Adaptive Fuzzy Decentralized Control for Fractional-Order Nonlinear Large-Scale Systems with Unmodeled Dynamics

This paper addresses the decentralized control issue for the fractional-order (FO) nonlinear large-scale strict-feedback systems with unmodeled dynamics. The unknown nonlinear functions are identified by fuzzy logic systems (FLSs) and the FO dynamic signals are introduced to dominate the unmodeled dynamics. Additionally, the fractional-order dynamic surface control (FODSC) design technique is introduced into the adaptive backstepping control algorithm to avoid the issue of “explosion of complexity”. Then, an adaptive fuzzy decentralized control scheme is developed via the FO Lyapunov stability criterion. It is proved that the controlled FO systems are stable and the tracking errors can converge to a small neighborhood of zero. The simulation example is provided to confirm the validity of the put forward control scheme.


I. INTRODUCTION
F RACTIONAL-ORDER nonlinear systems (FONSs) can be treated as the extension of integer-order nonlinear systems (IONSs), which is a research hotspot developed on the basis of fractional-order calculus theory. In practice, some real systems can be modeled by utilizing FONSs, for example, micro-electro-mechanical resonators [1], hyperchaotic economic systems [2] and lithiumion batteries [3]. Recently, a growing number of scholars have begun to focus on the control issues of FONSs, and obtained some results [4]- [6]. A smooth adaptive backstepping control scheme was put forward in [4], which ensures the globally asymptotically stable of commensurate FONSs. The authors in [5] designed a backstepping-based adaptive tracking controller for the FONSs. Then, this approach was further studied in [6] for FONSs with states immeasurable. However, the above literatures only consider the case that the nonlinear functions in the controlled objects are known.
As we all know, FLSs and neural networks (NNs) have ability to approximate unknown nonlinear functions. Consequently, in references [7]- [16], they are utilized to identify unknown nonlinear functions in FONSs. The authors in [7]- [11] put forward several adaptive intelligent (fuzzy and NN) control strategies for FONSs with measurable or unmeasured states, respectively. Furthermore, the conventional backstepping control design technique adopted in the abovementioned intelligent adaptive control strategies, which may lead to the issue of "explosion of complexity" since the repeated derivation of the virtual control functions. To solve this issue, several intelligent adaptive control strategies have been put forward in [12]- [13] by introducing fractional-order dynamic surface filters (FODSFs). Subsequently, to solve the issue of unmeasured states, the authors in [14]- [16] put forward some observer-based adaptive intelligent DSC strategies via designing state observers. Nevertheless, the abovementioned control methods are limited to the single-input and singleoutput (SISO) FONSs.
In real life, such as traffic systems, network transportation systems, aerospace systems are often described by interconnected systems composed of many lower-dimensional subsystems. Because of the complex structure and high dimensions of interconnected systems, the decentralized control is often used. The decentralized control only relies on the local information of the subsystems, so compared with the central control, it has the characteristics of reducing the computational complexity and enhancing the robustness and reliability of the interactive operation fault [17]. Consequently, the intelligent decentralized control of nonlinear interconnected systems is favored by many scholars and has achieved a great progress in the past decades [18]- [22]. Recently, the adaptive decentralized controller and adaptive sliding-mode decentralized controller have been put forward by [23]- [24] for the FO nonlinear large-scale systems. Furthermore, the authors in [25]- [27] put forward intelligent adaptive backstepping decentralized control schemes for the uncertain non witched or switched FO nonlinear large-scale systems. Later, an observer-based intelligent adaptive decentralized DSC control strategy has been developed by the authors in [28]. However, the unmodeled dynamics are not considered in the above controlled objects.
The unmodeled dynamics refers to some dynamic characteristics lost when modeling the systems. It widely exists in practical systems and has strong uncertainty, which may cause the poor performance or even instability of the controlled systems. Because of this, this problem has also been paid attention by the majority of scholars. An output feedback tracking controller has been put forward in [29] to ensure the stability for the FO interconnected systems with unmodeled dynamics. Later, the authors in [10] put forward an intelligent adaptive method for SISO FONSs with unmodeled dynamics. Reference [10] gave the notion of Mittag-Leffler input-to-state practical stable (ISpS) Lyapunov function and a necessary condition ensuring that FONSs have Mittag-Leffler (ISpS) Lyapunov functions. Then, a FO dynamic signal has been introduced to dominate the unmodeled dynamics. However, [10] is limited to SISO FONSs. It is necessary to design a control scheme for the FO nonlinear large-scale systems with unmodeled dynamics. This paper considers the fuzzy adaptive decentralized control design issue for the FO nonlinear large-scale systems with unmodeled dynamics. The main features of this study are as below: i) For the issue of unmodeled dynamics, firstly, the systems are required to have Mittag-Leffler ISpS Lyapunov functions, and then some FO dynamic signals are introduced to solve this problem. ii) By applying FODSC technique, the put forward control strategy avoids the issue of "explosion of complexity" in the previous methods [7]- [11] and [24]- [25].

