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Abstract
This paper focuses on the stability analysis and controller synthesis of continuous-time Markovian jump time-delay systems with incomplete transition rate descriptions. A general stability criterion is formulated first for state- and input-delay Markovian jump time-delay systems with fully known transition rates. On the basis of the proposed condition, an equivalent condition is given under the assumption of partly known/unknown transition rates. A new design technique based on a projection inequality has been applied to design both state feedback and static output feedback controllers. All conditions can be readily verified by efficient algorithms. Finally, illustrative examples are provided to show the effectiveness of the proposed approach.
... For instance, a decentralized Markovian jump ∞ control routing strategy for mobile multi-agent networked systems were investigated in [13] . The adaptive fuzzy decentralized tracking control for large-scale interconnected nonlinear networked control systems was studied in [14,15] , and a Lyapunov-function based event-triggered control was adopted to develop nonlinear discrete-time cyber-physical systems [16] . However, the decentralized control of interconnected systems is still an open field to be developed, and there are still many problems to be discussed. ...
... Next, a definition and some lemmas will be innovated to deduce the subsequent results of this paper. Definition 1( [16] ): Suppose ( ( ), , ≥ 0) is a functional candidate, then the infinitesimal operator ( ) is represented as ...
... , it is deduced that (16), (17) are equivalent to Π < 0 and (18) ...
This paper investigates the issue of decentralized control for interconnected semi-Markovian systems with partially accessible transition rates (TRs). Firstly, a dynamic system model with a memory event-triggered mechanism (METM) is designed, which can effectively improve the fault tolerance of the event-triggering mechanism by employing the historical trigger data. Then a state feedback control model with dynamic METM is constructed, in which the semi-Markovian parameters with completely unknown and partially known transition probabilities are considered. Some sufficient conditions that insure the stochastic stability of the interconnected semi-Markovian systems can be obtained by utilizing the Lyapunov function and suitable model transformations method. Meanwhile, the parameters and the controller gain matrices of dynamic METM are also solved simultaneously by applying the linear matrix inequalities (LMIs). Finally, a simulation example is given to verify the effectiveness of the proposed method.
... By using appropriately slack matrices, the stabilization problem of MJSs with partly unknown TRs has been studied in [20] . And in [21] , the control problem of MJSs with time delay has been further considered. More general than the two cases is generally uncertain TRs. ...
... The inequality (21) implies that lim t→∞ E t 0 x T (t ) x(t ) dt| φ(0) , r 0 = 0. Thus, it follows from Definition 1 that the system (10) is stochastically stable. This completes the proof. ...
This paper focuses on the problems of stability analysis and controller synthesis for a class of nonhomogeneous Markovian jump systems with both state and input delays. The time-varying transition rate matrix is described as a polytope set. By constructing a Lyapnouv-Krasovskii functional and introducing some appropriately slack matrices, a stochastic stability condition is proposed in term of infinite matrix inequalities. Based on this, a new stochastic stability condition is further derived in form of a finite set of LMIs. Then, in terms of linear matrix inequalities (LMIs) techniques, the sufficient condition on the existence of state-feedback controller is presented and proved. Finally, three numerical examples are provided to illustrate the effectiveness and usefulness of the obtained results.
... Recently, MJSs with partly unknown transition probabilities have attracted much attention, and many research results were published, such as the problems of stability analysis and controller synthesis were studied in [2][3][4][5][6][7], stochastic synchronization problem was discussed in [8] and so on. Especially references [4][5][6][7], MJSs with time delay were investigated because time delay is a source of instability in many cases. ...
... Recently, MJSs with partly unknown transition probabilities have attracted much attention, and many research results were published, such as the problems of stability analysis and controller synthesis were studied in [2][3][4][5][6][7], stochastic synchronization problem was discussed in [8] and so on. Especially references [4][5][6][7], MJSs with time delay were investigated because time delay is a source of instability in many cases. In the past years, various methods were proposed to handle the issue of time delay, for instance, Jensen's inequality, free-weighting matrix method, reciprocally convex approach, Wirtinger's inequality, Beseel-Legendre (B-L) inequality and so on, the corresponding results were proposed in references [10][11][12][13]15] respectively. ...
... In practice, deficient mode information in the Markov chain often happens because of the complexity of measuring the transition probabilities. The stochastic stabilization of Markovian jump time-delay systems with partly known transition probabilities is studied in Du et al. (2013), and the design condition for the state feedback controller is obtained in terms of solutions to a set of linear matrix inequalities (LMIs). The H ∞ control problem for the MJSs with partly known transition probabilities is investigated in Zhang et al. (2013b). ...
This study considers the designing of the H ∞ sliding mode controller for a singular Markovian jump system described by discrete-time state-space realization. The system under investigation is subject to both matched and mismatched external disturbances, and the transition probability matrix of the underlying Markov chain is considered to be partly available. A new sufficient condition is developed in terms of linear matrix inequalities to determine the mode-dependent parameter of the proposed quasi-sliding surface such that the stochastic admissibility with a prescribed H ∞ performance of the sliding mode dynamics is guaranteed. Furthermore, the sliding mode controller is designed to assure that the state trajectories of the system will be driven onto the quasi-sliding surface and remain in there afterward. Finally, two numerical examples are given to illustrate the effectiveness of the proposed design algorithms.
... Many results on Itô systems with Markovian jumps have been presented, for example H 2 control [5], Stackelberg strategy [6], H − index in fault detection [7] and so on. However, it is well known that in practice, due to technical limitations, it is difficult to fully measure the exact value of transition rates of Markovian jump process [8]. Markovian jump system with PKTR has been paid much concerns [9]. ...
This paper focuses on the problem of finite-time H∞ static output feedback control for Ito^ stochastic systems with Markovian jumps (MJs). First of all, by introducing a new state vector and a novel signal, several sufficient conditions for the existence of static output feedback controllers are established for the considered systems with completely known transition rates (CKTRs) and partially known transition rates (PKTRs), respectively. Then the static output feedback controllers are designed via solving linear matrix inequalities (LMIs), which ensure the closed-loop systems are stochastic H∞ finite-time boundedness. The validity of the developed method was demonstrated through two examples.
... Considering that time delay is a widespread phenomenon, it often causes the bad behavior of MJSs, even worsens the dynamic properties. The stability analysis and controller synthesis problems for Markovian jump time delay systems (MJTDSs) with partly unknown transition probabilities were studied in [5]− [8], and stochastic synchronization for Markovian coupled neural networks with partial information on transition probabilities was discussed in [9] and so on. ...
... Hence, it is important and urgent to study the MSSs with partly known TRs. Recently, some effects have been devoted to address the control synthesis problem for those kinds of systems (Du et al., 2013;Guo, 2016;Song et al., 2014;Zhang & Boukas, 2009). In addition to this, an interesting extension is to further consider the case that the TRs of the Markovian process are assumed to be time-varying, i.e. the Markovian process is nonhomogeneous. ...
