Mean square stabilization of discrete‐time switching Markov jump linear systems

This paper considers a special class of hybrid system called switching Markov jump linear system. The system transition is governed by two rules. One is Markov chain and the other is a deterministic rule. Furthermore, the transition probability of the Markov chain is not only piecewise but also orchestrated by a deterministic switching rule. In this paper, the mean square stability of the systems is studied when the deterministic switching is subject to two different dwell time conditions, ie, having a lower bound and having both lower and high bounds. The main contributions of this paper are two relevant stability theorems for the systems under study. A numerical example is provided to demonstrate the theoretical results.     

H2 filter design for discrete‐time Markov jump linear systems with partly unknown transition probabilities

The H₂ filter design problem for discrete‐time Markov jump linear systems with partly unknown transition probabilities is addressed in this paper. The so‐called partly unknown transition probabilities cover two cases: one is that some unknown elements have known lower and upper bounds, the other is that some unknown elements have no information available. By employing Finsler's lemma and linear matrix inequality (LMI) technique, sufficient conditions are developed in the LMI setting to design an H₂ filter such that the filtering error system is mean‐square stable and at the same time satisfies a prescribed H₂ performance index. Numerical examples are presented to illustrate the effectiveness of the developed theoretical results. Copyright © 2011 John Wiley & Sons, Ltd.     

A note on the robust control of Markov jump linear uncertain systems

This note addresses a robust control problem of continuous‐time jump linear Markovian systems subject to norm‐bounded parametric uncertainties. The problem is expressed in terms of a H_∞ control problem as in the purely deterministic case. The present formulation is simpler and it contains previous results in the literature as particular cases. Robust state feedback controllers are parameterized by means of a set of linear matrix inequalities. The result is illustrated by solving some examples numerically. Copyright © 2002 John Wiley & Sons, Ltd.     

H∞ state‐feedback controller design for continuous‐time nonhomogeneous Markov jump systems

This paper studies the problem of H_∞ state‐feedback controller design for continuous‐time nonhomogeneous Markov jump systems. The time‐varying transition rate matrix in continuous‐time domain is considered to lie in a convex bounded domain of polytopic type. By constructing a parameter‐dependent Lyapunov function and fully considering the information about the rate of change of time‐varying parameters, a new sufficient condition on the existence of an H_∞ state‐feedback controller is provided in the form of a parameter‐dependent matrix inequality. Moreover, based on the structure characteristics of Lyapunov matrix and transition matrix, the parameter‐dependent matrix inequality is converted into a finite set of linear matrix inequalities, which can be readily solved. Two numerical examples are provided to demonstrate the effectiveness of the theoretical results. Copyright © 2016 John Wiley & Sons, Ltd.     

Decision–control mechanism for Markovian jump linear systems with Gaussian noise

This paper investigates the decision–control mechanism for Markovian jump linear systems with Gaussian noise. The mechanism here consists of two parts: decision to govern the mode transition rate matrix and output‐feedback controller to govern system state. Motivated by this, a joint index is put forward to evaluate system performance, which is a combination of traditional jump linear quadratic Gaussian cost and additional decision cost because extra expenses will be taken for adopting decision to mode transition rate matrix. For the minimization of joint index, the designing of optimal decision–control pair is deduced to the seeking of optimal decision. Meanwhile, the optimal decision can be obtained via an iterative with its convergence proved. Numerical examples illustrate the validity of the proposed mechanism. Copyright © 2015 John Wiley & Sons, Ltd.     

Stochastic linear quadratic optimal control problem for systems driven by fractional Brownian motions

This paper is concerned with stochastic linear control systems driven by fractional Brownian motions (fBms) with Hurst parameter H∈(1/2,1) and the cost functional is quadratic with respect to the state and control variables. Here, the integrals with respect to fBms are the type of Stratonovich integrals. A stochastic maximum principle as a necessary condition of the optimal control is derived. The adjoint backward stochastic differential equation (BSDE) is driven by the fBms and its underlying standard Brownian motions. The existence and uniqueness of the solution of adjoint BSDE is proved. The explicit form of the unique optimal control is obtained.     

Model predictive and linear quadratic Gaussian control of a wind turbine

Model Predictive and linear quadratic Gaussian controllers are designed for a 5MW variable‐speed pitch‐regulated wind turbine for three operating points – below rated wind speed, just above rated wind speed, and above rated wind speed. The controllers are designed based on two different linear dynamic models (at each operating point) of the same wind turbine to study the effect of utilising different control design models (i.e. the model used for designing a model‐based controller) on the control performance. The performance of the LQG controller is enhanced by improving the robustness, achieved by replacing the Kalman filter with a modified Luenberger observer, whose gain is obtained to minimise the effect of uncertainty and disturbance. Copyright © 2016 John Wiley & Sons, Ltd.     

