Showing papers on "Riccati equation published in 2000"
TL;DR: In this article, an extended tanh-function method is proposed for constructing multiple travelling wave solutions of nonlinear partial differential equations (PDEs) in a unified way, and the key idea of this method is to take full advantage of a Riccati equation involving a parameter and use its solutions to replace the tanh function.
1,830 citations
TL;DR: In this article, it is shown that the chemical Langevin equation can be derived from the microphysical premise from which the chemical master equation is derived, which leads directly to an approximate time-evolution equation of the Langevin type.
Abstract: The stochastic dynamical behavior of a well-stirred mixture of N molecular species that chemically interact through M reaction channels is accurately described by the chemical master equation. It is shown here that, whenever two explicit dynamical conditions are satisfied, the microphysical premise from which the chemical master equation is derived leads directly to an approximate time-evolution equation of the Langevin type. This chemical Langevin equation is the same as one studied earlier by Kurtz, in contradistinction to some other earlier proposed forms that assume a deterministic macroscopic evolution law. The novel aspect of the present analysis is that it shows that the accuracy of the equation depends on the satisfaction of certain specific conditions that can change from moment to moment, rather than on a static system size parameter. The derivation affords a new perspective on the origin and magnitude of noise in a chemically reacting system. It also clarifies the connection between the stochas...
1,537 citations
TL;DR: The problem of solving LQR models with system diffusions dependent on both state and control variables, which is left open in part I, is tackled and an optimal control is explicitly constructed.
Abstract: In part I of this paper [S. Chen, X. Li, and X. Zhou, SIAM J. Control Optim., 36 (1998), pp. 1685--1702], an optimization model of stochastic linear quadratic regulators (LQRs) with indefinite control cost weighting matrices is proposed and studied. In this sequel, the problem of solving LQR models with system diffusions dependent on both state and control variables, which is left open in part I, is tackled. First, the solvability of the associated stochastic Riccati equations (SREs) is studied in the normal case (namely, all the state and control weighting matrices and the terminal matrix in the cost functional are nonnegative definite, with at least one positive definite), which in turn leads to an optimal state feedback control of the LQR problem. In the general indefinite case, the problem is decomposed into two optimal LQR problems, one with a forward dynamics and the other with a backward dynamics. The well-posedness and solvability of the original LQR problem are then obtained by solving these two subproblems, and an optimal control is explicitly constructed. Examples are presented to illustrate the results.
385 citations
TL;DR: In this paper, the authors presented separation results for globally bounded stabilizing state feedback controllers when the high-gain observer is designed using pole-placement so as to create a closed-loop system with two-time-scale structure.
245 citations
28 Jun 2000
TL;DR: In this article, the state-dependent Riccati equation method of nonlinear regulation is used to control the position and attitude of a spacecraft in the proximity of a tumbling target.
Abstract: Spacecraft which are required to remove space debris or collect disabled satellites must be able to achieve the attitude of the target while being positioned at a desired distance from the target. The six degree of freedom motion of a spacecraft performing rotational and translational maneuvers has nonlinear equations of motion. The state-dependent Riccati equation method of nonlinear regulation is used to control the position and attitude of a spacecraft in the proximity of a tumbling target. A six degree of freedom simulation of the spacecraft and target are utilized to demonstrate the effectiveness of the controller.
169 citations
TL;DR: In this article, a mixed sensitivity /spl Hscr/sub /spl infin/ problem is solved for dead-time systems, and the central controller is shown to be a feedback interconnection of a finite-dimensional system and a finite memory system.
Abstract: A mixed sensitivity /spl Hscr//sub /spl infin// problem is solved for dead-time systems. It is shown that for a given bound on the /spl Hscr//sub /spl infin//-norm causal stabilizing controllers exist that achieve this bound if and only if a related finite-dimensional Riccati equation has a solution with a certain nonsingularity property. In the case of zero time delay, the Riccati equation is a standard Riccati equation and the nonsingularity condition is that the solution be nonnegative definite. For nonzero time delay, the nonsingularity condition is more involved but still allows us to obtain controllers. All suboptimal controllers are parameterized, and the central controller is shown to be a feedback interconnection of a finite-dimensional system and a finite memory system, both of which can be implemented. Some /spl Hscr//sub /spl infin// problems are rewritten as pure rational /spl Hscr//sub /spl infin//, problems using a Smith predictor parameterization of the controller.
165 citations
TL;DR: The standard H/sub /spl infin// problem is solved for LTI systems with a single, pure input lag, based on state-space analysis, mixing a finite-dimensional and an abstract evolution model.
