Showing papers on "Riccati equation published in 2001"

Dynamic Mean-Variance Portfolio Selection with No-Shorting Constraints

Abstract: This paper is concerned with mean-variance portfolio selection problems in continuous-time under the constraint that short-selling of stocks is prohibited. The problem is formulated as a stochastic optimal linear-quadratic (LQ) control problem. However, this LQ problem is not a conventional one in that the control (portfolio) is constrained to take nonnegative values due to the no-shorting restriction, and thereby the usual Riccati equation approach (involving a "completion of squares") does not apply directly. In addition, the corresponding Hamilton--Jacobi--Bellman (HJB) equation inherently has no smooth solution. To tackle these difficulties, a continuous function is constructed via two Riccati equations, and then it is shown that this function is a viscosity solution to the HJB equation. Solving these Riccati equations enables one to explicitly obtain the efficient frontier and efficient investment strategies for the original mean-variance problem. An example illustrating these results is also presented.

A survey of spectral factorization methods

Abstract: Spectral factorization is a crucial step in the solution of linear quadratic estimation and control problems. It is no wonder that a variety of methods has been developed over the years for the computation of canonical spectral factors. This paper provides a survey of several of these methods with special emphasis on clarifying the connections that exist among them. While the discussion focuses primarily on scalar-valued rational spectra, extensions to nonrational and vector-valued spectra are briefly noted. Copyright c 2001 John Wiley & Sons, Ltd.

Indefinite Stochastic Linear Quadratic Control and Generalized Differential Riccati Equation

Abstract: A stochastic linear quadratic (LQ) control problem is indefinite when the cost weighting matrices for the state and the control are allowed to be indefinite. Indefinite stochastic LQ theory has been extensively developed and has found interesting applications in finance. However, there remains an outstanding open problem, which is to identify an appropriate Riccati-type equation whose solvability is {\it equivalent} to the solvability of the indefinite stochastic LQ problem. This paper solves this open problem for LQ control in a finite time horizon. A new type of differential Riccati equation, called the generalized (differential) Riccati equation, is introduced, which involves algebraic equality/inequality constraints and a matrix pseudoinverse. It is then shown that the solvability of the generalized Riccati equation is not only sufficient, but also necessary, for the well-posedness of the indefinite LQ problem and the existence of optimal feedback/open-loop controls. Moreover, all of the optimal controls can be identified via the solution to the Riccati equation. An example is presented to illustrate the theory developed.

Discrete-time indefinite LQ control with state and control dependent noises

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. We show that the well-posedness and the attainability of the LQ problem are equivalent. Furthermore, the set of all optimal controls is identified in terms of the solution to a generalized difference Riccati equation.

Solvability and asymptotic behavior of generalized Riccati equations arising in indefinite stochastic LQ controls

Abstract: The optimal control problem in a finite time horizon with an indefinite quadratic cost function for a linear system subject to multiplicative noise on both the state and control can be solved via a constrained matrix differential Riccati equation. In this paper, we provide general necessary and sufficient conditions for the solvability of this generalized differential Riccati equation. Furthermore, its asymptotic behavior is investigated along with its connection to the generalized algebraic Riccati equation associated with the linear quadratic control problem in finite time horizon. Examples are presented to illustrate the results established.

