Switched-observer-based adaptive neural networks tracking control for switched nonlinear time-delay systems with actuator saturation
Introduction
Time-delay phenomena have received substantial research attention over the past decades since they exist in many engineering application scenarios, such as chemical reactor systems [1], a single-link robotic manipulator [2], the pendulum systems [3], [4], etc. Time delays often cause a serious deterioration in the control performance, and if it is not well-addressed, the instability of practical control systems even occurs. There exists two classical methods to analyse the stability of time-delay systems, i.e., the Lyapunov-Razumikhin (L-R) method and the Lyapunov–Krasovskii (L-K) functionals method, as presented in [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16]. In addition, adaptive control of non-switched nonlinear systems with time delays has attracted considerable attention in the field of nonlinear control [2], [17], [18], [19]. In particular, adaptive control has proven its significant capability in compensating for non-switched nonlinear systems involving based on a parameter separation technique. Due to the systematic recursive design, the backstepping technique for adaptive control has been widely applied to stability analysis for uncertain nonlinear systems. Notably, as universal approximations, fuzzy logic systems (FLSs) or neural networks (NNs) have been widely cooperated with the backstepping technique to approximate uncertain nonlinear functions for non-switched systems [20], [21], [22]. However, in the practical scenarios, since system states are rarely fully measurable, the adaptive NNs or FLSs output-feedback control becomes an effective approach for the stability and stabilization of the nonlinear time-delay systems in the non-switched version, and many related results, e.g., [4], [10], [12], [23], [24], [25], [26], [27], [28], [29], [30], have been reported.
Switched systems have also attached tremendous attention in recent years since many natural and physical systems display switching dynamics, such as surface vehicles, chemical reactor systems, a one-link manipulator with motor dynamics and many other artificial systems [8], [9], [14], [31], [32], [33], [34], [35], [36]. In [7], the adaptive NN output-feedback controller was designed for the switched nonlinear time-delay systems under ADT, which is relaxed to the nonlower triangular form. The authors of [8] proposed a NN prescribed performance tracking scheme for unknown switched nonlinear time-delay nonaffine systems by the common Lyapunov function method. Further, based on the multiple Lyapunov functions (MLFs) approach, the authors of [9] investigated the adaptive FLS-based output-feedback prescribed control strategy for switched nonlinear time-delay systems via backstepping technique under ADT with dynamic surface control method. In [37], the problem of output feedback model predictive control for interval Type-2 Takagi–Sugeno fuzzy systems with bounded disturbance was investigated including an off-line design of the state observer and an online optimization for the controller gains. However, the restrictive assumptions for unknown time-delay nonlinearities are only considered in the first system state variable in [9], [13] or the system with the strict feedback form in [2], [29], [38]. There exist only a few results for some switched time-delay systems in the non-strict feedback or nonaffine form [7], [16]. Therefore, it is still challenging to establish a novel recursive design approach for avoiding nonstrict feedback or nonaffine terms in the switching framework.
Moreover, the phenomenon of actuator saturation in practical dynamic systems inevitably exists. It is a potential control challenge for the switched uncertain nonlinear time-delay system and generally severely limits system performance, resulting in undesirable quality or maybe instability. A myriad of results about the actuator saturation-based stability analysis and stabilization for non-switched systems and switched systems have been reported, for instance, [2], [23], [38], [39], [40]. In [2], for nonlinear time-delay high-order systems, the authors investigated the adaptive fuzzy tracking control to solve the problem of full-state constraints and input saturation. For uncertain nonlinear time-delay systems in the non-strict feedback form subjected to input and output constraints, the adaptive fuzzy observer-based controller method was presented in [23]. In [39], for the uncertain nonlinear systems with external disturbance, an auxiliary subsystem was designed to compensate for input saturation with the Nussbaum gain technique. To our best knowledge and in our humble opinion, for switched uncertain nonlinear time-delay systems with the non-strict feedback form, the saturated NN-based switched output-feedback tracking control strategy has not been presented by now. Consequently, some exciting questions naturally arise: Can we rigorously solve the output-feedback tracking control problem with actuator saturation if time delays are considered in all the system states for switched uncertain nonlinear systems? For each subsystem, how do design the appropriate MLKFs and constructively design the suitable adaptive output-feedback controllers under the ADT method?