Abstract: The standard H/sub /spl infin// problem is solved for LTI systems with a single, pure input lag. The solution is based on state-space analysis, mixing a finite-dimensional and an abstract evolution model. Utilizing the relatively simple structure of these distributed systems, the associated operator Riccati equations are reduced to a combination of two algebraic Riccati equations and one differential Riccati equation over the delay interval. The results easily extend to finite time and time-varying problems where the algebraic Riccati equations are substituted by differential Riccati equations over the process time duration.
122 citations
TL;DR: In this article, the authors explored the use of a discrete singular convolution algorithm as a unified approach for numerical integration of the Fokker-Planck equation and demonstrated that different implementations of the present algorithm, such as global, local, Galerkin, collocation and finite difference, can be deduced from a single starting point.
Abstract: This paper explores the use of a discrete singular convolution algorithm as a unified approach for numerical integration of the Fokker-Planck equation. The unified features of the discrete singular convolution algorithm are discussed. It is demonstrated that different implementations of the present algorithm, such as global, local, Galerkin, collocation and finite difference, can be deduced from a single starting point. Three benchmark stochastic systems, the repulsive Wong process, the Black-Scholes equation and a genuine nonlinear model, are employed to illustrate the robustness and to test the accuracy of the present approach for the solution of the Fokker-Planck equation via a time-dependent method. An additional example, the incompressible Euler equation, is used to further validate the present approach for more difficult problems. Numerical results indicate that the present unified approach is robust and accurate for solving the Fokker-Planck equation.
87 citations
TL;DR: In this paper, a symmetry-based approach to solving a given ordinary difference equation is described, and a Lie algebra of symmetry generators that is isomorphic to sl(3) is shown to achieve successive reductions of order.
Abstract: This paper describes a new symmetry-based approach to solving a given ordinary difference equation. By studying the local structure of the set of solutions, we derive a systematic method for determining one-parameter Lie groups of symmetries in closed form. Such groups can be used to achieve successive reductions of order. If there are enough symmetries, the difference equation can be completely solved. Several examples are used to illustrate the technique for transitive and intransitive symmetry groups. It is also shown that every linear second-order ordinary difference equation has a Lie algebra of symmetry generators that is isomorphic to sl(3). The paper concludes with a systematic method for constructing first integrals directly, which can be used even if no symmetries are known.
80 citations
TL;DR: In this article, the basic concepts of factorizable problems in one-dimensional Quantum Mechanics, as well as the theory of Shape Invariant potentials are reviewed, and the relation of this last theory with a generalization of the classical Factorization Method presented by Infeld and Hull is analyzed in detail.
Abstract: The basic concepts of factorizable problems in one-dimensional Quantum Mechanics, as well as the theory of Shape Invariant potentials are reviewed. The relation of this last theory with a generalization of the classical Factorization Method presented by Infeld and Hull is analyzed in detail. By the use of some properties of the Riccati equation the solutions of Infeld and Hull are generalized in a simple way.
TL;DR: In this paper, a relationship between the massless a2(2) integrable quantum field theory and a certain third-order ordinary differential equation was established, and a non-linear integral equation for the generalised spectral problem was derived, and some numerical checks were performed.
01 Jul 2000
TL;DR: In this article, the authors deal with reliable LQG controller design for linear systems with sensor failures and propose a model of sensor failures more practical than outage, where sensor failures are modeled as outage rather than outage.
Abstract: The paper deals with reliable LQG controller design for linear systems with sensor failures A model of sensor failures more practical than outage is adopted Two procedures of designing reliable LQG controllers are presented: stabilising solutions of three coupled Riccati equations; and solutions to one Riccati equation and two linear matrix inequalities The resulting control systems are guaranteed to be robustly stable and with a known cost bound on the LQG cost against sensor failures Algorithms are developed to obtain a solution and illustrated via numerical examples
01 Sep 2000
TL;DR: This work applies and extends the available theory for Newton's method endowed with exact line search to accelerate convergence and proposes to solve Lyapunov equations using iterative schemes for computing the matrix sign function.
Abstract: We investigate the numerical solution of continuous-time algebraic Riccati equations via Newton's method on serial and parallel computers with distributed memory. We apply and extend the available theory for Newton's method endowed with exact line search to accelerate convergence. We also discuss a new stopping criterion based on recent observations regarding condition and error estimates. In each iteration step of Newton's method a stable Lyapunov equation has to be solved. We propose to solve these Lyapunov equations using iterative schemes for computing the matrix sign function. This approach can be efficiently implemented on parallel computers using ScaLAPACK. Numerical experiments on an ibm sp 2 multicomputer report on the accuracy, scalability, and speed-up of the implemented algorithms.
