Showing papers on "Riccati equation published in 1983"
TL;DR: In this paper, a quantum linear problem is constructed which permits the investigation of the sine-Gordon equation within the framework of the inverse scattering method in an arbitrary representation of algebra and geometry.
Abstract: A quantum linear problem is constructed which permits the investigation of the sine-Gordon equation within the framework of the inverse scattering method in an arbitrary representation of algebra
. The corresponding R-matrix is found, satisfying the Yang-Baxter equation (the condition for the factorization of the multiparticle matrices of the scattering of particles on a straight line).
537 citations
01 Jan 1983
TL;DR: Through a sequence of coordinate transformations it is proven that the optimal control can be found by solving a reduced order Riccati equation.
Abstract: Linear systems of the form E\dot{x} = Ax + Bu with E singular are treated. It is desired to find a control which drives the system asymptotically to the origin, minimizing a quadratic cost functional. No restrictions are placed on initial conditions. The cost associated with the impulsive behavior of the system is examined as well as existence and uniqueness of the optimal control. Through a sequence of coordinate transformations it is proven that the optimal control can be found by solving a reduced order Riccati equation.
253 citations
TL;DR: In this article, the authors considered linear systems of the form E\dot{x} = Ax + Bu with E singular and proved that the optimal control can be found by solving a reduced order Riccati equation.
Abstract: Linear systems of the form E\dot{x} = Ax + Bu with E singular are treated. It is desired to find a control which drives the system asymptotically to the origin, minimizing a quadratic cost functional. No restrictions are placed on initial conditions. The cost associated with the impulsive behavior of the system is examined as well as existence and uniqueness of the optimal control. Through a sequence of coordinate transformations it is proven that the optimal control can be found by solving a reduced order Riccati equation.
246 citations
TL;DR: In this paper, the linear integral equation for the solutions of the Korteweg-de Vries (KdV) equation is derived from the direct linearization of a general nonlinear difference-difference equation.
242 citations
TL;DR: In this paper, the Riccati equations are applied to the linear-quadratic optimal control problem for hereditary differential systems, and it is shown that, for most such problems, the operator solutions are of trace class (i.e., nuclear).
Abstract: Recent theory of infinite dimensional Riccati equations is applied to the linear-quadratic optimal control problem for hereditary differential systems, and it is shown that, for most such problems, the operator solutions of the Riccati equations are of trace class (i.e., nuclear). With special attention to trace-norm convergence, an abstract approximation theory is developed and applied to a particular approximation scheme. Numerical examples are given.Problems on both finite and infinite time intervals are studied. For both the hereditary system and the approximating systems in the infinite time problem, characteristic equations are derived for the closed-loop eigenvalues, and formulas for the corresponding eigenvectors are given.
201 citations
TL;DR: In this article, it was shown that the set of real symmetric solutions of the Riccati equation is isomorphic to the algebraic variety of invariant subspaces of a related matrix.
Abstract: We prove that the set of real symmetric solutions of the algebraic Riccati equation is isomorphic to the algebraic variety of invariant subspaces of a related $n \times n$ matrix. By characterizing the structure of this variety, we obtain a detailed description of the geometric properties of the solution set of the algebraic Riccati equation.
101 citations
TL;DR: In this article, a method is developed for the design of Luenberger-type observers for linear time-invariant control systems whose state equation is of the form Ex= Ax+Bu where E is a singular square matrix.
Abstract: In this paper a method is developed for the design of Luenberger-type observers for linear time-invariant control systems whose state equation is of the form Ex= Ax+ Bu where E is a singular square matrix. The method is based on the singular-value decomposition of the matrix E, and on the reduction of the equation Ex= Ax +Bu to a system consisting of a differential equation of the form w1, = F1w1 + F2 + w2 +G1u and an algebraic equation of the form H1w1 +H2w2 + G1u = 0. If w2 can be eliminated from the differential equation by the aid of the algebraic equation and the original output equation of the system, the method yields a reduced-order observer for the generalized state space system.
82 citations
TL;DR: In this article, a Riccati type feedback synthesis of optimal control for Dirichlet boundary parabolic equations is considered, where the functional cost penalizes the L 2 -energy over [0,T] of state and control action u and also final state y(T) at t=T.
