Jump Linear Systems Research Articles

JLS-PPC: A Jump Linear System Framework for Networked Control

We present a unified formalism for multiagent networked control and estimation with scheduling, delays, and packet loss in the communication channels between the controller and distributed sensors and actuators. The modular framework is a combined construction of a stochastic jump linear system (JLS) description of the plant and network effects, a Kalman filter-based estimator, and packetized predictive control, a receding horizon optimization technique with buffering at the actuator. Integration of these elements enables the synthesis of a novel estimation technique that generalizes prior approaches for control and measurement packet loss to the case with schedules, delays, control buffering, and most importantly, delayed and lossy control packet acknowledgments (ACKs). The JLS framework allows a clean separation of jump variable estimation and a posteriori state estimation using a backup-and-rerun strategy, and can handle variable-length ACK histories for multiple independent control communication channels. Finally, we derive modified covariance priors for the filter that account for uncertainty in the control action applied at the vehicle when ACKs are not available and control buffers are used. Simulations with single vehicle and multivehicle systems demonstrate the methods and show the benefits of utilizing delayed and lossy ACKs.

Stochastic Stability and Stabilization of Semi-Markov Jump Linear Systems with Uncertain Transition Rates

This paper investigates the problems of robust stochastic stability and stabilization for a general class of continuous-time semi-Markovian jump linear systems (S-MJLSs). The main contribution of the research is to eliminate the limitations of the traditional S-MJLSs with precisely available information by introducing a system with uncertain, time-varying transition rates (TRs) of the jump process, in addition to the imperfect information on the system dynamic matrices. The new system is called the general uncertain semi-Markov jump linear system (GUS-MJLS); it does not contain certain values of the transition rates, but includes nominal time-dependent values in addition to bounded deviations. It is suitable to describe a broader class of dynamical systems with estimated information and modeling errors and also covers the concepts of Markov jump linear system (MJLSs) with time-constant and certain TRs. For this system, the stability is firstly analyzed through the multiple stochastic Lyapunov function approach. Then, based on the stability results, a robust state-feedback controller is formulated. To deal with the time-dependent TRs, a sojourn-time fractionizing technique is used and numerically testable conditions are developed. Finally, discussions on reducing the conservativeness of the robust theorems are provided. The theoretical results are successfully tested on an industrial continuous stirred tank reactor (CSTR) subject to stochastically varying environmental conditions. Comparative simulations are also provided to show the superiority of the presented framework and design method to the existing ones.DOI: http://dx.doi.org/10.5755/j01.itc.46.1.13881

Almost Sure Stability of Switching Markov Jump Linear Systems

Recently a special hybrid system called Switching Markov Jump Linear System (SMJLS) is studied. A SMJLS is subject to a deterministic switching and a stochastic Markovain switching. To extend the results already obtained and to investigate some new aspects of such systems, our main contributions in this paper are: i) Transient analysis of Markov process, i.e., the expectations of the sojourn time, the activation number of any mode, and the number of switchings between any two modes and ii) two sufficient conditions of the exponential almost sure stability for a general SMJLS. Different from previous work, which is a special case of our study, the transition rate matrix for the random Markov process in our study is not fixed, but varies when a deterministic switching takes place.

Networked Control Systems Application: Minimization of the global number of interactions, transmissions and receptions in Multi-Hop network using Discrete-Time Markovian Jump Linear Systems

This paper analyzes the Trade-Off between the global number of interactions, composed by the number of transmissions and receptions in the network, and the H-infinity norm degradation. Specifically on this study we implemented an optimum H-infinity filter in a semi-reliable Multi-Hop network modeled by Discrete-Time Markovian Jump Linear Systems. For this particular case, it was possible guarantee the filter stability with a global number the interaction of only 17.93% of packet that would be necessary for the same task using the classic filter.

