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In this paper, we improve an earlier iterative successive approximation method for solving the Hamilton–Jacobi equation (HJE) arising in deterministic optimal control of affine nonlinear systems. T...

In this paper, we develop a new framework for designing state estimators/filters and output measurement feedback controllers for affine nonlinear systems in state space. The problems are formulated as zero-sum differential games, and sufficient conditions for their solvability are given in terms of Hamilton–Jacobi–Isaacs equations (HJIEs). These HJIEs are new, in the sense that they are both state-dependent and measurement output dependent. This allows for the filter and observer gains to be optimized over all possible nonlinear gains. Examples and simulation results are also presented to support the theory.

In this paper, iterative or successive approximation methods for the Hamilton-Jacobi-Bellman-Isaacs equations (HJBIEs) arising in both deterministic and stochastic optimal control for affine nonlinear systems are developed. Convergence of the methods are established under fairly mild assumptions, and examples are solved to demonstrate the effectiveness of the methods. However, the results presented in the paper are preliminary, and do not yet imply in anyway that the solutions computed will be stabilizing. More improvements and experimentation will be required before a satisfactory algorithm is developed.

In this paper, we propose an iterative relaxation method for solving the Hamilton-Jacobi-Bellman-Isaacs equation U+0028 HJBIE U+0029 arising in deterministic optimal control of affine nonlinear systems. Local convergence of the method is established under fairly mild assumptions, and examples are solved to demonstrate the effectiveness of the method. An extension of the approach to Lyapunov equations is also discussed. The preliminary results presented are promising, and it is hoped that the approach will ultimately develop into an efficient computational tool for solving the HJBIEs.

In this paper, we present iterative or successive approximation methods for solving the coupled Hamilton–Jacobi–Isaacs equations (HJIEs) arising in nonzero-sum differential game for affine nonlinear systems. We particularly consider the ones arising in mixed $${\mathcal H}_{2}/{\mathcal H}_{\infty }$$ control. However, the approach is perfectly general and can be applied to any others including those arising in the N-player case. The convergence of the method is established under fairly mild assumptions, and examples are solved to demonstrate the utility of the method. The results are also specialized to the coupled algebraic Riccati equations arising typically in mixed $${\mathcal H}_{2}/{\mathcal H}_{\infty }$$ linear control. In this case, a bound within which the optimal solution lies is established. Finally, based on the iterative approach developed, a local existence result for the solution of the coupled-HJIEs is also established.

In this paper, we discuss computational methods for optimal control problems which preserve some geometric properties of the system. A Hamiltonian perturbation method is developed, which does not involve any discretization of either the cost function, or the dynamic equations. Instead, the approach relies mainly on an iterative successive approximation of the value-function, which converges uniformly to the solution. The approach is most effective when the Hamiltonian of the system is a polynomial function of the phase coordinates, and an example is presented to demonstrate this.

In this paper, we consider the H∞-filtering problem for affine nonlinear singular (or descriptor systems). Two types of filters are discussed, namely, 1) singular and 2) normal, and sufficient conditions for the solvability of the problem in terms of Hamilton-Jacobi-Isaacs equations (HJIEs) are presented. The results are also specialized to linear systems in which case the HJIEs reduce to a system of bilinear-matrix-inequalities (BLMIs) which can still be solved efficiently. Some simple examples are also given to illustrate the approach.

The developmental dynamics of multicellular organisms is a process that takes place in a multi-stable system in which each attractor state represents a cell type, and attractor transitions correspond to cell differentiation paths. This new understanding has revived the idea of a quasi-potential landscape, first proposed by Waddington as a metaphor. To describe development, one is interested in the ‘relative stabilities’ of N attractors ( N > 2). Existing theories of state transition between local minima on some potential landscape deal with the exit part in the transition between two attractors in pair-attractor systems but do not offer the notion of a global potential function that relates more than two attractors to each other. Several ad hoc methods have been used in systems biology to compute a landscape in non-gradient systems, such as gene regulatory networks. Here we present an overview of currently available methods, discuss their limitations and propose a new decomposition of vector fields that permits the computation of a quasi-potential function that is equivalent to the Freidlin–Wentzell potential but is not limited to two attractors. Several examples of decomposition are given, and the significance of such a quasi-potential function is discussed.

In this paper, we consider the Kalman (or \(\mathcal{H}_{2}\))-filtering problem for discrete-time affine nonlinear descriptor systems. Two types of filter are discussed, namely, (i) singular; and (ii) normal. Sufficient conditions for the solvability of the problem in terms of discrete-time Hamilton–Jacobi–Bellman equations (DHJBEs) are presented. The results are also specialized to linear systems in which case the DHJBEs reduce to a system of linear matrix-inequalities (LMIs). Examples are also presented to illustrate the results.

