Research articleEvent-triggered observer-based sliding mode control of nonlinear systems
Introduction
Takagi–Sugeno (T–S) model [1] is currently attracting rapid interest in the field of nonlinear systems for the issue of stability and control. Through a set of IF–THEN rules, T–S fuzzy system could be worked as an effective tool to represent complex nonlinear plant by local linear models that are smoothly connected via membership functions. Recently, due to the birth of T–S fuzzy frameworks and modern control techniques, fruitful results have been given on T–S fuzzy systems regarding stability and stabilization. For instance, the stability analysis and synthesis of nonlinear retarded systems via linear T–S fuzzy model approach was proposed in [2]; By employing the T–S fuzzy models, the filtering problem for multirate nonlinear sampled-data systems was studied in [3]; And in [4], the problem of designing fuzzy controllers for a class of nonlinear dynamic systems with fewer fuzzy rules was investigated. In the application field, a near-space hypersonic vehicle was presented in [5] to study the fault-tolerant control based on T–S fuzzy models; The classification problems of training images was proposed by a self-organizing T–S fuzzy network with support vector learning in [6]; See more details, please refer to [7], [8], [9], [10], [11] and references therein.
Since its appearance [12], the sliding mode control (SMC) has shown its unique superiority over others in the field of robust control. Practically, the SMC has been widely applied to dynamics in theory and practical for their good properties, for instance, fast response and good transient performance, strong robustness to system perturbations and uncertainties. Therefore, plenty of results have been reported on SMC of complex systems. Such as, the SMC in electro-mechanical systems was proposed in [13]; A non-singular terminal sliding mode controller for rigid manipulators was designed in [14]; Observer-based SMC for nonlinear fractional-order systems was reported in [15]; the SMC approach is also attractive in the field of discrete-time systems [16] and Markovian jump systems in [17], [18], [19], [20]. Besides, the methodology appeared in T–S fuzzy models for complex nonlinear plant could also well combined with SMC, for instance, novel integral SMC for nonlinear stochastic systems by T–S fuzzy models was proposed in [21], [22], [23]; Dissipativity-based fuzzy integral SMC of continuous-time T–S fuzzy systems with matched/unmatched uncertainties and external disturbance was investigated in [24]; The output feedback SMC problem for discrete-time nonlinear systems by T–S fuzzy dynamic models was addressed in [25]; SMC of nonlinear descriptor systems was proposed based on T–S fuzzy-affine model in [26]; In [27], reliable SMC for nonlinear descriptor systems by T–S fuzzy models was concerned. On the other hand, it is also noted that the state components of practical system needed for control purpose are sometimes not available due to a variety of reasons, such as high cost and environment limitations. Therefore, the observer design is necessary, which leads the issue of observer-based SMC for T–S fuzzy systems. However, few results were found to report on this issue except to the sliding mode observer design for T–S fuzzy systems with semi-Markov parameters [28], from which it is observed that the problem observer-based SMC for T–S fuzzy systems is interesting and worth researchers’ attention.
Nowadays, the networked control systems have achieved considerable attention due to their superiorities over traditional point-to-point systems. Since the time-triggered control was executed in a periodic way, which resulted in low efficiency utilization of network resources. While by employing event-triggered mechanism, a selected threshold determines whether transmission of signals happens based on event-triggered conditions, which largely reduces waste of computation and communication resources. Therefore, the event-triggered control strategy is attractive and becoming more and more popular in engineering. Recently, many nice works have been reported on the issue of event-triggered control, for instance, the event-triggered control for networked control systems with quantization and time-varying network delays was studied in [29]; A novel event-triggering scheme was proposed for controller design in [30], etc. [31], [32]. With respect to event-triggered based SMC, it also has witnessed some researches, but not enough. For example, the event-triggered SMC for stochastic systems and discrete-time systems was investigated in [33], [34]; A sliding mode observer design issue was proposed for discrete-time nonlinear systems through dynamic event-triggered method in [35]; An aperiodic sampled-data SMC of fuzzy systems were studied in [36]; The event-triggered fuzzy SMC of networked control systems regulated by semi-Markov switching parameters was given in [37]. However, the issue of event-trigger control of fuzzy mode-based systems through sliding mode observer has been seldomly touched. Studying such an issue is challenging since there are some difficulties, such as how to select appropriate switching surface function coordinates to the fuzzy models and how to deal with the impact from the network delays and how to design the sliding mode controller to accommodate the sliding motion, which partially motivate us to give this study.