Critical thresholds in euler-poisson equations

Abstract: We present a preliminary study of a new phenom- ena associated with the Euler-Poisson equations — the so called critical threshold phenomena, where the answer to questions of global smoothness vs. finite time breakdown depends on whether the initial configuration crosses an intrinsic,ON1O critical thresh- old. We investigate a class of Euler-Poisson equations, ranging from one-dimensional problems with or without various forcing mech- anisms to multi-dimensional isotropic models with geometrical symmetry. These models are shown to admit a critical thresh- old which is reminiscent of the conditional breakdown of waves on the beach; only waves above certain initial critical threshold experience finite-time breakdown, but otherwise they propagate smoothly. At the same time, the asymptotic long time behavior of the solutions remains the same, independent of crossing these initial thresholds. A case in point is the simple one-dimensional problem where the unforced inviscid Burgers' solution always forms a shock dis- continuity, except for the non-generic case of increasing initial profile,u 0 0. In contrast, we show that the corresponding one- dimensional Euler-Poisson equation with zero background has global smooth solutions as long as its initialN0;u 0O-configuration satisfiesu 0 p 2k0 - see (2.11) below, allowing a finite, crit- ical negative velocity gradient. As is typical for such nonlinear convection problems, one is led to a Ricatti equation which is balanced here by a forcing acting as a 'nonlinear resonance', and which in turn is responsible for this critical threshold phenom- ena.

Nonsymmetric Algebraic Riccati Equations and Wiener--Hopf Factorization for M -Matrices

Abstract: We consider the nonsymmetric algebraic Riccati equation for which the four coefficient matrices form an M-matrix. Nonsymmetric algebraic Riccati equations of this type appear in applied probability and transport theory. The minimal nonnegative solution of these equations can be found by Newton's method and basic fixed-point iterations. The study of these equations is also closely related to the so-called Wiener--Hopf factorization for M-matrices. We explain how the minimal nonnegative solution can be found by the Schur method and compare the Schur method with Newton's method and some basic fixed-point iterations. The development in this paper parallels that for symmetric algebraic Riccati equations arising in linear quadratic control.

Solving a Quadratic Matrix Equation by Newton's Method with Exact Line Searches

Abstract: We show how to incorporate exact line searches into Newton's method for solving the quadratic matrix equation AX2 + BX + C = 0, where A, B and C are square matrices. The line searches are relatively inexpensive and improve the global convergence properties of Newton's method in theory and in practice. We also derive a condition number for the problem and show how to compute the backward error of an approximate solution.

On the extended applications of Homogenous Balance Method

Abstract: In this paper, we study the travelling wave reductions for certain (2+1)- and (3+1)-dimensional physically important nonlinear evolutionary equations by using the recently proposed Homogenous Balance Method (HBM). Through this analysis we explore certain new solutions for the equations we have studied.

Contributions to the Theory of Optimal Control

Abstract: This chapter contains sections titled: Introduction Notation and Terminology Statement of Problem Relations With the Calculus of Variations Controllability Solution of the Linear Regulator Problem Stability of the Riccati Equation General Solution of the Riccati Equation Bibliography Appendix: The Generalized Inverse of a Matrix

Decomposition of the (2 + 1)- dimensional Gardner equation and its quasi-periodic solutions

Abstract: To decompose the (2 + 1)-dimensional Gardner equation, an isospectral problem and a corresponding hierarchy of (1 + 1)-dimensional soliton equations are proposed. The (2 + 1)-dimensional Gardner equation is separated into the first two non-trivial (1 + 1)-dimensional soliton systems in the hierarchy, and in turn into two new compatible Hamiltonian systems of ordinary differential equations. Using the generating function flow method, the involutivity and the functional independence of the integrals are proved. The Abel-Jacobi coordinates are introduced to straighten out the associated flows. The Riemann-Jacobi inversion problem is discussed, from which quasi-periodic solutions of the (2 + 1)-dimensional Gardner equation are obtained by resorting to the Riemann theta functions.

Linear-Quadratic Control of Backward Stochastic Differential Equations

Abstract: This paper is concerned with optimal control of linear backward stochastic differential equations (BSDEs) with a quadratic cost criteria, or backward linear-quadratic (BLQ) control. The solution of this problem is obtained completely and explicitly by using an approach which is based primarily on the completion-of-squares technique. Two alternative, though equivalent, expressions for the optimal control are obtained. The first of these involves a pair of Riccati-type equations, an uncontrolled BSDE, and an uncontrolled forward stochastic differential equation (SDE), while the second is in terms of a Hamiltonian system. Contrary to the deterministic or stochastic forward case, the optimal control is no longer a feedback of the current state; rather, it is a feedback of the entire history of the state. A key step in our derivation is a proof of global solvability of the aforementioned Riccati equations. Although of independent interest, this issue has particular relevance to the BLQ problem since these Riccati equations play a central role in our solution. Last but not least, it is demonstrated that the optimal control obtained coincides with the solution of a certain forward linear-quadratic (LQ) problem. This, in turn, reveals the origin of the Riccati equations introduced.

