Showing papers on "Riccati equation published in 1992"

Link between solitary waves and projective Riccati equations

Abstract: Many solitary wave solutions of nonlinear partial differential equations can be written as a polynomial in two elementary functions which satisfy a projective (hence linearizable) Riccati system. From that property, the authors deduce a method for building these solutions by determining only a finite number of coefficients. This method is much shorter and obtains more solutions than the one which consists of summing a perturbation series built from exponential solutions of the linearized equation. They handle several examples. For the Henon-Heiles Hamiltonian system, they obtain several exact solutions; one of them defines a new solitary wave solution for a coupled system of Boussinesq and nonlinear Schrodinger equations. For a third order dispersive equation with two monomial nonlinearities, they isolate all cases where the general solution is single valued.

Kalman filtering and Riccati equations for descriptor systems

Abstract: A general formulation of a discrete-time filtering problem for descriptor systems is considered. It is shown that the nature of descriptor systems leads directly to the need to examine singular estimation problems. Using a dual approach to estimation, the authors derive a so-called 3-block form for the optimal filter and a corresponding 3-block Riccati equation for a general class of time-varying descriptor models which need not represent a well-posed system in that the dynamics may be either over or under constrained. Specializing in the time-invariant case, they examine the asymptotic properties of the 3-block filter, and in particular analyze in detail the resulting 3-block algebraic Riccati equation. The noncausal nature of discrete-time descriptor dynamics implies that future dynamics may provide some information about the present state. A modified form for the descriptor Kalman filter that takes this information into account is presented. >

H∞ estimation for uncertain systems

Abstract: SUMMARY This paper deals with the problem of H, estimation for linear systems with a certain type of time-varying norm-bounded parameter uncertainty in both the state and output matrices. We address the problem of designing an asymptotically stable estimator that guarantees a prescribed level of H, noise attenuation for all admissible parameter uncertainties. Both an interpolation theory approach and a Riccati equation approach are proposed to solve the estimation problem, with each method having its own advantages. The first approach seems more numerically attractive whilst the second one provides a simple structure for the estimator with its solution given in terms of two algebraic Riccati equations and a parameterization of a class of suitable H, estimators. The Riccati equation approach also pinpoints the ‘worst-case’ uncertainty.

Memoryless stabilization of uncertain linear systems including time-varying state delays

Abstract: The problem of stabilizing a class of uncertain time-delay systems via memoryless linear feedback is examined. The systems under consideration are linear systems with time-varying state delays. They also contain uncertain parameters (possibly time-varying) whose values are known only to within a prescribed compact bounding set. The main contribution given is to enlarge the class of time-delay systems for which one can construct a stabilizing memoryless linear feedback controller. Within this framework, a novel notion of robust memoryless stabilizability is first introduced via the method of Lyapunov functionals. Then a sufficient condition for the stabilizability is proposed. It is shown that solvability of a parameterized Riccati equation can be used to determine whether the time-delay system satisfies the sufficient condition. If there exists a positive definite symmetric solution satisfying the Riccati equation, a suitable memoryless linear feedback law can be derived. >

Recursive structure of noncausal Gauss-Markov random fields

Abstract: An approach is developed for noncausal Gauss-Markov random fields (GMRFs) that enables the use of recursive procedures while retaining the noncausality of the field. Recursive representations are established that are equivalent to the original field. This is achieved by first presenting a canonical representation for GMRFs that is based on the inverse of the covariance matrix, which is called the potential matrix. It is this matrix rather than the field covariance that reflects in a natural way the MRF structure. From its properties, two equivalent one-sided representations are derived, each of which is obtained as the successive iterates of a Riccati-type equation. For homogeneous fields, these unilateral descriptions are symmetrized versions of each other, the study of only one Riccati equation being required. It is proven that this Riccati equation converges at a geometric rate, therefore the one-sided representations are asymptotically invariant. These unilateral representations make it possible to process the fields with well-known recursive techniques such as Kalman-Bucy filters and two-point smoothers. >

H ∞ optimal sampled-data control in continuous time systems

Abstract: We consider the design of H ∞ optimal discrete-time (digital) controllers in continuous-time systems. An apparent difficulty, especially in utilizing modern transform-domain analysis in this context, stems from the absence of an appropriate (transfer function) model for the hybrid-time (discrete and continuous) closed-loop system. This difficulty is overcome through the introduction of an equivalent difference-equation model for the continuous-time system, with distributed inputs and outputs; equivalence being in the sense that the continuous-and discrete-time inputs and outputs are essentially identical. Using the interplay between the discrete and the continuous time models, solutions of the well-known purely continuous-time and purely discrete-time standard problems extend to solutions of the problem at hand. They comprise Riccati equation characterizations of feasible combinations of sampling rates and bounds on the closed-loop induced input-output norm, and parameterization of compensators. We consid...

