This study provides an overview and in-depth analysis of recent advances in stability of linear systems with time-varying delays. First, recent developments of a delay convex analysis approach, a reciprocally convex approach and the construction of Lyapunov–Krasovskii functionals are reviewed insightfully. Second, in-depth analysis of the Bessel–Legendre inequality and some affine integral inequalities is made, and recent stability results are also summarised, including stability criteria for three cases of a time-varying delay, where information on the bounds of the time-varying delay and its derivative is totally known, partly known and completely unknown, respectively. Third, a number of stability criteria are developed for the above three cases of the time-varying delay by employing canonical Bessel–Legendre inequalities, together with augmented Lyapunov–Krasovskii functionals. It is shown through numerical examples that these stability criteria outperform some existing results. Finally, several challenging issues are pointed out to direct the near future research.

This study proposes a novel saturated prescribed performance (PP) controller for Euler–Lagrange dynamic systems. The design of bounded-input controller is facilitated by means of introduction of auxiliary dynamics in the closed-loop system. The controller takes advantages of using an adaptive multilayer neural network and a robust term to deal with uncertainties of the system. Through the stability analysis, it is shown that PPwith saturated control command is feasible having certain constraints on initial conditions and external disturbances. The proposed controller is validated through experimental and simulation studies conducted on a lower-limb robotic exoskeleton.

The event-based model predictive control (MPC) problem for discrete-time non-linear control systems with external disturbances is studied. Two triggering conditions are designed based on whether the system state is within the terminal region or not. The first one is triggered outside the terminal region when the difference between the actual system state trajectory and the corresponding optimal state trajectory violates a relative threshold, while the second one is triggered inside the terminal region if the difference between the actual system state trajectory and the predicted nominal state trajectory violates a desired relative threshold. The feasibility analysis and stability analysis are given in detail. Sufficient conditions for feasibility relating to the triggering threshold and disturbance bound are obtained. Sufficient conditions for stability of the event-based control system referring to system parameters, triggering threshold and disturbance bound are given. Besides, the upper bound of the state trajectory is obtained and a robust invariant set that the system state will enter in finite time is given. Finally, simulation examples are given to validate the effectiveness of the proposed method.

This study considers the synchronisation of stochastic-coupled systems with time-varying delays and Lévy noise on networks without strong connectedness (SCSTLNC) through periodically intermittent control. Also, here, internal delays, coupling delay, white noise, and Lévy noise are considered in SCSTLNC. Then, to guarantee synchronisation of SCSTLNC, several useful criteria are obtained by applying some techniques of inequalities, such as a graph-theoretic approach, a hierarchical approach, and the theory of asymptotically autonomous systems. The intensity of control is closely related to the coupling strength and the perturbed intensity of noise. In particular, the synchronisation of stochastic-coupled oscillators with time-varying delays and Lévy noise on networks without strong connectedness as a practical application of the authors' theoretical results are investigated. Finally, a numerical example of oscillator networks is provided to demonstrate the validity and feasibility of their analytical results.

This work develops a novel disturbance observer and integral sliding mode technique based fault tolerant attitude control scheme for spacecraft, which is subject to external disturbance torques and actuator failures. More specifically, a simple and novel finite-time disturbance observer is first designed to reconstruct the synthetic uncertainty deriving from actuator failures and disturbances, by which the synthetic uncertainty is also compensated or restricted. Then, an integral sliding mode based finite-time fault tolerant attitude stabilisation controller incorporating with an adjusting law is investigated to ensure the closed-loop attitude control system converge to the stable region in finite time. And also the finite-time stability of the closed-loop attitude control system driven by the proposed attitude control scheme is analysed and proved utilising Lyapunov methodology. Finally, a simulation example for a rigid spacecraft model is carried out to verify the effectiveness and superiority of the proposed attitude control approach.

This study proposes a trajectory planning approach to achieve minimum energy consumption and zero residual vibration for flexible servomotor systems with state constraints. Differing from existing approaches, the proposed approach firstly constrains residual vibration to zero, and total energy consumption as an objective is minimised under the condition of residual vibration suppression and meanwhile, the state constraints of velocity, acceleration, and jerk are satisfied. By the convexification of the minimum energy consumption objective function and the discretisation of relevant terms, the trajectory generation is successfully formulated into a quadratic programming problem, which is solved by convex optimisation technique. Comparative simulations and experiments are carried out, and the results indicate the effectiveness of the proposed method.

This study deals with the protocol design for achieving robust global consensus tracking of multi-agent systems. Agent dynamics are described as general linear systems with actuator saturation and input additive uncertainties and disturbances. Via developing a scheduled low-and-high gain design technique, the state feedback protocol is proposed, under which global consensus tracking and disturbance rejection for such systems can be achieved under the mild assumptions on agent dynamics and network topology. This result is further extended to the case of the reduced-order observer-based protocol design, with the help of the special coordinate basis approach. Finally, the advantages of the proposed protocols are illustrated by a numerical simulation.

In this study, the problem on the globally almost surely attractive sets is addressed for a class of discrete-time Markov jump systems with stochastic disturbances via impulsive control. Based on the Lyapunov function methods and the Markov's inequality, the authors derive some sufficient conditions guaranteeing the existence of the global almost surely attractive sets of the considered systems. Meanwhile, the estimation of the attractive sets is also given out. It is shown that the unbounded discrete-time Markov jump stochastic system without attractive sets can turn into the bounded one with attractive sets via proper impulsive control strategies. An example is also presented to illustrate the main results.

