Showing papers on "Riccati equation published in 1993"

Backward stochastic differential equations and applications to optimal control

Abstract: We study the existence and uniqueness of the following kind of backward stochastic differential equation, $$x(t) + \int_t^T {f(x(s),y(s),s)ds + \int_t^T {y(s)dW(s) = X,} }$$ under local Lipschitz condition, where (Ω, ℱ,P, W(·), ℱt) is a standard Wiener process, for any given (x, y),f(x, y, ·) is an ℱt-adapted process, andX is ℱt-measurable. The problem is to look for an adapted pair (x(·),y(·)) that solves the above equation. A generalized matrix Riccati equation of that type is also investigated. A new form of stochastic maximum principle is obtained.

On the generalized pantograph functional-differential equation

Abstract: The generalized pantograph equation y′(t) = Ay(t) + By(qt) + Cy′(qt), y(0) = y0, where q ∈ (0, 1), has numerous applications, as well as being a useful paradigm for more general functional-differential equations with monotone delay. Although many special cases have been already investigated extensively, a general theory for this equation is lacking–its development and exposition is the purpose of the present paper. After deducing conditions on A, B, C ∈ ℂd×d that are equivalent to well-posedness, we investigate the expansion of y in Dirichlet series. This provides a very fruitful form for the investigation of asymptotic behaviour, and we duly derive conditions for limt⋅→∞y(t) = 0. The behaviour on the stability boundary possesses no comprehensive explanation, but we are able to prove that, along an important portion of that boundary, y is almost periodic and, provided that q is rational, it is almost rotationally symmetric. The paper also addresses itself to a detailed analysis of the scalar equation y′(t) = by(qt), y(0) = 1, to high-order pantograph equations, to a phenomenon, similar to resonance, that occurs for specific configurations of eigenvalues of A, and to the equation Y′(t) = AY(t) + Y(qt) B, Y(0) = Y0.

Kalman filtering with random coefficients and contractions

Abstract: The Riccati transformation of linear filtering/control theory is shown to be a contraction on the space of positive symmetric matrices. This is used to describe the asymptotic behavior of the filter for systems with stochastic stationary parameters.

A state-space description for GPC controllers

Abstract: Generalized predictive control (GPC)-type control algorithms traditionally derived in the polynomial domain are derived in this paper in the state-space domain, but following the polynomial approach due to Clarke et al. (1987). Relations between the polynomial and state-space parameters are presented. Some possible state-space representations which were used earlier in different publications are discussed. The problem of deriving the GPC algorithm in the state-space domain is solved for the unrestricted case as well as for the case of restricted control and output horizons. Some properties of the state estimate for this problem are presented; in particular, two methods of Kalman filtering—optimal and asymptotic—are proposed. The solution is valid for any possible (minimal or non-minimal) state-space representation. Another approach to this problem is by the ‘dynamic programming method’ and solving the Riccati equation (Bitmead et al. 1990). This approach is also presented in this paper but the me...

