Showing papers on "Riccati equation published in 1971"
TL;DR: In this paper, the optimal control of linear systems with respect to quadratic performance criteria over an infinite time interval is treated, and the integrand of the performance criterion is allowed to be fully quadratically in the control and the state without necessarily satisfying the definiteness conditions which are usually assumed in the standard regulator problem.
Abstract: The optimal control of linear systems with respect to quadratic performance criteria over an infinite time interval is treated. Both the case in which the terminal state is free and that in which the terminal state is constrained to be zero are treated. The integrand of the performance criterion is allowed to be fully quadratic in the control and the state without necessarily satisfying the definiteness conditions which are usually assumed in the standard regulator problem. Frequency-domain and time-domain conditions for the existence of solutions are derived. The algebraic Riccati equation is then examined, and a complete classification of all its solutions is presented. It is finally shown how the optimal control problems introduced in the beginning of the paper may be solved analytically via the algebraic Riccati equation.
1,436 citations
271 citations
TL;DR: In this article, the Riccati equation with quadratic loss was studied for nonnegative definite and positive definite matrices Q"1 and Q"2 and the results were used to study the possible number of stationary solutions.
187 citations
TL;DR: In this article, a general theory for the optimal regulation of linear systems, comprised of infinite identical objects, is developed, where the bilateral z transform generating function is used to eliminate the object indices, which leads to a Riccati equation in the z domain.
133 citations
TL;DR: In this paper, the qualitative behavior of solutions near an equilibrium point of an autonomous functional differential equation of finitely retarded type is investigated, under certain appropriate hypotheses, and it is shown that, as a certain parameter varies in a prescribed way, a family of one or more periodic orbits of the given equation bifurcates from the given equilibrium point.
86 citations
TL;DR: In this paper, a complete stability theory for discrete-time linear systems with respect to general quadratic costs, including singular cases, is presented, and several tests for perfect observability are also given.
Abstract: Optimization of discrete-time linear systems with respect to general quadratic costs, including singular cases, is examined. By introduction of the concept of perfect observability, a complete stability theory is obtained. Several tests for perfect observability are also given, and application to the dual correlated noise filtering problem is made.
69 citations
TL;DR: In this paper, a nonlinear equation is derived for the equilibrium solutions in the player's allowed operator feedback spaces, and the results obtained from the study of this equation are then used to compute the solution to a class of differential games and to establish the uniqueness of the solution.
Abstract: Problems concerning equilibrium strategies for linear games are treated in a Hilbert space setting. First a nonlinear equation is derived for the equilibrium solutions in the player’s allowed operator feedback spaces. The results obtained from the study of this equation are then used to compute the solution to a class of differential games and to establish the uniqueness of the solution.
68 citations
TL;DR: In this paper, a closed-form solution to the Riccati equation for the optimal error regulation of a string of moving vehicles is found, where the use of lead and follow vehicles to prevent drifting of the string is included.
Abstract: A closed-form solution to the Riccati equation for the optimal error regulation of a string of moving vehicles is found. The use of lead and follow vehicles to prevent drifting of the string is included in the solution.
47 citations
TL;DR: In this article, the authors examined the relationship between the results obtained when the dimensionality of a regulator problem is reduced in the statement of the problem and when this reduction is made in the Riccati equation for the nonreduced problem.
Abstract: This correspondence examines the relationship between the results obtained when the dimensionality of a regulator problem is reduced in the statement of the problem and when this reduction is made in the Riccati equation for the nonreduced problem.
39 citations
TL;DR: It is shown that there exist both RK-methods and multistep methods, the error equation of which is exact, and the lowest order term of the global discretization error of the numerical solution to a system of ordinary differential equations satisfies this equation.
Abstract: The lowest order term of the global discretization error of the numerical solution to a system of ordinary differential equations satisfies a well-known differential equation. It is observed that the integration of this differential equation and thus the estimation of the discretization error becomes almost trivial if it is an exact differential equation. It is shown that there exist both RK-methods and multistep methods, the error equation of which is exact.
TL;DR: In this paper, the Riccati equation of optimal control is corrected for the discrete time matrix. But this correction is limited to the case where the discrete-time matrix is a Gaussian distribution.
Abstract: (1971). “On the discrete time matrix Riccati equation of optimal control-a correction”. International Journal of Control: Vol. 14, No. 1, pp. 205-207.
TL;DR: In this paper, the problem of response theory in statistical mechanics involves the determination of the density matrix $\ensuremath{\rho}$ from the Liouville equation and the subsequent computation of the response $r$ from this density matrix.
Abstract: The problem of response theory in statistical mechanics involves the determination of the density matrix $\ensuremath{\rho}$ from the Liouville equation and the subsequent computation of the response $r$ from this $\ensuremath{\rho}$. Projection techniques are applied to avoid the entire complicated problem of the full dynamics of $\ensuremath{\rho}$ and to select only that part of $\ensuremath{\rho}$ which is relevant to the response $r$. The procedure replaces an inhomogeneous equation by a linear homogeneous integrodifferential equation for response theory. This is a very general equation which can be analyzed in different ways to yield a variety of results. It is shown that the Kubo theory of linear response emerges as the lowest-order approximation. The general equation is solved without approximations for a step-function stimulus, and it is discussed in the context of the steady state.
TL;DR: In this article, a method for reducing the multi-dimensional Schrodinger equation to a one-dimensional integral equation is proposed, and the resulting integral equation, although complicated, may be treated by any of a number of numerical methods.
TL;DR: In this paper, basic results concerning the mean-square differentiability of a random process are developed and an autonomous (zero-input) shaping filter may be easily determined.
