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In this study, the authors consider the ℋ2 (or Kalman)-filtering problem for singularly perturbed (two-timescale) non-linear systems. Two types of filters, namely (i) decomposition and (ii) aggregate, are discussed, and sufficient conditions for the solvability of the problem in terms of Hamilton–Jacobi–Bellman equations (HJBEs) are presented. The results are also specialised to linear systems in which case the HJBEs reduce to a system of bilinear-matrix-inequalities.

In this paper, we consider the ℋ∞-filtering problem for singularly perturbed (two time-scale) nonlinear systems. Two types of filters are discussed, namely, (i) decomposition and (ii) aggregate, and sufficient conditions for the solvability of the problem in terms of Hamilton–Jacobi–Isaac's equations (HJIEs) are presented. Reduced-order filters are also derived in each case, and the results are specialized to linear systems, in which case the HJIEs reduce to a system of linear-matrix-inequalities (LMIs). Based on the linearization of the nonlinear models, upper bounds e* of the singular parameter e that guarantee the asymptotic stability of the nonlinear filters can also be obtained. The mixed ℋ2/ℋ∞-filtering problem is also discussed. Copyright © 2010 John Wiley & Sons, Ltd.

This study investigates the H∞ control problem for a class of continuous-time Markovian jump singular systems. The authors' attention is focused on the design of state and static output feedback controllers which ensure that the closed-loop system is regular, impulse-free and stochastically stable, and satisfies a prescribed H∞ performance level. To investigate the problem of H∞ performance analysis for the system under consideration, a bounded real lemma (BRL) is provided in terms of linear matrix inequality (LMI) containing equality constraints, which may not be a problem from the theoretical point of view, but may cause great big trouble in checking the conditions numerically. To solve this problem, a new version of BRL is also provided in terms of strict LMIs. Based on this, two approaches are established to solve the problem of state feedback H∞ control, and then, the problem of static output feedback H∞ control is solved, which can also be used for solving the dynamic output feedback control problems. Finally, two numerical examples are given to illustrate the effectiveness of the obtained theoretical results.

In this paper, we consider the discrete-time mixed ℋ2/ℋ∞ filtering problem for affine nonlinear systems. Necessary and sufficient conditions for the solvability of this problem with a finite-dimensional filter are given in terms of a pair of coupled discrete-time Hamilton–Jacobi-Isaac's equations (DHJIE) with some side-conditions. For linear systems, it is shown that these conditions reduce to a pair of coupled discrete-time algebraic-Riccati-equations (DAREs) or a system of linear matrix inequalities (LMIs) similar to the ones for the control case. Both the finite-horizon and infinite-horizon problems are discussed. Moreover, sufficient conditions for approximate solvability of the problem are also derived. These solutions are especially useful for computational purposes, considering the difficulty of solving the coupled DHJIEs. An example is also presented to demonstrate the approach. Copyright © 2010 John Wiley & Sons, Ltd.

This paper deals with the state feedback controller design for a class of high-order feedforward (upper-triangular) nonlinear systems with delayed inputs. The uncertainties in the systems are assumed to be dominated by higher-order nonlinearities multiplying by a constant growth rate. The designed controller, which is a continuous but not smooth feedback, could achieve global asymptotical stability. Based on the appropriate state transformation of time-delay systems, the problem of controller design can be converted into the problem of finding a parameter, which can be obtained by appraising the nonlinear terms of the systems. The nonlinear systems considered here are more general than conventional feedforward systems and they could be viewed as generalized feedforward systems. Two examples are given to show the effectiveness of the proposed design procedure. Copyright © 2009 John Wiley & Sons, Ltd.

The stochastic stability and robust stochastic stabilisation for time-delay discrete Markovian jump singular systems with parameter uncertainties are discussed. Based on stochastic Lyapunov functional, a delay-dependent linear matrix inequalities (LMIs) condition for the time-delay discrete Markovian jump singular systems to be regular, causal and stochastically stable is given. With this condition, the problem of robust stochastic stabilisation is solved. A numerical example to illustrate the effectiveness of the method is given.

