Output regulation of switched nonlinear systems using incremental passivity

Nonlinear Analysis: Hybrid Systems 27 (2018) 239–257

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Nonlinear Analysis: Hybrid Systems journal homepage: www.elsevier.com/locate/nahs

Output regulation of switched nonlinear systems using incremental passivity✩ Hongbo Pang a,b , Jun Zhao a, * a

College of Information Science and Engineering, Northeastern University, State Key Laboratory of Synthetical Automation of Process Industries, Shenyang 110819, China b College of Science, Liaoning University of Technology, Jinzhou 121000, China

article

info

Article history: Received 25 September 2016 Accepted 21 August 2017

Keywords: Switched nonlinear systems Incremental passivity Error-dependent switching law Error-feedback output regulation Switched internal model

a b s t r a c t This paper studies the global output regulation problem for a class of switched nonlinear systems using incremental passivity, even though the output regulation problem for none of subsystems is solvable. Firstly, a concept of incremental passivity for switched nonlinear systems without the requirement of the incremental passivity property of individual subsystem is introduced. This incremental passivity property is shown to be invariant under compatible feedback interconnection. Secondly, a switched nonlinear system is rendered to be incrementally passive by the design of a set of dynamic error feedback controllers and a dynamic switching law. In particular, an error-dependent switching law is designed to guarantee a specific class of switched systems incrementally passive. Thirdly, the incremental passivity for switched nonlinear systems is applicated to solve the global output regulation problem by the dual design of the composite switching law and error feedback controllers. The key idea is to design an incrementally passive switched internal model. Two examples including a Switched Chua’s Circuit are presented to illustrate the effectiveness of the proposed approach. © 2017 Elsevier Ltd. All rights reserved.

1. Introduction The output regulation problem is one of the most fundamental problems in control theory and engineering. The output regulation, in general, aims to track exosystem-generated reference signals asymptotically and reject the effect of exosystemgenerated disturbances. This problem for nonlinear systems has been well investigated [1–3]. The passivity theory, proposed by Willems [4], is effective tool for solving the output regulation problem [5–7]. To investigate a more extensive class of physical systems with an equilibrium point or not, [8,9] extended the conventional passivity property to incremental passivity. A state space form of incremental passivity definition and some related preliminary results were given in [10–13]. For an incrementally passive system, the storage function can be chosen as an incremental Lyapunov function for incremental stability analysis [12,13] and convergence analysis [14]. In addition, according to the invariance of incremental passivity under feedback interconnection, once the incremental passivity property of a nonlinear system is assured, an incrementally passive feedback controller can be designed to drive the trajectories to converge to a steady-state solution. Therefore, incremental passivity was used to solve the output regulation problems [10,11]. Incremental passivity theory was applied to the synchronization analysis problem of coupled oscillators [12–14] and the analysis of electrical circuits [15]. ✩ This work was supported by the National Natural Science Foundation of China under Grant 61773098; 111 Project, Grant/Award Number: B16009. Corresponding author. E-mail addresses: [email protected] (H. Pang), [email protected] (J. Zhao).

*

https://doi.org/10.1016/j.nahs.2017.08.011 1751-570X/© 2017 Elsevier Ltd. All rights reserved.

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H. Pang, J. Zhao / Nonlinear Analysis: Hybrid Systems 27 (2018) 239–257

On the other hand, a great amount of attention has been paid to switched systems due to their wide existence in many practical fields [16–22], such as mechanical systems [23,24], and electrical systems [17] and so on. Switched systems are composed of a family of continuous-time subsystems and a rule that governs the switching among them [25]. The output regulation problem for switched systems is much more difficult than that for non-switched systems because of the interactions of continuous dynamics and discrete dynamics. Several methods were adopted to deal with switched system, such as the common Lyapunov function technique [26], the multiple Lyapunov functions method [27,28], the average dwell time approach [23,29–31] and so on. Passivity property is also useful for switched nonlinear systems as non-switched systems. Passivity concepts of switched nonlinear systems and the corresponding feedback passification and passivity-based stabilization problems were investigated in [24,32,33]. The incremental passivity was still important for switched nonlinear systems [34–36]. Incremental passivity theory and the incremental passivity-based output tracking for switched nonlinear systems were investigated using multiple storage functions and multiple supply rates in [34]. But the adjacent storage functions are connected at the switching times, which is a strong requirement. In [35], a more general incremental passivity concept of switched nonlinear systems which allow the storage functions to increase at the switching times was proposed. A state-dependent switching law is designed to render a switched system incrementally passive. Moreover, the output regulation problem for switched nonlinear systems was solved using the established incremental passivity. However, when the whole states information is unavailable for measurements, the output regulation problem for switched nonlinear systems has not been investigated using incremental passivity theory. The difficulty in solving the output regulation problem for switched nonlinear systems is the design of the internal model and error-dependent switching laws. Motivated by the above discussion, we will generalize the results on incremental passivity and global output regulation in [10] to switched nonlinear systems. Compared with the existing literatures, the results of this paper have four distinct features. First, when the whole states information is unavailable for measurements, a set of dynamic feedback controllers for subsystems and a dynamic switching law are designed using error information to render the resulting closed-loop system incrementally passive. In particular, an error-dependent switching law is designed to guarantee a specific class of switched systems incrementally passive. Second, a switched regulator and a composite switching law are designed to solve the output regulation problem. Compared with conventional regulator, the switched regulator is parallel interconnections of the incrementally passive switched internal model and a switched stabilizer rendering the controlled plant incrementally passive (together with the linear error feedback controllers). The internal model and the stabilizer can be designed independently. Third, compared with the switched internal model designed in [23], the incrementally passive switched internal model is allowed to switch asynchronously with the controlled plant, which increases the freedom of design greatly. The internal model property is not required for each subsystem of the switched internal model. Finally, compared with [23,28,29], once the incremental passivity property of the controlled plant is assured, the stabilizer is designed as a set of the linear error feedback controllers. Thus, this paper does not need to verify that all the solutions converge to the steady-state solution, while we only have to verify the regulated outputs converge to zero directly. 2. Problem formulation and preliminaries Consider a switched nonlinear system described by x˙ = fσ (x, uσ , ω) , e = hσ (x, ω) ,

(1)

where x ∈ Rn is system state, ui ∈ Rm is the input vector of the ith subsystem, e ∈ Rm is the measured regulated output/error and σ (t ) : [0, ∞) → I = {1, 2, . . . , M } is the switching signal which is assumed to be a piecewise constant function and has a finite number of switchings on any finite time interval [25]. The exogenous signal ω (t ) including exogenous commands, exogenous disturbances is generated by the exosystem

ω˙ = s (ω) ,

ω (t0 ) ∈ W ,

(2)

where W ⊂ R is a given positively invariant set of initial conditions. It is assumed that any solution ω (t ) starting from ω (t0 ) ∈ W is bounded for all t ≥ t0 . fi , hi and s are smooth functions. Corresponding to the switching signal, the switching sequence is described by s

⏐ } { Σ = (i0 , t0 ) , (i1 , t1 ) , . . . , (ik , tk ) , . . . ⏐ik ∈ I , k ∈ N ,

(3)

where t0 is the initial time, and N denotes the set of nonnegative integers. When t ∈ [tk , tk+1 ) , σ (t ) = ik , that is, the ik -th subsystem is active. For any j ∈ I , let

{ } Σj = tj1 , tj2 , · · · tjn · · · ; ijq = j, q ∈ N

(4)

be the sequence of switching times when the jth subsystem is switched on, and thus tj1 +1 , tj2 +1 , . . . tjn +1 , . . . ; ijq = j, q ∈ N

{

}

(5)

H. Pang, J. Zhao / Nonlinear Analysis: Hybrid Systems 27 (2018) 239–257

241

is the sequence of switching times when the jth subsystem is switched off. In addition, we assume that the state of system (1) does not jump at the switching instants. In practice, the whole states information is often unavailable for measurements. Therefore, it is necessary to consider the error feedback output regulation problem. Error feedback regulation problem: Given system (1), design a set of dynamic error feedback controllers for the subsystems in the form

ξ˙ = ηi (ξ , e) , ui = θi (ξ , e) , i ∈ I ,

(6)

where ξ ∈ Rq is the measured state, ηi (ξ , e) and θi (ξ , e) are smooth functions and a switching signal σ (t ) such that all solutions of the closed-loop system (1), (2) and (6) starting from (x (t0 ) , ξ (t0 )) ∈ Rn+q and ω (t0 ) ∈ W are bounded for t ≥ t0 and e (t ) = hσ (t ) (x (t ) , ω (t )) → 0, as t → ∞. Remark 1. The switching signal σ (t ) and the controllers (6) comprise a switched regulator. Moreover, if there exists the unknown constant parameter p in system (1), then p can be considered as a part of the vector ω. In fact, any constant parameter p from a certain set P can be seen as a solution of the equation p˙ = 0 with the initial condition p (t0 ) = p. Therefore, p˙ = 0 can be considered as a part of the corresponding exosystem dynamics (2). Thus, what we will investigate is the globally robust output regulation problem. Now, a necessary condition of solvability of the output regulation problem for system (1) is given as follows. Assumption 1. For any solution of the exosystem (2) starting from ω (t0 ) ∈ W , there exist a bounded solution xω (t ) and a bounded input uiω (t ) , i ∈ I on R+ satisfying x˙ ω (t ) = fi (xω (t ) , uiω (t ) , ω (t )) , ∀t ≥ t0 , 0 = hi (xω (t ) , ω (t )) .