This paper is devoted to the problems of exponential stability and stabilization for piecewise-homogeneous Markovian switching complex-valued neural networks with incomplete transition rates (TRs). Both the time-varying delays and the coefficient matrices are switched among finite modes governed by a piecewise-homogeneous Markov process, where the TRs of the two-level Markov processes are assumed to be time-varying during different intervals. On the basis of an appropriately chosen Lyapunov–Krasovskii functional, some mode-dependent sufficient conditions are presented to guarantee the unforced network to be exponentially mean-square stable. Then, by proposing certain mode-dependent state feedback controller, stabilization criteria are derived through strict mathematical proofs. At the end of the paper, numerical examples are provided to illustrate the effectiveness of the theoretical results.
... So the study of Markovian jump systems with general unknown transition rates has appealed to a great many scholars. Stability, stabilization, and robust control of Markovian jump systems with partially unknown transition have been reported in [29][30][31][32][33]. Stability analysis for neutral Markovian jump systems with partially unknown transition probabilities has been proposed in [34,35]. ...
This paper deals with the problem of stochastic stability for a class of neutral distributed parameter systems with Markovian jump. In this model, we only need to know the absolute maximum of the state transition probability on the principal diagonal line; other transition rates can be completely unknown. Based on calculating the weak infinitesimal generator and combining Poincare inequality and Green formula, a stochastic stability criterion is given in terms of a set of linear matrix inequalities (LMIs) by the Schur complement lemma. Because of the existence of the neutral term, we need to construct Lyapunov functionals showing more complexity to handle the cross terms involving the Laplace operator. Finally, a numerical example is provided to support the validity of the mathematical results.
... For example, in networked control system, it will cost a lot to obtain the total transfer probability due to packet loss and delay. At present, some scholars have been engaged in the related research of partial unknown probability transfer and time-varying transfer rates, and have achieved some research results [9][10][11][12][13]. On the other hand, time delays occur in most engineering fields, which will not only destroy the actual performance of the system, but also make the system unstable, and even lead to the collapse of the whole system. ...
Time-varying disturbances not only exist in the external delay of Markovian jump system, but also may exist in the transfer rate of internal probability transfer. This paper put forward an improved interactive convex inequality method for Markovian jump system(MJS) with multi-time-varying disturbances: time-varying delays and time-varying transition rates of probability transfer. On this basis, an improved Lyapunov function is quoted to more approximate the actual system, and the optimization of positive definite conditions of the matrix in Lyapunov functional is discussed, the specific positive definite condition is improved. At the end of the paper, the effectiveness of the above methods is verified by several simulations.
... In [18], the control design problem with a guaranteed stability domain for local stabilization of quadratic discrete-time systems is investigated by using the so-called Finsler's lemma. Moreover, time delay is frequently encountered in control systems [19][20][21][22][23]. It is well recognized that the time delay is a source of instability, and degrades the system performance. ...
In this note, we deal with the exponential stability and stabilization problems for quadratic discrete-time systems with time delay. By using the quadratic Lyapunov function and a so called ‘Finsler's lemma', delay-independent sufficient conditions for local stability and stabilization for quadratic discrete-time systems with time delay are derived in terms of linear matrix inequalities (LMIs). Based on these sufficient conditions, iterative linear matrix inequality algorithms are proposed for maximizing the stability regions of the systems. Finally, two examples are given to illustrate the effectiveness of the methods presented in this paper.
... Considering that time delay is a widespread phenomenon, it often causes the bad behavior of MJSs, even worsens the dynamic properties. The stability analysis and controller synthesis problems for Markovian jump time delay systems (MJTDSs) with partly unknown transition probabilities were studied in [5]− [8], and stochastic synchronization for Markovian coupled neural networks with partial information on transition probabilities was discussed in [9] and so on. ...
Abstract—This paper focuses on the delay-dependent stability for a kind of Markovian jump time-delay systems (MJTDSs), whose transition rates are incompletely known. In order to reduce the computational complexity and achieve better performance, auxiliary function-based double integral inequality is combined
with extended Wirtinger’s inequality and Jensen inequality to deal with the double integral and the triple integral in augmented Lyapunov-Krasovskii function (ALKF) and their weak infinitesimal generator respectively, the more accurate approximation bounds with a fewer variables are derived. As a result, less
conservative stability criteria are proposed in this paper. Finally, numerical examples are given to show the effectiveness and the merits of the proposed method.
... In recent years, Markovian jump systems have received considerable attention from researchers because of its wide range of applications in many areas such as mobile robots, modeling production systems, networked control systems, manufacturing systems, and communication systems [5][6][7][8]. In the existing literature, a large number of works have been focused on the analysis and synthesis of linear Markovian jump systems [9][10][11][12]. Moreover, few results are proposed for cases involving Markovian jump systems with completely known jump probabilities or partially known jump probabilities [13,14]. ...
This paper investigates the problem of robust reliable dissipative filtering for a class of Markovian jump nonlinear systems with uncertainties and time-varying transition probability matrix described by a polytope. Our main attention is focused on the design of a reliable dissipative filter performance for the filtering error system such that the resulting error system is stochastically stable and strictly dissipative. By introducing a novel augmented Lyapunov–Krasovskii functional, a new set of sufficient conditions is obtained for the existence of reliable dissipative filter design in terms of linear matrix inequalities (LMIs). More precisely, a sufficient LMI condition is derived for reliable dissipative filtering that unifies the conditions for filtering with passivity and H∞ performances. Moreover, the filter gains are characterized in terms of solution to a set of linear matrix inequalities. Finally, two numerical examples are provided to demonstrate the effectiveness and potential of the proposed design technique. Copyright
... Zhang et al. firstly proposed the concept of partially known transition probabilities in [10]. Recently, MJSs with time delay and partly unknown transition probabilities have been investigated, such as the problems of stability analysis and controller synthesis were studied in [11][12][13][14], stochastic synchronization problem was discussed in [15], and so on. Unlike References [10,16,17], which only take the influence of transition probabilities into account, how to separate the unknown transition probabilities or transition rates was a focus ering the different effect of delay on output observations and mode observations, Matei and Baras designed the optimal state estimators for discrete-time MJSs in [25]. ...
This paper focuses on the full-order state estimation problem for continuous-time Markovian jumping systems with time delay and incomplete transition probabilities. Based on the Lyapunov stability theory, an extended Wirtinger's inequality is used to value the upper bound of integral term by considering the relationship among time delay, its upper, and their difference. At the same time, an extended vector inequality is applied to deal with the accompanying fractions that contain the information of time delay. As a result, an improved stability criterion is derived, and a full-order state estimates design method is proposed. Finally, two examples are provided to illustrate the merits of the proposed method. Copyright
A mean-field game based on a static output feedback (SOF) strategy for delay stochastic systems is considered. First, after defining the stabilization problem for a single decision maker, the problem of minimizing the upper bound of the cost function is given via the cost guarantee cost control theory. For this problem, the KKT condition establishes a necessary condition that satisfies the suboptimality by using a stochastic large-scale matrix equations. Then, using the obtained preliminary results, we apply it to the Pareto optimal strategy, which is a cooperative game, for the mean-field stochastic system involving a large number of decision makers. The main contribution is to derive a design method for deriving a centralized strategy. To obtain the strategy set, the Newton's method is considered. Furthermore, a new decomposition algorithm is discussed to avoid the huge dimensional problem. Finally, a simple numerical example is performed to demonstrate the usefulness and effectiveness of the proposed set of strategies.