This paper deals with the class of discrete-time linear systems with random abrupt changes and unknown transition probabilities but varying between known bounds for each mode. The ℋ∞ control problem of this class of systems is revisited and new sufficient conditions are developed in the linear matrix inequality (LMI) setting to design the state-feedback controller that stochastically stabilizes the system under consideration and at the same time guarantees the disturbance rejection with a desired level γ . Sufficient conditions for existence of the state-feedback controller are developed. It is shown that the addressed problem can be solved if the corresponding developed LMIs are feasible. Numerical examples are employed to show the usefulness of the proposed results. Copyright © 2008 John Wiley & Sons, Ltd.

In this study, the problems of stability and stabilisation of singular Markovian jump systems with external discontinuities and saturating inputs are addressed. A design procedure is proposed to determine simultaneously a state feedback controller and an associated domain of safe admissible states for which the regularity, the absence of impulsive behaviour and the stochastic stability in mean square sense of the closed-loop systems are guaranteed. This design procedure is based on linear matrix inequality technique (LMI) and the sequential linear programming matrix method (SLPMM). A numerical example is provided to demonstrate the effectiveness of the proposed methods.

A class of extended continuous-time Markov jump linear systems is proposed in this study. The generality lies in that the discrete dynamics of the class of systems is described by a Markov stochastic process, but with only partially known transition probabilities, which relax the traditional assumption in Markov jump systems that all the transition probabilities must be known a priori. Moreover, in contrast with the uncertain transition probabilities studied recently, no structure (polytopic ones), bounds (norm-bounded ones) or ‘nominal’ terms (both) are required for the partially unknown elements in the transition rate matrix. The sufficient conditions for H∞ control are derived via the linear matrix inequality formulation such that the closed-loop system is stochastically stable and has a guaranteed H∞ noise-attenuation performance. A tradeoff can be built using our approach between the difficulties to obtain all the transition probabilities and the systems performance benefits. A numerical example is provided to show the validity and potential of the developed theoretical results.

In this paper, a new theory of two-degrees-of-freedom (2-DOF)-ℋ∞ and certainty-equivalent filters is presented. Exact and approximate solutions to the nonlinear ℋ∞ filtering problem using this class of filters are derived in terms of discrete-time Hamilton–Jacobi–Isaacs equations. The expressions for the filter gains are determined as functions of the filter state and the system's output in contrast to earlier results. Hence, it is shown that coupled with the additional degree-of-freedom, these filters are a substantial improvement over the earlier 1-DOF case. The theory presented is also generalized to n-DOF filters, which bore strong connections to linear infinite-impulse response filters and hence are generalizations of this class of filters to the nonlinear setting. Simulation results are also given to show the usefulness of the new approach. Copyright © 2009 John Wiley & Sons, Ltd.

In this paper, we consider the mixed ℋ2/ℋ∞ filtering problem for affine nonlinear systems. Sufficient conditions for the solvability of this problem with a finite-dimensional filter are given in terms of a pair of coupled Hamilton–Jacobi–Isaacs equations (HJIEs). For linear systems, it is shown that these conditions reduce to a pair of coupled Riccati equations similar to the ones for the control case. Both the finite-horizon and the infinite-horizon problems are discussed. Simulation results are presented to show the usefulness of the scheme, and the results are generalized to include other classes of nonlinear systems. Copyright © 2008 John Wiley & Sons, Ltd.

The robust H∞ filtering problem for mode-dependent time-delay discrete Markov jump singular systems with parameter uncertainties is discussed. Based on delay-dependent linear matrix inequalities, a Markov jump filter is designed, which guarantees that the filtering error system is regular, causal, stochastically stable and satisfies H∞ performance for all admissible uncertainties. The H∞ filter can be of full or of a reduced order. A numerical example is given to illustrate the effectiveness of the proposed method.

