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This study addresses the event-triggered fault detection for a class of discrete-time systems with unknown but bounded external disturbance and measurement noise. In order to save limited network bandwidth, an adaptively adjusted event-triggered scheme is designed by considering the probability of system failure and the fault detection progress. Based on the event-triggered system output, an fault detection observer is well designed in finite frequency ranges, such that the generated residual is sensitive to injected faults and robust against disturbance and noise. Moreover, a novel zonotope-based residual evaluation strategy is constructed to reduce conservatism by considering the impacts of external disturbance, measurement noise and measured error caused by event-triggered scheme. Finally, simulation results are provided to demonstrate the effective performance of the proposed event-triggered fault detection strategy.
In this study, the robust stability of continuous-time switched linear systems is investigated under the assumptions that the matrices of the associated linear subsystems are subjected to affine perturbations. The notion of structured stability radius of a switched linear system which is asymptotically exponentially stable w.r.t. arbitrary switchings is introduced. Some lower bounds and upper bounds for estimating this radius are established, by using the system's common quadratic Lyapunov functions and via an approach based on solutions comparison principle. When the nominal switched system is of special structures (for instance when all matrices of subsystems are normal) the obtained bounds yield easily computable formulas for calculating or estimating the system's stability radius. Several examples are provided to illustrate the authors' approach.
This study is concerned with the observer-based reliable control problem for switched systems subject to actuator faults via the parallel triggering mechanism. Exponential observers and error systems are constructed due to the immeasurable system state. Since system modes and sampling controller modes may be mismatched, no more than one switching and multiple switchings are both considered in one triggering dependent sampling interval. Combining the mode-dependent average dwell time technique with the parallel triggering mechanism, sampling controllers are designed to realise the reliable control for switched systems subject to actuator faults. By this mechanism, decay rates of observer states are added into triggering conditions; the execution number of control tasks is reduced with similar system performances in the meantime. At last, the effectiveness of proposed results is illustrated by a simulation example.
This study addresses the asymptotic tracking problem subjected to linear quadratic (LQ) constraints for linear discrete-time systems, where packet dropout occurs in actuating channels. To solve this objective control problem, the controller-coding co-design approach is adopted, i.e. the controller, encoder and decoder are designed for taking full advantage of the network resource collaboratively, thereby achieving better transmission of control signals. A stabilisability condition in the mean square sense that reveals the fundamental limitation among the norm of the plant, data arrival rates and coding matrices is first derived. Then, a solvability condition is conducted to handle the additional stochastic LQ control objective by a modified discrete-time algebraic Riccati equation, and an iterative algorithm is also given for designing the corresponding state feedback gain and coding matrices. Relied on such design, the asymptotic tracking constraint is further fulfilled through solving a Sylvester equation, and the feedforward gain related to tracking is parameterised. Finally, a simulation with the implementation of the design method on two cooperative robots is included to show the effectiveness of the current results.
This study proposes a novel robust predictive current control that is based on a discrete-time disturbance observer for an interior permanent magnet synchronous motor (IPMSM), does not require rotor flux information. To confirm the effects of the current control response on a parameter mismatch, the parameter sensitivity for the current prediction of a conventional deadbeat predictive current control (DPCC) is analysed. With the proposed method, disturbances owing to a parameter mismatch, rotor flux term, and unmodelled dynamics are estimated using a Luenberger observer in the discrete-time domain. The estimated disturbances are compensated with the predicted reference voltage model considering a digital delay. The stability of the proposed disturbance observer owing to a parameter mismatch of the stator resistance and d–q inductance is also analysed. The proposed method is robust against the stator resistance and an inductance variation, and an accurate predicted current control can be obtained without an offline or online estimation of the rotor flux. Compared with the conventional DPCC, the proposed method can eliminate a steady-state current and transient state error caused by disturbances of the system. Experimental results are presented to verify the proposed control scheme even with mismatched parameters of the IPMSM.
This study proposes a model-free distributed output feedback control scheme that achieves synchronisation of the outputs of the heterogeneous follower agents with that of the leader agent in a directed network. A distributed two degree of freedom approach is presented that separates the learning of the optimal output feedback and the feedforward terms of the local control law for each agent. The local feedback parameters are learned using the proposed off-policy Q-learning algorithm, whereas a gradient adaptive law is presented to learn the local feedforward control parameters to achieve asymptotic tracking of each agent. This learning scheme and the resulting distributed control laws neither require access to the local internal state of the agents nor do they need an additional distributed leader state observer. The proposed approach has the advantage over the previous state augmentation approaches as it circumvents the need of introducing a discounting factor in the local performance functions. It is shown that the proposed algorithm converges to the optimal solution of the algebraic Riccati equation and the output regulator equations without explicitly solving them as long as the leader agent is reachable directly or indirectly from all the follower agents. Simulation results validate the proposed scheme.
This study deals with the problem of state bounding for positive singular discrete-time systems with time-varying delay and bounded disturbances. First, the authors derive some new conditions for the existence of componentwise bounds of the state vector of the systems without disturbances. Then, in the case of systems with bounded disturbances, they obtain a sufficient condition for the existence of componentwise ultimate bounds. Since the conditions are given in terms of the spectral radius of the system matrices, they can check them easily and compute directly the smallest componentwise ultimate bound. In addition, the problem of state bounding for positive time-varying singular systems is also investigated in this study. Finally, numerical examples are provided to illustrate the effectiveness and advantages of their results.
A study on the simultaneous state and fault estimation for non-linear discrete-time stochastic systems subjected to unknown disturbances is presented. The augmented system approach, system reformation using the state-dependent coefficient (SDC) factorisation, and unknown input filtering method are integrated to simultaneously estimate the state of the system and actuator and/or sensor faults. To achieve this aim, the non-linear system with faults and unknown disturbances is first transformed into an equivalent augmented system by using delayed measurements and the SDC factorisation. Next, within the SDC factorised linear-like augmented system and inspired by the robust two-stage Kalman filter, a novel multi-step estimator named as the robust simultaneous state and fault estimator is proposed to yield a robust state and fault estimation with a multi-step delay. Moreover, a novel real-time state estimator is designed based on the proposed hybrid fault reconstruction model in order to address the inherent time-delay problem. A comparison of the performance of the proposed filters with those of existing methods from the literature is demonstrated using a non-linear two-link manipulator system with friction forces acting simultaneously at each joint.
In this study, the event-triggered distributed fault detection problem is investigated for a class of discrete-time uncertain systems in the finite frequency domain. A sensor network is utilised to collect the information of interest, and an event-triggered communication scheme is adopted to alleviate the communication burden. For the addressed problem, a distributed fault detection filter is designed based on the measurement information from its neighbouring nodes and itself by the given topology. In addition, a fashionable index, named as performance, is employed in order to simultaneously achieve the residual sensitivity to faults and the robustness against disturbances. By resorting to Euler's formula combined with Lyapunov stability theory, some sufficient conditions are established to satisfy the desired performance over a given finite-frequency domain, and the distributed fault detection filter gains are explicitly characterised by solving a series of linear matrix inequalities. A simulation example is conducted to illustrate the feasibility of the proposed filter design technique.
In this study, the distributed event-triggered control strategy is found to cope with the pinning control problem of stochastic discrete-time complex networks with time-varying delay. Under the devised mechanism, exponential mean-square synchronisation can be implemented in the directed networks. By utilising the Lyapunov functional method and the stochastic analysis, several sufficient conditions are established. Moreover, the issue of average synchronisation is investigated. For the stochastic complex networks without reference state, it is demonstrated that all nodes can converge to a time-varying weighted average exponentially by a suitable distributed event-triggered controller. Finally, some numerical examples are presented to indicate the effectiveness of the devised control strategies.
The containment problem of heterogeneous multi-agent systems with fixed time delay under directed network is considered for both continuous-time and discrete-time multi-agent systems. Leaders and followers have different dynamics, modelled by double-integrator and single-integrator, respectively. It is also assumed that all leaders are stationary. Unlike the current approach in the literature which utilises LMIs to derive sufficient condition for containment reaching, the authors employ Laplace transform/Z-transform and final value theorem to obtain the necessary and sufficient condition which guarantees the containment of follower agents in the presence of communication delays. This approach gives the exact maximum allowed time delay, which is larger than the bound obtained by previous approaches. This study also proposes a noble reduced-order dynamic method, by adding virtual agents, in order to use final value theorem for stability analysis in the presence of heterogeneity in the network. In the end, to illustrate the effectiveness of the theoretical results, simulation examples are provided.
