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This paper concerns the observer-based control problem for a class of two-time-scale cyber-physical systems (TTSCPSs) under dual-scale denial-of-service (DoS) attacks. A TTSCPSs model is first proposed with slow and fast transmission channels, because that slow and fast dynamics are operating and communicating with slow and fast sampling rates, respectively. Due to the time scale difference between slow and fast transmission channels, the model of dual-scale DoS attacks is constructed by characterizing the attack frequency and duration on different time scales. A novel $\varepsilon$ -dependent switched observer is formulated, and corresponding security controller is designed to stabilize the TTSCPSs under dual-scale DoS attacks. By constructing an $\varepsilon$ -dependent Lyapunov function, sufficient conditions for the desired observer-based controller are derived such that the closed-loop TTSCPSs can be input-to-state stable. Moreover, ill-conditioned numerical issue caused by the two-time-scale feature can be avoided in the proposed design methods. Finally, examples are given to verify the effectiveness of the proposed algorithm.

This paper investigates the security control problem for a class of two-time-scale cyber-physical systems (TTSCPSs) with multiple transmission channels under the denial-of-service (DoS) attacks. A linear TTSCPSs model is first proposed with slow and fast transmission channels, which correspond to slow and fast physical components in terms of their communicating capacities and sampling rates. The measurement data-packets are transmitted via slow and fast transmission channels which are compromised by asynchronous DoS attacks. A novel composite controller depending on the singular perturbation parameter (SPP) is formulated and corresponding switching laws are designed to achieve certain resilience against DoS attacks. Then, by establishing a SPP-dependent Lyapunov function, sufficient conditions are obtained on the duration and frequency of the DoS attacks, such that, for any SPP less than or equal to a predefined upper bound, the input-to-state stability can be guaranteed for the closed-loop TTSCPSs. Finally, a networked DC motor control system is employed to demonstrate the effectiveness of the proposed security control algorithm.

This paper develops a novel equivalent input disturbance (EID)-based adaptive sliding mode control (SMC) method for singularly perturbed systems (SPSs). Firstly, the block diagonalization approach is introduced to decompose the full-order SPSs exactly, and slow and fast subsystems are obtained by solving the upper and lower triangular matrices individually. Secondly, an EID is constructed to estimate the unknown disturbances with the observer gain and error system convergence analyzed. Then, depending on the decoupled reduced-order system models, a Lyapunov equation-based solution is adopted to construct a composite sliding surface. Finally, combined with the EID estimation, an adaptive SMC law is proposed to compensate the adverse effect of disturbances and the reachability condition is proven. The presented control strategy is free of any priori disturbances information while the satisfactory system performance can be guaranteed. Simulation results on two examples illustrate its superiority over the existing methods.

This paper proposes a composite control approach for a class of semi-coupled two-time-scale systems, whose model parameters are completely unknown. In the framework of adaptive dynamic programming (ADP), a novel model-free off-policy learning algorithm without ill-conditioned numerical issues is derived on the basis of input and state data. The convergence of the developed ADP algorithm is guaranteed under reasonable assumptions. Using singular perturbation theory, we prove that the closed-loop system with the obtained controller in each iteration is asymptotically stable. Simulation results on an industrial process demonstrate the feasibility and effectiveness of our proposed approach.

The problem of event-triggered H ∞ filter design for singularly perturbed systems (SPSs) with external disturbance is investigated in this paper. A scheme consisting of an event-triggered mechanism...

This paper presents a decentralized composite suboptimal control strategy to solve the optimal control problem for a class of two-time-scale networks, which consist of m lower-level subsystems interconnected through an upper-level main system with unknown dynamics. Firstly, the original optimal control problem is reformulated into the optimal control problems of separated subsystems on different time-scales by means of singular perturbation theory. Secondly, a decentralized composite suboptimal control is designed by combing model-based fast controller and data-based slow controller. Thirdly, the asymptotic stability of the closed-loop overall system and the sub-optimality of the proposed scheme are rigorously proved. As a consequence, the ill-conditioned numerical issues and high dimensionality associated with the full-order model are eliminated in controller design. Finally, a numerical simulation is provided to illustrate the effectiveness of the theoretical developments.

