Solution Of Algebraic Riccati Equation Research Articles

Stable-Optimum Gain Tuning for Designing Mobile Robot Controllers Using Incest Prevented Evolution

We present an evolution strategy (ES) algorithm - incest prevented evolution strategy (IPES) enhancing our novel evolution strategy (NES) algorithm. Validity of NES and IPES algorithms is compared with other evolutionary algorithms (EAs) and relative performances and also compared with test function results. The IPES algorithm shows the highest balance between exploration and exploitation over the NES algorithm on these test functions by achieving high-precision global results. Both algorithms are applied to solve stabilizing optimum gain tuning problems in mobile robot controllers. Two optimal servocontrollers are considered for a mobile robot with two independent drive wheels. A bidirectional fitness (cost) function is constructed for these controllers so that stable but optimum gains are tuned automatically evolutionarily instead of using a traditional algebraic Riccati equation solution. Two trajectory tracking control examples (straight line and circular) are considered for controllers. The superiority of the IPES algorithm over the NES algorithm is repeated in the application domain and the effectiveness of evolutionary gain tuning demonstrated by simulation results.

Benchmarks for the numerical solution of algebraic Riccati equations

Two collections of benchmark examples are presented for the numerical solution of continuous-time and discrete-time algebraic Riccati equations. These collections 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.

A Homeomorphic Characterization Of Minimal Spectral Factors

In this paper a new characterization of the class of all minimal square spectral factors of a given rational spectral density is presented. This characterization, which is established without assumptions on poles and zeroes of the spectral density, extends a result presented in [A. Ferrante, IEEE Trans. Automat. Control, 39 (1994), pp. 2122{2126]. The characterization consists of two bijective maps which relate the set of minimal square spectral factors to a set of invariant subspaces of a certain matrix and to a set of symmetric solutions of an algebraic Riccati equation (ARE). In the second part of the paper it is proven that these two maps are homeomorphisms. This result extends and applies to spectral factorization theory recent results of H. Wimmer [Integral Equations Operator Theory, 21 (1995), pp. 362{375], where it is proven that the well-known relation between solutions of ARE and invariant subspaces of a certain matrix is, in fact, a homeomorphism.

Non-Hermitian Solutions of Algebraic Riccati Equations

AbstractNon-hermitian solutions of algebraic matrix Riccati equations (of the continuous and discrete types) are studied. Existence is proved of non-hermitian solutions with given upper bounds of the ranks of the skew-hermitian parts, under the sign controllability hypothesis.

Nonnegative solutions of algebraic Riccati equations

Nonnegative Hermitian solutions of various types of continuous and discrete algebraic Riccati equations are studied. The Hamiltonian is considered with respect to two different indefinite scalar products. For the set of nonnegative solutions the order structure and the topology of the set and the stability of solutions is treated. For general Hermitian solutions a method to compute the inertia is given. Although most attention is payed to the classical types arising from LQ optimal control theory, the case where the quadratic term has an indefinite coefficient is studied as well.

Nonnegative solutions of algebraic Riccati equations

Nonnegative Hermitian solutions of various types of continuous and discrete algebraic Riccati equations are studied. The Hamiltonian is considered with respect to two different indefinite scalar products. For the set of nonnegative solutions the order structure and the topology of the set and the stability of solutions is treated. For general Hermitian solutions a method to compute the inertia is given. Although most attention is payed to the classical types arising from LQ optimal control theory, the case where the quadratic term has an indefinite coefficient is studied as well.

General matrix pencil techniques for the solution of algebraic Riccati equations: a unified approach

We present a unified theory of matrix pencil techniques for solving both continuous and discrete-time algebraic Riccati equations (AREs) under fairly general conditions on the coefficient matrices. The theory applies to a large class of AREs and Riccati-like equations arising from the singular H/sup /spl infin//- and H/sup 2/-control problems, singular linear quadratic control, the 4-block Nehari problem, or from singular J-spectral factorizations. The underlying concept is the so-called proper deflating subspace of a (possibly singular) matrix pencil in terms of which necessary and sufficient conditions for the solvability of Riccati equations are given. It is shown that these conditions can be checked and the solutions computed by a numerically sound algorithm.