Stability analysis is considered in this paper for linear neutral delay systems subject to two different delays in both the state variables and the retarded derivatives of state variables. By choosing a suitable state vector indexed by an integer
k
, a new augmented Lyapunov-Krasovskii functional (LKF) is constructed, and a stability criterion based on linear matrix inequalities is developed accordingly. It is shown that the proposed condition is less conservative than the existing methods due to the introduction of the delay-product-type integral terms in the LKF. The resulting stability criterion is then applied to the robust stability analysis of neutral delay systems with norm-bounded uncertainty. Moreover, a delay-independent stability criterion is developed based on the proposed LKF, and its frequency-domain interpretation is also given. These developed stability criteria indexed by an integer
k
exhibit a hierarchical character: the larger the integer
k
, the less conservatism of the resulting stability criterion. Finally, two numerical examples are carried out to illustrate the effectiveness of the proposed method.
In this paper, a new Lyapunov method is used to study the almost sure stabilization of positive Markovian jump linear systems (PMJLSs) under asynchronous Markovian switching. This method is established by a composite Lyapunov function. Moreover, the almost surely exponential stability criterion of the closed-loop PMJLSs under asynchronous condition is presented, and a controller is designed by means of bilinear matrix inequalities (BMIs). The quantitative relationship among the magnitude, stationary distribution and Markov chain generator of the detected switch signal delay can be given by the obtained results. Finally, a simulation example is given to verify the theoretical results.
In this paper, aperiodic sampled-data control for a class of Itô stochastic Markovian jump systems with switching transition rate matrix containing partially known elements is investigated. By introducing a loop-based Lyapunov functional, mean square exponential stability criteria are presented and mode-dependent aperiodic sampled-data controllers are designed. Due to the constraint on the upper bound of the sampling interval is no greater than the dwell time, the transition rate matrix may switch in a sampling interval. The issue concerning the asynchronization between the sampled-data controller mode and the system mode is hence considered. A practical RLC electrical circuit example is presented to illustrate the effectiveness of the method.
This paper focuses on the non-zero sum differential game problem for Markovian jump systems with partially unknown transition probabilities. Firstly, a suboptimal control problem is studied by the free-connection weighting matrix method, and then the non-zero sum differential game problem is investigated on this basis. Several sufficient conditions for the existence of ε-suboptimal control strategy and ε-suboptimal Nash equilibrium strategies are provided, and their explicit expressions are designed. Moreover, the precise form for the upper bound of the cost function is also given. To facilitate the calculation, all conditions are converted into the corresponding equivalent linear matrix inequalities or bilinear matrix inequalities form. Finally, two numerical examples are utilized to demonstrate the effectiveness of the main results.
This paper addresses the problem of event-triggered dynamic output feedback control for uncertain Markovian jump systems with partly unknown transition rates. An event-triggered dynamic output feedback control scheme is proposed to stabilize a class of uncertain networked Markovian jump systems with partly unknown transition rates, and a suitable Lyapunov–Krasovskii functional is then introduced to guarantee the asymptotical stability in mean-square sense of the closed-loop controlled system. It is shown that the proposed control scheme is relatively convenient to choose the control gain matrices by the use of the Matlab LMIs Toolbox, and thus it might be convenient to application in practice. Finally, numerical examples are presented to illustrate the effectiveness of the proposed methodology.
In this paper, the finite-time sliding mode control issue is studied for a series of semi-Markov jump systems subject to actuator faults, where an asynchronous control method is adopted to overcome the non-synchronous phenomenon between the system mode and controller mode. Additionally, the event-triggered protocol, which determines whether the transmission of data should be performed according to the threshold condition, is introduced to alleviate the burden of data transmission in the communication channel. This paper aims to devise an asynchronous event-triggered sliding mode control law so as to guarantee the trajectories of the resulting closed-loop system can be forced onto the predefined sliding surface in a finite-time interval. Thence, by means of the mode-dependent Lyapunov functions and the finite-time theory, sufficient conditions are derived to assure that the closed-loop system is mean-square finite-time bounded in both reaching and sliding motion phases. Eventually, a numerical example and a tunnel diode circuit model are presented to illustrate the availability and practicability of the proposed approach.
This paper investigates the stochastic finite-time stability and the state feedback controller design for the linear semi-Markovian jump systems with generally uncertain transition rates. Firstly, sufficient condition is established by using a stochastic Lyapunov functional. Secondly, based on the linear matrix inequality condition, the state feedback controller is obtained to guarantee the finite-time stabilisation. Finally, numerical examples are given to demonstrate the effectiveness of the proposed method.
This paper is concerned with the problems of stochastic stability and control design for singular Markovian jump systems (SMJSs) with time varying delay. The derivative coefficient is considered to be mode dependent. A parameter-dependent reciprocally convex matrix inequality (PDRCMI) is constructed, which covers some existing ones as its special cases. Different from some existing ones, the introduced parameters are completely known, which can be optimized by an iteration algorithm. To decrease the redundant decision variables, a mild assumption is given for the case of mode-dependent coefficient such that the considered system is decomposed into differential equations and algebraic ones. In this case, the components of state vectors of the subsystem are applied to construct a new augmented Lyapunov-Krasovkii functional (LKF). Based on the PDRCI and the new augmented LKF, a novel stochastic stability condition with both less computational demands and less conservativeness is derived. Based on the decomposed subsystems and the stability condition, a set of state feedback controller is designed in terms of linear matrix inequalities (LMIs). Some numerical examples are introduced to illustrate the effectiveness of the proposed results.
This article develops a predictive robust H∞ static output feedback control approach for networked control systems where random network‐induced delays in both forward and feedback communication channels are modeled as two mutually uncorrelated Markov chains. By making use of the system augmentation method, the closed‐loop system is formulated as a singular Markovian jump system with two modes, wherein the transition probability matrices of the underlying Markov chains are considered to be partially accessible. Necessary and sufficient conditions for the stochastic admissibility and robust H∞ performance of the closed‐loop system are given under the assumption of partially known transition probability matrices. A linear matrix inequality condition is proposed to determine the two‐mode‐dependent static output feedback controller gains to compensate for the random network‐induced delays efficiently and provide the desired control performance. Finally, a numerical example is provided to demonstrate the effectiveness of the proposed approach.