In this note, the stability analysis and stabilization problems for a class of discrete-time Markov jump linear systems with partially known transition probabilities and time-varying delays are investigated. The time-delay is considered to be time-varying and has a lower and upper bounds. The transition probabilities of the mode jumps are considered to be partially known, which relax the traditional assumption in Markov jump systems that all of them must be completely known a priori. Following the recent study on the class of systems, a monotonicity is further observed in concern of the conservatism of obtaining the maximal delay range due to the unknown elements in the transition probability matrix. Sufficient conditions for stochastic stability of the underlying systems are derived via the linear matrix inequality (LMI) formulation, and the design of the stabilizing controller is further given. A numerical example is used to illustrate the developed theory.

This paper deals with the class of continuous-time singular linear systems with random abrupt changes. The state feedback stabilization and its robustness for this class of systems with norm-bounded uncertainties are tackled. Sufficient conditions for designing either a stabilizing controller or a robust stabilizing controller are developed in the LMI setting. The developed sufficient conditions are used to synthesize the state feedback controller that guarantees that either the nominal system or the uncertain system is piecewise regular, impulse free and stochastically stable or robust stochastically stable.

This paper deals with the class of continuous-time singular linear systems with time-delay in the state vector. The stabilization problem of this class of systems using a state feedback controller is tackled. New delay-dependent sufficient conditions on ℋ∞ stabilization are developed. A design algorithm for a memoryless state feedback controller which guarantees that the closed-loop dynamics will be regular, impulse-free and stable with γ-disturbance rejection is proposed. It is shown that the addressed problem can be solved if the corresponding developed linear matrix inequalities (LMIs) with some constraints are feasible. A numerical example is employed to show the usefulness of the proposed results. Copyright © 2007 John Wiley & Sons, Ltd.

This article deals with the class of uncertain stochastic hybrid linear systems with noise. The uncertainties we are considering are of norm bounded type. The stochastic stabilization and robust stabilization problems are treated. Linear matrix inequality (LMI)-based sufficient conditions are developed to design the state feedback controller with constant gain that stochastically (robust stochastically) stabilizes the studied class of systems. Our results are mode independent and require only the complete access to the state vector. Numerical examples are given to show the effectiveness of the proposed results.

This paper deals with the class of continuous-time singular linear systems with Markovian jump parameters and time delays. Sufficient conditions on the stochastic stability and stochastic stabilizability are developed. A design algorithm for a state feedback controller which guarantees that the closed-loop dynamics will be regular, impulse free, and stochastically stable is proposed in terms of the solutions to linear matrix inequalities.

This paper deals with the class of uncertain continuous-time linear stochastic hybrid systems with Wiener process. The uncertainties that we are considering are of the norm-bounded type. The robust \({\cal {H}}_{\infty}\) stochastic stabilization problem is treated. LMIs based sufficient conditions are developed to design the state feedback controller that robustly and stochastically stabilizes the studied class of systems and at the same time rejects a disturbance of desired level. The minimum disturbance rejection is also determined. A numerical example is provided to show the validity of the proposed results.

This paper deals with the uncertain class of continuous-time linear systems with Markovian jumping parameters and multiplicative Brownian disturbance. A design method for a nonfragile robust controller for this class of systems is proposed when the uncertainties are of the norm-bounded type. An LMI based sufficient condition is developed. The methodology used is based mainly on the Lyapunov approach. A numerical example is presented to show the usefulness of the proposed results.

This paper deals with the uncertain class of continuous-time linear systems with Markovian jumping parameters. A design method for a nonfragile robust controller for this class of systems is proposed when the uncertainties in the system matrices are of the norm-bounded type. An LMI based sufficient condition is developed. The methodology used is based mainly on the Lyapunov approach. A numerical example is presented to show the usefulness of the proposed results.

This paper considers the stabilization problem of the class of continuous-time linear stochastic hybrid systems with Wiener process. Theℋ∞state feedback stabilization problem is treated. A state feedback controller with constant gain that does not require access to the system mode is designed. LMI-based conditions are developed to design the state feedback controller with constant gain that stochastically stabilizes the studied class of systems and, at the same time, achieve the disturbance rejection of a desired level. The minimum disturbance rejection is also determined. Numerical examples are given to show the usefulness of the proposed results.

