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Showing papers on "Riccati equation published in 1970"



Journal ArticleDOI
TL;DR: An attempt is made to apply a certain class of optimal control theory to obtain an optimal controller to improve the dynamic response of a power system.
Abstract: In recent years important research has been done in the area of system optimization by control engineers. Many theoretical results have been published but application examples have mainly been on low-order systems. An attempt is made to apply a certain class of optimal control theory, known as the state regulator problem, to obtain an optimal controller to improve the dynamic response of a power system. The system differential equations are written in the first-order state variable form. A cost functional is then chosen, and the matrix Riccati equation is solved. Puri's and Gruver's method is applied for the numerical computation, and the system is made initially stable by shifting the system eigenvalues.

218 citations


Journal ArticleDOI
TL;DR: In this article, a nonrecursive algebraic solution for the Riccati equation is presented, which allows direct determination of the transient solution for any particular time without proceeding recursively from the initial conditions.
Abstract: Equations for the optimal linear control and filter gains for linear discrete systems with quadratic performance criteria are widely documented. A nonrecursive algebraic solution for the Riccati equation is presented. These relations allow the determination of the steady-state solution of the Riccati equation directly without iteration. The relations also allow the direct determination of the transient solution for any particular time without proceeding recursively from the initial conditions. The method involves finding the eigenvalues and eigenvectors of the canonical state-costate equations.

189 citations


Dissertation
01 Jan 1970
TL;DR: The Riccati equation is studied from an algebraic point of view, and the results are applied on optimal control of linear time invariant systems with quadratic loss.
Abstract: The matrix Riccati equation appears in many optimal control and filtering problems. In this paper the Riccati equation is studied from an algebraic point of view, and the results are applied on optimal control of linear time invariant systems with quadratic loss.

104 citations


Journal ArticleDOI
TL;DR: In this article, the authors considered the discrete time matrix Riccati equation and proved the convergence of the policy space approximation technique, which is analogous to those known for the continuous-time Riccaci equation, but the techniques used are simpler.
Abstract: This paper is concerned with the discrete time matrix Riccati equation. The properties established are those of minimality, convergence, uniqueness and stability. Further the convergence of the policy space approximation technique is proved. These results are analogous to those known for the continuous-time Riccati equation, but the techniques used are simpler.

103 citations


Journal ArticleDOI
TL;DR: Riccati-like linear functional differential equation with quadratic cost, analyzing feedback control solution existence and uniqueness, and uniqueness as mentioned in this paper, was used to analyze feedback control solutions.
Abstract: Riccati-like linear functional differential equation with quadratic cost, analyzing feedback control solution existence and uniqueness

72 citations


Journal ArticleDOI
TL;DR: In this paper, the classical calculus of variations approach to the design of finite-dimensional optimal control systems is extended to a general class of fully non-linear distributed-parameter systems with distributed and/or boundary control inputs.
Abstract: The classical calculus of variations approach to the design of finite-dimensional optimal control systems is extended to a general class of fully non-linear distributed-parameter systems with distributed and/or boundary control inputs. As in lumped theory the result is a distributed-parameter two-time-point boundary-value problem in the canonical form of Hamilton, which in the linear case reduces to a partial differential equation of the Riccati type. A computational approximation technique for integrating the canonical equations is given which is based on the Riccati equation and does not require any hill-climbing procedure. The results are finally extended to an important class of mixed distributed-parameter and lumped-parameter systems.

35 citations


Journal ArticleDOI
TL;DR: In this article, the Riccati equation associated with a class of discrete-time correlated noise problems is examined, and the concept of invariant directions for this equation is introduced.
Abstract: The Riccati equation associated with a class of discrete-time correlated noise problems is examined, and the concept of invariant directions for this equation is introduced. For single-output systems the set of such directions is completely characterized. Deletion of these directions by an appropriate transformation of the Riccati equation results in a minimal order equation for computation. This transformation also reveals the underlying structure of the optimal filter for the correlated noise problem.

34 citations


Journal ArticleDOI
TL;DR: In this article, sufficient conditions for nonnegativity of the second variation in singular and nonsingular control problems are presented; these conditions are in the form of equalities and differential inequalities.
Abstract: Sufficient conditions for nonnegativity of the second variation in singular and nonsingular control problems are presented; these conditions are in the form of equalities and differential inequalities. Control problem examples illustrate the use of the new conditions. The relationships of the new conditions to existing necessary conditions of optimality for singular and nonsingular problems are discussed. When applied to nonsingular control problems, it is shown that the conditions are sufficient to ensure the boundedness of the solution of the well-known matrix Riccati differential equation.

