H∞ control of continuous-time Markov jump linear systems with detector-based mode information†

This paper features new results on H_∞ analysis and control of linear systems with Markov jump disturbances, in a scenario of partial observations of the jump process. We consider the situations in which the jump process can only be measured through a suitable detector. A distinctive feature of the approach here is that it is general enough to encompass particular scenarios such as that of perfect information, no information (mode independent) and cluster observations of the Markov jump process. The main results, comprising a new bounded real lemma and a condition for state feedback synthesis, are expressed via linear matrix inequality-based optimisation problems. The method devised for the design of H_∞ controllers is applied to the control of an unmanned aerial vehicle model.     

A new look at the robust control of discrete-time Markov jump linear systems

In this paper, we make a foray in the role played by a set of four operators on the study of robust H₂ and mixed H₂/H_∞ control problems for discrete-time Markov jump linear systems. These operators appear in the study of mean square stability for this class of systems. By means of new linear matrix inequality (LMI) characterisations of controllers, which include slack variables that, to some extent, separate the robustness and performance objectives, we introduce four alternative approaches to the design of controllers which are robustly stabilising and at the same time provide a guaranteed level of H₂ performance. Since each operator provides a different degree of conservatism, the results are unified in the form of an iterative LMI technique for designing robust H₂ controllers, whose convergence is attained in a finite number of steps. The method yields a new way of computing mixed H₂/H_∞ controllers, whose conservatism decreases with iteration. Two numerical examples illustrate the applicability of the proposed results for the control of a small unmanned aerial vehicle, and for an underactuated robotic arm.     

Reduced-order filtering for linear systems with Markovian jump parameters

This paper addresses the reduced-order H_∞ filtering problem for continuous-time Makovian jump linear systems, where the jump parameters are modelled by a discrete-time Markov process. Sufficient conditions for the existence of the reduced-order H_∞ filter are proposed in terms of linear matrix inequalities (LMIs) and a coupling non-convex matrix rank constraint. In particular, the sufficient conditions for the existence of the zero-order H_∞ filter can be expressed in terms of a set of strict LMIs. The explicit parameterization of the desired filter is also given. Finally, a numerical example is given to illustrate the proposed approach.     

H 2 state feedback controller design for continuous Markov jump linear systems with partly known information

This article addresses the H ₂ control problem for continuous Markov jump linear systems with partly known information. The considered partly known transition probabilities cover the cases where the transition probabilities are exactly known, unknown and unknown but with known bounds. By decoupling the unknown transition probabilities from the Lyapunov matrices, new sufficient conditions for the H ₂ performance analysis of the considered systems are derived in terms of linear matrix inequalities (LMIs). Based on the result, an LMI-based method for designing H ₂ controllers is given. Two numerical examples are presented to illustrate the effectiveness of the proposed methods.     

Robust H∞ control of uncertain stochastic Markovian jump systems with mixed time-varying delays

In this paper, robust H_∞ control for a class of uncertain stochastic Markovian jump systems (SMJSs) with interval and distributed time-varying delays is investigated. The jumping parameters are modelled as a continuous-time, finite-state Markov chain. By employing the Lyapunov-Krasovskii functional and stochastic analysis theory, some novel sufficient conditions in terms of linear matrix inequalities are derived to guarantee the mean-square asymptotic stability of the equilibrium point. Numerical simulations are given to demonstrate the effectiveness and superiority of the proposed method comparing with some existing results.     

H2 control of discrete-time periodic systems with Markovian jumps and multiplicative noise

This paper addresses the problem of optimal and robust H₂ control for discrete-time periodic systems with Markov jump parameters and multiplicative noise. To analyse the system performance in the presence of exogenous random disturbance, an H₂ norm is firstly established on the basis of Gramian matrices. Further, under the condition of exact observability, a necessary and sufficient condition is presented for the solvability of H₂ optimal control problem by means of a generalised Riccati equation. When the transition probabilities of jump parameter are incompletely measurable, an H₂-guaranteed cost norm is exploited and the robust H₂ controller is designed through a linear matrix inequality (LMI) optimisation approach. An example of a networked control system is supplied to illustrate the proposed results.     

Robust H∞ estimation of stationary discrete-time linear processes with stochastic uncertainties

The problem of H_∞ filtering of stationary discrete-time linear systems with stochastic uncertainties in the state space matrices is addressed, where the uncertainties are modeled as white noise. The relevant cost function is the expected value, with respect to the uncertain parameters, of the standard H_∞ performance. A previously developed stochastic bounded real lemma is applied that results in a modified Riccati inequality. This inequality is expressed in a linear matrix inequality form whose solution provides the filter parameters. The method proposed is applied also to the case where, in addition to the stochastic uncertainty, other deterministic parameters of the system are not perfectly known and are assumed to lie in a given polytope. The problem of mixed H₂/H_∞ filtering for the above system is also treated. The theory developed is demonstrated by a simple tracking example.     

