Decentralized uniform input-to-state stabilization of hierarchically interconnected triangular switched systems with arbitrary switchings

Automatica 94 (2018) 300–306

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Automatica journal homepage: www.elsevier.com/locate/automatica

Brief paper

Decentralized uniform input-to-state stabilization of hierarchically interconnected triangular switched systems with arbitrary switchings✩ Svyatoslav Pavlichkov a , Chee Khiang Pang b, * a b

Faculty of Science and Engineering, University of Groningen, Nijenborgh 4, 9747 AG Groningen, Netherlands Engineering Cluster, Singapore Institute of Technology, 10 Dover Drive, Singapore 138683

article

info

Article history: Received 15 December 2016 Received in revised form 26 January 2018 Accepted 17 March 2018

Keywords: Stabilization methods Large-scale systems Decentralized control

a b s t r a c t We consider a class of large-scale systems composed of hierarchically interconnected switched nonlinear triangular form subsystems affected by external disturbances with arbitrarily varying switching signals. For any system of this class, we design a decentralized feedback controller which renders the entire largescale closed-loop system globally ISS with respect to the external disturbances uniformly and regardless of the unknown switching signals. To solve the problem, we use a certain modification of the classical small gain theorems formulated in terms of the ISS Lyapunov functions and combine it with our version of the backstepping approach with a suitable gain assignment. © 2018 Elsevier Ltd. All rights reserved.

1. Introduction The theory of control and stability of nonlinear large-scale and multi-agent systems has been gaining popularity over the past 15 years and it is important in many applications (Abdessameud, Tayebi, & Polushin, 2012; Dashkovskiy, Kosmykov, Mironchenko, & Naujok, 2012; Dashkovskiy, Rüffer, & Wirth, 2007; Liu & Jiang, 2013). The essential characteristics of multi-agent control systems are autonomy and decentralization: each agent should be selfaware whereas it is not always possible to observe the entire system due to its complexity, for instance. This naturally leads to the problem of decentralized control (Krishnamurthy & Khorrami, 2003; Mehraeen, Jagannathan, & Crow, 2011a, b). One efficient tool for solving such problems is the small gain approach based on small gain theorems (Liu & Jiang, 2013). In Jiang, Teel, and Praly (1994), the classical small gain theorems for two interconnected systems were obtained, which led to many fruitful results, e.g. Ito, Pepe, and Jiang (2010) and Karafyllis and Tsinias (2004). Next significant breakthrough was getting new small gain theorems for N ≥ 2 interconnected nonlinear systems (Dashkovskiy et al., 2007; Dashkovskiy, Rüffer, & Wirth, ✩ The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Warren E. Dixon under the direction of Editor Daniel Liberzon. Corresponding author. E-mail addresses: [email protected] (S. Pavlichkov), [email protected] (C.K. Pang).

*

https://doi.org/10.1016/j.automatica.2018.04.029 0005-1098/© 2018 Elsevier Ltd. All rights reserved.

2010; Jiang & Wang, 2008), which were extended in various directions (Dashkovskiy et al., 2012; Geiselhart & Wirth, 2015; Liu & Jiang, 2013). Another topic, which has been gaining popularity over the past decade, is motivated by the theory of switched systems (Efimov, Loria, & Panteley, 2011; Liberzon, Hespanha, & Morse, 1999; Liberzon & Morse, 1999; Sun & Ge, 2005) and by some fundamental results on the uniform stability of switched systems (MancillaAguilar & Garcia, 2001). It is known that some trajectories of a switched system can diverge while each constant switching signal produces some globally asymptotically stable system of ODE (Liberzon & Morse, 1999). Then, it is natural to explore whether the classical designs of stabilizers (e.g. backstepping) can be extended to the problem of uniform stabilization of switched systems by means of switching-independent stabilizers (Dashkovskiy & Pavlichkov, 2012; Long & Zhao, 2014; Ma & Zhao, 2010). Dashkovskiy and Pavlichkov (2012) deal with ‘‘centralized’’ uniform switching-independent stabilization of largescale interconnected switched systems in general triangular form, i.e., each agent should know all the components of all the states of the other agents. For switched systems in strict-feedback form with dynamic uncertainties, the problem of stabilization was tackled in Long and Zhao (2014). However, first, Long and Zhao (2014) deal with interconnections of two switched subsystems only similarly to the classical result from Jiang et al. (1994) devoted to the ODE case, and, second, Long and Zhao (2014) address stabilization under certain dwell-time conditions. Contrarily, the main result of our current work provides the uniform stabilization in presence of arbitrary switching signals without any dwell-time conditions and

S. Pavlichkov, C.K. Pang / Automatica 94 (2018) 300–306

our tool will be a suitable modification of Dashkovskiy et al. (2010) for N ≥ 2 interconnected switched systems. As in many other works devoted to decentralized control (Krishnamurthy & Khorrami, 2003; Mehraeen et al., 2011a, b), we assume that our network has a certain specific structure of interconnections. This structure is the same as in Mehraeen et al. (2011a) and its engineering motivation is given, for instance, in Mehraeen et al. (2011b). However, while Mehraeen et al. (2011a, b) deal with ODE systems without switchings, our problem formulation addresses the case of uniform, switching-independent ISS stabilization of interconnected switched systems, which is more general, and our main tool differs from Mehraeen et al. (2011a, b). 2. Preliminaries and main definitions N