Stability and Disturbance Attenuation for Markov Jump Linear Systems with Time-Varying Transition Probabilities

We address stability and disturbance attenuation for Markov jump linear systems where transition probabilities vary in a finite set. The time variation of the transition probabilities may be a priori known or unknown. Necessary and sufficient conditions for uniform stochastic stability and uniform stochastic disturbance attenuation are reported. In both cases, conditions are expressed as a set of finite-dimensional linear matrix inequalities that can be solved efficiently.

Average Reachability of Continuous-time Markov Jump Linear Systems and the Linear Minimum Mean Square Estimator

In this paper we study the average reachability gramian for continuous-time linear systems with additive noise and jump parameters driven by a general Markov chain. We define a rather natural reachability concept by requiring that the average reachability gramian be positive definite. Aiming at a testable condition, we introduce a set of reachability matrices for this class of systems and employ invariance properties of the null space of the noise coefficient matrices to show that the system is reachable if and only if these matrices are of full rank. We also show for reachable systems that the state second moment is positive definite. One consequence of this result in the context of linear minimum mean square state estimation for reachable systems is that the expectation of the error covariance matrix is positive definite. Moreover, the average boundedness of the error covariance matrix is invariant to a type of perturbation in the noise model, meaning that the estimates are not overly sensitive, which consists in a property that is desirable in applications and sometimes referred to as stability of the estimator.

Leakage-delay-dependent stability analysis of Markovian jumping linear systems with time-varying delays and nonlinear perturbations

This study is concerned with the problem of leakage-delay-dependent stability analysis of Markovian jumping linear systems with time-varying delays and nonlinear perturbations. Mixed time-varying delays, which include leakage, discrete, and distributed delays, pertaining to the proposed system are addressed. Based on an improved Lyapunov–Krasovskii functional with triple integral terms and by employing the model transformation technique and the reciprocal convex method, the sufficient conditions for the delay-dependent stability of the considered system are derived. Moreover, the sufficient conditions obtained are formulated in terms of linear matrix inequalities to achieve the global asymptotical stability in mean square of the considered delayed system. Finally, numerical examples are provided to demonstrate the less conservatism and effectiveness of the proposed results.

Mean Stability of Positive Markov Jump Linear Systems With Homogeneous and Switching Transition Probabilities

This brief investigates the mean stability problem of positive Markov jump linear systems (PMJLSs) in the discrete- time domain. First, some sufficient and necessary conditions are presented for PMJLSs with homogeneous transition probability (TP) by analyzing the time evolution of the first-order moment of the state. Then, by using a copositive Lyapunov function approach, a computable sufficient condition for the PMJLSs with switching TPs is proposed in the framework of dwell time to guarantee the mean stability. Finally, some numerical examples are given to demonstrate the effectiveness of the obtained theoretical results.

LQ Nash Games With Random Entrance: An Infinite Horizon Major Player and Minor Players of Finite Horizons

We study Dynamic Games with randomly entering players, staying in the game for different lengths of time. Particularly, a class of Discrete Time Linear Quadratic (LQ) Games, involving a major player who has an infinite time horizon and a random number of minor players is considered. The number of the new minor players, entering at some instant of time, is random and it is described by a Markov chain. The problem of the characterization of a Nash equilibrium, consisting of Linear Feedback Strategies, is reformulated as a set of coupled finite and infinite horizon LQ optimal control problems for Markov Jump Linear Systems (MJLS). Sufficient conditions characterizing Nash equilibrium are then derived. The problem of Games involving a large number of minor players is then addressed using a Mean Field (MF) approach and asymptotic ε-Nash equilibrium results are derived. Sufficient conditions for the existence of a MF Nash equilibrium are finally stated.