The H2-control for jump linear systems: cluster observations of the Markov state

The H₂-norm control problem of discrete-time Markov jump linear systems is addressed in this paper when part of, or the total of the Markov states is not accessible to the controller. The non-observed part of the Markov states is grouped in a number of clusters of observations; the case with a single cluster retrieves the situation when no Markov state is observed. The control action is provided in linear feedback form, which is invariant on each cluster, and this restricted complexity setting is adopted, aiming at computable solutions. We explore a recent result by de Oliveira, Bernussou, and Geromel (Systems Control Lett. 37 (1999) 261) involving an LMI characterization to establish a H₂ solution that is stabilizing in the mean square sense. The novelty of the method is that it can handle in LMI form the situation ranging from no Markov state observation to complete state observation. In addition, when the state observation is complete, the optimal H₂-norm solution is retrieved.     

Further results on H∞ control for discrete-time Markovian jump time-delay systems

This paper studies the problem of H_∞ control for a class of discrete-time Markovian jump systems with time delay. The purpose is to improve the existing results on H_∞ controller design for Markovian jump systems. A novel summation inequality is presented and an improved stability criterion for the system is derived by utilising the new inequality, which is proved to be less conservative than most results in the literature. Then the state feedback controller is designed, which guarantees the stochastic stability of the closed-loop system with a given disturbance attenuation. Numerical examples are provided to illustrate the effectiveness and advantages of the proposed techniques.     

Reliable H∞ control for nonlinear discrete-time systems with multiple intermittent faults in sensors or actuators

This study proposes a fault-tolerant control method for stochastic systems with multiple intermittent faults (IFs) and nonlinear disturbances, and both sensor and actuator faults are considered. The occurrence and disappearance of IFs are governed by Markov chain, and its transition probabilities are partly known. Hence, the faulty system can be described by a Markovian jump system (MJS). In order to ensure that the MJS is stochastically stable and satisfies H_∞ performance index, mode-dependent output feedback controllers are modelled using linear matrix inequalities. Numerous sufficient conditions for stochastic stability are obtained on the basis of Lyapunov stability theory. Finally, the effectiveness of the developed method is evaluated on the three-tank system.     

Mixed H2/H∞ pitch control of wind turbine with a Markovian jump model

This paper proposes a Markovian jump model and the corresponding H₂/H_∞ control strategy for the wind turbine driven by the stochastic switching wind speed, which can be used to regulate the generator speed in order to harvest the rated power while reducing the fatigue loads on the mechanical side of wind turbine. Through sampling the low-frequency wind speed data into separate intervals, the stochastic characteristic of the steady wind speed can be represented as a Markov process, while the high-frequency wind speed in the each interval is regarded as the disturbance input. Then, the traditional operating points of wind turbine can be divided into separate subregions correspondingly, where the model parameters and the control mode can be fixed in each mode. Then, the mixed H₂/H_∞ control problem is discussed for such a class of Markovian jump wind turbine working above the rated wind speed to guarantee both the disturbance rejection and the mechanical loads objectives, which can reduce the power volatility and the generator torque fluctuation of the whole transmission mechanism efficiently. Simulation results for a 2 MW wind turbine show the effectiveness of the proposed method.     

estimation for discrete-time piecewise homogeneous Markov jump linear systems

This paper concerns the problem of H_∞ estimation for a class of Markov jump linear systems (MJLS) with time-varying transition probabilities (TPs) in discrete-time domain. The time-varying character of TPs is considered to be finite piecewise homogeneous and the variations in the finite set are considered to be of two types: arbitrary variation and stochastic variation, respectively. The latter means that the variation is subject to a higher-level transition probability matrix. The mode-dependent and variation-dependent H_∞ filter is designed such that the resulting closed-loop systems are stochastically stable and have a guaranteed H_∞ filtering error performance index. Using the idea in the recent studies of partially unknown TPs for the traditional MJLS with homogeneous TPs, a generalized framework covering the two kinds of variations is proposed. A numerical example is presented to illustrate the effectiveness and potential of the developed theoretical results.     

<Emphasis Type="BoldItalic">H</Emphasis><Subscript>2</Subscript>-Control and the Separation Principle for Discrete-Time Markovian Jump Linear Systems

In this paper we consider the H₂-control problem of discrete-time Markovian jump linear systems. We assume that only an output and the jump parameters are available to the controller. It is desired to design a dynamic Markovian jump controller such that the closed-loop system is mean square stable and minimizes the H₂-norm of the system. As in the case with no jumps, we show that an optimal controller can be obtained from two sets of coupled algebraic Riccati equations, one associated with the optimal control problem when the state variable is available, and the other associated with the optimal filtering problem. This is the principle of separation for discrete-time Markovian jump linear systems. When there is only one mode of operation our results coincide with the traditional separation principle for the H₂-control of discrete-time linear systems. Date received: June 1, 2001. Date revised: October 13, 2003.     