Multivariable Frequency Response Methods for Coloured Noise Optimal Discrete-Time Kalman Filter

This paper is concerned with the problem of multi-input multi-output, infinite time, linear time-invariant optimal coloured-noise discrete-time Kalman filter in frequency domain. A simple new algorithm is given for the solution to the steady state gain of the coloured noise of the optimum filter based on a transfer function approach. The algorithm is based on spectral factorization of the observed spectral density matrix of the filter which generates directly the return-difference matrix of the optimal filter. This leads to the proof of necessary condition of optimality of filter. The method seems to be more direct than by algebraic Riccati equation solution and can easily be implemented on a digital computer.

Continuity properties of solutions to H2 and H∞ Riccati equations

In H 2 and H ∞ optimal control (semi-)stabilizing solutions of algebraic Riccati equations play an essential role. It is well-known that these solutions might have discontinuities as a function of the system parameters. The paper shows that these discontinuities are directly linked to non-left-invertibility and, in contrast to what one might think, unrelated to zeros on the imaginary axis.

A Multilevel Technique for the Approximate Solution of Operator Lyapunov and Algebraic Riccati Equations

We consider multigrid or, more appropriately, multilevel techniques for the numerical solution of operator Lyapunov and algebraic Riccati equations. The Riccati equation, which is quadratic, plays an essential role in the solution of linear-quadratic optimal control problems. The linear Lyapunov equation is important in the stability theory for linear systems and its solution is the primary step in the Newton–Kleinman (NK) algorithm for the solution of algebraic Riccati equations. Both equations are operator equations when the underlying linear system is infinite dimensional. In this case, finite-dimensional discretization is required. However, as the level of discretization increases, the convergence rate of the standard iterative techniques for solving high order matrix Lyapunov and Riccati equations decreases. To deal with this, multileveling is introduced into the iterative NK method for solving the algebraic Riccati equation and Smith’s method for solving matrix Lyapunov equations. Theoretical result...

Lattice properties of sets of semidefinite solutions of continuous-time algebraic riccati equations

The set of negative-semidefinite solutions of the algebraic Riccati equation A∗X + XA + XBB∗X − C∗C = 0 is studied under the most general hypothesis, namely that (A, B) is stabilizable modulo the undetectable subspace of (A, C). If is a lattice, then is isomorphic to 0 ⊕ S where T0 is the direct sum of copies of the set of nonnegative real numbers and is a closure of which is order isomorphic to a system of well-defined A-invariant subspaces. Necessary and sufficient conditions are given such that (or) is a lattice.

Backward error, sensitivity, and refinement of computed solutions of algebraic Riccati equations

AbstractIn this paper, a new backward error criterion, together with a sensitivity measure, is presented for assessing solution accuracy of nonsymmetric and symmetric algebraic Riccati equations (AREs). The usual approach to assessing reliability of computed solutions is to employ standard perturbation and sensitivity results for linear systems and to extend them further to AREs. However, such methods are not altogether appropriate since they do not take account of the underlying structure of these matrix equations. The approach considered here is to first compute the backward error of a computed solution X̂ that measures the amount by which data must be perturbed so that X̂ is the exact solution of the perturbed original system. Conventional perturbation theory is used to define structured condition numbers that fully respect the special structure of these matrix equations. The new condition number, together with the backward error of computed solutions, provides accurate estimates for the sensitivity of solutions. Optimal perturbations are then used in an iterative refinement procedure to give further more accurate approximations of actual solutions. The results are derived in their most general setting for nonsymmetric and symmetric AREs. This in turn offers a unifying framework through which it is possible to establish similar results for Sylvester equations, Lyapunov equations, linear systems, and matrix inversions.