This work strives for the issue of generalised dissipative asynchronous output feedback control for Markov jump repeated scalar non-linear systems with time-varying delay. The objective is to design an asynchronous output feedback controller, which ensures that the closed-loop system is generalised stochastically dissipative. Meanwhile, such an asynchronous controller covers not only the asynchronous controller but also the mode-independent one. By means of a stochastic analysis technique and a modified matrix decoupling method, sufficient conditions are given for deriving the desired controller. At length, an illustrative example is provided to demonstrate the availability of the presented approach.
In this study, the stability problem of continuous-time switched systems composed fully of unstable subsystems is considered. Unlike the hybrid conditions derived in previous literature, this study used a novel time-dependent quadratic Lyapunov function approach and a convex sufficient condition for switched linear systems is proposed in the framework of bounded maximum average dwell time (BMADT). The resulting condition in this study is enforced using the sum of squares programming. Then, the condition for switched linear systems is extended to uncertain systems and the robust stability condition is derived. Moreover, a new stability result of continuous-time switched systems with all subsystems unstable is investigated based on the BMADT conditions. The simulation shows that it is better than the results in previous literature. Two numerical examples are proposed to illustrate the authors' approach.
The authors present a simulation-based bounded-horizon verification framework for hybrid systems with Lipschitz continuity on the continuous dynamics. In this framework, the bounded initial set is covered by a finite set of representative states, whose forward simulations are used to generate an over-approximation of all the reachable states. A novel feature of the proposed approach is that the representative states are generated dynamically, on-the-fly, along with the forward simulations. This key innovation refines the current 'reachability-face' by a new partition only when needed. This approach works for hybrid systems with state-triggered discrete jumps and allows piecewise constant bounded inputs, extending the existing work applied to switched systems with neither state-triggered discrete jumps nor inputs. Additionally, when the continuous dynamics is incremental (input-to-state) stable, the algorithm uses a simple Lipschitz-based discrepancy function to provide a constant error bound of over-approximation. This is of practical significance since a Lipschitz-based discrepancy function is easily computable, while a more precise discrepancy function may not be available. Because of the constant error bound, the number of representative simulations also converges to a constant. A prototype verifier, HS 3 V, has been developed, implementing the proposed algorithms and providing verification results from several benchmarks for performance demonstration.
The present study mainly focuses on designing a stochastic sampled-data controller for chaotic Takagi–Sugeno (T–S) fuzzy systems. Distinct to the existing controller schemes, in this work, the random time delay is introduced into the proposed control scheme that ensures the exponential stabilisation of T–S fuzzy models. In addition, the input delays are assumed to be randomly time-varying, which copes with the traditional uncorrelated Bernoulli distributed sequences. Based on the proposed Lyapunov–Krasovskii functional and using new weighted integral inequalities, the stability and stabilisation conditions are derived and expressed in terms of linear matrix inequalities, which ensure the exponential stability of the states. Finally, in the simulation results, the chaotic nature of two dynamical systems are considered for validation of the derived conditions. From the simulation results, it is concluded that the proposed method can provide better stability performance and less conservative results.
The work is interested in the extended dissipative filtering issue for switched singular discrete-time systems with sensor failures. The persistent dwell-time switching regularity is utilised to describe the successive occurrence of slow switching and fast switching in a switched singular system. A set of variables, which take values in a fixed interval, are utilised for the quantitative analysis of the sensor failures. The main objective is centred on designing a reliable mode-dependent filter which can ensure that the resulting filtering error system is exponentially admissible and simultaneously satisfies the extended dissipative property in the presence of sensor failures. By adopting suitable decoupling methods and introducing reasonable matrix transformation techniques, some relaxed sufficient conditions are derived. The filter gains can be obtained by solving a convex optimisation problem in terms of linear matrix inequalities. Finally, an illustrative example is presented to verify the applicability of the designed filter.
This study addresses the consensus problem for linear multi-agent systems subject to external disturbances under the leaderless framework. A novel distributed dynamic output feedback control protocol is proposed, which utilises not only relative output information of neighbouring agents but also relative state information of neighbouring controllers. Through model transformation, the consensus control problem of multi-agents network is reduced to a set of independent stabilisation subproblems for n-dimensional linear systems. Sufficient analysis conditions are derived using the Lyapunov method. An important contribution of this work lies in that the leaderless output-feedback consensus synthesis conditions are convexified without introducing any conservatism and formulated as linear matrix inequalities, which can be solved efficiently via convex optimisation. This is achieved by using a novel dynamic output-feedback controller structure. A numerical example has been used to demonstrate the advantage of theoretical results.
In this study, the problem of finding state bounds is considered, for the first time, for a class of positive time-delay coupled differential-difference equations (CDDEs) with bounded disturbances. First, the authors present a novel method, which is based on non-negative matrices and optimisation techniques, for computing a like-exponential componentwise upper bound of the state vector of the CDDEs without disturbances. The main idea is to establish bounds of the state vector on finite-time intervals and then, by using the solution comparison method and the linearity of the system, extend to infinite time horizon. Next, by using state transformations, they extend the obtained results to a class of CDDEs with bounded disturbances. As a result, componentwise upper bounds, ultimate bounds and invariant set of the perturbed system are obtained. The feasibility of the obtained results is illustrated through a numerical example.
In this study, the problem of output consensus control for high-order continuous-time linear multi-agent systems with interval time-varying delays is investigated. The observability decomposition technique is employed to design the output consensus protocols which are the collections of delayed output information from neighbouring agents. On the basis of invertible transformations, output consensus for the concerned dynamical agents is transformed into the problem of asymptotical stability analysis for some lower dimensional subsystems. By introducing the prescribed convergence rate scalar into the constructed delay-dependent Lyapunov functionals, the framework for output consensus analysis is theoretically derived. Benefitting from the method of slack matrix variables, sufficient conditions are obtained in terms of linear matrix inequalities to design the protocol gain matrices which can guarantee the property of output consensus. Moreover, the output consensus function determined merely by the initial states of dynamic agents and consensus protocol is presented without the influences of time-varying delays. Numerical examples are exploited to illustrate the effectiveness of the derived results.
Finite-time asynchronous control problem is discussed for positive Markovian jump systems in this study. The non-synchronous behaviours generated between the system modes and controller modes are fully considered. To ensure the closed-loop system positivity and finite-time boundedness with a guaranteed performance level, a sufficient condition on the existence of an asynchronous controller is first established by applying Lyapunov–Krasovskii functional approach and recursive matrix inequality methods. Then, with the aid of matrix conversions, the specific form of controller gain matrices can be constructed by solving linear matrix inequality (LMI) conditions. A numerical example is presented and application is illustrated to validate the proposed results by employing a pest's age-structured population dynamic model.
Inspired by the results obtained in Liu et al. (2008, 2009), this study extends the constrained control problem to singular linear positive discrete-time systems with delay. By using the singular value decomposition approach, delay-dependent sufficient conditions for the regularity, causality, positivity and exponential stabilisation with a given decay rate of the system are established in terms of linear programming problem. A numerical example to demonstrate the effectiveness of the proposed method is given.
Discovering early signatures of voltage instability can enhance situational awareness and facilitate emergency control. Such signatures usually appear as decaying trends in load voltages and increase in reactive power output of generators. Discrete control actions such as frequent switching of load tap changer (LTC) and capacitor banks, that are otherwise unmonitored, can also be crucial indicators of a voltage problem. Hence, the authors propose a unified early warning scheme (EWS) for detecting voltage instability. To quantify and correlate spatial, temporal (spatio-temporal) trends and level changes, the authors use a multivariate, adaptive trend and level (T&L) filter. A unique feature of the proposed scheme is the signature authentication, by verifying simultaneity in T&L changes across multivariate time series. The authors show that, by extrapolating trends, limit violations in bus voltages and generator reactive power output can be estimated well in advance. Further, the level-change detection logic is shown to have the ability to detect frequent LTC-actions directly from the bus voltage time series. While the T&L filtering scheme is validated on field phasor measurement unit (PMU) datasets, EWS and emergency control is demonstrated on the classical 10-bus voltage stability test system.