Stochastic Linear Quadratic Optimal Control Problems

Abstract: This paper is concerned with the 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. Some intrinsic relations among the LQ problem, the stochastic maximum principle, and the (linear) forward—backward stochastic differential equations are established. Some results involving Riccati equation are discussed as well.

Reduced order controllers for Burgers' equation with a nonlinear observer

Abstract: A method for reducing controllers for systems described by partial differential equations (PDEs) is applied to Burgers’ equation with periodic boundary conditions. This approach differs from the typical approach of reducing the model and then designing the controller, and has developed over the past several years into its current form. In earlier work it was shown that functional gains for the feedback control law served well as a dataset for reduced order basis generation via the proper orthogonal decomposition (POD). However, the test problem was the two-dimensional heat equation, a problem in which the physics dominates the system in such a way that controller efficacy is difficult to generalize. Here, we additionally incorporate a nonlinear observer by including the nonlinear terms of the state equation in the differential equation for the compensator.

Approaches to quadratic stabilization of uncertain fuzzy dynamic systems

Abstract: A new approach to quadratic stabilization of uncertain fuzzy dynamic systems is developed in this paper. This uncertain fuzzy dynamic model is used to represent a class of uncertain continuous time complex nonlinear systems which include both linguistic information and system uncertainties. It is shown that the uncertain fuzzy dynamic system can be stabilized if a suitable Riccati equation or a set of Riccati equations has solutions. Constructive algorithms are also developed to obtain the stabilization feedback control laws. Finally, an example is given to illustrate the application of the proposed method.

Quadratic stabilization of uncertain discrete-time fuzzy dynamic systems

Abstract: New approaches to quadratic stabilization of uncertain discrete-time fuzzy dynamic systems are developed in this paper. This uncertain fuzzy dynamic model is used to represent a class of uncertain discrete-time complex nonlinear systems which include both linguistic information and system uncertainties. It is shown that the uncertain fuzzy dynamic system is stabilizable if a suitable Riccati equation or a set of Riccati equations have solutions. Constructive algorithms are also developed to obtain the stabilization feedback control laws. Finally, an example is given to illustrate the application of the proposed method.

Experimental real-time SDRE control of an underactuated robot

Abstract: In this paper, State-Dependent Riccati Equation (SDRE) nonlinear control is used to regulate the Pendubot, a two-link underactuated robot developed at UIUC, at one of its unstable equilibrium positions. The plant is not fully feedback linearizable due to weakly nonminimum phase zero-dynamics. However, this does not present a problem for SDRE control. Presented results show that SDRE control outperforms LQR control even when the same design parameters are used. The real-time experiments are implemented using a specialized 60 MHz digital signal processor running customized code for SDRE control computation, which involves solving the Algebraic Riccati Equation online. For this highly nonlinear 4-state system, the implementation was successful at a sampling time sufficient for the particular electromechanical plant. To the authors' best knowledge, this is the first work in which SDRE control is implemented on a physical plant in realtime, where the SDRE control is repeatedly computed online.

Existence, Uniqueness, and Parametrization of Lagrangian Invariant Subspaces

Abstract: The existence, uniqueness, and parametrization of Lagrangian invariant subspaces for Hamiltonian matrices is studied. Necessary and sufficient conditions and a complete parametrization are given. Some necessary and sufficient conditions for the existence of Hermitian solutions of algebraic Riccati equations follow as simple corollaries.

