Academic literature on the topic 'Algebraic Riccati equations'

In this paper we develop new and improved results in the numerical solution of the coupled algebraic Riccati equations. First we provide improved matrix upper bounds on the positive semidefinite solution of the unified coupled algebraic Riccati equations. Our approach is largely inspired by recent results established by Liu and Zhang. Our main results tighten the estimates of the relevant dominant eigenvalues. Also by relaxing the key restriction our upper bound applies to a larger number of situations. We also present an iterative algorithm to refine the new upper bounds and the lower bounds and numerically compute the solutions of the unified coupled algebraic Riccati equations. This construction follows the approach of Gao, Xue and Sun but we use different bounds. This leads to different analysis on convergence. Besides, we provide new matrix upper bounds for the positive semidefinite solution of the continuous coupled algebraic Riccati equations. By using an alternative primary assumption we present a new upper bound. We follow the idea of Davies, Shi and Wiltshire for the non-coupled equation and extend their results to the coupled case. We also present an iterative algorithm to improve our upper bounds. Finally we improve the classical Newton's method by the line search technique to compute the solutions of the continuous coupled algebraic Riccati equations. The Newton's method for couple Riccati equations is attributed to Salama and Gourishanar, but we construct the algorithm in a different way using the Fr\'echet derivative and we include line search too. Our algorithm leads to a faster convergence compared with the classical scheme. Numerical evidence is also provided to illustrate the performance of our algorithm.

In this dissertation we first derive a new unified upper solution bound for the continuous coupled algebraic Riccati equation, which arises from the optimal control of a Markovian jump linear system. In particular, we address the issue of rank deficiency with the control matrices. In the case of rank deficiency the existing matrix upper bounds are inapplicable. Moreover, our new result is not restricted to rank deficiency cases only. It now contains the existing results as special cases. Next, an iterative refinement is presented to improve our new unified matrix upper solution bounds. In particular, this iterative refinement determines a monotonically decreasing sequence of upper bounds for the solution of the continuous coupled algebraic Riccati equation. We formulate a new iterative algorithm by modifying this iterative refinement. We also prove that this new algorithm generates a monotonically decreasing sequence of matrix upper solution bounds that converges to the maximal solution of the continuous coupled algebraic Riccati equation. Furthermore, we prove the convergence of an accelerated Riccati iteration which computes a positive semidefinite solution of the continuous coupled algebraic Riccati equation. In particular, we establish sufficient conditions for the convergence of this algorithm. We also prove that for particular initial values this algorithm determines a monotonically increasing sequence of positive semidefinite matrices that converge to the minimal solution of the continuous coupled algebraic Riccati equation. Additionally, we show that for specific initial values this algorithm generates a monotonically decreasing sequence that converges to the maximal solution of the continuous coupled algebraic Riccati equation. In addition, we prove that this accelerated Riccati iteration converges faster than the Riccati iteration. Finally, we formulate a weighted modified accelerated Riccati iteration which is a more generalized Riccati type iteration. All of the existing Riccati iterations are now the special cases of this algorithm. Furthermore, we establish sufficient conditions for the convergence of this algorithm and we prove the monotonic convergence of the sequence generated by this algorithm. We also discuss how the weight and other quantities affect the rate of convergence of this algorithm. Illustrative numerical examples are also presented.

A collection of benchmark examples is presented for the numerical solution of continuous-time algebraic Riccati equations. This collection may serve for testing purposes in the construction of new numerical methods, but may also be used as a reference set for the comparison of methods.

This is the second part of a collection of benchmark examples for the numerical solution of algebraic Riccati equations. After presenting examples for the continuous-time case in Part I, our concern in this paper is discrete-time algebraic Riccati equations. This collection may serve for testing purposes in the construction of new numerical methods, but may also be used as a reference set for the comparison of methods.

In this paper introduce new approach for unified theory for continuous and discrete time (optimal) control problems based on the generalized Cayley transformation. We also relate the associated discrete and continuous generalized algebraic Riccati equations. We demonstrate the potential of this new approach proving new result for discrete algebraic Riccati equations. But we also discuss where this new approach as well as all other approaches still is non-satisfactory. We explain a discrepancy observed between the discrete and continuous cse and show that this discrepancy is partly due to the consideration of the wrong analogues. We also present an idea for a metatheorem that relates general theorems for discrete and continuous control problems.

