulated output of the switched system.

σ

ED

2.2

2

1.6

1.4

CE

switching signal σ

PT

1.8

1.2

AC

1

0.8

0

5

10

15

20

25

30

time(sec)

Figure 8: The switching signal of the switched system.

24

35

40

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the dilution rate. −∆Hk (x1 , x2 ) denotes the heat of reaction, and γk > 0 is a 425

heat transfer coefficient. x2c is the coolant temperature. Variables d1k and d2k are external disturbances in feed temperature and feed composition respectively.

CR IP T

vr denotes the external disturbance or as a reference signal d11 = ω 3 ω 3T , d12 =

2ωω T , d21 = 7ωω T , d22 = 3.5ωω T , vr = 2ωω T are generated by the exosystem   0 −1  , ω = (ω1 , ω2 ) . ω˙ = Sω, S =  (43) 1 0

The regulated error is thus defined to be e = x2 − vr . The control objective is 430

to control the reactor temperature x2 by manipulating the coolant temperature

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uk .

Let system parameters for the simulation F1 = 3.5L/s, γ1 = K1 = 1, x1in2 = x2in2 = 0K,V = 1L, x2c = 0K, x1in1 = x2in1 = 1mol/L, −∆H1 = x1 , φ1 = 3.5 (x2 − x1 ) − x31 − 3.5, F2 = 1L/s, γ2 = K2 = 0.5, φ2 = 8x1 + 4x2 + 6, −∆H2 = 435

0.5x2 .

x˙ 1 = −x31 − 7x1 + 3.5x2 + d11 ,

x˙ 2 = −4.5x2 − x41 + 3.5(x1 x2 − x21 − x1 ) + u1 + d12 .

ED

Subsystem1

M

Thus, we have

x˙ 1 = 3x1 + 2x2 − d21 ,

and V2 (x, x′ ) =

1 2

(44)

x˙ 2 = 1.5x2 + (4x1 x2 + 2x22 ) + 0.5u2 − d22 .

We choose the storage functions of subsystems as  V1 (x, x′ )=

CE

440

PT

Subsystem2

T

(x − x′ ) P2 (x − x′ ), where P1 = 

3 0

1 2

T

′ (x − x ) P1 (x− x′ )

 and P2 = 

1

0

0 1 0 2 Design the controllers as u1 = x41 −3.5(x2 x2 −x21 )+5.5x1 +w1 , u2 = −8x1 x2 −

AC

4x22 − 6x2 − 4x1 + w2 . Thus, (44) are transformed into the following system. Subsystem1

x˙ 1 = −x31 − 7x1 + 3.5x2 + d11 ,

x˙ 2 = −4.5x2 − x41 + 3.5(x1 x2 − x21 − x1 ) + u1 + d12 .

Subsystem2

x˙ 1 = 3x1 + 2x2 − d21 ,

x˙ 2 = 1.5x2 + (4x1 x2 + 2x22 ) + 0.5u2 − d22 . 25

(45)

.

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Differentiating Vi gives T T V˙ 1 ≤ −4V1 + (w1 − w ˆ1 ) (e − eˆ) , V˙ 2 ≤ 7.5V2 + (w2 − w ˆ2 ) (e − eˆ) .

( ( ))T (a) When ω is available for measurements,¯ xω (t) = ω12 + ω22 , 2 ω12 + ω22 is ( 2 ) 2 ¯2ω = 5 ω1 + ω2 . steady-solution of the regulator equations (10) with w ¯1ω = w

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According to Theorem 1, the output regulation problem is solved by wi = w ¯iω −e under a class of switching signal with the average dwell τa ≥ 4, λ n = 7.5.

ln µ λp γp −λn (1−γp ) , λp

=

Let µ = 3.5,the passivity rate γp = 0.7 and the average dwell time τa =

450

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2.5 > 2.28. The simulation results are depicted in Figs 9-12 for the initial states x (0) = (3.1, 2.7) , ω (0) = (0.5, 0.8) The state response of closed-loop system

(41) with wi = w ¯iω − e is shown in Fig. 9, which tell us that the state of closed-loop system is bounded. 3.5

2

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2.5

state response x

x

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3

x1

2

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1.5

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1

AC

0.5

455

0

5

10

15

20

25

30

35

40

time(sec)

Figure 9: State response of the switched system.

