Probabilistic-constrained tracking control for stochastic time-varying systems under deception attacks: A Round-Robin protocol

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Abstract

The probabilistic-constrained tracking control issue is investigated for a class of time-varying nonlinear stochastic systems with sensor saturation, deception attacks and limited bandwidth in an unified framework. The saturation of sensors is quantified by a sector-bound-based function satisfying certain conditions, and the random deception attacks are considered and modeled by a random indicator variable. To gain more efficient utilization of communication channels, a Round-Robin (RR) protocol is utilized to orchestrate the transmission order of measurements. The main purposes of this study aim to plan an observer-based tracking controller to achieve the following goals: (1) the related performance indicators of the estimation error is less than given bound at each time step; and (2) the violation probability of the tracking error confined in a predefined scope is supposed to be higher than a prescribed scalar and the area is minimized at each instant. In order to reach these requirements, a group of recursive linear matrix inequalities (RLMIs) are developed to estimate the state and design the tracking controller at the same time. Finally, two simulation examples are exploited to illustrate the availability and flexibility of the proposed scheme.

Introduction

Today, tracking control problem is of immense significance for nonlinear systems [1], [2], [3]. It has been witnessed a major breakthrough over the past decades and many outstanding tracking methods have been proposed (see [4], [5], [6], [7]). The key problem of tracking control is to design an appropriate controller such that the state/output of the considered system approaches the reference value as closely as possible. To date, in virtue of the advantages (such as low cost and simple maintenance) of the network control systems (NCSs), it has been paid increasing attention, and a variety of substantial publications focusing on trajectory tracking issues have been reported. For example, in [8], a novel way on the basis of linear matrix inequality has been proposed to ensure that the state vector of the plant can approximate a desired signal fairly. A learning-based tracking controller has been adopted in [9] for linear systems and when further thinking about limited bandwidth, event-driven scenario and quantized-constraint scheme for tracking control issues have been put forward in [10], [11], [12].

It should be noted that there is an implicit assumption in most existing literature that all sensors or control components are in good working conditions. This assumption, however, does not always hold in practical systems [13]. Limited by physical restrictions and energy constraints, sensors can not provide signals with large or even unlimited amplitudes in practical systems, which is called sensor saturation [14]. The phenomenon of sensor saturation will jeopardize the system performance or even destabilize the system if not dealt properly. To gain better security and sustainability of dynamic nonlinear systems, reliable tracking control strategy should be adopted such that the plant can operate normally even if there exists sensor saturation. With respect to sensors saturation, many outstanding results have been obtained for the controller or filtering design issues (see [15]).

On the other hand, in NCSs with limited bandwidth, data collisions and channel congestion may occur [16] if all sensor nodes are allowed to get access to deliver multiple packets. In order to address the need for more efficient utilization of communication channels with limited bandwidth, an active way is to apply the communication protocol orchestrating the transmission order of sensor nodes. Generally speaking, three types of communication protocols, e.g., Round-Robin (RR) protocol, Weighted-Try-Once-Discard (WTOD) protocol and stochastic communication protocol (SCP) have been utilized widely in communication work (see [17], [18], [19], [20], [21]). Among them, the RR protocol is well deployed in industry, under which each sensor node is assigned with equal access to the communication network based on the law of a fixed circular order. The widely employed RR protocol is a typical time-division multiple-access (TDMA) protocol, in which the scheduling of sensors is predetermined according to a fixed circular order. This kind of communication protocol is adopted in underwater communication network, Vehicular Ad Hoc Networks (VAHN) and embedded systems, and the network burden is cut down greatly and the risk of data collisions can be decreased. This law could be expressed by a periodic function which reflects the assignment nature. Different from the static scheduling of the RR protocol, the WTOD protocol is a dynamic scheduling protocol. The communication access assignment is achieved through competition. The stochastic communication protocol (SCP), under which each node displays a stochastic switch behavior, has been widely adopted in the automation industry. In view of above mentioned characteristic, some outcomes have been published in [22], [23]. In [24], a novel moving-horizon estimation method has been proposed to account for both the networked time-delay and the RR protocol. Moreover, a fixed-circulation-based approach has been adopted in [25], [26], [27] to describe the communication scheduling manner clearly. As such, it is practical to take sensor saturation and RR protocol into consideration when coping with tracking problems for NCSs. However, little advancement has been made for the time-varying nonlinear systems with both sensor saturation and RR protocol, thus it remains as an undergoing research project. This is the first motivation of the current research.

