Showing papers on "Riccati equation published in 2002"
TL;DR: This book discusses Classical and Modern Control Optimization Optimal Control Historical Tour, Variational Calculus for Discrete-Time Systems, and more.
Abstract: INTRODUCTION Classical and Modern Control Optimization Optimal Control Historical Tour About This Book Chapter Overview Problems CALCULUS OF VARIATIONS AND OPTIMAL CONTROL Basic Concepts Optimum of a Function and a Functional The Basic Variational Problem The Second Variation Extrema of Functions with Conditions Extrema of Functionals with Conditions Variational Approach to Optimal Systems Summary of Variational Approach Problems LINEAR QUADRATIC OPTIMAL CONTROL SYSTEMS I Problem Formulation Finite-Time Linear Quadratic Regulator Analytical Solution to the Matrix Differential Riccati Equation Infinite-Time LQR System I Infinite-Time LQR System II Problems LINEAR QUADRATIC OPTIMAL CONTROL SYSTEMS II Linear Quadratic Tracking System: Finite-Time Case LQT System: Infinite-Time Case Fixed-End-Point Regulator System Frequency-Domain Interpretation Problems DISCRETE-TIME OPTIMAL CONTROL SYSTEMS Variational Calculus for Discrete-Time Systems Discrete-Time Optimal Control Systems Discrete-Time Linear State Regulator Systems Steady-State Regulator System Discrete-Time Linear Quadratic Tracking System Frequency-Domain Interpretation Problems PONTRYAGIN MINIMUM PRINCIPLE Constrained Systems Pontryagin Minimum Principle Dynamic Programming The Hamilton-Jacobi-Bellman Equation LQR System using H-J-B Equation CONSTRAINED OPTIMAL CONTROL SYSTEMS Constrained Optimal Control TOC of a Double Integral System Fuel-Optimal Control Systems Minimum Fuel System: LTI System Energy-Optimal Control Systems Optimal Control Systems with State Constraints Problems APPENDICES Vectors and Matrices State Space Analysis MATLAB Files REFERENCES INDEX
1,259 citations
TL;DR: In this paper, a general solution for a fractional diffusion-wave equation defined in a bounded space domain is given, where the response expressions are written in terms of the Mittag-Leffler functions.
Abstract: A general solution is given for a fractional diffusion-wave equation defined in a bounded space domain. The fractional time derivative is described in the Caputo sense. The finite sine transform technique is used to convert a fractional differential equation from a space domain to a wavenumber domain. Laplace transform is used to reduce the resulting equation to an ordinary algebraic equation. Inverse Laplace and inverse finite sine transforms are used to obtain the desired solutions. The response expressions are written in terms of the Mittag–Leffler functions. For the first and the second derivative terms, these expressions reduce to the ordinary diffusion and wave solutions. Two examples are presented to show the application of the present technique. Results show that for fractional time derivatives of order 1/2 and 3/2, the system exhibits, respectively, slow diffusion and mixed diffusion-wave behaviors.
470 citations
TL;DR: It is demonstrated that the efficient frontier in the mean-standard deviation diagram is still a straight line or, equivalently, risk-free investment is still possible, even when the interest rate is random.
Abstract: This paper concerns the continuous-time, mean-variance portfolio selection problem in a complete market with random interest rate, appreciation rates, and volatility coef.cients. The problem is tackled using the results of stochastic linear-quadratic (LQ) optimal control and backward stochastic differential equations (BSDEs), two theories that have been extensively studied and developed in recent years. Specifically, the mean-variance problem is formulated as a linearly constrained stochastic LQ control problem. Solvability of this LQ problem is reduced, in turn, to proving global solvability of a stochastic Riccati equation. The proof of existence and uniqueness of this Riccati equation, which is a fully nonlinear and singular BSDE with random coefficients, is interesting in its own right and relies heavily on the structural properties of the equation. Efficient investment strategies as well as the mean-variance efficient frontier are then analytically derived in terms of the solution of this equation. In particular, it is demonstrated that the efficient frontier in the mean-standard deviation diagram is still a straight line or, equivalently, risk-free investment is still possible, even when the interest rate is random. Finally, a version of the Mutual Fund Theorem is presented.