(7)

Remark 2. Assumption 1 has been adopted for non-switched systems [10,11]. (7) implies that the regulator equation of each subsystem is solvable and xω (t ) is a common solution of the regulator equations of all subsystems. Thus, xω (t ) is a steady-state solution of system (1) corresponding to the inputs uiω (t ). Remark 3. Similar to [23,28,29], if there exist differentiable maps π (ω) and ci (ω) defined on a set W such that xω = π (ω) and uiω = ci (ω)then the regulator equation

∂π s (ω) = fi (π (ω) , ci (ω) , ω) , ∂ω 0 = hi (π (ω) , ω)

(8)

is equivalent to (7). In this paper, we will investigate incremental passivity for switched nonlinear systems and solve the global output regulation problem for system (1) using established incremental passivity theory. 3. Incremental passivity of switched nonlinear systems In this section, to solve the output regulation, we will introduce a definition of incremental passivity and obtain some basic results on incremental passivity for switched nonlinear systems. First, the definition of class GK function and the incremental passivity for switched nonlinear systems that will be used in the sequel are introduced as follows. Definition 1 ([37]). A function α : R+ → R+ is called a class GK function if it is right continuous at the origin with α (0) = 0 and increasing. Definition 2 ([35]). System x˙ = Fσ (x, uσ , ω) , y = Hσ (x, ω) , Fi , Hi ∈ C 1

(9)

is said to be incrementally passive under a given switching signal σ (t ) if there exists a nonnegative function S(σ (t), t , x1 , x2 ), called a storage function, and class GK function α such that for all t ≥ t0 , any bounded signal ω (t ), any two inputs u1σ and u2σ , any two solutions of system (9) x1 (t ) and x2 (t ) corresponding to these inputs, the respective outputs y1 (t ) = Hσ (x1 (t ) , ω (t )) and y2 (t ) = Hσ (x2 (t ) , ω (t )) satisfy the inequality S (σ (t ) , t , x1 (t ) , x2 (t )) − S (σ (t0 ) , t0 , x1 (t0 ) , x2 (t0 ))

∫

t

( ) (y1 (τ ) − y2 (τ ))T u1σ (τ ) (τ ) − u2σ (τ ) (τ ) dτ + α (∥x1 (t0 ) − x2 (t0 )∥) .

≤ t0

(10)

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H. Pang, J. Zhao / Nonlinear Analysis: Hybrid Systems 27 (2018) 239–257

Fig. 1. Feedback interconnection of switched systems.

Remark 4. Compared with [34], the storage functions in Definition 2 are allowed to increase at the switching times. Moreover, the active subsystem may be the ‘‘shortage’’ of incremental passivity. The item α (∥(x1 (t0 ) − x2 (t0 ))∥) is used to measure the total change of ‘‘energy’’ at the switching times and the shortage of incremental energy for the active subsystem on any finite time interval. When system (9) has only one subsystem and α ≡ 0,Definition 2 degenerates into the incremental passivity definition for nonlinear system in [10,13]. Next, we will show that incremental passivity property is invariant under compatible feedback interconnection as the non-switched case. Consider switched nonlinear systems x˙ 1 = Fσ11 x1 , u1σ1 , ω (t ) ,

)

(

H1 :

(11)

y1 = Hσ11 x1 , ω (t )

)

(

1 n with the signal σ1 (t ) :} [0, ∞) → I1 = {1, 2, . . . M1 } and the switching sequence { state x ∈ R 1 , the ( switching )

Σ1 =

i0 , t01 , i11 , t11 , . . . i1k1 , tk11 , . . . ⏐i1k1 ∈ I1 , k1 ∈ N and

(1

) (

⏐

)

x˙ 2 = Fσ22 x2 , u2σ2 , ω (t ) ,

)

(

H2 :

y2 = Hσ22

(

(12)

) x2 , ω (t )

2 n with the signal σ2 (t ) :} [0, ∞) → I2 = {1, 2, . . . M2 } and the switching sequence { state x ∈ R 2 , the ( switching )

Σ2 =

i0 , t02 , i21 , t12 , . . . i2k2 , tk22 , . . . ⏐i2k2 ∈ I2 , k2 ∈ N .

(2

) (

⏐

)

The feedback interconnection of H1 and H2 depicted in Fig. 1can be described by

)) ( 1) ( 1 ( 1 1 Fσ1 x , uσ1 , ω (t ) x˙ ) = Fσ (x, uσ , ω (t )) , ( x˙ = = x˙ 2 Fσ22 x2 , u2σ2 , ω (t ) ( 1) ( 1 ) Hσ1 y = Hσ (x, ω (t )) , y= = 2 2

(13)

Hσ2

y

2T T

where x = x , x

, uσ =

(

rσ1

)

, dim rσ22 = dim Hσ11 = dim u2σ2 ,dim rσ11 = dim Hσ22 = dim u1σ1 , u1σ1 = rσ11 − y2 ,u2σ2 = )⏐ {( } σ rσ22 + y1 ,σ = σ1 : [0, ∞) → I = I1 × I2 = i1 , i2 ⏐i1 ∈ I1 ; i2 ∈ I2 is the composite switching signal of system (13). Thus, 2 system (13) has M1 × M2 subsystems. Corresponding to the composite switching signal, we have the switching sequence ⏐ { } Σ = (i0 , t0 ) , (i1 , t1 ) , . . . , (ik , tk ) , . . . ⏐ik ∈ I , k ∈ N , (14) ( ) where t0 = t01 = t02 , ik = (σ1 (tk ) , σ2 (tk )) = i1k1 , i2k2 .

(

1T

)

( )

1 rσ2 2

Lemma 1. Suppose systems (11) are incrementally passive under the given switching signal σ1 and σ2 for any bounded signal ω (t ). Then, system (13) is incrementally passive under the switched signal σ (t ) = (σ1 (t ) , σ2 (t )) . Proof. Let S σ1 (t ) , t , x11 , x12 and S σ2 (t ) , t , x21 , x22 be the candidate storage functions of systems (11) and (12), respectively. The storage function of system (13) is chosen as

(

)

(

)

S (σ (t ) , t , x1 , x2 ) = S σ1 (t ) , t , x11 , x12 + S σ2 (t ) , t , x21 , x22 .

(

)

(

)

According to the definition of incremental passivity, for all t ≥ t0 , there exist class GK functions α1 and α2 such that S ((σ (t ) , t , x1 (t ) , x2 (t ))) − S (σ ( (t0 ) , t0 , x1 1(t0 ) , x2 1(t0 )) ) = S σ(1 (t ) , t , x11 (t ) , x12 (t ) − S ) σ(1 (t0 ) , t0 , x1 (t0 ) , x2 (t0 ) ) +S σ2 (t ) , t , x21 (t ) , x22 (t ) − S σ2 (t0 ) , t0 , x21 (t0 ) , x22 (t0 )

H. Pang, J. Zhao / Nonlinear Analysis: Hybrid Systems 27 (2018) 239–257 t

∫

)T (

)

)T (

)

u11σ1 (τ ) (τ ) − u12σ1 (τ ) (τ )

y11 (τ ) − y12 (τ )

(

≤

∫ t0t

dτ + α1 x11 (t0 ) − x12 (t0 )

)

(

(15)

u21σ2 (τ ) (τ ) − u22σ2 (τ ) (τ ) dτ + α2 x21 (t0 ) − x22 (t0 ) .

y21 (τ ) − y22 (τ )

(

+

243

t0

(

)

Substituting u1σ1 = rσ11 − y2 , u2σ2 = rσ22 + y1 into (15) gives S (σ (t ) , t , x1 (t ) , x2 (t )) − S (σ (t0 ) , t0 , x1 (t0 ) , x2 (t0 )) t

∫

)T (

y11 (τ ) − y12 (τ )

(

≤ t0

r11σ1 (τ ) (τ ) − r21σ1 (τ ) (τ ) dτ +

)

∫

t

y21 (τ ) − y22 (τ )

(

)T (

t0

r12σ2 (τ ) (τ ) − r22σ2 (τ ) (τ ) dτ

)