Parallel Algorithms for Optimal Control of Large Scale Linear Systems

Abstract: One - Theoretical Concepts.- 2. Linear-Quadratic Control Problems.- 2.1 Introduction.- 2.2 Recursive Methods for Singularly Perturbed Linear Continuous Systems.- 2.2.1 Parallel Algorithm for Solving Algebraic Lyapunov Equation.- 2.2.2 Parallel Algorithm for Solving Algebraic Riccati Equation.- 2.2.3 Case Study: Magnetic Tape Control Problem.- 2.3 Recursive Methods for Weakly Coupled Linear Continuous Systems.- 2.3.1 Parallel Algorithm for Solving Algebraic LyapIIDov Equation.- 2.3.2 Parallel Algorithm for Solving Algebraic Riccati Equation.- 2.4 Approximate Linear Regulator Problem for Continuous Systems.- 2.5 Recursive Methods for Singularly Perturbed Linear Discrete Systems.- 2.5.1 Parallel Algorithm for Solving Algebraic Lyapunov Equation.- 2.5.2 Case Study: An F-8 Aircraft.- 2.5.3 Parallel Algorithm for Solving Algebraic Riccati Equation.- 2.6 Approximate Linear Regulator for Discrete Systems.- 2.6.1 Case Study: Discrete Model of An F-8 Aircraft.- 2.7 Recursive Methods for Weakly Coupled Linear Discrete Systems.- 2.7.1 Parallel Algorithm for Solving Discrete Algebraic Lyapunov Equation.- 2.7.2 Case Study: Discrete Catalytic Cracker.- 2.7.3 Parallel Algorithm for Solving Algebraic Riccati Equation.- 2.7.4 Case Study: Discrete Model of a Chemical Plant.- 2.8 Notes and Comments.- 3. Decoupling Transformations.- 3.1 Introduction.- 3.2 Decoupling Transformation for Singularly Perturbed Linear Systems.- 3.3 Decoupling Transformation for Weakly Coupled Linear Systems.- 3.4 New Versions of Decoupling Transformations.- 3.4.1 New Decoupling Transformation for Linear Weakly Coupled System.- 3.4.2 New Decoupling Transformation for Linear Singularly Perturned Systems.- 3.5 Decomposition of the Differential Lyapunov Equations.- 3.6 Boundary Value Problem for Linear Continuous Weakly Coupled System.- 3.7 Boundary Value Problem for Linear Discrete-Time Weakly Coupled System.- 4. Output feedback control.- 4.1. Introduction.- 4.2 Output Feedback for Singularly Perturbed Linear Systems.- 4.3 Case Study: Fluid Catalytic Cracker.- 4.4 Output Feedback for Linear Weakly Coupled Systems.- 4.5 Case Study: Twelve Plate Absorption Column.- 5. Linear Stochastic Systems.- 5.1 Recursive Approach to Singularly Perturbed Linear Stochastic Systems.- 5.2 Case Study: F-S Aircraft LQG Controller.- 5.3 Recursive Approach to Weakly Coupled Linear Stochastic system.- 5.4 Case Study: Electric Power System.- 5.5 Parallel Reduced-Order Controllers for Stochastic Linear Discrete Singularly Perturbed Systems.- 5.6 Case Study: Discrete Steam Power System.- 5.7 Linear-Quadratic Gaussian Control of Discrete Weakly Coupled Systems at Steady State.- 5.8 Case Study: Distillation Column.- Appendix 5.1.- 6. Open-Loop Optimal Control Problems.- 6.1 Open-Loop Singularly Perturbed Control Problem.- 6.2 Case Study: Magnetic Tape Control.- 6.3 Open-Loop Weakly Coupled Optimal Control Problem.- 6.4 Case Study: Distillation Column.- 6.5 Open-Loop Discrete Singularly Perturbed Control Problem.- 6.6 Case Study: F-8 Aircraft Control Problem.- 6.7 Open-Loop Discrete Weakly Coupled Control Problem.- 6.8 Numerical Example.- 6.9 Conclusion.- Appendix 6.1.- Appendix 6.2.- Appendix 6.3.- Appendix 6.4.- 7. Exact Decompositions of Algebraic Riccati Equations.- 7.1 The Exact Decomposition of the Singularly Perturbed Algebraic Riccati Equation.- 7.2 Numerical Example.- 7.3 The Exact Decomposition of the Weakly Coupled Algebraic Riccati Equation.- 7.4 Case Study: A Satellite Control Problem.- 7.5 Conclusion.- Appendix 7.1.- Appendix 7.2.- Appendix 7.3.- 8. Differential and Difference Riccati Equations.