Sliding mode control of time‐varying delay Markov jump with quantized output

This paper investigates the quantized sliding mode control of Markov jump systems with time‐varying delay. A dynamical adjustment law is explored to quantize the system output. By constructing an observer‐based integral sliding surface, a sliding mode controller is designed to take over the dynamical motion of state estimation and ensure the reachability of sliding surface. A new scaling manner is developed to build the bound between the system output and quantized error. With the help of separation strategies for controller synthesis and general transition probabilities and a lower bound theorem for nonlinear integral terms, a new synthesis method to ensure the required stability and meet the required performance is proposed in the form of linear matrix inequalities. The validity of the proposed control method is illustrated by a numerical example.     

On linear quadratic optimal control of discrete-time complex-valued linear systems

We study in this paper the linear quadratic optimal control (linear quadratic regulation, LQR for short) for discrete-time complex-valued linear systems, which have several potential applications in control theory. Firstly, an iterative algorithm was proposed to solve the discrete-time bimatrix Riccati equation associated with the LQR problem. It is shown that the proposed algorithm converges to the unique positive definite solution (bimatrix) to the bimatrix Riccati equation with appropriate initial conditions. With the help of this iterative algorithm, the LQR problem for the antilinear system, which is a special case of complex-valued linear system, was carefully examined and three different Riccati equations–based approaches were provided, namely, bimatrix Riccati equation, anti-Riccati equation, and normal Riccati equation. The established approach is then used to solve the LQR problem for a discrete-time time-delay system with one-step state delay, and a numerical example was used to illustrate the effectiveness of the proposed methods.     

Improved approach for passive stability of discrete‐time Markovian jump linear systems via mode‐dependent time‐delayed controllers

In this paper, we investigate the problem of passive stability for discrete‐time Markovian jump linear systems via mode‐dependent time‐delayed controllers by employing an improved free‐weighting matrix approach. A Markov process as discrete‐time, discrete‐state Markov process is considered. First, a new result of mode‐dependent stability analysis is first established for error systems without ignoring any terms in the derivative of Lyapunov–Krasovskii function by considering the relationship between the time‐varying delay and its upper bound. Then, the delay‐dependent passivity criterion is provided and a mode‐dependent time‐delayed controller is designed in terms of linear matrix inequalities. Finally, two numerical examples are given to illustrate the effectiveness and less conservativeness of our proposed method. Copyright © 2011 John Wiley & Sons, Ltd.     

Dissipative control for singular Markovian jump systems with time delay

This paper focuses on the problem of dissipative control for continuous‐time singular time‐delayed systems with Markovian jump parameters. Different from usually mode‐dependent or mode‐independent controller design methods, a partially mode‐dependent dissipative controller is firstly proposed via using a mode‐dependent Lyapunov function. The stochastic property of the mode available to a controller operation is taken into consideration in the corresponding controller design. Moreover, the existence of the established controller is given in terms of strict linear matrix inequalities. Finally, numerical examples are used to demonstrate the effectiveness of the given theoretical results. Copyright © 2011 John Wiley & Sons, Ltd.     

Partially observed linear quadratic control problem with delay via backward separation method

In this paper, we study a partially observed linear quadratic optimal control problem derived by stochastic differential delay equations. Combining backward separation method with stochastic filtering, we obtain optimal feedback regulators in some special cases. Some filtering results for anticipated backward stochastic differential equations are also developed by expressing the solutions of the anticipated backward stochastic differential equations as some Itô's processes. Copyright © 2016 John Wiley & Sons, Ltd.     

A multiobjective optimization approach for linear quadratic Gaussian/loop transfer recovery design

This article bestows the linear quadratic Gaussian (LQG)/Loop Transfer Recovery (LTR) optimal controller design for a perturbed linear system having insufficient information about systems states through a multiobjective optimization approach. A Kalman filter observer is required to estimate the unknown states at the output from the noisy data. However, the main downside of the LQG controller's is that its robustness cannot be guaranteed because it consists of linear quadratic regulator (LQR) and Kalman observer, and due to observer incorporation within the LQR framework results in loss of robustness which is undesirable. Therefore, it is necessary to recover the robustness by tuning the controller which further plays havoc with system performance and control effort for certain plants. The present work addresses the investigation of the trade-off between multiobjective indexes (formulated on the basis of robustness, optimal control, and performances) through three multiobjective optimization algorithms as NSGA-II, multiobjective simulated annealing and multiobjective particle swarm optimization. The tuned parameters meet the competitive multiobjective performance indexes that are verified through simulation results. The Pareto front with multiple solutions helps to design a robust controller depending on the weightage given to the respective performance indexes. Simulation results reveal that the proposed multiobjective control strategy helps in recovering the characteristics of LQG/LTR.     