In this paper, we consider the H2 (or Kalman)-filtering problem for discrete-time singularly-perturbed (two time-scale) nonlinear 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 new discrete-time Hamilton-Jacobi-Bellman equations (DHJBEs) are presented. For each type of filter above, first-order approximate filters are derived, and reduced-order filters are also derived in each case. The results are also specialized to linear systems, in which case the DHJBEs reduce to a system of linear-matrix-inequalities (LMIs) which are efficient to solve. Examples are also presented to illustrate the results.

SUMMARY In a recently published paper, the authors considered the -filtering problem for singularly perturbed (two time-scale) nonlinear systems. Two types of filters were discussed; namely, decomposition and aggregate, and sufficient conditions for the solvability of the problem in terms of the Hamilton–Jacobi–Isaac's equations were presented. In this note, we provide a modification to the HJIEs that make them more formal and amenable to computations. Copyright © 2011 John Wiley & Sons, Ltd.

In this article, we introduce, formulate and solve the W1,2, W1,∞ nonlinear optimal control problems as extensions of H2, H∞ optimal control problems, respectively. As these spaces contain less smooth functions, a larger number of problems could be solved in this framework, and by a suitable choice of weighting functions, additional design objectives could be achieved using the present formulation. Moreover, any solution of the W1, p, p = 2, ∞ problem, is automatically a solution of the corresponding Hp-problem. Sufficient conditions for the solvability of the problems are given in terms of new Hamilton-Jacobi equations (HJEs). These new HJEs may also be easier to solve because of the additional degrees of freedom offered by the current norms. Both the state-feedback and output-feedback problems are discussed. The results are then specialised to linear systems, in which case the solutions are characterised in terms of new algebraic-Riccati equations.

In this article, we introduce, formulate and solve the W1,2 and W1,∞ estimation problems. We propose proportional, proportional-derivative and proportional-integral (PI) filters for each problem, and we derive sufficient conditions for the existence of optimal filter gains in terms of new Hamilton-Jacobi-Bellman and Hamilton-Jacobi-Isaacs equations. The output-feedback W1,2 and W1,∞ control problems are also re-solved using the separation principle. We show that, by combining an optimal estimator with state-feedback control, a viable optimal output-feedback controller can be synthesised.

SUMMARY In this paper, a new theory of two-degrees-of-freedom (2hbox − DOF) ℋ∞ filters as counterparts of 2-DOF controllers is presented. The theory is also extended to n − DOF filters which bore strong resemblance to linear finite-impulse-response filters and hence generalizes this class of filters to the nonlinear continuous-time case. Sufficient conditions for the solvability of the filter gains are derived in terms of new Hamilton–Jacobi–Isaacs equations which do not involve the system states. This is an improvement over earlier results in which the filter gains are functions of the system states. Simulation results are also presented to support the theory. Copyright © 2011 John Wiley & Sons, Ltd.

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 Kalman (or \(\mathcal{H}_{2}\))-filtering problem for affine nonlinear descriptor systems. Two types of filters are discussed, namely, (i) singular; (ii) normal, and sufficient conditions for the solvability of the problem in terms of Hamilton–Jacobi–Bellman equations (HJBEs) are presented. The results are also specialized to linear systems in which case the HJBEs reduce to a system of linear-matrix-inequalities (LMIs). Examples are also presented to illustrate the results.

In this paper we present sufficient conditions for the solvability of the static-output feedback stabilization problem for affine nonlinear systems using a factorization approach. We extend the results of (IEEE Trans. Autom. Control, Vol. 47, No. 12, pp. 2038–2041 (2002)) to include disturbance-attenuation and singular control. The sufficient conditions given are also less stringent than the ones given in (IEEE Trans. Autom. Control Vol. 47, No. 12, pp. 2038–2041 (2002)). The usefulness of the results are also illustrated with some examples. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

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.

In this article, we present the one-degree-of-freedom (1-DOF) or proportional (P), the 2-DOF or proportional-integral (PI) and the 3-DOF or proportional-integral-derivative (PID) ℋ∞ nonlinear filters as counterparts of the popular P, PI and PID controllers, respectively. We use the idea of certainty-equivalence principle in which the worst-case disturbance or noise is determined as a function of the estimated state and under the assumption that this converges to the actual value asymptotically. Sufficient conditions for the solvability of the various filter gains are derived in terms of new Hamilton–Jacobi–Isaacs equations, which do not involve the system states. Simulation results are also presented to support the theory.

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.

In this paper, bounded-real conditions for affine nonlinear state-delayed systems are derived using the concept of dissipativeness. Necessary and sufficient conditions for the system to be dissipative and to have finite L"2-gain also referred to as the bounded-real condition are given. The implications on the stability of the system and feedback interconnections of such systems are also considered. Finally, an equivalent of the positive-real lemma is derived and its implications on stability of the system and feedback interconnections of such systems are similarly discussed.

