Showing papers on "Riccati equation published in 1972"

Adaptive guaranteed cost control of systems with uncertain parameters

Abstract: Guaranteed cost control is a method of synthesizing a closed-loop system in which the controlled plant has large parameter uncertainty This paper gives the basic theoretical development of guaranteed cost control, and shows how it can be incorporated into an adaptive system The uncertainty in system parameters is reduced first by either: 1) on-line measurement and evaluation, or 2) prior knowledge on the parametric dependence of a certain easily measured situation parameter Guaranteed cost control is then used to take up the residual uncertainty It is shown that the uncertainty in system parameters can be taken care of by an additional term in the Riccati equation A Fortran program for computing the guaranteed cost matrix and control law is developed and applied to an airframe control problem with large parameter variations

Singular perturbation of linear regulators: Basic theorems

Abstract: The behavior of the solution of the Riccati equation for the linear regulator problem with a parameter whose perturbation changes the order of the system is analyzed. Sufficient conditions are given under which the solution of the original problem tends to the solution of a low-order problem. This result can be used for the decomposition of a high-order problem into two low-order problems.

Computer Solutions to the Schrödinger Equation

Abstract: Numerical solution to the Schrodinger equation can be introduced to undergraduate students at the junior level. Numerov process is discussed in detail for the homogeneous differential equation and an extension of the method to inhomogeneous equations and self-consistent field calculations is briefly mentioned. An application to the simple harmonic oscillator is given as an example.

On the discrete time algebraic Riccati equation

Abstract: A detailed account of the properties of a class of algebraic Riccati equations which arise in discrete time control and filtering problems is given. It is shown that a generalized notion of detectability plays an important role in classifying solutions of these equations. This concept is also related to a minimum phase condition.

A dichotomy in linear control theory

Abstract: A dichotomy transformation is constructed using the positive and negative definite solutions of a Riccati system. The solution of a fixed end-point optimal control problem is then approximately found by the superposition of the solutions of two independent stable initial value problems.

Some Chandrasekhar-type algorithms for quadratic regulators

Abstract: The by-now classical method for the quadratic regulator problem is based on the solution of an n × n matrix nonlinear Riccati differential equation, where n is the dimension of the state-vector. Care has to be exercised in numerical solution of the Riccati equation to ensure nonnegative-definiteness of its solution, from which the optimum m × n feedback gain matrix K(?) is calculated by a further matrix multiplication. For constant-parameter systems, we present a new algorithm that requires only the solution of n(m + p) simultaneous equations: the nm elements of the feed-back gain matrix K(?) and the np elements of a rank-p square-root of the derivative of P(?), where p is the rank of the nonnegative-definite weighting matrix Q that measures the contribution of the state trajectory to the cost functional. If n is large compared with p and m, our algorithm can provide considerable computational savings over direct solution of the Riccati equation, where n(n + 1)/2 simultaneous equations have to be solved. Also the square-root feature means that with reasonable care the automatic nonnegative-definiteness of the derivative matrix-P(?) can be carried over to P(?) itself. Similar results can be obtained for indefinite Q matrices, but with n(m + 2p) equations rather than n(m + p). The equations of our algorithm have the same form as certain famous equations introduced into astrophysics by S. Chandrasekhar, which explains our terminology. The method used in the paper can also be applied to Lyapunov differential equations, as discussed in an Appendix, and to the linear least-squares estimation of stationary processes, as discussed elsewhere.

Continuous-time state estimation under disturbances bounded by convex sets

Abstract: A linear continuous-time system is given whose input and output disturbances and initial conditions are unknown but bounded by known convex sets. These sets, together with the system dynamics and any available observation, determine at any time a set of all possible states, containing the true state of the system. An ellipsoidal bound on this set is obtained. The positive-definite matrix and the center which describe the bounding ellipsoid are found to obey two coupled differential equations: a Riccati matrix differential equation and a vector differential equation. They are similar in structure to the Kalman filter equations except that the matrix part of the solution is not precomputable. A precomputable bound can be obtained, however. The cases with no output and no input disturbances are discussed. An "almost-precomputable" bound is described. Computational results show the applicability and the limitation of the approach.

Iterative method of computing the limiting solution of the matrix Riccati differential equation

Abstract: The paper describes an iterative algorithm for computing the limiting, or steady-state, solution of the matrix Riccati differential equation associated with quadratic minimisation problems in linear systems. It is shown that the positive-definite solution of the algebraic equation PF + F?P?PGR?1G?P + S = 0, provided that it exists and is unique, can be obtained as the limiting solution of a quadratic matrix difference equation that converges from any nonnegative definite initial condition. The algorithm is simple, and, at least for moderate dimensions of the solution matrix, competitive in computational effort with other current techniques for obtaining the limiting solution of the Riccati equation.

A priori bounds for the Riccati equation

Abstract: Since the appearance of [6] and [7], the theory of linear filtering has experienced a renaissance. This theory, although evidently well known to statisticians in terms of "least squares estimates," has found many applications in the early sixties, largely because of the realization and synthesis methods provided in [6] and [7]. For an indication ofsome of the aerospace applications to guidance of spacecraft, the interested reader may find detailed information in [3]. Although the theory of linear filtering has changed little from that given in [7] for the continuous time problem, the practical realization of the so-called "correlated noise problem" as treated mathematically in [6], has recently found a solution in [4]. The full solution of this discrete time filtering problem and its meaning is described in detail in [5]. For readers desirous of a survey of recent results in linear and nonlinear filtering, it is available in [5], while more detailed information can be found in [3]. In this paper, our interest will center on the discrete matrix Riccati equation with emphasis on the study of the asymptotic behavior of its covariance matrix solution. A major tool in this study will be the Duffin parallel resistance of two nonnegative definite matrices A and B denoted by A B. This operation is described in detail in [1] and provides for us a link between the Riccati equation and the classical continued fraction theory described in [9] and [10]. We have undertaken to study the discrete Riccati equation from the point of view of continued fractions because this technique provides considerable generality in that the nonsingular theory becomes a rather special case (see [3], Chapter 5) and much deeper results are obtained for singular problems; also the methods are striking generalizations of classical continued fraction methods. We will be concerned with the cone C of d x d real entry symmetric nonnegative definite matrices. The cone C induces a natural partial ordering as for A e C and BE C, A _ B when and only when A -B E C. The object of study will be the map of C;