Throughout the paper, ⟨·, ·⟩ denotes the scalar product in R 1 and |ξ | := ⟨ξ , ξ ⟩ 2 denotes the quadratic norm of ξ ∈ RN . All vectors from RN are treated as columns, i.e., RN ∼ = RN ×1 . A function α : R+ → R+ is said to be of class K, if it is continuous, strictly increasing and α (0) = 0, and it is said to be of class K∞ if it is of class K and unbounded. A continuous function α : R+ → R+ is said to be positive definite, if α (r) = 0 implies r = 0. A continuous function β : R+ × R+ → R+ is said to be of class KL if for each fixed t ≥ 0 we have β (·, t) ∈ K and for each fixed s ≥ 0 we have β (s, t) → 0 as t → +∞ and t ↦→ β (s, t) is strictly decreasing for each s > 0 and ∀t ≥ 0 β (0, t) = 0. We say that V : Rn → R+ is positive definite if V (x) = 0 ⇔ x = 0 ∈ Rn , and it is radially unbounded, if, in addition, ∃α ∈ K∞ s.t. ∀ x ∈ Rn V (x) ≥ α (|x|). Consider the following nonlinear switched system x˙ (t) = Fσ (t) (t , x(t), D(t)),

t ∈ R,

(1)

with states x ∈ R , piecewise constant switching signals R ∋ t ↦ → σ (t) ∈ {1, . . . , M }, and external disturbance inputs D(·) ∈ L∞ (R; RN ), where each Fσ is continuous w.r.t. (t , x, D) and is locally Lipschitz continuous w.r.t. (x, D) for each fixed σ ∈ {1, . . . , M }. Let us note that our main result and its proof are the same in two cases: piecewise constant switching signals and just measurable switching signals. For all (t0 , x0 ) ∈ R × Rn , D(·) ∈ L∞ , and piecewise constant σ (·) by t ↦ → x(t , t0 , x0 , D(·), σ (·)) we denote the trajectory of (1) with x(t0 ) = x0 , D = D(t), σ = σ (t). n

Definition 1. System (1) is said to be uniformly input-to-state stable (UISS) (at the origin x∗ = 0 ∈ Rn ) if there are β ∈ KL, and γ ∈ K such that for each t0 ∈ R, each x0 ∈ Rn , each D(·) ∈ L∞ (R; RN ) and each piecewise constant t ↦ → σ (t) ∈ {1, . . . , M } we obtain

|x(t , t0 , x0 , D(·), σ (·))| ≤ max {β (|x0 |, t − t0 ), γ (∥D(·)∥L∞ [t0 ,+∞[ )}

for all t ≥ t0 .

(2)

For ODE case, the notion of ISS was introduced in Sontag (1989) and Definition 1 for switched systems was given in MancillaAguilar and Garcia (2001). Remark 1. It is easy to show that one of the sufficient conditions for the UISS property is as follows: there are a positive definite and radially unbounded ISS Lyapunov function V (x) of class C 1 and a gain γˆ (·) ∈ K s.t.

∀x ∈ R n

∀t ∈ R

∀σ ∈ {1, . . . , M }

∀D ∈ R N

V (x) ≥ γˆ (|D|) ⇒

∇ V (x)Fσ (t , x, D) ≤ −α (V (x))

(3)

with some continuous and positive definite α (·) : [0, +∞[ → [0, +∞[. If each Fσ is time-invariant, i.e., ∀σ Fσ = Fσ (x, u), then one should omit the quantifier ∀t ∈ R in (3) and in Definition 1, and one can put t0 = 0 in (2).

301

Fig. 1. Structure of interconnections of system (4) for νi = ν .

Remark 2. By Hadamard’s lemma, we call the following simple fact: if F ∈ C µ+1 (RN ; R), then F (ξ ) ∫− F (η) = Φ (ξ , η)(ξ − η), 1 ξ ∈ RN , η ∈ RN , where Φ (ξ , η) = 0 ∇ F (η + s(ξ − η))ds is of class C µ . (Because F (ξ ) − F (η) =

∫1[d 0

ds

F (η + s(ξ − η)) ds.)

]

3. Main results We consider a large-scale switched control system in the following form x˙ i,j = fi,j (xi,1 , . . . , xi,j+1 ) + ∆i,j,σ (t) (θ, X j , D(t)), j = 1, . . . , νi − 1, x˙ i,νi = fi,νi (xi,1 , . . . , xi,νi , ui ) + ∆i,νi ,σ (t) (θ, X νi , D(t)); i = 1, . . . , N ,

(4)

(with νi equations in each ith subsystem) with state vector components xi,j ∈ Rmi,j (with mi,j ≤ mi,j+1 ), controls ui = xi,νi +1 ∈ Rmi,νi +1 , external disturbances D(·) ∈ L∞ (R; Rl0 ), piecewise constant switching signals R ∋ t ↦ → σ (t) ∈ {1, . . . , M } and unknown ⊤ ⊤ parameters θ ∈ R~ , where Xi,p := [x⊤ for all p = i,1 , . . . , xi,p ] 1, . . . , νi , i = 1, . . . , N , and X1,min{p,ν1 }

[ X p :=

...

XN ,min{p,νN }

] for all p = 1, . . . , max νi .

(5)

1≤i≤N

As we mentioned above, some specific restrictions for the structure of interconnections are needed in such problems. For instance, if the dynamics of xi,p were allowed to depend on xj,p+1 (j ̸ = i) in (4) then we could not deal even with the problem of asymptotic stabilization. As a counterexample consider the system x˙ 1,1 = x1,2 − x2,2 , x˙ 1,2 = u1 , x˙ 2,1 = x2,2 − x1,2 , x˙ 2,2 = u2 . This system cannot be asymptotically stabilized even by a ‘‘centralized’’ feedback because its any trajectory satisfies x1,1 (t) + x2,1 (t) = const. The structure of interconnections in (4) for any two subsystems of (4) is depicted in Fig. 1 and it is the same as in Mehraeen et al. (2011a, b). (Note that Mehraeen et al. (2011a, b) address ODE systems without any switching signals.) Mehraeen et al. (2011b) provide an explicit engineering and physical motivation both for our problem formulation and for Mehraeen et al. (2011a, b). We assume that system (4) satisfies the following conditions: (I) All the functions fi,j , ∆i,j,σ are of class C ν+1 , where ν := max1≤i≤N {νi } and fi,j (0) = ∆i,j,σ (θ, 0) = 0 ∈ Rmi,j for every