271 citations
TL;DR: A design method of the reduced-order observer that is dependent on the solution of the Riccati equation is presented and an example is given to illustrate effects of the design method.
Abstract: This note deals with the design of reduced-order observers for Lipschitz nonlinear systems. It shows that the conditions under which a full-order observer exists also guarantee the existence of a reduced-order observer. A design method of the reduced-order observer that is dependent on the solution of the Riccati equation is then presented and an example is given to illustrate effects of the design method.
250 citations
27 Feb 2002
TL;DR: In this article, the Routh-Hurwitz and Schur-Cohn Criterion Nonlinear Systems and Stability Limit Sets and Invariant Manifolds Dissipative Maps Stability of Difference Equations Semicycle Analysis Dynamica Session on Semicycles Exercises InvARIANTS and RELATED LYAPUNOV FUNCTIONS Introduction Invariants for Linear Equations and Systems Invarantants and Corresponding Lyapunov Functions for Nonlinear systems Invariance under Lie Group Transformations Exercise DYNAMICS of Three-DIMENSION
Abstract: DYNAMICS OF ONE-DIMENSIONAL DYNAMICAL SYSTEMS Introduction Linear Difference Equations with Constant Coefficients Linear Difference Equations with Variable Coefficients Stability Stability in the Non-Hyperbolic Case Bifurcations Dynamica Session Symbolic Dynamics for One-Dimensional Maps Dissipative Maps and Global Attractivity Parametrisation and Poincare Functional Equation Exercises DYNAMICS OF TWO-DIMENSIONAL DYNAMICAL SYSTEMS Introduction Linear Theory Equilibrium Solutions The Riccati Equation Linearized Stability Analysis Dynamica Session Period Doubling Bifurcation Lyapunov Numbers Box Dimension Semicycle Analysis Stable and Unstable Manifold Dynamica Session on Henon's Equation Invariants Lyapunov Functions, Stability, and Invariants Dynamica Session on Lyness' Map Dissipative Maps and Systems Dynamica Session on Rational Difference Equations Area-preserving Maps and Systems Biology Applications Projects Applications in Economics Exercises SYSTEMS OF DIFFERENCE EQUATIONS, STABILITY, AND SEMICYCLES Introduction Linear Theory Stability of Linear Systems The Routh-Hurwitz and Schur-Cohn Criterion Nonlinear Systems and Stability Limit Sets and Invariant Manifolds Dissipative Maps Stability of Difference Equations Semicycle Analysis Dynamica Session on Semicycles Exercises INVARIANTS AND RELATED LYAPUNOV FUNCTIONS Introduction Invariants for Linear Equations and Systems Invariants and Corresponding Lyapunov Functions for Nonlinear Systems Invariants of Special Class of Difference Equations Applications Dynamica Session on Invariants Dynamica Session on Lyapunov Functions Invariance under Lie Group Transformations Exercises DYNAMICS OF THREE-DIMENSIONAL DYNAMICAL SYSTEMS Introduction Dynamica Session on Third Order Difference Equations Dissipative Difference Equation of Third Order Dynamica Session on Local Asymptotic Stability of Period-Two Solution Dynamica Session on Todd's Difference Equation Biology Applications Projects Exercises FRACTALS GENERATED BY ITERATED FUNCTIONS SYSTEMS Introduction Basic Definitions and Results Iterated Function System Basic Results on Iterated Functions Systems Calculation of Box Dimension for IFS Dynamica Session Exercises BIBLIOGRAPHY INDEX
208 citations
08 May 2002
TL;DR: State-dependent Riccati equation (SDRE) techniques are general design methods which provide a systematic and effective means of designing nonlinear controllers, observers, and filters as discussed by the authors.
Abstract: State-dependent Riccati equation (SDRE) techniques are general design methods which provide a systematic and effective means of designing nonlinear controllers, observers, and filters. The paper provides an overview of the capabilities of SDRE control and goes into detail concerning the art of carrying out effective SDRE designs for both systems that conform and do not conform to the basic structure and conditions required by the method. The paper is centered around the SDRE nonlinear regulator. The following situations which prevent a straightforward application of the SDRE method to the control problem at hand are addressed: the existence of state-independent terms, the existence of state-dependent terms which do not go to zero as the state vector goes to zero, the existence of nonlinearity in the controls, and the existence of uncontrollable and unstable but bounded state dynamics.