On Control of the Solution of a Stochastic Integral Equation with Degeneration

Abstract: This paper is devoted to the derivation of Bellman's differential equation in the payoff function v(x) for a broad class of cases (Theorems 1 and 2). We prove that v(x) is the smallest solution of this equation (Theorem 3).

Some relations among RKHS norms, Fredholm equations, and innovations representations

Abstract: We first show how reproducing kernel Hilbert space (RKHS) norms can be determined for a large class of covariance functions by methods based on the solution of a Riccati differential equation or a Wiener-Hopf integral equation. Efficient numerical algorithms for such equations have been extensively studied, especially in the control literature. The innovations representations enter in that it is they that suggest the form of the RKHS norms. From the RKHS norms, we show how recursive solutions can be obtained for certain Fredholm equations of the first kind that are widely used in certain approaches to detection theory. Our approach specifies a unique solution: moreover, the algorithms used are well suited to the treatment of increasing observation intervals.

Asymptotic formulae for the eigenvalues of a two-parameter ordinary differential equation of the second order

Abstract: We consider a two-point boundary value problem associated with an ordinary differential equation defined over the unit interval and containing the two parameters A and p. If for each real p. we denote the zzith eigenvalue of our system by Am(/j.), then it is known that Am(/j.) is real analytic in — co co, and indeed obtain such a development to an accuracy determined by the coefficients of our differential equation. With suitable conditions on the coefficients of our differential equation, the asymptotic formula for ¡\\m(p) may be further developed using the methods of this paper. These results may be modified so as to apply to A„,(/j.) as p.-*— co if the coefficients of our differential equation are also suitably modified.

Rapid Numerical Solution of the One-Dimensional Schrödinger Equation

Abstract: It is shown that, contrary to common experience and opinion, the exact solutions to Schrodinger's equation in one dimension (or any similar ordinary linear second-order differential equation) can be numerically computed at a speed characterized by the variations of the potential function, i.e. at effectively the speed of solving Hamilton's equations. The method of solution depends upon the existence and calculation of certain "fundamental" and unique special solutions, closely associated with the JWKB series. These solutions have exponential form with exponents which are everywhere finite, nonoscillatory functions varying smoothly with the potential ("quantal action"); their existence does not seem to have previously been known. They are generated when a novel iterative technique is employed to solve certain Riccati equations. The expressions obtained appear to be asymptotic representations, but are numerically essentially convergent (error < 1 × 10−6) whenever the JWKB approximation is reasonably valid. ...

Perturbation guidance for minimum time flight paths of spacecraft.

Abstract: The problem of transferring a rocket vehicle from a given circular orbit to a larger coplanar circular orbit in minimum time, using a constant low-thrust rocket engine, is considered. Parameters are chosen to correspond to a transfer from the earth's orbit in heliocentric space to the orbit of Mars. A path satisfying the first order necessary conditions of variational calculus is shown to be locally minimizing by application of a set of second order conditions. A physical explanation is offered to justify the retrothrust period occurring during the flight. A neighboring optimum feedback control law, based on estimated time-to-go, is applied to this problem. State variable and terminal constraint feedback gains are calculated while one of the second order conditions, involving the backward integration of a matrix Riccati equation, is being tested.

Basic Concepts in System Theory.

Abstract: : ;Contents: The systems; Some fundamentals; Passivity and positive reality in time-varying linear systems; Sensitivity reduction in linear time-varying systems; Optimization in Hilbert spaces; Concepts in modern control theory; Frequency domain methods; Spectra and inverses; Positiveness in Banach spaces; On casuality; On nonlinear systems; The matrix Riccati equation; Miscellaneous nonlinear results; Factoring I+T*T.

A simplified method for solving the matrix Riccati equation

Abstract: A simplified method of solving the matrix Riccati equation is discussed and illustrated in the context of optimal control theory. The method utilizes a preliminary solution with no requirement on boundary condition or sign definiteness. Utilizing this preliminary solution the desired solution is obtained by solving a linear problem of the same dimension as the state vector. One advantage of the method is that the boundary conditions for the terminal control problem which force some state variables to zero at the terminal time are easily implemented.

A Negative Definite Equilibrium and Its Induced Cone of Global Existence for the Riccati Equation

Abstract: The domain of global existence for the Riccati equation is shown to include a cone with vertex at the negative definite equilibria and containing the positive definite equilibria, under regularity conditions.

Iterative solution of the Riccati equation

Abstract: A new convergence proof of an iterative method for the steady solution of the Riccati equation is presented and its geometric nature and close resemblance to the Newton method are emphasized. Uniqueness, rate of convergence and initialization of the iterative process are discussed.

Riccati transformations for control optimization using the second variation

Abstract: Two new numerical methods which may be used to calculate solutions to optimal control problems are developed. These methods involve guessing initial values for unknown Lagrange multipliers and a control sequence. They are similar to the successive sweep method in that Riccati equations are used to calculate corrections to these guessed variables. They integrate Riccati equations, however, which differ from the one integrated by the standard sweep method. The effectiveness of these methods along with a standard method based on the integration of linear equations are compared for two example problems, the Brachistochrone and an Earth-to-Mars low thrust transfer problem.