This study is concerned with dwell time stability and stabilisation problems of switched positive linear systems (SPLSs). The dwell time refers to minimum dwell time and constant dwell time. Several stability conditions for primal and transpose SPLSs with dwell time are presented, and the relation between these conditions is illustrated. Some of these conditions are given in terms of infinite-dimensional linear programming (LP), which cannot be solved directly. Then, by utilising the piecewise linear approach, new alternative convex conditions are formulated in terms of finite-dimensional LP. Compared to the existing literature, results with lower or at least the same conservatism can be obtained under the new conditions for the same discretised order. An algorithm is given to reduce the computational cost. Meanwhile, it is proved that there exists a relation between these convex and non-convex conditions if the discretised order is sufficiently large. By utilising the transpose conditions, alternative convex conditions on stabilisation of SPLSs with dwell time are also presented. The controller gain matrices can be computed by solving a set of LP directly. Finally, the correctness and superiority of the results are verified by numerical examples.
The coherent robust control problem for a class of linear quantum passive systems with model uncertainties is considered in this study, where both the plant and the controller are described by quantum stochastic differential equations (QSDEs). In the framework of language, the model uncertainties are translated into the uncertainties in the Hamiltonian, the coupling operators, and the scattering matrices. Based on the quantum bounded real lemma, the robust controller design of the original uncertain plant is transformed into the controller design of a new scaled system without model uncertainties. Further, the latter is transformed into the solving problem of a couple of Riccati equations and therefore the numerical solutions of the coefficient matrices of QSDEs associated with the controller dynamics are obtained. Finally, numerical simulations on an optical system coupled to three optical channels are performed to verify the effectiveness of the method proposed in this study.
In this study, the authors propose a new algorithm for the flocking problem of multi-agent systems with the dynamic virtual leader under the sampled-data frameworks. A necessary time-varying condition is derived on the sampling period for preserving connectivity of the network and avoiding collision among agents. Then, defining a new energy function, it is proved that using a connectivity-preserving potential function and under the sampling period which satisfies the necessary condition, an asymptotic flocking motion is achievable. Under the proposed sampled-data flocking algorithm, the connectivity preservation of the network and collision avoidance among agents are guaranteed as well as the velocity convergence of all agents to that of the virtual leader is ensured. Moreover, employing the upper bound of the energy function, they specify an upper bound of the sampling period which could satisfy the time-varying necessary condition. Finally, they present a numerical simulation to illustrate the theoretical results.
This study is concerned with the problem of finite-time filter design for a class of Itô stochastic systems with Markovian switching and distributed time-varying delays. Firstly, a partially mode-dependent filter is designed to accommodate to unreliable network transmission. The attention is focused on deriving sufficient conditions for the filtering error system to ensure the finite-time boundedness and to satisfy a prescribed disturbance attenuation. Then based on stochastic functional theory, the existence of filter is presented by solving existing linear matrix inequalities optimisation problems. Furthermore, the result is extended to the case where the mode information is completely transmitted. Finally, a numerical example is provided to show the effectiveness of the proposed results.
This study is concerned with the problem of designing a robust model predictive control (MPC) for a class of uncertain discrete-time Markov jump linear systems. The main contribution is a set of linear matrix inequality (LMI) conditions obtained under new control policies for the unconstrained as well as the constrained MPC when uncertainties are present both in the system's matrices and in the transition probabilities of the modes. For the constrained MPC, hard constraints are considered over the input control and the states and results are extended to the so-called multi-step mode-dependent state-feedback control design. To illustrate the improvements obtained with the new set of LMI conditions, numerical simulations are carried out and compared with a recent reference in the literature.
This study is concerned with the problem of the functional observer-based sliding mode control (SMC) design for parametric uncertain discrete-time delayed stochastic systems includes mismatched parameter uncertainty in the state matrix and in the delayed state matrix. Stability analysis of sliding function is presented in the time delayed stochastic system with a linear matrix inequality approach. Moreover, it is shown that the state trajectories can be driven onto the specified sliding surface despite the presence of state delay, unmatched parameter uncertainty and stochastic noise in the system. The research is motivated by the fact that the system states are not always accessible for the state feedback. Therefore, SMC is estimated using the functional observer technique. To mitigate the side effect of the parameter uncertainty on the estimation error, a sufficient condition of stability is proposed based on Gershgorin disc theorem. The claims made are validated through numerical simulations.
In this study, the authors propose a novel sampled-data fuzzy chaotic synchronisation scheme under the input constraints consideration. The sampled-data chaos synchronisation controller feedbacks the synchronisation error between the state vectors of both the drive chaotic system and the response chaotic system at a constant sampling period. The chaotic synchronisation is achieved by stabilising the synchronisation error dynamics based on the H-infinity criterion. Linear matrix inequality-based sufficient conditions for synchronising two identical chaotic systems are derived based on the newly proposed time-dependent fuzzy Lyapunov–Krasovskii functional. Unlike previous approaches, the modelling error term is fully addressed so as to enhance the synchronisation performance. Finally, the effectiveness of the proposed method is validated through a numerical example.
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.
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.
In this study, a composite robust control approach with non-singular terminal sliding mode control and fixed-time disturbance observer (FxTDO) is proposed for a class of second-order non-linear systems in the presence of matched uncertain disturbances. A bi-limit-weighted homogeneous FxTDO is designed based on the efficient utilisation of power functions obtained by the differentiation error amplification strategy. The FxTDO enjoys a fast convergence rate and simple parameters tuning process. By introducing the disturbance estimation to directly compensate the total disturbance in the composite control law, only a particular small gain-based discontinuous control term is needed to guarantee the existence of sliding mode, and chattering phenomenon can be attenuated. The simulation results revealed the effectiveness of the proposed method.
Sensor fault detection of discrete-time descriptor systems with bounded disturbances is studied. The authors propose a new structure of a fault detection observer by augmenting the output, which can broaden the application scope. A criterion is used to achieve the fault sensitivity and disturbance attenuation ability of the residual simultaneously. The residual generation and time-varying threshold calculation are integrated together based on analysis. Furthermore, sufficient conditions for the performance of the proposed observer are formulated in terms of linear matrix inequalities. Numerical simulations of a direct current motor and a flight vehicle are conducted to verify the applicability of the proposed fault detection observer design method.
In this study, the problem of designing a stochastic optimal controller for sampled-data systems whose sampling interval is subjected to a certain probability distribution is addressed. To design the controller, the Kronecker product operation and the Vandermonde matrix were introduced. A design method of the stochastic optimal controller is proposed. It is shown that the controller guarantee that the closed-loop system has exponentially mean square stability. Finally, the simulation results illustrate the effectiveness and practicability of the proposed method.
In this study, an integrated fault tolerant preview tracking control framework is proposed based on reduced-order simultaneous state and fault estimation, robust preview control and fault signal compensation. In general, this work consists of three key design parts. Firstly, the analysis and synthesis conditions of a novel reduced-order simultaneous state and fault estimator are given based on the equivalent model reconstruction and optimisation of the parametric solution. Note that such a reduced-order design method can avoid traditional output equation reduction requirement, and has a wider application scope. Secondly, the robust preview tracking control policy is constructed by integrating state-feedback, preview action, and integral operation. Relying on augmentation modelling technique, a linear quadratic preview control design problem is transformed into an equivalent augmented state-feedback control design problem. Thirdly, the fault compensation is achieved by using estimated fault to accommodate fault influence. These involved designs are performed at the optimisation level, and thus guarantee the robust tracking performance of closed-loop systems. The effectiveness of the above conclusions is finally verified via two case studies.
This study focuses on the stability analysis and design of output feedback controllers for a non-linear discrete-time switched system with uncertain subsystems. The considered system involves time-varying delay, non-linear terms, and affine parametric uncertainties. The main goals of this study are first to derive sufficient conditions to guarantee the stability of the unforced switched system and then, to construct two stabilising output feedback controllers (static and dynamic) to stabilise the closed-loop system with considering arbitrary switching signals. In this regard, based on switched Lyapunov functions, an appropriate Lyapunov–Krasovskii functional is constructed to establish the sufficient conditions; these conditions depend only on the upper bounds of the time-delay and uncertain parameters. Furthermore, in the proposed method, the admissible bound of time-delay and the maximum non-linearity of subsystems could be calculated. Finally, numerical examples are provided to verify the theoretical results.