Based on the above analysis, in this paper, we will concern with the problem of observer-based SMC for nonlinear system represented by T–S fuzzy model through event-triggered mechanism. In the output channel to the observer, the signal is transmitted through a communication network and suffers from transmission delays [38]. In order to save bandwidth and resource, an event-triggered condition is proposed to decide whether output signals should be transmitted to the designed observer. The main contributions of this paper can be concluded as: (1) In order to accommodate to physical system without constraint on the input matrices that are full column rank, a novel transformation is provided such that the designed state-observer system is stabilizable; (2) In the observer design, an event-triggered mechanism is constructed between the output channel of original system to the observer, which could save limited network bandwidth and improved network using efficiency. In addition, the network-induced delay is considered in the communication; (3) A novel sliding surface function is proposed, based on which the obtained sliding motion with good property of dynamics; (4) Based on a delay system model approach, the signal transmission delay is incorporated into a time-varying delay system by segmenting method based on sampling intervals. The rest of this paper is organized as follows: Problem statement and preliminaries are presented in Section 2; Section 3 gives the main results, which include the design of observer based on event-triggered condition, design of sliding surface, finite-time reachability analysis of predefined sliding surface and stability analysis of sliding mode dynamics; Section 4 shows two numerical examples to verify the proposed method and Conclusion is drawn in Section 5.
Notions: In the article, () denotes a symmetric positive definite(semi-positive definite) matrix. and represents the zero matrix and identity matrix, respectively. means a minimum eigenvalue of a matrix. The symmetric elements of a symmetric matrix are denoted by . represents .
Section snippets
Problem formulation and preliminaries
Consider the following T–S fuzzy model:
Plant Rule : IF is and is and and is
THEN where denotes the state vector; , , are also acted as the premise variables; are fuzzy sets. is the control input; is the controlled output; is the initial condition. is a known nonlinear function and reserved as the nonlinear part
Main results
In this paper, the output measurement is sampled periodically by a sampler with sampling period as shown in Fig. 1. is the current sampled measurement and (, is the time for triggering) is the latest transmitted one. Here, the following triggering condition is provided to determine whether to retrieve the transmission: where is a positive definite weighting matrix,
Numerical examples
Example 1 Consider the system (1) with the following parameters (the input matrix is plan-rules independent case): The membership functions are
In the designed observer (13), the gain matrices are chosen as and . In the sliding surface function, the gain matrices are given by and
Conclusions
The paper has investigated the sliding mode control of T–S fuzzy model-based nonlinear systems with one-sided Lipschitz condition. First, in order to construct a fuzzy observer, an event-triggered mechanism has been proposed to decide whether the output signals should be transmitted or not in the network communication. Then, with the aid of the designed state observer, an integral sliding surface has been proposed, based on which sliding mode dynamics has been obtained by the equivalent control
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.
Acknowledgments
This work was supported in part by the NSFC under grant nos. (62003231, 61803279, 61703226); partially supported by the Natural Science Foundation of Shandong Province under grant ZR2019MF027; funded by Open Foundation of The Suzhou Smart City Research Institute under grant nos. (ZSCR2019001,SZSCR2019006); funded by the China Postdoctoral Science Foundation under Grant (ZSCR2019001), and funded by Qing Lan Project of Jiangsu .
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