Jump Markov Nonlinear Systems Research Articles

Rao–Blackwellised particle filtering and smoothing for jump Markov non‐linear systems with mode observation

This study is concerned with the problem of filtering and fixed-lag smoothing for jump Markov non-linear systems when the mode information can be extracted from an image sensor. Based on the idea of Rao–Blackwellisation, the authors present a general theoretical framework to derive the recursive estimates by employing the particle filtering method. A suboptimal image-enhanced Rao–Blackwellised particle filter is proposed, in which the mode state is estimated by using random sampling and the continuous state as well as the relevant likelihood function are approximated as Gaussian distributions. The one-step fixed-lag smoothing result is also obtained for such systems with lagged mode observations. Performance comparison of the proposed algorithms with the existing methods is provided through a manoeuvring target tracking simulation study.

Optimal finite-time passive controller design for uncertain nonlinear Markovian jumping systems

This paper studies the optimal finite-time passive control problem for a class of uncertain nonlinear Markovian jumping systems (MJSs). The Takagi and Sugeno (T–S) fuzzy model is employed to represent the nonlinear system with Markovian jump parameters and norm-bounded uncertainties. By selecting an appropriate Lyapunov-Krasovskii functional, it gives a sufficient condition for the existence of finite-time passive controller such that the uncertain nonlinear MJSs is stochastically finite-time bounded for all admissible uncertainties and satisfies the given passive control index in a finite time-interval. The sufficient condition on the existence of optimal finite-time fuzzy passive controller is formulated in the form of linear matrix inequalities and the designed algorithm is described as an optimization one. A numerical example is given at last to illustrate the effectiveness of the proposed design approach.

Full-State Linearization and Stabilization of SISO Markovian Jump Nonlinear Systems

This paper investigates the linearization and stabilizing control design problems for a class of SISO Markovian jump nonlinear systems. According to the proposed relative degree set definition, the system can be transformed into the canonical form through the appropriate coordinate changes followed with the Markovian switchings; that is, the system can be full-state linearized in every jump mode with respect to the relative degree setn,…,n. Then, a stabilizing control is designed through applying the backstepping technique, which guarantees the asymptotic stability of Markovian jump nonlinear systems. A numerical example is presented to illustrate the effectiveness of our results.

Bayesian Filtering for Non-linear Markov Jump Systems with Non-homogeneous Transition Probabilities

摘要: 针对具有时变不确定转移概率的非线性非齐次Markov跳变系统, 提出一种贝叶斯状态估计方法.该方法首次采用带约束高斯概率密度函数来刻画转移概率的真实特性. 然后,基于参考概率空间法, 将实际的概率测度投影到理想概率空间, 得出信息变量的递归表达式. 同时, 在贝叶斯框架内给出转移概率矩阵的最大后验估计式. 进一步, 采用粒子逼近法求解转移概率矩阵的最大后验估计, 解决非线性函数的多重积分问题, 进而获取状态估计值. 最后, 通过一个仿真示例表明该方法的有效性. 关键词: 非线性非齐次Markov跳变系统 / 贝叶斯估计 / 参考概率空间 / 粒子逼近

Improved fuzzy control design for nonlinear Markovian-jump systems with incomplete transition descriptions

This paper addresses the state feedback control of nonlinear continuous-time, Markovian-jump systems. The nonlinearity is represented by Takagi–Sugeno fuzzy models and the transition probability matrix is assumed to be partly known: some elements in the matrix are known, some are unknown but with known lower and upper bounds, and some are completely unknown. By making full use of the continuous property of the transition probability matrix, new sufficient conditions for the stochastic stability of the system are obtained in terms of linear matrix inequalities. We show that the conditions given are less conservative than or at least the same as those for existing results. Moreover, using the conditions obtained, we establish a method for design of a H∞ state feedback controller. Numerical examples illustrate the effectiveness of the proposed method.

Stochastic Admissibility for a Class of Nonlinear Singular Markovian Jump Systems with Time-Delay

This paper is concerned with the problem of the stochastic admissibility for a class of nonlinear singular Markovian jump systems with time-delay by employing the free-weighting matrix method and the Lyapunov functional method. The singular matrices Er(t)in the discussed system are mode-dependent. A sufficient condition is presented to guarantee the considered system to be stochastically admissible. The proposed result is formulated in terms of linear matrix inequalities(LMIs). A numerical example is provided to verify the effectiveness of the obtained method.