Abstract: The problem of determining a shaping filter for nonstationary colored noise is considered. The shaping filter transforms white noise into a possibly nonstationary random process (having no white noise component) with a specified covariance function. A set of conditions to be satisfied by the covariance function leads to the determination of a shaping filter. The shaping filter coefficients are simply related to the solution of a matrix Riccati equation. In order to formulate the Riccati equation, basic results concerning the mean-square differentiability of a random process are developed. If the Riccati equation can not be defined, an autonomous (zero-input) shaping filter may be easily determined.
TL;DR: In this paper, the Riccati equation is reduced to two especially simple forms, one of them being the Riemann equation, and examples of solutions are given for a perfect fluid sphere in general relativity.
TL;DR: In this article, the Riccati equation is converted to a canonical form corresponding to Luenberger's canonical representation for controllable multivariable systems, and three special matrices closely associated with the canonical form are defined and two related lemmas are established.
Abstract: A new method is presented for the numerical deterruination of the solution of the steady-state matrix Riccati equation. The equation is converted to a canonical form corresponding to Luenberger's canonical representation for controllable multivariable systems. Three special matrices closely associated with Luenberger's canonical form are defined and two related lemmas are established. These results are used to obtain concise expressions for the eigenvectors of the Hamiltonian matrix associated with the canonical Riccati equation in terms of the solutions of a much simpler reduced Hamiltonian system. Using a theorem due to Potter the solution of the Riccati equation is written in terms of the concise eigenvector expressions. The method is particularly well suited to problems in which the ratio of system states to system inputs is large and it can lead to a 26 to 1 reduction in the computational effort required to solve the Riccati equation.
01 Jan 1971
TL;DR: In this article, the authors derived an a posteriori error bound for an approximate solution of a system of ordinary differential equations as the solution of the Riccati equation, which is used to control step length in a numerical method.
Abstract: An a posteriori error bound, for an approximate solution of a system of ordinary differential equations, is derived as the solution of a Riccati equation. The coefficients of the Riccati equation depend on an eigenvalue of a matrix related to a Jacobian matrix, on a Lipschitz constant for the Jacobian matrix, and on the approximation defect. An upper bound is computable as the formal solution of a sequence of Riccati equations with constant coefficients. This upper bound may sometimes be used to control step length in a numerical method.
01 Jan 1971
TL;DR: Bellman and Kalaba as discussed by the authors proposed the method of invariant imbedding, which is a powerful method of analyzing linear systems, based on the principle of invariance, which was first used by Ambarzumian and Chandrasekhar.
Abstract: The principle of invariance, more widely known as invariant imbedding, is a powerful method of analyzing linear systems. The principle of invariance was first used by Ambarzumian [1], and Chandrasekhar [7] who later extended the work of Ambarzumian. The principle of invariance provided a stimulus for Bellman and Kalaba [3] to further pursue the ideas of Ambarzumian and Chandrasekhar. Bellman and Kalaba called the invariance principle the method of invariant imbedding, see [3] – [6].
TL;DR: In this paper, the numerical solution of the biharmonic equation in a rectangular domain is presented in the context of continuous dynamic programming techniques and the equations are specialized to the solution of elastic rectangular plates.
Abstract: The numerical solution of the biharmonic equation in a rectangular domain is presented in the context of continuous dynamic programming techniques. The equations are specialized to the solution of elastic rectangular plates. A suitable approximate expression of a certain functional equation containing derivatives only in one direction is used to derive equations for the stiffness and flexibility matrices of the plate. It is shown that those matrices satisfy matrix Riccati equations subject to suitable initial conditions. It is also shown that the condition of optimality in the Hamilton-Jacobi-Bellman equation directly expresses a classical variational principle, i.e. the principle of complementary energy. Some numerical examples are finally presented.
TL;DR: Combined optimal estimation and control techniques are applied for the first time to satellite tracking systems of NASA, resulting in an estimated state of the satellite and of the tracking system.
Abstract: Combined optimal estimation and control techniques are applied for the first time to satellite tracking systems. Both radio antenna and optical tracking systems of NASA are considered. The optimal estimation is accomplished using an extended Kalman filter, resulting in an estimated state of the satellite and of the tracking system. This estimated state constitutes an input to the optimal controller. The optimal controller treats a linearized system with a quadratic performance index. The maximum principle is applied and a steady-state approximation to the resulting Riccati equation is obtained. A computer program, RATS, implementing this algorithm is described. A feasibility study of real-time implementation, tracking simulations, and parameter sensitivity studies are also reported.
TL;DR: In this article, an extension of Reguli Falsi involving a parabolic approximation is presented, which has convergence exponents 2, 2.414, and 2.732.
Abstract: An extension is made to Reguli Falsi involving a parabolic approximation. Formulae are derived which have convergence exponents 2, 2.414, 2.732, respectively.
IBM1
TL;DR: The Ritz-Trefftz algorithm is applied to the computer solution of the state regulator problem and is designed to improve the speed of solution and the storage requirements to the point where real-time implementation becomes feasible.
Abstract: In this paper the Ritz-Trefftz algorithm is applied to the computer solution of the state regulator problem. The algorithm represents a modification of the Ritz direct method and is designed to improve the speed of solution and the storage requirements to the point where real-time implementation becomes feasible. The modification is shown to be more stable computationally than the tradiational Ritz approach. The first concern of the paper is to describe the algorithm and establish its properties as a valid and useful numerical technique. In particular such useful properties as definiteness and reasonableness of condition are established for the method. The second part of the paper is devoted to a comparison of the new techniques with the standard procedure of numerically integrating a matrix Riccati equation to determine a feedback matrix. The new technique is shown to be significantly faster for comparable accuracy.