12 Dec 2000
TL;DR: In this article, a closed form solution to the estimation-theoretic simultaneous localisation and map building (SLAM) problem is presented by explicit solution of the differential Riccati equation associated with the n-landmark SLAM problem.
Abstract: Presents a closed form solution to the estimation-theoretic simultaneous localisation and map building (SLAM) problem. The solution is obtained by explicit solution of the differential Riccati equation associated with the n-landmark SLAM problem. The solution describes and explains the many experimental and theoretical results obtained so far in the study of the SLAM problem. Further, the solution, for the first time, allows a precise means of analysing the performance of different SLAM algorithms and enables the design of efficient SLAM systems.
TL;DR: The relation between the Chern and Tenenblat approach to conservation laws of equations describing pseudo-spherical surfaces and the more familiar Riccati equation approach (conservation laws from associated linear problems) is investigated in this paper.
Abstract: The relation between the Chern and Tenenblat approach to conservation laws of equations describing pseudo-spherical surfaces (conservation laws from pseudo-spherical structure) and the more familiar “Riccati equation” approach (conservation laws from associated linear problem) is investigated. Two examples [cylindrical Korteweg–de Vries (KdV) and Lund–Regge equations] are presented. Chern and Tenenblat’s point of view is then connected with the theory of soliton surfaces. A generalization of the original Chern–Tenenblat construction of conservation laws results, and a reasonable family of large deformations for scalar equations describing pseudo-spherical surfaces, the “equations describing Calapso–Guichard surfaces,” can be introduced. It is shown that these equations are also the integrability condition of linear problems.
TL;DR: In this article, a numerical method to invert the dielectric permittivity profile from the Riccati equation using the Newton-Kantorovich iterative scheme is described.
Abstract: A numerical method to invert the dielectric permittivity profile from the Riccati equation using the Newton-Kantorovich iterative scheme is described. Instead of handling the equations in terms of usual geometrical depth, we determine the profile as a function of the electromagnetic path length since the convergence and the stability of the solution are found to be significantly better in this case. The initial profile used as a starting point for the inversion is obtained by another method employing successive reconstruction of dielectric interfaces and homogeneous layers in a step-like form. This method, though not always accurate, is fast and well suited for the approximate reconstruction of the profile, thus creating ideal starting conditions for the previous approach. As a result, the computation time is considerably reduced, without using any a priori information. The approach is applicable to both continuous and discontinuous profiles of high contrast and exhibits a good stability of the solution with respect to noisy input data. A lossy medium profile can also be inverted provided the overall thickness of the inhomogeneous slab and the background permittivity are known.
Journal Article•
TL;DR: In this paper, a software reliability growth model based on a discrete analog of a Gompertz equation that has an exact solution is proposed, which is able to predict in the early test phase when software can be released.
Abstract: SUMMARY I describe a software reliability growth model that yields accurate parameter estimates even with a small amount of input data. The model is based on a proposed discrete analog of a Gompertz equation that has an exact solution. The difference equation tends to a differential equation on which the Gompertz curve model is defined, when the time interval tends to zero. The exact solution also tends to the exact solution of the differential equation when the time interval tends to zero. The discrete model conserves the characteristics of the Gompertz model because the difference equation has an exact solution. Therefore, the proposed model provides accurate parameter estimates, making it possible to predict in the early test phase when software can be released.
TL;DR: In this article, the problem of finding a class of shape-invariant potentials which depend on n parameters transformed by translation is solved using the properties of the Riccati equation.
Abstract: Shape-invariant potentials in the sense of Gendenshtein (1983 JETP Lett. 38 356) which depend on more than two parameters are not known to date. Cooper et al (1987 Phys. Rev. D 36 2458) posed the problem of finding a class of shape-invariant potentials which depend on n parameters transformed by translation, but it was not solved. We analyse the problem using some properties of the Riccati equation and find the general solution.
TL;DR: In this article, a generalized algebraic Riccati equation (GARE) was introduced, which involves a matrix pseudo-inverse and two additional algebraic equality/inequality constraints.
Book•
01 Jan 2000
TL;DR: In this paper, the Riccati Equation in the classical Calculus of Variations has been used to solve the Minimization Problem for a Multiple Integral (MIMO) problem.