This paper deals with the class of continuous-time linear systems with Markovian jumps and multiple time delays. The systems that we are treating are assumed to have time-varying delays in their dynamics which can be different and also have uncertainties in the system parameters. The time-varying structure of the bounded uncertainties is considered. Delay-dependent conditions for stochastic stability and stochastic stabilizability and their robustness are considered. A design algorithm for a stabilizing memoryless controller is proposed. All the results are given in the LMI formalism.

This paper considers the class of discrete-time jump linear systems with time-delay and polytopic uncertain parameters. The problems of delay-independent robust stability, stabilization and H∞ control are cast into the framework of linear matrix inequality (LMI) and thus solved by LMI Toolbox of Matlab. By extending the system state, the system with time-delay is converted into a higher dimension Markov jump system without time-delay, and thus can be handled as a standard jump linear system with uncertain parameters. Numerical examples are provided to show the usefulness of the theoretical results.

This paper deals with the class of uncertain systems with multiple time delays. The stability and stabilizability of this class of systems are considered. Their robustness are also studied when the norm-bounded uncertainties are considered. Linear matrix inequality (LMIs) delay-dependent sufficient conditions for both stability and stabilizability and their robustness are established to check if a system of this class is stable and/or is stabilizable. Some numerical examples are provided to show the usefulness of the proposed results.

This paper addresses the guaranteed cost control problem of jump linear systems with norm-bounded uncertain parameters. A time-multiplied performance index is considered. The performance is calculated first and an LMI-based algorithm is developed to design a state feedback control law with constant gain matrices which robustly stabilizes the system in the mean-square quadratically stable sense.

This paper deals with the class of uncertain systems with multiple time delays. The stability and stabilizability of this class of systems are considered. System robustness is also studied when the norm-bounded uncertainties are considered. LMI delay-dependent sufficient conditions for stability, stabilizability, and system robustness are established to check whether a system of this class is stable and/or stabilizable. Numerical examples are provided to show the usefulness of the proposed results.

SUMMARY In Section 5 of Aliyu and Boukas (Int. J. Robust Nonlinear Control 2009; 19:394–417), the authors have presented certainty-equivalent filters for the mixed ℋ2/ℋ∞ filtering problem for affine nonlinear systems. Sufficient conditions for the solvability of the problem with a finite-dimensional filter are given in terms of a pair of coupled Hamilton–Jacobi–Isaacs equations (HJIEs). In this note, we supply a correction to these HJIEs. Moreover, for linear systems this correction is not necessary. Copyright © 2011 John Wiley & Sons, Ltd.

This work develops a class of stochastic optimization algorithms. It aims to provide numerical procedures for solving threshold-type optimal control problems. The main motivation stems from applications involving optimal or suboptimal hedging policies, for example, production planning of manufacturing systems including random demand and stochastic machine capacity. The proposed algorithm is a constrained stochastic approximation procedure that uses random-direction finite-difference gradient estimates. Under fairly general conditions, the convergence of the algorithm is established and the rate of convergence is also derived. A numerical example is reported to demonstrate the performance of the algorithm.

This paper studies the class of uncertain linear systems with time delay and Markov jump disturbance, in which the time delay is assumed to be dependent on the system mode. An LMI-based condition for this class of systems to be robustly stable is established. Sufficient conditions for the robust stabilizability under a state feedback controller are developed, and an LMI-based method to design the state feedback is proposed. Numerical examples are worked out to show the usefulness of the theoretical results.

This paper is concerned with scheduling of a marketing-production system in which the demand rate changes in accordance with marketing actions. Our goal is to provide illuminating descriptions and insight on the optimal control policies. A basic building-block model involving a single machine is developed. The objective is to choose the rates of production and marketing to maximize the overall profit. Consisting of a pair of hedging points, the optimal policy is attractive in practice because of its simplicity and can be computed by solving algebraic equations. Aiming at obtaining analytic solutions, various control regions of the optimal policy are depicted; the structure of the hedging-point policy is fully analyzed.

