Observer-based adaptive fuzzy output constrained control for uncertain nonlinear multi-agent systems

doi:10.1016/j.ins.2018.08.025

Information Sciences

Volume 467, October 2018, Pages 446-463

https://doi.org/10.1016/j.ins.2018.08.025 Get rights and content

Abstract

In this paper, consensus control problem is studied for uncertain nonlinear multi-agent systems with output constraint. Fuzzy logic systems (FLSs) and fuzzy state observer are employed to approximate unknown nonlinear functions and estimate unmeasured states, respectively. Barrier Lyapunov Function (BLF) is introduced to handle with the problem of output constraint. By combining adaptive backstepping and dynamic surface control (DSC) technique, a distributed adaptive fuzzy output feedback control scheme is proposed. It is proved that the semi-globally uniformly ultimately boundedness (SGUUB) of all the signals in the closed-loop can be guaranteed and all followers’ outputs can be well synchronized to the leader's output while maintaining consensus tracking errors to be bounded. The simulation example is provided to show the effectiveness of the presented control method.

Introduction

Consensus is one of the basic problems in the cooperative control of multi-agent systems (MASs), which contains multiple followers and one leader. There exists an active leader or a virtual leader, who only provides command to a small portion of the followers. Its aim is to make all followers track the trajectory of the leader successfully. During the past few years, several consensus control schemes have been proposed [3,6,8,10,14,33,35] for MASs. Among them, the consensus control methods require the system functions either known or parameterized, that is, the unknown parameters appear linearly with respect to the unknown nonlinear functions.

However, the presented results in [3,6,8,10,14,33,35] are invalid when the controlled systems contain uncertain nonlinear functions. To eliminate this restriction, neural networks (NNs) and FLS because of good approximation capability have been combined with distributed adaptive backstepping controller for nonlinear strict-feedback MASs with uncertainties. Adaptive fuzzy/NNs control schemes were proposed [25,27,28] for nonlinear systems, whose unknown dynamics are nonlinear functions. A distributed adaptive NNs control scheme was proposed [8] for uncertain nonlinear MASs. The literature [7] presented a fuzzy adaptive consensus control method for uncertain controlled systems, in which the considered nonlinear functions have unknown dynamics. Peng and Wang [16] were concerned with distributed maneuvering of multiple autonomous surface vehicles. Peng et al. [17] developed an output feedback path-following control scheme such that under-actuated autonomous underwater vehicles moving in a vertical plane without employing surge, heave, and pitch velocities. The author in [32] proposed a distributed adaptive NNs consensus control approach for unknown nonlinear MASs in strict-feedback form. [22], [26], [34] presented adaptive fuzzy/NNs control methods, which not only can make tracking errors converge to be a neighborhood of the origin, but also can determine the exact size of tracking errors. Adaptive fuzzy consensus control approaches were presented for uncertain nonlinear systems having initial-state learning [11] and input saturation [31]. Adaptive NNs schemes were proposed [5,18] for uncertain nonlinear MASs. The adaptive NNs control strategies were presented for uncertain nonlinear MASs having state time-delay [2] and unknown dynamic [1]. Shen et al. [20] studied the problem of output constraint for uncertain nonlinear MASs, in which an adaptive fuzzy control scheme was presented to address the problem of actuator faults. Wang et al. [30] developed a fuzzy adaptive output feedback control method for uncertain nonlinear systems.

Apparently, all proposed schemes in above literatures [1–[3], [5]–8,10,11,14,18,20,21,24–28,30,31–35] did not consider the problem of output constraint, which is commonly exists in the real world, such as electrostatic micro actuators, robotic systems, thruster assisted position mooring systems, the temperature of chemical reactor, physical stoppages, flexible marine riser and crane systems. It is well known, the design of BLF can effectively solve output constraint problem. Hence, [4,9,12,13,15,19,22,23,29] proposed several adaptive fuzzy and NNs control schemes for uncertain nonlinear systems with output constraint. However, the proposed control schemes in [4,9,12,13,15,19,22,23,29] were not for uncertain nonlinear MASs. To the best of our knowledge, until now, no results on adaptive NNs or fuzzy consensus control are developed for uncertain nonlinear MASs with output constraint and unmeasured states. Therefore, adaptive fuzzy control problem of uncertain nonlinear MASs needs to be further studied.

