Showing papers on "Riccati equation published in 1998"
01 Jan 1998
TL;DR: In this article, the authors present a guided tour of robust stability and robust stability of time-delay systems, including polynomials, quasipolynomials, and nonlinear delay systems.
Abstract: Stability and robust stability of time-delay systems: A guided tour.- Convex directions for stable polynomials and quasipolynomials: A survey of recent results.- Delay-independent stability of linear neutral systems: A riccati equation approach.- Robust stability and stabilization of time-delay systems via integral quadratic constraint approach.- Graphical test for robust stability with distributed delayed feedback.- Numerics of the stability exponent and eigenvalue abscissas of a matrix delay system.- Moving averages for periodic delay differential and difference equations.- On rational stabilizing controllers for interval delay systems.- Stabilization of linear and nonlinear systems with time delay.- Nonlinear delay systems: Tools for a quantitative approach to stabilization.- Output feedback stabilization of linear time-delay systems.- Robust control of systems with a single input lag.- Robust guaranteed cost control for uncertain linear time-delay systems.- Local stabilization of continuous-time delay systems with bounded inputs.
574 citations
TL;DR: In this article, a structured interior-point method for the efficient solution of the optimal control problem in model predictive control is presented. But the cost of this approach is linear in the horizon length compared with cubic growth for a naive approach.
Abstract: We present a structured interior-point method for the efficient solution of the optimal control problem in model predictive control. The cost of this approach is linear in the horizon length, compared with cubic growth for a naive approach. We use a discrete-time Riccati recursion to solve the linear equations efficiently at each iteration of the interior-point method, and show that this recursion is numerically stable. We demonstrate the effectiveness of the approach by applying it to three process control problems.
570 citations
TL;DR: In this paper, the capabilities of the state-dependent Riccati equation (SDRE) technique are illustrated in producing two control designs for the benchmark problem, and the first design shows that in the absence of disturbances and uncertainties, the SDRE nonlinear feedback solution compares very favorably to the optimal open-loop solution of the posed nonlinear regulator problem.
Abstract: A nonlinear control problem has been posed by Bupp et al. to provide a benchmark for evaluating various nonlinear control design techniques. In this paper, the capabilities of the state-dependent Riccati equation (SDRE) technique are illustrated in producing two control designs for the benchmark problem. The SDRE technique represents a systematic way of designing nonlinear regulators. The design procedure consists of first using direct parameterization to bring the nonlinear system to a linear structure having state-dependent coefficients (SDC). A state-dependent Riccati equation is then solved at each point x along the trajectory to obtain a nonlinear feedback controller of the form u=−R-1(x)BT(x)P(x)x, where P(x) is the solution of the SDRE. Analysis of the first design shows that in the absence of disturbances and uncertainties, the SDRE nonlinear feedback solution compares very favorably to the optimal open-loop solution of the posed nonlinear regulator problem, the latter being obtained via numerical optimization. It is also shown via simulation that the closed-loop system has stability robustness against parametric variations and attenuates sinusoidal disturbances. In the second design it is demonstrated how a hard bound can be imposed on the control magnitude to avoid actuator saturation. © 1998 John Wiley & Sons, Ltd.
349 citations
TL;DR: It has been shown that the above robust H/sub /spl infin//-filtering problem can be solved in terms of differential Riccati inequalities with finite discrete jumps.
Abstract: The paper is concerned with the problem of robust H/sub /spl infin// filtering for a class of systems with parametric uncertainties and unknown time delays under sampled measurements. The parameter uncertainties considered are real time-varying and norm-bounded, appearing in the state equation. An approach has been proposed for the designing of H/sub /spl infin// filters, using sampled measurements, which would guarantee a prescribed H/sub /spl infin// performance in the continuous-time context, irrespective of the parameter uncertainties and unknown time delays. Both cases of finite and infinite horizon filtering are studied. It has been shown that the above robust H/sub /spl infin//-filtering problem can be solved in terms of differential Riccati inequalities with finite discrete jumps.
