Robust adaptive stabilization of nonlinear systems with mismatched time delays
Introduction
Time delay is a common phenomenon in control engineering, and it exists widely in biological system, networked control system, process control system, and so on, see, e.g. [1], [2]. Due to the existence of modelling error, measurement error and external disturbance, the significant uncertainties are usually encountered in practical systems, see, e.g. [3], [4]. Specially, [5] studied the asymptotic stability of two kinds of strict-feedback nonlinear delayed systems. Adaptive output-feedback control scheme was developed for linear and nonlinear dynamic systems with time delays within the model reference adaptive control framework in [6], [7]. Adaptive fuzzy control was applied to nonlinear delayed systems in [8], [9]. The dynamic gain based backstepping approach was developed for strict-feedback nonlinear delayed systems, cascade nonlinear delayed systems and high-order nonlinear delayed systems in [10], [11], respectively. By adding a dynamic subsystem, smooth output-feedback controller was designed for multiple delayed uncertain systems in [12]. A novel Lyapunov–Krasovskii functional with two integral-type functions was applied to the adaptive control problem of a class of large-scale delayed systems with dead-zone inputs in [13]. In [14], effective adaptive memoryless controller was designed for uncertain delayed nonlinear systems.
In this paper, we consider the following nonlinear time-delay system: where is the state variable; is the control vector; , are the delayed state perturbations; , , are the time-varying delays satisfying with unknown constants , and , ; , are bounded disturbances; is the initial continuous function; , , , are known and continuous.
Assumption 1 The uncertain functions satisfy: where , , denotes Euclidean norm,
are unknown constant vectors, and , , are known, continuous, nondecreasing and nonnegative functions.
Assumption 2 There exists a function , -class functions , , and -class function , such that the following inequalities hold:
Assumption 3 For , , the functions , , , , satisfy , , , which imply that and are the same order infinitely small with , that is, where are constants.
Assumption 4 If is bounded, the functions , , , are bounded for all .
Instead of dynamic gain approach, we propose the static gain function method to deal with the delayed state perturbations. New control gain function and Lyapunov–Krasovskii functional are proposed. When the delayed state perturbations are counteracted by one term in the derivative of the Lyapunov–Krasovskii functional, another nonnegative term in the derivative of the Lyapunov–Krasovskii functional appears. In view of this, we introduce control gain function and embed it into the Lyapunov–Krasovskii functional. Based on the construction of , the stability of the closed-loop delayed system can be achieved.
Section snippets
Adaptive control design
We first develop a continuous function , . When ; when , where , are design constants, and are defined in Assumption 3. The control gain is defined by with positive design constant .
Assumption 5 The design constants , , satisfy:
Simulation study
A. Example 1
We consider an uncertain nonlinear system: where . Let us choose For simulation, we set the
Conclusion
Under some system assumptions, the control law and adaptive laws with explicit control gain function are constructed. By employing appropriate Lyapunov–Krasovskii functional with explicit control gain function, asymptotic convergence of the closed-loop system states is proved. Simulation results are given to demonstrate the design procedure for the presented memoryless state feedback controller. For further research, it is interesting to extend the proposed control strategy to large-scale
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.
Acknowledgements
This work was supported by the National Natural Science Foundation of China under Grant 61873330, the Taishan Scholarship Project of Shandong Province, China under Grant tsqn20161032 and Central Government Guides Local Funds for Science and Technology Development of Shandong Province, China under Grant YDZX20203700001633.
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