Original articlesA new robust discrete-time sliding mode control design for systems with time-varying delays on state and input and unknown unmatched parameter uncertainties
Introduction
Time-delays frequently occur in various applications covering a broad range of domains such as telecommunication networks [22], [30], teleoperation and missile guidance processes [9], and chemical fields [21]. Although the presence of delays can have, sometimes, stabilizing effects as mentioned in [23], but is often perilous as it may cause undesirable system transient response, dwindle system performances and even lead to instability [6], [20], [28]. Furthermore, the presence of parameter uncertainties and disturbances may induce instability and result in performances degradation too [38]. Therefore, considerable research has been devoted to the stability analysis and the control design of uncertain Time-Delay Systems (tds) [29]. In this context, several control strategies have been proposed to deal with uncertainties and delays. Among these approaches the Sliding Mode Control (smc) has been successfully applied thanks to its prominent features such as, insensitivity to dynamic variations and robustness against parameter uncertainties and external disturbances [5].
Recently, there has been a growing focus on the design of continuous-time smc for linear uncertain systems subject to constant delay which affects only the state [12], [32], or only the input [16]. While, the case of time-varying delay affecting the state has been discussed in [16]. The design of continuous-time smc has been also investigated for linear tds under the assumption of no uncertainty: In fact, in [15], the considered system is subject to known constant delays that affect both the state and the input. We have to mention that in [15], the state delay equals to the input delay. While in [34], the system is subject to time-varying delays affecting the state and the input. In such work, the state delay is different from the input delay.
From the perspective of practical implementation, and seeing the widespread use of dsp chips, the digital controllers are more effective and convenient than continuous-time controllers. Therefore, the design of Discrete-Time Sliding Mode Controller (dt-smc) witnesses a growing interest. Numerous investigations have addressed the design of dt-smc for linear uncertain tds subject to constant delay which affects only the state [19], [31], [35] or only the input [33]. The case of time-varying delay has been studied too: In fact, the considered systems in [14], [17], [18], [36] are subject to only state delay whereas in [3] the system is affected by only input delay. In [4], a discrete-time quasi sliding mode control scheme has been proposed for discrete linear tds with parameter uncertainties, input nonlinearity and process disturbance. The considered delay is time-varying and affects only the state.
However, in practise, the delays occur in both the state and the input. In [24], a dt-smc scheme has been investigated for linear uncertain multi-input tds subject to Unknown But norm Bounded (ubb) uncertainty which affects only the system state. The considered delays are assumed to be known and constant and appear in the state and the input as well. In [25] a discrete-time neuro-sliding mode control algorithm has been developed to achieve robust tracking and model following of discrete linear tds. The considered system is subject to time-varying matched uncertainties, matched external disturbances and known constant delays that affect the state and the input. In [27], a discrete-time adaptive global sliding mode control scheme combined with a state observer has been designed for the robust stabilization of a nonlinear tds. The system is subject to matched uncertainty and known constant delays on the state and the input. Such study has been accomplished under the assumption that the system is affected by input nonlinearity that should satisfy a sector condition.
We have to mention that, the works [24], [25], [27] have focused on the case of known constant delays. However, in practice these delays are known to be time-varying. So far, this aspect is indeed scarcely studied. Therefore, requires an in-depth investigation and more research should be dedicated to the design of dt-smc schemes for systems with time-varying delays on the state and the input. In [1] and [2], using the lmi technique and the Cone Complementarity Linearization (ccl), dt-smc algorithms have been developed for linear tds. The considered delays in both references are assumed to be known, time-varying and bounded. It is worth mentioning that the state delay has a different value from the input delay. Furthermore, the authors in [2] have worked under the assumption that the system is affected by parameter uncertainties. Whereas, the authors in [1] have neglected the presence of the uncertainty.
We have to point out that, except for [2], the design of dt-smc scheme for discrete linear uncertain tds with time-varying delays affecting the state and the input has never been anyway else investigated.
The main contributions are summarized as follows:
- 1.
A novel robust discrete-time sliding mode controller is designed for linear uncertain discrete-time systems with time-varying delays affecting both the state and the input. We note that the proposed control scheme in the current study is different from [2]. Although we recognize that the system features in both works is similar, and even the class of system considered in the present work is a particular case of the one studied in [2], we would like to stress the fact that the methodology adopted in the design of the new control algorithm is different from [2]. First, in our current work, the expression of the sliding surface depends on the delayed state and the delayed input vectors, which is not the case in [2]. Second, the conception of the surface sliding in [2] is conducted through the ccl algorithm whereas in our case, it is realized via the lmi technique. Hence, some drawbacks associated with the use of ccl algorithm are avoided in the current study.
- 2.
Robustness against uncertainties is investigated in this paper by considering unmatched Unknown But norm-Bounded (UBB) uncertainties. These uncertainties appear not only on the block matrix multiplying the state vector but are also present in the block matrix multiplying the delayed state vector.
- 3.
Based on a new Lyapunov functional candidate, the current work provides sufficient conditions, ensuring the asymptotic stability of the closed loop system. With reference to these conditions, and using results developed in [24], a new sliding surface has been designed. In fact, its design is achieved using the Linear Matrix Inequality (lmi) approach combined with the slack variable approach.