279 citations
TL;DR: In this article, the Hyers-Ulam stability of the quadratic functional equation (1) on a restricted domain was investigated, and the result was applied to the study of an asymptotic behavior of that equation.
258 citations
TL;DR: A Newton-like method for solving algebraic Riccati equations that uses an exact line search to improve the sometimes erratic convergence behavior of Newton's method and accelerates convergence when Newton steps are too small or too long.
Abstract: We present a Newton-like method for solving algebraic Riccati equations that uses an exact line search to improve the sometimes erratic convergence behavior of Newton's method. It avoids the problem of a disastrously large first step and accelerates convergence when Newton steps are too small or too long. The additional work to perform the line search is small relative to the work needed to calculate the Newton step.
119 citations
TL;DR: In this article, a notion of nonlinear stabilizability is introduced, which is a necessary condition for global closed-loop stability and is illustrated by application to a dual-spin spacecraft.
Abstract: Inthelastfewyears,algorithmsusingstate-dependentRiccatiequations (SDREs)havebeenproposedforsolving nonlinear control problems Under state feedback, pointwise solutions of an SDRE must be obtained along the system trajectory To ensure the control is well dee ned, global controllability and observability of state-dependent system factorizations are commonly assumed Here connections between controllability of the state-dependent factorizations and truesystem controllability are rigorously established Itisshown thata localequivalencealways holdsfortheclassofsystemsconsidered,andspecialcasesthatimplyglobalequivalencearealsogivenAdditionally, a notion of nonlinear stabilizability is introduced, which is a necessary condition for global closed-loop stability The theory is illustrated by application to a e ve-state nonlinear model of a dual-spin spacecraft
99 citations
TL;DR: The principal result involves sufficient conditions in terms of a modified Riccati equation for characterizing state feedback controllers that enforce a bound on H"~ performance and guarantee closed-loop stability in the face of system state delay.
96 citations
TL;DR: In this article, a linear quadratic closed-loop optimal controller is obtained from a steady-state solution of the matrix Riccati equation and a Kalman filter reconstructs the state variables and the unknown perturbations from a reduced number of measured variables.
Abstract: Linear quadratic optimal control theory is applied to the automatic control of two different eight-pool irrigation canals. The model used to design the controller is derived from the Saint-Venant equations discretized through the Preissmann implicit scheme. The linear quadratic closed-loop optimal controller is obtained from steady-state solution of the matrix Riccati equation. A Kalman filter reconstructs the state variables and the unknown perturbations from a reduced number of measured variables. Both perturbation rejection and tracking aspects are incorporated in the controller. Known offtake withdrawals and future targets are anticipated through an open-loop scheme utilizing time varying solutions of the linear quadratic optimization problem. The controller and Kalman filter are tested on a full nonlinear model and prove to be stable, robust, and precise.
94 citations
TL;DR: In this article, a new transformation between two integrable hierarchies of the Camassa-Holm equation and the Hunter-Saxton equation is presented, which is the high-frequency limit of the CAMASHA-HOLM equation.
Abstract: In this article we present a new transformation between two integrable hierarchies of the Camassa-Holm equation and the Hunter-Saxton equation. For instance we present a transformation between the Harry-Dym equation and the extended Harry-Dym equation. Moreover, we describe a relationship between the Hunter-Saxton equation, which is the high-frequency limit of the Camassa-Holm equation, and the Sinh-Gordon equation by a reciprocal transformation.
87 citations
TL;DR: In this paper, a linearly perturbed version of the well-known matrix Riccati equations which arise in certain stochastic optimal control problems is studied and the results are derived under relatively weaker assumptions and include, inter alia, an extension of Theorem 4.1 of [26] to handle systems not necessarily observable.