Motivated by the discussions mentioned above, the main contributions of this paper are summarized below.
- 1.
The MLKFs are designed to overcome the influence of time delays in all the system states. Adaptive NN-based output-feedback tracking control problem is solved for the switched uncertain nonlinear systems in non-strict feedback form under ADT and the backstepping technique.
- 2.
Due to unavailability of state measurements, the switched NNs output feedback controllers cooperated with the switched delay-independent state observer for individual subsystems are designed, compared with the existing controllers for non-switched nonlinear time-delay systems or switched nonlinear time-delay systems in [2], [32]. The proposed control strategy can achieve the tracking control objective for the switched system and greatly ensure that the output tracking error remains within a small neighbourhood around the origin.
- 3.
The Nussbaum gain technique is used to compensate for the nonlinear term arising from actuator saturation in the novel auxiliary switched subsystems, which is flexible in the last backstepping iteration when compared to [2], [38].
For the current state-of-the-art, the distinctive merits of the proposed control strategy are elaborated below. In this paper, we work with the switched uncertain nonlinear time-delay systems with actuator saturation. In contrast, for non-switched or switched time-delay systems, this paper takes the more general feedback framework containing unknown varying time delays in all the system states into account, while time delays have merely relied on the first system state in [9], [13] or the system with the strict feedback form in [29], [38]. The constructive switched-observer-based adaptive NNs controllers for individual subsystems are designed, while none of [7], [25] considered the actuator saturation or the unmeasurable states.
The remainder of the paper is organized as follows. Section 2 introduces the notation used throughout this paper. Section 3 presents problem statement and preliminaries. Section 4 proposes the main results of this paper. Section 5 describes and analyses the effectiveness of the proposed control strategy. Finally, this paper is summarized in Section 6.
Section snippets
Notation
Assumption 1 There exist , such that the continuous nonlinear functions and satisfywhere is the estimation of . Assumption 2 The given desired signal and the derivatives exist and are continuous and bounded, and with known compact set , where is a constant. Assumption 3 Given a constant , the initial conditions are satisfying
Switched Uncertain Nonlinear Time-Delay Systems
Consider the following switched uncertain nonlinear time-delay systems in the non-strict feedback formwhere vector and are the state and the output of the switched system ((1a), (1b), (1c), (1d)), respectively. is a piecewise right continuous function defined as a switching signal, and
Main Results
In this section, a constructive adaptive NNs output-feedback control strategy for the switched system ((1a), (1b), (1c), (1d)) will be proposed by employing the MLKFs method, backstepping, and ADT method. First, a switched delay-independent state observer will be constructed to compensate for the unmeasurable states. Then, the switched adaptive NNs output-feedback controllers for individual subsystems will be designed recursively by the backstepping technique. Finally, boundedness analysis will
Simulation Results
In this section, two examples will be given to illustrate the effectiveness of the proposed control strategy. Example 1 Consider the CSTR system with two different feed streams (see Fig. 2). The temperature control of the CSTR system has been extensively studied in [45], [1], [32], and it is clear to find the physical meaning of the corresponding parameters of the CSTR process and select its appropriate values. Based on the mass and energy balances, the CSTR system at each operating subsystem is described
Conclusion
This paper presented an adaptive NNs output-feedback tracking control strategy for a category of switched uncertain nonlinear time-delay systems with actuator saturation. By introducing the proper MLKFs and the Nussbaum technique, the proposed controller can guarantee SGUUB of the switched uncertain nonlinear time-delay systems under the ADT method, and the output tracking error stays in a small neighbourhood around the origin. Because of boundedness analysis, the simulation results achieve the
CRediT authorship contribution statement
Zhenhua Li: Conceptualization, Methodology, Software, Investigation, Writing - original draft. Gang Cao: Writing - review & editing. Wei Xie: Writing - review & editing, Validation, Supervision. Rui Gao: Writing - review & editing. Weidong Zhang: Supervision, Writing - review & editing.
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 Key R&D Program of Guangdong (Grant No.: 2020B1111010002), the Key R&D Program of Hainan (Grant No.: ZDYF2021GXJS041), the National Natural Science Foundation of China (Grant No.: U2141234), and the Hainan Special PhD Scientific Research Foundation of Sanya Yazhou Bay Science and Technology City (Grant No.: HSPHDSRF-2022–01-005).
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