Showing papers on "Riccati equation published in 1990"
TL;DR: In this article, it was shown that necessary and sufficient conditions for the existence of sub-optimal solutions to the model matching problem associated with H∞ control are that two coupled J-spectral factorizations exist.
Abstract: It is shown that necessary and sufficient conditions for the existence of sub-optimal solutions to the model matching problem associated with H∞ control are that two coupled J-spectral factorizations exist. The second J-spectral factor provides a parameterization of all solutions to the model matching problem. The existence of the J-spectral factors is then shown to be equivalent to the existence of stabilizing, non-negative definite solutions to two algebraic Riccati equations, allowing a state-space formula for a linear fractional representation of all controllers to be given.
229 citations
TL;DR: In this paper, a review of large-N expansion methods is presented, including perturbed oscillator methods, Riccati equation methods, pseudospin methods and collective field methods.
189 citations
TL;DR: In this paper, the linear heat equation in materials with memory was studied by reducing it to an abstract Volterra equation, and results of regularity, asymptotic behavior, and positivity were given.
Abstract: The linear heat equation in materials with memory is studied by reducing it to an abstract Volterra equation. Results of regularity, asymptotic behavior, and positivity are given.
118 citations
05 Dec 1990
TL;DR: In this paper, the Riccati equation formulation of the positive real lemma is used to guarantee robust stability in the presence of positive real (but otherwise unknown) plant uncertainty.
Abstract: The properties of positive real transfer functions are used to guarantee robust stability in the presence of positive real (but otherwise unknown) plant uncertainty. These results are then used for controller synthesis to address the problem of robust stabilization in the presence of positive real uncertainty. One of the principal motivations for these results is to utilize phase information in guaranteeing robust stability. In this sense these results go beyond the usual limitations of the small gain theorem and quadratic Lyapunov functions, which may be conservative when phase information is available. The results of this study are based upon a Riccati equation formulation of the positive real lemma and thus resemble certain Riccati-based approaches to bounded real (H/sub infinity /) control. >
109 citations
TL;DR: Compared to the classical single-rate sampled-data feedback in which the state is always sampled at the same rate, the multirate system can provide a better response with a considerable reduction in the optimal cost.
Abstract: The optimal multirate design of linear, continuous-time, periodic and time-invariant systems is considered. It is based on solving the continuous linear quadratic regulation (LQR) problem with the control being constrained to a certain piecewise constant feedback. Necessary and sufficient conditions for the asymptotic stability of the resulting closed-loop system are given. An explicit multirate feedback law that requires the solution of an algebraic discrete Riccati equation is presented. Such control is simple and can be easily implemented by digital computers. When applied to linear time-invariant systems, multirate optimal feedback optimal control provides a satisfactory response even if the state is sampled relatively slowly. Compared to the classical single-rate sampled-data feedback in which the state is always sampled at the same rate, the multirate system can provide a better response with a considerable reduction in the optimal cost. In general, the multirate scheme offers more flexibility in choosing the sampling rates. >
100 citations
TL;DR: In this paper, it was shown that the ideas developed by Byers in [Proc. Summer Research Conference, AMS Vol. 47, Contemporary Math., American Mathematical Society, Providence, RI, pp. 35-49] on the sensitivity of the algebraic Riccati equation can be sharpened and extended to norms other than the Frobenius norm.
Abstract: In this paper it is shown that the ideas developed by Byers in [Proc. Summer Research Conference, AMS Vol. 47, Contemporary Math., American Mathematical Society, Providence, RI, 1984, pp. 35–49] on the sensitivity of the algebraic Riccati equation can be sharpened and extended to norms other than the Frobenius norm. This extension is crucial from an interpretive point of view because use of the spectral norm allows an identification between the condition number of the algebraic Riccati equation and the damping properties of the closed-loop dynamical system. Moreover, this approach has the pleasant feature that it carries over to a completely parallel theory for the sensitivity of the differential Riccati equation, an area that has not been considered previously.
82 citations
TL;DR: In this paper, conditions for the existence of a suitable state feedback are formulated in terms of a quadratic matrix inequality, reminiscent of the dissipation inequality of singular linear QoE optimal control.