This article is concerned with the problem of the containment control with respect to continuous-time semi-Markovian multiagent systems with semi-Markovian switching topologies. Two kinds of classical control schemes, which are dynamic containment control and static containment control schemes, are utilized to address the proposed synthesis problem. Based on the linear matrix inequality (LMI) method, the dynamic containment controller and static containment controller are designed to plunge into the studied semi-Markovian multiagent systems, respectively. Moreover, the random switching topologies with the semi-Markovian process, the partly unknown transition rates, and the generally uncertain transition rates are taken into account, which can be applicable to more practical situations. Finally, the simulation results are provided to illustrate the effectiveness of the proposed theoretical results.
This paper addresses the problem of establishing whether a matrix with rational dependence on a complex parameter and its conjugate is Hurwitz (i.e., has all eigenvalues with negative real part) over the complex unit circumference. A necessary and sufficient condition is proposed, which requires testing stability of a constant matrix and feasibility of a linear matrix inequality (LMI). Moreover, the numerical complexity of the proposed approach is investigated in terms of size and number of free scalar variables of the LMI by deriving their formulas as functions of the problem data. Also, it is shown that the numerical complexity may be significantly reduced without introducing approximations or conservatism whenever some symmetry properties are satisfied. Lastly, the extension of the proposed approach to the case of Schur stability (i.e., eigenvalues with magnitude smaller than one) and D-stability (i.e., eigenvalues on special regions of the complex plane) are presented. The proposed approach is illustrated by some numerical examples that also show its application to stability analysis of 2D systems with mixed signals.
This paper deals with the quantized control problem for nonlinear semi-Markov jump systems subject to singular perturbation under a network-based framework. The nonlinearity of the system is well solved by applying Takagi-Sugeno (T-S) fuzzy theory. The semi-Markov jump process with the memory matrix of transition probability is introduced, for which the obtained results are more reasonable and less limiting. In addition, the packet dropouts governed by a Bernoulli variable and the signal quantization associated with a logarithmic quantizer are deeply studied. The major goal is to devise a fuzzy controller, which not only assures the mean-square σ-error stability of the corresponding system but also allows a higher upper bound of the singularly perturbed parameter. Sufficient conditions are developed to make sure that the applicable controller could be found. The further examination to demonstrate the feasibility of the presented method is given by designing a controller of a series DC motor model.
This paper presents a stabilisation problem for singular Markovian jump system with time-varying transition rate based on a reduced-order observer. To design the reduced-order observer, a canonical equivalent form for singular Markovian jump system is given. By the output variables of the equivalent form of singular Markovian jump system and the state of the reduced-order observer, an integral sliding surface is designed. Then, the sufficient condition for the stochastic admissibility of sliding motion is given, in which the time-varying transition rate is handled by the quantized method. Furthermore, a sliding mode controller is developed such that the system can be driven to the sliding surface in finite time. Finally, the effectiveness and superiority of the obtained results are illustrated by two examples, a numerical example and a practical example of DC motor model respectively.
The paper deals with controller design for stochastic Markovian switching systems with time-varying delay and actuator saturation by implying a new criterion for the domain of attraction firstly. By constructing more appropriate Lyapunov–Krasovskii functional, some new conditions for verifying stochastic stability of the plant are established. Then, the state feedback controller is designed to expand the domain of attraction of the corresponding closed-loop system. The procedure of deriving controller gain matrices is converted into an optimisation problem with constraints of a set of linear matrix inequalities. The mathematical model of RLC series circuit illustrates the validity of the obtained results.
This paper addresses the stochastic stability problem of positive Markov jump linear systems (PMJLSs). A necessary and sufficient condition of stochastic stability for PMJLSs is given which can be checked by solving linear programming feasibility problems due to the positivity property. A common co-positive Lyapunov function is constructed to solve the problem, and the equivalence among stochastic stability, 1-moment stability and exponential mean stability is proved afterwards. Considering the uncertain transition rates situation, PMJLSs with partially known transition rate matrix are investigated, and a necessary and sufficient condition for robust stochastic stability is proposed in linear programming form. Numerical examples are presented to show the effectiveness of the proposed results.
The paper is devoted to the investigation of observer-based H∞ sliding mode controller design for a class of nonlinear uncertain neutral Markovian switching systems (NUNMSSs) with general uncertain transition rates. In this case, each transition rate can be completely unknown or only its estimate value is known. Firstly, an H∞ non-fragile observer subjected to the general uncertain transition rates of the NUNMSSs is constructed. By some specified matrices, the connections among the designed sliding surfaces corresponding to every mode are founded. Secondly, the state-estimation-based sliding mode control law is designed to guarantee the reachability of the sliding surface in a finite time interval. Thirdly, an asymptotically stochastic stability criterion is established by utilizing a new-type Lyapunov function to guarantee the error system and sliding mode dynamics to be asymptotically stochastic stable with a given disturbance attenuation level. Finally, an example is provided to illustrate the efficiency of the proposed method.
In this paper, a passivity and fault alarm-based hybrid controller is designed for a Markovian jump delayed system with actuator failures. First, a passive condition is given, and a type of hybrid controller that combines with robust and fault-tolerant controllers is presented to ensure that both the normal system and the fault system are robustly stochastically passive. Next, a fault alarm signal is proposed by choosing the alarm threshold, and this signal is used to invoke the fault-tolerant controller. Finally, a numerical example is provided to show the effectiveness of the method.
This chapter introduced the research background and significance of MJS, as well as the current research status of MJS, so as to provide a basis of reference for further research of S-MJS. The main differences between S-MJS and MJS have been provided, followed by a description of the advantages of the S-MJS and its broad application prospects. Additionally, we have mentioned several problems that are yet to be solved, methods that require refinements and the main research contents of this dissertation.
This paper mainly focuses on the reachable set estimation problem of a time-varying delayed inertial Markov jump bidirectional associative memory (BAM) neural network with bounded disturbance inputs. The disturbances are assumed to be either unit-energy bounded or unit-peak bounded. Different from systems of the past studies, this paper is for inertial Markov jump BAM neural network with both time-varying delay and time-varying transition rates. The time-varying character of the considered transition rates is assumed to be piecewise-constant. In order to reduce the conservatism, the delay-partitioning technique is utilized to solve this reachable set estimation problem. As a result, it is obtained that the ellipsoid defined in this paper contains the reachable set Rup, which indicates the reachable set Rue is included. Further, we extend the results to the uncertain Markov jump BAM neutral network with partially unknown transition probabilities. Numerical examples are proposed to show the effectiveness of the given results.
In this work, the finite-time dissipative control problem is considered for singular discrete-time Markovian jumping systems with actuator saturation and partly unknown transition rates. By constructing a proper Lyapunov–Krasonski functional and the method of linear matrix inequalities (LMIs), sufficient conditions that ensure the systems singular stochastic finite-time stability and singular stochastic finite-time dissipative are obtained. Then, the state feedback controllers are designed, and in order to get the optimal values of the dissipative level, the results are extended to LMI convex optimization problems. Finally, numerical examples are given to illustrate the validity of the proposed methods.
