Model reduction of discrete Markovian jump systems with time‐weighted H2 performance

This paper is concerned with the optimal time‐weighted H₂ model reduction problem for discrete Markovian jump linear systems (MJLSs). The purpose is to find a mean square stable MJLS of lower order such that the time‐weighted H₂ norm of the corresponding error system is minimized for a given mean square stable discrete MJLSs. The notation of time‐weighted H₂ norm of discrete MJLS is defined for the first time, and then a computational formula of this norm is given, which requires the solution of two sets of recursive discrete Markovian jump Lyapunov‐type linear matrix equations. Based on the time‐weighted H₂ norm formula, we propose a gradient flow method to solve the optimal time‐weighted H₂ model reduction problem. A necessary condition for minimality is derived, which generalizes the standard result for systems when Markov jumps and the time‐weighting term do not appear. Finally, numerical examples are used to illustrate the effectiveness of the proposed approach. Copyright © 2015 John Wiley & Sons, Ltd.     

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.     

一类离散时间非齐次马尔可夫跳跃系统最优控制

本文研究了一类离散时间非齐次马尔可夫跳跃线性系统的线型二次高斯(linear quadratic Gaussian,LQG)问题,其中系统模态转移概率矩阵随时间随机变化,其变化特性由一高阶马尔可夫链描述.对于该系统的LQG问题,文中首先给出了线性最优滤波器,得到最优状态估计;其次,验证分离定理成立,并利用利用动态规划方法设计了系统最优控制器;最后,数值仿真结果验证了所设计控制器的有效性.     

Stochastic stability and robust stabilization of semi‐Markov jump linear systems

The semi‐Markov jump linear system (S‐MJLS) is more general than the Markov jump linear system (MJLS) in modeling some practical systems. Unlike the constant transition rates in the MJLS, the transition rates of the S‐MJLS are time varying. This paper focuses on the robust stochastic stability condition and the robust control design problem for the S‐MJLS with norm‐bounded uncertainties. The infinitesimal generator for the constructed Lyapunov function is first derived. Numerically solvable sufficient conditions for the stochastic stability of S‐MJLSs are then established in terms of linear matrix inequalities. To reduce the conservativeness of the stability conditions, we propose to incorporate the upper and lower bounds of the transition rate and meanwhile apply a new partition scheme. The robust state feedback controller is accordingly developed. Simulation studies and comparisons demonstrate the effectiveness and advantages of the proposed methods. Copyright © 2012 John Wiley & Sons, Ltd.     

Fault detection and isolation for discrete-time switched linear systems based on average dwell-time method

This article is concerned with the problem of fault detection and isolation (FDI) for discrete-time switched linear systems based on the average dwell-time method. The proposed FDI framework consists of a bank of FDI filters, which are divided into N groups for N subsystems. The FDI filters belonging to one group correspond to the faults for a subsystem, and generate a residual signal to guarantee the fault sensitivity performance for the subsystem, the fault attenuation performance for other subsystems and the disturbance attenuation performance for all subsystems. Different form employing the weighting matrices to restrict the frequency ranges of faults for each subsystem, the finite-frequency H _? performance for switched systems is first defined. Sufficient conditions are established by linear matrix inequalities (LMIs), and the filter gains are characterised in terms of the solution of a convex optimisation problem. Two examples are used to demonstrate the effectiveness of the proposed design method.     

Fault detection and isolation for networked control systems with finite frequency specifications

The fault detection and isolation (FDI) problem with finite frequency specifications is addressed in this paper under the framework of geometric approach for networked control systems subject to communication constraints and packet losses. The considered communication constraint is that only one of the transmission nodes is allowed to gain access to the shared channel. Also, those transmission nodes are scheduled to transmit data according to a specified stochastic protocol. Then by virtues of the common unobservable subspace and the finite frequency stochastic H _? index, a novel FDI scheme is developed in which a set of FDI filters that perform the FDI task with only partially available measurements are designed such that each residual is only sensitive to one fault in certain frequency domain and decoupled from the others. Further, less conservative conditions including some previous existing results have been presented to construct the FDI filters. Finally, an example is given to illustrate the effectiveness of the proposed method.Copyright © 2012 John Wiley & Sons, Ltd.     

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.     

Robust fault detection for networked systems with communication delay and data missing

In this paper, the robust fault detection problem is investigated for a class of discrete-time networked systems with unknown input and multiple state delays. A novel measurement model is utilized to represent both the random measurement delays and the stochastic data missing phenomenon, which typically result from the limited capacity of the communication networks. The network status is assumed to vary in a Markovian fashion and its transition probability matrix is uncertain but resides in a known convex set of a polytopic type. The main purpose of this paper is to design a robust fault detection filter such that, for all unknown inputs, possible parameter uncertainties and incomplete measurements, the error between the residual signal and the fault signal is made as small as possible. By casting the addressed robust fault detection problem into an auxiliary robust H_∞ filtering problem of a certain Markovian jumping system, a sufficient condition for the existence of the desired robust fault detection filter is established in terms of linear matrix inequalities. A numerical example is provided to illustrate the effectiveness and applicability of the proposed technique.     