A. SYSTEM DESCRIPTIONS
The FO nonlinear large-scale systems are as follows: where α ∈ (0, 1), , · · · M, j = 1, 2, · · · , m i ) are the system state vectors. u i are the control inputs of the system and Y = [y 1 , y 2 , ...y M ] T ∈ R M is the system output. f i,j (.) are smooth unknown nonlinear functions. z i ∈ R mi are unmodeled dynamics and H i,j (z i , Y, t) are the dynamical disturbances. q i (·) and H i,j (·) are uncertain functions. Remark 1: If the issue of unmodeled dynamics is ignored and there are no interconnected terms, the control objects (1) will become M independent SISO FONSs in strict-feedback form. For SISO FONSs in strict-feedback form have been extensively studied in [7]- [15]. However, this paper studies FO strict-feedback nonlinear large-scale systems, and considering the existence of unmodeled dynamic problem, it is more complex than the controlled systems in the previous references [7]- [15]. Therefore, the control design of this paper is more difficult.
Assumption 1 [12][13]: The given reference signals y i,d are sufficiently smooth functions of t and y i,d , C 0 D α t y i,d and Assumption 3 [17,[19][20]: There exist a unknown constant q * i,j ≥ 0 and known smooth functions ϑ i,j,0 ≥ 0 and ϑ i,j,l ≥ 0 such that where ϑ i,j,0 (0) = 0. Remark 2: Noting that Assumptions 1-3 are common assumptions and can be found in references [10,12,13,17,19,20], respectively. The purpose of Assumption 2 is to explain the existence of Mittag-Leffler ISpS Lyapunov functions in the z i − systems. The meaning of Assumption 3 is to deal with dynamical disturbances. However, the ways for addressing dynamical disturbances are not unique, such q k i,j,l (||y l || k )) in [18,22], where q k i,j,0 and q k i,j,l are unknown constants and p = {p i,j |1 ≤ i ≤ M, 1 ≤ j ≤ m i } is a known integer. In other words, Assumption 3 in the put forward control strategy can be replaced by other assumptions with the same function, and the control objectives can still be achieved after some modifications.
Control objectives: On the basis of FO Lyapunov stability criterion, an adaptive fuzzy decentralized controller is designed for systems (1), which makes all signals in the controlled systems are bounded as well as the tracking errors as small as possible.
Remark 3: Note the inequality |x| ≤ xg(x) + ε, where g(x) represents a class of smooth functions with g(0) = 0. The selection of g(x) is not unique, such as x/4ε in [34] and tanh(x/ε) in [37]. Therefore, the inequality will hold by choosing an appropriate g(x).
Lemma 4 [10]: If system (1) has a Mittag-Leffler ISpS Lyapunov function, then for any constantsc i in (0, c i ), any initial condition z i,0 = z i (t 0 ) and r i0 > 0, for a function such that D i (t 0 , t) for all t ≥ t 0 + T 0 and for all t ≥ t 0 . A FLS in [31] is defined bŷ In (9), A l i and G l are fuzzy sets, µ A l i (x i ) and µ G l (y) are their membership functions.ȳ l = max y∈R µ G l (y). Let Denote Lemma 5 [31]: Suppose that f (x) is a continuous function defined on a bounded compact set Ω, then there exists a FLS where δ > 0 is a given constant. (1) are unknown nonlinear functions, according to Lemma 5, it can be approximated by where w * i,j ideal parameter vectors defined over bounded sets Ω i,j , and δ i,j satisfies |δ i,j | ≤ δ * i,j with δ * i,j > 0. Remark 4: It is worth noting that, like references [9,12,35,39], this paper selects FLSs as the approximator to address unknown nonlinear functions in the controlled systems (1). However, there are some other nonlinear approximators, such as NNs [10,11,13,38,40] and neuro-fuzzy network system [15,36], which can replace FLSs and achieve the same purpose.