A new method based on fault chains and the Markov process for analyzing time-delay power system stability is proposed in this article. First, reasonable fault chains are formed using the fault chain generation method based on network power flow. Second, by combining the fault chains and Markov processes, a class of Lyapunov–Krasovskii functionals considering the Markov jump is established. Free weighting items constructed with the Markov process transfer rate matrix and Newton–Leibniz formula, respectively, are then introduced to the weak infinitesimal operators in the Lyapunov–Krasovskii functional. Meanwhile, to lower the conservativeness of the method, the whole time-varying time-delay interval has been divided into two subintervals. Finally, the time-delay upper bound that the power system can bear in the case of cascading faults is calculated by means of the generalized eigenvalue problem. The efficiency and feasibility of the proposed method in solving the maximum time delay in the case of cascading faults have been validated by the dynamic time-domain simulation results. Comparative results with other methods show the proposed method has low conservativeness.
Finite-time synchronization is studied for delayed complex networks with stochastic disturbance. Based on the finite-time stability theorem, some sufficient conditions are obtained to ensure finite-time synchronization for Markovian jump complex networks with time delays and partially unknown transition rates. Finally, the effectiveness of the proposed method is demonstrated by illustrative examples.
The problem of design of anti-windup gains for Markovian jump systems subject to parametric uncertainty and actuator saturation is addressed in this paper. The system under study is assumed to have partially unknown transition rates. A set of anti-windup compensation gains are designed for dynamic output feedback controllers that have been designed in advance for stabilization in the absence of control saturation. It is shown that the redesigned controllers can achieve stabilization of the system regardless the presence or absence of actuator saturation. A convex optimization method is developed to obtain the largest possible estimation of the domain of attraction for the closed-loop system. Simulation results are presented to illustrate the effectiveness of the proposed method.
This paper studies the problem of stochastic stability and stabilization for positive Markov jump systems with distributed time delay and incomplete known transition rates. By using delay partitioning technique, a new form of Lyapunov functional is constructed and a stochastic stability criterion is formulated for positive Markov jump systems with fully known transition rates. Based on this, the case of incomplete known transition rates is discussed. Then a state feedback controller is designed to guarantee the corresponding closed-loop system positive and stochastically stable. The results in this paper are expressed in the form of linear programming problems, and can be easily solved by standard software.
This paper addresses the problem of fault estimation for a class of nonlinear Markov jump systems with Lipschitz-type nonlinearities and general transition rates allowed to be uncertain and unknown. First, by introducing a mode-dependent intermediate variable, an intermediate estimator is proposed to estimate faults and state simultaneously. Then, a vertex separator is exploited to develop a sufficient condition for the fault estimator design in terms of linear matrix inequalities. The design guarantees the boundedness in probability of the estimation errors if the derivations of the faults are bounded. Further, it is proved that the proposed approach is less conservative than the traditional methods. Finally, examples are given to show the advantages and effectiveness of the results.
In this paper, the local synchronization problem of Markovian nonlinearly coupled neural networks with uncertain and partially unknown transition rates is investigated. Each transition rate in this Markovian nonlinearly coupled neural networks model is uncertain or completely unknown because the complete knowledge on the transition rates is difficult and the cost is probably high. By applying the Lyapunov–Krasovskii functional, a new integral inequality combining with free-matrix-based integral inequality and further improved integral inequality, the less conservative local synchronization criteria are obtained. The new delay-dependent local synchronization criteria containing the bounds of delay and delay derivative are given in terms of linear matrix inequalities. Finally, a simulation example is provided to illustrate the effectiveness of the proposed method.
The static output feedback stabilisation problems for singular Markovian jump system are studied. Transition rates are assumed to be generally uncertain, namely, they could be unknown or only their estimate values are known. A criterion on the existence of an output feedback controller is established in terms of bilinear matrix inequalities (BMIs) to guarantee the closedloop systems are stochastically admissible. To solve the BMIs in the criteria, the other criterion is presented by using a less conservative linearisation method, and an algorithm is proposed to solve the related conditions. Finally, three numerical examples are given to show the effectiveness of their methods.
This article presents a fuzzy dynamic reliable sampled-data control design for nonlinear Markovian jump systems, where the nonlinear plant is represented by a Takagi–Sugeno fuzzy model and the transition probability matrix for Markov process is permitted to be partially known. In addition, a generalised as well as more practical consideration of the real-world actuator fault model which consists of both linear and nonlinear fault terms is proposed to the above-addressed system. Then, based on the construction of an appropriate Lyapunov–Krasovskii functional and the employment of convex combination technique together with free-weighting matrices method, some sufficient conditions that promising the robust stochastic stability of system under consideration and the existence of the proposed controller are derived in terms of linear matrix inequalities, which can be easily solved by any of the available standard numerical softwares. Finally, a numerical example is provided to illustrate the validity of the proposed methodology.
This paper is concerned with the problem of stochastic stability analysis of discrete-time two-dimensional (2-D)
Markovian jump systems (MJSs) described by the Roesser model with interval time-varying delays. The transition probabilities of the jumping process/Markov chain are assumed to be uncertain, that is, they are not exactly known but can be estimated. A Lyapunov-like scheme is first extended to 2-D MJSs with delays. Based on some novel 2-D summation inequalities proposed in this paper, delay-dependent stochastic stability conditions are derived in terms of linear matrix inequalities (LMIs) which can be computationally solved by various convex optimization algorithms. Finally, two numerical examples are given to illustrate the effectiveness of the obtained results.
In this paper, we develop a novel optimal tracking control scheme for a class of nonlinear discrete-time Markov jump systems (MJSs) by utilizing a data-based reinforcement learning method. It is not practical to obtain accurate system models of the real-world MJSs due to the existence of abrupt variations in their system structures. Consequently, most traditional model-based methods for MJSs are invalid for the practical engineering applications. In order to overcome the difficulties without any identification scheme which would cause estimation errors, a model-free adaptive dynamic programming (ADP) algorithm will be designed by using system data rather than accurate system functions. Firstly, we combine the tracking error dynamics and reference system dynamics to form an augmented system. Then, based on the augmented system, a new performance index function with discount factor is formulated for the optimal tracking control problem via Markov chain and weighted sum technique. Neural networks are employed to implement the on-line ADP learning algorithm. Finally, a simulation example is given to demonstrate the effectiveness of our proposed approach.
This paper provides estimation and control design methods for linear discrete-time systems with multiplicative noise. First, we present the design of the state feedback controller, such that the closed loop system is mean square stable. Second, we provide sufficient conditions for the existence of the state estimator; these conditions are expressed in terms of linear matrix inequalities (LMIs), and the parametrization of all admissible solutions is addressed. Finally, an estimator design approach is formulated using LMIs, and the performance of the estimator is examined by means of numerical examples.
