Showing papers on "Riccati equation published in 1984"
01 Dec 1984
TL;DR: The approach presented uses the generalized eigenproblem formulation for the solution of general forms of algebraic Riccati equations arising in both continuous- and discrete-time applications.
Abstract: Numerical issues related to the computational solution of the algebraic matrix Riccati equation are discussed. The approach presented uses the generalized eigenproblem formulation for the solution of general forms of algebraic Riccati equations arising in both continuous- and discrete-time applications. These general forms result from control and filtering problems for systems in generalized (or implicit or descriptor) state space form. A Newtontype iterative refinement procedure for the generalized Riccati solution is given. The issue of numerical condition of the Riccati problem is addressed. Balancing to improve numerical condition is discussed. An overview of a software package, RICPACK, coded in portable, reliable Fortran is given. Results of numerical experiments are reported.
496 citations
TL;DR: In this article, a new and direct approach to stochastic model reduction is developed by establishing an equivalence between canonical correlation analysis and solutions to algebraic Riccati equations.
Abstract: A new and direct approach to stochastic model reduction is developed. The order reduction algorithm is obtained by establishing an equivalence between canonical correlation analysis and solutions to algebraic Riccati equations. Also the concept of balanced stochastic realization (BSR) plays a fundamental role. Asymptotic stability of the reduced-order realization is established, and spectral domain interpretations for the BSR are given.
242 citations
TL;DR: A new time-domain method of quadratic-optimum control synthesis for systems described by finite-memory output predictors is presented, which leads to algorithms which are numerically robust and therefore suitable for real-time computation using microprocessors with reduced wordlength.
225 citations
TL;DR: In this article, an approximation framework is presented for computation of Riccati operators that can be guaranteed to converge to the RICCati operator in feedback control for abstract evolution systems in a Hilbert space.
Abstract: An approximation framework is presented for computation (in finite imensional spaces) of Riccati operators that can be guaranteed to converge to the Riccati operator in feedback controls for abstract evolution systems in a Hilbert space. It is shown how these results may be used in the linear optimal regulator problem for a large class of parabolic systems.
206 citations
TL;DR: In this article, the convergence and properties of the Riccati difference equation are studied for systems which are not necessarily stabilizable (in the filtering sense), particularly those having uncontrollable roots on the unit circle.
Abstract: This paper studies the convergence and properties of the solutions of the Riccati difference equation Special emphasis is given to systems which are not necessarily stabilizable (in the filtering sense), particularly those having uncontrollable roots on the unit circle Besides generalizing and unifying previous work, the results have application to a number of important problems including filtering and control of systems with purely deterministic disturbances such as sinusoids and drift components
180 citations
TL;DR: In this article, the authors consider the problem of controlling an ordinary differential equation, subject to positive switching costs, and show in particular that the value functions form the "viscosity solution" of the dynamic programming quasi-variational inequalities.
Abstract: We consider the problem of controlling an ordinary differential equation, subject to positive switching costs, and show in particular that the value functions form the “viscosity solution” (cf. [6], [7]) of the dynamic programming quasi-variational inequalities. This interpretation allows for a rigorous application of various dynamic programming techniques.
146 citations
TL;DR: In this paper, a vector field is defined to determine a cooperative system of differential equations provided all the off-diagonal terms of the Jacobian matrix are nonpositive (or nonnegati...
Abstract: A vector field in $\mathbb{R}^n $ determines a competitive (or cooperative) system of differential equations provided all the off-diagonal terms of the Jacobian matrix are nonpositive (or nonnegati...
126 citations
TL;DR: This paper collects the bounds that have been presented up to now and summarizes them in an unified form to prove particularly convenient for those wishing to get a ready estimate of the solution while solving the equations numerically or to develop theoretical results that rely on these bounds.
Abstract: In recent years, several bounds have been reported for different measures of the ‘ extent’ or ‘ size ’ of the solution of the algebraic matrix equations arising in control theory, such as the Riccati equation and the Lyapunov equation. This paper collects the bounds that have been presented up to now and summarizes them in an unified form. This will prove particularly convenient for those wishing to get a ready estimate of the solution while solving the equations numerically or to develop theoretical results that rely on these bounds.