θ s.t. |θ| ≤ θ ∗ . (II) For each i = 1, . . . , N and each j = 1, . . . , νi , the function xi,j+1 ↦ → fi,j (xi,1 , . . . , xi,j , xi,j+1 ) is right invertible, i.e., there is a map (xi,1 , . . . , xi,j , w ) ↦→ αi,j (xi,1 , . . . , xi,j , w) of class C ν with αi,j (0) = 0 such that fi,j (xi,1 , . . . , xi,j , αi,j (xi,1 , . . . , xi,j , w )) = w for all xi,1 ∈ Rmi,1 , . . . , xi,j ∈ Rmi,j , w ∈ Rmi,j . (III) There exists some known θ ∗ ≥ 0 such that |θ| ≤ θ ∗ . Our main result is summarized in the following theorem. Theorem 1. Suppose that system (4) satisfies Assumptions (I)–(III). Then there exists a decentralized feedback controller in the form ui = uˆ i (xi,1 , . . . , xi,νi ) of class C 1 such that uˆ i (0) = 0 and such that the

302

S. Pavlichkov, C.K. Pang / Automatica 94 (2018) 300–306

closed-loop system (4) with this feedback ui = uˆ i (xi,1 , . . . , xi,νi ) is UISS at the origin X ∗ = 0 with respect to the external disturbances D(t) and regardless of the unknown θ. For constructive design of each uˆ i (·) we need to know only the dynamics of ith subsystem of (4) for all possible σ = 1, . . . , M . Remark 3. Assumption (II), which we call ‘‘right invertibility’’ being motivated by Respondek (1990), is a bit more general than the conditions for the strict-feedback forms (Korobov, 1973). Actually it means the same control authority for xi,j+1 as in the strict-feedback forms. Let us note that the class of the strictfeedback forms hierarchically interconnected as in (4) and Fig. 1 is also an important special case of (4) and appears in applications (Mehraeen et al., 2011b). However there are applications and physical models where xi,j+1 are vectors and the right invertibility Assumption (II) appear naturally. We start with the following simple model: let xi ∈ R2 , i = 1, . . . , N , be the positions of N pursuers or ‘‘hunters’’ on a plane, let x ∈ R2 be the position of the corresponding evader and suppose that ξi ∈ R2 and ξ ∈ R2 are their corresponding velocities and their corresponding dynamics is given by x˙ i = ξi ,

ξ˙i = ui

x˙ = ξ ,

ξ˙ =

N ∑

In all these cases Condition (II) is satisfied and the corresponding implicit function providing the desired (virtual or final) controller can be easily found analytically and numerically. Remark 4. Let us note that when dealing with uniform stabilization of a single switched system with arbitrary switchings, one needs knowledge of all possible switching subsystems, see e.g. Ma and Zhao (2010). This is natural because, if, for instance, we have no information about the dynamics for σ = 1 and our ‘‘arbitrary switching signal’’ is ∀t σ (t) = 1, then it is not clear which controller can stabilize the system uniformly. In our case, for each ith subsystem of (4), we also need to know only the dynamics of this ith subsystem for each σ = 1, . . . , M , but we do not need to know the dynamics of any jth subsystem with j ̸ = i. Below (Proof of Theorem 3 and Example) we will see that, if the dynamics of the ith subsystem of (4) depends only on limited number of ‘‘neighbors’’, i.e., the dynamics of xi,p depends not on the entire X¯p but on some limited number of Xj,p (j ̸ = i), then the numerical complexity of the designed feedback is satisfactory and does not depend on how large the number N of the subsystems in (4) is. 4. Small gain theorems Our tool will be the following modification of the small gain theorems from Dashkovskiy et al. (2010).

ϕj,σ (t) (xj − x − lj )

j=1

respectively with ϕj,σ (0) = 0 and with ϕj,σ (·) ∈ C 2 for all j = 1, . . . , N , σ = 1, . . . , M. Terms ϕj,σ (xj − x − lj ) characterize the runaway efforts of the evader concerned with the ith pursuer; if the distance between them is ‘‘safe’’ then ϕi,σ = 0. (Of course, one can consider the case φj,σ (|xj − x|) = 0, whenever |xj − x| ≥ λj,σ > 0, but it would be more convenient to fix a ‘‘long enough’’ vector li ∈ R2 such that |li | ≥ λj,σ for all σ = 1, . . . , M and replace ϕj,σ (|xj −x|) with ϕj,σ (xj −x−lj ) as we assume above). The piecewise constant t ↦ → σ (t) can be interpreted as instant changes of the policy of the evader. In addition assume that the pursuers are not just point masses, but can also perform a rotational (planar) motion

ψ˙ i = ωi ,

where ψi , ωi are the angle and the angular velocity for the ith pursuer and vi denotes its control torque (for simplicity the mass moment of inertia equals 1). We denote ej := xj − x − lj ; let χj be the angle of xj − x = ej + lj in polar coordinates and define φj := ψj −χj . Our decentralized stabilizing controller should provide ej → 0 as t → +∞, which can be interpreted as bringing the evader to some desirable position, and φj → 0 as t → +∞, which can be interpreted as ‘‘aiming’’ of the jth pursuer. This yields the following large-scale interconnection of control systems satisfying (I)–(III):

φ˙ i = ωi − Fi (ζi , ei ) ω˙ i = vi .