III. FUZZY ADAPTIVE DECENTRALIZED CONTROL DESIGN AND STABILITY ANALYSIS
By adopting the backstepping-based control algorithm, the detailed design processes of the fuzzy adaptive decentralized controller will be given.

Substituting (50) and (55) into (49) results in
Design the decentralized contorller u i and the adaptation laws C 0 D α t w i,mi and C 0 D α t υ i,mi as where c i,mi > 0, τ i,mi > 0 andτ i,mi > 0 are design parameters. By substituting (57)-(59) into (56), one has where Theorem 1: For the FO controlled systems (1), if Assumptions 1-3 satisfied, the devised actual decentralized controller (57), the virtual decentralized controllers (25) and (41) and the FO parameter adaptation laws (26), (27), (42), (43), (58) and (59) are adopted, then whole control scheme can assure that all signals of the closed-loop system are bounded. Additionally, the tracking errors can be made as small as possible.
Proof: On the basis of Assumption 1 with a constant π i > 0, By the Young's inequality, the following inequalities hold From (60)-(62), one can obtain Then, (78) can be rewritten as In view of [7], [13] and [33], from (64), we can obtain where Φ(t) > 0. Using Laplace transform on (65) yields For (66), utilizing the inverse Laplace transform results in where * is the convolution operator. Since Φ(t) and t α−1 E α,α (−ηt α ) are nonnegative functions, the term Φ(t) * t α−1 E α,α (−ηt α ) ≥ 0 in (67). Then, the following inequality holds Utilizing Lemma 1 gives where d is a positive constant. Therefore A repeated utilize of Lemma 1, we can obtain Finally, (67) and the tracking error can be written as follows where r is a positive constant. From (73), we further have Based on (72), it is concluded that the controlled plant is stable. In view of (74), when t aproaches infinity, we can get lim t→∞ |ζ i,1 | ≤ 2φd η . Hence, it is concluded that all the signlas of the closed-loop system are bounded and the tracking errors can achieve satisfactory performance. This completes the proof.

V. CONCLUSIONS
This paper has investigated the tracking control problem for FO nonlinear large-scale systems with unmodeled dynamics. The FLSs and the FO dynamic signals have been adopted to cope with the unknown nonlinear continuous functions and unmodeled dynamics, respectively. By combining adaptive backstepping recursive design algorithm with the FO Lyapunov stability criterion, a stable fuzzy adaptive decentralized DSC strategy has been developed. The designed control scheme has the main features of ensuring the controlled system stable and also making the tracking errors to be smaller. The validity of the developed control approach has been confirmed via simulated example. For future work, the problem of unmeasurable state will be further considered,   such as references [14,15,41,42] or the control design of fractional-order nonlinear multi-agent systems.