We present new Fortran 77 subroutines which implement the Schur method and the matrix sign function method for the solution of the continuous­time matrix algebraic Riccati equation on the basis of LAPACK subroutines. In order to avoid some of the well­known difficulties with these methods due to a loss of accuracy, we combine the implementations with block scalings as well as condition estimates and forward error estimates. Results of numerical experiments comparing the performance of both methods for more than one hundred well­ and ill­conditioned Riccati equations of order up to 150 are given. It is demonstrated that there exist several classes of examples for which the matrix sign function approach performs more reliably and more accurately than the Schur method. In all cases the forward error estimates allow to obtain a reliable bound on the accuracy of the computed solution.

In this thesis we present theoretical and numerical results for a feedback control problem defined by a thermal fluid. The problem is motivated by recent interest in designing and controlling energy efficient building systems. In particular, we show that it is possible to locally exponentially stabilize the nonlinear Boussinesq Equations by applying Neumann/Robin type boundary control on a bounded and connected domain. The feedback controller is obtained by solving a Linear Quadratic Regulator problem for the linearized Boussinesq equations. Applying classical results for semilinear equations where the linear term generates an analytic semigroup, we establish that this Riccati-based optimal boundary feedback control provides a local stabilizing controller for the full nonlinear Boussinesq equations. In addition, we present a finite element Galerkin approximation. Finally, we provide numerical results based on standard Taylor-Hood elements to illustrate the theory.<br>Ph. D.

This work is devoted to investigating the algebraic and analytic integrability of first order polynomial partial differential equations via an understanding of the well-developed area of local and global integrability of polynomial vector fields. In the view of characteristics method, the search of first integrals of the first order partial differential equations P(x,y,z)∂z(x,y) ∂x +Q(x,y,z)∂z(x,y) ∂y = R(x,y,z), (1) is equivalent to the search of first integrals of the system of the ordinary differential equations dx/dt= P(x,y,z), dy/dt= Q(x,y,z), dz/dt= R(x,y,z). (2) The trajectories of (2) will be found by representing these trajectories as the intersection of level surfaces of first integrals of (1). We would like to investigate the integrability of the partial differential equation (1) around a singularity. This is a case where understanding of ordinary differential equations will help understanding of partial differential equations. Clearly, first integrals of the partial differential equation (1), are first integrals of the ordinary differential equations (2). So, if (2) has two first integrals φ1(x,y,z) =C1and φ2(x,y,z) =C2, where C1and C2 are constants, then the general solution of (1) is F(φ1,φ2) = 0, where F is an arbitrary function of φ1and φ2. We choose for our investigation a system with quadratic nonlinearities and such that the axes planes are invariant for the characteristics: this gives three dimensional Lotka– Volterra systems x' =dx/dt= P = x(λ +ax+by+cz), y' =dy/dt= Q = y(µ +dx+ey+ fz), z' =dz/dt= R = z(ν +gx+hy+kz), where λ,µ,ν 6= 0. v Several problems have been investigated in this work such as the study of local integrability and linearizability of three dimensional Lotka–Volterra equations with (λ:µ:ν)–resonance. More precisely, we give a complete set of necessary and sufficient conditions for both integrability and linearizability for three dimensional Lotka-Volterra systems for (1:−1:1), (2:−1:1) and (1:−2:1)–resonance. To prove their sufficiency, we mainly use the method of Darboux with the existence of inverse Jacobi multipliers, and the linearizability of a node in two variables with power-series arguments in the third variable. Also, more general three dimensional system have been investigated and necessary and sufficient conditions are obtained. In another approach, we also consider the applicability of an entirely different method which based on the monodromy method to prove the sufficiency of integrability of these systems. These investigations, in fact, mean that we generalized the classical centre-focus problem in two dimensional vector fields to three dimensional vector fields. In three dimensions, the possible mechanisms underling integrability are more difficult and computationally much harder. We also give a generalization of Singer’s theorem about the existence of Liouvillian first integrals in codimension 1 foliations in Cnas well as to three dimensional vector fields. Finally, we characterize the centres of the quasi-homogeneous planar polynomial differential systems of degree three. We show that at most one limit cycle can bifurcate from the periodic orbits of a centre of a cubic homogeneous polynomial system using the averaging theory of first order.