Abstract: In this paper the standard $H_\infty $ control problem using state feedback is considered. Given a linear, time-invariant, finite-dimensional system, this problem consists of finding a static state feedback such that the resulting closed-loop transfer matrix has $H_\infty $ norm smaller than some a priori given upper bound. In addition it is required that the closed-loop system is internally stable. Conditions for the existence of a suitable state feedback are formulated in terms of a quadratic matrix inequality, reminiscent of the dissipation inequality of singular linear quadratic optimal control. Where the direct feedthrough matrix of the control input is injective, the results presented here specialize to known results in terms of solvability of a certain indefinite algebraic Riccati equation.
80 citations
TL;DR: In this paper, the energy method was used to obtain a desired decay estimate for the linearized Boltzmann equation with respect to the existence of global solutions near a Maxwellian.
Abstract: The initial value problem for the nonlinear Boltzmann equation is studied. For the existence of global solutions near a Maxwellian, it is important to obtain a desired decay estimate for the linearized equation. In previous works, such a decay estimate was obtained by a method based on the spectral theory for the linearized Boltzmann operator. The aim of this paper is to show the same decay estimate by a new method. Our method is the so-called energy method and makes use of a Ljapunov function for the ordinary differential equation obtained by taking the Fourier transform. Our Ljapunov function is constructed explicitly by using some property of the equations for thirteen moments.
79 citations
TL;DR: In this article, an infinite-dimensional linear time-varying system on the interval $( - ∞, ∞ )$ is considered and three quadratic problems: the infinite horizon problem, and one-sided and two-sided average cost problems.
Abstract: An infinite-dimensional linear time-varying system on the interval $( - \infty ,\infty )$ is considered. We introduce three quadratic problems: the infinite horizon problem, and one-sided and two-sided average cost problems. A Riccati equation on $( - \infty ,\infty )$ is considered first and sufficient conditions for the existence and uniqueness of a bounded solution are given. Then by dynamic programming the quadratic problems are solved. Similar problems in the stochastic case are considered.
78 citations
TL;DR: An adaptive excitation controller for a synchronous generator based on the linear optimal control theory is proposed, which exhibits better performances than an automatic voltage regulator (AVR) with a conventional power system stabilizer (PSS).
Abstract: An adaptive excitation controller for a synchronous generator based on the linear optimal control theory is proposed. The generator operating conditions are tracked by a model whose parameters are identified every sampling interval using the actual input and output of the generator. The control is computed by solving a third-order Riccati equation and the identified model parameters. Studies on a single-machine infinite-bus system and a three-machine infinite-bus system show that the proposed controller exhibits better performances than an automatic voltage regulator (AVR) with a conventional power system stabilizer (PSS). >
TL;DR: In this paper, the suboptimality of some parameter for H/sub infinity /-optimization by dynamic state-feedback is characterized in terms of the solvability of Riccati inequalities.
Abstract: The suboptimality of some parameter for H/sub infinity /-optimization by dynamic state-feedback is characterized in terms of the solvability of Riccati inequalities. This is done without restricting the finite zero structure of the plant. If there are no system zeros on the imaginary axis, the H/sub infinity /-problem can be treated in a complete and satisfactory way. Explicit characterizations optimum to be achieved are provided, and a closed formula for the optimal value is derived in terms of the H/sub infinity /-norm of some fixed transfer matrix. If the optimum is not attained, any sequence of controllers of bounded size which is constructed to approach the infimal norm must necessarily be high-gain. A globally and quadratically convergent algorithm to compute the optimal value is proposed. This algorithm is generalized to the H/sub infinity /-optimization problem by measurement feedback. >
TL;DR: New matrix-valued algorithms based on a matrix generalization of the BDFs are proposed for solving stiff Riccatti differential equations, where the amount of work required to compute the solution per time step is only O(n/sup 3/) flops by using the matrix- valued algorithms, whereas the classical approach requires O( n/sup 6/) flop per timestep.