H∞ filtering for piecewise homogeneous Markovian jump nonlinear systems

This paper concerns the problem of H_∞ filtering for piecewise homogeneous Markovian jump nonlinear systems. Different from the existing studies in the literatures, the existence of variations in transition rates for Markovian jump nonlinear systems is considered. The purpose of the paper is to design mode-dependent and mode-independent filters, such that the dynamics of the filtering errors are stochastic integral input-to-state stable with H_∞ performance index. Using the linear matrix inequality method and the Lyapunov functional method, sufficient conditions for the solution to the H_∞ filtering problem are derived. Finally, three examples are proposed to illustrate the effectiveness of the given theoretical results.     

Sampled-data H∞ filtering for Markovian jump singularly perturbed systems with time-varying delay and missing measurements

In this paper, sampled-data H_∞ filtering problem is considered for Markovian jump singularly perturbed systems with time-varying delay and missing measurements. The sampled-data system is represented by a time-delay system, and the missing measurement phenomenon is described by an independent Bernoulli random process. By constructing an ?-dependent stochastic Lyapunov–Krasovskii functional, delay-dependent sufficient conditions are derived such that the filter error system satisfies the prescribed H_∞ performance for all possible missing measurements. Then, an H_∞ filter design method is proposed in terms of linear matrix inequalities. Finally, numerical examples are given to illustrate the feasibility and advantages of the obtained results.     

线性Markov 切换系统的随机Nash 微分博弈及混合H2/H∞ 控制

研究线性Markov切换系统的随机Nash微分博弈问题。首先借助线性Markov切换系统随机最优控制的相关结果,得到了有限时域和无线时域Nash均衡解的存在条件等价于其相应微分(代数) Riccati方程存在解,并给出了最优解的显式形式;然后应用相应的微分博弈结果分析线性Markov切换系统的混合H2/H∞控制问题;最后通过数值算例验证了所提出方法的可行性。     

H ∞ model reduction for discrete-time Markov jump linear systems with partially known transition probabilities

In this article, the H _∞ model reduction problem for a class of discrete-time Markov jump linear systems (MJLS) with partially known transition probabilities is investigated. The proposed systems are more general, relaxing the traditional assumption in Markov jump systems that all the transition probabilities must be completely known. A reduced-order model is constructed and the LMI-based sufficient conditions of its existence are derived such that the corresponding model error system is internally stochastically stable and has a guaranteed H _∞ performance index. A numerical example is given to illustrate the effectiveness and potential of the developed theoretical results.     

Stabilisation and H∞ control of neutral stochastic delay Markovian jump systems

This paper investigates the exponential stabilisation and H_∞ control problem of neutral stochastic delay Markovian jump systems. First, a delay feedback controller is designed to stabilise the neutral stochastic delay Markovian jump system in the drift part. Second, sufficient conditions for the existence of feedback controller are proposed to ensure that the resulting closed-loop system is exponentially stable in mean square and satisfies a prescribed H_∞ performance level. Finally, numerical examples are provided to show the effectiveness of the proposed design methods.     

基于不完全转移率的连续Markov 跳变奇异系统的H∞ 控制

研究一类不完全转移率信息的Markov跳变奇异系统的H∞控制问题,提出连续Markov跳变奇异系统的新型有界实引理,并将其推广到不完全转移率条件.进一步设计H∞状态反馈控制器,使得闭环系统在转移率部分未知的条件下随机可容许,且满足H∞性能H.所得结论涵盖了奇异矩阵模态依赖情形,且表示为严格线性矩阵不等式形式,利于工程实现.最后,通过仿真算例表明了所提出方法的有效性和优越性.     

Optimal time-weighted H 2 model reduction for Markovian jump systems

This article addresses the optimal time-weighted H ₂ model reduction problem for Markovian jump linear systems. That is, for a given mean square stable Markovian jump system, our aim is to find a mean square stable jump system of lower order such that the time-weighted H ₂ norm of the corresponding error system is minimised. The time-weighted H ₂ norm of the system is first defined, and then a computational method is constructed. The computation requires the solution of two sets of recursive Lyapunov-type linear matrix equations associated with the Markovian jump system. To solve the optimal time-weighted H ₂ model reduction problem, we propose a gradient flow method for its solution. A necessary condition for minimality is derived, and a computational procedure is provided to obtain the minimising reduced-order model. The necessary condition generalises the standard result for systems when Markov jumps and the time-weighting term do not appear. Finally, two numerical examples are given to demonstrate the effectiveness of the proposed approach.