In this paper, we present a transformation approach for solving the Hamilton-Jacobi-Bellman equations (HJBEs) arising in H"2 or quadratic-cost control of nonlinear deterministic and stochastic systems. We show that the HJBE can be solved analogously to a scalar quadratic equation, and we give a parameterization of solutions to the HJBE characterizing the solution of the optimal control problem. The procedure is also generalized to include nonsmooth or viscosity solutions of the (S)HJBE.

In this paper, the state-feedback mixed H2/H∞ control problem for state-delayed linear systems is considered. Sufficient conditions for the solvability of this problem are given in terms of the solution to a pair of algebraic Riccati equations similar to the nondelayed case. However, these Riccati equations are more difficult to solve than those arising in the pure H2, H∞ problems, and an alternative approach is to solve a pair of linear matrix inequalities (LMIs).

In this paper, we present an approach to the solution of the Hamilton-Jacobi-Isaacs equation (HJIE) arising in the H"~ control problem for nonlinear systems. We show that the HJIE can be solved analogously to a scalar quadratic equation with some additional side conditions, and present a computational procedure for determining symmetric solutions. Examples are given for second-order affine nonlinear systems to illustrate the procedure, and the method can be extended to higher-order systems.

1 College of Information and Electrical Engineering, Shandong University of Science and Technology, Qingdao 266590, China 2 Department of Mechanical Engineering, Ecole Polytechnique de Montreal, Station Centre-Ville, P.O. Box 6079, Montreal, QC, Canada H3C 3A7 3 School of Control Science and Engineering, Shandong University, Jinan 250061, China 4 Institute of Automation, Qufu Normal University, Qufu 273165, China

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.

In this paper, a guaranteed cost control for non-linear uncertain systems is developed. The relationship between the idea of guaranteed cost control and dissipative feedback is first discussed which have been independently studied in the literature on robust control. It turns out that the two approaches are equivalent, and both presume the existence of a storage function (a Lyapunov function candidate) which guarantees the stability of the closed-loop system. The idea of guaranteed cost control is then applied to solve the robust control problem for non-linear systems with both drift vector field and input vector field uncertainties. Necessary and sufficient conditions are given for the solvability of this problem in terms of certain Hamilton-Jacobi Isaac's inequalities. The robustness properties of the controllers are also discussed. Finally, a passivity-based approach to the synthesis of guaranteed cost controllers is discussed.

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.

A multi-item model of a production-inventory system incorporating deterioration, shortages and capacity/budget constraints is considered. An optimal control policy for the model is developed using linear quadratic (LQ) theory for the case of deterministic demands. The problem of controlling large-scale production-inventory facilities is also considered, and the interaction prediction method is used to develop optimal policies. Results of simulations show that using the developed policy, any desired inventory levels can be maintained while minimizing costs and satisfying demand without violating capacity constraints.

The discrete-time inventory control problem for deteriorating items with deterministic or stochastic demand is considered. The control problem is stated as an optimization problem with constrained control variables. The linear quadratic (LQ) criterion is used to derive the optimal control policy for both deterministic and stochastic demand cases. In the presence of bounded uncertainties, a guaranteed cost control law is also proposed. A numerical example is provided to show the usefulness of the theoretical results.

An inventory model of a single-item production system incorporating deterioration and incorporating shortages and back orders is presented. The linear quadratic theory is used to develop an optimal policy which minimizes a quadratic cost criterion of production, inventory holding and shortages. The cases of both deterministic (not necessarily constant) and stochastic demands are considered. In the stochastic case, the demand is modeled as a Gaussian process with a certain mean and variance. Results of simulations are also presented and show very good agreement with the theory developed.

The problem of disturbance attenuation with internal stability for nonlinear systems with Markovian jumping parameters is considered. It is shown that this problem is solvable if there exists a set of smooth positive semi-definite functions satisfying certain Hamilton-Jacobi-Isaacs inequalities. Furthermore, it is shown that if this solution exists, it represents a stochastic Lyapunov function for the closed-loop nonlinear system.

In this paper, the problem of disturbance attenuation with internal stability for uncertain nonlinear systems satisfying certain matching conditions is considered. The uncertainties are assumed to be due to both parametric and higher order nonlinearities neglected as modelling errors. It is shown that this problem is solvable if there exist a smooth positive semi-definite function satisfying a certain Hamilton-Jacobi-Isaacs equation (inequality) with some appropriate scaling function. Furthermore, it is shown that if this solution exists, it represents a Lyapunov function for the closed-loop nonlinear system.