(6)

j=1

System (6) is just a simplified demonstrating model, which is a special case of (4) satisfying (II). However, we can mention alternative multi-agent systems, for instance motivated by Fossen (2011), see eqs. (7.10), (13.388), (13.389)

η˙ i = R(ψ )νi ,

Mi ν˙ i + Ci (νi )νi + Di (νi )νi = τi

˙ = 0, η = [N , E , ψ]⊤ ∈ R3 , ν = with M = M ⊤ ∈ R3×3 , M ⊤ 3 [u, v, r ] ∈ R , and

( R(ψ ) =

cos ψ sin ψ 0

− sin ψ cos ψ 0

0 0 1.

)

X˙ i (t) = Φi,σ (t) (X1 (t), X2 (t), . . . , XN (t), D(t)), 1 ≤ i ≤ N ,

(7)

with Xi ∈ R , Φi ∈ C . Suppose that there are γi,j (·) ∈ K, γi (·) ∈ K, αi (·) ∈ K and positive definite and radially unbounded Lyapunov functions Vi (Xi ), i = 1, . . . , N , in C 1 such that ni

1

Vi (Xi ) ≥ max{max γi,j (Vj (Xj )); γi (|D|)} ⇒ ∀σ = 1, M j̸ =i

(8)

∇ Vi (Xi )Φi,σ (X1 , . . . , Xn , D) ≤ −αi (Vi (Xi )). for all i = 1, . . . , N . Assume that

∀r > 0

ω˙ i = vi ,

⎧ e˙ = ζi , ⎪ ⎪ ⎨ i N ∑ ˙ ⎪ ζ = ϕj,σ (t) (ej ), i ⎪ ⎩

Theorem 2. Consider the switched system

(

) γi1 ,i2 ◦ γi2 ,i3 ◦ . . . ◦ γik ,i1 (r) < r

(9)

holds true for all { } ⊂ {1, . . . , N } with il ̸= il′ if l ̸= l . Then (7) is UISS w.r.t. D(·) as the input. il kl=1

′

The proof of Theorem 2 is based on the same construction as in Dashkovskiy et al. (2010). In comparison with Dashkovskiy et al. (2010), the dynamics is affected by the unknown switching signal σ (·) whereas all the properties should be uniform w.r.t. σ ∈ {1, . . . , M }. However all the techniques from Dashkovskiy et al. (2010) are based on the behavior of the ‘‘trajectories’’ in the (V1 , . . . , VN )-space of the Lyapunov functions Vi along (8). This allows us to argue as in Dashkovskiy et al. (2010) (with adding σ ). Hence we do not consider Theorem 2 as our main result, it is our auxiliary tool needed for the proof of our main Theorem 1. Based on Theorem 2, we will prove Theorem 1 by a backstepping design with a certain gain assignment to satisfy (9) at each step of the backstepping process. 5. Adding an integrator The main part of the proof of Theorem 1 is based on the following Theorem 3 given below. Consider a switched nonlinear system of the form z˙ = g(z , w ) + ϕσ (t) (θ, z , ξ (t)),

t ∈ R,

(10)

with states z ∈ Rk , controls w ∈ Rq , external disturbances ξ (·) ∈ L∞ (R; RN ), switching signals R ∋ t ↦→ σ (t) ∈ {1, . . . , M }

S. Pavlichkov, C.K. Pang / Automatica 94 (2018) 300–306

and some unknown parameter θ ∈ R~ . Also consider the following dynamic extension z˙ = g(z , w ) + ϕσ (t) (θ, z , ξ (t))

{

t∈R

w ˙ = h(z , w, v ) + φσ (t) (θ, z , w, ξ (t), η(t)),

(11)

with states y = (z , w ) ∈ Rk × Rq , controls v ∈ Rm , where q ≤ m, with disturbances ζ (·) = (ξ (·), η(·)) ∈ L∞ (R; RN +l0 ), with the same switching signal σ (·) and uncertainty θ ∈ R~ . We assume that:

∈ C µ+1 (Rk × Rq ; Rk ); h ∈ C µ+1 (Rk+q × Rm ; Rq ); ϕσ ∈ C µ+1 (R~ × Rk × RN ; Rk ) and φσ ∈ C µ+1 (R~ × Rk+q × RN +l0 ; Rq ) for all σ = 1, . . . , M with some µ ∈ N; and g(0, 0) = ϕσ (θ, 0, 0) = 0 ∈ Rk and h(0, 0, 0) = φσ (θ , 0, 0, 0) = 0 ∈ Rq for all θ ∈ R~ , σ ∈ {1, . . . , M }. (C2) There is v(·, ·, ·) ∈ C µ+1 (Rk × Rq × Rq ; Rm ) such that h(z , w, v(z , w, ω)) = ω for all (z , w, ω) ∈ Rk × Rq × Rq . (C3) There is some known θ ∗ ≥ 0 such that |θ| ≤ θ ∗ .

Hence, by (12), dV ⏐

We denote ζ := (ξ , η) and write ζ and y instead of ξ , η and z , w respectively in all the next estimates. Define the following Lyapunov functions for (10), (11): W (z) := |z |2 , V (y) := |y|2 , U(ξ ) := |ξ |2 , Ω (ζ ) := |ζ |2 . Given feedbacks w ¯ (·, ·) : R × Rk → Rm and v¯ (·, ·, ·) : R × Rk+q → Rm , denote the derivatives of W (z) and V (y) w.r.t. (10) and (11) with and without these feedbacks by

⏐ dW ⏐ ⏐ dt

(10)

⏐ dV ⏐ := ⟨2z , g(y) + ϕσ (θ, z , ξ )⟩ and ⏐ dt

:=

(11)

⟨2z , g(y) + ϕσ (θ , z , ξ )⟩ + ⟨2w, h(y, v ) + φσ (θ, y, ζ )⟩; dW ⏐ dt

⏐ ⏐

(10),w=w ¯ (t ,z)

:= ⟨2z , g(t , z , w ¯ (t , z)) + ϕσ (θ, z , ξ )⟩;

⏐ dV ⏐ and ⏐ dt

(12)

(11),v=¯v (t ,y)

Theorem 3. Assume that (C1)–(C3) hold, and there are γ (·) ∈ K∞ , λ∗ > 0 such that γ (R) = γ0 R, whenever R ∈ [0, R0 ] with some γ0 > 0, R0 > 0 and such that for all (z , ξ ) ∈ Rk × RN , σ ∈ {1, . . . , M }, θ ∈ R~ with |θ| ≤ θ ∗ we have:

≤ −λ∗ W (z).

