Hidden Markov model based control for singular Markov jump systems

A hidden Markov model based control strategy is proposed to ensure the exponential mean-square admissible (EMS-admissible) of singular Markov jump systems (SMJSs) with mode-dependent singular matrix in this paper. The discontinuities caused by the mode-dependent singular matrix and Markov jump switching are investigated to make the results more practical. Concrete controller designing process is presented to tackle the instabilities brought by the hidden Markov model and the discontinuities. A new kind of Lyapunov function is constructed to characterize the switching features presented by Markov jump system and hidden Markov model based controller, which derives the stability conditions of the SMJSs. Then, a linearization strategy is proposed to reduce the coupled terms generated by the hidden Markov model. The hidden Markov model based controller is further designed based on the established conditions to ensure the EMS-admissible of the considered SMJSs. A numerical example and a direct current (DC) motor example are given to illustrate the effectiveness of the control approach.     

Reliable control for event-triggered singular Markov jump systems with partly unknown transition probabilities and actuator faults

In this paper, the problem of reliable controller design for event-triggered singular Markov jump systems with partly known transition probabilities, nonlinear perturbations and actuator faults is studied. To mitigate the burden of data transmissions over network, two event-triggered schemes with different triggering conditions are introduced. The switch law between the two event-triggered schemes is governed by a random variable with Bernoulli distribution. Taking nonlinear perturbations and actuator faults into account, the resulting closed-loop system is converted into a time-delay singular Markov jump system with partly known transition probabilities. Sufficient conditions of stochastically admissible for the resulting closed-loop system are obtained in terms of a group of linear matrix inequalities. The co-design of desirable reliable controller and weighting matrices of event-triggered schemes is presented. Finally, two numerical examples are given to show the effectiveness of the developed results.     

Observer-based control for singular nonhomogeneous Markov jump systems with packet losses

This paper is concerned with the observer-based H_∞ control for a class of singular Markov jump systems over a finite-time interval, where the transition probability (TP) is time-varying and is limited to a convex hull. Due to the limited capacity of network medium, packet losses are presented in the underlying systems. Firstly, using a stochastic Lyapunov functional, a sufficient condition on singular stochastic H_∞ finite-time boundedness for the corresponding closed-loop error systems is provided. Subsequently, a linear matrix inequality (LMI) condition on the existence of the H_∞ observer-based controller is developed from a new perspective. Finally, three numerical examples are provided to illustrate the effectiveness of the proposed controller design method, wherein it is shown that the proposed method yields less conservative results than those in the literature.     

Fuzzy-model-based tracking control of Markov jump nonlinear systems with incomplete mode information

This paper considers the tracking control problem for nonlinear Markov jump systems based on T–S fuzzy model approach with incomplete mode information. It is assumed that the mode transition rate matrix is not a priori knowledge and only partial information is available. Moreover, the mode where the system stays when operating is not fully accessible to the designed controller. In this incomplete mode information scenario, a hidden Markov model based mechanism is modified to simulate the mode deficiency mapping. The incomplete transition rate matrix is well defined in the form of a polynomial. Based on this, by constructing a polynomially parameter-dependent Lyapunov matrices and linear matrix techniques, sufficient conditions are established to ensure the stochastic stability and a prescribed tracking performance. The controller design scheme are presented by solving a series of LMIs. Examples are given in the end to illustrate the effectiveness of our proposed results.     

Improved asynchronous stabilization method for nonhomogeneous Markovian jump fuzzy systems with incomplete transition rates

This paper proposes an improved asynchronous stabilization method for nonhomogeneous Markovian jump fuzzy systems (MJFSs) with incomplete transition rates via a nonparallel distributed compensation (non-PDC) control scheme. To cover a more realistic situation in the continuous-time MJFS domain, this paper focuses on addressing the following issues differently from previous studies: 1) emergence of a nonhomogeneous Markov process and asynchronous control mode, 2) separation of the Lyapunov matrix from the control gain within stabilization conditions, and 3) relaxation of stabilization conditions subject to multiple time-varying parameters. Specifically, the effects of nonhomogeneity and asynchronism are simultaneously reflected in the stabilization conditions, and separation and relaxation techniques are additionally developed as needed. Finally, the advantages of our method are illustrated through two examples: the first is provided to show the effectiveness of the proposed separation technique, and the second is provided to show that the relaxed stabilization conditions are less conservative than previous results.     