Motivated by the aforementioned observations, the output constraint problems are investigated in this paper for uncertain nonlinear MASs. In control design, FLSs are applied to approximate the unknown nonlinear functions and the fuzzy state observer is constructed. BLF is utilized to handle with the problem of output constraint. Combining adaptive backstepping design with DSC, a distributed observer-based fuzzy adaptive control strategy is presented, which can avoid “explosion of complexity”. The major contributions can be described in two aspects: (1) In [13,29], BLF is used to solve the problem of output constraint for general nonlinear systems. However, in this paper, the design of BLF is firstly used to deal with output constraint problem for uncertain nonlinear MASs with unmeasured states. 2) By designing a fuzzy state observer, the observer-based fuzzy adaptive control scheme can remove the limitation of the measurable states required by previous literatures [11,21,32].

The rest of this paper is organized as follows. The problem formulation and preliminaries are shown in Section 2. Section 3 gives the fuzzy state observer design. In Section 4, the distributed consensus controller design and stability analysis are given. The simulation example is demonstrated to show the effectiveness of the approach in Section 5. Finally, the conclusion is drawn in Section 6.

Section snippets

Problem formulations

Consider the following a class of uncertain nonlinear MASs, which are made up of one leader and N follower agents. The dynamics of the ith follower can be expressed as ${\begin{matrix} {\dot{x}}_{i, l} = x_{i, l} + f_{i, l} ({\bar{x}}_{i, l}) \\ ⋮ \\ {\dot{x}}_{i, n} = u_{i} + f_{i, n} ({\bar{x}}_{i, n}) \\ y_{i} = x_{i, 1} 1 \leq i \leq N l = 1, \dots n - 1 \end{matrix}$ where ${\bar{x}}_{i, n} = {[x_{i, 1}, \dots, x_{i, n}]}^{T} \in R^{n}$ is the state vector and $f_{i, l} (\cdot) (l = 1, \dots, n)$ is unknown nonlinear function. u_i and y_i ∈ R denote control input and control output of system (1), respectively. It is assumed that the only measurable variable is $y_{i} = x_{i, 1}$ .

Control objective: This

Fuzzy state observer design

Since the states $x_{i, k} (k = 2, \dots, n)$ in system (1) can not be measured directly, fuzzy state observer needs to be constructed for estimating the unmeasurable states. Rewritten (1) as follows $\begin{matrix} {\dot{x}}_{i} = A x_{i} + K_{i} y_{i} + \sum_{l = 1}^{n} B_{i, l} f_{i, l} ({\bar{x}}_{i, l}) + b_{i} u_{i} \\ y_{i} = C_{i} x_{i} \end{matrix}$ where $\begin{matrix} A_{i} = {[\begin{matrix} - k_{i, 1} \\ ⋮ & I_{n - 1} \\ - k_{i . n} & 0 & \dots & 0 \end{matrix}]}_{n \times n}, K_{i} = [\begin{matrix} k_{i, 1} \\ ⋮ \\ k_{i, n} \end{matrix}], B_{i, l} = {[{\underset{︸}{0, \dots, 1,}}_{l} \dots, 0]}_{n \times 1}^{T}, b_{i} = {[0, \dots, 0, 1]}_{1 \times n}^{T}, \\ C_{i} = {[1, 0, \dots, 0]}_{1 \times n}^{T} . \end{matrix}$

The vector K_i is chosen such that A_i is a strict Hurwitz matrix. Thus, for any matrix $Q_{i} = Q_{i}^{T} > 0$ , and satisfies $A_{i}^{T} P_{i} + P_{i} A_{i} = - 2 Q_{i}$ where P_i > 0 denotes the definite