Abstract: In this paper a linearly perturbed version of the well-known matrix Riccati equations which arise in certain stochastic optimal control problems is studied. Via the concepts of mean square stabilizability and mean square detectability we improve previous results on both the convergence properties of the linearly perturbed Riccati differential equation and the solutions of the linearly perturbed algebraic Riccati equation. Furthermore, our approach unifies, in some way, the study for this class of Riccati equations with the one for classical theory, by eliminating a certain inconvenient assumption used in previous works (e.g., [10] and [26]). The results are derived under relatively weaker assumptions and include, inter alia, the following: (a) An extension of Theorem 4.1 of [26] to handle systems not necessarily observable. (b) The existence of a strong solution, subject only to the mean square stabilizability assumption. (c) Conditions for the existence and uniqueness of stabilizing solutions for systems not necessarily detectable. (d) Conditions for the existence and uniqueness of mean square stabilizing solutions instead of just stabilizing. (e) Relaxing the assumptions for convergence of the solution of the linearly perturbed Riccati differential equation and deriving new convergence results for systems not necessarily observable.
TL;DR: It turns out that the optimal controller has an observer-based structure well-known in the state-space case, so that the results in this paper are natural extensions of thestate-space results to the descriptor systems.
TL;DR: The authors deal with the problem of stabilizing a class of uncertain linear systems with time-varying multi-state delay and subject to norm-bounded parameter uncertainty via memoryless linear state feedback, and derive sufficient conditions for the robust stabilizability.
Abstract: The authors deal with the problem of stabilizing a class of uncertain linear systems with time-varying multi-state delay and subject to norm-bounded parameter uncertainty via memoryless linear state feedback. Some sufficient conditions for the robust stabilizability are derived fur this class of uncertain systems. If there exists a positive-definite symmetric solution satisfying the algebraic Riccati equation (or inequality), a suitable memoryless state feedback law can be derived also. Moreover, all such parametric algebraic Riccati inequalities have been transformed into some linear matrix inequality problems, so there is no tuning of the parameters to gain a stabilizing solution.
TL;DR: In this article, new perturbation results for the two different algebraic Riccati equations (continuous time and discrete time) are derived in a uniform manner, illustrated by numerical examples.
Abstract: New perturbation results for the two different algebraic Riccati equations (continuous time and discrete time) are derived in a uniform manner. The new results are illustrated by numerical examples.
01 Jan 1998
TL;DR: There are a broad variety of topics related to partial differential equations, ranging from abstract functional analytic framework to aspects of modelling, with the main emphasis, however, on theory and numerics of optimal control for nonlinear distributed parameter systems.
Abstract: The volume here presented contains the Proceedings of the International Conference on Control of Distributed Parameter Systems, held in Graz (Austria) from July 15–21, 2001. It was the one eighth in a series of conferences that began in 1982. The book includes are a broad variety of topics related to partial differential equations, ranging from abstract functional analytic framework to aspects of modelling, with the main emphasis, however, on theory and numerics of optimal control for nonlinear distributed parameter systems. The proceedings contain 16 articles written by 27 authors, each of the papers containing new research results, not published before. They give a very useful overview to many of the current theoretical and industrial problems. The upto-date references at the end of the articles are also very helpful, and the nice, uniform TeX style of the book will be appreciated by the readers. In what follows, I describe briefly the papers contained in this collection. 1 H.T. Banks, S.C. Beeler and H.T. Tran, State estimations and tracking control of nonlinear dynamical systems. Based on the ”state-dependent Riccati equation”, nonlinear estimators and nonlinear feedback tracking controls are constructed for a wide class of systems. An application to a flight dynamics simulation shows that the corresponding computational methods are easily implementable and efficient. H.T. Banks, H. Tran and S. Wynne, The well-posedness results for a shear wave propagation model. Existence and uniqueness results are established for a nonlinear model for propagation of shear waves in viscoelastic tissue. R. Becker and B. Wexler, Mesh adaptation for parameter identification problems. The authors consider automatic mesh refinement for parameter identification problems involving PDEs. The idea is to solve the inverse problem on a ”cheap” discrete model, which still captures the ”essential” features of the physical model. To this end, a posteriori error estimator is used to successively
TL;DR: This paper developed some group theoretical methods which are shown to be very useful for a better understanding of the properties of the Riccati equation and discuss some of its integrability conditions from a group theoretical perspective.
