Adaptive inverse optimal consensus control for uncertain high-order multiagent systems with actuator and sensor failures

doi:10.1016/j.ins.2022.05.021

Information Sciences

Volume 605, August 2022, Pages 119-135

https://doi.org/10.1016/j.ins.2022.05.021 Get rights and content

Abstract

This paper addresses a neuroadaptive inverse optimal consensus problem of uncertain nonlinear multiagent systems (MASs) subject to actuator and sensor faults simultaneously. Unlike traditional adaptive dynamic programming methods, the proposed control mechanism minimizes a loss function without solving the Hamilton-Jacobi-Bellman equation, which simplifies the computational workload. In addition, a compensation strategy for actuator and sensor faults is considered and a novel fault-tolerant adaptive inverse optimal protocol incorporating the Lyapunov design is constructed. It is demonstrated that the system is input-to-state stabilizable (ISS) under the designed inverse optimal controller and the tracking errors of the MASs can converge to a predefined range. A simulation example is presented to illustrate the effectiveness of the control design.

Introduction

During recent years, the consensus tracking for nonlinear MASs, as implemented in sensor networks, mobile robots and spacecrafts [1], [2], [3], has received sustained attention in control field, and a range of control designs on sliding mode control [4], [5], adaptive control [6], [7], [8] and even adaptive finite-time control [9], [10], [11] are developed. In addition, optimal control is proposed for most of the controlled models such as linear cyber-physical systems [12], via minimizing a predetermined loss function to resolve the optimal performance. Technically, the difficulty in achieving optimal control lies in solving the Hamilton-Jacobi-Bellman equation (HJBE), and this problem is handled by a typical solution- adaptive dynamic programming (ADP). Results on ADP for MASs can be found in [13], [14], [15], [16], [17]. Specifically, a fuzzy ADP-based method is investigated for MASs differential games in [13], where the consensus issue of MASs is addressed. In [14], the distributed optimal controllers are designed for nonlinear MASs with input delay. To ensure leader-to-formation stsbility for MASs, a data-driven approach with ADP is proposed in [15]. In [16], [17], the backstepping-based adaptive optimal control schemes are considered for MASs and stochastic MASs, respectively.

It should be pointed out that the performance of the system via ADP technology relies critically on the approximation accuracy of the neural networks (NNs), since two NNs would be utilized to solve the complex HJBE. Hence, the approximation error in NNs is required to be small enough, otherwise the system stability is at risk of spoiling. To remove above drawback, inverse optimal control, committed to stabilize an auxiliary system constructed rather than solve for HJBE, is first investigated as an alternative optimal control mechanism in [18]. Afterwards, works on inverse optimal approach for nonlinear system have been considered in [19], [20] and applied to the attitude control for spacecraft [21]. Moreover, the inverse optimal design is considered for linear MASs in [22]. Nevertheless, a common assumption of knowable system dynamics hinders the practicality of these previous algorithms. Fortunately, with exploiting the properties of adaptive control in handling uncertainties, several adaptive neural/fuzzy approaches have been introduced to deal with the inverse optimal issue for the uncertain nonlinear systems. For example, the authors in [23] have proposed an inverse optimal scheme via a discrete-time neural method for nonlinear system, and the controlled system is proved to be ISS. By using fuzzy logic systems (FLSs), fuzzy adaptive state feedback and output feedback are investigated in [24], [25] to solve the design of inverse optimal controllers for uncertain nonlinear system, respectively. These protocols, however, are still restricted by the approximation capability of the NNs or FLSs. Note that better approximation accuracy of the online parameters updated implies increasing the NNs nodes or fuzzy rules, which severely burdens the computation as same as the ADP-based methods. To obviate the workload of the computation, direct adaptive strategy is introduced in [26] and the result on the direct adaptive inverse optimal method have been proposed in [27]. However, these cited works in [19], [20], [21], [22], [23], [24], [25] including [27] treat the actuators and sensors of the considered system as ideal, which implies apparent restrictions during practical implementations.