This study addresses the problems of passivity analysis and disturbance observer-based adaptive integral sliding mode control (ISMC) for uncertain singularly perturbed systems (SPSs) with input non-linearity. Firstly, the passivity condition enabling system exponentially stable is presented in terms of linear matrix inequalities (LMIs). Secondly, a singular perturbation parameter e-dependent disturbance observer is constructed, and the estimate is incorporated in the design of the adaptive ISMC with the reachability condition guaranteed. Then, a set of LMIs are obtained to determine the controller gain of the ISMC, and an e-bound estimation approach is derived, such that the closed-loop SPSs are passive and exponentially stable. Compared with the existing results, the proposed approaches have less conservatism because of adopting a more general storage function, which leads to a superior transient performance and a bigger e-bound. Finally, the advantages and effectiveness of the proposed methods are demonstrated by three examples.

This paper presents a novel integral sliding mode control (ISMC) for singularly perturbed systems (SPSs) with mismatched disturbances. A singular perturbation parameter $$\varepsilon $$ -dependent disturbance observer is constructed to estimate the disturbances. An ISMC which incorporates the disturbance vector estimate is designed, such that the reachability condition is guaranteed. In order to compensate the effect of mismatched components of the disturbances extracted by the projection matrix theory, $${H_\infty }$$ control theory is employed to determine the gain of ISMC by solving a set of linear matrix inequalities. As a result, the closed-loop SPSs under the ISMC are internally exponentially stable with the $${H_\infty }$$ performance index guaranteed. Finally, the effectiveness and advantages of the proposed method are demonstrated by two examples.

This paper considers the problem of anti-windup (AW) control for nonlinear singularly perturbed switched systems with actuator saturation. An AW controller consisting of a dynamic state fee...

In this paper, a novel adaptive-gain sliding mode observer (SMO) is designed for sensorless control of permanent magnet linear synchronous motors (PMLSMs). By the proposed adaptive-gain algorithm, the gain can dynamically adapt to a changing operation condition of the PMLSMs and reach an appropriate value in finite time. Stability of the SMO is proved by using Lyapunov stability theory. Compared with the traditional SMO, the adaptive-gain SMO can improve the control precision, especially when the operation condition is not invariant. Simulation results are given to show the advantages of the proposed method.

This paper presents a disturbance observer-based integral sliding mode control (ISMC) for singularly perturbed systems (SPSs) with mismatched disturbances. First, a linear state feedback control law for the slow subsystem of the SPS is designed, and the fast subsystem is established. Second, a disturbance observer and an ISMC incorporating the disturbance estimate are constructed. The gain of ISMC is obtained by applying the H∞ control theory, and the resulting ISMC can effectively attenuate the mismatched disturbances. Finally, the proposed method is applied to an electric system, which illustrates the effectiveness and feasibility. The proposed design methods are based on the reduced-order subsystems and thus free of the high dimensionality and possible numerical ill-conditioned problems.

In this note, the infinite horizon optimal output tracking control problem for two-time-scale systems is investigated. The problem is transformed into an optimal regulator problem of the augmented system which is constructed based on the command generator and the original system. The adaptive dynamics programming technique is utilized to learn the optimal solution in real time without relying on the knowledge of system dynamics. By the structured cost function parameter matrix for a full-order model, ill-conditioned numerical issue of two-time-scale systems is overcome. The proposed algorithm has a rigorous convergence proof. Finally, a DC system is given to show the feasibility of the proposed scheme by simulation.

This paper is concerned with the design of fuzzy controller with guaranteed H∞ performance for a class of Takagi-Sugeno (T-S) fuzzy singularly perturbed switched systems. First, by using the average dwell time approach together with the piecewise Lyapunov function technique, a state feedback controller that depends on the singular perturbation parameter ε is developed. This controller is shown to work well for all ε∈(0,ε0]. Then, for sufficiently small ε, an ε-independent controller design method is proposed. Furthermore, under the ε-independent controller, the ε-bound estimation problem of the overall switched closed-loop system is solved. Finally, an inverted pendulum system is used to evaluate the feasibility and effectiveness of the obtained results.