This study addresses the problem of robust exponential stability and stabilisation of sampled-data linear parameter varying (LPV) systems with an aperiodic sampling rate. Utilising the input delay approach and taking the distance between real and measured parameters into account, new stability and stabilisation conditions are derived for LPV systems with arbitrary dependence on the parameters. By means of a modified parameter dependent Lyapunov–Krasovskii functional, stability conditions are formulated as a set of parameter-dependent linear matrix inequalities (PLMIs) which are suitable to investigate the effect of the sampling rate on the closed-loop stability. Furthermore, sufficient conditions to guarantee the feasibility of PLMIs over the set of whole parameters are derived that lead to the feasibility of a finite number of linear matrix inequalities. Applying the projection lemma new stability analysis conditions are obtained and shown to be suitable for the stabilisation problem. Under the new stability criteria, an efficient procedure developed for robust sampled-data controller design in the presence of uncertainty on the varying parameters and unknown time varying sampling rate. The proposed method is applied to a sampled-data fuzzy control problem of a non-linear system and multi-rate sampled-data LPV systems. Several examples show the efficiency of the method.
In this study, the robust estimation problem is investigated for discrete-time switched linear systems with unknown persistent excitation. The robust current estimator is proposed to estimate the real state of the switched systems without the stability restriction. The estimator input is constructed based on the current and previous information of a switching signal and output measurement. Owing to the reference system and its estimator, the exponential stability of the estimation error systems is presented under the average dwell time switching and arbitrary switching. The upper bound or the property of attenuation is unnecessary for the unknown input considered here. In addition, neither the conventional decoupling principle nor non-singular transformation is employed. At last, the effectiveness of established results is illustrated by a numerical example and a DC–DC boost converter.
In this study, first, a new definition is proposed for the state of the boost converter. In this approach, the state is defined based on the relative value of two successive samples of the output voltage with respect to the reference voltage. Then a finite-state-machine (FSM) model, which takes into account the possible variations of the input voltage and load resistance, is proposed. This model is discrete-time and represents the evolution of states implicitly based on the events like load and input voltage variation. One advantage of this model is that it is valid both in continuous conduction mode and discontinuous conduction mode. At the next step, a voltage-mode controller, which can regulate the output voltage at the presence of load and input voltage variations, is proposed. Stability of the resulting closed-loop system is studied using the FSM model and the discrete-time Lyapunov method. Simulations and experimental results are presented.
This study investigates an asynchronous switching state feedback control problem for discrete impulsive switched delay systems with average dwell time. A novel time-varying multi-Lyapunov–Krasovskii functional approach is adopted to establish the control synthesis conditions that guarantee the exponential stability of the closed-loop systems. The relation between the upper bound of asynchronous switching delay and the stability conditions is also explicitly given, and the overall conservativeness of the stability results is improved over the existing results. A numerical example is provided to illustrate the effectiveness of our theoretical results, where the comparison with the existing results demonstrates significant improvement using the proposed method.
This study investigates the stabilisation problem for a class of discrete-time two-dimensional (2D) switched systems. Here the 2D systems mean that the state evolves following two independent directions, which are usually represented as horizontal states and vertical states. The considered systems are represented by Roesser model and incorporate both stable and unstable modes. Based on a switching signal regulation approach, which restricted the switching number as well as the running time ratio between the unstable and stable modes, the exponential stability condition is established for the 2D switched systems with synchronous switching. Besides, solving the proposed stabilisation results in linear matrix inequality form, the stabilising controller gains can be obtained. Moreover, considering the time delay in the controllers when the switching occurs, the proposed results are extended to the stabilisation problem for the 2D switched systems under asynchronous switching. A numerical example further demonstrates the validity of the developed results.
This study presents a rigorous mathematical framework for the global asymptotic synchronisation of memristive neural networks comprising multiple time-varying delays (MTVDs) through sampled-data control. First, a novel Lyapunov–Krasovskii functional (LKF) is constructed with some new terms, which can fully capture the information on lower and upper bounds of each MTVD. Second, extended convex combination method is presented, which can successfully solve the combination of MTVDs. Third, based on the LKF and employing the extended convex combination technique, synchronisation criterion is derived. In comparison with existing results, the established criterion is more appropriate since it fully utilises the lower and upper bounds of each MTVD. Finally, simulation results are presented to validate the theoretical models.
This work studies the delayed output feedback (DOF) of discrete-time systems with input and output delays that are arbitrarily large, bounded, and exactly known. The significance of the DOF is that only the present and delayed inputs and outputs are used for feedback. For systems with a single input delay and multiple output delays, the idea of state prediction is utilised to construct the DOF controller. While for systems with multiple input and output delays, the DOF controller is designed by firstly reducing the original system to a delay-free one. It is shown in both cases that the closed-loop systems controlled by DOF are asymptotically stable and act like delay-free systems with the same dimensions as open-loop systems. Compared with the continuous-time case, the DOF controllers for discrete-time systems do not contain distributed delay terms and the implementation problems can be avoided. The DOF is used to stabilise the spacecraft rendezvous system with input and output delays and simulations show the effectiveness of the proposed approach.
A class of discrete-time non-linear positive time-delay switched systems with sector-type non-linearities is studied. Sufficient conditions for the existence of common and switched diagonal Lyapunov–Krasovskii (L–K) functionals for this system class are derived; these are expressed as feasibility conditions for systems of linear algebraic inequalities. Corresponding spectral conditions for the existence of common L–K functionals are also described. Furthermore, it is shown that the proposed approaches can be applied to discrete-time models of digital filters and neural networks. Finally, a numerical example is given to illustrate the effectiveness of theoretical results.
In this study, a geometrically incremental dissipativity is proposed for switched discrete-time non-linear systems using multiple storage functions and multiple supply rates. The uniform incremental stability conditions are derived for geometrically incrementally dissipative discrete-time switched systems by using multiple Lyapunov functions approach and average dwell-time technique. The geometrically incremental dissipativity is preserved for the feedback interconnected switched non-linear systems with a composite switching law, while uniform incremental stability is preserved under some certain conditions. Two numerical examples are given to illustrate the validity of the authors' results.
This study is concerned with the problem of stability analysis of two-dimensional (2D) discrete-time Roesser systems with time-varying delays. First, based on an augmented Lyapunov–Krosovskii functional, a less conservative stability criterion incorporating time-varying terms is established by utilising a general free-matrix-based inequality. Next, in order to eliminate the time-varying terms without introducing redundant constraints, a quasi-convex combination method is proposed. Then, compared with the criteria derived via the other inequalities, the conservatism analysis is given to prove the proposed criterion can lead to a better result theoretically. Finally, a numerical example is presented to illustrate the advantage of the presented method.
This study addresses the problem of distributed sampled-data fuzzy controller design for a class of non-linear distributed parameter systems which are described by first-order hyperbolic partial differential equations (PDEs). To achieve this goal, first, the non-linear system is modelled by a continuous-time Takagi–Sugeno first-order hyperbolic PDE fuzzy model. Subsequently, the authors design a new distributed sampled-data fuzzy controller that generates a zero-order hold sampled-data control signal appropriate for the PDE systems. Then, a new Lyapunov–Krasovskii functional is suggested to provide the stability analysis conditions of the closed-loop control system. Moreover, the stabilisation conditions are obtained and converted to linear matrix inequalities using some new null terms. The proposed technique has removed the structural constraints on the convection and Lyapunov matrices. Finally, the proposed approach is applied on a biological system and a non-isothermal plug flow reactor .
This study presents a linear matrix inequality (LMI) approach to the sampled-data control of Takagi–Sugeno fuzzy systems, based on the intelligent digital redesign (IDR) technique. The objective of the IDR is to design a digital control system whose trajectory closely matches that of a given well-constructed analogue control system by minimising the state-matching error. In this study, state-matching performance is enhanced by using a continuous-time state-matching criterion, which guarantees that the state-matching error is minimised through the entire time interval. Unlike previous studies, mismatched information of membership functions for both analogue and digital control systems is directly manipulated. Moreover, the authors introduce an improved fuzzy Lyapunov functional that consists of both membership functions for analogue and digital control systems, which relaxes the conservativeness of LMI conditions. Finally, two examples demonstrating the effectiveness of the authors' method are provided.