+ α1 (∥x1 (t0 ) − x2 (t0 )∥) + α2 (∥x1 (t0 ) − x2 (t0 )∥) t

∫

( ) (y1 (τ ) − y2 (τ ))T u1σ (τ ) (τ ) − u2σ (τ ) (τ ) dτ + α (∥x1 (t0 ) − x2 (t0 )∥) ,

= t0

where α = α1 + α2 is class GK function. Therefore, system (13) is incrementally passive under the switched signal σ (t ) = (σ1 (t ) , σ2 (t )) . Definition 3 ([10]). A storage function S (t , x1 , x2 ) is called regular if for any sequence (tk , x1k (tk ) , x2k (tk )) , k = 1, 2, . . . such that x2k is bounded, tk tends to infinity and ∥x1k ∥ → +∞, it holds that S (tk , x1k , x2k ) → +∞, as k → +∞. To obtain the main result of this paper, we need the following result. Lemma 2. Consider system (9). Let ω (t ) and ω ˙ (t ) be bounded on R+ . Suppose system (9) with a storage function S (σ (t ) , t , x1 , x2 ) ∆Sσ (t ) (t , x1 , x2 ) is incrementally passive for a given switching signal σ (t ), where Si (t , x1 , x2 ) are regular. If there exists a bounded(solution x )(t ) for system (9) with uσ (t ) = 0 such that the output y (t ) = Hσ (x (t ) , ω (t )) = 0 for t ≥ t0 and limk→∞ tjk +1 − tjk ̸ = 0 for all j ∈ I then, all solutions of system (9) with ui = −Ki y are defined and bounded for all t ≥ t0 and y (t ) → 0 as t → ∞, where Ki are positive definite matrices. Proof. For ∀t ≥ t0 , there exists k ∈ N such that t ∈ [tk , tk+1 ) , since system (9) is incrementally passive, we have S (σ (t ) , t , x1 (t ) , x2 (t )) − S (σ (t0 ) , t0 , x1 (t0 ) , x2 (t0 )) = Sik (t , x1 (t ) , x2 (t )) − Si0 (t0 , x1 (t0 ) , x2 (t0 )) ∫ t ) ( ≤ (y1 (τ ) − y2 (τ ))T u1σ (τ ) (τ ) − u2σ (τ ) (τ ) dτ + α (∥x1 (t0 ) − x2 (t0 )∥) .

(16)

t0

According to the conditions of the lemma, the closed-loop system (9) with uσ (t ) (t ) = −Kσ (t ) y (t ) has a bounded solution x (t ) and the output y (t ) = 0 for t ≥ t0 . The input corresponding to the solution x (t ) is uσ (t ) (t ) = −Kσ (t ) y (t ) ≡ 0. Substituting x (t ), y (t ) = 0 and uσ (t ) (t ) = 0 for x2 (t ), y2 (t ), u2σ (t ) (t ) and x (t ), y (t ) and uσ (t ) (t ) = −Kσ (t ) y (t ) for x1 (t ), y1 (t ), u1σ (t ) (t ) gives Sik (t , x (t ) , x (t )) − Si0 (t0 , x (t0 ) , x (t0 )) t

∫

y(τ )T Kσ y (τ ) dτ + α (∥x (t0 ) − x (t0 )∥)

≤− t0

t

∫

λy(τ )T y (τ ) dτ + α (∥x (t0 ) − x (t0 )∥) ,

≤−

(17)

t0

where λ = mini∈I {λmin (Ki )}, λmin (Ki ) > 0 denotes the minimum eigenvalue of Ki . From (17), we have Sik (t , x (t ) , x (t )) +

∫

t

λy(τ )T y (τ ) dτ ≤ Si0 (t0 , x (t0 ) , x (t0 )) + α (∥x (t0 ) − x (t0 )∥) ,

(18)

t0

which implies Sik (t , x (t ) , x (t )) ≤ Si0 (t0 , x (t0 ) , x (t0 )) + α (∥x (t0 ) − x (t0 )∥)

(19)

for t ≥ t0 . Since functions Si are regular, the boundedness of both Sik (t , x (t ) , x (t )) and x (t ) implies the boundedness of x (t ).

244

H. Pang, J. Zhao / Nonlinear Analysis: Hybrid Systems 27 (2018) 239–257

Next, { ⏐we will show lim } t →∞ y (t ) = 0. For each j satisfying limk→∞ tjk +1 − tjk ̸= 0, there exists δ > 0 such that the set Π = k⏐tjk +1 − tjk ≥ δ is infinite. Let the auxiliary function

(

{ ˜ Hj (t ) =

Hj (x (t ) , ω (t )) , t ∈ ∪

0,

otherwise.

tjk , tjk +1 ,

[

k∈Π

)

)

(20)

From (19) and (20), we have

∫

t

λ˜ HjT (τ ) ˜ Hj (τ ) dτ ≤

∫

t

λy(τ )T y (τ ) dτ ≤ Si0 (t0 , x (t0 ) , x (t0 )) + α (∥x (t0 ) − x (t0 )∥) .

(21)

t0

t0

Therefore, lim H  t →∞ ˜  j (t ) = 0. Suppose this is wrong, then there exist ε0 > 0 and a sequence of time q1 , q2 , . . . , qk → ∞such that Hj (qi ) ≥ ε0 , ∀i. Seen from the equation of the closed-loop system (9) with uσ (t ) (t ) = −Kσ (t ) y (t ), x˙ (t ) is bounded. This, in turn, together with the boundedness of )ω (t ) and ω ˙ (t ) on R+ implies that H˙ j ∫(x (t ) , ω (t )) is bounded on [ ∞ + ˜ HjT (τ ) ˜ R . Hence, Hj (t ) is uniformly continuous over ∪ tjk , tjk +1 . Since tjk +1 −tjk ≥ δ, k ∈ Π , we have t λ˜ Hj (τ ) dτ = ∞, k∈Π

0

which contradicts (21). Therefore, limt →∞ Hj (x (t ) , ω (t )) = 0 for all j ∈ I, which implies y (t ) → 0 as t → ∞. Remark 5. If each subsystem of system (9) has a common output y = H ∫(x, ω) then the condition limk→∞ tjk +1 − tjk ̸ = 0 ∞ for j = 1, 2 · · · M is not needed in Lemma 2. In fact, since (21) holds, t λy(τ )T y (τ ) dτ < ∞. Similar to the proof of 0 + Lemma 2, y (t ) is uniformly continuous on R . According to Barbalat’s lemma, we have limt →∞ y (t ) = 0. Therefore, Lemma 2 generalizes the result in [10,34].

(

)

4. Incremental passivity-based output regulation In this section, we will give incremental passivity condition and solve the global output regulation problem for system (1) using the incremental passivity theory for switched nonlinear systems. 4.1. Sufficient conditions for incremental passivity First, we will design a switching law and error feedback controllers for system (1) to render the corresponding closed-loop system incrementally passive, when the whole states information is unavailable for measurements. Theorem 1. Consider system (1). Suppose that there exist a set of feedback controllers ui = usi + vi , usi = θsi (ζ , e) , ζ˙ = ηsi (ζ , e) ,

(22)

where ηsi (ζ , e) and θsi (ζ , e) are smooth functions with ηis (0, 0) = 0, θsi (0, 0) = 0, ζ ∈ Rr , nonnegative smooth functions Vsi (ζ1 , ζ2 ) and V (x1 , x2 , t ), which are regular, continuous functions βij (ζ1 , ζ2 ) ≤ 0 and δij (ζ1 , ζ2 ) ≤ 0,smooth functions µij (ζ1 − ζ2 ) with µii (ζ1 − ζ2 ) = 0 and µij (0) = 0 such that for any ω (t ) ∈ W ⊂ Rs and ∀i, j, k ∈ I, the closed-loop system x˙ = fσ (x, θsσ (ζ , e) + vσ , ω (t )) , ζ˙ = ηsσ (ζ , e) e = hσ (x, ω)

(23)

satisfies

∂V ∂V ∂V ∂ Vsi + fi (x1 , θsi (ζ1 , e1 ) + v1i , ω) + fi (x2 , θsi (ζ2 , e2 ) + v2i , ω) + ηsi (ζ1 , e1 ) ∂t ∂ x1 ∂ x2 ∂ζ1 M ∑ ( ) ∂ Vsi + ηsi (ζ2 , e2 ) + βij (ζ1 , ζ2 ) Vsi (ζ1 , ζ2 ) − Vsj (ζ1 , ζ2 ) + µij (ζ1 − ζ2 ) ≤ (v1i − v2i )T (e1 − e2 ) , ∂ζ2

(24)

j=1

M ∑ ) ( ∂µij ∂µij ηsi (ζ1 , e1 ) + ηsi (ζ2 , e2 ) + δij (ζ1 , ζ2 ) Vsi (ζ1 , ζ2 ) − Vsj (ζ1 , ζ2 ) + µij (ζ1 − ζ2 ) ≤ 0, ∂ζ1 ∂ζ2

(25)

µij (ζ1 − ζ2 ) + µjk (ζ1 − ζ2 ) ≤ min {0, µik (ζ1 − ζ2 )} , ∀i, j, k.