This paper proposes a T -S fuzzy model-based sliding mode control for permanent magnet synchronous motors(PMSM). Firstly, the PMSM is modeled as a T-S fuzzy singularly perturbed system(SPS). Secondly, the parameter perturbations and local torque changes are taken into account in the design of T-S fuzzy model-based sliding mode controller. In order to reduce the chattering problem, the fuzzy control method is used to replace the discontinuous sliding mode controller. Finally, the feasibility and effectiveness of the proposed methods are demonstrated by the simulation results.

Summary This paper investigates the problems of controller design and stability analysis for singularly perturbed switched systems subject to actuator saturation. A set of well-defined sufficient conditions for the existence of state feedback controllers is proposed, under which the closed-loop system is locally asymptotically stable for arbitrary switching law as long as the singular perturbation parameter is sufficiently small. With the obtained controller, the estimation problem of stability bound and basin of attraction of the closed-loop system is reduced to solving a convex optimization problem. A numerical example and a hydraulic servo position system are used to illustrate the obtained results. Copyright © 2016 John Wiley & Sons, Ltd.

This paper considers the problems of $$H_\infty $$H? control and $$\varepsilon $$?-bound estimation of discrete-time singularly perturbed systems. A set of well-defined conditions for the existence of state feedback controllers are proposed, under which the resulting closed-loop system is asymptotically stable while satisfying a prescribed $$H_\infty $$H? norm bound when the singular perturbation parameter $$\varepsilon $$? is lower than a pre-defined upper bound. It is shown that the proposed controller design method is less conservative than the existing ones. Furthermore, a method of estimating the $$\varepsilon $$?-bound is proposed, which leads to less conservative results and requires lower computational burden than the existing methods for a wide class of singularly perturbed systems. Finally, examples are given to show the advantages and effectiveness of the obtained results.

This paper studies the problems of tip position regulation and vibration suppression of flexible manipulators without using the model. Because of the two-timescale characteristics of flexible manipulators, applying the existing model-free control methods may lead to ill-conditioned numerical problems. In this paper, the dynamics of a flexible manipulator is decomposed into two subsystems which are linear and controllable at different timescales by singular perturbation (SP) theory and a model-free composite controller is designed to alleviate the ill-conditioned numerical problems. To do this, a model-free composite control strategy is constructed which facilitates in designing the controller in slow and fast timescales. In the slow timescale, the slow subsystem controller is designed by adaptive dynamic programming (ADP) based on the measurements of the slow inputs and the position, while the vibration in the slow timescale is estimated by the least square method. In the fast timescale, the vibration is reconstructed based on the measurements of vibration and its estimate in the slow timescale, by which the fast controller is designed using ADP. Stability of the closed-loop system is proved by SP theory. Finally, simulations are given to show the feasibility and effectiveness of the proposed methods.

We will consider the problem of fast sampling control for singularly perturbed systems subject to actuator saturation andL2disturbance. A sufficient condition for the existence of a state feedback controller is proposed. Under this controller, the boundedness of the trajectories in the presence ofL2disturbances is guaranteed for any singular perturbation parameter less than or equal to a predefined upper bound. To improve the capacity of disturbance tolerance and disturbance rejection, two convex optimization problems are formulated. Finally, a numerical example is presented to demonstrate the effectiveness of the main results of this paper.

In this paper, the problem of sampled-data observer-based anti-windup (AW) control for singularly perturbed systems with actuator saturation is considered. A sampled-data observer-based AW controller consisting of a sampled-data observer, controller and AW compensator is proposed for the first time. Based on an e-dependent Lyapunov-Krasovskii functional and linear matrix inequalities, the three components of sampled-data observer-based AW controller are designed simultaneously, which can reduce the conservatism. Furthermore, a convex optimization algorithm is formulated to obtain a desired stability bound and enlarge the basin of attraction at the same time. Finally, examples are provided to demonstrate the effectiveness and merits of the obtained results.

The issue of the sampled-data control for singularly perturbed systems (SPSs) with input saturation is considered. Based on the Lyapunov-Krasovskii functional and linear matrix inequality (LMI) approach, the design method of sampled-data controller for SPSs with actuator saturation is proposed. Then, the estimation of stability region is given by solving the convex optimization problem. Finally, numerical examples are provided to demonstrate the merits of the obtained results.

