Riccati equation

A novel optimization approach based on the Newton-Kantorovich iterative scheme applied to the Riccati equation describing the reflection from the inhomogeneous half-space was proposed recently [7]. The method works well with complicated highly contrasted dielectric profiles and retains stability with respect to the noise in the input data. However, this algorithm like others needs the measurement data to be given in a broad frequency band. In this work, the method is improved to be valid for the input data obtained in an essentially restricted frequency band, i.e. when both low and high frequency data are not available. This... 

No a priori information about the unknown profile is used in this algorithm, and the initial profile to start the iterative process is chosen as (z) = 1. Moreover, the solution of the forward problem at each iteration can be obtained with the use of the scattering matrices concept [8] instead of a numerical solution of the Riccati equation (4). This allows to perform reconstruction in a few seconds of a microcomputer time. The whole algorithm can be summarized as follows ... 

It can be shown that the constrained functional minimization of equation (9.48) yields again the matrix Riccati equations (9.23) and (9.25) obtained for the LQR, combined with the additional set of reverse-time state tracking equations... 

Using the recursive equations (9.29) and (9.30), solve, in reverse time, the Riccati equation commencing with P(A ) = 0. 

Burns, R.S. (1989) Application of the Riccati Equation in the Control and Guidance of Marine Vehicles. In The Riccati Equation in Control, Systems, and Signals, Pitagora Editrice, Bologna, Italy, pp. 18-23. 

Laub, A.J. (1979) A Schur Method for Solving Riccati Equations, IEEE Trans, on Automat. Contr., AC-24, pp. 913-921. 

Payne, H.J. and Silverman, L.M. (1973) On the Discrete Time Algebraic Riccati Equation, IEEE Trans. Automatic Control, AC-18, pp. 226-234. 

Riccati, J.F. (1724) Animadversiones In Aequationes Differentiales, Acta Eruditorum Lipsiae. Re-printed by Bittanti, S. (ed.) (1989) Count Riccati and the Early Days of the Riccati Equation, Pitagora Editrice, Bologna, Milano. 

S. Bittanti, P. Coaneri, and G.D. Nicolao. The difference periodic Riccati equation for the periodic prediction problem. IEEE Trans. Automat. Contr., 33 706-712, 1988. 

The remaining task lies in the determination of the control matrix X and observer matrix Z such that the sufficient condition for robust performance, Eq. (22.28), holds. A Lyapunov-based approach is employed to obtain these two matrices. After some lengthy and complicated manipulations of Eq. (22.29) and the control structure shown in Fig. 22.3, the following two Riccati equations are derived, whose positive-definite solutions correspond to the control and observer matrices, X and Z. 

The gain factors c, i=l,...,4 are obtained by minimizing a time-invariant quadratic performance index by the solution of the steady state matrix RICCATI-equation. 

The success of MPC is based on a number of factors. First, the technique requires neither state space models (and Riccati equations) nor transfer matrix models (and spectral factorization techniques) but utilizes the step or impulse response as a simple and intuitive process description. This nonpara-metric process description allows time delays and complex dynamics to be represented with equal ease. No advanced knowledge of modeling and identification techniques is necessary. Instead of the observer or state estimator of classic optimal control theory, a model of the process is employed directly in the algorithm to predict the future process outputs. 

Systems II and IV can be reduced to Riccati equations for which the solutions may be represented in terms of Bessel and Haenkel functions. System IV has been analyzed in some detail by A. D. Stepukhovich and L. M. Timonin, Zhur, Fix, Khim., 26, 143 (1951). System III had been earlier reported on by A. A. Balandin and L. S. Leibenson, CompL rend, acad, sci, U.R,S,S.f 39, 22 (1943). 

Nonlinear Difference Equations Riccati Difference Equation The Riccati equation yx-t-iyx + 1 + byx + c = 0 is a nonhnear... 

Formidable though these may seem it is actually possible to find a solution in finite terms. Dividing (21) by (26) we have a Riccati equation for f as a function of a. If /3(a) denotes (6 — a)/ra, then... 

For numerical investigations of stress localizations in laminates, the discretizational effort can be reduced significantly if only the boundary needs to be discretized, as it is for e -ample the case in the classical boundary element method (BEM). But in this method a fundamental solution is needed which is in many cases difficult to achieve or even unknown. The Boundary Finite Element Method (BFEM) to be presented here does not require such a fundamental solution, because the element formulation is based on the finite element method (FEM), Thus the BFEM can be characterized to be a finite element based boundary discretization method. This method was originally developed from Wolf and Song [10] under the name Consistent Finite Element Cell Method for time-dependent problems in soil-mechanics. The basic assumption of this method is that a stiffness matrix describing the force-displacement relation at discrete degrees of freedom at the boundary of the continuum is scalable with respect to one point in three-dimensional space, the so-called similarity center, if similar contours within the continuum are considered. In contrast to this, the current work deals with the case of equivalent cross-sectional properties, i.e., that cross-sections parallel to the boundary can be described by the same stiffness matrix, which is the appropriate formulation for the case of the free-edge effect and the matrix crack problem. The boundary stiffness matrix results from a Matrix-Riccati equation. The field quantities inside of the continuum can be calculated from an ordinary differential equation. 

Inserting the matrices (14) and the expression (24) for the inverse Kg/ in equation (22) and performing the limit /t 0 results in the following Matrix-Riccati equation for the unknown elastic stiffness matrix K°° ... 

K k] is derived from S by K[k] = R B[k" S + N ). Control scheme tracks values of variables and parameters that determine the operating point of systems and with any change in these parameters, a new linear controller gain is calculated based on the most recent operating point. It should be noted that computation time of every step for the combined state estimation and solving Riccati equation must be less that sampling time of the system. 

If we scale time by replacing 6 = k2t, the above expression becomes identical to the special form of the Riccati equation. Thus, on comparing with Eq. 2.57, we take... 

Very few nonlinear equations yield analytical solutions, so graphical or trial-error solution methods are often used. There are a few nonlinear finite difference equations, which can be reduced to linear form by elementary variable transformation. Foremost among these is the famous Riccati equation... 

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