(14)

Then for every ε ∈ ]0, λ2 ] and every εˆ (·) ∈ K∞ such that ∀R ∈ [0, R′0 ] εˆ (R) = ε0 R with some ε0 > 0, R′0 > 0 there is vˆ (·) ∈ C µ (Rk ×Rq ; Rm ) with vˆ (0) = 0 ∈ Rm such that γˆ (·) := γ (·) + εˆ (·) ∈ K∞ satisfies the following: ∗

V (y) ≥ γˆ (Ω (ζ )) ⇒

dV ⏐ dt

⏐ ⏐

(11),v=ˆv (y)

( ε) ≤ − λ∗ − V (y)

(15)

2

⏐ ⏐

dt

(11)

(10),w=0

⟨

+ 2w, h(y, v ) + φσ (θ, y, ζ )

⟩

⟨ ⟩ + 2w, J ⊤ (y)z for all (y, ζ ), (σ , θ ).

Define εmin (·) ∈ C ([0, +∞[; [0, +∞[) by

∀R > 0

εmin (R) :=

1 εγ (γˆ −1 (R)) 2

γˆ −1 (R)

(17)

and by εmin (0) := limR→+0 εmin (R). Since γ (·), γ −1 (·), εˆ (·), γˆ (·), γˆ −1 (·) are strictly increasing, continuous and locally linear around R = 0, εmin (·) is continuous on [0, +∞[, positive definite and locally linear around R = 0. Step 2. The goal of this Step 2 is to get an upper estimate for ⟨2w, φσ (θ, y, ζ )⟩ eliminating unknown σ (and θ ) from (16). By Hadamard’s lemma,

φσ (θ, y, ζ ) = Φσz (θ, y, ζ )z + Φσw (θ, y, ζ )w + Φσζ (θ, y, ζ )ζ with Φσz (·), Φσw (·), Φσζ (·) defined from Remark 2. Hence, for each ε > 0,

√ √ ⏐ 4⏐ ⏐ ε ⏐ z ⏐w⏐ ⏐Φ (θ, y, ζ )z ⏐ + 2|w|× ⟨2w, φσ (θ, y, ζ )⟩ ≤ 2 σ ε 4 √ √ ⏐ 4 ⏐ ⏐ εmin (R) ⏐ ζ w ⏐ ⏐ ⏐Φ (θ, y, ζ )ζ ⏐. w |Φσ (θ, y, ζ )w| + 2 σ εmin (R) 4

(18)

Remark 7. Next we estimate the right-hand side of (18) to eliminate σ ∈ {1, . . . , M }. But, since backstepping requires taking derivatives at each inductive step, we cannot take maxσ ∈{1,...,M } |Φσz | or maxσ ∈{1,...,M } |Φσξ | to estimate (18).

max

M { ∑

|y|2 ≤ R; σ =1 |ζ |2 ≤ d; |θ| ≤ θ ∗

} ∥Φσz ∥2 + ∥Φσw ∥2 + ∥Φσζ ∥2 .

inequality a2 + b2 ≥ 2ab, we obtain for each ε > 0:

√ 2

√ ∗ ⏐ ⏐ ⏐w⏐ ε ⏐Φ z (θ, y, ζ )z ⏐ ≤ ε ⟨z , z ⟩ + 4M (R, d) ⟨w, w⟩, σ ε 4 4 ε

4⏐ ⏐

whenever |y|2 ≤ R, |ζ |2 ≤ d, |θ| ≤ θ ∗ . Applying similar estimates to all the terms of (18), we get for each ε > 0:

( ) ε |y|2 ≤ R, |ζ |2 ≤ d, |θ| ≤ θ ∗ ⇒ ⟨2w, φσ (θ, y, ζ )⟩ ≤ |z |2 4 ( )) ⟩ ⟨ ( 4 4 εmin (R)|ζ |2 + w, 1 + M∗ (R, d) 1 + + w + ε εmin (R) 4

(20) for all y, ζ , σ , θ. To eliminate all the unknown σ , ξ , θ from ⟨2z , ϕσ (θ, z , ξ )⟩, we note that by Hadamard’s lemma g(z1 , 0) − g(z2 , 0) = A(z1 , z2 )(z1 − z2 ), and

⏐ ⏐ Proof of Theorem 3. Step 1. First we estimate dV dt ⏐

ϕσ (θ, z1 , ξ1 ) − ϕσ (θ, z2 , ξ2 ) = Cσ (θ, z1 , z2 , ξ1 , ξ2 )(z1 − z2 )

by using

(12),(14). We note that, by Hadamard’s lemma, g(y) − g(z , 0) = J(y)w for all y ∈ Rk+q with J(·, ·) ∈ C µ defined from Remark 2.

(19)

(Φσz , Φσw , Φσζ are functions of (θ, y, ζ )). Combining (19) with the

for all (y, ζ ) ∈ Rk+q+N +l0 , σ ∈ {1, . . . , M }, θ ∈ R~ s.t. |θ| ≤ θ ∗ .

(11)

(16)

Remark 6. Note that (16) has unknown σ , (and unknown θ ), while our controller should be switching-independent. Consequently, our task is to eliminate σ (and θ ) from the final estimate.

(13)

for all (t , y, v, ζ , σ , θ ). Our main statement on adding an integrator is as follows.