(b) When ω is unavailable for measurements, a switched internal model with

each subsytem being incrementally passive is designed as follows.     [ ] 2 −1 2  e˜, v1 (τ ) = 2 1 τ, τ +  Subsystem1 τ˙ =  0.5 2 −4 26

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1.5 ω1 ω2

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exogenous signal ω

1

0.5

0

−0.5

−1.5

0

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−1

10

15

20

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time (sec)

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Figure 10: The exogenous signal ω (t).

e

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1.2

1

0.6

0.4

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regulated output

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0.8

0.2

AC

0

−0.2

0

5

10

15

20

25

30

time(sec)

Figure 11: The regulated output of the switched system.

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2.2

σ

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switching signal σ

1.8

1.6

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1

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1.2

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Figure 12: The switching signal of the switched system.

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0

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−100 −200

AC

40

20

0

−300

−20

−400 −500

2

τ

80

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state response x

100

τ1

x2

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200

100

x1

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300

0

10

20

30

40

−40

0

time (sec)

10

20

time (sec)

Figure 13: State response of the switched system.

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2

ω1 ω2

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exogenous signal ω

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Figure 14: The exogenous signal ω (t).

2

1.8

PT

switching signal sigma

σ

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2.2

1.6

AC

CE

1.4

1.2

1

0.8

0

10

20

30

40

time(sec)

Figure 15: The switching signal of the switched system.

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5000

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regulated output

0

−5000

−10000

−20000

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−15000

30

40

50

60

time(sec)

Figure 16: The regulated output of the switched system.

Subsystem2 τ˙ = 

−1 1

2





τ + 

M



−2

1

1.5



 e˜, v1 (τ ) =

ED

The common storage are chosen as Wk (τ1 , τ 2 ) =  function 

1, 2, where P = 

1 2

1 3

]

τ.

(46)

T

(τ1 − τ2 ) P (τ1 − τ2 ) , k =

PT

 . Therefore, each subsystem of the switched model is 0 2 incrementally passive. ))T ( ) ( ( is a steady¯ω (t) = ω12 + ω22 , 2 ω12 + ω22 τω = 2ω12 + 2ω22 , ω12 + ω22 , x

solution of closed-loop system (45) with the controllers     [ ] −1 2 2 τ −   e, v1 (τ ) = 2 1 τ, w1 = v1 (τ ) − e, τ˙ =  2 −4 0.5     [ ] −1 2 1 τ −   e, v1 (τ ) = 1 3 τ. w2 = v2 (τ ) − 2e, τ˙ =  1 −2 1.5

AC

CE

460

1 0

[

Therefore, the output regulation problem is solvable under the switching signal σ ′ = (σ, στ ) with the average dwell time τa ≥

ln µ , λp = 4, λ n = 7.5. λp γp − λn (1 − γp ) 30

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and ττ a >

ln µτ = 0. λτ

Let µ = 3.5,the passivity rate γp = 0.7 and the average dwell time τa = ττ a =

465

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2.3 > 2.28. The simulation results are depicted in Figs. 13-16. It can be seen

from Figs. 13-15 that all the solutions of closed-loop system starting from the initial states x (0) = (54.1, 30.7) , τ (0) = (69, 50), ω (0) = (1, 0.8) are bounded

and the output converges to zero. The switching signal is given by Fig.16. 470

Therefore, the output regulation problem is solvable under average dwell time

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τa = 2. Thus, the simulation results well illustrate the theory presented.

6. Conclusions

We have solved the output regulation problem for a class of switched nonlinear systems with at least a feedback incrementally passive subsystem via average 475

dwell time method. First of all, we solve the output regulation via full informa-

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tion feedback. Then, the error feedback regulators and a composite switching law with average dwell time are designed to solve the error feedback regulation

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