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Random terms in many natural and social science systems have distinct Markovian characteristics, such as Markov jump-taking values in a finite or countable set, and Wiener process-taking values in a continuous set. In general, these systems can be seen as Markov-process-driven systems, which can be used to describe more complex phenomena. In this paper, a discrete-time stochastic linear system driven by a homogeneous Markov process is studied, and the corresponding linear quadratic (LQ) optimal control problem for this system is solved. Firstly, the relations between the well-posedness of LQ problems and some linear matrix inequality (LMI) conditions are established. Then, based on the equivalence between the solvability of the generalized difference Riccati equation (GDRE) and the LMI condition, it is proven that the solvability of the GDRE is sufficient and necessary for the well-posedness of the LQ problem. Moreover, the solvability of the GDRE and the feasibility of the LMI condition are established, and it is proven that the LQ problem is attainable through a certain feedback control when any of the four conditions is satisfied, and the optimal feedback control of the LQ problem is given using the properties of homogeneous Markov processes and the smoothness of the conditional expectation. Finally, a practical example is used to illustrate the validity of the theory. [ABSTRACT FROM AUTHOR]

This article investigates the stability of random impulsive delay differential systems with the kind of random impulsive intensity determined by an arbitrary random sequence or an irreducible aperiodic Markov chain. Moreover, the continuous dynamics are described by random delay differential equations whose random disturbances are driven by second‐order moment processes. Based on the approaches of average impulsive interval and mode‐dependent average impulsive interval, the criteria of global asymptotic stability in probability and exponential stability in mth moment are constructed, respectively. In the end, the feasibility of the proposed theoretical results is verified by an example. [ABSTRACT FROM AUTHOR]

This paper is concerned with the robust $$H_{\infty }$$ filtering problem of nonlinear stochastic delayed systems with Markov jumps, where time delay exists in state equation, measurement equation and controlled output. Firstly, by introducing a nonlinear stochastic bounded real lemma, we present the $$H_{\infty }$$ filter design via solving a set of Hamilton-Jacobi inequalities (HJIs). Secondly, by using fuzzy approach, the $$H_{\infty }$$ filter can be designed via solving a set of Linear Matrix inequalities instead of HJIs. Finally, two numerical examples are given to show the effectiveness of our results. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

• For random impulsive delay systems with multiple random impulsive intensity, the criteria of the existence and uniqueness of the global solution with general conditions are firstly given. • In the random impulsive delay systems, there are multiple kinds of random impulses mixing effect on the random nonlinear systems, which are more general than the case with the signal impulse and can more accurately describe some complex situations in practical problems. • With the random impulsive intensity at each impulsive moment being driven by a homogeneous irreducible aperiodic Markov chain, some criteria of noise-to-state stability are proposed for ran- dom impulsive delay systems based on mode-dependent average impulsive interval and average impulsive interval. This paper investigates the noise-to-state stability of random impulsive delay systems with multiple random impulses, whose random impulsive amplitude is driven by a homogeneous irreducible aperiodic Markov chain. Firstly, some sufficient conditions for the existence-and-uniqueness of the global solution are given for random impulsive systems with time-varying delay. Secondly, the noise-to-state stability criteria are proposed based on the approaches of mode-dependent average impulsive interval and the average impulsive interval respectively. Finally, the effectiveness is verified by the simulation results. [ABSTRACT FROM AUTHOR]

The wind turbine generator system (WTGS) is usually linearised around operating points, and traditional control techniques concentrate on local dynamic response nearby each operating point. However, the actual wind speed is switching frequently between different operating points, which could cause a serious negative influence on the stability and dynamic response of WTGS. To overcome this disadvantage, this study proposes a systematic method to combine the switching rule into the control design to improve the dynamic response of the mechanical side of WTGS. Through modelling the WTGS driven by the switching wind speed into a class of linearised Markovian jump controlled systems, the regional pole placement technique is developed for such a class of systems and applied on improving the dynamic response of WTGS. Combined with the H∞ control, the proposed method can achieve better wind energy conversion efficiency and reduce the power volatility efficiently. Through combining the analysis on the actual historical wind speed into the control design, simulation results for a 2 MW wind turbine show the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

This paper aims to develop the stability theory for singular stochastic Markov jump systems with state-dependent noise, including both continuous-time and discrete-time cases. The sufficient conditions for the existence and uniqueness of a solution to the system equation are provided. Some new and fundamental concepts such as non-impulsiveness and mean square admissibility are introduced, which are different from those of other existing works. By making use of the [ABSTRACT FROM AUTHOR]

This study investigates the finite horizon H2 /H∞ control of time‐varying stochastic Markov jump systems (SMJSs) with state, control and disturbance‐dependent noise. Firstly, the stochastic bounded real lemma of SMJSs is established, which by itself has theoretical importance. Secondly, several necessary and sufficient conditions for H2 /H∞ control of SMJSs are proposed by means of coupled generalised differential Riccati equations. Finally, two numerical examples are given to show the effectiveness of the obtained results. [ABSTRACT FROM AUTHOR]