(10),w=0

dW (θ, z , ξ , σ ) ⏐

M∗ (R, d) :=

+ ⟨2w, h(y, v¯ (t , y)) + φσ (θ, y, ζ )⟩.

dt

≤

Based on this remark define for all R ≥ 0, d ≥ 0:

:= ⟨2z , g(y) + ϕσ (θ, z , ξ )⟩

⏐ dW ⏐ W (z) ≥ γ (U(ξ )) ⇒ ⏐

⏐ ⏐

dt

(C1) g

Remark 5. Condition (C2) is similar to Assumption (II) and has the same motivation as in Remark 3.

303

+ χσ (θ, z1 , z2 , ξ1 , ξ2 )(ξ1 − ξ2 )

(21)

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S. Pavlichkov, C.K. Pang / Automatica 94 (2018) 300–306

for all (z1 , ξ1 ), (z2 , ξ2 ) in Rk+N with A(·), Cσ (·), χσ (·) defined from Remark 2. Define for all R ≥ 0, d ≥ 0: M(R, d) := M { ∑

{

max

|zi |2 ≤R, |ξi |2 ≤d, |θ |≤θ ∗

2 ∥A(z1 , z2 )∥+

6. Proof of Theorem 1 In the Base Case of our backstepping design, (4) becomes x˙ i,1 = fi,1 (xi,1 , xi,2 ) + ∆i,1,σ (t) (θ, xi,1 , ξi ), i = 1, . . . , N ,

∥Cσ (θ , z1 , z2 , ξ1 , ξ2 )∥ + ∥χσ (θ, z1 , z2 , ξ1 , ξ2 )∥2

}}

.

(22)

σ =1

Arguing as above for (20), we obtain from (22):

( ) |y|2 ≤ R, |ζ |2 ≤ d, |θ| ≤ θ ∗ ⇒ ∀σ ∈ {1, . . . , M } |⟨2z , g(z , 0) + ϕσ (θ , z , ξ )⟩| ≤ 4M(R, d)|z |2 + |ξ |2

(23)

for all (y, ζ , θ ). Define M(R, d) := M (R, d) + M(R, d), then from (16),(20) we obtain for all σ ∈ {1, . . . , M }:

with states xi,1 controls xi,2 , and external disturbances ξi := ⊤ ⊤ ⊤ ⊤ ⊤ [ x⊤ 1,1 , . . . , xi−1,1 , xi+1,1 , . . . , xN ,1 , D ] . For each xi,1 -subsystem of (29), define its Lyapunov function by Vi,1 (xi,1 ) := ⟨xi,1 , xi,1 ⟩. Fix any λ∗ > 0. Lemma 1. For each locally linear γˆ0 (·) ∈ K∞ , s.t. ∀R ∈ [0, R′0 ] γˆ0 (R) = γ0 R with some R′0 > 0, γ0 > 0, there is w ˆ i (· ) ∈ C ν (Rmi,1 ; Rmi,2 ) with w ˆ i (0) = 0 s.t. for all (xi,1 , ξi , θ ), if |θ| ≤ θ ∗ , and Vi,1 (xi,1 ) ≥ γˆ0 (|ξi |2 ), then

∗

dV ⏐ dt

⏐ ⏐

≤ (11)

dW ⏐ dt

⏐ ⏐

(10),w=0

)(

+ M(R, d)

1+

4

ε

+

εmin (R)

σ from

) ⟩ ε εmin (R) 2 w + |z |2 + |ζ | , 4

4

whenever |y|2 ≤ R, |ζ |2 ≤ d, |θ| ≤ θ ∗ .

(24)

Step 3. We obtained (24), which is switching-independent (for ˙ |(10),w=0 in (24) see (14)). In Step 3, we define the desired conW troller to obtain (15). Take any H(·), λ(·) in C ∞ ([0, +∞[; ]0, +∞[) s.t. for all R > 0

(

)(

H(R) ≥ 1 + M(R, γ −1 (R))

λ(R) ≥ λ∗ + +

εmin (R) 4

1

εˆ (γˆ −1 (R))

1+

4

ε

+

8

)

εmin (R)

,

(25)

(

)

γˆ −1 (R) + γ −1 (R) .

(26)

By Assumption (C2), there is vˆ (·, ·) ∈ C µ (Rk × Rq ; Rm ) with vˆ (0, 0) = 0 such that for all y = (z , w) ∈ Rk × Rq h(y, vˆ (y)) + J ⊤ (y)z + H(|y|2 )w = −λ(|y|2 )w.

(27)

Then from (24) and from (25), (26), (27), we obtain: dt

(11),v=ˆv (y)

( { } ε) ≤ − λ∗ − V (y) + ⟨2z , g(z , 0) + ϕσ (θ, z , ξ )⟩ + λ∗ ⟨z , z ⟩ 2

−

ε 4

( ( ε )) εmin (|y|2 ) ⟨w, w⟩ + ⟨ζ , ζ ⟩ ⟨z , z ⟩ − λ(|y|2 ) − λ∗ − 2

dVi,1 ⏐ dt

⏐ ⏐

(29),xi,2 =w ˆ i (xi,1 )

≤ −λ∗ Vi,1 (xi,1 ).

dVi,1 dt

⏐ ⏐

(29)

, we apply Hadamard’s lemma to ∆i,1,σ (θ , xi,1 , ξi )

and argue as in Proof of Theorem 3, Step 2 (see (18),(19),(20)). Finally, we note that Theorem 3 and Lemma 1 yield Theorem 1. For νi = 1, one takes any suitable γˆ0 (·) and finds w ˆ i (·) by Lemma 1 to make (29) with xi,2 = w ˆ i (xi,1 ) satisfying Theorem 2. Then we take the standard backstepping transformation z = xi,1 , w = xi,2 − wi,1 (xi,1 ), augment gains to keep (9) satisfied, apply Theorem 3, and prove Theorem 1 for νi = 2. Arguing by induction we obtain Theorem 1 for all νi . 7. Example Consider the following interconnected system