The problem of delay-dependent stability in the mean square sense for stochastic systems with time-varying delays, Markovian switching and nonlinearities is investigated. Both the slowly time-varying delays and fast time-varying delays are considered. Based on a linear matrix inequality approach, delay-dependent stability criteria are derived by introducing some relaxation matrices which can be chosen properly to lead to a less conservative result. Numerical examples are given to illustrate the effectiveness of the method and significant improvement of the estimate of stability limit over some existing results in the literature.
This note concerns the problem of the robust stability of a linear system with a time-varying delay and polytopic-type uncertainties. In order to construct a parameter-dependent Lyapunov functional for the system, we first devised a new method of dealing with a time-delay system without uncertainties. In this method, the derivative terms of the state, which is in the derivative of the Lyapunov functional, are retained and some free weighting matrices are used to express the relationships among the system variables, and among the terms in the Leibniz-Newton formula. As a result, the Lyapunov matrices are not involved in any product terms of the system matrices in the derivative of the Lyapunov functional. This method is then easily extended to a system with polytopic-type uncertainties. Numerical examples demonstrate the validity of the proposed criteria.
In the past few years, a lot of research has been dedicated to the stability of interval systems as well as the stability of systems with Markovian switching. However, little research has been on the stability of interval systems with Markovian switching, which is the topic of this paper. The system discussed is the stochastic delay interval system with Markovian switching. It is a very advanced system and takes all the features of interval systems, Ito equations, and Markovian switching, as well as time lag, into account. The theory developed is applicable in many different and complicated situations so the importance of the paper is clear.
This paper deals with the problem of stochastic optimal control for a class of nonlinear systems subject to Markovian jump parameters. The nonlinearities in the different jump modes are initially parameterized by multilayer neural networks (MNNs), which lead to neural Markovian jump systems. A stochastic neural Lyapunov function (NLF) is used to analyze the stability of the resulting neural control MJSs. Then, based on this stochastic NLF and the neural model, a linear state feedback controller is designed to stabilize the closed-loop nonlinear system and guaranteed an upper bound of the system performance for all admissible approximation errors of the MNNs. The control gains can be derived by solving a set of linear matrix inequalities. Finally, a single link robot arm is demonstrated to show the effectiveness of the proposed design techniques. ICIC International
This paper deals with the problem of robust fault detection for Markovian jump linear systems with polytopic uncertainties. Using a generalized form of observer-based fault detection filter (FDF) as a residual generator, the design of robust FDF is formulated in the framework of stochastic H∞ filtering. Based on analyzing the robust mean square stability and stochastic H∞performance of the FDF, sufficient conditions on the existence of both mode-dependent and mode-independent H∞ FDFs are respectively derived and solutions to the H∞ FDFs are given in terms of linear matrix inequalities. A numerical example is given to show the effectiveness of the proposed method.
In this paper, the problem of finite-time stabilization for a class of uncertain Markov jump systems with partially known transition probabilities is investigated. The main aim of this paper is to derive the finite-time stabilization criteria for the underlying systems when the transition probabilities are partially known and to design a state feedback stabilizing controller such that the trajectories of the system stay within a given bound in a fixed time interval. Sufficient conditions for the existence of the desired controller are established with the linear matrix inequalities framework. A numerical example is used to illustrate the effectiveness of the developed theoretic results.
This paper focuses on stability and H ∞ performance analysis for a class of continuous-time Markovian jump systems with unknown transition probability and time-varying delay. Based on the free-connection weighting matrix method, the less conservative stability criterion and bounded real lemma (BRL) of Markovian jump systems with time-varying delay are obtained by considering the influence of partly unknown transition probability or completely unknown transition probability. Finally, numerical examples are given to illustrate the effectiveness and the merits of the proposed method.
In this paper, we develop a descriptor approach to simultaneous H2/H∞ control design for a class of uncertain time-delay systems with Markovian jump parameters. The rationale behind this approach is to exhibit the delay-dependence dynamics in the design procedure. In the system under consideration, the delay factor is constant, the jumping parameters are modelled as a continuous-time, discrete-state Markov process and the uncertainties are assumed to be real, time-varying and norm-bounded. A state-feedback control is derived for both the nominal and uncertain systems such that the H2-performance measure is minimized while guaranteeing a prescribed H∞-norm bound on the controlled system. All the developed results are cast in the format of linear matrix inequalities (LMIs) and numerical examples are presented.
This paper considers the class of continuous-time jump linear systems with time-delay and polytopic uncertain parameters. When the time-delay is a known constant, by using the linear matrix inequality (LMI) technique, we first establish delay dependent sufficient conditions for robust stability and robust H∞ control of the class of systems under study. The problem of determining the maximum time-delay under which the system will remain stable is cast into a generalized eigenvalue problem and thus solved by LMI techniques. When the control input contains a time-delay, an algorithm to design a state feedback controller with constant gain matrix is developed. Two examples are given to show the validness of the theoretical results.
In this paper, the problem of continuous gain-scheduled fault detection (FD) is studied for a class of stochastic nonlinear systems which possesses partially known jump rates. Initially, by using gradient linearization approach, the nonlinear stochastic system is described by a series of linear jump models at some selected working points. Subsequently, observer-based residual generator is constructed for each jump linear system. Then, a new observer-design method is proposed for each re-constructed system to design H∞H∞ observers that minimize the influences of the disturbances, and to formulate a new performance index that increase the sensitivity to faults. Finally, continuous gain-scheduled approach is employed to design continuous FD observers on the whole nonlinear stochastic system. Simulation example is given to show the effectiveness and potential of the developed techniques.
The problem of robust mode-dependent delayed state feedback control is investigated for a class of uncertain time-delay systems with Markovian switching parameters and mixed discrete, neutral, and distributed delays. Based on the Lyapunov-Krasovskii functional theory, new required sufficient conditions are established in terms of delay-dependent linear ma- trix inequalities for the stochastic stability and stabilization of the considered system using some free matrices. The desired control is derived based on a convex optimization method such that the resulting closed-loop system is stochastically stable and satisfies a prescribed level of performance, simultaneously. Finally, two numerical examples are given to illustrate the effectiveness of our approach.
In this paper, the problem of designing a fixed static output feedback control law which minimizes an upper bound on linear quadratic (LQ) performance measures for r distinct MIMO plants is addressed using linear matrix inequality (LMI) technique. An iterative LMI algorithm is proposed to obtain the solution. Examples are used to demonstrate its effectiveness. Copyright ©1999 John Wiley & Sons, Ltd.
This paper is concerned with delay-dependent exponential stability for stochastic Markovian jump systems with nonlinearity and time-varying delay. An improved exponential stability criterion for stochastic Markovian jump systems with nonlinearity and time-varying delay is proposed without ignoring any terms by considering the relationship among the time-varying delay, its upper bound and their difference, and using both Itô's differential formula and Lyapunov stability theory. A numerical example is given to illustrate the effectiveness and the benefits of the proposed method. Copyright © 2009 John Wiley & Sons, Ltd.