31 citations


Journal ArticleDOI
TL;DR: In this article, the problem of expanding the matrix equation PA + A'P = -Q into a system of algebraic equations is considered and an alternate approach which is rather simple and very effective in computer applications is suggested.
Abstract: The problem of expanding the matrix equation PA + A'P = - Q into a system of algebraic equations is considered. Starting from the previously published results, it suggests an alternate approach which is rather simple and very effective in computer applications. This approach is particularly useful when the given equation has to be solved iteratively which is the case in solution of the algebraic matrix Riccati equation appearing in optimal control problems.

29 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that (1.3) is necessary and sufficient for the boundedness of all solutions of the Ricatti equation with initial conditions in the first quadrant (third quadrant) will intersect the positive (negative) x-axis.
Abstract: is necessary and sufficient for the global asymptotic stability of the equilibrium point (0, 0) of (1.2). The idea of the proof is to show that (1.3) is necessary and sufficient for the boundedness of all solutions of (1.2). In doing this the crucial step is determining whether a solution with initial conditions in the first quadrant (third quadrant) will intersect the positive (negative) x-axis. In this paper we investigate global asymptotic stability of the origin (0, 0) of (1.2) for the case when a ? 1. In addition to the above idea, the main feature of our approach is to reduce (1.2) to a first order generalized Ricatti equation and then determine conditions under which it has and does not have positive solutions for large values of the independent variable. Using a different method, Willet and Wong [4] have considered this same problem. Their approach is more general and (1.2) is considered as a special case. If a _ 1, they obtain a necessary and sufficient condition (which is the same as Burton's for cx 1, however, it will be shown here that their condition is included in one of those obtained here.


Journal ArticleDOI
TL;DR: In this article, sufficient conditions are given for a matrix Riccati differential equation to have a bounded solution; these conditions are less stringent than those known heretofore, and the conditions are in the form of linear, algebraic and differential inequalities.

Journal ArticleDOI
TL;DR: The null distribution of the Hotelling-Lawley generalized T~-' statistic has been shown to satisfy a homogeneous linear differential equation (d.c.). The lat ter has been used to tabulate some exact percentage points of To\" by analytic continuation of Constantine's [3] series, and a table for the 5-variate case is presented in this paper.
Abstract: The null distribution of the Hotelling-Lawley generalized T~-' statistic has been shown [4] to satisfy a homogeneous linear differential equation (d.c.). The lat ter has been used to tabulate some exact percentage points of To\" by analytic continuation of Constantine's [3] series, and a table for the 5-variate case is presented in this paper. The Ito-Siotani [9], [16] asymptot ic expansions for the distribution function and percentage points of T: are also extended.



Journal ArticleDOI
01 Aug 1970
TL;DR: In this article, the optimal control of continuous-time linear multivariable systems with finite-and infinite-time quadratic performance is investigated using the maximum principle, and a solution is obtained in terms of the components of the transition matrix associated with the linear system of 2n differential equations in the n state variables and the n adjoint variables.
Abstract: The optimal control of continuous-time linear multivariable systems with finite- and infinite-time quadratic performance is investigated using the maximum principle. The time-varying optimal-control law is shown to be related to the matrix Riccati differential equation, and a solution is obtained in terms of the components of the transition matrix associated with the linear system of 2n differential equations in the n state variables and the n adjoint variables. The time-varying transition-matrix formulation is also used to develop a solution of the steady-state matrix Riccati and associated Lyapunov equations. The infinite-time optimal-control solution is derived in terms of partitioned eigenvector components related to the stable modes of the augmented system, and the technique is shown to be similar in principle to the method previously developed for reducing the order of a set of matrix differential equations by neglecting high-order modes. The results are also shown to be related to the solution of the algebraic Riccati equation and are used to develop an eigenvector solution for the Lyapunov equation. The techniques are extended to consider the optimal control of the linear system containing input derivatives. Results are obtained for the optimal control of a 1st- and 2nd-order linear system and for a 6th-order model representing the dynamics of a superheater system. The various techniques available for obtaining a solution of the linear optimal-control problem, namely the recurrence-type control algorithm based on dynamic programming, the eigenvector solution of the steady-state Riccati equation and the solution obtained by numerical integration of the time-varying Riccati equation, are compared.

Journal ArticleDOI
TL;DR: Navier-Stokes equation approximation for nonhomogeneous case obtained by relating Burgers equation to Riccati equation through similarity transformation as discussed by the authors, using similarity transformation for similarity transformation.
Abstract: Navier-Stokes equation approximation for nonhomogeneous case obtained by relating Burgers equation to Riccati equation through similarity transformation

Journal ArticleDOI
TL;DR: In this paper, a non-autonomous n-th order differential equation for a function x(t) ∈ Rm was investigated, and the authors proved the existence of at least one periodic solution (with the same period as the external forcing).
Abstract: We investigate a non-autonomous n-th order differential equation for a function x(t) ∈ Rm supposing that the equation contains one nonlinear term only depending on x. Our aim is to prove the existence of at least one periodic solution (with the same period as the external forcing). The conditions developed for the nonlinear term are rather general and do not imply the global boundedness of the solutions.