116 citations
TL;DR: In this article, the total hierarchy of the Kadomtsev-Petviashvili (KP) equation is transformed to a system of linear partial differential equations with constant coefficients, and complete integrability of the KP equation is proved by using this linear system.
102 citations
TL;DR: In this paper, the authors studied the stabilization and regulation of linear discrete-time systems whose coefficients depend on one or more parameters and derived closed-form expressions for a collection of stabilizing gains in terms of the reachability Gramian.
Abstract: The stabilization and regulation of linear discrete-time systems whose coefficients depend on one or more parameters is studied. For linear systems whose coefficients are continuous functions of real or complex parameters (respectively, analytic or rational functions of real parameters), it is shown that reachability of the system for all values of the parameters implies that the system can be stabilized using gains that are also continuous (analytic, rational) functions of the parameters. Closed-form expressions for a collection of stabilizing gains are given in terms of the reachability Gramian. For systems which are stabilizable for all values of the parameters, it is shown that continuous (analytic, rational) stabilizing gains can be computed from a finite-time solution to a Riccati difference equation whose coefficients are functions of the parameters. These results are then applied to the problem of tracking and disturbance rejection in the case when both the plant and the exogenous signals contain parameters.
97 citations
TL;DR: In this paper, optimal control problems for bilinear systems are studied and solved with a view to approximating analogous problems for general nonlinear systems, and a power-series approach is presented which requires offline calculations as in the linear case (Riccati equation).
Abstract: Optimal control problems for bilinear systems are studied and solved with a view to approximating analogous problems for general nonlinear systems. For a given bilinear optimal control problem, a sequence of linear problems is constructed, and their solutions are shown to converge to the desired solution. Also, the direct solution to the Hamilton-Jacobi equation is analyzed. A power-series approach is presented which requires offline calculations as in the linear case (Riccati equation). The methods are compared and illustrated. Relations to classical linear systems theory are discussed.
TL;DR: In this paper, it was shown that the stabilizing solution P * to the algebraic Riccati equation in (A, B, C ) depends analytically on ( A, B, C ).
Abstract: The central result of this correspondence is Lemma 1.1, which states that the stabilizing solution P * to the algebraic Riccati equation in ( A, B, C ) depends analytically on ( A, B, C ). In the remainder of the correspondence, various control- and system-theoretic ramifications of the analyticity lemma are considered. An important consequence of Lemma 1.1 is that any smooth-state feedback control law for a finite-dimensional plant may be implemented using dynamic input-output compensators whose transfer functions depend smoothly on the plant transfer function.
TL;DR: For periodically time-varying matrix Riccati equations, controllability and observability (in the usual sense) are sufficient for the existence of a unique positive definite periodic solution as discussed by the authors.
Abstract: For periodically time-varying matrix Riccati equations, controllability and observability (in the usual sense) are shown to be sufficient for the existence of a unique positive definite periodic solution.
TL;DR: A spline approximation method for computation of the gain operators in feedback controls is proposed and tested numerically and compares with a method based on “averaging” approximations.
Abstract: We consider the infinite interval regulator problem for systems with delays. A spline approximation method for computation of the gain operators in feedback controls is proposed and tested numerically. Comparison with a method based on “averaging” approximations is made.
TL;DR: In this paper, the authors proved global existence for the solution of a Riccati differential equation connected with the synthesis of a boundary control problem governed by parabolic partial differential equations.
Abstract: Global existence is proved for the solution, of a Riccati differential equation connected with the synthesis of a boundary control problem governed by parabolic partial differential equations.
TL;DR: In this paper, the notion of local error as inherited from the theory of ordinary differential equations must be generalized for retarded problems along with a reliable basis for a step selection scheme is developed.
Abstract: Retarded initial value problems are routinely replaced by an initial value problem of ordinary differential equations along with an appropriate interpolation scheme Hence one can control the global error of the modified problem but not directly the actual global error of the original problem In this paper we give an estimate for the actual global error in terms of controllable quantities Further we show that the notion of local error as inherited from the theory of ordinary differential equations must be generalized for retarded problems Along with the new definition we are led to developing a reliable basis for a step selection scheme
TL;DR: In this article, the general solution of the matrix Riccati equation W W(t) = A + WB + CW + WDW (A(t, B(t), C, D, D(t)) in terms of 5 particular solutions, are applied to obtain numerical solutions of this coupled system of nonlinear equations.