This paper studies the problem of continuous gain-scheduled PI tracking control on a class of stochastic nonlinear systems subject to partially known jump probabilities and time-varying delays. First, gradient linearization procedure is used to construct model-based linear stochastic systems in the vicinity of selected operating states. Next, based on stochastic Lyapunov stabilization analysis, sufficient conditions for the existence of a PI tracking control are established for each linear model in terms of linear matrix inequalities. Finally, continuous gain-scheduled approach is employed to design continuous nonlinear PI tracking controllers on the entire nonlinear jump system. Simulation example is given to illustrate the effectiveness of the developed design techniques.
This paper discusses the H∞ filtering problem for a class of deterministic systems with time-varying delays, where the stochastic property of time-varying delays described by Markovian approach is taken into consideration in filter design. Firstly, the delay interval is separated into several subintervals, which can be described by Markov process. Then, a new H∞ filtering method for deterministic system with time-varying delay is given, whose filter can switch with time delay in terms of Markov process. Sufficient conditions for the existence of H∞ filter are obtained as linear matrix inequalities, where the mode transition rates are known exactly or inexactly. Finally, numerical examples are used to demonstrate the utility of the given methods.
An effective approach is introduced to study the stability of continuous systems with multiple timevarying delay components. By employing a new Lyapunov¿Krasovskii functional form based on delay partitioning, delay-dependent stability criteria are established for cases with or without the information of the delay rates. The contribution of the paper is 2-fold. First, it provides an improvement, as well as generalisation, of the existing stability criteria for continuous systems with multiple time-varying delay components. Second, it is illustrated numerically that the approach can be applied to estimate the delay bound for system stability in the single delay case with reduction both in conservatism and computational complexity when compared with the existing methods.
In this note, the stability analysis and stabilization problems for a class of discrete-time Markov jump linear systems with partially known transition probabilities and time-varying delays are investigated. The time-delay is considered to be time-varying and has a lower and upper bounds. The transition probabilities of the mode jumps are considered to be partially known, which relax the traditional assumption in Markov jump systems that all of them must be completely known
a
priori
. Following the recent study on the class of systems, a monotonicity is further observed in concern of the conservatism of obtaining the maximal delay range due to the unknown elements in the transition probability matrix. Sufficient conditions for stochastic stability of the underlying systems are derived via the linear matrix inequality (LMI) formulation, and the design of the stabilizing controller is further given. A numerical example is used to illustrate the developed theory.
This study is concerned with the problem of robust finite-time filtering for a class of non-linear Markov jump systems (MJSs) with partially known information on the transition jump rates. The non-linearities in the system are parameterised by multilayer neural networks. Our attention is focused on the design of a modedependent full-order H∞ filter to ensure the finite-time boundedness of the filtering error system and a prescribed H∞ attenuation level for all admissible uncertainties and approximation errors of the networks. Sufficient conditions of filtering design are developed in terms of solvability of a set of linear matrix inequalities. A tunnel diode circuit is used to show the effectiveness and potentials of the proposed techniques.
This technical note is concerned with exploring a new approach for the analysis and synthesis for Markov jump linear systems with incomplete transition descriptions. In the study, not all the elements of the transition rate matrices (TRMs) in continuous-time domain, or transition probability matrices (TPMs) in discrete-time domain are assumed to be known. By fully considering the properties of the TRMs and TPMs, and the convexity of the uncertain domains, necessary and sufficient criteria of stability and stabilization are obtained in both continuous and discrete time. Numerical examples are used to illustrate the results.
There has recently been intense interest in spreading sequences of
Markov chains for asynchronous SSMA (spread-spectrum multiple-access)
communication systems. It is experimentally found that the
auto-correlation values of M-phase spreading sequences of some Markov
chains always take real numbers. In this paper, we theoretically give a
necessary and sufficient condition that the auto-correlation values of
M-phase spreading sequences of Markov chains always take real numbers
This paper focuses on the problem of delay-dependent robust stochastic stability analysis and controller synthesis for Markovian jump systems with state and input delays. It is assumed that the delays are constant and unknown, but their upper bounds are known. By constructing a new Lyapunov–Krasovskii functional and introducing some appropriate slack matrices, new delay-dependent stochastic stability and stabilization conditions are proposed by means of linear matrix inequalities (LMIs). An important feature of the results proposed here is that all the robust stability and stabilization conditions are dependent on the upper bound of the delays. Memoryless state feedback controllers are designed such that the closed-loop system is robustly stochastically stable. Some numerical examples are provided to illustrate the effectiveness of the proposed method.
This Letter is concerned with the robust state estimation problem for uncertain time-delay Markovian jumping genetic regulatory networks (GRNs) with SUM logic, where the uncertainties enter into both the network parameters and the mode transition rate. The nonlinear functions describing the feedback regulation are assumed to satisfy the sector-like conditions. The main purpose of the problem addressed is to design a linear estimator to approximate the true concentrations of the mRNA and protein through available measurement outputs. By resorting to the Lyapunov functional method and some stochastic analysis tools, it is shown that if a set of linear matrix inequalities (LMIs) is feasible, the desired state estimator, that can ensure the estimation error dynamics to be globally robustly asymptotically stable in the mean square, exists. The obtained LMI conditions are dependent on both the lower and the upper bounds of the delays. An illustrative example is presented to demonstrate the feasibility of the proposed estimation schemes.
In this paper, the problem of worst case (also called H∞) Control for a class of uncertain systems with Markovian jump parameters and multiple delays in the state and input is investigated. The jumping parameters are modelled as a continuous-time, discrete-state Markov process and the parametric uncertainties are assumed to be real, time-varying and norm-bounded that appear in the state, input and delayed-state matrices. The time-delay factors are unknowns and time-varying with known bounds. Complete results for instantaneous and delayed state feedback control designs are developed which guarantee the weak-delay dependent stochastic stability with a prescribed H∞-performance. The solutions are provided in terms of a finite set of coupled linear matrix inequalities (LMIs). Application of the developed theory to a typical example has been presented.
A sufficient condition for the exponential estimates of a class of Markovian systems with time delay is established for the first time in terms of the linear matrix inequality (LMI) technique. The estimating procedure is implemented by solving an LMI. The proposed condition is also extended to the uncertain case. Moreover, a state feedback stabilizing controller which makes the closed-loop system exponentially stable with a prescribed lower bound of decay rate is designed. Numerical examples are provided to illustrate the effectiveness of the theoretical results.
We examine the H∞ control analysis, the output feedback stabilization, and the output feedback H∞ control synthesis problems for discrete-time state-space symmetric systems. We obtain explicit analytical solutions for the H∞ norm of such systems, and an explicit parametrization of the output feedback stabilizing controllers and H∞ controllers. Extensions to robust and positive real control of such systems are also examined. These results are obtained from the linear matrix inequality formulations of the stabilization and the H∞ control synthesis problems for discrete-time systems using simple matrix algebraic tools. The results have obvious computational advantages, especially for large scale symmetric systems.
