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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.

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.

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 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.

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.

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 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 numerical methods for the optimization of piecewise, stationary and deterministic systems. We mainly extend the methods used by Kushner and by Gonzales and Roffman to deal with a different kind of optimization problem and provide a comparison between these extended methods. The basic idea behind the numerical approximation methods is to build a discrete Markov decision process with finite state space and finite control space which is readily solvable and approximates the optimization problem for the class of piecewise deterministic systems. To illustrate the usefulness of the proposed methods, a numerical example in the field of production systems is presented