TL;DR: An exact analytical solution for travelling waves of the Fisher equation with a general nonlinearity is found in this article, where boundary values, the boundedness and the stability of the solution are discussed.
TL;DR: In this article, the Dirac equation is solved for a linear potential and the complete asymptotic behavior of the solution in the nonrelativistic regime is developed by means of the comparison differential equation technique.
Abstract: The Dirac equation is solved for a linear potential The complete asymptotic behavior of the solution in the nonrelativistic regime is developed by means of the comparison differential equation technique
Book•
31 Dec 1984
TL;DR: In this paper, the authors present an approach to the problem of finding a solution for the discrete approximation problem in the context of the Calculus of Variations, which is a generalization of the calculus of variations.
Abstract: I/Monotone Convergence and Positive Operators.- 1. Introduction.- 2. Monotone Operators.- 3. Monotonicity.- 4. Convergence.- 5. Differential Equations with Initial Conditions.- 6. Two-Point Boundary Conditions.- 7. Nonlinear Heat Equation.- 8. The Nonlinear Potential Equation.- Bibliography and Comments.- II/Conservation.- 1. Introduction.- 2. Analytic and Physical Preliminaries.- 3. The Defining Equations.- 4. Limiting Differential Equations.- 5. Conservation for the Discrete Approximation.- 6. Existence of Solutions for Discrete Approximation.- 7. Conservation for Nonlinear Equations.- 8. The Matrix Riccati Equation.- 9. Steady-State Neutron Transport with Discrete Energy Levels.- 10. Analytic Preliminaries.- 11. Reflections, Transmission, and Loss Matrices.- 12. Existence and Uniqueness of Solutions.- 13. Proof of Conservation Relation.- 14. Proof of Nonnegativity.- 15. Statement of Result.- Bibliography and Comments.- III / Dynamic Programming and Partial Differential Equations.- 1. Introduction.- 2. Calculus of Variations as a Multistage Decision Process.- 3. A New Formalism.- 4. Layered Functionals.- 5. Dynamic Programming Approach.- 6. Quadratic Case.- 7. Bounds.- Bibliography and Comments.- IV / The Euler-Lagrange Equations and Characteristics.- 1. Introduction.- 2. Preliminaries.- 3. The Fundamental Relations of the Calculus of Variations.- 4. The Variational Equations.- 5. The Eulerian Description.- 6. The Lagrangian Description.- 7. The Hamiltonian Description.- 8. Characteristics.- Bibliography and Comments.- V / Quasilinearization and a New Method of Successive Approximations.- 1. Introduction.- 2. The Fundamental Variational Relation.- 3. Successive Approximations.- 4. Convergence.- Bibliography and Comments.- VI / The Variation of Characteristic Values and Functions.- 1. Introduction.- 2. Variational Problem.- 3. Dynamic Programming Approach.- 4. Variation of the Green's Function.- 5. Justification of Equating Coefficients.- 6. Change of Variable.- 7. Analytic Continuation.- 8. Analytic Character of Green's Function.- 9. Alternate Derivation of Expression for ?(x).- 10. Variation of Characteristic Values and Characteristic Functions.- 11. Matrix Case.- 12. Integral Equations.- Bibliography and Comments.- VII / The Hadamard Variational Formula.- 1. Introduction.- 2. Preliminaries.- 3. A Minimum Problem.- 4. A Functional Equation.- 5. The Hadamard Variation.- 6. Laplace-Beltrami Operator.- 7. Inhomogeneous Operator.- Bibliography and Comments.- VIII / The Two-Dimensional Potential Equation.- 1. Introduction.- 2. The Euler-Lagrange Equation.- 3. Inhomogeneous and Nonlinear Cases.- 4. Green's Function.- 5. Two-Dimensional Case.- 6. Discretization.- 7. Rectangular Region.- 8. Associated Minimization Problem.- 9. Approximation from Above.- 10. Discussion.- 11. Semidiscretization.- 12. Solution of the Difference Equations.- 13. The Potential Equation.- 14. Discretization.- 15. Matrix-Vector Formulation.- 16. Dynamic Programming.- 17. Recurrence Equations.- 18. The Calculations.- 19. Irregular Regions.