173 citations
TL;DR: In this paper, a generalized difference Riccati equation is introduced and it is proved that its solvability is necessary and sufficient for the existence of an optimal control which can be either of state feedback or open-loop form.
Abstract: This paper deals with the discrete-time stochastic LQ problem involving state and control dependent noises, whereas the weighting matrices in the cost function are allowed to be indefinite. In this general setting, it is shown that the well-posedness and the attainability of the LQ problem are equivalent. Moreover, a generalized difference Riccati equation is introduced and it is proved that its solvability is necessary and sufficient for the existence of an optimal control which can be either of state feedback or open-loop form. Furthermore, the set of all optimal controls is identified in terms of the solution to the proposed difference Riccati equation.
166 citations
TL;DR: In this paper, a simplified version of the Gross-Pitaevskii equation is used to describe the growth of vortex lattices and nonlinear losses in a hydrogen condensate.
Abstract: We show how to adapt the ideas of local energy and momentum conservation in order to derive modifications to the Gross-Pitaevskii equation which can be used phenomenologically to describe irreversible effects in a Bose-Einstein condensate. Our approach involves the derivation of a simplified quantum kinetic theory, in which all processes are treated locally. It is shown that this kinetic theory can then be transformed into a number of phase-space representations, of which the Wigner function description, although approximate, is shown to be the most advantageous. In this description, the quantum kinetic master equation takes the form of a Gross-Pitaevskii equation with noise and damping added according to a well defined prescription - an equation we call the stochastic Gross-Pitaevskii equation. From this, a very simplified description we call the phenomenological growth equation can be derived. We use this equation to study?(i) the nucleation and growth of vortex lattices, and?(ii) nonlinear losses in a hydrogen condensate, which it is shown can lead to a curious instability phenomenon.
165 citations
TL;DR: It is shown that the open-loop solution to the two- person leader-follower stochastic differential game admits a state feedback representation if a new stochastically Riccati equation is solvable.
Abstract: A leader-follower stochastic differential game is considered with the state equation being a linear Ito-type stochastic differential equation and the cost functionals being quadratic. We allow that the coefficients of the system and those of the cost functionals are random, the controls enter the diffusion of the state equation, and the weight matrices for the controls in the cost functionals are not necessarily positive definite. The so-called open-loop strategies are considered only. Thus, the follower first solves a stochastic linear quadratic (LQ) optimal control problem with the aid of a stochastic Riccati equation. Then the leader turns to solve a stochastic LQ problem for a forward-backward stochastic differential equation. If such an LQ problem is solvable, one obtains an open-loop solution to the two-person leader-follower stochastic differential game. Moreover, it is shown that the open-loop solution admits a state feedback representation if a new stochastic Riccati equation is solvable.
158 citations
TL;DR: In this article, the authors survey recent and also older results on nonsymmetric matrix Riccati differential equations and their corresponding algebraic variants, and cite various applications connected with matrix RICCati equations.
147 citations
TL;DR: In this paper, a direct and unified algorithm for constructing multiple travelling wave solutions of nonlinear partial differential equations (PDEs) is presented and implemented in a computer algebraic system.
Abstract: In this paper, a direct and unified algorithm for constructing multiple travelling wave solutions of nonlinear partial differential equations (PDEs) is presented and implemented in a computer algebraic system. 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 in the tanh method. It is quite interesting that the sign of the parameter can be used to exactly judge the number and types of such travelling wave solutions. In this way, we can successfully recover the previously known solitary wave solutions that had been found by the tanh method and other more sophisticated methods. More importantly, for some equations, with no extra effect we also find other new and more general solutions at the same time. By introducing appropriate transformations, our method is further extended to the nonlinear PDEs whose balancing numbers may be any nonzero real numbers. The efficiency of the method can be demonstrated on a large variety of nonlinear PDEs such as those considered in this paper, Burgers-Huxley equation, coupled Korteweg-de Vries equation, Caudrey-Dodd-Gibbon-Kawada equation, active-dissipative dispersive media equation, generalized Fisher equation, and nonlinear heat conduction equation.