Wheel slip control in ABS brakes using gain scheduled constrained LQR

Abstract: A wheel slip controller for ABS brakes is formulated using an explicit constrained LQR design. The controller gain matrices are designed and scheduled on the vehicle speed based on local linearizations. A Lyapunov function for the nonlinear control system is dervied using the Riccati equation solution in order to prove stability and robustness with respect to uncertainty in the road/tyre friction characteristic. Experimental results from a test vehicle with electromechanical brake actuators and brake-by-wire show that high performance and robustness are achieved.

Model predictive neural control with applications to a 6 DOF helicopter model

Abstract: We present a method for optimal control of MIMO non-linear systems based on a combination of a neural network (NN) feedback controller and a state-dependent Riccati equation (SDRE) controller. Optimization of the NN is performed within a receding horizon model predictive control (MPC) framework, subject to dynamic and kinematic constraints. The SDRE controller augments the NN controller by providing an initial feasible solution and improving stability. The resulting technique is applied to a 6 degree of freedom (DOF) model of an autonomous helicopter.

Stochastic Linear-Quadratic Control via Semidefinite Programming

Abstract: We study stochastic linear-quadratic (LQ) optimal control problems over an infinite time horizon, allowing the cost matrices to be indefinite. We develop a systematic approach based on semidefinite programming (SDP). A central issue is the stability of the feedback control; and we show this can be effectively examined through the complementary duality of the SDP. Furthermore, we establish several implication relations among the SDP complementary duality, the (generalized) Riccati equation, and the optimality of the LQ control problem. Based on these relations, we propose a numerical procedure that provides a thorough treatment of the LQ control problem via primal-dual SDP: it identifies a stabilizing feedback control that is optimal or determines that the problem possesses no optimal solution. For the latter case, we develop an $\epsilon$-approximation scheme that is asymptotically optimal.

An approach for solving general switched linear quadratic optimal control problems

Abstract: This paper successfully addresses an important class of hybrid optimal control problems of practical significance. It provides a viable general approach to hybrid optimal control based on nonlinear optimization and it shows that when this approach is applied to linear quadratic problems it leads to computationally attractive algorithms. Unlike conventional optimal control problems, optimal control problems for switched systems require the solutions of not only optimal continuous inputs but also optimal switching sequences. Many practical problems only involve optimization where the number of switchings and the sequence of active subsystems are given. This is stage 1 of the two stage optimization method proposed by the authors in previous papers. In order to solve stage 1 problems using efficient nonlinear optimization techniques, the derivatives of the optimal cost with respect to the switching instants need to be known. In this paper, we focus on and solve a special class of optimal control problems, namely, general switched linear quadratic problems. The approach first transcribes a stage 1 problem into an equivalent problem parameterized by the switching instants and then obtains the derivative values based on the solution of an initial value ordinary differential equation formed by the general Riccati equation and its differentiations. Examples illustrate the results.

Conditional similarity reduction approach: Jimbo-Miwa equation

Abstract: The direct method developed by Clarkson and Kruskal (1989 J. Math. Phys. 30 2201) for finding the symmetry reductions of a nonlinear system is extended to find the conditional similarity solutions. Using the method of the Jimbo-Miwa (JM) equation, we find that three well-known (2+1)-dimensional models - the asymmetric Nizhnik-Novikov-Veselov equation, the breaking soliton equation and the Kadomtsev-Petviashvili equation - can all be obtained as the conditional similarity reductions of the JM equation.

Sound propagation in rigid bends: a multimodal approach.

Abstract: The sound propagation in a waveguide with bend of finite constant curvature is analyzed using multimodal decomposition. Two infinite first-order differential equations are constructed for the pressure and velocity in the bend, projected on the local transverse modes. A Riccati equation for the impedance matrix is then derived, which can be numerically integrated after truncation at a sufficient number of modes. An example of validation is considered and results show the accuracy of the method and its suitability for the formulation of radiation conditions. Reflection and transmission coefficients are also computed, showing the importance of higher order mode generation at the junction between the bend and the straight ducts. The case of varying cross-section curved ducts is also considered using multimodal decomposition.