Riccati-type Differential Equations Research Articles

Optimal sensor location problem for a linear distributed parameter system

This paper studies an optimal sensor location problem for a linear distributed parameter system. It is assumed that a criterion for the optimal sensor location is to minimize the trace of the optimal filtering error covariance function. The existence and uniqueness theorem concerning a solution of the optimal filtering error covariance function for the pointwise observation case is considered. Then by using the existence and uniqueness theorem, the comparison theorem for the partial differential equations of Riccati type is proved. By using the theorems obtained here, the existence theorem concerning a solution of the optimal sensor location problem is proved and the necessary and sufficient conditions for optimality are derived. Finally, some numerical examples for the optimal sensor location problem are illustrated.

Optimal Sensor Location in a Linear Distributed Parameter System

In this paper, we treat the optimal sensor location problem by using functional analysis. Thus, the partial differential equations are embedded into the ordinary differential equations in Hilbert spaces. It is assumed that a criterion for the sensor location is to minimize the trace of the estimation error covariance operator described by the operator-valued differential equations of Riccati type plus the measurement cost. We prove the existence and uniqueness theorem concerning the solution of the estimation error covariance operator by using the theory of the evolution operator and Picard's method. Furthermore, we prove the comparison theorem for the operator-valued differential equations. Based on the theorems, we derive the sufficient condition for the optimal sensor location and then derive the necessary condition. Finally, some numerical examples for the optimal sensor locations are shown.

Sufficiency Conditions for Finite Escape Times in Systems of Quadratic Differential Equations

Sufficient conditions are derived for solutions of a general autonomous system of quadratic ordinary vector differential equations to exhibit finite escape times. These results are extended t o a matrix Riccati-type differential equation using Kronecker matrix products and a stacking operator. Finally, these results are applied to a class of models used extensively in population dynamics. GENERAL procedures are well-known for analysing the stability properties of systems of non-linear ordinary differential equations. Important among these procedures are: (i) the various Lyapunov-type theorems (Hahn, 1967; Cesari, 1963, chapter 3) and (ii) the analysis of the stability properties of the corresponding variational system, viz. the non-linear system linearized about one of its stationary points (Willems, 1970, chapter 5). Stability results pertaining to specific systems are known for many different types of non-linear differential equations. Often these results are derived by finding a suitable Lyapunov function for the system concerned and accordingly the applicability of the results is limited to that system. However, few explicit results exist for the stability properties of quadratic differential equations, especially since suitable Lyapunov functions for general quadratic systems are difficult to find. The primary reason for this is that the time derivative of a scalar function o f even order is odd and it is therefore difficult t o comment upon its sign without knowing more about the structure of the solutions of the quadratic differential equations. In this paper we derive sufficiency conditions for solutions of autonomous systems of quadratic differential equations to have a finite escape time. An upper bound for the escape time is also given. Using the Kronecker matrix product and an operator which stacks columns of a matrix into an extended vector, we show that the matrix Riccati-equation that arises in the linear-quadratic regulator problem of optimal control theory, can be rewritten into a format such that our results complement known sufficient conditions of the Kalman-type (Jacobson, 1970). Finally, we demonstrate that the results derived in this paper can be applied to a general class of systems of autonomous quadratic differential equations used to model interacting populations. The Lotka-Volterra model of predator-prey dynamics is a special case of this class. 25 377

Wave impedance of induction devices using the scalar Riccati equation

A frequently recurring problem in induction device theory involves the determination of the input wave impedance to a device. This is commonly done by solving for the field components. In this paper, a completely different method based on the scalar Riccati equation is described. It is shown that the wave impedance can be obtained by the numerical integration of a Riccati-type differential equation, having wave impedance as the unknown variable. This is illustrated with three simple models each having an unknown field distribution in one direction only. Extensive computer testing was carried out and the results obtained were found to agree very closely with published analytical methods. The method is very simple, easily programmed and avoids the use of higher transcendental functions.

