Showing papers on "Riccati equation published in 1991"

The Riccati Equation

Abstract: 1 Count Riccati and the Early Days of the Riccati Equation.- 2 Solutions of the Continuous and Discrete Time Algebraic Riccati Equations: A Review.- 3 Algebraic Riccati Equation: Hermitian and Definite Solutions.- 4 A Geometric View of the Matrix Riccati Equation.- 5 The Geometry of the Matrix Riccati Equation and Associated Eigenvalue Methods.- 6 The Periodic Riccati Equation.- 7 Invariant Subspace Methods for the Numerical Solution of Riccati Equations.- 8 The Dissipation Inequality and the Algebraic Riccati Equation.- 9 The Infinite Horizon and the Receding Horizon LQ-Problems with Partial Stabilization Constraints.- 10 Riccati Difference and Differential Equations: Convergence, Monotonicity and Stability.- 11 Generalized Riccati Equation in Dynamic Games.

Differential and Algebraic Riccati Equations With Application to Boundary/Point Control Problems: Continuous Theory and Approximation Theory

Abstract: 1. Introduction: Two abstract classes statement of main problems.- 2. Abstract differential Riccati equation for the first class subject to the analyticity assumption (H.1)=(1.5).- 3. Abstract differential Riccati equations for the second class subject to the trace regularity assumption (H.2)=(1.6).- 4. Abstract differential Riccati equations for the second class subject to the regularity assumptions (H.2R)=(1.8).- 5. Abstract algebraic Riccati equations: Existence and uniqueness.- 6. Examples of partial differential equation problems satisfying (H.1).- 7. Examples of partial differential equation problems satisfying (H.2).- 8. Example of a partial differential equation problem satisfying (H.2R).- 9. Numerical approximations of the solution to the abstract differential and algebraic Riccati equations.- 10. Examples of numerical approximation for the classes (H.1) and (H.2).- 11. Conclusions.

Memoryless stabilization of uncertain dynamic delay systems: Riccati equation approach

Abstract: The authors present a procedure for obtaining the memoryless linear state feedback control of uncertain dynamic delay systems. The uncertainties are time varying and within a given compact set. This method is an extension of the Riccati equation approach proposed by I.R. Petersen and C.V. Hollot (1986). The extension is straightforward. Also the uncertainties do not need to satisfy the matching conditions. >

H∞ estimation for discrete-time linear uncertain systems

Abstract: SUMMARY This paper is concerned with the problem of Hm estimation for linear discrete-time systems with timevarying norm-bounded parameter uncertainty in both the state and output matrices. We design an estimator such that the estimation error dynamics is quadratically stable and the induced operator norm of the mapping from noise to estimation error is kept within a prescribed bound for all admissible uncertainties. A Riccati equation approach is proposed to solve the estimation problem and it is shown that the solution is related to two algebraic Riccati equations.

The Periodic Riccati Equation

Abstract: The history of the time-varying Riccati equation can be traced back to Riccati’s original manuscripts of 1715–1725. Indeed, the major concern of Count Riccati was to study the problem of the separation of variables in quadratic and time-varying scalar differential equations [1]. The equation has been the subject of several contributions in the subsequent centuries. In recent times, the importance of the Riccati equation in Control, Systems, and Signals has led to the development of a considerable research activity on the subject, see e.g., [2], [3], [4] for the time-varying matrix Riccati equation.

Invariant Subspace Methods for the Numerical Solution of Riccati Equations

Abstract: In this tutorial paper, an overview is given of progress over the past ten to fifteen years towards reliable and efficient numerical solution of various types of Riccati equations. Our attention will be directed primarily to matrix-valued algebraic Riccati equations and numerical methods for their solution based on computing bases for invariant subspaces of certain associated matrices. Riccati equations arise in modeling both continuous-time and discrete-time systems in a wide variety of applications in science and engineering. One can study both algebraic equations and differential or difference equations. Both algebraic and differential or difference equations can be further classified according to whether their coefficient matrices give rise to so-called symmetric or nonsymmetric equations. Symmetric Riccati equations can be further classified according to whether or not they are definite or indefinite.

On the validity of the Ginzburg-Landau equation

Abstract: The famous Ginzburg-Landau equation describes nonlinear amplitude modulations of a wave perturbation of a basic pattern when a control parameterR lies in the unstable regionO(e2) away from the critical valueRc for which the system loses stability. Heree>0 is a small parameter. G-L's equation is found for a general class of nonlinear evolution problems including several classical problems from hydrodynamics and other fields of physics and chemistry. Up to now, the rigorous derivation of G-L's equation for general situations is not yet completed. This was only demonstrated for special types of solutions (steady, time periodic) or for special problems (the Swift-Hohenberg equation). Here a mathematically rigorous proof of the validity of G-L's equation is given for a general situation of one space variable and a quadratic nonlinearity. Validity is meant in the following sense. For each given initial condition in a suitable Banach space there exists a unique bounded solution of the initial value problem for G-L's equation on a finite interval of theO(1/e2)-long time scale intrinsic to the modulation. For such a finite time interval of the intrinsic modulation time scale on which the initial value problem for G-L's equation has a bounded solution, the initial value problem for the original evolution equation with corresponding initial conditions, has a unique solutionO(e2) — close to the approximation induced by the solution of G-L's equation. This property guarantees that, for rather general initial conditions on the intrinsic modulation time scale, the behavior of solutions of G-L's equation is really inherited from solutions of the original problem, and the other way around: to a solution of G-L's equation corresponds a nearby exact solution with a relatively small error.