(26)

j=1

Design the switching law as

σ (t ) = i if

( ) σ t − = i and

(ζ1 (t ) , ζ2 (t )) ∈ int Ωi ,

H. Pang, J. Zhao / Nonlinear Analysis: Hybrid Systems 27 (2018) 239–257

245

} ( ) { ⏐ ˜ij , if σ t − = i and (ζ1 (t ) , ζ2 (t )) ∈ Ω σ (t ) = min arg Ωj ⏐ (ζ1 (t ) , ζ2 (t )) ∈ Ωj ⏐ } { where Ωi = (ζ1 , ζ2 ) ⏐Vsi (ζ1 , ζ2 ) − Vsj (ζ1 , ζ2 ) + µij (ζ1 − ζ2 ) ≤ 0, j ∈ I and

(27)

⏐ } { ˜ij = (ζ1 , ζ2 ) ⏐Vsi (ζ1 , ζ2 ) − Vsj (ζ1 , ζ2 ) + µij (ζ1 − ζ2 ) = 0, i ̸= j . Ω

(28)

Then, system (23) is incrementally passive under the switching law (27). Proof. Similar to [37], we can show that Ωi ⏐, i ∈ I makes a partition of R2r . The candidate storage function is chosen as

{

⏐

}

S (σ (t ) , t , x1 , x2 , ζ1 , ζ2 ) = Sσ (t ) (t , x1 , x2 , ζ1 , ζ2 ) = V (x1 , x2 , t ) + Vsσ (t ) (ζ1 , ζ2 ) . For fixed function ω (t ) ∈ W , when (ζ1 , ζ2 ) ∈ Ωi , since (24) holds, differentiating Si along the system trajectory of (23) gives S˙i = V˙ (x1 , x2 , t ) + V˙ si (ζ1 , ζ2 ) ≤ (v1i − v2i )T (e1 − e2 ) .

(29)

˜ij According to the switching law (27), {once (ζ1 , ζ2 ) enters Ωi it will stay in Ωi until it hits the boundary in Ω ⏐ the trajectory } ˜ij for some j. Thus, we obtain ˜ij . That is to say, switching only happens on Ω and then enters ΩL , where L = min j⏐Ωi ∩ Ω the switching sequence (3) and Sik+1 (tk+1 , x1 (tk+1 ) , x2 (tk+1 ) , ζ1 (tk+1 ) , ζ2 (tk+1 )) − Sik (tk+1 , x1 (tk+1 ) , x2 (tk+1 ) , ζ1 (tk+1 ) , ζ2 (tk+1 )) = Vsik+1 (ζ1 (tk+1 ) , ζ2 (tk+1 )) − Vsik (ζ1 (tk+1 ) , ζ2 (tk+1 )) = µik ik+1 (ζ1 (tk+1 ) − ζ2 (tk+1 )) .

(30)

(25) means that µ ˙ ij ≤ 0 on Ωi , which tell us that µik j (ζ1 (t ) − ζ2 (t )) are decreasing on [tk , tk+1 ) . For t0 ≤ t < ∞, there exist k ∈ N such that t ∈ [tk , tk+1 ) , from (22), we have S (σ (t ) , t , x1 (t ) , x2 (t ) , ζ1 (t ) , ζ2 (t )) − S (σ (t0 ) , t0 , x1 (t0 ) , x2 (t0 ) , ζ1 (t0 ) , ζ2 (t0 ))

= Sik (t , x1 (t ) , x2 (t ) , ζ1 (t ) , ζ2 (t )) − Sik (tk , x1 (tk ) , x2 (tk ) , ζ1 (tk ) , ζ2 (tk ))

+

k ∑ (

+

k−1 ∑ (

Sip tp , x1 tp , x2 tp , ζ1 tp , ζ2 tp

( ))

( )

( )

( )

(

( ))) ( ) ( ) ( ) ( − Sip−1 tp , x1 tp , x2 tp , ζ1 tp , ζ2 tp

p=1

Sip tp+1 , x1 tp+1 , x2 tp+1 , ζ1 tp+1 , ζ2 tp+1

(

(

(

)

)

(

)

(

))

( ))) ( ) ( ) ( ) ( − Sip tp , x1 tp , x2 tp , ζ1 tp , ζ2 tp

p=0

∫

t

≤

k ∑ ( )) ( ( ) ) ( µip−1 ip ζ1 tp − ζ2 tp (e1 (τ ) − e2 (τ ))T v1σ (τ ) (τ ) − v2σ (τ ) (τ ) dτ +

t0

p=1

⎧∫ t ( ) ⎪ T ⎪ ⎪ ⎨ (e1 (τ ) − e2 (τ )) v1σ (τ ) − v2σ (τ ) dτ if k is even t ≤ ∫ 0t ( ) ⎪ ⎪ T ⎪ ⎩ (e1 (τ ) − e2 (τ )) v1σ (τ ) − v2σ (τ ) dτ + µi0 i1 (ζ1 (t0 ) − ζ2 (t0 )) if k is odd t0

∫

t

( ) (e1 (τ ) − e2 (τ ))T v1σ (τ ) − v2σ (τ ) dτ + α (∥(ζ1 (t0 ) − ζ2 (t0 ))∥) ,

≤ t0

where α (s) = max∥ζ1 −ζ2 ∥≤s ⏐µij (ζ1 (t0 ) − ζ2 (t0 )) ⏐, i, j ∈ I under the switching law (27).

{⏐

⏐

}

is class GK function. Then, system (23) is incrementally passive

Example 1. Consider system (1) consisting of two subsystems described by f1 (x, u1 , ω) =

(

)

2x2 − x1 , 1 2 −x − 6x + 8 + u1 + sin ω

e = 2x2 + 2ω, x = x1 , x2 .

(

)

f2 (x, u2 , ω) =

(

3

)

2x2 − 3x1 , 1 −x − 8x2 + u2 + cos ω (31)

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H. Pang, J. Zhao / Nonlinear Analysis: Hybrid Systems 27 (2018) 239–257

Design the dynamic feedback controllers 1

ζ˙ 1 = −6ζ 1 + ζ 2 + e, ζ˙ 2 =

1 3

3

1

ζ 1 + 2ζ 2 + e, 3

3 ζ˙ 1 = −2ζ 1 + ζ 1 + ζ 2 + 10e,

(32)

ζ˙ 2 = 2ζ 1 − 20ζ 2 + 3e,

u1 = −ζ 1 − ζ 2 + v1 ,

u2 = −20ζ 1 − 3ζ 2 + v2 , ( 1 2) where ζ = ζ , ζ . The candidate storage functions of the resulting closed-loop system are chosen as Si (x1 , x2 , ζ1 , ζ2 ) = Vsi (ζ1 , ζ2 ) + V (x1 , x2 ) , i = 1, 2, where Vsi (ζ1 , ζ2 ) = (ζ1 − ζ2 )T Pi (ζ1 − ζ2 ) , i = 1, 2 and V (x1 , x2 ) =

[ P1 =

1 0

]

[

0 2 , P2 = 3 0

]

0 1

[ and P =

1 0

1 2

(x1 − x2 )T P (x1 − x2 ) ,

]

0 . 2

Its derivative along solutions of system (32) satisfies

)2

S˙1 ≤ −β12 (V1 − V2 ) − 3(e1 − e2 )2 + (v11 − v12 ) (e1 − e2 ) − x11 − x12 ,

(

)4

S˙2 ≤ −β21 (V2 − V1 ) − 4(e1 − e2 )2 + (v21 − v22 ) (e1 − e2 ) − 3 x11 − x12 ,

(

where β12 = −7, β21 = −10. According to Theorem 1, system (31) is incrementally passive under switching law (27). Remark 6. The conditions (24)–(26) are commonly used for switched nonlinear systems [29,35,37]. (24) means that each subsystem is not required to be incrementally passive. The incremental passivity inequality for each subsystem is only required to hold on Ωi , which is weaker than the condition in [9,10]. Next, we will give a sufficient condition to be incrementally passive for system (1) in the form x˙ = Fσ (x, ω) + Bσ uσ , e = Cx + H (ω) ,

(33)

where Bi , C are constant matrices and Fi , H are assumed to be continuous in ω and smooth in x for i ∈ I. An error-dependent switching law is to be designed to render system (33) incrementally passive in the following theorem. Theorem 2. Consider system (33). Suppose that there exist βij ≤ 0, δij ≤ 0, (βij , δij may depend on x), smooth functions µij (x1 − x2 ) = (x1 − x2 )T Γij (x1 − x2 ), νij (e1 − e2 ) = (e1 − e2 )T Λij (e1 − e2 ) with Γii = 0 and Λii = 0 , matrices Qi = QiT , positive definite matrices Pi and constants λij such that Pi