Novel stability results for aperiodic sampled-data systems

This paper is concerned with stability for aperiodic sampled-data systems. Firstly, for aperiodic sampled-data systems without uncertainties, a new Lyapunov-like functional is constructed by introducing the double integral of the derivative of the state, the integral of the state, and the integral of the cross term of the state and the sampled state. When estimating the derivative of the Lyapunov-like functional, superior integral inequalities to Jensen inequality are employed to get a tighter upper bound. By the Lyapunov-like functional principle, sampling-interval-dependent stability results are derived. Then, the stability results are extended to aperiodic sampled-data systems with polytopic uncertainties. Finally, some examples are listed to show the stability results have less conservatism than some existing ones.     

Sliding mode output-feedback control of discrete-time Markov jump systems using singular system method

The issue of adaptive sliding mode controller design via output knowledge is studied for discrete-time Markov jump systems in this paper by means of using singular system scheme. To force the system state onto the sliding motion, an appropriate switching surface depended on the system output is established. Meanwhile, the reachability of the sliding manifold is guaranteed by synthesizing the robust sliding mode controller and adaptive sliding mode controller for the accessible and inaccessible upper bounds of sliding patch, respectively. By using Lyapunov functional technique, sufficient criteria to guarantee the sliding motion to be stochastically admissible are proposed. Then the reachability conditions of the predesigned switching surface are developed. Finally, simulation results are provided to illustrate the effectiveness of the proposed approach.     

Finite and infinite horizon indefinite linear quadratic optimal control for discrete-time singular Markov jump systems

This paper concerns the indefinite linear quadratic (LQ) optimal control problem for discrete-time singular Markov jump systems (MJSs) with finite and infinite horizon, where the weight matrices for state and control of cost function are all indefinite. Firstly, the indefinite LQ problem for singular MJSs is equivalently transformed into indefinite LQ problem for MJSs under a series of equivalent transformations. Then, the sufficient and necessary condition is proposed for the solvability of finite horizon case, the optimal control and optimal cost value are given, and the resulting optimal closed-loop system is regular, casual. Next, some sufficient and necessary conditions are obtained to ensure the transformed equivalent LQ problem for MJSs to be definite one, which can guarantee the generalized algebraic Riccati equation with Markov jump has a unique semi-positive definite solution. Meanwhile, the optimal control and nonnegative optimal cost value in infinite horizon are acquired, and the resulting optimal closed-loop system is stochastically admissible. Finally, three examples are presented to illustrate the theoretical results.     

Dissipative filtering for singular Markov jump systems with generally uncertain transition rates via new integral inequality approach

This paper considers the problem of dissipative filtering problem for singular Markov jump systems with time-varying delay and generally uncertain transition rates. Firstly, by tuning the improved integral inequality and Wirtinger-based integral inequalities, a sufficient condition is derived to guarantee that the considered system is regular, impulse-free, stochastically stable with the dissipation performance. Then, based on the derived condition, and applying linear matrix inequalities (LMIs) techniques, the filter is synthesized. Finally, some numerical examples are given to illustrate the effectiveness of the obtained theoretic results.     

Event-triggered passive control for Markovian jump discrete-time systems with incomplete transition probability and unreliable channels

This paper investigates the passivity of Markovian jump discrete-time systems (MJDTSs) with channel fading via event-triggered state feedback control. First, the concerned MJDTSs contain infinitely distributed delays and switching rules with partially known transition probability (TP) information. Next, the fading channel, as an unreliable channel, is introduced into MJDTSs to better reflect the engineering practice in networked environment. Due to the present of channel fading, a series of random variables satisfying some certain probability density functions (PDFs) will be obstacles in the process of proof. Then, an event-triggered controller is designed for MJDTSs with channel fading and incomplete transition probability (ITP) for the first time. Thanks to this event-triggered mechanism, the state feedback control could greatly reduce energy consumption during transmission. Subsequently, under the above controller, we obtain some novel sufficient criteria in the form of linear matrix inequalities (LMIs) to ensure the passivity of closed-loop system. Finally, some simulation results are provided to demonstrate the feasibility and effectiveness of the proposed theoretical method.     

Non-fragile guaranteed cost control for networked nonlinear Markov jump systems under multiple cyber-attacks

This paper is concerned with the problem of non-fragile guaranteed cost control (GCC) for networked nonlinear Markov jump systems subject to multiple cyber-attacks, which are characterized by Takagi–Sugeno (T–S) fuzzy model with time-varying delay. Specifically, a variety of cyber-attacks, including deception attacks and Denial-of-Service (DoS) attacks, are considered, which occur in the forward and feedback communication links, respectively. To achieve stochastic stability under guaranteed cost function (GCF), the paper proposes a Lyapunov–Krasovskii (L–K) function approach. The approach derives sufficient conditions for stochastic stability, and obtains non-fragile controller gains and the uniform upper bound of the GCF using linear matrix inequalities (LMIs) technique. Finally, the effectiveness of the proposed algorithm is evaluated by simulation experiment.     