In general, actuators and sensors in the real control situation can fail individually or simultaneously during operation owing to the complex environmental factors. Then the regular action of the system would be interrupted and the system performance may be seriously damaged. Thus, it is necessary to compensate for the undesirable effects from these faults. Results on fault-tolerant control (FTC) schemes are constantly being reported. For example, in [28], an FTC adaptive decentralized approach is considered for a nonlinear large-scale system. Based on [28], authors in [29], [30] have been devoted to the actuator compensation with finite-time controller and prescribed performance controller. To compensate for sensor faults, the FTC methods are proposed for discrete-time nonlinear systems in [31] and the Internet-based three-tank system in [32]. Furthermore, FTC-based optimal approaches can be found in [33], [34], [35], [36] with the expectation of reaching optimal performance. Note that only single category of fault is considered in these works. Control strategies aimed at compensating for actuator and sensor failures simultaneously have been reported in [37], [38], [39] recently. However, these nonoptimal approaches cannot work in achieving optimal performance for nonlinear systems, let alone MASs with uncertain topology. Expanding schemes [37], [38], [39] to inverse optimal FTC method for MASs is challenging, reflected in how to assure that the auxiliary system is asymptotically stable while compensating for faults in MASs. To our best knowledge, the problem of designing FTC-based inverse optimal controllers for MASs with actuator and sensor failures still remains unsolved.

Inspired by the above observations, an adaptive inverse optimal method for MASs subject to actuator and sensor faults is investigated in this article. The main contributions are summarized as follows:

1.
An FTC adaptive inverse optimal mechanism is established firstly via a new compensation strategy for nonlinear uncertain MASs. Hence, inverse optimal gain assignment for MASs with actuator and sensor failures is addressed and the considered system is guaranteed to be ISS.
2.
Unlike existing ADP methods for MASs, the requirement of solving complex HBJE is avoided under the performance-optimal solution proposed in this paper.
3.
Different from the non-FTC inverse optimal methods, inverse optimal strategy incorporating direct adaptive techniques alleviate the heavy computational burden for the consensus of MASs.

Section snippets

Graph theory

A directed graph is typically denoted as $G (R, V, A)$ , where $R = (r_{1}, r_{2}, \dots, r_{N})$ is the set of nodes and the $V \subseteq R \times R$ represents the set of edges in the graph. Then, the edge $(r_{j}, r_{i}) \in V$ reveals that ith node can receive the information sent by jth node and therefore, $N_{i} = {r_{j} | (r_{j}, r_{i}) \in V}$ is defined as the neighbor nodes set of the ith node. $A = [a_{ij}] \in R^{N \times N}$ is defined as an adjacency matrix to represent the communication relationships among the nodes. Specifically, if $(r_{j}, r_{i}) \in V$ , then $a_{ij} > 0$ ; otherwise, $a_{ij} = 0$ . In

Construction of the auxiliary system

The prerequisite for solving the inverse optimal issue of the system (17) is to solve the asymptotic control for the auxiliary system. Select $ς (y) = \frac{1}{η} y^{2}$ in which $η$ denotes the design parameter, and we obtain $h_{ς} (2 y) = η y^{2}$ . Then, the auxiliary system of (17) is given as $[\begin{matrix} {\dot{x}}_{i, 1} \\ {\dot{x}}_{i, 2} \\ ⋮ \\ {\dot{x}}_{i, n_{i}} \end{matrix}] = [\begin{matrix} {\hat{x}}_{i, 2} + ψ_{i, 1} \\ {\hat{x}}_{i, 3} + ψ_{i, 2} \\ ⋮ \\ ψ_{i, n_{i}} \end{matrix}] + [\begin{matrix} η {(L_{d_{i, 1}} (V_{i, 1} - V_{i, 0}))}^{T} \\ η {(L_{d_{i, 2}} (V_{i, 2} - V_{i, 1}))}^{T} \\ ⋮ \\ η {(L_{d_{i, n_{i}}} (V_{i, n_{i}} - V_{i, n_{i - 1}}))}^{T} \end{matrix}] + [\begin{matrix} 0 \\ 0 \\ ⋮ \\ ϑ_{i} \end{matrix}] u_{i}$ where V is the CLFs ( $V_{i, 0} = 0$ ) and ${(L_{d_{i, l}} (V_{l} - V_{l - 1}))}^{T} = (| ξ_{i, l} | - σ_{i, l}) {sg}_{i, l} K_{i, l} (1 - \sum_{q = 1}^{l - 1} \frac{\partial α_{i, l - 1}}{\partial x_{i, q}}) d_{i, l}^{T}$

Remark 4

The key element

Stability and performance of the MASs

From the design process introduced above, the obtained results are summarized into the theorems below.