This paper studies the inventory control problem for a production system with uncertain processing time and delay in control. First, the stabilization of the delayed system is analysed. Then, a controller is designed such that a disturbance attenuation of the system is achieved. The problem of robust control of the system with parametric uncertainty is also investigated. Linear matrix inequality approach is employed to solve the above problems. A numerical example is given to show the potential of the proposed techniques. Copyright © 2000 John Wiley & Sons, Ltd.

This paper deals with the robust H∞-control problem for uncertain continuous-time linear time-delay systems with Markovian jumping parameters. Based on Lyapunov functional approach, a sufficient condition that assures the robust stochastic stabilizability and preserves the H∞-disturbance attenuation for the uncertain class of linear time-delay systems with Markovian jumping parameters is established. The uncertainties considered in this paper are of the norm-bounded type. A numerical example is given to demonstrate the usefulness of the results.

A class of estimation/learning algorithms using stochastic approximation in conjunction with two kernel functions is developed. This algorithm is recursive in form and uses known nominal values and other observed quantities. Its convergence analysis is carried out; the rate of convergence is also evaluated. Applications to a nonlinear chemical engineering system are examined through simulation study. The estimates obtained will be useful in process operation and control, and in on-line monitoring and fault detection.

For a flexible manufacturing system, determining the feasible production set at a given time is essential for production management. In this paper, we propose methods for estimating the production set for a general multi-part type failure prone flexible manufacturing system. We also discuss how to get an open loop control that can ensure the system producing at the margin of its expected production set and how to estimate the corresponding variance of the expected production set. Some special flexible manufacturing systems are discussed where we also suggest some simpler approximations, Numerical examples are presented to illustrate our algorithm.

This paper deals with the production planning problem for discrete-time manufacturing systems with deteriorating items. A minimax approach is presented throughout the paper for the case where the demand is unknown. Both the cases of finite horizon and infinite horizon are discussed. Moreover, the case of production planning for manufacturing systems with failure-prone machines is also considered. For this case, a stochastic approach is used in which the states of the machines are represented by a two-state Markov process with transition rates determined by the availability of the machines. Finally, a robust control policy to take care of plant uncertainties is developed. Simulation results are also presented to show the usefulness of the approaches.

In this paper, a robust control law for constrained manipulators with parametric uncertainties is derived. Two schemes are presented; the first, based on The Corless-Leitmann approach, will guarantee ultimate uniform stability of the system, and hence uniform boundedness errors convergence. As a second approach, a class of continuous feedback controls is proposed to guarantee asymptotic stability of the uncertain system. The analysis is based on a theoretical result of asymptotic stability. In this approach, due to the continuity of the control and asymptotic stability of the system, we can achieve acceleration convergence and, thus, constraint force convergence.

Studies the problem of Kalman filtering for a class of uncertain linear continuous-time systems with Markovian jumping parameters. The system under consideration is subjected to time-varying norm-bounded parameter uncertainties in the state and measurement equations. Stochastic quadratic stability of the above system is analyzed. A state estimator is designed such that the covariance of the estimation error is guaranteed to be within a certain bound for all admissible uncertainties, which is in terms of solutions of two sets of coupled algebraic Riccati equations.

In this paper, we study the problem of robust stabilizability of the class of uncertain linear systems with Markovian jumping parameters. Under the assumption of complete access to the continuous state, the stochastic stabilizability of the nominal system and the boundedness of the system's uncertainties, sufficient conditions which guarantee the robust stability of the uncertain systems are presented, which are in terms of a set of coupled algebraic Riccati equations. A numerical example is given to illustrate the potential of the proposed technique.

This paper studies the problem of control for discrete time delay linear systems with Markovian jump parameters. The system under consideration is subjected to both time-varying norm-bounded parameter uncertainty and unknown time delay in the state, and Markovian jump parameters in all system matrices. We address the problem of robust state feedback control in which both robust stochastic stability and a prescribed H/sub /spl infin// performance are required to be achieved irrespective of the uncertainty and time delay. It is shown that the above problem can be solved if a set of coupled linear matrix inequalities has a solution.