Abstract: 1. Classical Calculus of Variations.- 2. Riccati Equation in the Classical Calculus of Variations.- 3. Lie Groups and Lie Algebras.- 4. Grassmann Manifolds.- 5. Matrix Double Ratio.- 6. Complex Riccati Equations.- 7. Higher-Dimensional Calculus of Variations.- 8. On the Quadratic System of Partial Differential Equations Related to the Minimization Problem for a Multiple Integral.- Epilogue.- Appendix to the English Edition.- References.
TL;DR: The convergence of the approximated control law to the optimal one is proved and the approximation scheme developed for the solution of the linear quadratic Gaussian control on a finite time interval for hereditary systems with multiple noncommensurate delays and distributed delay is proved.
Abstract: In this paper an approximation scheme is developed for the solution of the linear quadratic Gaussian (LQG) control on a finite time interval for hereditary systems with multiple noncommensurate delays and distributed delay. The solution here proposed is achieved by means of two approximating subspaces: the first one to approximate the Riccati equation for control and the second one to approximate the filtering equations. Since the approximating subspaces have finite dimension, the resulting equations can be implemented. The convergence of the approximated control law to the optimal one is proved. Simulation results are reported on a wind tunnel model, showing the high performance of the method.
TL;DR: In this paper, an infinite horizon H∞ state-feedback control problem for singularly perturbed linear systems with a small state delay is considered, and a simplified controller with parameter-independent gain matrices, solving the original problem for all sufficiently small values of this parameter, is obtained.
Journal Article•
TL;DR: A deep investigation is made on some system theoretical aspects of periodic systems and control, including the notions of H 2 and H∞ norms, the parametrization of stabilizing controllers, and the existence of periodic solutions to Riccati differential equations and/or inequalities.
Abstract: The paper is divided, in two parts. In the first part a deep investigation is made on some system theoretical aspects of periodic systems and control, including the notions of H 2 and H∞ norms, the parametrization of stabilizing controllers, and the existence of periodic solutions to Riccati differential equations and/or inequalities. All these aspects are useful in the second part, where some parametrization and control problems in H 2 and H∞ are introduced and solved.
TL;DR: In this paper, it was shown that the sequences of the solutions of the decoupled algebraic Lyapunov equations are monotonic under proper initialization, and that these sequences converge from above to the positive-semidefinite stabilizing solution of the system of coupled algebraic Riccati equations of the optimal control problem of jump parameter linear systems.
TL;DR: In this article, the group-theoretic methods for calculating the solution of Burgers' equation with appropriate boundary-and initial-conditions are presented, and the obtained differential equation is solved analytically and the solution obtained in closed form, for a specific choice of boundary condition.
TL;DR: In this article, a stochastic linear quadratic optimal control problem (LQ problem) is studied, where the coefficients are allowed to be random and the cost functional is allowed to have a negative weight on the square of the control variable.
Abstract: This paper studies a stochastic linear quadratic optimal control problem (LQ problem, for short), for which the coefficients are allowed to be random and the cost functional is allowed to have a negative weight on the square of the control variable. The authors introduce the stochastic Riccati equation for the LQ problem. This is a backward SDE with a complicated nonlinearity and a singularity. The local solvability of such a backward SDE is established, which by no means is obvious. For the case of deterministic coefficients, some further discussions on the Riccati equations have been carried out. Finally, an illustrative example is presented.
TL;DR: In this paper, a transition matrix method is used to transform a test for the unboundedness of an n £ n matrix solution of a Riccati equation into a test of a scalar being zero.
Abstract: A recent advance in sufe cient conditions for a weak local minimum in the Bolza optimal control problem is used to develop a practical procedure for applying second-order necessary conditions and sufe cient conditions. For a system with n state variables, a transition matrix method is used to transform a test for the unboundedness of an n £ n matrix solution of a Riccati equation into a test for a scalar being zero. This allows routine testing of second-order conditions, including the Jacobi no-conjugate-point necessary condition. Four example problems are analyzed: a simple minimum-time problem, the shortest path between two points on a sphere, a multiobjective spacecraft trajectory optimization, and an application of Hamilton’ s Principle to a circular orbit in an inversesquare gravitational e eld. In those examples for which second-order conditions are violated and an analytical solution does not exist, a genetic algorithm is used to determine a near-optimal solution.
TL;DR: In this paper, the Picone identity and Riccati equation were derived for disconjugacy of a linear Hamiltonian system on an arbitrary time scale, and the Riccaci equation was shown to be sufficient for positive definiteness of the quadratic functional associated with the system.