Numerical integration of the differential Riccati equation and some related issues

Abstract: . In this paper the problem of direct numerical integration of differential Riccati equations (DREs) and some related issues are considered. The DRE is an expression of a particular change of variables for a linear system of ordinary differential equations. The error that an approximate solution of the DRE induces on the original variables of the system is considered, and it is related to geometrical properties of the system itself. Sharp bounds on the global error for the computed solution are also given in terms of local errors and geometrical properties of the original system. Nonstiff and stiff DREs of unsymmetric and symmetric type are considered. A useful matrix interpretation is given for many integration schemes (such as the backward differentiation formulas, BDF), when applied to the DRE. This allows the matrix structure of the problem to be exploited. In particular, for stiff DREs, the resulting strategy allows for a saving of three orders of magnitude with respect to the standard reform...

The H∞ control problem

Abstract: Part 1 Introduction: robustness analysis the infinity control problem stabilization of uncertain systems the graph topology the mixed-sensitivity problem main items of this book - singular systems, differential game, the minimum entropy infinity control problem, the finite horizon infinity control problem, discrete time systems. Part 2 Notation and basic properties: introduction linear systems - continuous time, discrete time rational matrices geometric theory the Hardy and Lebesgues spaces - continuous time, discrete time (almost) disturbance decoupling problems. Part 3 The regular full-information infinity control problem: introduction problem formulation and main results intuition for the formal proof solvability of the Riccati equation existence of a suitable controller. Part 4 The general full-information infinity control problem: introduction problem formulation and main results solvability of the quadratic matrix inequality existence of state feedback laws the design of a suitable compensator a direct feed through matrix from disturbance to output invariant zeros on the imaginary axis - frequency domain loop shifting, cheap control conclusion. Part 5 The infinity control problem with measurement feedback: introduction problem formulation and main results reduction of the original problem to an almost disturbance decoupling problem the design of a suitable compensator characterization of achievable closed loop systems no assumptions on any direct feedthrough matrix - an extra direct-feedthrough matrix from disturbance to output, an extra direct-feedthrough matrix from control to measurement conclusion. Part 6 The singular zero-sum differential game with stability: introduction problem formulation and main results existence of almost equilibria necessary conditions for the existence of almost equilibria the regular differential game conclusions. Part 7 The singular minimum entropy infinity control problem: introduction problem formulation and results properties of the entropy function a system transformation (almost) disturbance decoupling and minimum entropy conclusion. Part 8 The finite horizon finite control problem: introduction problem formualtion and main results completion of the squares existence of compensators concluding remarks.

The discrete time H∞ control problem with measurement feedback

Abstract: This paper is concerned with the discrete time $H_\infty $ control problem with measurement feedback. It follows that, as in the continuous time case, the existence of an internally stabilizing controller that makes the $H_\infty $ norm strictly less than 1 is related to the existence of stabilizing solutions to two algebraic Riccati equations. However, in the discrete time case, the solutions of these algebraic Riccati equations must satisfy extra conditions.

Matrix Riccati Equations

Abstract: This chapter presents matrix Riccati equations applicable to systems of quadratic ordinary differential equations. It yields an exact solution. The chapter highlights that there is an exact solution available for matrix Riccati equations. If the given ordinary differential equations can be put in the form of a matrix Riccati equation, the solution can be found. Matrix Riccati equations arise naturally in a number of physical settings. The gains in a Kalman–Bucy filter satisfy a matrix Riccati equation. Also, the deflection of a beam can be described by such equations. They also appear quite often in the context of control theory and invariant embedding solutions. Kerner has also shown that nonlinear differential systems of arbitrary order, ζ i = X i (ζ i , ζ 2 ,…. ζ k , t ) for t = 1,2,…, k , may often be reduced to Riccati systems.

