Uniform asymptotic stability of linear time-varying singularly perturbed systems

An upper bound for the singular perturbation parameter is found for the uniform asymptotic stability of singularly perturbed linear time-varying systems.     

Finite-Time sliding mode control for singularly perturbed PDE systems

This article proposes a sliding mode control strategy for hyperbolic PDE systems under the requirement of finite-time boundedness. First, the singular perturbation theory is introduced to model multi-time scales phenomena, and a quantized measurement method is employed to save the communication resources in network. In addition, by considering the effect of the singular perturbation phenomenon in PDE systems, a sliding surface dependent on spatial position and singular perturbation parameter is constructed, then a sliding mode control law is developed to drive state trajectories to the designed sliding surface in finite time. Moreover, a partitioning strategy is introduced to ensure that the system is finite-time bounded in the reaching phase and the sliding motion phase, respectively. Finally, some sufficient conditions are given to ensure that the system is finite-time bounded in both reaching phase and sliding motion phase, and a simulation example of the chemical tubular reactor demonstrates the effectiveness of the proposed method.     

Nonstationary quantized control for discrete-time Markov jump singularly perturbed systems against deception attacks

This paper addresses the nonstationary quantized control problem for the discrete-time Markov jump singularly perturbed systems (MJSPSs) subject to deception attacks (DAs). The control inputs are characterized by randomly occurring DAs and nonstationary quantization simultaneously, where the DAs are depicted by means of a Bernoulli distributed sequence. By applying a multi-layer structure methodology (MLSM), the nonstationary controllers are devised for MJSPSs. Meanwhile, the correlation among system mode, controller mode, and quantizer mode are portrayed via the nonstationary Markov process. Based on a mode-dependent Lyapunov functional, sufficient criteria are established such that the resulting closed-loop system (CLS) is stochastic mean square exponential ultimately bounded (SMSEUB), and the desired controller is designed. Ultimately, two simulation examples are offered to elaborate on the validity and superiority of the proposed theoretical results.     

Exponential stability of a class of nonlinear singularly perturbed systems with delayed impulses

A class of nonlinear singularly perturbed systems with delayed impulses is considered. By delayed impulses we mean that the impulse maps describing the state's jumping at impulsive moments are dependent on delayed state variables. Assuming that each of two lower order subsystems possesses a Lyapunov function, exponential stability criteria for all small enough values of singular perturbation parameter are obtained. It turns out that the achieved exponential stability is robust with respect to small impulse input delays. A stability bound on perturbation parameter is also derived through using those Lyapunov functions. Additionally, for a class of singularly perturbed Lur'e systems with delayed impulses, an LMI-based method to determine stability and an upper bound of the singular perturbation parameter is presented. The results are illustrated by an example for the position control of a dc-motor with unmodelled dynamics.     

Global Mittag-Leffler consensus for fractional singularly perturbed multi-agent systems with discontinuous inherent dynamics via event-triggered control strategy

In this paper, the global Mittag-Leffler consensus tracking issue is considered for fractional singularly perturbed multi-agent systems (FSPMASs) based on event-triggered control strategy, where the inherent dynamic is modeled to be a discontinuous function with nondecreasing property. Firstly, a differential inequality with respect to fractional-order derivative of convex function is developed. As the special cases, the inequalities about fractional-order derivative of three known functions are also addressed. Secondly, a distributed event-triggered control scheme is designed to guarantee that the considered FSPMASs can achieve the global Mittag-Leffler consensus. Moreover, the Mittag-Leffer convergence speed of tracking the leader for followers can be adjusted to any desired values in advance. In addition, under fractional Filippov differential inclusion framework, by applying Lur’e Postnikov-type Lyapunov functional with variable upper limit integral item and Clarke’s non-smooth analysis technique, the global Mittag-Leffler consensus conditions are addressed in terms of matrix inequalities (MIs). Finally, two numerical simulations are provided to illustrate the validity of the proposed design method and theoretical results.     