The dissipative analysis and control problems for a class of Markov jump non-linear stochastic systems (MJNSSs) are investigated. A sufficient condition for the dissipativity of MJNSSs is given in terms of coupled non-linear Hamilton–Jacobi inequalities (HJIs). Generally, it is difficult to solve the coupled HJIs. In this paper, based on T–S fuzzy model, the dissipative analysis and controller design for MJNSSs is proposed via solving a set of linear matrix inequalities (LMIs) instead of HJIs. Finally, a numerical example is presented to show the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, an infinite horizon $$H_2/H_\infty $$ control problem is addressed for a broad class of discrete-time Markov jump systems with ( $$x,u,v$$)-dependent noises. First of all, under the condition of exact detectability, the stochastic Popov-Belevich-Hautus (PBH) criterion is utilized to establish an extended Lyapunov theorem for a generalized Lyapunov equation. Further, a necessary and sufficient condition is presented for the existence of state-feedback $$H_2/H_\infty $$ optimal controller on the basis of two coupled matrix Riccati equations, which may be solved by a backward iterative algorithm. A numerical example with simulations is supplied to illustrate the proposed theoretical results. [ABSTRACT FROM AUTHOR]

This paper applies the spectrum technique to investigate the general stability of stochastic Markov jump linear systems ( SMJLSs). First, the notion of spectrum for SMJLSs is defined and the relationship between the spectrum and the asymptotical mean square stability is revealed. Second, by analyzing the spectrum distribution in the complex plane, some new concepts of stability, such as critical stability, regional stability, and essential instability, are introduced for SMJLSs. Necessary and sufficient conditions on such general stability are established. Finally, several numerical examples are carried out to illustrate the developed theory. [ABSTRACT FROM AUTHOR]

This paper mainly studies the notions of detectability and observability for discrete-time stochastic Markov jump systems with state-dependent noise. Two concepts, called ' W-detectability' and ' W-observability,' for such systems are introduced, and are shown to coincide with the other concepts on detectability and observability reported recently in literature. Moreover, some criteria and interesting properties for both W-detectability and W-observability are obtained. [ABSTRACT FROM AUTHOR]

This study is devoted to investigating the inherent relationship among the H2, H∞ and H2 /H∞ control designs for discrete‐time Markov jump systems with multiplicative noise. From a new perspective of non‐zero‐sum Nash game, we show that the Nash equilibrium solution may present a unified treatment approach for these three types of control design problems by taking adequate parameters in the quadratic performance indices. Moreover, a numerical example is supplied to illustrate the proposed results. [ABSTRACT FROM AUTHOR]

This study is concerned with the stability analysis and stabilisation of linear stochastic systems with Markovian switching by using the spectral technique. The notion of spectrum for Markovian jump linear stochastic systems (MJLSSs) is introduced and the relationship between the spectrum and the asymptotical mean‐square stability is revealed. As applications, a necessary and sufficient condition for regional stability of MJLSSs and a result on generalised Lyapunov equations are derived. Examples are presented to illustrate the results established. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we consider the infinite horizon control problem for discrete-time time-varying linear systems subject to Markov jump parameters and state-multiplicative noise. A stochastic version of a bounded real lemma is firstly developed for a general class of discrete-time time-varying Markov jump systems with state- and disturbance-multiplicative noise. By which we present a necessary and sufficient condition for the solvability of the control problem in terms of four coupled discrete-time Riccati equations. Moreover, the obtained design is applied to a macroeconomic problem to verify its effectiveness. [Copyright &y& Elsevier]

This paper addresses observability and detectability of discrete-time periodic systems with nonhomogeneous Markov jump parameter. Popov–Belevitch–Hautus (PBH)-type criteria are proposed for the concerned structural properties. It is shown that, different from stochastic systems with constant coefficients or homogeneous Markov chain, the spectral criteria of considered plants do not rely on a single operator but on a finite sequence of linear evolution operators. By use of the obtained detectability criterion, an extended Lyapunov theorem is established, which relates asymptotic mean square stability to a periodic Lyapunov equation. Further, a difference Riccati equation with periodic coefficients is studied and some sufficient conditions are presented for the existence of stabilizing solution. [ABSTRACT FROM AUTHOR]

In this paper, the H ∞ control problem is studied for nonlinear stochastic Markov jump systems with state, control and external disturbance-dependent noise ( ( x , u , v ) -dependent noise for short). A sufficient condition is derived for the infinite horizon H ∞ control of such systems in terms of a set of coupled second-order Hamilton–Jacobi inequalities (HJIs). In general, it is difficult to solve these coupled HJIs. By using fuzzy approach, the infinite horizon H ∞ control design for nonlinear stochastic Markov jump systems is developed via solving a set of linear matrix inequalities (LMIs) instead of HJIs. Two numerical examples are presented to illustrate the effectiveness of the proposed design methods. [ABSTRACT FROM AUTHOR]

Abstract: This paper considers the stabilization problems for interconnected nonlinear stochastic Markovian jump systems from the viewpoint of dissipativity theory. Based on the strongly stochastic passivity theory, the feedback equivalence and global stabilization problems are studied for interconnected nonlinear stochastic Markovian jump systems. The strongly stochastic -dissipativity sustains a direct control for this class of systems instead of solving coupled Hamilton–Jacobi inequalities. [Copyright &y& Elsevier]