A Detector-Based Approach for the <inline-formula> <tex-math notation="TeX">$H_{2} $</tex-math></inline-formula> Control of Markov Jump Linear Systems With Partial Information

In this paper, we study the $H_{2} $ -control for discrete-time Markov Jump Linear Systems (MJLS) with partial information . We consider the case in which we do not have access to the Markov jump parameter but, instead, there is a detector that emits signals which provides information on this parameter. A salient feature of our formulation is that it encompasses, for instance, the cases with perfect information, no information and cluster observations of the Markov parameter, which were previously analyzed in the Markov jump control literature. The goal is to derive a feedback linear control using the information provided by the detector in order to stochastically stabilize the closed loop system. We present two Lyapunov like equations for the stochastic stability of the system. In addition, we show that a Linear Matrix Inequalities (LMI) formulation can be obtained in order to design a stochastically stabilizing feedback control. In the sequel we deal with the $H_{2} $ control problem and we show that, again, an LMI optimization problem can be formulated in order to design a stochastically stabilizing feedback control with guaranteed $H_{2} $ -cost. We also present two special cases, one of them always satisfied for the limit case in which the detector provides perfect information on the Markov parameter, and the Bernoulli jump case, under which LMI conditions become necessary and sufficient for the stochastic stabilizability of the system and the LMI optimization problems provide the optimal $H_{2} $ cost. For the Bernoulli jump case we show that our formulation generalizes previous ones. The case with convex polytopic uncertainty on the parameters of the system and on the transition probability matrix is also considered. The paper is concluded with some numerical examples.

Optimal [formula omitted] state feedback sampled-data control design of Markov Jump Linear Systems

This paper is entirely devoted to analyze and solve the H2 optimal state feedback sampled-data control design problem of Markov Jump Linear Systems. Several theoretical aspects are addressed, in particular, mean square stability and performance optimization. The numerical procedure follows by embedding the system under consideration into a more general class of dynamical systems named Hybrid Markov Jump Linear System. Hence, the state feedback sampled-data control gains are calculated from the solution of a new and specific two-point boundary value problem. A global uniformly convergent algorithm based exclusively on linear matrix inequality is proposed. The theory is illustrated by means of practical examples.

A New Approach to Static Output Control of Uncertain Continuous Markov Jump Linear Systems

This paper is concerned with the static output stabilization of uncertain continuous Markov jump linear systems with partly known transition probabilities. Uncertainties in system output matrices are addressed in terms of norm-bounded and polytopic formulations. Adopting a new separation technique, sufficient conditions for designing static output controllers are established in terms of linear matrix inequalities. Compared with the existing results, the proposed methods do not impose equality constraints and calculate the null space of output matrices. A numerical example is given to show the effectiveness of the proposed method.

Robust Stabilization of Markovian Jump Linear Singular Systems with Wiener Process and Generally Incomplete Transition Rates

This paper is devoted to the investigation of robust stability and robust stabilizability for a class of uncertain Markovian jump continuous-time singular systems with Wiener process and generally uncertain transition rates (GUTRs). This new uncertain model is more general than the existing ones and can be applicable to more practical situations because of its uncertain parameters and transition rates. Each transition rate can be completely unknown, or only its estimate value is known. Our interests are focused on establishing sufficient criteria on robust stability and the design of a state feedback controller such that the robust stochastic stability is guaranteed, even if the singular system incorporates GUTRs, Wiener process disturbance and norm-bounded uncertainties. These sufficient criteria are derived in terms of linear matrix inequalities. Finally, a numerical example is presented to illustrate the effectiveness and applicability of the proposed method.

Stabilization of Markov Jump Linear Systems with Input Quantization

This paper addresses the feedback stabilization problem for Markov jump linear systems with quantized control input. A simple but powerful input controller for eliminating the input quantization in Markov jump linear systems is used. The proposed mode-dependent input controller comprises two parts: the linear part and the nonlinear part, where the linear part determines the fundamental characteristics of the systems, and the nonlinear part eliminates the effect of the input quantization. The stability is analyzed in detail for both the discrete-time and continuous-time Markov jump linear systems with input quantization. The sufficient conditions and the specific mode-dependent input controller to guarantee the stability of the systems are obtained. Numerical examples are given to illustrate the effectiveness of the results.

Co-design of controller and routing redundancy over a wireless network

In this paper we investigate the exploitation of redundancy when routing actuation data to a discrete-time LTI system connected to the controller via a wireless network affected by packet drops. We assume that actuation packets can be delivered from the controller to the actuator via multiple paths, each associated with a delay and a packet loss probability. We show that the joint design of controller gain and routing redundancy exploitation can tremendously improve the control performance. To achieve this goal we set up and solve a LQR problem for a class of systems that extends discrete-time Markov Jump Linear Systems, in that both continuous and discrete control signals can be actuated.