Abstract: Riccati type feedback synthesis of optimal controls for Dirichlet boundary parabolic equations is considered. The functional cost penalizes the L2 -energy over [0,T] of state and control action u and also final state y(T) at t=T. This latter fact, makes the functional cost discontinuous on the space of admissible controls: L2(Σ); Σ=[0T] × Γ. After overcoming some technical difficulties related to the above mentioned discontinuity, we prove that the optimal control uO can be written in the desired feedback form: uO (t)=−CP (t) yO (t) for all 0≤t
75 citations
Journal Article•
72 citations
01 Jan 1983
TL;DR: In this paper, the Riccati equation is a matrix quadratic equation that arises in the theory of optimal and stochastic control and its relationship with linear algebra is reviewed.
Abstract: The algebraic Riccati equation is a matrix quadratic equation that arises in the theory of optimal and stochastic control. The relationship with symplectic linear algebra is reviewed. An investigation of the perturbation theory leads to a condition number for the problem. A new numerical method, the Hamiltonian QR algorithm, preserves special structure and is numerically stable and efficient.
70 citations
TL;DR: In this paper, a method for the numerical solution of ordinary differential equations is analyzed that is explicit and yet can conserve the quadratic quantities conserved by the equations, which can be a useful alternative to the usual leapfrog technique, in that it does not suffer from the occurrence of blowup phenomena.
01 Dec 1983
TL;DR: In this paper, it was shown that the stabilizing solution of P* to the Riccati Equation in (A,B,C) depends analytically on the Taylor series expansion.
Abstract: The central result of this paper is that the stabilizing solution P*, to the Algebraic Riccati Equation in (A,B,C) depends analytically on (A,B,C). A variant of this result was first presented in an earlier paper ([13]) by the author, but the simplicity of the proof was obscured by the context; moreover, the version given in §1 is more general. In addition, §1 contains recursive formulas for the terms in the Taylor series expansion of P* about a point (A,B,C). In the remainder of the paper, various control - and system-theoretic ramifications of the analyticity lemma are considered.
TL;DR: In this paper, an upper bound on the maximum eigenvalue of the solution matrix K of the algebraic Riccati equation is established, and several lower bounds are derived for some of the largest eigenvalues of K.
Abstract: In this note, an upper bound on the maximum eigenvalue of the solution matrix K of the algebraic Riccati equation is established. The approach outlined also results in several lower bounds, which are more general than those derived in [1], for some of the largest eigenvalues of K .
DOI•
01 Jul 1983TL;DR: In this paper, a fast state-space self-tuner is developed for suboptimal control of linear stochastic multivariable systems, which is determined by utilizing both the standard recursive-extended-least-squares parameter estimation algorithm and the recently developed matrix sign algorithm, which gives a fast solution of the steady-state discrete Riccati equation.
Abstract: A fast state-space self-tuner is developed for suboptimal control of linear stochastic multivariable systems. The suboptimal self-tuner is determined by utilising both the standard recursive-extended-least-squares parameter estimation algorithm and the recently developed matrix sign algorithm, which gives a fast solution of the steady-state discrete Riccati equation. The developed suboptimal state-space self-tuner can be applied to a class of stable/unstable and minimum/non-minimum phase linear stochastic multivariable systems, in which the pair (A, C) is block observable and the pair (A, B) is stablisable. Also, the pair (A, Q?) with Q?Q?T = Q is detectable where A, B and C are system, input and output matrices, respectively, and Q is a weighting matrix in a quadratic performance index.
TL;DR: In this paper, a systematic study of the general behavior of solutions to the Bernoulli equation which governs the evolution of acceleration waves in nonlinear systems is made, including the case of weak discontinuity in a general quasilinear hyperbolic system.
Abstract: A systematic study is made of the general behaviour of solutions to the Bernoulli equation which governs the evolution of acceleration waves in nonlinear systems. The theorems obtained contain all the known results and, in some instances, when specialized to the case of an existing theorem they provide a sharper results obtained here for the Bernoulli equation, and the corresponding results for the propagation of weak discontinuity in a general quasilinear hyperbolic system
TL;DR: In this paper, the authors prove the existence of a set of initial data to which correspond solutions of the nonlinear Klein-Gordon equation with a polynomial nonlinear term, which converge asymptotically, when t→+∞, to solutions of a linear Klein Gordon equation.
Abstract: We prove the existence of a set of initial data to which correspond solutions of the nonlinear Klein-Gordon equation with a polynomial nonlinear term, which converge asymptotically, when t→+∞, to solutions of the linear Klein-Gordon equation.