In this study, a predictor-based output feedback control design is proposed for sampled systems with input delay subject to disturbance. An extended state observer (ESO) is first constructed to simultaneously estimate the system state and disturbance based on only the output measurement. Then a filtered predictor is constructed by using the estimated state and disturbance to compensate the input delay so as to improve the disturbance rejection performance. For the presence of a constant or asymptotically stable disturbance, a robust H infinity feedback control design based on state prediction and disturbance estimation is proposed to realise no steady-state output error, despite mismatched disturbance. When there exists a time-varying disturbance with deterministic dynamics, such as a sinusoidal type, the proposed predictor can achieve a small prediction error by properly tuning the filter, such that the existing observer-based feedback control methods for delay-free systems can be directly adopted to stabilise such a system with input delay. Moreover, the output error bounds in the presence of time-varying disturbance are quantitatively analysed for using the ESO-based predictor feedback and anti-disturbance predictor feedback, respectively. Two illustrative examples are given to demonstrate the effectiveness and merit of the proposed method.
This study addresses the control synthesis of positive switched systems in both continuous- and discrete-time contexts. First, a novel design approach to the state-feedback controller of continuous-time positive switched systems without time delay is presented by virtue of multiple linear copositive Lyapunov functions associated with linear programming. Then, the design approach is extended to discrete-time positive switched systems. It is shown that the presented approach is less conservative than existing ones through comparisons. Furthermore, the approach is developed to the stabilisation of positive switched systems with time delay. Some discussions on the presented approach are provided to show potential applications in corresponding control issues of positive switched systems. Finally, two examples are given to illustrate the theoretical findings.
Passivity theory and the positive real lemma have been recognised as two of the cornerstones of modern systems and control theory. As digital control is pervasive in virtually all control applications, developing a general theory on the discrete-time positive real lemma appears to be an important issue. While for minimal realisations the relations between passivity, positive-realness and existence of solutions of the positive real lemma equations is very well understood, it seems fair to say that this is not the case in the discrete-time case, especially when the realisation is non-minimal and no conditions are assumed on left- and/or right-invertibility of the transfer function. The purpose of this study is to present a necessary and sufficient condition for existence of solutions of the positive real equations under the only assumption that the state matrix A is asymptotically stable.
In this study, the exponential stability problem of impulsive system is investigated via sampled-data control in the presence of random time-varying delays and non-linear perturbations. In particular, the time delays are considered to be randomly time varying and they obey the Bernoulli distributions. The authors' main attention is focused on the design of a sampled-data controller to ensure an exponential stability for the closed-loop system. By extending the first- and second-order reciprocal convex approach, an efficient method called third-order reciprocal convex technique is used to manipulate the main results. Through the construction of a suitable Lyapunov–Krasovskii functional combined with input delay approach and Briat lemma, several delay-dependent sufficient conditions for the concerned system are derived in the form of linear matrix inequalities which can be readily solved by utilising the valid software packages. Some numerical examples are given to illustrate the effectiveness of the developed control technique.
This study develops a novel event-triggered control design methodology for discrete-time dynamic systems characterised by linear parameter-varying (LPV) models. To this end, an event-triggering mechanism is first introduced for LPV systems aiming at reducing the data transmission of states, scheduling variables and controller outputs, i.e. at both the sensor and the controller nodes. Then, an event-based LPV state feedback controller is proposed to achieve a reference tracking objective. Sufficient conditions for simultaneous design of the controller parameters and event-triggering conditions are provided in terms of linear matrix inequality conditions to guarantee the stability of the closed-loop system and to track a desired step reference signal. The experimental results are finally presented to demonstrate and validate the properties and performance of the proposed control design approach on a laboratory tank system.
This study is concerned about the quantitative exponential stability (QES) and stabilisation of discrete-time Markov jump systems with multiplicative noises. First, the defects of exponential stability in practical applications are analysed. Based on this analysis, a concept of the QES is given, and two stability criteria are derived. By utilising an auxiliary definition of general finite-time stability (GFTS), the relations among QES, GFTS and finite-time stability are established. Moreover, the quantitative exponential stabilisation is studied, and state feedback controller and the observer-based controller are designed. Subsequently, the relation between the states' upper bound and states' decay rate of considered systems is quantitatively shown by a searching method. Finally, an example is used to illustrate the effectiveness of the authors' obtained results.
A direct adaptive tuning strategy is proposed for model predictive controllers. Parameter tuning is essential for a satisfactory control performance. Various tuning methods are proposed in the literature which can be categorised as heuristic, numerical and analytical methods. The proposed tuning methodology is based on an analytical model predictive control tuning approach for plants described by first-order plus dead time models. For a fixed tuning scheme, the tuning performance deteriorates in dealing with unknown or time varying plants. To overcome this problem, an adaptive tuning strategy is utilised. It is suggested to employ a discrete-time model reference adaptive control with recursive least squares estimations for controller tuning. The proposed method is also extended to multivariable systems. The stability and convergence of the proposed strategy is proved using the Lyapunov approach. Finally, simulation and experimental studies are used to show the effectiveness of the proposed methodology.
This study deals with switched non-linear systems subject to sampled-data control. The states and switching signals of controllers in this study are both considered to be sampled. The corresponding closed-loop systems are modelled as asynchronously switched T–S fuzzy systems with time-varying delay. Based on asynchronous switching approach and time-varying delay approach, the state feedback controllers are designed in terms of a strict linear matrix inequality. Moreover, the switching strategy is studied to guarantee the exponential stability of the systems. Finally, a numerical example is given to illustrate the effectiveness of the proposed method.
In this study, a novel event-based model-free adaptive control (MFAC) algorithm for discrete-time non-linear systems is presented. Different from the traditional MFAC scheme which calculates the control signal at fixed sampling instants, an event-based sampling scheme is given to calculate the new control signal only when the input/output (I/O) data sufficiently changes. The event-triggered MFAC can obviously reduce the computational load and network communication. The closed-loop system is proven to be ultimately bounded by using the Lyapunov technique. Finally, the simulation examples indicate the effectiveness and applicability of the proposed event-trigger model-free adaptive control algorithm.
This study will investigate the distributed linear quadratic (LQ) control problem for discrete identical uncoupled multi-agent systems with a global performance index coupling the behaviour of the multiple agents. An existence condition to the optimal distributed LQ controller is given first. In general, such condition can be checked by solving a discrete algebraic Riccati equation through a numerical method. When the condition fails to hold, a suboptimal distributed controller design method is proposed for a class of LQ performance. The solution can be obtained by solving two local algebraic Riccati equations whose dimension is the same as a single agent. The stability condition is given in terms of the spectrum of a matrix representing the desired sparsity pattern of the distributed controller. Comparing to the centralised control, the computation and communication complexity is much lesser. Finally, the suboptimality is parameterised, and can be measured by solving a Lyapunov equation.
This study investigates the cooperative containment control problem for heterogeneous discrete-time linear multi-agent systems. The structures of the systems are uncertain, and there exists more than one leader in the authors' systems. For these leaders, it is unnecessary to establish links exchanging information between them. Instead, the authors assume that at least one leader has directed paths to all the follower nodes. A distributed discrete-time compensator is presented to estimate the convex hull information of the leaders. Then, based on the estimation of the convex hull information of the reference outputs, they design a novel distributed internal model compensator to tackle the uncertain parts of the dynamics. Finally, a distributed dynamic output feedback approach is utilised to study the distributed systems with external disturbance under the directed communication topology. A numerical example is provided to illustrate the validity of the theoretical results.
The authors are concerned with the stability of hybrid stochastic differential equations by feedback controls based on discrete-time state observations. Under some reasonable conditions, they establish an upper bound on the duration between two consecutive state observations. Moreover, we can design the discrete-time state feedback control to stabilise the given hybrid stochastic differential equations in the sense of pth moment exponential stability by developing a new theory. In comparison to the results given in the previous literature, this study has two new characteristics: (i) the stability criterion concerns pth moment exponential stability, which is different from the existing works; (ii) discrete-time state observations depend on time delays.
An implicit iterative algorithm is established for solving a class of Lyapunov matrix equations appearing in the discrete-time stochastic systems with multiplicative noise. This algorithm contains a tuning parameter which can be appropriately chosen such that it has better convergence performance. Some convergence conditions have been derived for the proposed iterative algorithm, and for a special case an approach is given to choose the optimal tuning parameter such that the algorithm has the best convergence performance. In addition, the properties of boundedness and monotonicity of the proposed algorithm are also investigated when the corresponding stochastic system is asymptotically mean-square stable.