Further Results on Stability Analysis of Discrete-Time Markov Jump Linear Systems with Time-Varying Delay and Partly Known Transition Probabilities

This paper is concerned with the stability analysis of discrete-time Markov jump linear systems (MJLSs) with time-varying delay and partly known transition probabilities. The time delay is varying between lower and upper bounds, and the partly known transition probabilities cover the cases of known, uncertain with known lower and upper bounds, and completely unknown, which is more general than the existing result. Via constructing an appropriate Lyapunov function and employing a new technique to separate Lyapunov variables from unknown transition probabilities, a novel stability criterion is obtained in the framework of linear matrix inequality. A numerical example is given to show the effectiveness of the proposed approach.

Stabilisation of Discrete-Time Piecewise Homogeneous Markov Jump Linear System with Imperfect Transition Probabilities

This paper is devoted to investigating the stability and stabilisation problems for discrete-time piecewise homogeneous Markov jump linear system with imperfect transition probabilities. A sufficient condition is derived to ensure the considered system to be stochastically stable. Moreover, the corresponding sufficient condition on the existence of a mode-dependent and variation-dependent state feedback controller is derived to guarantee the stochastic stability of the closed-loop system, and a new method is further proposed to design a static output feedback controller by introducing additional slack matrix variables to eliminate the equation constraint on Lyapunov matrix. Finally, some numerical examples are presented to illustrate the effectiveness of the proposed methods.

Stochastic Nash Games for Markov Jump Linear Systems with State- and Control-Dependent Noise

This paper investigates Nash games for a class of linear stochastic systems governed by Ito’s differential equation with Markovian jump parameters both in finite-time horizon and infinite-time horizon. First, stochastic Nash games are formulated by applying the results of indefinite stochastic linear quadratic (LQ) control problems. Second, in order to obtain Nash equilibrium strategies, cross-coupled stochastic Riccati differential (algebraic) equations (CSRDEs and CSRAEs) are derived. Moreover, in order to demonstrate the validity of the obtained results, stochastic H 2/H ∞ control with state- and control-dependent noise is discussed as an immediate application. Finally, a numerical example is provided.

Almost sure stability of discrete‐time Markov jump linear systems

This paper deals with transient analysis and almost sure stability for discrete-time Markov Jump Linear System (MJLS). The expectation of sojourn time and activation number of any mode, and switching number between any two modes of discrete-time MJLS are presented firstly. Then a result on transient behavior analysis of discrete-time MJLS is given. Finally a new deterministically testable condition for the exponential almost sure stability of discrete-time MJLS is proposed.

An auto-framing method for stochastic process signal by using a hidden Markov model based approach

In this paper, an “auto-framing” method, an algorithmic method to divide stochastic time-series process data into appropriate intervals, is developed based on the approach of hidden Markov model (HMM). While enormous amounts of process time-series data are being measured and collected today, their use is limited by the high costs to gather, store, and analyze them. “Data-framing” refers to the task of dividing stochastic signal data into time frames of distinct patterns so that the data can be stored and analyzed in an efficient manner. Data-framing is typically carried out manually, but doing so can be both laborious and ineffective. For the purpose of automating the data-framing task, stochastic signals of switching patterns are modeled using a hidden Markov model (HMM) based jump linear system (JLS), which switches the stochastic model probabilistically in accordance with the underlying Markov chain. Based on the model, an estimator is constructed to estimate from the collected signal data the state sequence of the underlying Markov chain, which is subsequently used to decide on the framing points. An Expectation Maximization (EM) algorithm, which is composed of two optimal estimators, fixed interval Kalman smoother and Viterbi algorithm, is used to estimate for the state estimation. We demonstrate the effectiveness of the HMM-based approach for auto-framing using simulated data constructed based on real industrial data.