Quadratic stabilisation with H∞-norm bound of non-linear discrete-time uncertain systems with bounded control

The paper addresses the problem of quadratic stabilisability with H_∞-norm bound of uncertain discrete-time control-affine systems by norm-bounded controls. Both structured parameter uncertainties and unstructured exogenous disturbances are taken into account. The given definition of quadratic stabilisability is a generalisation of that used for linear systems so far. A necessary condition of the stabilisability is formulated. A state feedback control satisfying an a priori constraint is proposed for the solution of the formulated H_∞ problem. The proposed method may be applicable even in such cases when the linearisation technique cannot be used.     

Nonstationary H∞ dynamic output feedback control for discrete‐time Markov jump linear systems with actuator and sensor saturations

This paper is concerned with the nonstationary dynamic output feedback control problem for a class of discrete‐time Markov jump linear systems (MJLSs) under simultaneous consideration of actuator and sensor saturations. The so‐called nonstationary controllers dominated by two other different piecewise‐stationary Markov chains are introduced, making the designed controllers not only mode‐dependent but also dependent on other variations associated with the mode transitions in the original MJLSs. The sufficient conditions formulated in terms of bilinear matrix inequalities for the H_∞ control problem are established such that the resulting closed‐loop system is stochastically stable and achieves a prescribed H_∞ noise attenuation level. A suboptimal algorithm with line search is exploited to solve for the associated controller gains. Effectiveness of the developed theoretical results is verified via a numerical example. Copyright © 2015 John Wiley & Sons, Ltd.     

Robust fault detection and isolation of time-delay systems using a geometric approach

This paper investigates the development of Fault Detection and Isolation (FDI) filters for both retarded and neutral time-delay systems with unknown time-varying delays. Using a geometric framework, the notion of a finite unobservability subspace is introduced for time-delay systems and an algorithm for its construction is presented. A bank of residual generators is then designed so that each residual is affected by one fault and is partially decoupled from the others while the H_∞ norm of the transfer function between the disturbances and the uncertainties in delays and the residuals are guaranteed to remain less than a prescribed value. Furthermore, it is shown that in the case of known delays it is possible to generate residuals that enjoy perfect decoupling properties among faults. Simulation results presented demonstrate the effectiveness of our proposed FDI algorithms.     

Delay-dependent H ∞ stabilisation criterion for continuous-time networked control systems with random delays

This article mainly addresses the H _∞ stabilisation problem for a class of networked control systems (NCSs) with random input delays in the continuous-time domain. A Markov jump linear system (MJLS) model with two jump parameters is developed which takes both the sensor-to-controller and controller-to-actuator delays into account. Based on such a MJLS, a new Lyapunov–Krasovskii functional is proposed to establish a delay-dependent H _∞ stabilisation criterion in terms of linear matrix inequalities (LMIs), and a mode-dependent state-feedback controller of NCSs can be determined. Two numerical examples are included to illustrate the derived result.     

Structured computation of optimal controls for constrained cascade systems

ABSTRACT

Constrained finite-horizon linear-quadratic optimal control problems are studied within the context of discrete-time dynamics that arise from the series interconnection of subsystems. A structured algorithm is devised for computing the Newton-like steps of primal-dual interior-point methods for solving a particular re-formulation of the problem as a quadratic program. This algorithm has the following properties: (i) the computation cost scales linearly in the number of subsystems along the cascade; and (ii) the computations can be distributed across a linear processor network, with localised problem data dependencies between the processor nodes and low communication overhead. The computation cost of the approach, which is based on a fixed permutation of the primal and dual variables, scales cubically in the time horizon of the original optimal control problem. Limitations in these terms are explored as part of a numerical example. This example involves application of the main results to model data for the cascade dynamics of an automated irrigation channel in particular.     

Consensus of networked multi-agent systems with communication delays based on the networked predictive control scheme

The consensus problem of discrete-time networked multi-agent systems (NMASs) with a communication delay is investigated in this article, where the dynamics of agents described by discrete-time linear time-invariant systems can be either uniform or non-uniform. For the NMASs with a directed topology and constant delay, a novel protocol based on the networked predictive control scheme is proposed to compensate for communication delay actively. Using algebraic graph theories and matrix theories, necessary and/or sufficient conditions of achieving consensus are obtained, which indicates that, under the proposed protocol, the consensus is independent of the network delay and only dominated by agents' dynamics and communication topology. Meanwhile, the protocol design and consensus analysis are also presented in the case of no network delay. Simulation results are further presented to demonstrate the effectiveness of theoretical results.     