- Bibliography and Comments.- IX / The Three-Dimensional Potential Equation.- 1. Introduction.- 2. Discrete Variational Problems.- 3. Dynamic Programming.- 4. Boundary Conditions.- 5. Recurrence Relations.- 6. General Regions.- 7. Discussion.- Bibliography and Comments.- X / The Heat Equation.- 1. Introduction.- 2. The One-Dimensional Heat Equation.- 3. The Transform Equation.- 4. Some Numerical Results.- 5. Multidimensional Case.- Bibliography and Comments.- XI / Nonlinear Parabolic Equations.- 1. Introduction.- 2. Linear Equation.- 3 The Non-negativity of the Kernel.- 4. Monotonicity of Mean Values.- 5. Positivity of the Parabolic Operator.- 6. Nonlinear Equations.- 7. Asymptotic Behavior.- 8. Extensions.- Bibliography and Comments.- XII / Differential Quadrature.- 1. Introduction.- 2. Differential Quadrature.- 3. Determination of Weighting Coefficients.- 4. Numerical Results for First Order Problems.- 5. Systems of Nonlinear Partial Differential Equations.- 6. Higher Order Problems.- 7. Error Representation.- 8. Hodgkin-Huxley Equation.- 9. Equations of the Mathematical Model.- 10. Numerical Method.- 11. Conclusion.- Bibliography and Comments.- XIII / Adaptive Grids and Nonlinear Equations.- 1. Introduction.- 2. The Equation ut =-uux.- 3. An Example.- 4. Discussion.- 5. Extension.- 6. Higher Order Approximations.- Bibliography and Comments.- XIV / Infinite Systems of Differential Equations.- 1. Introduction.- 2. Burgers' Equation.- 3. Some Numerical Examples.- 4. Two-Dimensional Case.- 5. Closure Techniques.- 6. A Direct Method.- 7. Extrapolation.- 8. Difference Approximations.- 9. An Approximating Algorithm.- 10. Numerical Results.- 11. Higher Order Approximation.- 12. Truncation.- 13. Associated Equation.- 14. Discussion of Convergence of u (N).- 15. The Fejer Sum.- 16. The Modified Truncation.- Bibliography and Comments.- XV / Green's Functions.- 1. Introduction.- 2. The Concept of the Green's Function.- 3. Sturm-Liouville Operator.- 4. Properties of the Green's Function for the Sturm-Liouville Equation.- 5. Properties of the ? Function.- 6. Distributions.- 7. Symbolic Functions.- 8. Derivative of Symbolic Functions.- 9. What Space Are We Considering?.- 10. Boundary Conditions.- 11. Properties of Operator L.- 12. Adjoint Operators.- 13. n-th Order Operators.- 14. Boundary Conditions for the Sturm-Liouville Equation.- 15. Green's Function for Sturm-Liouville Operator.- 16. Solution of the Inhomogeneous Equation.- 17. Solving Non-Homogeneous Boundary Conditions.- 18. Boundary Conditions Specified on Finite Interval [a, b].- 19. Scalar Products.- 20. Use of Green's Function to Solve a Second-order Stochastic Differential Equation.- 21. Use of Green's Function in Quantum Physics.- 22. Use of Green's Functions in Transmission Lines.- 23. Two-Point Green's Functions - Generalization to n-point Green's Functions.- 24. Evaluation of Arbitrary Functions for Nonhomogeneous Boundary Conditions by Matrix Equations.- 25. Mixed Boundary Conditions.- 26. Some General Properties.- 1. Nonnegativity of Green's Functions and Solutions.- 2. Variation-Diminishing Properties of Green's Functions.- Notes.- XVI / Approximate Calculation of Green's Functions.- XVII / Green's Functions for Partial Differential Equations.- 1. Introduction.- 2. Green's Functions for Multidimensional Problems in Cartesian Coordinates.- 3. Green's Functions in Curvilinear Coordinates.- 4. Properties of ? Functions for Multi-dimensional Case.- XVIII / The Ito Equation and a General Stochastic Model for Dynamical Systems.- XIX / Nonlinear Partial Differential Equations and the Decomposition Method.- 1. Parametrization and the An Polynomials.- 2. Inverses for Non-simple Differential Operators.- 3. Multidimensional Green's Functions by Decomposition Method.- 4. Relationships Between Green's Functions and the Decomposition Method for Partial Differential Equations.- 5. Separable Systems.- 6. The partitioning Method of Butkovsky.- 7. Computation of the An.- 8. The Question of Convergence.