- 8.1 Recursive Solution of the Singularly Perturbed Differential Riccati Equation.- 8.2 Case Study: A Synchronous Machine Connected to an Infinite Bus.- 8.3 Recursive Solution of the Riccati Differential Equation of Weakly Coupled Systems.- 8.4 Case Study: Gas Absorber.- 8.5 Reduced-Order Solution of the Singularly Perturbed Matrix Difference Riccati Equation.- 8.6 Case Study: Linearized Discrete Model of an F-8 Aircraft.- 8.7 Reduced-Order Solution of the Weakly Coupled Matrix Difference Riccati Equation.- 8.8 Numerical Example.- Appendix 8.1.- Appendix 8.2.- Appendix 8.3.- Appendix 8.4.- Two - Applications.- 9. Quasi Singularly Perturbed and Weakly Coupled Linear Systems.- 9.1 Linear Control of Quasi Singularly Perturbed Hydro Power Plants.- 9.2 Case Study: Hydro Power Plant.- 9.2.1 Weakly Controlled Fast Modes Structure.- 9.2.2 Strongly Controlled Slow Modes Structure.- 9.2.3 Weakly Controlled Fast Modes and Strongly Controlled Slow Modes Structure.- 9.3 Reduced-Order Design of Optimal Controller for Quasi Weakly Coupled Linear System.- 9.4 Case Studies.- 9.4.1 Chemical Reactor.- 9.4.2 F-4 Fighter Aircraft.- 9.4.3 Multimachine Power System.- 9.5 Reduced-Order Solution for a Class of Linear-Quadratic Optimal Control Problems.- 9.5.1 Numerical Example.- 9.6 Case Studies.- 9.6.1 Case Study 1: L-1011 Fighter Aircraft.- 9.6.2 Case Study 2: Distillation Column.- Notes.- Appendix 9.1.- 10. Singularly Perturbed Weakly Coupled Linear Control Systems.- 10.1 Introduction.- 10.2 Singularly Perturbed Weakly Coupled Linear Control Systems.- 10.3 Case Studies.- 10.3.1 Case Study 1: A Model of Supported Beam.- 10.3.2 Case Study 2: A Satellite Control Problem.- 10.4 Quasi Singularly Perturbed Weakly Coupled Linear Control Systems.- 10.5 Case Studies.- 10.6 Conclusion.- Appendix 10.1.- 11. Stochastic Output Feedback of Linear Discrete Systems.- 11.1 Introduction.- 11.2 Output Feedback of Quasi Weakly Coupled Linear Stochastic Discrete Systems.- 11.3 Case Study: Flight Control System for Aircrafts.- 11.4 Output Feedback of Singularly Perturbed Stochastic Discrete Systems.- 11.4.1 Problem Formulation.- 11.4.2 Slow-Fast Lower Order Decomposition.- 1111.5 Case Study: Discrete Model of a Steam Power System.- 12. Applications to Differential Games.- 12.1 Weakly Coupled Linear-Quadratic Nash Games.- 12.2 Solution of Coupled Algebraic Riccati Equations.- 12.2.1 Zeroth-Order Approximation.- 12.2.2 Solution of Higher Order of Accuracy.- 12.3 Numerical Examples.- Appendix 12.1.- Appendix 12.2.- 13. Recursive Approach to High Gain and Cheap Control Problems.- 13.1 Linear-Quadratic Cheap and High Gain Control Problems.- 13.1.1 High Gain Feedback Control.- 13.1.2 Cheap Control Problem.- 13.1.3 Parallel Algorithm for Solving Algebraic Riccati Equations for Cheap Control and High Gain Feedback.- 13.2 Case Study: Large Space Structure.- 13.3 Decomposition of the Open-Loop Cheap Control Problem.- 13.4 Numerical Example.- 13.5 Exact Decomposition of the Algebraic Riccati Equation for Cheap Control Problem.- 13.6 Numerical Example.- Appendix 13.1.- 14. Linear Approach to Bilinear Control Systems.- 14.1 Introduction.- 14.2 Reduced-Order Open Loop Optimal Control of Bilinear Systems.- 14.3 Reduced-Order Closed Loop Optimal Control of Bilinear Systems.- 14.3.1 Composite Near-Optimal Control of Bilinear Systems.- 14.4 Case Study: Induction Motor Drives.- 14.5 Near-Optimal Control of Singularly Perturbed Bilinear Systems.- 14.6 Optimal Control of Weakly Coupled Bilinear Systems.- 14.6.1 Open-Loop Control of Weakly Coupled Bilinear Systems.- 14.6.2 Closed-Loop Control of Weakly Coupled Bilinear Systems.- 14.7 Case Study: A Paper Making Machine.- 14.8 Conclusion.