Stochastic Admissibility for a Class of Nonlinear Singular Markovian Jump Systems with Time-Delay and Partially Unknown Transition Probabilities

The problem of the stochastic admissibility for a class of nonlinear singular Markovian jump systems with time-delay and partially unknown transition probabilities is discussed in this note. The considered singular matrices Er(t) in the discussed system are mode-dependent. By using the free-weighting matrix method and the Lyapunov functional method, a sufficient condition which guarantees the considered system to be stochastically admissible is presented in the form of linear matrix inequalities(LMIs). Finally, a numerical example is given to show the effectiveness of the presented method.

H ∞ filtering for nonlinear singular Markovian jumping systems with interval time-varying delays

The problem of H ∞ filtering for nonlinear singular Markovian jumping systems with interval time-varying delays is investigated. The delay factor is assumed to be time-varying and belongs to a given interval, which means that the lower and upper bounds of the interval time-varying delays are available. Furthermore, the derivative of the time-varying delay function can be larger than one. With partial knowledge of the jump rates of the Markov process, a new delay-range-dependent bounded real lemma for the solvability of the jump system is obtained based on the Lyapunov–Krasovskii functional, which is in terms of strict linear matrix inequalities (LMIs). When these LMIs are feasible, an expression of a desired H ∞ filter is given. Numerical examples are given to illustrate the effectiveness of the developed techniques.

Consensus-Based Distributed Multiple Model UKF for Jump Markov Nonlinear Systems

This note studies the problem of distributed estimation for jump Markov nonlinear systems (JMNLSs) in a not fully connected sensor network. Based on the consensus theory, a distributed unscented Kalman filter (UKF) is first derived for nonlinear systems without jumping parameters and then it is extended to develop a distributed multiple model UKF for JMNLSs. The proposed filtering algorithm is illustrated via a simulation example involving tracking a maneuvering target.

Exponential Passification of Markovian Jump Nonlinear Systems with Partially Known Transition Rates

The problems of delay‐dependent exponential passivity analysis and exponential passification of uncertain Markovian jump systems (MJSs) with partially known transition rates are investigated. In the deterministic model, the time‐varying delay is in a given range and the uncertainties are assumed to be norm bounded. With constructing appropriate Lyapunov‐Krasovskii functional (LKF) combining with Jensen’s inequality and the free‐weighting matrix method, delay‐dependent exponential passification conditions are obtained in terms of linear matrix inequalities (LMI). Based on the condition, desired state‐feedback controllers are designed, which guarantee that the closed‐loop MJS is exponentially passive. Finally, a numerical example is given to illustrate the effectiveness of the proposed approach.

H∞ Filtering for a Class of Piecewise Homogeneous Markovian Jump Nonlinear Systems

H∞ filtering problem for a class of piecewise homogeneous Markovian jump nonlinear systems is investigated. The aim of this paper is to design a mode‐dependent filter such that the filtering error system is stochastically stable and satisfies a prescribed H∞ disturbance attenuation level. By using a new mode‐dependent Lyapunov‐Krasovskii functional, mixed mode‐dependent sufficient conditions on stochastic stability are formulated in terms of linear matrix inequalities (LMIs). Based on this, the mode‐dependent filter is obtained. A numerical example is given to illustrate the effectiveness of the proposed main results.

Risk-sensitive filtering for nonlinear Markov jump systems on the basis of particle approximation

SUMMARY In this paper, the risk-sensitive filtering method that relaxes the dependence on model accuracy is extended to nonlinear Markov jump systems (MJSs). In the method, the so-called reference probability technique together with particle approximation is utilized to derive the risk-sensitive filter in nonlinear non-Gaussian framework. The novelty of the proposed approach is that a ‘risky’ interacting resampling step is performed to both moderate the modeling uncertainties and to solve the problem of particle explosion. A designer-chosen parameter named risk-sensitive parameter allows us to make a trade-off between the filtering accuracy for the nominal model and the robustness to uncertainties. With a meaningful example, it shows that the developed method can outperform the widely used method-particle filter and interacting multiple model-particle filter in nonlinear MJSs with uncertainties. Copyright © 2011 John Wiley & Sons, Ltd.

Rao-Blackwellised unscented particle filtering for jump Markov non-linear systems: an H ∞ approach

In this study, a robust Rao-Blackwellised particle filter (RBPF) is proposed for jump Markov non-linear systems (JMNLSs) with unknown noise statistics. A non-linear filter is presented by applying the unscented transform technique in the H∞ setting, which is used to update the continuous-state particles in the RBPF framework. Moreover, a way to adaptively adjust the disturbance tolerance level for performance requirement is presented. Simulation results using the proposed approach are also presented.