This study considers the control problem for a class of non-linear singularly perturbed systems (SPSs) subject to actuator saturation. A sufficient condition for the existence of state-feedback controllers to achieve a prescribed stability bound is proposed and the corresponding basin of attraction is estimated. Then a convex optimisation problem is formulated, by which an optimal controller can be obtained to achieve a prescribed stability bound and simultaneously maximise the estimate of the basin of attraction of the SPSs for any allowable singular perturbation parameter. Furthermore, a stability condition is established, which improves the existing stability bound analysis methods for a class of non-linear SPSs. Finally, examples are given to show the advantages and effectiveness of the obtained results.

This paper addresses the problems of stability bound analysis and synthesis for singularly perturbed systems with time-varying delay. First, by constructing an appropriate Lyapunov-Krasovskii functional, a sufficient condition is derived for the system to be stable when the singular perturbation parameter is lower than a predefined upper bound which is referred to as the stability bound of the singularly perturbed system. The proposed criterion needs less computational cost than the existing ones. Then, a state feedback controller design method is proposed to achieve a prescribed stability bound, which can be applied to both standard and nonstandard singularly perturbed systems with time-varying delay. Finally, the effectiveness and merits of the proposed approach are shown through numerical examples.

This paper investigates the problem of fuzzy controller design for a class of Takagi-Sugeno (T-S) fuzzy singularly perturbed switched systems. By using the average dwell time approach together with the piecewise Lyapunov function technique, a set of well-conditioned sufficient conditions for the existence of controller is proposed, under which the overall switched closed-loop system is asymptotically stable. A state feedback controller depending on the singular perturbation parameter e, which is shown to work well for all e ∈ (0, e 0 ), where e 0 is the stability bound of singularly perturbed systems, is developed. In addition, when e is sufficiently small, the e-dependent controller can be reduced to an e-independent one. Then, an e-independent state feedback stabilization controller design method is proposed in terms of linear matrix inequalities. Furthermore, under the controller, the stability bound estimation problem of the overall switched closed-loop system is solved. Finally, an inverted pendulum system is used to show the feasibility and effectiveness of the obtained results.

This paper considers observer design problem of singularly perturbed systems (SPSs) with multirate sampled and delayed measurements. The outputs are classified into two sets which are measured at different sampling rates and subject to transmission delays. The error system is modeled as a continuous-time SPS with a slow-varying delay and a fast-varying delay. A new Lyapunov functional taking the delay properties into account is constructed. Based on the Lyapunov–Krasovskii functional, sufficient conditions for stability of the error system are proposed by which an observer design method is proposed. A realistic example is used to illustrate the obtained results.

This paper addresses the observer design problem for a class of nonlinear descriptor systems whose nonlinear terms are slope-restricted. The full-order observer is formulated as a nonlinear descriptor system. A linear matrix inequality (LMI) condition is derived to construct the full-order observer. The existence and uniqueness of the solution to the obtained observer system are guaranteed. Furthermore, under the same LMI condition and a common assumption, a reduced-order observer is designed. Finally, the design methods are reduced to a strict LMI problem and illustrated by a numerical example.

In this paper, we consider the strongly absolute stability problem of Lur'e descriptor systems (LDSs). First, we define a generalized Lur'e Lyapunov function (GLLF) and show that the negative-definite property of the derivative of the GLLF guarantees strongly absolute stability of LDSs. As a result, the existing Popov-type criteria are reduced to sufficient conditions for the existence of the GLLF. Then, we propose a necessary and sufficient condition for the existence of the GLLF to guarantee the strongly absolute stability of LDSs. This criterion is shown to be less conservative than the existing ones. Finally, we discuss the computational issues and present two numerical examples to illustrate the effectiveness of the obtained results. Copyright © 2008 John Wiley & Sons, Ltd.

This paper considers a Lur’e-type descriptor system (LDS) which is a feedback system whose forward path is a single-input and single-output linear time-invariant descriptor system and feedback path is an uncertain nonlinearity with sector constraint. The notion of strongly absolute stability is defined for LDS and some sufficient conditions are presented. If the forward path of LDS is impulsive free, the classical circle criterion (CC) for Lur’e-type standard systems (LSS) is shown to be still valid for LDS. If the forward path of LDS is not impulsive free, a new CC is derived by using a modified Nyquist stability condition. Finally, numerical examples are given to illustrate the effectiveness of the obtained results.