H∞ model reduction for discrete-time singular systems

This paper investigates the problem of H_∞ model reduction for linear discrete-time singular systems. Without decomposing the original system matrices, necessary and sufficient conditions for the solvability of this problem are obtained in terms of linear matrix inequalities (LMIs) and a coupling non-convex rank constraint set. When these conditions are feasible, an explicit parametrization of the desired reduced-order models is given. Particularly, a simple LMI condition without rank constraint is derived for the zeroth-order H_∞ approximation problem. Finally, an illustrative example is provided to demonstrate the applicability of the proposed approach.     

Derandomisation-based multiple frequency control for stochastic Markov jump systems

Under the framework of derandomisation approach, the state feedback control issues for both continuous-time and discrete-time Markov jump linear systems (MJLSs) are investigated to meet multiple performance objectives over multiple frequency ranges. Because of the stochastic jumping among different modes, the generalised Kalman–Yakubovic–Popov lemma-based finite-frequency controller design approach cannot be directly applied to MJLSs. To overcome this limitation, a derandomisation approach is established by transforming the original stochastic multiple modes systems to deterministic ones. Then the multiple frequency controllers for both discrete-time and continuous-time MJLSs are designed to guarantee the multiple performances of the closed-loop systems. To verify the effectiveness of the developed algorithms, examples are presented, where the performance requirements include specifications in low-frequency and high-frequency ranges, respectively.     

control for fast sampling discrete-time singularly perturbed systems

This paper is concerned with the H_∞ control problem via state feedback for fast sampling discrete-time singularly perturbed systems. A new H_∞ controller design method is given in terms of solutions to linear matrix inequalities (LMIs), which eliminates the regularity restrictions attached to the Riccati-based solution. A method for evaluating the upper bound of singular perturbation parameter with meeting a prescribed H_∞ performance bound requirement is also given. Furthermore, the results are extended to robust controller design for fast sampling discrete-time singularly perturbed systems with polytopic uncertainties. Numerical examples are given to illustrate the validity of the proposed methods.     

State and mode feedback control strategy for discrete‐time Markovian jump linear systems with time‐varying controllable mode transition probability matrix

In this article, the state and mode feedback control strategy is investigated for the discrete‐time Markovian jump linear system (MJLS) with time‐varying controllable mode transition probability matrix (MTPM). This strategy, consisting of a state feedback controller and a mode feedback controller, is proposed to ensure MJLS's stability and meanwhile improve system performance. First, a mode‐dependent state feedback controller is designed to stabilize the MJLS based on the time‐invariant part of the MTPM such that it can still keep valid even if the MTPM is adjusted by the mode feedback control. Second, a generalized quadratic stabilization cost is put forward for evaluating MJLS's performance, which contains system state, state feedback controller, and mode feedback controller. To reduce the stabilization cost, a mode feedback controller is introduced to adjust each mode's occurrence probability by changing the time‐varying controllable part of MTPM. The calculation of such mode feedback controller is given based on a value‐iteration algorithm with its convergence proof. Compared with traditional state feedback control strategy, this state and mode feedback control strategy offers a new perspective for the control problem of general nonhomogeneous MJLSs. Numerical examples are provided to illustrate the validity of the proposed strategy.     

Distributed robust control of linear multi-agent systems with parameter uncertainties

This article considers the distributed robust control problems of uncertain linear multi-agent systems with undirected communication topologies. It is assumed that the agents have identical nominal dynamics while subject to different norm-bounded parameter uncertainties, leading to weakly heterogeneous multi-agent systems. Distributed controllers are designed for both continuous- and discrete-time multi-agent systems, based on the relative states of neighbouring agents and a subset of absolute states of the agents. It is shown for both the continuous- and discrete-time cases that the distributed robust control problems under such controllers in the sense of quadratic stability are equivalent to the H _∞ control problems of a set of decoupled linear systems having the same dimensions as a single agent. A two-step algorithm is presented to construct the distributed controller for the continuous-time case, which does not involve any conservatism and meanwhile decouples the feedback gain design from the communication topology. Furthermore, a sufficient existence condition in terms of linear matrix inequalities is derived for the distributed discrete-time controller. Finally, the distributed robust H _∞ control problems of uncertain linear multi-agent systems subject to external disturbances are discussed.     

Robust static output feedback control for linear discrete-time systems with time-varying uncertainties

This paper is concerned with the problem of designing robust static output feedback controllers for linear discrete-time systems with time-varying polytopic uncertainties. Sufficient conditions for robust static output feedback stabilizing controller designs are given in terms of solutions to a set of linear matrix inequalities, and the results are extended to H₂ and H_∞ static output feedback controller designs. Numerical examples are given to illustrate the effectiveness of the proposed design methods.