In this study, an sampled-data fuzzy control problem is considered for non-linear parabolic partial differential equation (PPDE) systems with control inputs missing. It is assumed that the actuator to the plant is set to zero if the control inputs missing which occurs in a random way satisfies a Bernoulli distribution. With the aid of the modal decomposition technique, a non-linear ordinary differential equation (ODE) system is initially obtained to represent the dominant dynamics of the PPDE system. Subsequently, a Takagi–Sugeno fuzzy model is utilised to accurately describe the resulting non-linear ODE system. Then, using a novel time-dependent functional and the stochastic analysis technique, an sampled-data fuzzy controller with the stochastic missing data via linear matrix inequalities is proposed to stabilise exponentially the non-linear PPDE system in the mean square sense and achieve an performance for the derived non-linear ODE system. Finally, an example on a temperature cooling fin is given to verify the proposed design strategy.
This study is concerned with the dwell-time stability for impulsive systems under periodic or aperiodic impulses. A Lyapunov-like functional approach is established to study the stability for impulsive systems. The Lyapunov-like functional is time-varying and decreasing but is not imposed definite positive nor continuous. A specific Lyapunov-like functional is constructed by introducing the integral of state together with the cross-terms of the integral and impulsive states. A tight bounding is obtained for the derivative of this functional with the help of the improved Jensen inequality reported recently and the integral equation of the impulsive system. On the basis of the Lyapunov-like functional approach, new dwell-time-dependent stability results with ranged dwell-time, maximal dwell-time and minimal dwell-time are derived for periodic or aperiodic impulsive systems. The stability results have less conservatism than some existing ones, which are illustrated by numerical examples.
The problem of distributed filtering for discrete-time periodic non-linear systems over sensor networks in the presence of jumping uncertainties and unreliable channels is investigated in this study. A random variable which is subject to a Markov chain is introduced to govern the jumping uncertainties. Considering the unreliable channels between the system and filters, a discrete-time fading model is introduced to describe this channel fading phenomenon with the discrete probability distribution of each channel coefficients known a priori. On the basis of a fixed network topology, by establishing the periodic Lyapunov function, sufficient conditions on the existence of distributed filter are proposed to ensure the stability and dissipativity of the filtering error system. The distributed filter for discrete-time periodic non-linear systems is designed via solving a set of linear matrix inequalities. The merits and validity of the proposed design techniques are verified by means of a numerical simulation example.
In this study, the finite-time control problem is studied for discrete time-varying systems with randomly occurring non-linearity and missing measurements. The randomly occurring non-linearity is modelled according to a Bernoulli distributed white sequence with a known conditional probability. The missing measurements phenomenon is assumed to occur in a random way and the missing probabilities are time-varying with known upper and lower bounds. Two sufficient conditions are established for the existence of the state feedback and output feedback controllers, which guarantee the finite-time stochastic stability of the closed-loop systems. The recursive linear matrix inequality approach is employed to design the desired controller gains. A numerical example is provided to illustrate the effectiveness of the obtained results.
This study is concerned with the iterative learning controller design for a class of non-linear switched discrete-time repetitive systems. First, the definitions of exponential stability and average dwell time in iterative learning control (ILC) systems are introduced, and sufficient conditions for exponential stability of non-linear switched discrete-time ILC systems are established by using the methods of common two-dimensional (2D) Lyapunov function and multiple 2D Lyapunov functions, respectively. Then, the proposed results are used to design the iterative learning controller. Finally, an example is given to illustrate the effectiveness of the proposed results.
The robust guaranteed cost consensus problem of high-order discrete-time linear multi-agent systems (MASs) with parameter uncertainties and time-varying delays is studied, and a linear consensus protocol of it is designed. Norm-bounded uncertainties and polytopic uncertainties are considered. First, the idea of robust guaranteed cost control is introduced into consensus problems for the MASs, where a cost function is defined based on state errors among neighbouring agents and control inputs of all the agents. Second, by constructing suitable Lyapunov functions and using the stability theory of discrete-time linear systems, two sufficient linear matrix inequality conditions are derived to insure that high-order discrete-time linear MASs with the two types of parameter uncertainties and time-varying delays reach robust guaranteed cost consensus. At the same time, two upper bounds of the guaranteed cost function are also given. Third, convergence results are given as final consensus values of the MASs with parameter uncertainties and time-varying delays. Finally, two numerical comparisons are given to illustrate the correctness and availability of the theoretical results.
Nanopositioning systems with piezoelectric actuation have found extensively applications in micro-/nanomanipulation domain. This paper reports on the design, realisation and testing of a new output-based discrete-time terminal sliding mode control (DTSMC) scheme, which is applied to achieve precision motion control of a piezoelectric nanopositioning system by overcoming the unmolded non-linearity and disturbance. The controller is easy to implement because it does not require a hysteresis model and a state observer. It is able to deliver a precise tracking without chattering phenomenon. The stability of the control system is proved and the effectiveness of the control scheme is demonstrated through experimental studies. Results show that the non-linear DTSMC robust control is superior to the conventional linear discrete-time sliding-mode control in motion tracking application. The DTSMC enables the generation of a larger bandwidth than the conventional approaches. Moreover, the DTSMC endows the system a well robustness feature in the presence of model uncertainty and external disturbances.
This study considers the output consensus problem of high-order leader-following multi-agent systems with unknown non-linear dynamics, in which the delayed and sampled outputs of the system are the only available data. The unknown non-linear dynamics are assumed to satisfy the Lipschitz condition and the interconnected topologies are assumed to be undirected and connected. A distributed observer-based output feedback controller is proposed for the system to reach output consensus. Both of the bounds of the allowable delay and sampling period are also obtained. Stability analysis shows that the considered systems are globally exponentially stable under the output feedback controller. Finally, a simulation example is given to validate our theoretical results.
In this study, the authors aim to study explicit iterative algorithms for solving coupled discrete-time Lyapunov matrix equations. First, an explicit iterative algorithm based on fixed point theory of dynamic equations is presented via adding a tuning parameter. Second, a necessary and sufficient condition is provided for the convergence of the proposed algorithm. Moreover, the optimal value of the tuning parameter is derived for the fastest convergence of the algorithm. Third, by using the latest updated information, a modified version of the presented explicit iterative algorithm is also established with a necessary and sufficient condition being provided to guarantee the convergence of the modified algorithm. Finally, a numerical example is given to demonstrate the effectiveness of the proposed algorithms.
This study proposes a sampled-data design method for robust control of open two-level quantum systems with operator errors. The required control performance is characterised using the concept of a sliding mode domain related to fidelity, coherence or purity. The authors have designed a control law offline and then utilise it online for a two-level system subject to decoherence with operator errors in the system model. They analyse three cases of approximate amplitude damping decoherence, approximate phase damping decoherence and approximate depolarising decoherence. They design specific sampling periods for these cases that can guarantee the required control performance.
The bounded-input bounded-output stability of fractional delay systems of neutral type is investigated in this study. The proposed method calculates the number of unstable poles of the system for each delay value. All time-delay intervals in time-delay space where system is stable are precisely determined. The stability of systems with not only indefinitely large values of time-delay, but also with multiple poles on the imaginary axis has been studied. The proposed method is employed for investigating the stability of a fractional delay system when its fractional orders approach to the integer numbers or zero. It is proved that simple case of neutral types and time-delay systems have the same conditions for the independent stability. Two numerical examples are provided to illustrate the proficiencies of the proposed method.
In this study, the authors study the input delay compensation problem for discrete-time linear systems with both state and input delays. Under the assumption that the original time-delay system without input delay can be stabilised by state feedback, a nested predictor feedback controller is established to predict the future states such that the arbitrarily large yet exactly known input delay in the original system is completely compensated. Consequently, it is shown that the closed-loop system consisting of the original time-delay system and the nested prediction feedback controller is asymptotically stable. Under an additional assumption, an explicit nested predictor feedback controller without involving any nested summations is also established. Finally, two numerical examples are carried out to illustrate the obtained theoretical results.
In this study, two novel iterative algorithms are presented to solve the Lyapunov matrix equations appearing in discrete-time periodic linear systems. In both algorithms, a weighted combination of the estimation in the last and the current steps is used to update the estimation of the unknown matrices. It is shown that the sequences generated by the proposed algorithms with zero initial conditions monotonically converge to the unique positive definite solution of the periodic Lyapunov matrix equation if the associated system is asymptotically stable. Finally, a numerical example is used to illustrate the effectiveness of the proposed algorithms.