01 Dec 1983
TL;DR: In this paper, a linear-quadratic fixed-order dynamic compensation in the presence of disturbance and observation noise is considered and necessary conditions for the optimization problem are derived in a new and highly simplified form.
Abstract: We consider steady-state, linear-quadratic fixed-order dynamic compensation in the presence of disturbance and observation noise. First-order necessary conditions for the optimization problem are derived in a new and highly simplified form. These necessary conditions constitute a system of two modified Riccati equations and two modified Lyapunov equations coupled by a projection which plays an essential role in defining the geometric structure of the compensator. When the order of the compensator is equal to the dimension of the plant, the classical linear-quadratic-Gaussian results are immediately obtained.
TL;DR: For an ordinary differential equation and an equation of parabolic type, with a small parameter in the highest derivative, schemes are constructed, which uniformly converge with respect to the parameter as discussed by the authors.
Abstract: For an ordinary differential equation and an equation of parabolic type, with a small parameter in the highest derivative, schemes are constructed, which uniformly converge with respect to the parameter. For the ordinary equation, an algorithm is given for constructing a scheme of high order of accuracy.
TL;DR: In this article, a new linear integral equation is proposed, yielding an exact linearization of the integrable discrete versions of e.g. the nonlinear Schrodinger equation, the isotropic Heisenberg spin chain and the complex sine-Gordon equation.
TL;DR: In this paper, elementary techniques from linear algebra and elementary properties of the Grassmann manifolds are used to prove the existence of periodic orbits and to study the equilibrium structure of Riccati differential equations.
Abstract: In this paper, elementary techniques from linear algebra and elementary properties of the Grassmann manifolds are used to prove the existence of periodic orbits and to study the equilibrium structure of Riccati differential equations.
TL;DR: In this paper, the generic behavior of nodal patterns of the eigenfunctions of the Schrodinger equation depending on parameters is discussed and numerical examples are presented for the case of a hard-walled parallelogram.
TL;DR: An example is given in this paper where an unstable neutral differential-difference equation whoae spectrum lies in the left half-plane of the spectrum, and the spectrum is shown to be stable.
Abstract: An example is given of an unstable neutral differential-difference equation whoae spectrum lies in the left half-plane.
TL;DR: In this article, a new representation for the Schrodinger equation for the three-body problem in the molecular state approach is built, which “kills” cross derivatives in the Schröter equation, and this new representation has a simple form and good asymptotic properties.
TL;DR: In this paper, a description of maximal A-invariant neutral subspaces is provided for a matrix A which is selfadjoint with respect to an indefinite scalar product.
Abstract: For a matrix A which is selfadjoint with respect to an indefinite scalar product, a description of maximal A-invariant neutral subspaces is provided. This description is motivated by the characterization of hermitian solutions of an algebraic Riccati equation.
TL;DR: In this article, the authors studied the bound, critical length, and extension of solutions of a dissipative Riccati equation on a Hilbert space, and showed that no bounded solution can be extended beyond the critical length.
TL;DR: In this paper, the isospectral eigenvalue problem for the Herry-Dym equation is derived using a prolongation technique, which is then exploited to find a recursion operator.
Abstract: The isospectral-eigenvalue problem for the Herry-Dym equation is derived using a prolongation technique. The eigenvalue equation is then exploited to find a recursion operator.
01 Mar 1983
TL;DR: In this paper, a number of known and conjectured nonoscillation criteria for sublinear Emden-Fowler equations are shown to be equivalent, and one of these criteria is then extended to cover cases in which a growth condition on the coefficient of the equation is not satisfied.
Abstract: A number of known and a conjectured nonoscillation criteria for sublinear Emden-Fowler equations are shown to be equivalent. One of these criteria is then extended to cover cases in which a growth condition on the coefficient of the equation is not satisfied.
IBM1
TL;DR: In this paper, a new formulation of the algebraic Riccati equation has been presented and a closed form solution has been obtained under the hypothesis that certain commutativity conditions are fulfilled on a transformed space.
Abstract: In a recent paper [3], by utilizing the square root of a matrix approach a new formulation of the algebraic Riccati equation has been presented and a closed form solution has been obtained under the hypothesis that certain commutativity conditions are fulfilled on a transformed space. In this note the imposed conditions are weakened and the formulation is extended to a wider class of Riccati problems.