A new truncation in Painleve analysis

Abstract: The author observes that when the expansion variable in Painleve analysis satisfies a system of Riccati equations, truncation at a level higher than constant level is allowed. This extends the range of exact solutions to nonlinear partial differential equations that one is able to obtain using truncated Painleve expansions.

Dynamics of Short-Term Univariate Forecast Error Covariances

Abstract: The covariance equation based on second-order closure for dynamics governed by a general scalar nonlinear partial differential equation (PDE) is studied. If the governing dynamics involve n space dimensions, then the covariance equation is a PDE in 2n space dimensions. Solving this equation for n = 3 is therefore computationally infeasible. This is a hindrance to stochastic-dynamic prediction as well as to novel methods of data assimilation based on the Kalman filter. It is shown that the covariance equation can be solved approximately, to any desired accuracy, by solving instead an auxiliary system of PDEs in just n dimensions. The first of these is a dynamical equation for the variance field. Successive equations describe, to increasingly high order, the dynamics of the shape of either the covariance function or the correlation function for points separated by small distances. The second-order equation, for instance, describes the evolution of the correlation length (turbulent microscale) field...

Assimilation of Sea Surface Topography into an Ocean Circulation Model Using a Steady-State Smoother

Abstract: The central issue of how to obtain useful, approximate, uncertainty estimates for assimilation methods using full general circulation models is addressed. Such estimates would be used with assimilation done by either sequential methods or Pontryagin principle/adjoint techniques. The problem of computing the error covariances of realistic oceanic general circulation models is explored by finding the asymptotic solutions to the Riccati equations governing Kalman filters and related smoothers. Existence of the steady-state is established through applying the concepts of controllability and observability to a coarse-resolution primitive equation model in the presence of altimetric observations. A “doubling algorithm” is then used to solve the Riccati equation. The methodology has the added benefit of rendering sequential estimation methods much less costly. Results are presented for a “twin experiment” and for Geosat altimeter data from the North Atlantic Ocean. A realistic altimetric system improves...

Towed array shape estimation using Kalman filters-theoretical models

Abstract: The dynamical behavior of a thin flexible array towed through the water is described by the Paidoussis equation. By discretizing this equation in space and time a finite-dimensional state-space representation is obtained where the states are the transverse displacements of the array from linearity in either the horizontal or vertical plane. The form of the transition matrix in the state-space representation describes the propagation of transverse displacements down the array. The outputs of depth sensors and compasses located along the array are shown to be related in a simple, linear manner to the states. From this state-space representation a Kalman filter which recursively estimates the transverse displacements and hence the array shape is derived. It is shown how the properties of the Kalman filter reflect the physics of the propagation of motion down the array. Solutions of the Riccati equation are used to predict the mean square error of the Kalman filter estimates of the transverse displacements. >

Universal properties of the two-dimensional Kuramoto-Sivashinsky equation

Abstract: We investigate the properties of the Kuramoto-Sivashinsky equation in two spatial dimensions. We show by an explicit, numerical, coarse-graining procedure that its long-wavelength properties are described by a stochastic, partial differential equation of the Kardar-Parisi-Zhang type. From the computed parameters in our effective, stochastic equation we argue that the length and time scales over which the correlation functions cross over from linear diffusive to those of the full nonlinear equation are very large. The behavior of the three-dimensional equation is also discussed.

A multishift algorithm for the numerical solution of algebraic riccati equations

Abstract: We study an algorithm for the numerical solution of algebraic matrix Riccati equa- tions that arise in linear optimal control problems. The algorithm can be considered to be a multishift technique, which uses only orthogonal symplectic similarity transformations to compute a Lagrangian invariant subspace of the associated Hamiltonian matrix. We describe the details of this method and compare it with other numerical methods for the solution of the algebraic Riccati equation.

Linear quadratic problems with indefinite cost for discrete time systems

Abstract: This paper deals with the discrete-time, infinite-horizon linear quadratic problem with indefinite cost criterion. Given a discrete-time linear system, an indefinite cost-functional and a linear subspace of the state space, the problem of minimizing the cost-functional over all inputs that force the state trajectory to converge to the given subspace is considered. A geometric characterization of the set of all Hermitian solutions of the discrete-time algebraic Riccati equation is given. This characterization forms the discrete-time counterpart of the well-known geometric characterization of the set of all real symmetric solutions of the continuous-time algebraic Riccati equation as developed by Willems [IEEE Trans. Automat. Control, 16 (1971), pp. 621–634] and Coppel [Bull. Austral. Math. Soc., 10 (1974), pp. 377–401]. In the set of all Hermitian solutions of the Riccati equation the solution that leads to the optimal cost for the above-mentioned linear quadratic problem is identified. Finally, necessary ...