A transfer function approach to the problems of discrete-time systems: H/sub infinity /-optimal linear control and filtering

Abstract: A solution is derived to the problems of H/sub infinity /-optimal linear state regulation and filtering. The solution method for both problems is based on a transfer function approach which applies standard spectral factorization. Return difference relations are given which are extensions of the relations associated with the linear quadratic problem. >

Stability evaluation and track life of the PDAF for tracking in clutter

Abstract: The authors present an effective approach of a hybrid nature to the nonsimulation performance evaluation of the probabilistic data association filtering (PDAF) method for tracking in clutter. In this approach, a continuous-valued covariance, which is a function of a discrete-valued random variable (the number of validated measurements), is used to characterize the tracking errors in an average sense. This covariance can be calculated offline recursively from a modified Riccati equation, which can be obtained by replacing the measurement-dependent terms in the original stochastic equation with their conditional expectations. This approach has the merit that it yields a quantification of the transients of tracking divergence as well as substantially better accuracy than previous work. Such an approach is particularly suitable for stability evaluation of tracking filters. In addition, a quantitative study of the track-life problem is made in which the number of validated measurements plays a central role. >

Approximation theory for linear-quadratic-Guassian optimal control of flexible structures

Abstract: This paper presents approximation theory for the linear-quadratic-Gaussian optimal control problem for flexible structures whose distributed models have bounded input and output operators. The main purpose of the theory is to guide the design of finite-dimensional compensators that approximate closely the optimal compensator, which is infinite-dimensional. Design of the optimal compensator separates into an optimal linear-quadratic control problem and a dual optimal state estimation problem; the solution to each problem lies in the solution to an infinite-dimensional Riccati operator equation. The approximation scheme in the paper approximates the infinite-dimensional LQG problem with a sequence of finite-dimensional LQG problems defined for a sequence of finite-dimensional, usually finite-element or modal, approximations of the distributed model of the structure. Two Riccati matrix equations determine the solution to each approximating problem.The finite-dimensional equations for numerical approximation ...

On the structure of H/sup infinity / control systems and related extensions

Abstract: An existence condition of a H/sup infinity / controller which achieves a prescribed norm bound of the closed-loop transfer function is derived in the frequency domain based on a generalization of the notion of J-lossless systems. This condition is regarded as a frequency-domain representation of the well-known existence condition in the state space represented in terms of the two algebraic Riccati equations. A notion of J-orthogonal complement, which is introduced as a generalization of the usual orthogonal complement, play an important role in clarifying the fundamental frequency domain structure of the model-matching problem and simplifying the computation of controllers. The results are extended to the nonstandard case where the direct feedthrough from the input to the error or from the exogenous signal to the output is no longer of full rank. It is shown that the proper controller achieving the prescribed norm bound exists even in this case. In nonstandard cases, the controller order can be smaller than the plant order. >

The Dissipation Inequality and the Algebraic Riccati Equation

Abstract: Undoubtedly one of the most important concepts in linear systems and control, both from a theoretical as well as from a practical point of view, is the algebraic Riccati equation. Since its introduction in control theory by Kaiman [16] the beginning of the sixties, the algebraic Riccati equation has known an impressive range of applications, such as linear quadratic optimal control, stability theory, stochastic filtering and stochastic control, stochastic realization theory, synthesis of linear passive networks, differential games and, most recently, H ∞ optimal control and robust stabilization. The purpose of the present paper is to give an expository survey of the main concepts, results and applications related to the algebraic Riccati equation.

A dynamic games approach to controller design: disturbance rejection in discrete-time

Abstract: It is shown that the discrete-time disturbance rejection problem, formulated in finite and infinite horizons, and under perfect state measurements, can be solved by making direct use of some results on linear-quadratic zero-sum dynamic games. For the finite-horizon problem an optimal (minimax) controller exists, and can be expressed in terms of a generalized (time-varying) discrete-time Riccati equation. The existence of an optimum also holds in the infinite-horizon case, under an appropriate observability condition, with the optimal control, given in terms of a generalized algebraic Riccati equation, also being stabilizing. In both cases, the corresponding worst-case disturbances turn out to be correlated random sequences with discrete distributions, which means that the problem (viewed as a dynamic game between the controller and the disturbance) does not admit a pure-strategy saddle point. Results for the delayed state measurement and the nonzero initial state cases are presented. >

A numerical algorithm for optimal feedback gains in high dimensional linear quadratic regulator problems

Abstract: A hybrid method for computing the feedback gains in linear quadratic regulator problems is proposed. The method, which combines use of a Chandrasekhar type system with an iteration of the Newton–Kleinman form with variable acceleration parameter Smith schemes, is formulated to efficiently compute directly the feedback gains rather than solutions of an associated Riccati equation. The hybrid method is particularly appropirate when used with large dimensional systems such as those arising in approximating infinite-dimensional (distributed parameter) control systems (e.g., those governed by delay-differential and partial differential equations). Computational advantages of the proposed algorithm over the standard eigenvector (Potter, Laub–Schur) based techniques are discussed, and numerical evidence of the efficacy of these ideas is presented.