Trilayer Stackelberg game for nonlinear systems using adaptive dynamic programming

This paper considers a trilayer Stackelberg game problem for nonlinear system with three players. A novel performance function is defined for each player, which depends on the coupling relationships with the other two players. The coupled Hamilton–Jacobi–Bellman (HJB) equations are built from the performance functions, and the optimal control polices of three players are obtained based on the Bellman’s principle of optimality. Because of the nonlinearity and coupling characteristics, a policy iteration (PI) algorithm with a three-layer decision-making framework is developed to online learn the coupled HJB equations. In order to implement the algorithm, we construct a critic-action neural network (NN) structure and design a NN approximation-based iteration algorithm. Finally, a simulation example is presented to verify the effectiveness of the proposed method.     

Stability robustness of perturbed structured systems

In this paper, the robust stability problem of structured linear systems is analyzed. In order to preserve the structure of the initial system, the structure of the admissible perturbations is characterized. Frobenius and infinity matrix norms are used to study the robust stability of the system affected by different admissible perturbations. Upper bounds for the perturbations are obtained, which guarantee that the perturbed system remains stable. Finally, some examples are shown to illustrate the results.     

一类线性切换系统的鲁棒跟踪控制

文章研究了线性切换系统的鲁棒跟踪控制,并提出可解性的充分条件。设计切换控制规则使得切换线性系统满足加权H∞参考模型,并采用平均驻留时间法和Lyapunov函数来处理稳定性分析和控制器设计。通过使用线性矩阵不等式,控制器设计问题可以得到很好的解决。     

Impulsive fixed-time observer for linear time-delay systems

This paper presents a fixed-time observer for a general class of linear time-delay systems. In contrast to many existing observers, which normally estimate system’s trajectory in an asymptotic fashion, the proposed observer estimates system’s state in a prescribed time. The proposed fixed-time observer is realized by updating the observer in an impulsive manner. Simulation results are also presented to illustrate the convergence behavior of the proposed fixed-time observer.     

Improved closed loop identification of transfer function model for unstable systems

The method of identifying first order plus time delay transfer function model proposed for unstable systems by Ananth and Chidambaram [Closed loop identification to transfer function model for unstable systems, J. Franklin Inst. 336 (1999) 1055-1061] is modified to avoid the stability problems [Cheres, Parameter estimation of an unstable system with a PID controller in a closed loop configuration, J. Franklin Inst., 2005, accepted for publication] of the method. Two modifications are proposed. In the first modification of the method, the under-determined algebraic equations problem is converted into an optimization problem for calculation of the three parameters of the first order plus time delay (FOPTD) model. A simple method is given for the initial guess values of the model parameters. In the second approach, from the definition of Laplace transform of the output response, a third equation is formulated. The resulted three equations, in terms of the three parameters of the transfer function model, are then numerically solved. Simulation results are given for the second order plus time delay transfer function considered by Cheres 2005 [Parameter estimation of an unstable system with a PID controller in a closed loop configuration, J. Franklin Inst., 2005, accepted for publication]. The responses of the identified models with the same PID controllers are compared with that of the actual system. PID controllers are designed based on the identified models. The closed loop responses of the controllers on the original system are evaluated and compared. The present methods give better control performances.     

Unknown input observer for linear non-minimum phase systems

An unknown input observer is to estimate the system state of a dynamic system subject to unknown input excitations. In this note, by assuming that at each time instant, the unknown input can be approximated by a polynomial over a local time interval, a finite-time observer is proposed to achieve approximate joint state and input estimation. Both the obtained state and input estimates are moving averages of the present and past output signals. The advantage of the proposed design is that it can be applied to non-minimum phase systems or systems with non-unity relative degree. Notice that most previous unknown input observer designs require the system to be minimum-phase and relative degree one.     