A Common Generalization of Functional Equations Characterizing Normed and Quasi-Inner-Product Spaces

Abstract: We determine the general solutions of the functional equation Â(x + y)+f2(x-y)=f3(x)+f4(y), x,yeG for fi. G —¥ F (i = 1,2,3,4), where G is a 2-divisible group and F is a commutative field of characteristic different from 2. The motivation for studying this equation came from a result due to Dry gas (4) where he proved a Jordan and von Neumann type char­ acterization theorem for quasi-inner products. Also, this equation is a generalization of the quadratic functional equation investigated by several authors in connection with inner product spaces and their generalizations. Special cases of this equation include the Cauchy equation, the Jensen equation, the Pexider equation and many more. Here, we determine the general solution of this equation without any regularity assumptions onft.

The exact slow-fast decomposition of the algebraic Ricatti equation of singularly perturbed systems

Abstract: The algebraic Riccati equation for singularly perturbed control systems is completely and exactly decomposed into two reduced-order algebraic Riccati equations corresponding to the slow and fast time scales. The pure-slow and pure-fast algebraic Riccati equations are asymmetric ones, but their O( epsilon ) perturbations are symmetric. It is shown that the Newton method is very efficient for solving the obtained asymmetric algebraic Riccati equations. The method presented is very suitable for parallel computations. Due to the complete and exact decomposition of the Riccati equation, this procedure might produce new insight into the two-time-scale optimal filtering and control problems. >

On the time-varying Riccati difference equation of optimal filtering

Abstract: This paper studies the time-varying Riccati difference equation (RDE) for the filtering problem. In particular, existence, stabilizability, and attractiveness properties of the real symmetric solutions that remain bounded on $( - \infty , + \infty )$ (infinite-time solutions) are investigated. Under the assumption of uniform detectability, conditions for the existence of the maximal and stabilizing solutions are given. Analogous results are worked out for the minimal and antistabilizing solutions by making reference to the uniform antidetectability notion. Moreover, it is shown that, under uniform observability, the set of all symmetric infinite-time solutions constitute an infinite number of lattices with common minimal and maximal elements.

On the inverse scattering problem for the Helmholtz equation in one dimension

Abstract: A scheme is presented for the solution of inverse scattering problems for the one-dimensional Helmholtz equation. The scheme is based on a combination of the standard Riccati equation for the impedance function with a new trace formula for the derivative of the potential, and can be viewed as a frequency-domain version of the layer-stripping approach. The principal advantage of the procedure is that if the scatterer is be reconstructed has m>or=1 continuous derivatives, the accuracy of the reconstruction is proportional to 1/am, where a is the highest frequency for which scattering data are available. Thus a smooth scatterer is reconstructed very accurately from a limited amount of available data. The scheme has an asymptotic cost O(n2), where n is the number of features to be recovered (as do classical layer-stripping algorithms), and is stable with respect to perturbations of the scattering data. The performance of the algorithm is illustrated by several examples.

Hidden symmetries associated with the projective group of nonlinear first-order ordinary differential equations

Abstract: Hidden symmetries, those not found by the classical Lie group method for point symmetries, are reported for nonlinear first-order ordinary differential equations (ODEs) which arise frequently in physical problems. These are for the special class of the eight nonAbelian, two-parameter subgroups of the eight-parameter projective group. The first-order ODEs can be transformed by non-local transformations to new separable first-order ODEs which then can be reduced to quadratures. The first-order ODEs include Riccati equations and equations which in particular cases are of the form of Abel's equation. The procedure demonstrates the feasibility of integrating nonlinear ODEs that do not show any apparent Lie group point symmetry. Applications to the Vlasov characteristic equation and the reaction-diffusion equation are given.

Theory of chiral multilayers

Abstract: We investigate reflection from and transmission through chiral multilayers with discrete and continuous variations in material characteristics. Both boundary-value and initial-value approaches are used. The S-parameter matrix and associated copolarized and cross-polarized reflection and transmission coefficients are derived from the chiral constitutive relations, Maxwell’s equations, and boundary conditions. A generalized matrix Riccati equation is found for the reflection and transmission coefficients of arbitrary chiral multilayers by using an initial-value approach and Ambarzumian’s principle of invariant embedding. All results are exact and applicable to both normal and oblique incidence. Special emphasis is given to the physical principles involved, to special cases, and to salient features.