Abstract: In the time-varying case, a classical approach which has been widely used to compute the solution of the Riccati matrix equation of, for example, size n*n, is to unroll the matrices into vectors and integrate the resulting system of n/sup 2/ vector differential equations directly. If the system of vectorized differential equations is stiff, the cost (computation time and storage requirements) of applying the popular backward differentiation formulas (BDFs) to the stiff equations will be very high for large n because a linear system of algebraic equations of size n/sup 2/*n/sup 2/ must be solved at each time step. New matrix-valued algorithms based on a matrix generalization of the BDFs are proposed for solving stiff Riccatti differential equations. The amount of work required to compute the solution per time step is only O(n/sup 3/) flops by using the matrix-valued algorithms, whereas the classical approach requires O(n/sup 6/) flops per time step. >
TL;DR: In this article, a short new proof for the comparison theory of the matrix valued Riccati equation with singular initial values is given, and applications to Riemannian geometry are briefly indicated.
Abstract: We give a short new proof for the comparison theory of the matrix valued Riccati equationB′+B2+R=0 with singular initial values. Applications to Riemannian geometry are briefly indicated.
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04 Apr 1990
TL;DR: In this paper, the authors proposed a method for estimating Econometric Models with Measurement Errors in State-Space Formulation, which is based on the Kalman Filter and Chandrasekhar Equation.
Abstract: 1. Introduction.- 2. Formulation of Econometric Models in State-Space.- 2.1. Structural Form, Reduced Form and State-Space Form.- 2.2. Additional Remarks.- 3. Formulation of Econometric Models with Measurement Errors.- 3.1. Model of the Exogenous Variables.- 3.2. State-Space Formulation.- 4. Estimation of Econometric Models with Measurement Errors.- 4.1. Evaluation of the Likelihood Function.- 4.2. Maximization of the Likelihood Function.- 4.3. Initial Conditions.- 4.4. Gradient Methods and Identification.- 4.5. Asymptotic Properties.- 4.6. Numerical Considerations.- 4.7. Model Verification.- 5. Extensions of the Analysis.- 5.1. Missing Observations and Contemporaneous Aggregation.- 5.2. Temporal Aggregation.- 5.3. Correlated Measurement Errors.- 6. Numerical Results.- 7. Conclusions.- Appendices.- A. Kalman Filter and Chandrasekhar Equations.- A.1. Kalman Filter.- A.2. Chandrasekhar Equations.- B. Calculation of the Gradient.- C. Calculation of the Hessian.- D. Calculation of the Information Matrix.- E. Estimation of the Initial Conditions.- F. Solution of the Lyapunov and Riccati Equations.- F.1. Lyapunov Equation.- F.2. Riccati Equation.- G. Fixed-Interval Smoothing Algorithm.- References.- Author Index.
TL;DR: In this article, the authors discuss the influence on the solutions of finite-difference schemes of using a variety of denominator functions in the discrete modeling of the derivative for any ordinary differential equation.
Abstract: This paper discusses the influence on the solutions of finite-difference schemes of using a variety of denominator functions in the discrete modeling of the derivative for any ordinary differential equation. The results obtained are a consequence of using a generalized definition of the first derivative. A particular example of the linear decay equation is used to illustrate in detail the various solution possibilities that can occur.
01 Mar 1990
TL;DR: In this paper, two linear quadratic regulators are developed for placing the closed-loop poles of linear multivariable continuous-time systems within the common region of an open sector, bounded by lines inclined at +/- pi/2k (for a specified integer k not less than 1) from the negative real axis, and the left-hand side of a line parallel to the imaginary axis in the complex s-plane.
Abstract: Two linear quadratic regulators are developed for placing the closed-loop poles of linear multivariable continuous-time systems within the common region of an open sector, bounded by lines inclined at +/- pi/2k (for a specified integer k not less than 1) from the negative real axis, and the left-hand side of a line parallel to the imaginary axis in the complex s-plane, and simultaneously minimizing a quadratic performance index. The design procedure mainly involves the solution of either Liapunov equations or Riccati equations. The general expression for finding the lower bound of a constant gain gamma is also developed.
01 Jan 1990
TL;DR: In this paper, for an infinite-dimensional linear quadratic control problem in Hilbert space, approximation of the solution of the algebraic Riccati operator equation in the strong operator topology is considered under conditions weaker than uniform exponential stability of the approximating systems.