M ∑ ) ( ∂F T ∂ Fi βij C T Qi − Qj + Λij C ≤ 0, (x, ω) + i (x, ω) Pi + ∂x ∂x

Pi Bi = C T , ∀x ∈ Rn ,

(34)

j=1

Γij

M ∑ ( ) ∂F T ∂ Fi δij C T Qi − Qj + Λij C ≤ 0, Γij Bi = 0, ∀x ∈ Rn , (x, ω) + i (x, ω) Γij + ∂x ∂x

(35)

j=1

Γij + Γjk − Γik ≤ 0, Γij + Γjk ≤ 0, Λij + Λjk − Λik ≤ 0, Λij + Λjk ≤ 0,

(36)

Pi − Pj + Γij = λij C T Qi − Qj + Λij C ,

(37)

(

)

where Γij and Λij are symmetric matrices, hold for any ω ∈ W , ∀i, j, k ∈ I . The switching law is designed as

( ) σ (t ) = i if σ t − = i and

(e1 (t ) , e2 (t )) ∈ intΩi ,

{ ⏐ } ( ) ˜ij , σ (t ) = min arg Ωj ⏐ (e1 (t ) , e2 (t )) ∈ Ωj if σ t − = i and (e1 (t ) , e2 (t )) ∈ Ω ⏐ { ( ) } where Ωi = (e1 , e2 ) ⏐(e1 − e2 )T Qi − Qj + Λij (e1 − e2 ) ≤ 0, j ∈ I and ⏐ ( ) } { ˜ij = (e1 , e2 ) ⏐(e1 − e2 )T Qi − Qj + Λij (e1 − e2 ) = 0, i ̸= j . Ω Then, system (33) is incrementally passive under switching law (38).

(38)

(39)

H. Pang, J. Zhao / Nonlinear Analysis: Hybrid Systems 27 (2018) 239–257

247

Proof. The candidate storage function is chosen as S (σ (t ) , x1 , x2 ) ≜ Sσ (t ) (x1 , x2 ) , where Si (x1 , x2 ) =

1

(x1 − x2 )T Pi (x1 − x2 ) , i ∈ I . { ⏐ } Similar to the proof of Theorem 1, we can show that Ωi ⏐i ∈ I makes a partition of R2m . When (e1 , e2 ) ∈ Ωi , differentiating Si (x1 , x2 ) along the system trajectory of (33) gives 2

S˙i = (x1 − x2 )T Pi (Fi (x1 , ω) − Fi (x2 , ω)) + (x1 − x2 )T Pi Bi (u1i − u2i ) . Similar to [38], according to (34), we obtain 1

(x1 − x2 )T Ji (x, ω) (x1 − x2 ) + (Cx1 − Cx2 )T (u1i − u2i ) 2 ≤ (Cx1 + H (ω) − Cx2 − H (ω))T (u1i − u2i )

S˙i =

≤ (e1 − e2 )T (u1i − u2i ) , ∂F T

∂F

where Ji (x, ω) = pi ∂ xi (x, ω) + ∂ xi (x, ω) pi , x is some point between x1 and x2 . ˜ij for some j. Thus, we obtain the switching sequence According to the switching law (38), switching only takes place on Ω (3) and

) ( (e1 (tk+1 ) − e2 (tk+1 ))T Qik+1 − Qik + Λik+1 ik (e1 (tk+1 ) − e2 (tk+1 )) = 0,

(40)

which implies Sik+1 (x1 (tk+1 ) , x2 (tk+1 )) − Sik (x1 (tk+1 ) , x2 (tk+1 ))

= =

1 2 1 2

(x1 (tk+1 ) − x2 (tk+1 ))T Γik ik+1 (x1 (tk+1 ) − x2 (tk+1 ))

(41)

µik ik+1 (x1 (tk+1 ) − x2 (tk+1 )) .

Similarly, according to (35), we have µ ˙ ij ≤ 0 on Ωi , which implies that µik j (x1 (t ) − x2 (t )) are decreasing on tk, tk+1 . For any t ≥ t0 , there exist k such that t ∈ [tk , tk+1 ) , since (41) holds, we have

[

]

S (σ (t ) , x1 (t ) , x2 (t )) − S (σ (t0 ) , x1 (t0 ) , x2 (t0 ))

= Sik (x1 (t ) , x2 (t )) − Sik (x1 (tk ) , x2 (tk ))

+

k−1 ∑ (

Sip x1 tp+1 , x2 tp+1

( (

)

(

))

k ( ( ) ( ))) ( )) ( ( ) ( ))) ∑ ( ( ( ) + Sip x1 tp , x2 tp − Sip−1 x1 tp , x2 tp − Sip x1 tp , x2 tp p=1

p=0

∫

t

≤

k ( ) ( ( ) ( )) 1∑ µip−1 ip x1 tp − x2 tp (e1 (τ ) − e2 (τ ))T u1σ (τ ) (τ ) − u2σ (τ ) (τ ) dτ +

2

t0

p=1

⎧∫ t ( ) ⎪ T ⎪ ⎪ ⎨ (e1 (τ ) − e2 (τ )) u1σ (τ ) (τ ) − u2σ (τ ) (τ ) dτ if k is even t ≤ ∫ 0t ( ) ⎪ 1 ⎪ T ⎪ ⎩ (e1 (τ ) − e2 (τ )) u1σ (τ ) (τ ) − u2σ (τ ) (τ ) dτ + µi0 i1 (x1 (t0 ) − x2 (t0 )) if k is odd t0

∫

2

t

( ) (e1 (τ ) − e2 (τ ))T u1σ (τ ) (τ ) − u2σ (τ ) (τ ) dτ + α (∥x1 (t0 ) − x2 (t0 )∥) ,

≤ t0

where α (s) = max∥x1 −x2 ∥≤s the switching law (38).

⏐ { 1⏐ } ⏐µij (x1 − x2 ) ⏐, i, j ∈ I is class GK function. Then, system (33) is incrementally passive under 2

Remark 7. Sometimes it is difficult or even impossible to get the whole information of states. Therefore, the switching law (38) which is dependent on the regulated error of system (33) is more realistic and can be implemented easily in practice.

248

H. Pang, J. Zhao / Nonlinear Analysis: Hybrid Systems 27 (2018) 239–257

4.2. Incrementally passive switched internal model Similar to non-switched case, a switched internal model for system (1) is defined as follows. Definition 4. Consider a switched system described by

) ( eστ , vτ = γστ (τ ) , τ˙ = φστ τ ,˜

(42)

eσ ′ = dim hσ ,dim γστ = where τ ∈ R is state, στ (t ) : [0, ∞) → Iτ = {1, 2, . . . Mτ } is the switching signal and dim ˜ dim uσ . For any solution of the exosystem ω (t ) starting from ω (t0 ) ∈ W , there exists τ ω (t ) that is bounded on R+ satisfies d

τ˙ ω (t ) = φτστ (τ ω (t ) , 0) , uσ ω (t ) = γστ (τ ω (t )) , ∀t ≥ t0 .

(43)

Therefore, system (42) is said to have the internal model property. Corresponding to the switching signal, we have the switching sequence

∑

⏐ } { = τ0 ; (iτ 0 , t0 ) , (iτ 1 , tτ 1 ) , . . . , (iτ 1 , tτ 1 ) , . . . ⏐iτ l ∈ Iτ , l ∈ N .

τ

Remark 8. In general, if there exist a map τ ω = T (ω) : W → Rd satisfying

∂T s (ω) = φστ (τ ω , 0) , uσ ω = γστ (τ ω ) , ∂ω

(44)

then, (44) is equivalent to (43). It is clear that for system (1), (42) is an internal model in the usual sense given in [10]. In fact, each subsystem of system (42) is not required to have internal model property. When switching occurs, system (43) can generate a set of steady-state controllers. Therefore, system (42) is called a switched internal model of system (1). Moreover, the switching signal of system (42) is allowed to be different from the switching signal of system (1). Therefore, the internal model may switch asynchronously with system (1). This gives more design freedom of the design of switched internal model systems. And the steady control is common i.e., uiω (t ) = γ (τ ω (t )) , ∀t ≥ t0 . If σω = στ then uiω (t ) = γi (τ ω (t )) , ∀t ≥ t0 . It is more general than the switched internal model designed in [23]. The standing assumption on system (42) is given as follows. ′

Assumption 2. There exist regular nonnegative smooth functions Wk (τ1 , τ2 ), functions λkl (τ1 , τ2 ) ≤ 0, λkl (τ1 , τ2 ) ≤ 0 and smooth functions ρkl (τ1 − τ2 ) with ρkl (0) = 0 and ρkk (τ1 − τ2 ) = 0 for k, l ∈ Iτ , a map τ ω = T (ω) : W → Rd such that uiω = γ (τ ω ) and the following conditions hold