Journal ArticleDOI
TL;DR: In this article, a z transform factorisation result from the discrete-time matrix Riccati equation is developed from the matrix Ricciati equation, in a similar way to the well known spectral factorization result of continuous time systems.
Abstract: A z transform factorisation result is developed from the discrete-time matrix Riccati equation, in a similar way to the well known spectral-factorisation result of continuous time systems.

Journal ArticleDOI
TL;DR: In this paper, an imbedding equation is used to solve linear regulator problems depending on a parameter α. From this equation, the optimum matrix K(α) is obtained for all α in [α 0, α 1 ].
Abstract: An imbedding equation is used to solve linear regulator problems depending on a parameter α. From this equation the optimum matrix K(\alpha) is obtained for all \alpha \in [\alpha_{0},\alpha_{1} ]. Two typical applications of this equation are discussed.

Journal ArticleDOI
TL;DR: Using projection operator techniques and the Liouville formalism, a derivation of an exact evolution equation for the internal degrees of freedom of a molecule in a temperature bath is presented in this article.
Abstract: Using projection operator techniques and the Liouville formalism, a derivation of an exact evolution equation for the internal degrees of freedom of a molecule in a temperature bath is presented. This non‐Markovian “master equation” gives the time evolution of the diagonal part of the reduced density matrix of the internal degrees of freedom. Since the collision term in this equation depends explicitly on the intensive variables of the reservoir, the simultaneous limits of low reservoir density and long time may be used to reduce the exact equation to a Markovian master equation. Using an identity which connects the Liouville formalism to scattering theory, it is shown that the collisional transition probabilities which occur in this master equation can be written in terms of scattering cross sections. Finally, it is demonstrated that the transition probabilities satisfy the condition of detailed balance and that the master equation agrees with that obtained by the usual physical derivations.

Journal ArticleDOI
TL;DR: In this article, a generalized master equation for a system described in a phase space of generalized coordinates w and momenta J was derived, which gives the time evolution of the reduced density distribution function ρ(t, w, J) for the momenta.
Abstract: For a system described in a phase space of generalized coordinates w and momenta J, the generalized master equation gives the time evolution of the reduced‐density distribution function ρ(t, J) for the momenta. A generalization of the generalized master equation, having a similar non‐Markoffian form, is derived for the full distribution function ρ(t, w, J). This equation is an alternate form of the Liouville equation. The derivation is an extension of a previous derivation of the generalized master equation from the Liouville equation utilizing projection operators in a Hilbert space. The time‐evolution equation for the reduced distribution function ρr(t, wr, J), depending on the subset wr of the set of coordinates w, is derived. The approach to a stationary state for t → ∞ is discussed.

Book ChapterDOI
01 Jan 1970
TL;DR: Structural properties of equilibrium solutions of quadratic matrix equation, using variational interpretation of associated Riccati equation, transform techniques and Parseval formula, were studied in this article.
Abstract: Structural properties of equilibrium solutions of quadratic matrix equation, using variational interpretation of associated Riccati equation, transform techniques and Parseval formula

Journal ArticleDOI
TL;DR: In this paper, a near-optimal feedback control law for a distributed parameter system with a quadratic performance index is obtained by the method of trajectory approximation, where the system equation is reduced to an approximate system of ordinary differential equations by the Galerkin-Kantorovich method, and then Pontryagin's maximum principle is applied to show that the feedback control is linear and can be obtained as the solution of a matrix Riccati equation.
Abstract: A near-optimal feedback control law for a distributed parameter system with a quadratic performance index is obtained by the method of trajectory approximation The system equation is reduced to an approximate system of ordinary differential equations by the Galerkin—Kantorovich method, and then Pontryagin's maximum principle is applied to show that the feedback control law is linear and can be obtained as the solution of a matrix Riccati equation Numerical computations performed using Chebyshev polynomials as the Galerkin weighting functions on the equation for a heat exchanger with wall flux forcing indicate that four thermocouples are enough to attain virtual optimality

Journal ArticleDOI
TL;DR: Liapunov functions construction through conversion of differential equation with polynomial nonlinearities into auxiliary exact differential equation using algorithm using algorithm as discussed by the authors, which can be used to construct Liapunova functions.
Abstract: Liapunov functions construction through conversion of differential equation with polynomial nonlinearities into auxiliary exact differential equation using algorithm




Journal ArticleDOI
TL;DR: An eigenvector matrix is used in deriving an explicit expression for the solution of the matrix Riccati equation and the eigenvectors are not normalized in the usual sense of the word but must be scaled in one of two particular ways.
Abstract: An eigenvector matrix is used in deriving an explicit expression for the solution of the matrix Riccati equation. A misleading situation arises from the derivation of this expression as originally published [1], which it is the purpose of this correspondence to clarify. For the expression to be valid the eigenvectors are not normalized in the usual sense of the word but must be scaled in one of two particular ways.