Abstract: For an infinite-dimensional linear quadratic control problem in Hilbert space, approximation of the solution of the algebraic Riccati operator equation in the strong operator topology is considered under conditions weaker than uniform exponential stability of the approximating systems. As an application, strong onvergence of the approximating Riccati operators in case of a previously developed spline approximation scheme for delay systems is established. Finally, convergence of the transfer-functions of the approximating systems is investigated.
TL;DR: In this article, the authors describe a method that can be used to approximate the solution of the stochastic growth model by approximating the return and transition functions of the original problem by taking second-order and first-order Taylor expansions around the steady state of the system.
Abstract: I describe a method that can be used to approximate the solution of the stochastic growth model. The method relies on approximating the return and transition functions of the original problem by taking second-order and first-order Taylor expansions around the steady state of the system. The result is the optimal linear regulator problem.
TL;DR: In this paper, the periodic problem is reformulated as a prediction problem relative to a suitable time-invariant system, which enables one to establish a one-to-one correspondence between the solutions of the Discrete-time Periodic Riccati Equation (DPRE) and the solution of a suitable discrete-time and time-inariant RICE.
TL;DR: In this paper, the Riccati reaction-diffusion problem is solved by solving a linear algebraic system at each time step, as opposed to solving a nonlinear system which frequently happens when solving nonlinear partial differential equations.
Abstract: Two numerical methods, which do not bring contrived chaos into the solution, are proposed for the solution of the Riccati (logistic) equation. Though implicit in nature, with the resulting improvements in stability, the methods are applied explicitly. When extended to the numerical solution of Fisher’s equation, in which the quadratic polynomial representing the derivative in the Riccati equation appears as the reaction term, the solution is found by solving a linear system of algebraic equations at each time step, as opposed to solving a nonlinear system which frequently happens when solving nonlinear partial differential equations. The approaches adopted are extended to an ordinary differential equation in which the derivative is expressed as a cubic polynomial in the dependent variable. The solution of this initial-value problem is not available in closed form for finite values of the independent variable t . Under the conditions stated, numerical solutions are seen to converge to the correct steady-state solution. A nonlinear partial differential equation which governs the conduction of electrical impulses along a nerve axon and which has the aforementioned cubic polynomial as its reaction term, is solved by applying the numerical methods developed for solving the ordinary differential equation. The solution to this nonlinear reaction-diffusion equation is determined by solving a linear algebraic system at each time step.
TL;DR: In this article, the existence and construction of stabilizing compensators for linear time-invariant systems defined on Hilbert spaces are discussed and an existence result is established using Galerkin-type approximations in which independent basis elements are used instead of the complete set of eigenvectors.
Abstract: In this paper existence and construction of stabilizing compensators for linear time-invariant systems defined on Hilbert spaces are discussed. An existence result is established using Galerkin-type approximations in which independent basis elements are used instead of the complete set of eigenvectors. A design procedure based on approximate solutions of the optimal regulator and optimal observer via Galerkin-type approximation is given and the Schumacher approach is used to reduce the dimension of compensators. A detailed discussion for parabolic and hereditary differential systems is included.
05 Dec 1990
TL;DR: In this paper, a continuous-valued covariance, which is a function of a discrete-valued random variable, is used to characterize the tracking errors in an average sense, and the covariance can be calculated offline recursively from a modified Riccati equation, which can be obtained by replacing the measurementdependent terms in the original stochastic equation with their conditional expected values.
Abstract: An effective hybrid approach to the performance evaluation of the probabilistic data association (PDA) method for tracking in clutter is presented. In this approach, a continuous-valued covariance, which is a function of a discrete-valued random variable, is used to characterize the tracking errors in an average sense. This covariance can be calculated offline recursively from a modified Riccati equation, which can be obtained by replacing the measurement-dependent terms in the original stochastic equation with their conditional expected values. This approach has the merit of yielding a quantification of the transients of tracking divergence, as well as better accuracy than previous work. Such an approach is particularly suitable for stability studies of tracking filters. In addition, a quantitative study of the track life problem is conducted, in which the number of validated measurements play a central role. >
TL;DR: In this paper, the existence of maximal solution is proved for a generalized version of the standard algebraic Riccati equations which arise in certain stochastic optimal control problems, where the maximal solution can be obtained by a generalization of the Riemannian problem.