Abstract: In this paper we develop some group theoretical methods which are shown to be very useful for a better understanding of the properties of the Riccati equation and we discuss some of its integrability conditions from a group theoretical perspective. The nonlinear superposition principle also arises in a simple way.
TL;DR: In this article, the nonlinear Schrodinger equation is derived in a general intrinsic geometric setting involving a normal congruence originally investigated by Marris and Passman in 1969 in a hydrodynamical context.
Abstract: The nonlinear Schrodinger equation is derived in a general intrinsic geometric setting involving a normal congruence originally investigated by Marris and Passman in 1969 in a hydrodynamical context. Geometric properties of members of the normal congruence are adduced and an intrinsic representation of the auto-Backlund transformation is obtained for the nonlinear Schrodinger equation.
TL;DR: In this paper, the problems of robust filtering, robust prediction, and robust smoothing for a class of continuous time uncertain systems are defined, and non-conservative solutions are given in terms of Riccati differential equations.
Abstract: This paper is concerned with a class of continuous time uncertain systems which satisfy a certain Integral Quadratic Constraint. The problems of robust filtering, robust prediction, and robust smoothing for such systems are defined, and nonconservative solutions are given in terms of Riccati differential equations. This paper also addresses a problem of robust observability for this class of uncertain systems.
TL;DR: In this article, the solution of the matrix Riccati equation with a terminal boundary condition is given by using the algebraic form of the Riccaci equation, and an illustrative example for the proposed method is given.
TL;DR: The H/sub 2/ guaranteed cost control problem for a singularly perturbed norm-bounded uncertain system is addressed using the quadratic stabilizability framework and a control minimizing an upper bound on the H/ sub 2/ norm of a certain transfer matrix is shown.
Abstract: The H/sub 2/ guaranteed cost control problem for a singularly perturbed norm-bounded uncertain system is addressed using the quadratic stabilizability framework. After defining the corresponding slow and fast uncertain subsystems, the set of quadratic stabilizing composite controls is characterized. Two Riccati equations have to be solved, one for the slow subsystem and the other for the fast subsystem. Choosing appropriately the weighting matrices, it is shown how to pick up in the set of quadratic stabilizing composite controls, a control minimizing an upper bound on the H/sub 2/ norm of a certain transfer matrix. The case of the guaranteed cost control problem for the reduced system is also investigated.
TL;DR: In this paper, group theoretical methods are used to study some properties of the Riccati equation, which is the only differential equation admitting a nonlinear superposition principle, and the Wei-Norman method is applied to obtain the associated differential equation in the group SL(2, ℝ).
Abstract: Group theoretical methods are used to study some properties of the Riccati equation, which is the only differential equation admitting a nonlinear superposition principle. The Wei–Norman method is applied to obtain the associated differential equation in the group SL(2, ℝ). The superposition principle for first order differential equation systems and Lie–Scheffers theorem are also analyzed from this group theoretical perspective. Finally, the theory is applied in the solution of second order differential equations like time independent Schrodinger equation.
TL;DR: In this paper, a necessary and sufficient condition is established for mean-square quadratic stability and mean square quadratically stabilizability of continuous-time linear systems with Markovian jumps and norm-bound uncertainties in the parameters.
Abstract: In this paper, we investigate the quadratic stability and quadratic stabilizability of the class of continuous-time linear systems with Markovian jumps and norm-bound uncertainties in the parameters. Under some appropriate assumptions, a necessary and sufficient condition is established for mean-square quadratic stability and mean-square quadratic stabilizability of this class of systems. The quadratic guaranteed cost control problem is also addressed via a LMI optimization problem.
TL;DR: In this paper, the authors studied the relationship between algebraic Riccati equation and linear matrix inequality, and showed that only a subset of the rank-minimizing solutions of the linear matrix inequalities correspond to the corresponding solutions of algebraic Riemannians.
TL;DR: In this paper, the authors introduce certain nonlinear partially observable stochastic optimal control problems which are equivalent to completely observable control problems with finite-dimensional state space, where nonlinearities can enter the unobservable dynamics as gradients of potential functions.