Learning control using neural networks

Abstract: The basic features of the learning-type neural network (NN) controller are clarified. Analytical and experimental results show its stability, convergence and generalization ability compared with the adaptive-type NN and conventional learning control. As an application of the learning-type NN, a nonlinear optimum regulator is presented whose learning ability can obtain optimum conditions without solving a difficult Riccati equation. Moreover, it can be applied to a nonlinear control system because of its nonlinear mapping ability, although the conventional optimum regulator can only be applied to a linear system. Finally task planning is proposed in terms of skill acquisition using the learning-type NN, which implies the possibility of making an interface with an upper symbolic-level control. >

Initial boundary value problems and optimal control for nonautonomous parabolic systems

Abstract: A large class of linear nonautonomous parabolic systems in bounded domains is considered, with control acting on the boundary through Dirichlet or Neumann conditions, from the point of view of semigroup theory. The results from [Rend. Sem. Mat. Univ. Padova, 78 (1987), pp. 47–107], [On fundamental solutions for abstract parabolic equations, Lecture Notes in Math., Vol. 1223, Springer-Verlag, Berlin, Heidelberg, 1986, pp. 1–11] on abstract homogeneous parabolic Cauchy problems allow operators with varying domains and Holder continuous coefficients to be handled. A representation formula for solutions corresponding to square integrable control functions is derived and used to solve a linear-quadratic regulator problem over finite time horizon, by a direct study of the associated integral Riccati equation.

Riccati Difference and Differential Equations: Convergence, Monotonicity and Stability

Abstract: The main theme of this Chapter will be the connections between various Riccati equations and the closed loop stability of control schemes based on Linear Quadratic (LQ) optimal methods for control and estimation. Our presentation will encompass methods applicable both for discrete time and continuous time, and so we discuss concurrently the difference equations (discrete time) and the differential equations (continuous time) — the intellectual machinery necessary for the one suffices for the other and so it makes sense to dispense with both cases in one fell swoop.

Algebraic Riccati Equation: Hermitian and Definite Solutions

Abstract: Let A, B, and C be constant n × n matrices with entries in C, the field of complex numbers. Let B and C be hermitian, i.e., B = B* and C = C*, where an asterisk is used to denote the conjugate transpose of a matrix. The quadratic equation $$XA + A*X - XBX + C = 0$$ (3.1) for the n × n complex matrix X is called the algebraic Riccati equation.

The reflection of electromagnetic waves from stratified anisotropic media

Abstract: A direct approach to the investigation of the reflection of electromagnetic waves from a plane interface separating a vacuum half-space and a stratified anisotropic layer is presented. The formulation involves the generation of a Riccati differential equation for a certain 2*2 reflection matrix. The approach is already well established for scalar problems, but its implementation in cases which allow mode conversion is believed to be new. The reflection matrix concerned is unitary when the anisotropic layer is nondissipative, and an efficient numerical method to solve the equation is outlined. Sample results are presented to illustrate the theory. >

Compensator design for stability enhancement with collocated controllers

Abstract: A continuum model rather than a finite element model is used. The optimal compensator design is formulated as a stochastic regulator problem and is shown to be solvable by the general infinite-dimensional theory developed by the author despite the lack of exponential stabilizability. Infinite-dimensional steady-state Riccati equations characterizing the feedback control gain and the Kalman filter gain operators can be solved explicitly. The associated performance indexes including the mean-square control-effort are calculated in closed form. As a first approximation, the compensator transfer function can be realized as a bank of bandpass filters in parallel centered at the undamped mode frequencies. Numerical calculations for the gain and bandwidths for a typical configuration are presented. The performance of the compensator is evaluated when in fact in the true model there is no actuator noise. The theoretical problem involved is to show that the infinite-dimensional stochastic process is asymptotically stationary. It is possible to calculate the steady-state covariance in closed form and thereby calculate performance indexes of interest explicitly, facilitating the choice of optimal design parameters. >

On the linear quadratic minimum-time problem

Abstract: A nontraditional minimum-time problem that includes quadratic-state and control-weighting terms in the performance index is investigated. This formulation provides a convenient solution to the problem that uses the solution of the Riccati equation to compute the optimal feedback gain and the optimal time. In some cases the latter is simply found using the derivative of the Riccati equation solution. >