TL;DR: In this article, the discrete version of the Hirota equation, i ψ n +(1/2)[(α-β i )ψ n -1 +(α+β i ),ψn +1 ](1+γ|ψ | 2 | 2 )-αψ = 0, is proposed.
Abstract: The discrete version of the Hirota equation i ψ n +(1/2)[(α-β i )ψ n -1 +(α+β i )ψ n +1 ](1+γ|ψ n | 2 )-αψ n =0, is proposed. The 1-, 2- and N -soliton solutions for this equation are investigated.
TL;DR: The closed-loop Lyapunov operator is seen to be central to the question of whether Newton refinement will improve an approximate solution (region of convergence), as well as providing a means of bounding the actual error in terms of the residual error.
Abstract: Recent error bounds derived from the Schur method of solving algebraic Riccati equations (ARE) complement residual error bounds associated with Newton refinement of approximate solutions. These approaches to the problem of error estimation not only work well together but also represent the first computable error bounds for the solution of Riccati equations. In this paper the closed-loop Lyapunov operator is seen to be central to the question of whether Newton refinement will improve an approximate solution (region of convergence), as well as providing a means of bounding the actual error in terms of the residual error. In turn, both of these issues are related to the condition of the ARE and the damping of the associated closed-loop dynamical system. Numerical results are given for seven problems taken from the literature.
01 Jan 1990
TL;DR: In this paper, a geometric characterization of the set of all hermitian solutions of the discrete-time algebraic Riccati equation is given, and necessary and sufficient conditions for the existence of optimal control are given.
Abstract: This paper deals with the discrete-time infinite-horizon linear quadratic problem with indefinite cost criterion. Given a discrete-time linear system, an indefinite cost-functional and a linear subspace of the state space, we consider the problem of minimizing the cost-functional over all inputs that force the state trajectory to converge to the given subspace. We give a geometric characterization of the set of all hermitian solutions of the discrete-time algebraic Riccati equation. This characterization forms the discrete-time counterpart of the well-known geometric characterization of the set of all real symmetric solutions of the continuous-time algebraic Riccati equation as developed by Willems [IEEE Trans. Automat. Control, 16 (1971), pp. 621- 634] and Coppel [Bull. Austral. Math. Soc., 10 (1974), pp. 377-401]. In the set of all hermitian solutions of the Riccati equation we identify the solution that leads to the optimal cost for the above mentioned linear quadratic problem. Finally, we give necessary and sufficient conditions for the existence of optimal controls. Keywords: Discrete time optimal control, indefinite cost, algebraic Riccati equation, linear endpoint constraints.
TL;DR: Based on the generalized variational principle the analysis of a substructural chain is considered, and the 1:1 relationship between the structural analysis problem and the linear quadratic optimal control problem is introduced as mentioned in this paper.
TL;DR: In this article, a design methodology for the synthesis of a robust controller that accounts for both unmodeled dynamics and structured real-parameter uncertainty for multiple-input/multiple-output systems is presented.
Abstract: This paper presents a design methodology for the synthesis of a robust controller that accounts for both unmodeled dynamics and structured real-parameter uncertainty for multiple-input/multiple-output systems. The unmodeled dynamics are assumed to be characterized as a single block dynamic uncertainty at a point in the closed-loop system. In a design aimed at constraining both the H^ norm of a certain disturbance transfer matrix and a quadratic Gaussian performance index under their respective bounds, a surrogate system may be formed by modeling the structured real-parameter uncertainty as additional noise inputs and additional weights at the existing noise inputs and measurement outputs of the system. Application of a Riccati equation approach to this surrogate system then yields a robust controller that, when used in the actual system, will result in a closed-loop system that has the same H^ bound and quadratic Gaussian performance index bound as the surrogate system, even in the presence of given real-parameter variations.