Distributed consensus controller design and stability analysis

The distributed fuzzy adaptive control method will be developed in this part by backstepping DSC technique. Firstly, define coordinates transformation as follows: $\begin{matrix} s_{i, 1} & = & \sum_{j \in N_{i}} a_{i, j} (y_{i} - y_{j}) + a_{i 0} (y_{i} - y_{r} (t)) \\ s_{i, l} & = & {\hat{x}}_{i, l} - {\bar{α}}_{i, l} \\ ω_{i, l} & = & α_{i, l} - {\bar{α}}_{i, l} l = 2, \dots, n \end{matrix}$ where s_{i, 1} means the tracking error, s_{i, l} is the error surface, ω_{i, l} represents error between α_{i, l} and ${\bar{α}}_{i, l}$ . α_{i, l} is intermediate control function and ${\bar{α}}_{i, l}$ is a newly introduced state variable, which can be obtained by first-order filter.

Step 1: According

Simulation example

In this part, the simulation example is provided to depict the effectiveness of the presented fuzzy adaptive control approach. Considering the following nonlinear multi-agent systems, which contain a leader and four followers (see Fig. 2). ${\begin{matrix} {\dot{x}}_{i, 1} = x_{i, 2} + f_{i, 1} (x_{i, 1}) \\ {\dot{x}}_{i, 2} = u_{i} + f_{i, 2} ({\bar{x}}_{i, 2}) \\ y_{i} = x_{i, 1} \begin{matrix} i = 1, 2, 3, 4 \end{matrix} \end{matrix}$ where $f_{1, 1} = x_{1, 1} / (1 + x_{1, 1}^{4})$ , $f_{1, 2} = 0.1 \sin (x_{1, 1} - x_{1, 2}) e^{- (x_{1, 1}^{2} + x_{1, 2}^{4})}$ , $f_{2, 1} = x_{2, 1} / (1 + x_{2, 1}^{4})$ , $f_{3, 1} = x_{3, 1} / (1 + x_{3, 1}^{4})$ , $f_{2, 2} = 0.15 \sin (x_{2, 1} - x_{2, 2}) e^{- (x_{2, 1}^{2} + x_{2, 2}^{4})}$ , $f_{4, 1} = x_{4, 1} / (1 + x_{4, 1}^{4})$ , $f_{3, 2} = 0.15 \sin (x_{3, 1} - x_{3, 2}) e^{- (x_{3, 1}^{2} + x_{3, 2}^{4})}$

Conclusion

In this paper, a consensus adaptive fuzzy control scheme has been proposed for uncertain nonlinear MASs with output constraint. FLSs and BLF are utilized to approximate unknown nonlinear functions and address the problem of output constraint, respectively. Moreover, the presented control method does not require the states of the uncertain nonlinear controlled systems to be measured directly. It has demonstrated that the presented control strategy not only can achieves semi-globally uniformly

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^☆: This work was supported in part by the National Natural Science Foundation of China under Grant 61773188.

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Observer-based adaptive fuzzy output constrained control for uncertain nonlinear multi-agent systems☆

Abstract

Introduction

Section snippets

Problem formulations

Fuzzy state observer design

Distributed consensus controller design and stability analysis

Simulation example

Conclusion

Fuzzy Sets Syst.

Syst. Control Lett.

Automatica

Fuzzy Sets Syst.

Automatica

Syst. Control Lett.

Automatica

Automatica

Inf. Sci.

Inf. Sci.

Inf. Sci.

IEEE Trans. Syst. Man Cybern. Syst.

Neural-network-based adaptive leader-following control for multi-agent systems with uncertainties

IEEE Trans. Neural Netw.

Adaptive consensus control for a class of nonlinear multi-agent time-delay systems using neural networks

IEEE Trans. Neural Netw. Learn. Syst.

Information flow and cooperative control of vehicle formations

IEEE Trans. Autom. Control

Decentralized robust adaptive control for the multi-agent system consensus problem using neural networks

IEEE Trans. Syst. Man Cybern. Part B (Cybernetics)

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Nonlinear Dyn.