In this paper, the stability and stabilization problems of a class of continuous-time and discrete-time Markovian jump linear system (MJLS) with partly unknown transition probabilities are investigated. The system under consideration is more general, which covers the systems with completely known and completely unknown transition probabilities as two special cases — the latter is hereby the switched linear systems under arbitrary switching. Moreover, in contrast with the uncertain transition probabilities studied recently, the concept of partly unknown transition probabilities proposed in this paper does not require any knowledge of the unknown elements. The sufficient conditions for stochastic stability and stabilization of the underlying systems are derived via LMIs formulation, and the relation between the stability criteria currently obtained for the usual MJLS and switched linear systems under arbitrary switching, are exposed by the proposed class of hybrid systems. Two numerical examples are given to show the validity and potential of the developed results.
This paper is concerned with the problem of exponential stabilization for uncertain linear systems with Markovian jump parameters and mode-dependent input delays. Sufficient stabilization conditions are developed in terms of matrix inequalities, which can be solved by a proposed iterative algorithm based on the cone complementarity linearization (CCL) method. Memory controllers are also designed such that the closed-loop system is exponentially mean-square stable for all admissible uncertainties. Numerical examples are given to show that the developed method is efficient and less conservative.
This brief paper is concerned with the robust stabilization problem for a class of Markovian jump linear systems with uncertain switching probabilities. The uncertain Markovian jump system under consideration involves parameter uncertainties both in the system matrices and in the mode transition rate matrix. First, a new criterion for testing the robust stability of such systems is established in terms of linear matrix inequalities. Then, a sufficient condition is proposed for the design of robust state-feedback controllers. A globally convergent algorithm involving convex optimization is also presented to help construct such controllers effectively. Finally, a numerical simulation is used to illustrate the developed theory.
In this note, the static output feedback stabilization problem is addressed using the linear matrix inequality technique. A necessary and sufficient condition for static output feedback stabilizability for linear time-invariant systems is derived in the form of a matrix inequality. The extension of the result to H∞ control is studied. An iterative LMI (ILMI) algorithm is proposed to compute the feedback gain. Numerical examples are employed to demonstrate the effectiveness and the convergence of the algorithm.
In this note, we provided an improved way of constructing a Lyapunov-Krasovskii functional for a linear time delay system. This technique is based on the reformulation of the original system and a discretization scheme of the delay. A hierarchy of Linear Matrix Inequality based results with increasing number of variables is given and is proved to have convergence properties in terms of conservatism reduction. Examples are provided which show the effectiveness of the proposed conditions.
This paper deals with the problems of delay-dependent robust Hinfin control and filtering for Markovian jump linear systems with norm-bounded parameter uncertainties and time-varying delays. In terms of linear matrix inequalities, improved delay-dependent stochastic stability and bounded real lemma (BRL) for Markovian delay systems are obtained by introducing some slack matrix variables. The conservatism caused by either model transformation or bounding techniques is reduced. Based on the proposed BRL, sufficient conditions for the solvability of the robust Hinfin control and Hinfin filtering problems are proposed, respectively. Dynamic output feedback controllers and full-order filters, which guarantee the resulting closed-loop system and the error system, respectively, to be stochastically stable and satisfy a prescribed Hinfin performance level for all delays no larger than a given upper bound, are constructed. Numerical examples are provided to demonstrate the reduced conservatism of the proposed results in this paper.
This work deals with the stabilisation problem of a Markovian jump system with time-delay. Based on the application of the descriptor model transformation and a recent result on bounding of cross products of vectors, a new delay-dependent stabilisation condition using a dynamic output feedback controller is formulated in terms of matrix inequalities. A numerical algorithm is developed to construct a full-order output feedback controller guaranteeing a suboptimal maximal delay such that the system can be stochastically stabilised. In addition, the errors of the paper by Boukas et al. (IEE Proc.-Control Theory Appl., 2002, 149, (5), 379-386) are highlighted.
The stochastic stability and stabilisation problems of a Markov jump system (MJS) with time-delay under a dynamical output feedback control law is considered. LMI-based sufficient conditions are established for the system to be stochastic stable and exponentially stable. A sufficient and necessary condition for the MJS to be stabilisable in the mean square quadratic stable sense, which implies stochastic stability of the system, is established. An LMI-based control design method is proposed and a numerical example is included to show the usefulness of the theoretical results.
There has recently been intense interest in spreading sequences of Markov chains. It is experimentally found that the autocorrelation values of M-phase spreading sequences of some Markov chains always take real numbers. In this paper, we theoretically give a necessary and sufficient condition that the autocorrelation values of M-phase spreading sequences of Markov chains always take real numbers. We also discuss time synchronization using these sequences.
This note addresses the problem of H∞ filtering for continuous-time linear systems with Markovian jumping parameters. The main contribution of the note is to provide a method for designing an asymptotically stable linear time-invariant H∞ filter for systems where the jumping parameter is not accessible. The cases where the transition rate matrix of the Markov process is either exactly known, or unknown but belongs to a given polytope, are treated. The robust H∞ filtering problem for systems with polytopic uncertain matrices is also considered and a filter design method based on a Lyapunov function that depends on the uncertain parameters is developed. The proposed filter designs are given in terms of linear matrix inequalities.
We consider continuous time switched systems that are stabilized via a computer. Several factors (sampling, computer computation, communications through a network, etc.) introduce model uncertainties produced by unknown varying feedback delays. These uncertainties can lead to instability when they are not taken into account. Our goal is to construct a switched digital control for continuous time switched systems that is robust to the varying feedback delay problem. The main contribution of this note is to show that the control synthesis problem in the context of unknown time varying delays can be expressed as a problem of stabilizability for uncertain systems with polytopic uncertainties
This note concerns the robust guaranteed cost control problem for a class of continuous-time Markovian jump linear system with norm-bounded uncertainties and mode-dependent time-delays. The problem is to design a memoryless state feedback control law such that the closed-loop system is stochastically stable and the closed-loop cost function value is not more than a specified upper bound for all admissible uncertainties. Based on linear matrix inequality, delay-dependent sufficient conditions for the existence of such controller are derived by using a descriptor model transformation of the system and by applying Moon's inequality for bounding cross terms. Sufficient conditions which depend on the difference between the largest and the smallest time-delays are also presented. Two numerical examples are given for illustration of the proposed theoretical results.
This paper studies the problem of control for discrete time delay
linear systems with Markovian jump parameters. The system under
consideration is subjected to both time-varying norm-bounded parameter
uncertainty and unknown time delay in the state, and Markovian jump
parameters in all system matrices. We address the problem of robust
state feedback control in which both robust stochastic stability and a
prescribed H∞ performance are required to be achieved
irrespective of the uncertainty and time delay. It is shown that the
above problem can be solved if a set of coupled linear matrix
inequalities has a solution