Risk-Sensitive Cubature Filtering for Jump Markov Nonlinear Systems and Its Application to Land Vehicle Positioning⋆

In this paper, a risk-sensitive filter for a class of jump Markov nonlinear systems is proposed. By using a set of weighted cubature points to approximate the intractable risk-sensitive recursions, the proposed filter could exhibit improved numerical stability with respect to the derivation of the square-root covariance, for which the unscented transform technique might halt the operations due to the presence of non-positive definite covariance. The performance of the proposed filter is illustrated through a simulation study of tracking a land vehicle with maneuvers.

Offline Bayesian Identification of Jump Markov Nonlinear Systems

AbstractThis paper presents a framework for the offline identification of nonlinear switched systems with unknown model structure. Given a set of sampled trajectories, and under the assumption that they were generated by switching among a number of models, we estimate a set of vector fields and a stochastic switching mechanism that best describes the observed data. The switching mechanism is described by a position dependent hidden Markov model that provides the probabilities of the next active model given the current active model and the state vector. The vector fields and the stochastic matrix is obtained by interpolating a set of nodes distributed over a relevant region in the state space. The work follows a Bayesian formulation where the EM-algorithm is used for optimization.

Risk-sensitive filtering and smoothing for jump Markov non-linear systems based on unscented transform

This study is concerned with risk-sensitive filtering and smoothing for a class of discrete-time jump Markov non-linear systems. Using the so-called reference probability method, the authors present a general theoretical framework to yield recursions for deriving filtered and smoothed estimates through identifying the approximations made by the interacting multiple model (IMM) estimation approach. A suboptimal risk-sensitive filtering algorithm is developed by applying the unscented transform (UT) technique and the one-step fixed-lag smoothing result is also presented for such systems. The effectiveness of the proposed algorithms is demonstrated via a manoeuvering target tracking simulation study.

Robust Finite-Time H ∞ Control for Nonlinear Jump Systems via Neural Networks

This paper presents a neural network-based robust finite-time H ∞ control design approach for a class of nonlinear Markov jump systems (MJSs). The system under consideration is subject to norm bounded parameter uncertainties and external disturbance. In the proposed framework, the nonlinearities are initially approximated by multilayer feedback neural networks. Subsequently, the neural networks undergo piecewise interpolation to generate a linear differential inclusion model. Then, based on the model, a robust finite-time state-feedback controller is designed such that the nonlinear MJS is finite-time bounded and finite-time stabilizable. The H ∞ control is specified to ensure the elimination of the approximation errors and external disturbances with a desired level. The controller gains can be derived by solving a set of linear matrix inequalities. Finally, simulation results are given to illustrate the effectiveness of the developed theoretic results.

Switching Controller Design for a Class of Markovian Jump Nonlinear Systems Using Stochastic Small-Gain Theorem

Switching controller design for a class of Markovian jump nonlinear systems with unmodeled dynamics is considered in this paper. Based on the differential equation and infinitesimal generator of jump systems, the concept of Jump Input-to-State practical Stability (JISpS) in probability and stochastic Lyapunov stability criterion are put forward. By using backsetpping technology and stochastic small-gain theorem, a switching controller is proposed which ensures JISpS in probability for the jump nonlinear system. A simulation example illustrates the validity of this design.

Mode-independent robust stabilization for uncertain Markovian jump nonlinear systems via fuzzy control

This paper is concerned with the robust-stabilization problem of uncertain Markovian jump nonlinear systems (MJNSs) without mode observations via a fuzzy-control approach. The Takagi and Sugeno (T-S) fuzzy model is employed to represent a nonlinear system with norm-bounded parameter uncertainties and Markovian jump parameters. The aim is to design a mode-independent fuzzy controller such that the closed-loop Markovian jump fuzzy system (MJFS) is robustly stochastically stable. Based on a stochastic Lyapunov function, a robust-stabilization condition using a mode-independent fuzzy controller is derived for the uncertain MJFS in terms of linear matrix inequalities (LMIs). A new improved LMI formulation is used to alleviate the interrelation between the stochastic Lyapunov matrix and the system matrices containing controller variables in the derivation process. Finally, a simulation example is presented to illustrate the effectiveness of the proposed design method.

Efficient particle filter for jump Markov nonlinear systems

An efficient particle filtering algorithm is presented for state estimation of jump Markov nonlinear systems. The standard particle filter can easily suffer from particle degeneracy around mode changes. The new algorithm allows the user to fully control the number of particles in a certain mode, without distorting or violating the Markov property. This property can be exploited to prevent the filter degenerating.