On a Class of Differential Equations of Riccati Type

On a Class of Differential Equations of Riccati Type

Applications of a Riccati type differential equation to Riemannian manifolds with totally geodesic distributions

Applications of a Riccati type differential equation to Riemannian manifolds with totally geodesic distributions

Matrix Riccati equation in the electrical-network problem

The general electrical-network problem is shown to include a Riccati-type differential equation which represents a matrix analogue of the classical operator solution of the defining 2nd-order equations. The solution matrix represents a transformation between dual sets of variables which can be related to eigenvector components as in the linear optimal-regulator problem. A transition-matrix solution of the electrical-network problem is also discussed.

Position-Required and its Application to Guidance Problems

This paper presents and develops a concept that, compared with previous methods, will greatly improve the computational feasibility of second-order approximations to velocity-required for a wide variety of boundary-value problems arising in guidance. First, it is shown that a large category of boundary-value problems and trajectory shaping constraints may be included within a variable time-of-arrival constraint and a variable target position constraint. This observation is of paramount importance since it makes guidance equations much more explicit and may avoid extensive preflight shaping of powered flight trajectories. Then the concept of position-required is developed, and Riccati-type differential equations are derived which the first- and second-order partial derivatives of position-required obey. The number of such equations is quite small (between 33 and 87 depending upon the particular boundaryvalue problem) compared with 168 equations to obtain all of the first- and second-order miss (sensitivity) coefficients. A crucial property of the differential equations for the derivatives of position-required is that initial conditions at the target are known a priori, which is in contrast with the lack of initial conditions for similar equations derived previously for velocityrequired. In a region where velocity-required and its derivatives exist, algebraic equations are derived from which the derivatives of velocity-required can be calculated from the derivatives of position-required.

Matrix Quadratic Solutions

Matrix quadratic equation solution derivation applied in finding steady state solutions of Riccati differential equations with constant coefficients

Propagation of ELF waves below an inhomogeneous anisotropic ionosphere

The ionospheric anisotropy is considered with horizontal magnetic field either for transverse (East-West or West-East) or for longitudinal (South-North) propagation. For transverse propagation in a vertically stratified medium the differential equations of the various field components are uncoupled and a closed form solution is given for identical exponential height variation of the components of tensor conductivity. For arbitrary height variation of the tensor conductivity numerical solutions are obtained after expressing the surface impedance below the ionosphere in terms of a Riccati-type differential equation. The West-East direction of propagation exhibits a lower attenuation constant than the East-West direction for f cps. This is contrary to the expectations based on a model of a homogeneous anisotropic ionosphere. For longitudinal propagation the differential equations of the various field components are coupled, with the coupling being particularly strong above the D region. The differential equations are simplified by assuming no coupling in the lower ionosphere and strong coupling above a pre-selected altitude y_{1} . For exponential height variation of the tensor conductivity components the closed form solution differs negligibly from the isotropic case. For arbitrary height varition of the tensor conductivity numerical solutions are obtained similarly as for the transverse propagation. Over most of the frequency range the attenuation figures for South-North propagation are intermediate between the corresponding figures for West-East and East-West propagation.

Addendum to "A Matrix Differential Equation of Riccati Type"

Addendum to "A Matrix Differential Equation of Riccati Type"

Corrigenda: On the Partial Products of Infinite Products of Alephs

Professor J. Radon has called my attention to the fact that the presentation of the Legendre equation of a problem of Lagrange as a matrix differential equation of Riccati type, and the fundamental theorems on the integration of such an equation, are given in two earlier papers of his [ Uber die Oszillationstheoreme der konjugierten Punkte biem Probleme von Lagrange, Miinchner Sitzungsberichte (1927), pp. 243-257; Zum Problem von Lagrange, Abhandlungen aus dem Mathematischen Seminar IHamburg, vol. 6 (1928), pp. 273-299]. In particular, the results of Sections 3 and 4 of the indicated paper of the author are contained in these papers of Radon.

A Matrix Differential Equation of Riccati Type

A Matrix Differential Equation of Riccati Type