This study considers the problem of stability analysis of discrete-time two-dimensional (2D) Roesser systems with interval time-varying delays. New 2D finite-sum inequalities, which provide a tighter lower bound than the existing ones based on 2D Jensen-type inequalities, are first developed. Based on an improved Lyapunov–Krasovskii functional, the newly derived inequalities are then utilised to establish delay-range-dependent linear matrix inequality-based stability conditions for a class of discrete time-delay 2D systems. The effectiveness of the obtained results is demonstrated by numerical examples.
In this study, finite-time synchronisation between two chaotic systems with discrete and distributed delays is investigated by using periodically intermittent memory feedback control. Based on finite-time stability theory, some novel and effective synchronisation criteria of intermittent control are derived by means of linear matrix inequalities and differential inequality techniques. Furthermore, a necessary condition of finite-time synchronisation of intermittent control is given for neural networks with discrete and distributed delays. A numerical example on two chaotic neural networks shows the effectiveness and correctness of the derived theoretical results. In addition, a secure communication synchronisation problem is presented to demonstrate practical effectiveness of the proposed method.
This study is devoted to the problem of designing a robust output-feedback discrete-time sliding mode control (ODSMC) for the networked systems involving both measuring and actuating data packet losses. Packet losses in the networked control systems (NCSs) have been modelled by utilising the probability and the characteristics of the sources and the destinations. Here, the well-known Bernoulli random binary distribution is used to model consecutive packet losses in the NCSs. In this study, first, a robust observer-based discrete-time sliding mode control is proposed for the NCSs including random packet losses. The packet losses occur in the channels from the sensors to the controller and the channels from the controller to the actuators. Then, using the notion of exponential mean square stability, the boundedness of the obtained closed-loop system is analysed with a linear matrix inequality approach. Our proposed robust ODSMC can be applied to unstable NCSs, and there is no need to stabilise the underlying system in advance. Illustrative examples are presented to show the effectiveness of the proposed approach.
This study addresses the estimation problem for discrete-time measurement-delay systems with multiplicative noise and Markov jump. First, the state equation is converted into two equations according to the geometric arguments and flexible transformations. Then, based on the reorganised innovation approach, the finite-horizon linear minimum mean-square-error estimator is derived in terms of two Riccati difference equations and two Lyapunov difference equations. Finally, under the assumptions of mean square stability of the system and ergodicity of the associated Markov chain, a sufficient condition for the existence of the infinite-horizon estimator is presented. We provide a numerical example to manifest the efficiency of the proposed approach.
In this study, the problem of controlling integrating and unstable systems with long time delay is analysed in the discrete-time domain for digital implementation. Based on a generalised predictor-based control structure, where the plant time delay can be taken out of the control loop for the nominal plant, an analytical controller design is proposed in terms of the delay-free part of the nominal plant model. Correspondingly, further improved control performance is obtained compared with recently developed predictor-based control methods relying on numerical computation for controller parameterisation. The load disturbance rejection controller is derived by proposing the desired closed-loop transfer function, and another one for set-point tracking is designed in terms of the H2 optimal control performance specification. Both controllers can be tuned relatively independently in a monotonic manner, with a single adjustable parameter in each controller. By establishing the sufficient and necessary condition for holding robust stability of the closed-loop control system, tuning constraints are derived together with numerical tuning guidelines for the disturbance rejection controller. Illustrative examples taken from the literature along with temperature control tests for a crystallisation reactor are used to demonstrate the effectiveness and merit of the proposed method.
In this study, linear quadratic Gaussian (LQG) control is used for designing a wide area damping controller (WADC) in power system. The designed continuous and discrete mode LQG-based WADC is robust enough to deal with the problem of imperfect communication medium. The imperfect communication medium includes process noise, measurement noise, packet drop-out and signal latency in the remote control signal. The appropriate control input and the controller location are selected using the modal residue method. The imperfections in the communication network are modelled using Pade approximations for time delay and by inserting Gaussian random noise. Using two typical test power systems, the performance of robust LQG-based WADC as a power system stability agent is assessed in the frequency domain and via time-domain simulation of the non-linear model on MATLAB/Simulink and DIgSILENT software platform. Finally, a real-time implementation of the designed WADC is demonstrated on the Opal RT-Lab real-time digital simulator.
This study proposes a new summation inequality, called the free-matrix-based summation inequality, which further extends certain existing summation inequalities in the literature. Less conservative stability criteria are proposed for discrete-time systems with time-varying delays. Numerical examples are provided to demonstrate the improvement of the proposed approach.
Bell states play a central role in the field of quantum information. For two-qubit systems, this study proposes a new non-smooth control strategy within the theoretical framework of quantum continuous measurement, and achieves global feedback stabilisation in any target Bell state. For a target Bell state, the authors select one measured observable and two control channels, where the control law associated with one channel is kept constant. They design a non-smooth control law for the other control channel, and prove the stability of the whole closed-loop system. Some simulation experiments show that the control strategy in this paper has a good control effect.
In this study, a novel intelligent digital redesign (IDR) technique is proposed for the sampled-data fuzzy controller of the non-linear system with packet losses. For the IDR development, the non-linear system is represented in the Takagi–Sugeno fuzzy model and the packet loss is assumed to be an independent, identically distributed Bernoulli random process. After introducing the traditional IDR technique, the closed-loop system with state-matching error and the state-matching error cost function are achieved for the development of the novel IDR technique. In the discrete-time Lyapunov sense, the exponential mean-square stability is guaranteed while minimising the state-matching error, and its sufficient condition is derived into linear matrix inequalities. Two examples are provided to verify the effectiveness of the proposed technique by comparison.
The design of sliding switching surfaces consist in finding a manifold where the controlled system dynamics are asymptotically stable. This problem has been studied for a long time, during which a wide range of designing approaches has been proposed, which can be classified into two research methodologies; the first one corresponding to eigenvalues designing approaches and the second one is based on optimal approaches. In addition, some research approaches only can be applied to specific type of systems, e.g. single-input single-output systems or continuous time systems. Finally, the performance or the level of the achieved accuracy among the different approaches varies significantly. The proposed approach facilitates the task of designing sliding surfaces applied to multi-input multi-output (MIMO) linear systems. The presented methodology introduces a switching surface for MIMO linear systems through a simple equation that allows both eigenvalues placement design or optimal methods without any coordinate system representation transformation. In addition, a design parameter is also included for the adjustment of the non-ideal dynamics. Moreover, this approach guarantees high accuracy of the system dynamics, either for arbitrary eigenvalues assignment or for optimal switching surface definitions. In addition, these benefits are attained using a single switching surface equation. An illustrative example is included to show the advantages of the proposed design methodology.
In this study, an intelligent digital redesign (IDR) technique is proposed for an observer-based sampled-data fuzzy controller of non-linear systems. By using a Takagi–Sugeno fuzzy model, the pre-designed analog and sampled-data fuzzy controllers are supposed, and these discretised closed-loop systems are obtained, respectively. Based on the IDR problem, the authors guarantee both stability and state-matching conditions. Unlike the previous techniques, the proposed IDR not only improves the state-matching performance using the state-matching error cost function, but is also derived in the strict linear matrix inequality format. In a numerical example, the effectiveness of the proposed technique and the results of the improved performance are shown.
This study considers the functional observers design for a class of discrete-time switched singular systems simultaneously subject to state delays, unknown inputs (UIs) and arbitrary switching sequences. The singular matrix E is assumed to be switching-mode-dependent, and the UIs are present in both the state and the measurement channels. A mode-dependent, delay-type observer is first constructed, then, based on the unknown input decoupling and the switched Lyapunov theory, a method is proposed to design such observer which ensures both: the total elimination of the UIs and exponential estimate of the given function of the state vector. The conditions for the existence of the proposed observer are given through algebraic matrix inequalities, and exponential stability of the observation error dynamics is derived by using a properly constructed decay-rate-dependent switched Lyapunov function and the linear matrix inequality technique. Finally, an illustrative example is given to show the effectiveness of the obtained results.