It should be emphasized that the measurement or control signals are transmitted through communication networks, which is liable to be attacked by malicious adversaries [28], [29], [30], [31], [32], [33], [34], [35], [36]. For example, the security problem with respect to distributed filtering has been researched in [29]. Besides, when it come to power grid, H load frequency control has been investigated in [28]. Generally speaking, deception attacks are regarded as one of the noticeable cyber attacks, which undermine the integrity of the data severely by injecting the incorrect information [37], [38], [39], [40], [41]. For instance, the optimal linear deception attacks on remote state estimation has been discussed in [42]. As for the industrial process, in [43], a hybrid cyber-attack has been studied for multiarea power systems. Nevertheless, such deception attacks has not been considered for tracking control problems and corresponding results are rare, which stirs the current research.

Up to now, when considering the tracking control problem, the majority of the researches require that, for deterministic systems, the tracking error should trend to zero and for stochastic systems, the tracking error should trend to zero in mean square sense and/or the variance is minimized. More recently, the minimum-variance requirement is relaxed as variance-bounded requirements in [12]. That is, compared with minimizing the variance of the tracking error, the authors in [12] prefer to layout proper tracking controller such that the variance of the tracking error is limited, which offers extra freedom in designing the tracking controller. However, variance-constrained requirement still has some shortcomings, for example, if there are random occurring large or even unbounded noises in the system, it will be much conservative or even impossible to find the variance bounds. In this case, a probabilistic-constrained design aim is more realistic [44], [45]. It is easy to understand that the probabilistic-constrained performance has been applied in many related fields, such as tracking control, flight control, robot-path planning and missile control, and it is intuitive for us to evaluate the system performance by the numbers of violations of the tracking error concentrating in a given set. Regrettably, such kind of issue has not been paid enough attention. According to the authors, little progress has been made in this field, let alone considering tracking control task under the RR protocol, sensor saturation and deception attacks, which encourages us to explore deeply.

As discussed above, the main contributions of the current study are highlighted as shown. (1) Considering stochastic noises, stochastic nonlinear functions, deception attacks, RR protocol and sensor saturation, the system model is very comprehensive, which is in agreement with the actual situations. (2) A novel probabilistic-constrained tracking controller design approach is first presented for the stochastic nonlinear systems under RR protocol and deception attacks, and an optimization algorithm is proposed to design the observer-based controller together with the tracking controller.

The reminder of this paper is arranged as follows. In section II, the strategy of the tracking control issue is elaborated. Section III demonstrates the design steps of the concerned systems, the availability of the proposed method has been confirmed by several sufficient conditions. Simulation examples are operated in section IV and the obtained results show the effectiveness of the proposed approach.

Section snippets

Preliminaries

In this paper, the stochastic nonlinear time-varying system is described by:{xk+1=(Ak+i=1lρikAik)xk+f(k,xk)+Bkuk+Dkωk,yk=Sat(Ckxk)+Ekνk.where xkRn, ykRa, ukRq are the state, measured output and control input, respectively. ωk and νk are one-dimension noise sequence with features of E{ωk}=E{νk}=0 and E{ωk2}=W, E{νk2}=V. Ak,Bk,Ck,Dk and Ek are known matrices with appropriate dimensions. ρik(i=1,2,,l) are white stochastic noise sequence with independence, which satisfy E{ρik2}=1. f(k,xk) is a

Main results

Before proceeding further, the following two lemmas are employed in the proof of main results.

Lemma 1

[47] Let χ0(j),χ1(j),,χp(j) be quadratic functions of jRnχi(j)=jTYij,i=0,1,,mwith Yi=YiT. Thenχ1(j)0,,χp(j)0χ0(j)0holds if τ10,,τp0 such thatjT(Y0i=1pτiYi)j0.

Lemma 2

[44] For given matrix Y>0 and a prescribed vector e with proper dimensions, a constraint set B is established asB:={u|(ue)TY(ue)1},where u is a stochastic variable. If the following inequalityE{(ue)TY(ue)}1pholds, we havePr(uB

Two simulation examples

In this part, a numerical example and a three-tank-oriented paradigm are demonstrated to clarify the usefulness of the presented tracking controller implementation.

Conclusion

The observer-based tracking control issue has been studied for stochastic time-varying systems with sensor saturation and random deception attacks using Round-Robin protocol. An effective approach has been proposed to cope with sensor saturation which has been considered as an abnormal phenomenon in the communication network. The semi-definite program method and recursive linear matrix inequalities (RLMIs) are been applied in the design of the observer and tracking controller. Besides a

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.

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    This work was supported in part by the National Natural Science Foundation of China under Grants 62173231, 61903252, 61903254, 61773218, Natural Science Foundation of Shanghai (21ZR1444900) and in part by Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning.

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