This paper considers the problems of practical stability analysis and synthesis of linear descriptor systems subject to time-varying and norm-bounded exogenous disturbances. A sufficient condition for the systems to be regular, impulsive-free and practically stable is derived. Then the synthesis problem is addressed and a state feedback controller is designed. To deal with the computational issue, the conditions of the main results are converted into linear matrix inequality (LMI) feasibility problems. Furthermore, two optimization algorithms are formulated to improve the system performances. Finally, numerical examples are given to illustrate the obtained results.

This paper considers a class of nonlinear descriptor systems described by a linear time-invariant descriptor system with feedback-connected sector-constrained nonlinearities. First, we discuss the positive realness problem of descriptor systems and present a new version of positive real lemma. Second, we define the notion of strongly absolute stability (SAB) which is equivalent to the linear part is regular and impulsive-free and the overall feedback system is exponential stable and a SAB criteria in frequency domain is derived. Then, we address the problem of designing a state feedback controller such that the closed-loop feedback-connected system is SAB. To achieve this, we give a linear matrix inequality (LMI)-based SAB criteria, and the above problem is converted into an LMI feasibility problem. Finally, some numerical examples are given to illustrate our approach

Lur'e-type descriptor systems (LDS) described by a linear time-invariant descriptor system with feedback-connected nonlinearities are considered. Firstly, the notion of generalized absolute stability (GAS) is defined for LDS and a necessary and sufficient condition (NASC) is derived. It is shown that GAS for LDS implies that the nominal system is admissible and the overall system is globally exponentially stable. Then, a sufficient condition for a generalised absolutely stable LDS to be of index one and solvable is presented. Furthermore, a method to estimate the decay rate of the solutions to the system is obtained. Finally, the synthesis problem is addressed and an approach to designing a state-feedback controller that renders the closed-loop system is generalised absolutely stable with a given decay rate is given. A numerical example illustrates the approach.

In this paper, the absolute stability problem of Lur'e systems with multiple nonlinearities is investigated. Popov-type absolute stability criteria are surveyed and classified by distinguishing the Lur'e Lyapunov function upon which the criteria are based. A modified Lur'e Lyapunov function is presented. Some necessary and sufficient conditions for the existence of the Lyapynov function to guarantee the absolute stability of Lur'e systems are derived. By these conditions, LMI-based stability criteria are presented. The obtained criteria are expected to be less conservative than the existing ones. Finally, numerical examples are given to illustrate the advantages and effectiveness of our results. Copyright © 2006 John Wiley & Sons, Ltd.

Practical stability is of significant practical importance in scientific and engineering problems but less investigated for descriptor systems. This paper develops the concepts of practical stability to descriptor systems and by using Lyapunov functions and comparison principle, presents some techniques to specify admissible control sets, such that the motion of the closed-loop descriptor systems is practically stable and impulsive-free. Finally, as an application, this paper discusses linear time-invariant descriptor systems and derives a sufficient condition. To illustrate our approach, a simple circuit network and a numerical example are analysed.

This paper introduces the concepts of practical stability for descriptor systems with time delays in terms of two measurements. Some results of practical stability parallel to Lyapunov stability theorems are given. Based on Lyapunov functions and the comparison principle, a criterion, by which the problem of a descriptor system with time delay is reduced to that of a standard state-space system without time delays, is derived. Finally, the application of the results obtained is demonstrated through an analysis of a class of linear time-invariant descriptor systems with time delay.

This paper investigates the problem of fuzzy controller design for nonaffine-in-control singularly perturbed switched systems (NCSPSSs). First, the NCSPSS is approximated by Takagi-Sugeno (T-S) models which include not only state but also control variables in the premise part of the rules. Then, a dynamic state feedback controller design method is proposed in terms of linear matrix inequalities. Under the controller, stability bound estimation problem of the closed-loop system is solved. Finally, an example is given to show the feasibility and effectiveness of the obtained methods.