The spectral linear filter method for a stochastic optimal control problem of partially observable systems

In this paper, two spectral methods are presented to solve a stochastic optimal control problem of a partially observable system. These two methods work together to solve such problems. In fact, solving such problems involves two cases: obtaining the control function and simulating the partially observable system. At first, a spectral linear filter is defined as a function of time to obtain an appropriate solution for a partially observable system. This linear filter is equipped with an orthogonal basis and it is made to predict the future behavior of this system. In this method, the goal is to approximate the trend of the partially observable system. The second method is suggested to achieve the optimal control corresponding to each sample path. In this method, the spectral Fourier transform is used. These two methods are used together to solve linear and nonlinear cases. In fact, the innovative contents of this paper are both the spectral linear filter and the suggested spectral optimal control method.     

Mode feedback control strategy for continuous‐time Markovian jump linear systems with controllable mode transition rate matrix

This paper investigates mode feedback control strategy for continuous‐time Markovian jump linear systems with controllable mode transition rate matrix (MTRM). Based on the fact that system stability as well as its performance is dependent on MTRM, a mode feedback controller is proposed to perform control on MTRM such that system stability is achieved. Meanwhile, this strategy can help to improve system performance since it can adjust the occurrence probability of each subsystem. Firstly, taking into account that MTRM will totally determine system stability in absence of state feedback controller, the existence and restrictions on mode feedback controller are discussed where two cases are included. Then, based on a quadratic stabilization cost function that is a combination of state cost and control cost, a feasible solution of mode feedback control strategy is then obtained with its detailed algorithm given. Simulation results including comparisons with current state feedback mechanism are provided to illustrate the effectiveness of the proposed strategy.     

New delay‐dependent exponential stability for neutral Markovian jump systems with mixed delays and nonlinear perturbations

This paper deals with the problem of delay‐dependent exponential stability for neutral Markovian jump systems with mixed delays and nonlinear perturbations. Based on Lyapunov stability theory and linear matrix inequality method, some new exponential stability criteria are presented. The difference between this paper and other existing results is that the lower bounds of the neutral delay, the upper bounds of the neutral delay and discrete delay are considered, which will obtain some less conservative stability analysis results. Numerical examples are given to show that the proposed criteria improve the existing results.     

Observer design for stochastic time-delayed Markovian jump systems with incomplete transition rates and actuator saturation

This paper is concerned with observer design for stochastic time-delayed Markovian jump systems with incomplete transition rates and actuator saturation. By employing mode-dependent Lyapunov-Krasovskii functional, an observer-based feedback controller is designed to guarantee stochastic stability of the corresponding closed-loop saturated system and an estimation of the domain of attraction in the mean square is expanded. The procedure of deriving observer gain matrices is converted into an optimization problem with constraints of a set of linear matrix inequalities. Finally, the vertical take-off and landing helicopter model is given to demonstrate the validity of the main results.     

Singular linear quadratic optimal control for singular stochastic discrete‐time systems

The finite time horizon singular linear quadratic (LQ) optimal control problem is investigated for singular stochastic discrete‐time systems. The problem is transformed into positive LQ one for standard stochastic systems via two equivalent transformations. It is proved that the singular LQ optimal control problem is solvable under two reasonable rank conditions. Via dynamic programming principle, the desired optimal controller is presented in terms of matrix iterative form. One simulation is provided to show the effectiveness of the proposed approaches. Copyright © 2012 John Wiley & Sons, Ltd.     

Less conservative stabilization conditions for Markovian jump systems with incomplete knowledge of transition probabilities and input saturation

This paper proposes less conservative stabilization conditions for Markovian jump systems with incomplete knowledge of transition probabilities and input saturation. The transition rates associated with the transition probabilities are expressed in terms of three properties, which do not require the lower and upper bounds of the transition rates, differently from other approaches in the literature. The resulting conditions are converted into the second‐order matrix polynomial of the unknown transition rates. The polynomial can be represented as quadratic form of vectorized identity matrices scaled by one and the unknown transition rates. And then, the LMI conditions are obtained from the quadratic form. Also, an optimization problem is formulated to find the largest estimate of the domain of attraction in mean square sense of the closed‐loop systems. Finally, two numerical examples are provided to illustrate the effectiveness of the derived stabilization conditions. Copyright © 2016 John Wiley & Sons, Ltd.     

A new method for suboptimal control of a class of non‐linear systems

In this paper, a new non‐linear control synthesis technique (θ–D approximation) is discussed. This approach achieves suboptimal solutions to a class of non‐linear optimal control problems characterized by a quadratic cost function and a plant model that is affine in control. An approximate solution to the Hamilton–Jacobi–Bellman (HJB) equation is sought by adding perturbations to the cost function. By manipulating the perturbation terms both semi‐global asymptotic stability and suboptimality properties are obtained. The new technique overcomes the large‐control‐for‐large‐initial‐states problem that occurs in some other Taylor series expansion based methods. Also this method does not require excessive online computations like the recently popular state dependent Riccati equation (SDRE) technique. Furthermore, it provides a closed‐form non‐linear feedback controller if finite number of terms are taken in the series expansion. A scalar problem and a 2‐D benchmark problem are investigated to demonstrate the effectiveness of this new technique. Both stability and convergence proofs are given. Copyright © 2004 John Wiley & Sons, Ltd.