Theorem 1

The asymptotic control requirement for the auxiliary system (21) is satisfied by constructing the controller $u_{i}$ (60) and the adaptive law ${\dot{\hat{κ}}}_{i}$ (56) and ${\dot{\hat{π}}}_{i}$ (57) for the auxiliary system, so that the recursively derived controller $u_{i}^{*}$ (69) is capable of solving the inverse optimal problem of the MASs (17) and the loss function is minimized. Moreover, the MASs is ISS and the following

Simulation

A simulation example will be presented to illustrate the effectiveness of the proposed algorithm for MASs with actustor and sensor faults.

With the communication structure shown in Fig. 1, the MASs considered consist of one leader and four followers wherein $b_{1} = 1, b_{2} = 0, b_{3} = 0, b_{4} = 2, d_{1} = 1, d_{2} = 5, d_{3} = 2$ and $d_{4} = 3$ . Then the dynamics of each follower are presented as $\begin{matrix} {\dot{x}}_{i, 1} & = x_{i, 2} - x_{i, 1} \cos (x_{i, 1}) + x_{i, 1} \exp (- t) \\ {\dot{x}}_{i, 2} & = w_{i} - x_{i, 1} \exp (- x_{i, 1}) - x_{i, 2} \end{matrix}$ where $ψ_{i, 1} ({\bar{x}}_{i, 1}) = - x_{i, 1} \cos (x_{i, 1}), ψ_{i, 2} ({\bar{x}}_{i, 2}) = - x_{i, 1} \exp (- x_{i, 1}) - x_{i, 2}$ , and $w_{i} = ϑ_{i} u_{i} + δ (t)$

Conclusion

In this paper, we tackle the inverse optimal issue of adaptive FTC for nonlinear uncertain MASs with actuator and sensor failures. The usage of the inverse optimal framework avoids solving complex HBJEs in the sence that computation burden is reduced. In addition, compensation for simultaneous actuator and sensor failures is considered, which improves the practicality of the algorithm designed. Under the proposed inverse optimal controller, the MASs is ensured to be ISS and the system obtains

CRediT authorship contribution statement

Chengjie Huang: Conceptualization, Methodology, Writing – original draft. Shengli Xie: Conceptualization, Methodology, Writing – review & editing, Project administration. Zhi Liu: Visualization, Investigation. C.L. Philip Chen: Supervision. Yun Zhang: Project administration, Software, Validation.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgement

The authors thank the editors and anonymous reviewers for their feedback that has improved the quality of this work.

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^☆: This work was supported in part by the National Natural Science Foundation of China under Grant U1911401, 61703112 and 61973087, in part by State Key Laboratory of Synthetical Automation for Process Industries, China (2020-KF-21-02), and in part by the National Key Research and Development Program of China under Project 2020AAA0108303.

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Adaptive inverse optimal consensus control for uncertain high-order multiagent systems with actuator and sensor failures☆

Abstract

Introduction

Section snippets

Graph theory

Construction of the auxiliary system

Stability and performance of the MASs

Simulation

Conclusion

CRediT authorship contribution statement

Declaration of Competing Interest

Acknowledgement

Automatica

Inf. Sci.

Inf. Sci.

Automatica

Automatica

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IEEE Trans. Cybern.

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IEEE Trans. Autom. Control

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IEEE Trans. Cybern.

Distributed adaptive neural control for stochastic nonlinear multiagent systems

IEEE Trans. Cybern.

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IEEE Trans. Syst. Man Cybern.: Syst.

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