∂ Wk ∂ Wk φk (τ1 ,˜ e1k ) + φk (τ2 ,˜ e2k ) ∂τ1 ∂τ2 Mτ ∑ e1k − ˜ e2k ) , λkl (τ1 , τ2 ) (Wk (τ1 , τ2 ) − Wl (τ1 , τ2 ) + ρkl (τ1 − τ2 )) ≤ (v1τ − v2τ )T (˜ +

(45)

l=1 M

τ ∑ ∂ρkl ∂ρkl ′ φk (τ1 ,˜ e1k ) + φk (τ2 ,˜ e2k ) + λkl (τ1 , τ2 ) (Wk (τ1 , τ2 ) − Wl (τ1 , τ2 ) + ρkl (τ1 − τ2 )) ≤ 0, ∂τ1 ∂τ2

(46)

ρkj (τ1 − τ2 ) + ρjl (τ1 − τ2 ) ≤ min {0, ρkl (τ1 − τ2 )} , ∀k, j, l,

(47)

l=1

(

) { } ∂T s (ω) − φk (τ ω , 0) + max max {Wk (τ1 , τ ω ) − Wl (τ1 , τ ω ) + ρkl (τ1 − τ ω )} , 0 = 0. k̸ =l ∂ω

(48)

Remark 9. Define

⏐ { } Ωτ k = (τ1 , τ2 ) ⏐Wk (τ1 , τ2 ) − Wl (τ1 , τ2 ) + ρkl (τ1 − τ2 ) ≤ 0, l ∈ Iτ , ⏐ { } ˜τ kl = (τ1 , τ2 ) ⏐Wk (τ1 , τ2 ) − Wl (τ1 , τ2 ) + ρkl (τ1 − τ2 ) = 0, k ̸= l . Ω

(49)

(45) means that incremental passivity property holds on Ωτ k . When λkl = 0, each subsystem is incrementally passive. When (42) has only one subsystem, (45) degenerates into the incremental passivity conditions for nonlinear systems [11]. (48) implies that the internal model property for each subsystem is only required to hold on Ωτ k . Now, we give a sufficient condition for the internal model system (42) to be incremental passive as follows.

H. Pang, J. Zhao / Nonlinear Analysis: Hybrid Systems 27 (2018) 239–257

249

Lemma 3. Suppose that Assumption 2 holds. Design the state-dependent switching law

στ (t ) = k if

( ) στ t − = k and

(τ1 (t ) , τ2 (t )) ∈ Ωτ k ,

} ( ) { ⏐ στ (t ) = min arg Ωτ l ⏐ (τ1 (t ) , τ2 (t )) ∈ Ωτ l if στ t − = k and

˜τ kl . (τ1 (t ) , τ2 (t )) ∈ Ω

(50)

Then, system (42) is an incrementally passive internal model system under the switching law (50). Proof. Similar to the proof of Theorem 1, we obtain that

˙k = W

∂ Wk ∂ Wk φk (τ1 ,˜ e1k ) + φk (τ2 ,˜ e2k ) ≤ (v1τ − v2τ )T (˜ e1k − ˜ e2k ) ∂τ1 ∂τ2

(51)

∆

and ρ˙ ij ≤ 0 on Ωτ k . Let W (στ (t ) , τ1 , τ2 ) = Wστ (t ) (τ1 , τ2 ) . Using a similar method as in Theorem 1, we can obtain that system (42) is incrementally passive. On the other hand, let ˜ e2k = 0, τ2 = τ ω . Thus, according to (48) and (49), system (42) is incrementally passive internal model system. Remark 10. The switching law (50) implies that the adjacent storage functions are not necessarily connected at the switching points. The switching law (50) includes the ‘‘min-switching’’ law [34] as a special case ρij ≡ 0. ek and vτ = C τ , i.e. φk (τ , 0) = ϕk (τ ) , γk (τ ) = C τ . Suppose Corollary 1. Consider system (42) with φk (τ ,˜ ek ) = ϕk (τ ) + Mk˜ that there exist smooth functions ρkl (τ1 − τ2 ) = (τ1 − τ2 )T ρkl (τ1 − τ2 ), where ρkl are symmetric matrices with ρkk = 0, λkl ≤ 0, ′ ′ λkl ≤ 0 (λkl and λkl may dependent on τ ), matrices Qi = QiT , and positive definite matrices Pk such that (48), M

Pk

τ ∑ ∂ϕ T ∂ϕk λkl (Pk − Pl + ρkl ) ≤ 0, Pk Mk = C T , ∀τ ∈ Rd , (τ ) + k (τ ) Pk + ∂τ ∂τ

(52)

l=1 M

ρkl

τ ∑ ∂ϕk ∂ϕk ′ λkl (Pk − Pl + ρkl ) ≤ 0, ρkl Mk = 0, ∀τ ∈ Rd , (τ ) + (τ ) ρkl + ∂τ ∂τ

(53)

l=1

ρkj + ρjl − ρkl ≤ 0,

ρkj + ρjl ≤ 0

(54)

hold for any ω ∈ W and k, l, j ∈ Iτ . Then, system (42) is incrementally passive internal model system under switching law (50) with Wk (τ1 , τ2 ) = 12 (τ1 − τ2 )T Pk (τ1 − τ2 ). Proof. Similar to the proof of Theorem 2, according to Lemma 3, Corollary 1 holds. ∂ϕ T

∂ϕ

Remark 11. (52) means that Pk ∂τk (τ ) + ∂τk (τ ) Pk ≤ 0 is only required to hold in a subregion Ωτ k , which tell us that the incremental passivity property of i-th subsystem holds only on Ωτ k . When system (42) has only one subsystem, (52) degenerates into the incremental passivity condition [10]. An incrementally passive switched internal model depends on the structure of the exosystem (2) as well as on the properties of system (1). In particular on uiω defined by (43). Therefore, it is not easy to verify the existence of an incrementally passive internal model. However, it can be designed in a constructive way. Example 2. If T = Id, ϕk = s in (44) and switched internal model as follows

∂s ∂ω

s + ∂∂ω ≤ 0 then according to Corollary 1, we can construct incrementally passive T

τ˙ = φk (τ ,˜ ek ) = s (τ ) + Mk˜ ek , vk = Mk τ . In fact, the storage function of (55) is chosen as W (στ (t ) , τ1 , τ2 ) = passive under an arbitrary switching law.

(55) 1 2

(τ1 − τ2 )T (τ1 − τ2 ). Thus, system (55) is incrementally

4.3. Solution to the output regulation problem Next, we will show how to use incremental passivity theory to solve the output regulation problem for system (1). Theorem 3. Consider systems (1) and (2). Suppose that all conditions ( T Tof )Theorem 1 and Assumptions ( )T 1 and 2 hold. Then, the ˆ = eT , vτT are incrementally passive interconnected switched systems (23) and (42) with input vˆ˜ σ = vsσ , vM στ and output e under the composite switched law

⏐ { } I = I × Iτ = (i, k) ⏐i ∈ I ; k ∈ Iτ , (σ (t ) , στ (t )) : [0, ∞) → ˜

(56)

250

H. Pang, J. Zhao / Nonlinear Analysis: Hybrid Systems 27 (2018) 239–257

where σ (t ) and στ (t ) are defined by (27) and (50), respectively. If, in addition, limk→∞ tjk +1 − tjk ̸ = 0 for all j ∈ I, then, the output regulation problem is solved by the feedback controllers

(

)

ζ˙ = ηsi (ζ , e) , usi = θsi (ζ , e) τ˙ = φk (τ , −e) , uiM = γk (τ ) ui = usi + uiM − Ki e, k ∈ Iτ , i ∈ I

(57)

where Ki are positive definite matrices, under the switching law

( ) ˜ σ (t ) = σ0 (t ) , στ ω (t )

(58)

with

( ) σ0 t − = i and (ζ (t ) , 0) ∈ int Ω0i , { ⏐ } ( ) σ0 (t ) = min arg Ω0j ⏐ (ζ (t ) , 0) ∈ Ω0j , if σ0 t − = i and σ0 (t ) = i,

if

˜0ij , (ζ (t ) , 0) ∈ Ω

and

( ) στ ω (t ) = k, if στ ω t − = k and (τ (t ) , τ ω (t )) ∈ Ωτ ω k , ⏐ { } ( ) στ ω (t ) = min arg Ωτ ω l ⏐ (τ (t ) , τ ω (t )) ∈ Ωτ ω l if στ ω t − = k and ⏐ { } where Ω0i = (ζ , 0) ⏐Vsi (ζ , 0) − Vsj (ζ , 0) + µij (ζ ) ≤ 0, i ̸ = j , ⏐ } { ˜0ij = (ζ , 0) ⏐Vsi (ζ , 0) − Vsj (ζ , 0) + µij (ζ ) = 0, i ̸= j Ω and ⏐ } { Ωτ ω k = (τ , τ ω ) ⏐Wk (τ , τ ω ) − Wl (τ , τ ω ) + ρkl (τ − τ ω ) ≤ 0, l ∈ Iτ , ⏐ } { ˜τ kl = (τ , τ ω ) ⏐Wk (τ , τ ω ) − Wl (τ , τ ω ) + ρkl (τ − τ ω ) = 0, k ̸= l . Ω