TL;DR: In this article, the structure of discrete-time linear systems with stationary inputs in the geometric framework of splitting subspaces is studied, and the relation between models with and without noise in the observation channel is investigated.
Abstract: From a conceptual point of view, structural properties of linear stochastic systems are best understood in a geometric formulation which factors out the effects of the choice of coordinates. In this paper we study the structure of discrete-time linear systems with stationary inputs in the geometric framework of splitting subspaces set up in the work by Lindquist and Picci. In addition to modifying some of the realization results of this work to the discrete-time setting, we consider some problems which are unique to the discrete-time setting. These include the relations between models with and without noise in the observation channel, and certain degeneracies which do not occur in the continuous-time case. One type of degeneracy is related to the singularity of the state transition matrix, another to the rank of the observation noise and invariant directions of the matrix Riccati equation of Kalman filtering. We determine to what extent these degeneracies are properties of the output process. The geometric framework also accommodates infinite-dimensional state spaces, and therefore the analysis is not limited to finite-dimensional systems.
TL;DR: In this paper, a regulator problem for a system governed by a linear stochastic differential equation with unbounded coefficients in Hilbert spaces is studied, and the regulator problem is shown to be NP-hard.
Abstract: We study a regulator problem for a system governed by a linear stochastic differential equation with unbounded coefficients in Hilbert spaces.
TL;DR: The authors deduit de nouvelles conditions suffisantes pour la stabilite asymptotiques uniformes de la solution zero d'equations differentielles a retard lineaires non autonomes.
Abstract: On deduit de nouvelles conditions suffisantes pour la stabilite asymptotiques uniformes de la solution zero d'equations differentielles a retard lineaires non autonomes
TL;DR: In this article, the inertia theorems for the time-invariant Liapunov equation and Riccati equation were generalized to the case where the coefficient matrices are periodically time-varying.
TL;DR: In this paper, a singular perturbation method was developed to obtain approximate solutions in terms of an outer series and a final correction series for the Riccati difference equation in the singularly perturbed structure.
Abstract: The closed-loop optimal control of a linear, time-invariant, singularly perturbed discrete system is considered. The resulting matrix Riccati difference equation is formulated in the singularly perturbed structure. It is observed that the degeneration affects some of the final conditions of the Riccati equation. A singular perturbation method is developed to obtain approximate solutions in terms of an outer series and a final correction series. The outer series takes advantage of the order reduction associated with degeneration and the correction series takes care of the affected final conditions. Two examples are given to illustrate the proposed method.
TL;DR: It is shown how the regularity conditions of mathematical programming can be translated into controllability conditions and how the solution of the quadratic program is carried out in the standard Riccati equation.
TL;DR: In this paper, a comment by Turner and Chun on the solution of the matrix Riccati equation used in Ref. 1 was made, and it was shown that the advantage is minimal when the independent modal space control method is used.
Abstract: This reply is concerned with a comment by Turner and Chun on the solution of the matrix Riccati equation used in Ref. 1. Whereas the technique suggested by Turner and Chun may have some numerical advantage over Potter's method in the case of coupled large-order systems, the advantage is minimal when the independent modal-space control method is used.
TL;DR: In this article, unmixed solutions of the matrix equation XDX + XA + AX * − C = 0, D ⩾0 are studied, where C is the number of vertices in the matrix.
TL;DR: In this article, a recursion operator for the classical Boussinesq equation is given, which yields infinitely many symmetries and conservation laws, each of which is a hamiltonian system.
TL;DR: In this paper, the determinants of the positive definite solutions of the discrete algebraic Riccati and Lyapunov matrix equations are presented, and lower bounds for the product of the eigenvalues of the matrix solutions are given.
Abstract: Inequalities which are satisfied by the determinants of the positive definite solutions of the discrete algebraic Riccati and Lyapunov matrix equations are presented. The results give lower bounds for the product of the eigenvalues of the matrix solutions. Also for a discrete Lyapunov equation, an algorithm is presented to determine under what conditions a positive diagonal solution will exist. If all the conditions are satisfied, the algorithm also provides such a diagonal solution.