Abstract: This paper introduces certain nonlinear partially observable stochastic optimal control problems which are equivalent to completely observable control problems with finite-dimensional state space. In some cases the optimal control laws are analogous to linear-exponential-quadratic-Gaussian and linear-quadratic-Gaussian tracking problems. The problems discussed allow nonlinearities to enter the unobservable dynamics as gradients of potential functions. The methodology is based on explicit solutions of a modified Duncan--Mortensen--Zakai equation.
TL;DR: In this article, the Schrodinger equation was considered as a differential equation, disregarding the physical interpretation associated with solutions, by introducing the notion of A-related equations, A being a differential operator, and associated with it a Riccati equation.
Abstract: We consider the Schrodinger equation just as a differential equation, disregarding the physical interpretation associated with solutions. By introducing the notion of A-related equations, A being a differential operator, we associate with it a Riccati equation and study the solutions when the potential is a meromorphic function.
TL;DR: The optimal control law is derived for discrete-time linear stochastic systems with quadratic performance criterion and control-dependent noise using a generalized Riccati difference equation and of the asymptotic behavior of its solutions.
TL;DR: In this paper, group theoretical methods are used to study some properties of the Riccati equation, which is the only differential equation admitting a nonlinear superposition principle, and the Wei-Norman method is applied to obtain the associated differential equation in the group $SL(2,R)$.
Abstract: Group theoretical methods are used to study some properties of the Riccati equation, which is the only differential equation admitting a nonlinear superposition principle. The Wei-Norman method is applied to obtain the associated differential equation in the group $SL(2,R)$. The superposition principle for first order differential equation systems and Lie-Scheffers theorem are also analysed from this group theoretical perspective. Finally, the theory is applied in the solution of second order differential equations like time-independent Schroedinger equation
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TL;DR: In this paper, a new parametrisation of the Eilenberger equations of superconductivity in terms of the solutions to a scalar differential equation of the Riccati type is introduced.
Abstract: A new parametrisation of the Eilenberger equations of superconductivity in terms of the solutions to a scalar differential equation of the Riccati type is introduced. It is shown that the quasiclassical propagator, and in particular the local density of states, may be reconstructed, without explicit knowledge of any eigenfunctions and eigenvalues, by solving a simple initial value problem for the linearised Bogoliubov-de Gennes equations. The Riccati parametrisation of the quasiclassical propagator leads to a stable and fast numerical method to solve the Eilenberger equations. For some spatially varying model pair potentials exact solutions to the Eilenberger Equations are found.
TL;DR: In this article, the authors investigated the asymptotic behavior of the solutions as the distance from the finite end of a semi-infinite cylinder tends to infinity and thus established spatial decay results of the Saint-Venant type.
Abstract: One method of regularizing the initial value problem for the backward heat equation involves replacing the equation by a singularly perturbed hyperbolic equation which is equivalent to a damped wave equation with negative damping. Another regularization of this problem is obtained by perturbing the initial condition rather than the differential equation. For both of these problems, we investigate the asymptotic behavior of the solutions as the distance from the finite end of a semi-infinite cylinder tends to infinity and thus establish spatial decay results of the Saint-Venant type.
TL;DR: In this paper, the authors approximate the original infinite-dimensional system by a sequence of finite-dimensional systems and consider the corresponding finite dimensional disturbance-attenuation problems, making the same assumptions required in approximations for the classical linear quadratic regulator problem.
Abstract: As in the finite-dimensional case, the appropriate state feedback for the infinite-dimensional $H^\infty$ disturbance-attenuation problem may be calculated by solving a Riccati equation. This operator Riccati equation can rarely be solved exactly. We approximate the original infinite-dimensional system by a sequence of finite-dimensional systems and consider the corresponding finite-dimensional disturbance-attenuation problems. We make the same assumptions required in approximations for the classical linear quadratic regulator problem and show that the sequence of solutions to the corresponding finite-dimensional Riccati equations converge strongly to the solution to the infinite-dimensional Riccati equation. Furthermore, the corresponding finite-dimensional feedback operators yield performance arbitrarily close to that obtained with the infinite-dimensional solution.