This paper considers the stabilization problem for singularly perturbed systems (SPSs) subject to actuator saturation. A state feedback stabilization controller design method is proposed and a basin of attraction is constructed. The proposed method is expressed in terms of linear matrix inequalities. The numerical problems due to the ill conditioning are avoided by using a non-decomposing strategy. Finally, two examples are given to show the advantages and effectiveness of the the method proposed in this paper.

This paper considers the problem of passivity analysis of singularly perturbed systems with nonlinear uncertainties. By a novel storage function depending on the singular perturbation parameter, a new method for estimating the allowable upper bound of the singular perturbation parameter is proposed, under which the singularly perturbed system is passive when the singular perturbation parameter is lower than this upper bound. The proposed method is shown to be less conservative than the existing method because the adopted storage function is more general. Finally, a RLC circuit is presented to illustrate the advantages and effectiveness of the proposed method.

Input-to-state stability (ISS) has been shown to be a good way to describe the robust stability of nonlinear systems with exogenous disturbances. In this chapter, input-to-state stability of Lur’e singular systems with disturbances is investigated. The notion of input-to-state stability for nonlinear singular systems is defined based on the concept of ISS for standard state-space systems and the characteristics of singular systems. An LMI-based sufficient condition for ISS of Lur’e singular systems is derived by the classical ISS theory and a state feedback controller design method is proposed, such that the closed-loop system is ISS. Furthermore, ISS of Lur’e singular systems whose linear parts of the systems are allowed to be unstable is considered and a stability criterion is proposed.

This paper considers the problems of passivity analysis and synthesis of singularly perturbed systems with nonlinear uncertainties. By a novel storage function depending on the singular perturbation parameterε, a new method is proposed to estimate theε-bound, such that the system is passive when the singular perturbation parameter is lower than theε-bound. Furthermore, a controller design method is proposed to achieve a predefinedε-bound. The proposed results are shown to be less conservative than the existing ones because the adopted storage function is more general. Finally, an RLC circuit is presented to illustrate the advantages and effectiveness of the proposed methods.

Absolute stability of Lur’e system whose forward path is a linear timeinvariant system and the feedback path is a nonlinearity satisfying sector constraints is one of the basic problems in control theory. In this chapter, strongly absolute stability of Lur’e singular systems is studied. Circle criterion and Popov criterion are derived. First, the concept of strongly absolute stability of Lur’e singular systems is defined and the positive realness of singular systems is discussed. Second, single-input-single-output Lur’e singular systems are considered and the graphical representation of circle criterion is given. Third, multiple-input-multiple-output Lur’e singular systems are considered and an LMI-based circle criterion is derived by a generalized Lyaponov function and S-procedure. Then, Popov criterion for standard state-space systems is generalized to singular systems. Finally, we propose a generalized Lur’e Lyaponov function (GLLF), by which a Popov-like criterion is derived.

This paper investigates the problems of stability analysis and stabilization for singular linear systems subject to actuator saturation. A set invariance condition is derived to ensure that the closed-loop system is regionally regular and impulsive-free and an ellipsoid is contractively invariant. Using this set invariance condition, the estimation of the domain of attraction for a given feedback gain can be formulated as an optimization problem. By viewing the feedback gains as extra free parameters, the optimization problem can be used to design the feedback gain resulting in the largest contractively invariant ellipsoid.

This paper is devoted to the investigation of practical stability of a class of uncertain T-S fuzzy systems. The uncertainties satisfy the so-called ”matching conditions” and the bound is known. In this paper, a sufficient condition for practical stability of dynamic systems is first presented. Then a controller design method for uncertain T-S fuzzy systems is derived, which is based on the above sufficient condition and the quadratic stability condition for T-S fuzzy systems. The advantage of our approach is that the controller we present contains a tuning parameter which can improve systems performance and the main computation process can be conveniently performed by LMI toolbox. Finally, an example is given to illustrate the application of the obtained approach.

In this paper, we investigate the problems of local stability analysis and synthesis for T-S fuzzy systems with time-delay. Firstly, the notion of local stability for T-S fuzzy systems with time-delay is introduced. Then, by using non-quadratic Lyapunov functions and LMI technique, a sufficient condition for the open-loop system to be locally stable is derived, by which a nonlinear controller is designed. At last, an example is presented to illustrate our results.