Event-triggered sliding mode control for networked linear systems

This paper investigates the control-based event-triggered sliding mode control for a networked linear system whose feedback information is transmitted over a digital communication network. In this paper, a novel event-triggered mechanism based on control value is proposed. Different from traditional event-triggered mechanisms that are normally based on states, our mechanism pays more attention to the desired control input value of the system. When the deviation between the current control input and the control input being calculated on the basis of the previous system state exceeds a given threshold, an event is triggered. For the sake of reducing the information to be transmitted, a quantization policy is executed and only a few bits are needed to transmit the feedback symbol of each sample. The combination of the control-based event-triggered mechanism and the quantization policy can significantly reduce both the transmission frequency and the number of bits of each feedback packet. For the concerned system, sliding mode control is implemented. The reachability of the sliding mode surface and the robust stability of the system are analyzed by fully taking quantization effects into account. Moreover, the effects of transmission delay of feedback packets on the event-triggering mechanism are considered. Under the proposed mechanism, the lower bound of event intervals is proven to be non-zero, i.e., the Zeno behavior is excluded. Simulations of a mechanical system are done to further verify the superiority of the proposed mechanism.     

On linear copositive Lyapunov functions for switched positive systems

This paper addresses the existence of a common linear copositive Lyapunov function for discrete-time switched positive systems. Our results reveal that the existence of such a function is equivalent to the Schur stability of a kind of special matrices, these matrices consist of column vector of system matrices in an appropriate manner. A simple example is provided to illustrate the implication of our results.     

A delay-dependent stability criterion for linear neutral delay systems

A new stability criterion for linear neutral delay systems is developed in this note. Based on Park's inequality, a new delay-dependent stability criterion is derived. A numerical example is proposed to illustrate the less conservatism of the obtained results.     

Sliding manifold design for linear systems with unmatched disturbances

This paper offers an efficient sliding manifold design method that minimizes the impact of unmatched disturbances onto SM dynamics and system accuracy. System sensitivity upon unmatched constant or slowly varying external disturbance vector is evaluated by the steady-state dependent criterion function. An infinite set of sliding manifolds that minimize the chosen optimization criterion is determined and a way of selecting a manifold out of that set that provides adopted SM dynamics is suggested. The proposed approach has been demonstrated on numerical examples and verified by simulations.     

Advanced stability criteria for linear systems with time-varying delays

This paper investigates a stability problem for linear systems with time-varying delays. By constructing suitable augmented Lyapunov–Krasovskii functionals, improved stability criteria under various conditions of time-varying delays are derived within the framework of linear matrix inequalities (LMIs). Moreover, to reduce the computational burden caused by the non-convex term including h²(t), how to deal with it is applied by estimating it to the convex term including h(t). Finally, three illustrative examples are given to show the effectiveness of the proposed criteria.     

Bounds for characteristic exponents of discrete linear time varying systems

In this paper we introduce certain constants that characterize exponential stability or exponential unstability of discrete linear time varying systems. We also present bounds for Lyapunov and Perron exponents in terms of the introduced constants.     

Near insensitivity of linear feedback systems

In this paper, the notion of near insensitivity with respect to disturbance and parameter variations is introduced as a performance measure for linear feedback systems and conditions for near insensitivity are derived. It is shown that near insensitivity is attainable with high gain feedback provided that certain geometric conditions relating the system matrices and the parameter variation matrices are satisfied.     

Fault detectability analysis for linear systems within a set-based framework

This paper investigates the fault detectability issue for linear systems within a set-based framework. A set-based fault detectability analysis method is developed based on the geometric approach-based fault detection observer and zonotopes. Three concepts of fault detectability are established according to the relationship between the observer output sets in the presence and absence of faults. Necessary and sufficient conditions are derived corresponding to different detectability concepts. It can be found that the obtained conditions are related to system inputs in open-loop systems and system controllers in closed-loop systems. The impact of both system inputs and system controllers on fault detectability is analyzed. A numerical simulation is presented to illustrate the validity of the conclusions.