In this study, the stochastic sampled-data approach is developed to deal with the H ∞ control problem for a class of linear multi-agent systems. For each agent, signals should be transmitted to the control input, including not only its own measurements, but also the output information received from the neighbour agents. The relationships of information communications between the whole multi-agents are totally reflected by a weighted directed graph with a fixed topology structure. To the addressed problem, the author's focus is to design an output feedback controller for each agent such that, for the external energy bounded disturbances, the exponential stability in the mean square sense and interference rejection ability under consideration can be guaranteed. By assuming that the sampling period of each agent switches between two values in a probabilistic way, the stochastic sampling phenomenon is well characterised in every sampling tasks. In virtue of Lyapunov theory and semi-definite programming method, sufficient conditions are derived to ensure the exponential stability as well as H ∞ performance, and the feedback controllers are also designed by solving some certain matrix inequalities. Finally, a numerical simulation is given to demonstrate the validness of the proposed control strategy.
This study is concerned with stability and stabilisation of switched positive systems in both continuous- and discrete-time contexts. Several criteria of Metzler/Hurwitz and non-negative/Schur matrices are presented. By using these criteria and dual system theory, a sufficient condition for stability of switched positive systems is established. On the basis of the sufficient condition, a new controller design is proposed for switched positive systems. It is shown that the proposed design reduces the conservatism of the existing approaches in the literature where the controller gain matrices always have the property of being rank one. The results are also extended to discrete-time systems. Finally, two illustrative examples verify the validity of the theoretical findings.
This study discusses an intelligent digital redesign (IDR) for observer-based output-feedback controllers in Takagi–Sugeno form under immeasurable premise variables. A delta operator that asymptotically connects an analogue control system with its discrete-time model is adopted. The condition for the IDR problem with stability is proposed in terms of matrix inequality. The authors show that the proposed IDRed sampled-data controller recovers the analogue control performance from the perspective of the closed-loop trajectory under the fast sampling limit.
Two recent Lyapunov-based methods: free weighting matrix approach and Jensen inequality approach, have reduced the conservatism and the complexity of the stability result for one-dimensional (1D) time-delay systems, respectively. In this study, the authors further concern the analysis of delay-dependent stability and stabilisation for two-dimensional (2D) discrete systems with interval time-varying delays. By applying a new Lyapunov functional combining with the approaches of 2D Jensen inequalities and free weighting matrices, a new delay-dependent stability criterion is derived in terms of linear matrix inequalities (LMIs). Compared with the existing result, less decision variables are involved in the stability condition, so the burden of numerical computation is reduced greatly. It is also rigorously proved that the author's result is less conservative than some recent ones. On the basis of the stability criterion, state feedback is considered to realise the stability control and the state feedback gain can be solved by LMIs. Numerical examples show the effectiveness and advantage of their results.
This study deals with the current control of three-phase inverters connected to the grid by means of LCL filters. The control action is given by the feedback of the filter states, in coordinates αβ0, and of the internal states of resonant controllers. The control gains are computed by means of an optimal linear quadratic regulator that is robust to uncertain and time-varying parameters related to the grid impedance at the point of common coupling. As contributions, one has the detailed proof of a robust discrete linear quadratic control, showing its applicability also for the time-varying case, the experimental validation of the results in terms of the total harmonic distortion and harmonic limits for the grid injected current, according to the IEEE 1547 standard, and the analysis of the ℋ∞ norm of the closed-loop system for several values of grid inductance, indicating a value of grid inductance for which one has best performance for the time-invariant case. For comparison, a conventional state feedback controller is designed and implemented, showing the limitation of the non-robust strategy.
This study is concerned with robust quasi-min–max model predictive control (MPC) for a class of discrete-time singular systems with persistent disturbance and input constrains. To deal with the persistent disturbance, the authors introduce the notion of input-to-state stability (ISS) of discrete-time singular system for the first time. The optimal control can be obtained by solving a quasi-min–max optimal problem of a finite horizon cost function. On the basis of the proposed dual-mode MPC approach, it can be proved that the closed-loop discrete-time singular system is ISS, regular and causal. Finally, a numerical simulation shows the feasibility and effectiveness of the proposed method.
This study is concerned with the problem of quasi-sliding mode control design for a class of discrete-time systems with time-varying delay and unmatched disturbances. On the basis of the Lyapunov–Krasovskii method, combined with the reciprocally convex approach, sufficient conditions for the existence of a stable sliding surface are first derived in terms of matrix inequalities. These conditions also guarantee that the state trajectories of the reduced-order system are exponentially convergent within a ball whose radius can be minimised to deal with the effects of time-varying delay and disturbances. A robust quasi-sliding mode control scheme is then developed to drive the system state trajectories towards that ball in a finite time and maintain them therein after subsequent time. A numerical example is given to illustrate the effectiveness of the proposed approach.
In this study, the discrete-time antilinear systems with Markovian jumping parameters are investigated. The concept of stochastic stability is extended to the context of discrete-time Markovian jump (DTMJ) antilinear systems. By using stochastic Lyapunov approach, the authors derive some necessary and sufficient conditions for a DTMJ antilinear system to be stochastically stable in terms of coupled anti-Lyapunov matrix equations. In addition, two types of iterative algorithms are proposed to solve these coupled anti-Lyapunov matrix equations. Finally, some numerical examples are given to show the efficiency of the proposed algorithms and potential applications of the obtained results on antilinear systems.
This study investigates the containment control problem of second-order discrete-time multi-agent systems with Markovian missing data in actuators and one step network-induced time delay. The process of missing data from the controller to the actuator is modelled by a homogeneous, finite-state and discrete-time Markov chain. The authors first discuss the containment control problem for the case when all the elements of the transition probability matrix are completely known, then the result is extended to a more general case with only partially known transition probabilities. The distributed control protocol with one step time delay is proposed. Based on the stochastic Lyapunov–Krasovskii functional method, sufficient conditions in terms of a set of matrix inequalities are given to guarantee that the states of all the followers asymptotically converge to the convex hull formed by the corresponding states of the leaders in mean square sense. A cone complementary linearisation algorithm is used to obtain the control gains. Finally, two numerical simulations are provided to show the effectiveness of theoretical results.
This study addresses the output feedback stabilisation problem of discrete-time linear systems with input delay. By viewing the difference between the current input and the delayed input as a special disturbance, full- and reduced-order extended state observers are constructed, respectively, to estimate both the state and the disturbance simultaneously. Then the composite control laws are synthesised to actively compensate the input delay by using the state and disturbance estimates. Different from existing predictor-based control approaches, the proposed controllers are memoryless, and the past input information are thus not needed. Moreover, it can be observed that the information on the size of the input delay is not involved in either the observers or the controllers. Finally, simulation results are provided to illustrate the advantages and effectiveness of the proposed approaches.
This study is concerned with the finite-time synchronisation control (FTSC) of complex networks with discontinuous and continuous node dynamics. Two types of controllers (including continuous and discontinuous ones) are designed to ensure synchronisation of networks based on non-smooth analysis. Many sufficient criteria are given to guarantee FTSC by utilising the famous finite-time stability theorem. Compared the new obtained results with the previous literatures, the FTSC is discussed firstly when the node dynamics as well as the controllers are both discontinuous. Meanwhile, the upper bound of the settling time for synchronisation can be estimated. Finally, numerical examples are given to illustrate the effectiveness of the theoretical results.
V2 controlled buck converter with ramp compensation is studied, its two border voltages are derived and one-dimensional discrete-time model is established, upon which the dynamical effect of V2 control technique on stability control and mode shift of V2 controlled buck converter with ramp compensation is studied by using the bifurcation diagram, Lyapunov exponent spectrum, and parameter space map. The research results verified by simulation and experimental measurements indicate that by utilizing ramp compensation, V2 controlled buck converter can not only shift its operation mode from discontinuous conduction mode to continuous conduction mode, but also effectively stabilize its operation state from unstable subharmonic oscillation to stable periodic state.
This study investigates the problem of event-based synchronisation of linear discrete-time dynamical networks. Leader-following and leaderless synchronisations are achieved by a distributed event-trigger strategy. It is shown that feedback control updating is unnecessary until an event is triggered. The combinational-state variables and the Riccati equation are used to construct a Liapunov function and to design the event-triggering conditions. Numerical examples are provided to illustrate the theoretical results.
This study addresses semi-global practical asymptotic stabilisation and performance recovery of a class of sampled-data non-linear systems using a time-scale separation redesign. In particular, a high-gain filter is designed to estimate uncertainties in continuous-time and a fast variable coming from the filter is used to cancel the effect of the uncertainty, then the controller including the filter is descritised in Euler method. The authors provide conditions relating the sampling time and the filter gain, with which the results for the sampled-data system are the same as those that have been developed for the continuous-time system. As a verification of results, numerical simulations are given.