TL;DR: In this article, the global existence and uniqueness result for a general one-dimensional backward stochastic Riccati equation was obtained and applied to the mean-variance hedging problem.
TL;DR: Two numerical examples, the classical Earth-Mars orbit transfer in minimal time and the Rayleigh problem in electrical engineering, demonstrate that the Riccati equation approach provides a viable numerical test of second order sufficient conditions (SSC).
Abstract: Second order sufficient conditions (SSC) for control problems with control-state constraints and free final time are presented. Instead of deriving such SSC from first principles, we transform the control problem with free final time into an augmented control problem with fixed final time for which well-known SSC exist. SSC are then expressed as a condition on the positive definiteness of the second variation. A convenient numerical tool for verifying this condition is based on the Riccati approach, where one has to find a bounded solution of an associated Riccati equation satisfying specific boundary conditions. The augmented Riccati equations for the augmented control problem are derived, and their modifications on the boundary of the control-state constraint are discussed. Two numerical examples, (1) the classical Earth-Mars orbit transfer in minimal time and (2) the Rayleigh problem in electrical engineering, demonstrate that the Riccati equation approach provides a viable numerical test of SSC.
TL;DR: In this article, the authors obtain oscillation criteria for a second-order self-adjoint matrix differential equation on a measure chain in terms of the eigenvalues of the coefficient matrices and the graininess function.
TL;DR: In this article, the authors apply the theory of measure chains to investigate the oscillation and nonoscillation of the above equation on the basis of some well-known results, and in some sense, they show a method to unify the delay differential equation and delay difference equation.
TL;DR: In this article, a 4-vector ζ is constructed out of a combination of scalar and vector products of the vorticity ω and the vortex stretching vector ω ·∇ u = Sω.
TL;DR: In this article, a general method for the construction of kinetic schemes of evolutionary equations is illustrated with the simple example of the linear advection equation, where the role of the collision effect is clarified theoretically and numerically.
TL;DR: In this article, a linear program for an ordinary differential equation is presented, of which a feasible solution defines a continuous piecewise affine linear Lyapunov function for the differential equation.
Abstract: An algorithm that derives a linear program for an ordinary differential equation is presented, of which a feasible solution defines a continuous piecewise affine linear Lyapunov function for the differential equation. The linear program can be generated for an arbitrary region containing an equilibrium of the differential equation. The domain of the Lyapunov function is the region used in the generation of the linear program. The Lyapunov function secures the asymptotic stability of the equilibrium and gives a lower bound on its region of attraction.
TL;DR: In this article, an explicit procedure for obtaining the equation of motion for the Wigner distribution when the underlying governing equation is a linear ordinary or partial differential equation is presented, and the cases of constant and variable coefficients are considered.
10 Dec 2002
TL;DR: In this article, a nonlinear control synthesis technique (/spl theta/ - D approximation) is presented, which achieves suboptimal solutions to nonlinear optimal control problems in the sense that it solves the Hamilton-Jacobi-Bellman (HJB) equation approximately by adding perturbations to the cost function.
Abstract: In this paper, a new nonlinear control synthesis technique (/spl theta/ - D approximation) is presented. This approach achieves suboptimal solutions to nonlinear optimal control problems in the sense that it solves the Hamilton-Jacobi-Bellman (HJB) equation approximately by adding perturbations to the cost function. By manipulating the perturbation terms both semi-globally asymptotic stability and suboptimality properties can be obtained. The convergence and stability proofs are given. This method overcomes the large control for large initial states problem that occurs in some other Taylor expansion based methods. It does not need time-consuming online computations like the state dependent Riccati equation (SDRE) technique. A vector problem is investigated to demonstrate the effectiveness of this new technique.
TL;DR: In this article, the robust linear filtering of hybrid discrete-time Markovian jump linear systems is considered and a linear matrix inequalities (LMI) formulation is proposed to solve the problem.