˜τ ω kl , (τ (t ) , τ ω (t )) ∈ Ω

ω

Proof. We interconnect the close-loop system (23) with the switched internal model system (42) through vσ = vτ + vsσ ,˜ eστ = −e + vM στ , where dim ˜ eστ = dim vM στ , dim γστ = dim vsσ . According to Lemmas 1, 3, and Theorem 1, for any solution ω (t() of the exosystem (2) starting ) ( from)Tω (t0 ) ∈ W , the interconnected switched systems (23) and (42) with T T ˆ = eT , vτT are incrementally passive the input vˆ˜ with a regular storage function under the σ = vsσ , vM στ and output e ⏐ } { σ (t ) and στ (t ) are defined composite switched law (σ (t ) , στ (t )) : [0, ∞) → ˜ I = I × Iτ = (i, k) ⏐i ∈ I ; k ∈[Iτ , where ] Kσ 0 ˆ ˆ ˆ , where K˜σ = 0 0 . Namely, vsi = −Ki e, i ∈ I and by (27) and (50), respectively. Design the controllers as vˆ˜ σ = −K˜ σe

T vMk = 0, k ∈ Iτ . According to Assumption 1 and (43), (xω (t ) , 0, τ ω (t )) is a bounded solution of the interconnected switched system with input vˆ˜ σ = 0 and along this solution e (t ) ≡ 0. Moreover, since any solution ω (t ) starting from ω (t0 ) ∈ W is bounded for all t ≥ t0 and the functions s are assumed to be smooth, ω ˙ (t ) are bounded. Hence, according to Lemma 2, all ˆ σ eˆ are bounded and limt →∞ e (t ) = 0. solutions of the interconnection switched system (23) and (42) with vˆ˜ σ = −K˜ ( T T ) ˆ˜σ = 0,ζ1 = ζ , ζ2 = 0, τ1 = τ , τ2 = τ ω , we Remark 12. The switching law (56) depends on vˆ˜ σ = vsσ , vM στ . When v can obtain the switching law (58). The sets Ω0i × Ωτ ω k , i ∈ I , k ∈ Iτ form a partition of the combined state space R2(r +d) . According to the switching law (58), Ω0i × Ωτ ω k is the active region of the (i, k)-th subsystem of the interconnected system.

Remark 13. To design the switching law, we need to find appropriate positive definite functions Wk satisfying inequality (45). It is well-known that there is no effective way to solve PDIs. However, several approaches for finding numeral solutions may be applicable to solving (24) and (45) [39–42]. In particular, (34) and (52) can be solved by the recently developed approaches for solving the state-dependent matrix inequality (SDMI) [39,41,42]. We will design a switched internal model system which switch synchronously with controlled plant. Theorem 4. Consider systems (1) and (2). Suppose that all conditions of Theorem 1 and Assumptions 1 and 2 hold with M = Mτ , constants βij = λkl , i = k, j = l, ρkl = µij = 0, namely,

∂V ∂V ∂V + fi (x1 , θsi (ζ1 , e1 ) + v1i , ω) + fi (x2 , θsi (ζ2 , e2 ) + v2i , ω) ∂t ∂ x1 ∂ x2 ∂ Vsi ∂ Vsi + ηsi (ζ1 , e1 ) + ηsi (ζ2 , e2 ) ∂ζ1 ∂ζ2 M ∑ ( ) + βij Vsi (ζ1 , ζ2 ) − Vsj (ζ1 , ζ2 ) ≤ (v1i − v2i )T (e1 − e2 ) ,

(59)

j=1 M

∑ ( ) ∂ Wi ∂ Wi φi (τ1 ,˜ e1i ) + φi (τ2 ,˜ e2i ) + βij Wi (τ1 , τ2 ) − Wj (τ1 , τ2 ) ≤ (v1τ − v2τ )T (˜ e1i − ˜ e2i ) . ∂τ1 ∂τ2 j=1

(60)

H. Pang, J. Zhao / Nonlinear Analysis: Hybrid Systems 27 (2018) 239–257

251

( ) ( )T T Then, the interconnected switched systems (23) and (42) with input vˆ i = vsiT , vMi and output eˆ = eT , vτT are incrementally

passive under the switching law

( ) σ (t ) = i if σ t − = i and

(z1 (t ) , z2 (t ) , ) ∈ int Ωi ,

} ( ) { ⏐ ˜ij , (61) if σ t − = i and (z1 (t ) , z2 (t )) ∈ Ω σ (t ) = min arg Ωj ⏐ (z1 (t ) , z2 (t )) ∈ Ωj ⏐ } ( T T )T { where z = ζ , τ , Ωi = (z1 , z2 ) ⏐Si (z1 , z2 ) − Sj (z1 , z2 ) ≤ 0, j ∈ I and { } ˜ij = (z1 , z2 ) Si (z1 , z2 ) − Sj (z1 , z2 ) = 0, i ̸= j , Si (z1 , z2 ) = Vsi (ζ1 , ζ2 ) + Wi (τ1 , τ2 ) , i ∈ I . Ω ( ) If, in addition, limk→∞ tjk +1 − tjk ̸ = 0 for all j ∈ I, then, the output regulation problem is solved by the feedback controllers ζ˙ = ηsi (ζ , e) , usi = θsi (ζ , e) , τ˙ = φi (τ , −e) , uiM = γi (τ ) ui = usi + uiM − Ki e, i ∈ I

(62)

under the switching law (61) with z1 = z = ζ T , τ T

(

)T

( )T , z2 = z = 0, τ Tω , where Ki are positive definite matrices.

Proof. Similar to the proof of Theorem 3, Theorem 4 holds. 5. Examples In this section, we present two examples to demonstrate the effectiveness of our main results. Example 3. Consider system (1) described by

⎞ ( 2 ) 1 ⎛ ⎞ −x1 x1 + 4 + x2 + 3 + u11 + ω6 1 ⎜ ⎟ 2 x1 + x2 + 3 + u12 − 6ω2 ⎜ 1 ⎟ ⎜ ⎟ 2 ⎜ x 1 + x 2 + 1 + 1 u2 − 5 ω 2 − x 4 ⎟ 1 2 2 4⎟ ⎟ , f2 (x, u2 , ω) = ⎜ f1 (x, u1 , ω) = ⎜ ⎜2x − 10x + 20 + u2 − x ⎟ , 3 1 2 ⎜ 2 ⎟ ⎝ ⎠ ⎜ ⎟ 2x4 − x3 2x4 − 2x3 ⎝ ⎠ 3 4 2 2 1 −x − 8x + x + 8ω −x3 − 6x4 + x2 ⎛

2

[ 1] e=

e e2

[ =

x1 − ω2 x2 − 2ω2

where x = x1 , x2 , x3 , x4

(

)T

]

= Cx − Dω2 ,

,C =

[

1 0

0 1

(63)

]

0 0

0 0

,D =

[] 1 2

and the exosystem is described by ω ˙ = 0. ( ) ( 2 )T 2 2 2 = 20 −1 + ω , xω (t ) = ω , 2ω , 2ω , ω2 is solution of the regulator

= = 3ω − = 6 −1 + ω , Eq. (7). We choose the storage function S (σ (t ) , x1 , x2 ) = Sσ (t ) (x1 , x2 ) for (60), where 1 u1ω

2 u1ω

2

S1 (x1 , x2 ) =

⎡ with P1 =

1 ⎣0 0 0

0 3 0 0

1 2 0 0 1 0

1 3, u2ω

(

) 2

(x1 − x2 )T P1 (x1 − x2 ) ⎤ ⎡ 0 0⎦ 0 2

and P2 =

2 ⎣0 0 0

0 1 0 0

2 u2ω

and S2 (x1 , x2 ) = 0 0 1 0

1 2

(x1 − x2 )T P2 (x1 − x2 ) ,

⎤

0 0⎦ . 0 2

Differentiating Si along the system trajectory of (63) gives S˙1 ≤ β12 (V2 − V1 ) + (u11 − u12 )T (e1 − e2 ) , S˙2 ≤ β21 (V1 − V2 ) + (u21 − u22 )T (e1 − e2 ) , where

)2 3 ( )2 1 1( β12 = −3.5, β21 = −7, V1 = x11 − x12 + x21 − x22 = (e1 − e2 )T Q1 (e1 − e2 ) with 2 2 2 [ ] [ ] ( 1 ) ) 1( 2 1 1 0 2 0 T 1 2 2 2 Q1 = , V2 = x1 − x2 + x1 − x2 = (e1 − e2 ) Q2 (e1 − e2 ) with Q2 = . 0

3

2

2

0

1

Since C Qi − Qj C = Pi − Pj , i, j = 1, 2, according to Theorem 2, system (63) is incrementally passive under switching law (38). Therefore, the conditions of Theorem 2 are satisfied.