Rλ∗ + 4RM(R, γ −1 (R))

⏐ ( ) dV ⏐ |y|2 ≥ γ (|ζ |2 ) and |θ| ≤ θ ∗ ⇒ ⏐

∀σ ∈ {1, . . . , M }

Proof of Lemma 1 is a simple case of Proof of Theorem 3 and therefore is omitted due to space limits. To eliminate the unknown ⏐

⟨ ( + 2w, h(y, v ) + J ⊤ (y)z + 1 4

(29)

4

(28) for all (y, ζ , σ , θ ). It easy to check that (28) yields (15). We take and fix any y = (z , w ) ∈ Rk+q , and any ζ = (ξ , η) ∈ RN +l0 such that V (y) ≥ γˆ (Ω (ζ )), i.e., |y|2 ≥ γˆ (|ζ |2 ), and then consider the following two possible cases. Case 1. Assume that |w|2 ≤ εˆ (γˆ −1 (|y|2 )). Then (15) follows from (14), (17), (28). Case 2. Assume that |w|2 ≥ εˆ (γˆ −1 (|y|2 )). Then (15) follows from (23),(26),(28). Thus, in both the cases, the feedback v = vˆ (y) defined by (27) satisfies (15). ■ Remark 8. Following Remark 3 we note that (27) is resolvable in many applications (see e.g. Kim & Kim, 2003) and we assume by definition that numerical or analytic solving (27) is possible (see Condition (C2), and Remarks 3, 5).

x˙ i,1 = fi (xi,2 ) +

N ∑

ai,j,σ (t) (θ )xi,1 xj,1 + bi,σ (t) (θ )xi,1 D(t)

j=1

x˙ i,2 = ui + xi,1

N [∑

(30) ci,j,σ (t) (θ )xj,2 , i = 1, . . . , N ,

]

j=1

with switchings t ↦ → σ (t) ∈ {1, . . . , M }, states xi,1 , xi,2 , controls ui , disturbances D = D(t), and with |θ| ≤ θ ∗ < ∞. We take fi (xi,2 ) = xi,2 + x3i,2 ; functions ai,j,σ (θ ), bi,σ (θ ), ci,j,σ (θ ) are continuous. We design a decentralized feedback ui = ui (xi,1 , xi,2 ) which renders (30) UISS and provide some simulation. Consider the system x˙ i,1 = fi (xi,2 ) +

N ∑

ai,j,σ (θ )xi,1 xj,1 + bi,σ (θ )xi,1 D

(31)

j=1

with states xi,1 and controls xi,2 . Fix any gain coefficient γ (1) ∈ λ ]0, 1[, and any λi,1 > 0, λ∗ > 0, ε ∈ ]0, min 4i,1 [ s.t. Vi ≥ γ (1) max Vj ⇒ −(λi,1 − 2ε )Vi + 4ε j ̸ =i

∑

Vj ≤ −λ∗ Vi

(32)

j̸ =i

for all nonnegative V1 , . . . , VN . Then, applying the inequality A2 + B2 ≥ 2AB as in Section 5, we obtain V˙ i,1 |(31),xi,2 =vi,1 (xi,1 ) ≤ −λi,1 Vi,1 + ε

∑

Vj,1 + ε D2 ,

j ̸ =i

where vi,1 (xi,1 ) := −λi xi,1 − m∗i x3i,1 with λi := λi,1 +

m∗i :=

M N ∑ 1 ∑ ∗ (a∗i,j,σ + max ci2,j,σ (θ ))) (bi,σ + |θ |≤θ ∗ 2ε

σ =1

j=1

(33)

ε 2

,

(34)

(35)

S. Pavlichkov, C.K. Pang / Automatica 94 (2018) 300–306

305

8. Conclusion In this paper, we solved the problem of uniform ISS stabilization for a class of large-scale systems whose structure of interconnections is similar to those from Dashkovskiy and Pavlichkov (2012) and Mehraeen et al. (2011a, b). In contrast to Dashkovskiy and Pavlichkov (2012), our uniformly stabilizing controller is decentralized. In contrast to Mehraeen et al. (2011a, b), we address switched systems with unknown switchings and our stabilizing controller is switching-independent and uniform w.r.t. the switching signals. This is achieved by the following two novelties in comparison with Dashkovskiy and Pavlichkov (2012) and Mehraeen et al. (2011a, b): by application of a modification of small-gain theorems for large-scale switched systems and by a suitable backstepping design with gain assignment. Our design is constructive and our problem formulation is motivated by engineering applications (Mehraeen et al., 2011b). Fig. 2. Trajectories of the closed-loop (30).

References

where a∗i,j,σ := max a2i,j,σ (θ ), |θ |≤θ ∗

b∗i,σ := max b2i,σ (θ ). |θ |≤θ ∗

(36)

For (30), define (xi,1 , zi,2 ) := (xi,1 , xi,2 − vi,1 (xi,1 )) and Vi,2 := x2i,1 + zi2,2 , and then, from (33), again using the inequality A2 + B2 ≥ 2AB as in Section 5, we obtain: V˙ i,2 |(30) ≤ {−λi,1 Vi,1 + ε

∑

Vj,1 + ε D2 } + ε

j ̸ =i

(zj2,2 +

j=1

Vj,1 ) + 2zi,2 u + Φi (xi,1 , zi,2 ) + ε

(

N ∑

)

N (∑

Vj,1 + D2 + Γ,∗

)

j=1

where

Φi (xi,1 , zi,2 ) = xi,1 Qi − ωi,1 (xi,1 )fi (zi,2 + vi,1 (xi,1 )) N ∑ m∗j [ ] + zi,2 x2i,1 m∗i + m∗i [λj + (1) Vi,2 ]2 + m∗i ωi2,1 (xi,1 ) γ j=1

Qi = x2i,2 + xi,2 vi,1 (xi,1 ) + vi2,1 (xi,1 ) + 1, ωi,1 =

∂vi,1 . ∂ xi,1

(37)

and Γ ∗ = Γ ∗ (xi,1 , zi,2 ) satisfies Γ ∗ (xi,1 , zi,2 ) ≤ 0, if Vi,2 ≥ γ (1) maxj̸=i Vj,2 . The feedback uˆ (x1 , z2 ) = −λi zi,2 − Φi (xi,1 , zi,2 ) yields: V˙ i,2 |(30),u=ˆu(x1 ,z2 ) ≤ −(λi,1 − 2ε )Vi,2 + 3ε

∑

Vj , 2 + 2 ε D 2 .