A H/sub infinity / optimal theory-based generator control system

Abstract: A generator control system is designed on the basis of H/sub infinity / control theory, and the control performance of the system is investigated. A Riccati equation approach is first introduced. How to apply this approach to designing the generator control system is then presented. The control performance of the system based on H/sub infinity / control design is studied by numerical examples and is compared with that based on linear-quadratic control design. Some advantages and disadvantages of the generator control system based on the H/sub infinity / design are clarified. >

Stabilising controller and observer synthesis for uncertain large-scale systems by the Riccati equation approach

Abstract: The paper introduces a Riccati equation approach to synthesise of the full state observers and state feedback controllers for uncertain largescale systems. In this approach, if two given algebraic Riccati equations are solved, their solutions can be applied to synthesise the stabilising state feedback and observer gain matrices. The uncertainties considered in each subsystem may be time-varying and appear in the system matrices (matrix Ai), input connection matrices (matrix Bi), or/and output matrices (matrix Ci). However the values of those uncertainties are constrained to lie within some known admissable bounds. Furthermore, the so-called matching conditions are not needed in the paper.

Robust adaptive LQ control schemes

Abstract: The certainty equivalence principle is used to combine a robust adaptive law with a control structure derived from the linear quadratic (LQ) control problem. The resulting adaptive control scheme is applicable to minimum and nonminimum phase continuous-time plants and is robust with respect to unmodeled dynamics and bounded disturbances. The computational complexity of the continuous-time adaptive LQ control scheme is improved by using a hybrid adaptive law which requires the solution of an algebraic Riccati equation at each interval of time rather than at each time t. >

Wave Problems for Repetitive Structures and Symplectic Mathematics

Abstract: Based on the analogy between structural mechanics and optimal control theory, the eigensolutions of a symplectic matrix, the adjoint symplectic ortho-normalization relation and the eigenvector expansion method are introduced into the wave propagation theory for sub-structural chain-type structures, such as space structures, composite material and turbine blades. The positive and reverse algebraic Riccati equations are derived, for which the solution matrices are closely related to the power flow along the sub-structural chain. The power flow orthogonality relation for various eigenvectors is proved, and the energy conservation result is also proved for wave scattering problems.

Self-similar solutions of the generalized Emden-Fowler equation

Abstract: When, in the generalized Emden-Fowler equation y” + f ( x ) y n = 0, the function f ( x ) takes certain forms, the equation possesses one or two Lie point symmetries and in some cases closed-form solutions can be obtained.

Lower summation bounds for the discrete Riccati and Lyapunov equations

Abstract: Lower eigenvalue summation (including trace) bounds for the solution of the discrete algebraic Riccati and Lyapunov matrix equations are presented. These are tighter than, or supplement, existing results. >

Scaling of the discrete-time algebraic Riccati equation to enhance stability of the Schur solution method

Abstract: A simple scaling procedure for discrete-time Riccati equations is introduced. This procedure eliminates instabilities which can be associated with the linear equation solution step of the generalized Schur method without changing the condition of the underlying problem. A computable bound for the relative error of the solution of the Riccati equation is also derived. >

Some remarks on the Riccati equation arising in an optimal control problem with state- and control-dependent noise

Abstract: This paper solves a quadratic optimal control for a linear stochastic evolution equation with unbounded coefficients. It is assumed that the stochastic noise depends both on the state and on the control. The dynamic programming approach is used and attention is focused on the Riccati equation. In §§5 and 6 some attractivity and maximality properties of the solutions of the algebraic Riccati equation are proved and it is shown that, in some special cases, there exists a maximal solution.

Computational structural mechanics and optimal control—the simulation of substructural chain theory to linear quadratic optimal control problems†

Abstract: The theory of optimal control and the theory of a substructural chain in static structural analysis are mutually simulated issues. From the minimum potential energy variational principle of the substructural chain, the generalized variational principle with two kinds of variables is derived first. By comparing that generalized variational principle with the variational principle in LQ control theory, the simulation relation is established. Based on that relation, the potential energy and mixed energy formulation of the algebraic Riccati equations are derived, then iterative algorithms are proposed which give the upper and lower bounds to the solution matrix. By using the solutions of the positive and negative co-ordinate algebraic Riccati equations, the canonical transformation matrices for the eigenproblems of the substructural chain and LQ control are constructed respectively, which reduce the eigenproblem to half-size. The properties of the solutions are analysed, which establishes the basis for expansion solutions.

Game theory approach to state estimation of linear discrete-time processes and its relation to H∞ -optimal estimation

Abstract: A game theory approach to the state-estimation of linear discrete-time systems is presented. The resulting state estimation suggests an alternative to the Kalman filter, in cases where the exact statistics of the input and the measurement noise processes is not known. It turns out that the game-theoretic filter provides an H∞-optimal estimation. Moreover, it is shown that the covariance matrix of the estimation error is bounded, from above, by the solution of a modified Riccati equation.