This study focuses on the stability property of a class of distributed delay systems with constant coefficients. More precisely, the authors will discuss deeper the stability analysis with respect to the delay parameter. The authors' approach will allow to give new insights in solving the so-called complete stability problem. There are three technical issues need to be studied: First, the detection of the critical zero roots (CZRs); second, the analysis of the asymptotic behaviour of such CZRs; third, the asymptotic behaviour analysis of the critical imaginary roots (CIRs) with respect to the infinitely many critical delays. They extended their recently-established frequency-sweeping approach, with which these technical issues can be effectively solved. Based on these results, a procedure was proposed, with which the complete stability analysis of such systems was accomplished systematically. Moreover, the procedure represents a unified approach: Most of the steps required by the complete stability problem may be fulfilled through observing the frequency-sweeping curves. Finally, some examples illustrate the effectiveness and advantages of the approach.

This study focuses on the practical stochastic synchronisation of coupled harmonic oscillators subjected to heterogeneous noises, where the dissipative and restorative couplings are no longer required in the connected network topologies. By employing the variational approach in combination with Lyapunov-like analysis, some simple yet generic practical synchronisation criteria are established in the sense of probability distribution and of mean square for coupled harmonic oscillator with directed network topology. Three main issues on stochastic synchronisation, including practical distribution synchronisation, stochastic distribution synchronisation, and practical mean square synchronisation, as well as their differences and relationships are fully addressed. The developed practical synchronisation criteria are then applied to a representative model of electrical systems which are composed of LC oscillators with linear time-invariant (LTI) resistors and inductors. Finally, numerical simulations are provided to show the effectiveness of the developed methods.

The research in advanced driver assistance systems (ADAS) has progressed to design architectures for driver specific performance. Personalised ADASs in this aspect have been developed with adaptation to driver attributes, state, style, behaviour, skill and so on. For the lane keeping task, the driver driving style while navigating a high/low curvature track plays an important part in the design of a lane keeping assist system. Considering this aspect, a robust co-operative control approach is formulated to design a personalised lane keeping assist with adaptation to driver style. Based on statistical analysis of lateral jerk and steer feel, a fuzzy rule based identification procedure for the classification of the driver style as clam, moderate, aggressive or very aggressive is designed. Using the identified driving style, a modulation function is proposed to adapt the assist torque. The assist torque is generated based on a robust higher order sliding mode approach as a feedback control for the driver-vehicle system. Closed-loop stability of the proposed driver-vehicle design in the presence of disturbances is established. Co-operative control between human driver-autonomous controller for the lane keeping task over the Satory test track with adaption to driver style is then shown for validation of the proposed architecture.

In this study, the authors are interested in the mean square exponential stability of stochastic systems with variable and distributed delays. Different from the traditional methods, based on the well-known Perron–Frobenius theorem and Itô formula, a proof by contradiction to explore some new criteria for the mean square exponential stability of stochastic delay systems is introduced. In particular, the proposed novel stability criteria reduce the traditional restrictions imposed on variable delays and also provide an optimal upper bound for delays. Two examples are given as applications to verify the effectiveness of the obtained results.

This study addresses the distributed finite-time tracking control for multiple uncertain Euler–Lagrange systems with input saturations and error constraints. In order to achieve distributed finite-time tracking control in the model-independent way, neural networks are used to approximate the system unknown terms. Besides, the authors consider that there exists input saturation for each dimension of each follower in the systems. The input saturations are tackled by converting the systems with input saturations to the cases with unknown input gains. Then, based on backstepping technique, the distributed finite-time tracking control law is developed. Tan-type barrier Lyapunov function is utilised to guarantee that the error variables would not exceed the predefined bounds. Lyapunov theory and graph theory demonstrate the finite-time stability of the systems. By choosing a class of two-link manipulators consisting of one leader and four followers, numerical simulations are provided to show the effectiveness of the proposed control algorithm.

This work studies a new method to synthesise a robust observer to estimate the immeasurable state variables for an uncertain large-scale Takagi–Sugeno (T–S) fuzzy system, which has not been considered in the previous studies. This system consists of a set of T–S fuzzy subsystems that are interconnected, but these systems have certain uncertainties. An observer relying on the unknown input method is synthesised such that not only the unknown states are estimated but also the effects of the uncertainties are eliminated completely, even though the upper bounds of the uncertainties are not known. The conditions to design the observer are obtained in the main theorems by employing Lyapunov methodology, S-procedure, and linear matrix inequalities technique. Finally, the illustrative examples are provided to illustrate the advantages and success of the proposed method.

This study considers the design of encrypted control systems to secure data privacy when the control systems operate over a network. In particular, the authors propose to combine Paillier cryptosystem with a quantiser whose sensitivity changes with the evolution of the system. This allows the encrypted control system to balance between the cipher strength and processing time. Such an ability is essential for control systems that are expected to run real time. It also allows the closed-loop system to achieve the asymptotic stability for linear systems. Extensions to event-triggered control and non-linear control systems are also discussed.

In this study, the stability problem of discrete-time switched systems without stable subsystems is considered. Using the k-sample-like method, the authors construct a Lyapunov function whose value at the switching instant is less than the value at the last switching instant when the corresponding dwell time belongs to a special time span. According to whether the dwell time belongs to the time span, switchings are divided into two categories: the switching without divergence time and the switching with divergence time. Based on that, a new less conservative exponential stability theorem is established with the bounded maximum average dwell time. Furthermore, by jointly considering the dynamic characteristics of the subsystems before and after switching instants, they also give the stability result via the dwell time with the floating lower and upper bounds. Finally, some numerical examples are given to illustrate the effectiveness of the theoretical results.