A Riccati equation for block-diagonalization of ill-conditioned systems

A simple transformation, originally introduced for singularly perturbed systems, is now applicable to a larger class of time-invariant systems.     

A reformulation of the algebraic Riccati equation problem

The algebraic Riccati equation problem is reformulated so as to yield a simple solution when the system has only real roots, as may occur when using a spatially quantized distributed parameter model. A restriction is also placed on the choice of the synthetic output matrixC.     

A new formulation of the algebraic Riccati equation problem

A new formulation of the algebraic Riccati equation is presented and a closed form solution is obtained under the hypothesis that the coefficient matrices fulfill certain commutativity conditions on a transformed space. The formulation extends the results of [1] to a wider class of Riccati problems.     

A nonrecursive algebraic solution for the discrete Riccati equation

Equations for the optimal linear control and filter gains for linear discrete systems with quadratic performance criteria are widely documented. A nonrecursive algebraic solution for the Riccati equation is presented. These relations allow the determination of the steady-state solution of the Riccati equation directly without iteration. The relations also allow the direct determination of the transient solution for any particular time without proceeding recursively from the initial conditions. The method involves finding the eigenvalues and eigenvectors of the canonical state-costate equations.     

A note on eigenvalue bounds in algebraic Riccati equation

In this note, an upper bound on the maximum eigenvalue of the solution matrixKof the algebraic Riccati equation is established. The approach outlined also results in several lower bounds, which are more general than those derived in [1], for some of the largest eigenvalues ofK.     

A numerical solution of the stochastic discrete algebraic Riccati equation

This article proposes two algorithms for solving a stochastic discrete algebraic Riccati equation which arises in a stochastic optimal control problem for a discrete-time system. Our algorithms are generalized versions of Hewer’s algorithm. Algorithm I has quadratic convergence, but needs to solve a sequence of extended Lyapunov equations. On the other hand, Algorithm II only needs solutions of standard Lyapunov equations which can be solved easily, but it has a linear convergence. By a numerical example, we shall show that Algorithm I is superior to Algorithm II in cases of large dimensions. This work was presented in part at the 13th International Symposium on Artificial Life and Robotics, Oita, Japan, January 31–February 2, 2008     

A note on the solution of the algebraic Riccati equation

In this note, we consider the problem of solving the algebraic Riccati equation (ARE) arising in the optimal control theory, Several cases are studied. The solution here is, unlike the usual ones, related directly to the controllability and the observability matrices, the weighting matrices and the resulting closed-loop eigenvalues. As a result, it only requires few matrix operations to obtain the solution.     

Curvature properties of the algebraic Riccati equation

We prove that the solution to the algebraic Ricatti equation (ARE) is concave with respect to a nonnegative-definite symmetric state weighting matrix Q when the input weighting matrix R = R^T > 0. We also prove that the solution to the ARE is concave with respect to a positive-definite diagonal input weighting matrix R when Q = Q^T ≥ 0.     

The algebraic Riccati equation — A polynomial approach

Polynomial models are used to give a unified approach to the problem of classifying the set of all real symmetric solutions of the algebraic Riccati equation.     

Nonexistence of solutions to the Riccati equation

The Riccati equation does not necessarily give a positive definite solution when the plant has finite escape time in the time interval considered.     

Global aspects of the matrix Riccati equation

Matrix Riccati equations are interpreted as differential equations on Grassman manifolds. Necessary conditions for the Riccati equation to be a Morse-Smale system are given in the autonomous and periodic cases. Under this condition, the equation is structurally stable and has a unique asymptotically stable equilibrium point or periodic solution.     

A note on the maximal solution of the periodic Riccati equation

The periodic symmetric solutions of the periodic Riccati differential equation associated with the filtering problem are considered by the authors. It is proven that, under the sole assumption of detectability, there exists a maximal solution. Moreover, such a solution turns out to be strong, i.e. the characteristic multipliers of the associated closed-loop system belong to the closed unit disk. The proof relies on an iterative linearization technique, which calls for a sequence of periodic Lyapunov equations. Similar results are given for the minimal solution     

Bounds on eigenvalues of matrix products with an application to thealgebraic Riccati equation

Lower and upper summation bounds for the eigenvalues of the product XY are presented, under various restrictions on matrices X, Y∈R^n×n. An application to the algebraic Riccati equation yields a trace lower bound. It is observed that these bounds are tighter than those in the literature     

A stabilizing solution to the algebraic Riccati equation the resolvent method

A new approach to the problem of analytic representation of the stabilizing solution to the algebraic Riccati equation is proposed. The quadratic matrix equation is reduced to a linear one using the resolvent (sI _2n -H)^?1 of the Hamilton matrix. The symmetric solution to the obtained linear equation defines a stabilizing solution to the Riccati equation. Matrix coefficients of the linear equation are defined by the integral of resolvent in the complex domain over the closed contour which contains all its right poles. This construction of the solution to the problem gives rise to the development of important parts of the analysis and of the corresponding computing procedures.     

Numerical integration of the differential matrix Riccati equation

Two new Bernoulli substitution methods for solving the Riccati differential equation are tested numerically against direct integration of the Riccati equation, the Chandrasekhar algorithm, and the Davison-Maki method on a large set of problems taken from the literature. The first of these new methods was developed for the time-invariant case and uses the matrix analog of completing the square to transform the problem to a bisymmetric Riecati equation whose solution can be given explicitly in terms of a matrix exponential of ordern. This method is fast and accurate when the extremal solutions of the associated algebraic Riccati equation are well separated. The second new method was developed as a means of eliminating the instabilities associated with the Davison-Maki algorithm. By using reinitialization at each time step the Davison-Maki algorithm can be recast as a recursion which is over three times faster than the original method and is easily shown to be stable for both time-invariant and time-dependent problems. From the results of our study we conclude that the modified Davison-Maki method gives superior performance except for those problems where the number of observers and controllers is small relative to the number of states in which ease the Chandrasekhar algorithm is better.     

Solution bounds of the continuous Riccati matrix equation

A new approach is proposed for estimating the solution of the continuous algebraic Riccati equation (CARE). Upper and lower solution bounds of the CARE are presented. Comparisons show that the present bounds are more general and/or tighter than existing results.     

Strong solutions of the discrete-time algebraic Riccati equation

Conditions are given under which a solution of the DARE is positive semidefinite if and only if all the eigenvalues of its associated closed-loop matrix are in the closed unit disc.     

Numerical solution of the discrete-time periodic Riccati equation

In this paper we present a method for the computation of the periodic nonnegative definite stabilizing solution of the periodic Riccati equation. This method simultaneously triangularizes by orthogonal equivalences a sequence of matrices associated with a cyclic pencil formulation related to the Euler-Lagrange difference equations. In doing so, it is possible to extract a basis for the stable deflating subspace of the extended pencil, from which the Riccati solution is obtained. This algorithm is an extension of the standard QZ algorithm and retains its attractive features, such as quadratic convergence and small relative backward error. A method to compute the optimal feedback controller gains for linear discrete time periodic systems is dealt with