A note on the regularity of solutions of infinite dimensional Riccati equations

This note is concerned with the regularity of solutions of algebraic Riccati equations arising from infinite dimensional LQR control problems. We show that distributed parameter systems described by certain parabolic partial differential equations often have a special structure that smooths solutions of the corresponding Riccati equation. This analysis is motivated by the need to find specific representations for Riccati operators that can be used in the development of computational schemes for problems where the input and output operators are not Hilbert-Schmidt. This situation occurs in many boundary control problems and in certain distributed control problems associated with optimal sensor/actuator placement.

A Riccati equation approach to a constant I/O scaled [formula omitted] optimization problem

This paper studies a constant I/O scaled H ∞ optimization problem with a Riccati equation approach. An iterative algorithm is presented for searching for the optimal I/O scaling matrix. We show that a descent direction of the scaling matrix can be determined by algebraic Riccati equation solutions.

Numerical Solution of the Coupled Mode Equations in Duct Acoustics

We present a fast solver for the coupled mode equations in duct acoustics. It is based on a partitioning of the resulting system of ordinary differential equations into separate subsystems, the number of which increases by the separability of the problem. This structure is obtained by transformations of the dependent variables such that weakly interacting modes are separated. The decoupling process requires the numerical solution of algebraic Riccati equations. However, these computations can be done on a spatial scale much larger than the characteristic wavelength of the problem.

Relationship between three discrete-time H ∞ algebraic Riccati equation solutions

This paper presents a proof of a connection between the strong solutions to three algebraic Riccati equations which are associated with the solution of a discrete-time H ∞-optimal control problem. Of these solutions, denoted X∞, Y∞ and Z∞, two parametrize a generator of all γ-suboptimal output-feedback controllers and one is associated with a dual state-feedback problem. The relationship is proved by establishing the existence of a similarity transformation between two symplectic matrix pencils which are associated with two of the Riccati equations. A similar result is known to exist in the continuous-time setting.

Bounds on the solution of the algebraic Riccati equation under perturbations in the coefficients

This paper is concerned with bounds on symmetric solutions of the continuous algebraic matrix Riccati equation under perturbations on its coefficients. Special emphasis is given to the question of estimating the ‘size’ of the perturbations on the stabilizing solution of the Riccati equation when one or all its coefficients are subject to small perturbations. Upper bounds on the norm of the perturbations on the stabilizing solution are presented. Moreover, it is shown that these perturbations can be determined as a single-valued continuous function of the perturbations on the coefficient of the Riccati equation.

Simplifying theH∞ theory via loop-shifting, matrix-pencil and descriptor concepts

The 2-Riccati H∞ controller formulae and derivations are simplified via various ‘loop-shifting’ transformations that are naturally expressed in terms of a degree-one polynomial system matrix (PSM) closely related to the Luenberger descriptor form of a system. The technique enables one without loss of generality to restrict attention to the simple case in which D 11 =0, D 22 = 0, D T 12= [0 l], D 21 = [0 l],D l 12 C1=0 and B 1 D T 21 =0. Matrix-fraction descriptions (MFDs) for the algebraic Riccati equation solutions afford another change of variables, which brings the 2-Riccati H∞ controller formulae into a cleaner, more symmetric descriptor form having the important practical advantage that it eliminates the numerical difficulties that can occur in cases where one or both of the Riccati solutions, Pi and Q, blow up and in cases where l— QP is nearly singular. Numerical difficulties previously associated with verifying the existence conditions P ≥ 0, Q ≥ 0, and λmaxlpar;QP) < 1 are largely eliminated by e...

Orthogonal projections and solution of algebraic Riccati equations

A computational procedure is considered for constructing orthogonal projection operators by solving a suitable matrix algebraic Riccati equation (ARE). The expressions for the projections are written in terms of the sign-function of the original matrix.

Defect correction method for the solution of algebraic Riccati equations

The solution of discrete and continuous algebraic Riccati equations is considered. It is shown that if an approximate solution is obtained, then the defect for this solution again solves an algebraic Riccati equation of the same form, and that the system properties of detectability and stabilizability are inherited by this defect equation. On the basis of these results, a general defect-correction method is proposed and numerical examples are given for the use of this method in combination with A. Bunse-Gerstner and V. Mehrmann's (1986) SR method.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>