(38)

j ̸ =i

By (32),(38), Vi,2 ≥ max 4ελD∗ ; γ (1) maxj̸=i Vj,2 implies V˙ i,2 |(30),u=ˆu ∗ ≤ − λ2 Vi,2 . Since γ (1) ∈ ]0, 1[, by Theorem 2, uˆ (·) renders (30) UISS. We make simulations for M = 2 and for (30) composed of N = 3 subsystems. We take two permutations: j1 (1) = 2, j1 (2) = 3, j1 (3) = 1, and j2 (1) = 3, j2 (2) = 1, j2 (3) = 2; take θ ∗ = 1, and for σ = 1 we take: bi,σ (θ ) = 1 and ai,j,σ (θ ) = ci,j,σ (θ ) = θ for j = j1 (i) and ai,j,σ (θ ) = ci,j,σ (θ ) = 0 otherwise, while for σ = 2 we take: bi,σ (θ ) = 0 and ai,j,σ (θ ) = ci,j,σ (θ ) = θ for j = j2 (i) and ai,j,σ (θ ) = 2k 2k+1 ci,j,σ (θ ) = 0 otherwise. We take σ (t) = 1 if t ∈ [ 10 , 10 [ and

{

2

}

σ (t) = 2 if t ∈ [ 2k10+1 , 2k10+2 [ for all k ∈ Z. For D(t) = 0, feedback ui = uˆ (x1 , z2 ) renders (30) UGAS, and the results are depicted in Fig. 2.

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Svyatoslav Pavlichkov received the M.Sc. with distinction and Ph.D. in Mathematics from V.N.Karazin Kharkiv National University, Ukraine in 1997 and 2002 respectively. He served as an assistant lecturer, senior lecturer and docent at Faculty of Mechanical Engineering and Mathematics of V.N. Karazin Kharkiv National University and at Faculty of Computer Science and Mathematics of V.I. Vernadsky Taurida National University, and as a post-doctoral Research Fellow at the University of Bremen (Center of Industrial Mathematics), University of Applied Sciences Erfurt, University of Passau (Institute of Computer Science and Mathematics), University of Wuerzburg (Institute of Mathematics), and National University of Singapore (Department of Electrical and Computer Engineering). He was also visiting Louisiana State University (Department of Mathematics) in 2005 and University of Greifswald (Institute of Mathematics and Computer Science) in 2004 and 2008 as a visiting researcher. Since February 2018, he is a post-doctoral researcher at the Faculty of Science and Engineering, University of Groningen. His research interests include nonlinear control and systems theory.

Chee Khiang Pang received the B.Eng.(Hons.), M.Eng., and Ph.D. degrees in 2001, 2003, and 2007, respectively, all in electrical and computer engineering, from National University of Singapore (NUS). In 2003, he was a Visiting Fellow in the School of Information Technology and Electrical Engineering (ITEE), University of Queensland (UQ), St. Lucia, QLD, Australia. From 2006 to 2008, he was a Researcher (Tenure) with Central Research Laboratory, Hitachi Ltd., Kokubunji, Tokyo, Japan. In 2007, he was a Visiting Academic in the School of ITEE, UQ, St. Lucia, QLD, Australia. From 2008 to 2009, he was a Visiting Research Professor in the Automation & Robotics Research Institute (ARRI), University of Texas at Arlington (UTA), Fort Worth, TX, USA. From 2009 to 2016, he was an Assistant Professor in Department of Electrical and Computer Engineering (ECE), NUS, Singapore. Currently, he is an Assistant Professor (Tenure) in Engineering Cluster, Singapore Institute of Technology, Singapore. He is an Adjunct Assistant Professor of NUS, Senior Member of IEEE, and a Member of ASME. He is also the Chairman of IEEE Control Systems Chapter of Singapore. His research interests are on data-driven control and optimization, with realistic applications to robotics, mechatronics, and manufacturing systems. Dr. Pang is an author/editor of 4 research monographs including Intelligent Diagnosis and Prognosis of Industrial Networked Systems (CRC Press, 2011), HighSpeed Precision Motion Control (CRC Press, 2011), Advances in High-Performance Motion Control of Mechatronic Systems (CRC Press, 2013), and Multi-Stage Actuation Systems and Control (CRC Press, 2018). He is currently serving as an Executive Editor for Transactions of the Institute of Measurement and Control, an Associate Editor for Asian Journal of Control, IEEE Control Systems Letters, Journal of Defense Modeling & Simulation, and Unmanned Systems, and on the Conference Editorial Board for IEEE Control Systems Society (CSS). In recent years, he also served as a Guest Editor for Asian Journal of Control, International Journal of Automation and Logistics, International Journal of Systems Science, Journal of Control Theory and Applications, and Transactions of the Institute of Measurement and Control. He was the recipient of The Best Application Paper Award in The 8th Asian Control Conference (ASCC 2011), Kaohsiung, Taiwan, 2011, and the Best Paper Award in the IASTED International Conference on Engineering and Applied Science (EAS 2012), Colombo, Sri Lanka, 2012.