( T

)

252

H. Pang, J. Zhao / Nonlinear Analysis: Hybrid Systems 27 (2018) 239–257

Fig. 2. State response of the switched system.

We design a switched internal model as follows.

[

3 0 0

[

] [ 6 0 6 0 0 0 20 ˜ τ. e, v2 (τ ) = 0 20 0 0 0

Subsystem 1: τ˙ =

Subsystem 2: τ˙ =

[ 0 3 3 ˜ e, v1 (τ ) = 0 0 ]

0 3

]

0 τ, 0

]

The storage function is chosen as Wk (τ1 , τ2 ) = 21 (τ1 − τ2 )T (τ1 − τ2 ) , k = 1, 2. Therefore, the switched model is incrementally passive under arbitrary switching ( law. )T ( ) τω = ω2 − 1, ω2 − 1, ω2 − 1 , xω (t ) = ω2 , 2ω2 , 2ω2 , ω2 is a steady-solution of closed -loop system (60) with the controllers 3 u1 = v1 (τ ) − e, τ˙ = − 0 0

[

[ 0 3 3 e, v1 (τ ) = 0 0 ]

0 3

]

0 τ, 0

[ ] 6 0 6 0 0 u2 = v2 (τ ) − 2e, τ˙ = − 0 20 e, v2 (τ ) = τ. 0 20 0 0 0 [

]

According to Theorem 3, the output regulation problem is solvable under the switching law (38). The simulation results are depicted in Figs. 2–5 for the initial state ω (0) = 1.3, τ (0) = (2, 13, 11) , x (0) = (7.3, 26.4, 12.1, 21). Fig. 2 shows the state response of the switched system, which indicates that the closed-loop system is bounded. The exogenous signal ω (t ) presented in Fig. 3 is bounded. The switching signal is given by Fig. 4. From Fig. 5, we can see that the output converges to zero. Therefore, the global output regulation problem is solvable. Thus, simulations results demonstrate the effectiveness of the proposed regulator. Example 4. Consider a switching Chua’s circuit [43–46]. The dynamic equations are given as follows.

v˙ 1 = −

1

v˙ 2 =

v1 −

C1i R 1

C2 R 1

v1 +

1

1

C1i R 1

C2 R

v˙ 3 = v2 − vd , L L e = v1 − vr ,

v2 −

v2 −

1 C2

1 C1i

gi (v1 ) −

v3 ,

1 C1i

u, (64)

i = 1, 2 · · · N

where C1i and gi (v1 ) denote the i-th capacitor and the i-th current flowing through the resistor Rnli , respectively. v1 and v2 are the voltages at the ends of the capacitors C1i and C2 , respectively. iL is the current in the inductance L. u is the current

253

exogenous signal ω

H. Pang, J. Zhao / Nonlinear Analysis: Hybrid Systems 27 (2018) 239–257

switching signal σ

Fig. 3. The exogenous signal ω (t ).

regulated error

Fig. 4. The switching signal of the switched system.

Fig. 5. The regulated error of the switched system.

from generator as active control action of the circuit. The prescribed signalvr = 2ωT ω and external disturbance vd = ωT ω are generated by the exosystem

ω˙ = S ω, S =

[

0 1

] −1 0

, ω = (ω1 , ω2 ) .

The regulated error is thus defined to be e = v1 − vr .

(65)

254

H. Pang, J. Zhao / Nonlinear Analysis: Hybrid Systems 27 (2018) 239–257

Similar to [43] and [44], we assume that gi (v1 ) is a linear function gi (v1 ) = Gbi v1 + Gai − Gbi E with Gai , Gbi < 0, E > 0.

(

)

(66)

From (65), the control objective of output regulation is to design a switched feedback controller [45,46] and a switching law such that all solutions of closed-loop system are bounded and the voltage output v1 of the Chua’s circuit to track the prescribed (reference )signal vr , while rejecting the disturbance vd . Namely e (t ) → ∞, t → ∞. T Let x = x1 , x2 , x3 = (v1 , v2 , iL )T . The candidate storage function of the ith mode is chosen as Vi (x1 , x2 ) =

1

C1i x11 − x12

(

)2 1 ( )2 1 ( + C2 x21 − x22 + L x31 − xˆ 32 .

)2

2 2 2 Design an output feedback controller for ith mode ui = −Gbi e − Gai − Gbi E − wi .

(

)

(67)

Differentiating Vi together with (67) gives V˙ i (x1 , x2 ) ≤ −Gbi x11 − x12

(

)2

) ( − (ui1 − ui2 ) x11 − x12 ≤ (wi1 − wi2 ) (e1 − e2 ) .

Consider the case N = 2. Set the parameters as L = 19 H , C2 = 19 µF , C11 = 1.0µF , R = 1Ω , C12 = 0.5µF , 4 2 Ga1 = −0.53, Gb1 = −0.25, Ga2 = −0.43, Gb2 = −0.35, E = 1. Then, we have the following switched model of the Chua’s circuit: x˙ 1 = −x1 + x2 + 0.25 x1 − e + w1 ,

(

x˙ 2 = x˙ 3 =

2 19 4

x1 − x2 −

2 19 4

19 19 e = x1 − v r .

x2 −

)

2 19

x˙ 1 = −2x1 + 2x2 + 0.7 x1 − e + 2w2 ,

(

x3 ,

x˙ 2 =

vd ,

x˙ 3 =

2 19 4 19

x1 − x2 −

2 19 4 19

x2 −

2 19

)

x3 ,

(68)

vd ,

It is easy to verify two models are incrementally passive and w 1ω = 0.5 ω12 + ω22 , w 2ω = 0.3 ω12 + ω22 , xω =

(

2ω12 + 2ω22 , ω12 + ω22 , ω12 + ω22

(

Subsystem 1 τ˙ = Subsystem 2 τ˙ =

)T

)

(

)

is solution of the regulator Eq. (7). We design a switched internal model

[

0.3 ˜ e, v1 (τ ) = [0.3, 0.2] τ ; 0.2

[

0.1 ˜ e, v2 (τ ) = [0.2, 0.1] τ . 0.1

]

]

The candidate storage function is chosen as Wi (τ1 , τ2 ) =

1 2

(τ1 − τ2 )T Pi (τ1 − τ2 ) , i = 1, 2, P1 =

[

1 0

]

[

0 2 , P2 = 1 0

]

0 . 1

) ( )T τω = ω12 + ω22 , ω12 + ω22 , xω = 2ω12 + 2ω22 , ω12 + ω22 , ω12 + ω22 is a steady-solution of closed-loop system (68) and (

the controllers

w1 = v1 (τ ) − e, w2 = v2 (τ ) − 2e,

τ˙ = −

[

τ˙ = −

0.3 0.3 e, v1 (τ ) = 0.2 0.2

]

0.1 e, 0.1

[

]

[

v2 (τ ) =

]T

[

τ,

0.2 0.1

]T

τ.

,

)T

According to Theorem 4, the output regulation problem is solvable under the switching law (59), where z = eT , τ T , Si (z1 , z2 ) = 12 C1i (e1 − e2 )2 + Wi (τ1 , τ2 ) , i ∈ I. Simulations of a switching Chua’s circuit with the control controller (67) and the switching law (61) were performed using the initial state x (0) = (100.3, 71.4, 7.2) , τ (0) = (9, 8.2) , ω (0) = (0.6, 0.7) in MATLAB using the ‘ode45s’ solver. The simulation results are depicted in Figs. 6–9. Fig. 6 shows the state response of the switched system, which tell us that the state of closed-loop system is bounded. Seen from Fig. 7, the exogenous signal ω (t ) is bounded. The switching signal is given by Fig. 9. From Fig. 8, we can see that the output converges to zero. Therefore, the global output regulation problem is solvable. Thus, the simulation results will illustrate the theory presented.

(

6. Conclusions This paper has investigated incremental passivity and incremental passivity-based output regulation problem for switched nonlinear systems. A switched regulator with a composite switching law is designed to solve the output regulation

H. Pang, J. Zhao / Nonlinear Analysis: Hybrid Systems 27 (2018) 239–257

exogenous signal ω

Fig. 6. State response of the switched system.

Fig. 7. The exogenous signal ω (t ).

Fig. 8. The regulated error of the switched system.

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H. Pang, J. Zhao / Nonlinear Analysis: Hybrid Systems 27 (2018) 239–257

switching signal σ

256

Fig. 9. The switching signal of the switched system.

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