- 4.
The effectiveness of the proposed control algorithm is evaluated in an Autonomous Underwater Vehicle (auv). The simulation results are encouraging.
This paper is organized as follows: Section 2 is devoted to the system description and the problem formulation. Section 3 is dedicated to the conception of the dt-smc, this section is split into two parts: The first one focuses on the sliding surface design and the stability analysis, while the second deals with the synthesis of the new control law. Section 4 is reserved to the application of the proposed control strategy to the Autonomous Underwater Vehicle (auv). Finally, conclusion is given in Section 5.
Section snippets
System description and problem formulation
In this paper, we consider a discrete linear uncertain systems with time-varying delays described by: where is the state vector, the control input, , , and are real constant matrices of dimensions , , and respectively, and are the parameter uncertainties on matrices and respectively. is a time-varying delay that affects both the state and the input it is assumed to be bounded as
Design of robust DT-SM controller
In this section, we will design a novel robust Discrete-Time Sliding Mode Control (DT-SMC) algorithm for the system (1), following two steps: The first is devoted to the conception of the novel sliding surface , the second is dedicated to the synthesis of the new control law .
Application to the Autonomous Underwater Vehicle (auv)
To illustrate the effectiveness of our proposed control algorithm, simulation results are performed on the Autonomous Underwater Vehicle (auv) system. The objective of these simulations is to show the ability of the new controller to stabilize the AUV system at a desired position, and to evaluate the performances of the proposed control scheme in terms of precision, convergence and robustness. Such task should be achieved regardlessof the presence of ubb unmatched uncertainties and time-varying
Conclusion
In this paper, a new robust Discrete-Time Sliding Mode Control (dt-smc) algorithm has been investigated for linear uncertain systems with both state and input time-varying delays. The unmatched uncertainties, present into the block matrices multiplying the state and the delayed state vectors, have been assumed to be Unknown But norm-Bounded (ubb). Thanks to the proposition of a new Lyapunov functional candidate, sufficient delay-dependent stability conditions, have been derived with the LMIs
References (38)
- et al.
Robust adaptive sliding mode control for uncertain systems with unknown time-varying delay input
ISA Trans.
(2018) - et al.
Stabilization for state/input delay systems via static and integral output feedback
Automatica
(2010) - et al.
Sliding mode control in the presence of input delay: A singular perturbation approach
Automatica
(2012) - et al.
Robust sliding mode control for uncertain discrete singular systems with time-varying delays and external disturbances
Automatica
(2017) - et al.
Finite-time stabilization of input-delay switched systems
Appl. Math. Comput.
(2020) - et al.
Estimating the unknown time delay in chemical processes
Eng. Appl. Artif. Intell.
(2016) - et al.
Impulse-dependent settling-time for finite time stabilization of uncertain impulsive static neural networks with leakage delay and distributed delays
Math. Comput. Simulation
(2021) - et al.
New stability and stabilization conditions for stochastic neural networks of neutral type with Markovian jumping parameters
J. Franklin Inst. B
(2018) - et al.
Robust adaptive sliding mode control for uncertain discrete-time systems with time delay
J. Franklin Inst. B
(2010) - et al.
Optimal guaranteed cost control of discrete-time uncertain systems with both state and input delays
J. Franklin Inst. B
(2001)
Stochastic stability criterion of neutral-type neural networks with additive time-varying delay and uncertain semi-Markov jump
Neurocomputing
Quasi-sliding mode control for discrete-time uncertain systems with time-varying delay and stochastic disturbance
Internat. J. Control
Applications of Sliding Mode Contro
Advances in Linear Matrix Inequality Methods in Control
Nonlinear RISE-based control of an autonomous underwater vehicle
IEEE Trans. Robot.
Marine Control System-Guidance, Navigation and Control of Ships, Rigs and Underwater Vehicles
Lecture Notes: TTK4190 Guidance and Control of Vehicles
Cited by (9)
Robust stabilization and synchronization in a network of chaotic systems with time-varying delays
2022, Chaos, Solitons and FractalsCitation Excerpt :The control of a coupled differential-difference equation with bounded time delay is shown in [16]. In [17], a robust discrete time sliding mode controller with time-varying inputs in the states is presented in the presence of unknown unmatched parameter uncertainties. In [18], a nonlinear bilateral teleoperation system with time-varying delay is synchronized by robust observer-based adaptive control.
A Robust Adaptive Non-Singular Terminal Sliding Mode Control: Application to an Upper-Limb Exoskeleton with Disturbances and Uncertain Dynamics
2024, Information Technology and ControlRobust control strategy for networked vehicle system with fake data injection attack
2024, Proceedings of the Institution of Mechanical Engineers. Part I: Journal of Systems and Control EngineeringAdaptive Sliding Mode Control for Optimal ℒ <inf>2</inf>-gain Performance of Interconnected Large-scale Cyber Physical Systems Under FDIAs Environments
2023, International Journal of Control, Automation and Systems