Abstract: In this paper we consider the robust linear filtering of hybrid discrete-time Markovian jump linear systems. We assume that only an output of the system is available, and therefore the values of the jump parameter are not known. It is desired to design a dynamic linear filter such that the closed loop system is mean square stable and minimizes the stationary expected value of the square error. We consider uncertainties on the parameters of the possible modes of operation of the system. A linear matrix inequalities (LMI) formulation is proposed to solve the problem. For the case in which there are no uncertainties on the modes of operation of the system, we show that the LMI formulation provides a filter with the same stationary mean square error as the one obtained from the Riccati equation approach.
TL;DR: An analogy to barotropic Friedmann-Robertson-Lemaitre cosmology is shown where the expansion of the universe can be also shown to obey a Riccati equation.
Abstract: We discuss two applications of a Riccati equation to Newton's laws of motion. The first one is the motion of a particle under the influence of a power law central potential V(r)=kr(epsilon). For zero total energy we show that the equation of motion can be cast in the Riccati form. We briefly show here an analogy to barotropic Friedmann-Robertson-Lemaitre cosmology where the expansion of the universe can be also shown to obey a Riccati equation. A second application in classical mechanics, where again the Riccati equation appears naturally, are problems involving quadratic friction. We use methods reminiscent to nonrelativistic supersymmetry to generalize and solve such problems.
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TL;DR: In this paper, the authors introduced the concept of unbounded solutions to the Riccati equation and gave a complete description of its solutions associated with the spectral graph subspaces of the block operator matrix.
Abstract: We introduce a new concept of unbounded solutions to the operator Riccati equation $A_1 X - X A_0 - X V X + V^\ast = 0$ and give a complete description of its solutions associated with the spectral graph subspaces of the block operator matrix $\mathbf{B} = \begin{pmatrix} A_0 & V V^\ast & A_1 \end{pmatrix}$. We also provide a new characterization of the set of all contractive solutions under the assumption that the Riccati equation has a contractive solution associated with a spectral subspace of the operator $\mathbf{B}$. In this case we establish a criterion for the uniqueness of contractive solutions.
TL;DR: In this article, an exact multimodal formalism is proposed for acoustic propagation in three-dimensional rigid bends of circular cross-section, and two infinite systems of first-order differential equations are constructed for the components of the pressure and axial velocity in the bend, projected on the local transverse modes.
TL;DR: In this paper, the use of state dependent differential Riccati equations and numerical integration to propagate their solutions forward in time is explored, and examples illustrating the usefulness of these methods are given.
TL;DR: In this article, rational transformations of independent variables of linear matrix ODEs with the Schlesinger transformations (RS-transformations) are used to construct algebraic solutions of the sixth Painleve equation.
Abstract: Compositions of rational transformations of independent variables of linear matrix ordinary differential equations (ODEs) with the Schlesinger transformations (RS-transformations) are used to construct algebraic solutions of the sixth Painleve equation. RS-Transformations of the ranks 3 and 4 of 2 × 2 matrix Fuchsian ODEs with 3 singular points into analogous ODE with 4 singular points are classified.
TL;DR: In this paper, a qualitative analysis of a two-dimensional plane autonomous system which is equivalent to an approximate sine-Gordon equation is given using the qualitative theory of ordinary differential equations.
18 Oct 2002
TL;DR: For the nonsymmetric algebraic Riccati equation, the authors proved that the minimal non-negative solution is positive when the M-matrix is irreducible.
TL;DR: In this article, the authors study the 2D surface quasi-geostrophic equation for thermal active scalars, which is an equation arising in the study of fast rotating fluids, and consider the problem of convergence of solutions of the viscous problem to the ones of the inviscid problem.
Abstract: We study the 2D surface quasi-geostrophic equation for thermal active scalars, that is an equation arising in the study of fast rotating fluids. This equation is a model problem for the 3D Euler equation. We consider the problem of convergence of solutions of the viscous problem to the ones of the inviscid problem. We also consider the long time behavior and we discuss the various kind of attractors that make sense for the quasi-geostrophic equation.