Event-triggered H∞ control for network-based uncertain Markov jump systems under DoS attacks

This paper investigates the event-triggered $H_{\infty }$ control problem for network-based Markov jump systems subject to denial-of-service (DoS) attacks. In order to reduce the amount of signal transmission, the event-triggering scheme (ETS) is adopted between sensor and controller. Due to DoS attacks invalidating data over networks, a new switched time-delay Markov jump model with unstable subsystems is developed based on state feedback controller. Then with the help of piecewise Lyapunov-Krasovskii functional method, a set of sufficient conditions incorporating constraints of DoS attacks are provided, which guarantees that the resulting switched time-delay Markov jump system is stochastically stable with a certain $H_{\infty }$ performance. Subsequently, we present criterions to obtain the parameters of state feedback gain and ETS. Finally, an example is provided to show the effectiveness of the proposed method.     

Bayesian estimation for jump Markov linear systems with non-homogeneous transition probabilities

This paper considers the state estimation problem for a class of discrete-time non-homogeneous jump Markov linear systems (JMLSs), where the transition probability matrix (TPM) is assumed to be time-variant and takes value in a finite set randomly at each time step. To show the simplicity brought by the finite-valued hypothesis, the optimal recursion for the posterior TPM probability density functions conditioned on that the TPM belongs to a continuous set is firstly derived. Then, we naturally incorporate the proposed TPM estimation into the recursion of system state. Two interacting multiple-model (IMM)-type approximation stages are employed to avoid the exponential computational requirements. The resulting filter reduces to the IMM filter when the number of candidate TPMs is unity. A meaningful example is presented to illustrate the effectiveness of our method.     

Resilient sampled-data control design for singular networked systems with random missing data

This paper proposes an active resilient control strategy for singular networked control systems with external disturbances and missing data scenario based on sampled-data scheme. To characterize the missing data scenario, a stochastic variable satisfying Bernoulli distributed white sequence is introduced. Based on this scenario, in this paper, two different models are proposed. For both the models, by using Lyapunov–Krasovskii functional approach, which fully uses the available information about the actual sampling pattern, some sufficient conditions in terms of linear matrix inequalities (LMIs) are separately obtained to guarantee that the resulting closed-loop system is admissible and strictly dissipative with a prescribed performance index. In particular, Jensen’s and Wirtinger based integral inequalities are employed to simplify the integral terms which appeared in the derivation of stabilization results. Then, if the obtained LMIs are feasible, the corresponding parameters of the designed resilient sampled-data controller are determined. Finally, two numerical examples are presented to demonstrate the effectiveness of the proposed control design technique.     

Extended dissipativity analysis for T-S fuzzy systems based on reliable memory control and aperiodic sampled-data method

This paper studies the extended dissipativity (ED) issue for T-S fuzzy systems (TSFSs) via reliable memory control scheme and aperiodic sampled-data (ASD) method. First, considering the random variation of sampling interval and the time delays (TDs) of sampling signal transmission in the communication network, a reliable aperiodic memory sampled-data control (RAMSDC) strategy is proposed. Then, the developed delay-dependent Lyapunov-Krasovskii functional (LKF) with some two-sided looped-functional (TSLF) terms is constructed to fully utilize sampled state information. The introduced free matrices in the TSLF need not to be positive definite, which reduces the conservativeness of the obtained results. Next, a sufficient condition is given to ensure the ED, and the controller gain matrix is obtained by means of linear matrix inequality (LMI) technique. At last, the effectiveness of theoretical results in practical application is verified by the use of a truck-trailer model.     

A computational study of optimal control of Markov jump systems

This paper considers a class of optimal control problems governed by Markov jump systems. Our focus is to develop a computational method, based on the control parametrization approach, for solving this class of optimal control problems. Due to the existence of the continuous-time Markov chain, the optimal control problem under consideration is a stochastic optimal control problem, and hence the control parametrization technique cannot be applied directly. For this, a derandomization approach is introduced to obtain a representative deterministic optimal control problem. Then, the control parametrization method is applied to obtain an approximate finite dimensional optimization problem which can be computed numerically using the gradient-based optimization method. For this, the gradient formulas of the cost function and the constraint functions with respect to control variables are derived. Finally, a practical application involving a RLC circuit model is solved using the method proposed.     

Stability and stabilization of Markov jump systems with generally uncertain transition rates

This paper is focused on stability and stabilization of Markov jump systems with generally uncertain transition rates. By in-depth exploiting relationships among the transition rates, a new stability criterion is derived for the systems. The developed criterion is less conservative than some current ones. Moreover, it can be equivalently transformed into solvable condition for state-feedback stabilization while some current ones have to further introduce conservatism when establishing the solvable stabilization conditions. Examples show the validity and advantages of the developed results.