Hidden symmetries and linearization of the modified Painlevé–Ince equation

Abstract: The linearization and hidden symmetries of the modified Painleve–Ince equation, y‘+σyy’+βy3=0, where σ and β are constants, are presented. The linearization of this equation by a nonlocal transformation yields a damped (stable) or growing (unstable) harmonic oscillator equation for β≳0. Hidden symmetries are analyzed by transforming the modified Painleve–Ince equation to a third‐order ordinary differential equation (ODE) which, in general, is invariant under a three‐parameter group by a Riccati transformation. A type I hidden symmetry is found of a second‐order ODE found from the third‐order ODE where a symmetry is lost in the reduction of order by the non‐normal subgroup invariants. A type II hidden symmetry occurs in the third‐order ODE because the symmetries of a second‐order ODE, reduced from the third‐order ODE by another set of normal subgroup invariants, are increased.

A Fourier-Riccati approach to radiative transfer. I. Foundations.

Abstract: The three-dimensional equation of radiative transfer is formally solved using a Fourier-Riccati approach while calculations are performed on cloudy media embedded in a two-dimensional space. An extension to Stephens' work this study addresses the coupling between space and angle asserted by the equation of transfer. In particular, the accuracy of the computed radiation field as it is influenced by the angular resolution of the phase function and spatial discretization of the cloudy medium is discussed. The necessity of using a large number of quadrature points to calculate fluxes even when the phase function is isotropic for media exhibiting vertical and horizontal inhomogeneities is demonstrated. Effects of incorrect spatial sampling on both radiance and flux fields are also quantified by example. Radiance and flux comparisons obtained by the Fourier-Riccati model and the independent pixel approximation for inhomogeneous cloudy media illustrate the inadequacy of the latter even for tenuous clouds.

Mixed H 2 /H ∞ control with pole placement: state feedback case

Abstract: This paper considers state feedback mixed H2/H∞ control with pole placement via convex optimization. A generalized Riccati equation and a change of variable technique are employed in order to convexify the design problem. The resulting convex optimization problem is shown to be reducable to Generalized Eigenvalue Minimization Problem.

The standard H ∞ problem and the maximum principle: the general linear case

Abstract: The classical, relatively simple ideas of linear quadratic (LQ) optimization are used in a time domain treatment of the standard$H_\infty $problem. Given the power of time domain analysis, the problem can be solved in the general framework of linear time varying, possibly infinite-dimensional, finite as well as infinite horizon systems, under no structural restrictions (such as block dimensions, zero blocks in the D operator, etc.). In the spirit of recent finite-dimensional linear time invariant (LTI) results, the solution is given in terms of two coupled Riccati equations; it includes a criterion for suboptimality and a parametrization of all suboptimal compensators. Results pertinent to LTI, periodic, and asymptotic systems are obtained as corollaries.

Discretization and some qualitative properties of ordinary differential equations about equilibria.

Abstract: Discretizations and Grobman-Hartman Lemma, discretizations and the hierarchy of invariant manifolds about equilibria are considered. For one-step methods, it is proved that the linearizing conjugacy for ordinary differential equations in Grobman-Hartman Lemma is, with decreasing stepsize, the limit of the linearizing conjugacies of the discrete systems obtained via time-discretizations. Similar results are proved for all types of invariant manifolds about equilibria. The estimates are given in terms of the degree of smoothness of the original ordinary differential equation as well as in terms of the stepsize and of the order of the discretization method chosen. The results sharpen and unify those of Beyn [6], Beyn and Lorenz [7] and Feckan [17], [19].

Approximation of the algebraic Riccati equation in the Hilbert space of Hilbert-Schmidt operators

Abstract: This paper deals with the problem of approximating the infinite-dimensional algebraic Riccati equation, considered as an abstract equation in the Hilbert space of Hilbert–Schmidt operators. Two kin...

Linear dynamical systems, coherent state manifolds, flows, and matrix Riccati equation

Abstract: The classical motion and the quantum evolution determined by linear Hamiltonians on coherent state manifolds of compact and noncompact Hermitian symmetric space structure are considered. A matrix Riccati equation, with opposite signs of the quadratic terms in the compact and noncompact cases, determines the classical motion and the quantum evolution. The geometric meaning of equations of motion as flow on a complex Grassmann manifold and his noncompact dual is emphasized. Possibilities of generalizing the results to flag manifolds are pointed out.

Perturbation analysis of the discrete Riccati equation

Abstract: The sensitivity of the discrete-time matrix Riccati equation relative to perturbations in its coefficients is studied. Both local and non-local perturbation bounds are obtained. In particular the conditioning of the equation is determined.