On a fallacious conjecture about the stabilizability properties of solutions of the Riccati difference equation

In [1], among other results, some conjectures concerning the monotonicity and stabilizing properties of solutions of the difference Riccati equation were proven to be fallacious by means of suitable counterexamples. Moreover, a ‘possibly fallacious conjecture’ was formulated for which no counterexample had been found. In this letter, such a counterexample is provided together with an interpretation of the somewhat counterintuitive behaviour of the Riccati equation.     

Monotonicity and stabilizability- properties of solutions of the Riccati difference equation: Propositions, lemmas, theorems, fallacious conjectures and counterexamples

The problem considered is that of selecting an initial covariance matrix for the Kalman filter to ensure that the closed-loop filter at every subsequent time instant is exponentially asymptotically stable as a time-invariant filter. Sufficient conditions are derived based on monotonicity properties of the solution of the Riccati difference equation. The results have application in observer design, and the cases of filtering for nonstabilizable systems and systems with singular system matrices are included.     

Discrete linear Hamiltonian eigenvalue problems

This paper introduces general discrete linear Hamiltonian eigenvalue problems and characterizes the eigenvalues. Assumptions are given, among them the new notion of strict controllability of a discrete system, that imply isolatedness and lower boundedness of the eigenvalues. Due to the quite general assumptions, discrete Sturm-Liouville eigenvalue problems of higher order are included in the presented theory.     

On stability of a class of positive linear functional difference equations

We first give a sufficient condition for positivity of the solution semigroup of linear functional difference equations. Then, we obtain a Perron–Frobenius theorem for positive linear functional difference equations. Next, we offer a new explicit criterion for exponential stability of a wide class of positive equations. Finally, we study stability radii of positive linear functional difference equations. It is proved that complex, real and positive stability radius of positive equations under structured perturbations (or affine perturbations) coincide and can be computed by explicit formulae. Pham Huu Anh Ngoc and Toshiki Naito are supported by the Japan Society for Promotion of Science (JSPS) ID No. P 05049.     

CP methods of higher order for Sturm-Liouville and Schrödinger equations

The algorithm upon which the code SLCPM12, described in Computer Physics Communications 118 (1999) 259-277, is based, is extended to higher order. The implementation of the original algorithm, which was of order {12,10} (meaning order 12 at low energies and order 10 at high energies), was more efficient than the well-established codes SL02F, SLEDGE and SLEIGN. In the new algorithm the orders {14,12}, {16,14} and {18,16} are introduced. Besides regular Sturm-Liouville and one-dimensional Schrödinger problems also radial Schrödinger equations are considered with potentials of the form V(r)=S(r)/r+R(r), where S(r) and R(r) are well behaved functions which tend to some (not necessarily equal) constants when r→0 and r→∞. Numerical illustrations are given showing the accuracy, the robustness and the CPU-time gain of the proposed algorithms.     

Linear-quadratic optimal control revisited

We offer a new proof of the relationship between the solution of a matrix Riccati equation and the optimal solution of a linear-quadratic regulator problem in the presence of linear terminal constraints.     

Embedding of time-varying contractive systems in lossless realizations

The lossless embedding problem, also known as the Darlington synthesis or unitary extension problem, considers the extension of a given contractive system to become the partial input-output operator of a lossless system. In this paper the embedding problem is solved for discrete-time time-varying systems with finite but possibly time-varying state dimensions, for the strictly contractive as well as the boundary case. The construction is done in a state space context and gives rise to a time-varying Riccati difference equation which is shown to have a closed-form solution. As a corollary, a discrete-time Bounded Real Lemma is formulated, linking contractiveness of an input-output operator to conditions on its state realization.This research was supported in part by the commission of the EC under the ESPRITBRA program 6632 (NANA2).     

Asymptotic behavior of certain Riccati difference equations

Matrix Riccati difference equations are investigated on the infinite index set. Under natural assumptions an existence and uniqueness theorem is proven. The existence of the asymptotic expansion of the solution and computability of its coefficients are shown, provided the coefficients of the equation have such an expansion.     

Delta 域Riccati方程研究：连续与离散的统一方法

基于Ｄｅｌｔａ算子研究连续Ｒｉｃｃａｔｉ方程和离散Ｒｉｃｃａｔｉ方程的统一形式,得到Ｄｅｌｔａ域Ｒｉｃｃａｔｉ方程解的定界估计,本文结果与现有的结果相比,具有较小的保守性,在极限情形下可分别得到连续和离散Ｒｉｃｃａｔｉ方程的相关结论。     

A Lyapunov theorem for semigroups of operators

The well known condition for the existence of a unique solution to finite dimensional Lyapunov equations is extended to a family of Hilbert space operators.     

Normalized coprime factorizations for linear time-varying systems

In this paper we show that a finite dimensional linear time-varying continuous-time system admits normalized coprime factorizations if and only if it admits a stabilizable and detectable realization. We construct state-space formulas for these factorizations using the stabilizing solutions to standard Riccati differential equations. In the process, we give a simple proof that stabilizability and detectability are sufficient to ensure the existence of such solutions. Based on these results, and on recent advances in the theory of _∞ optimization, we present an algorithm to compute the distance between two systems in the gap metric.     

Nonlinear optimal control as quantum mechanical eigenvalue problems

A novel approach for approximating the nonlinear optimal feedback control of a system with a terminal cost is proposed. To lessen the difficulty due to nonlinearity, we try to treat the system in a framework of linear theories. For this, we assume a quantum mechanical linear wave associated with the system. Since the control system is constrained by state equations, we handle the system according to quantum mechanics of constrained dynamics. A Hamiltonian is represented as a linear operator acting on a function that describes behavior of waves. Subsequently, nonlinear feedback is calculated without any time integration in the backward direction. Using eigenvalues and eigenfunctions of the linear Hamiltonian operator, an optimal feedback law is given as a combination of analytic functions of time and state variables. We take as an example a system described by two scalar variables for state and control input. Simulation studies on the system by the eigenvalue analysis show that the proposed method reduces calculation time to nearly a tenth that of a numerical calculation of a Hamilton-Jacobi equation by a finite difference method.     

Robust stability of delay difference systems under fractional perturbations in infinite-dimensional spaces

In this article we study the robust stability of difference systems with delays under fractional perturbations in infinite-dimensional spaces. First, the estimates of the complex stability radius are addressed. Second, it is shown that for positive linear systems, the complex, real and positive stability radii coincide and can be computed by a simple formula. Finally, a simple example is given to illustrate the obtained results.     

LQ (optimal) control of hyperbolic PDAEs

The linear quadratic control synthesis for a set of coupled first-order hyperbolic partial differential and algebraic equations is presented by using the infinite-dimensional Hilbert state-space representation of the system and the well-known operator Riccati equation (ORE) method. Solving the algebraic equations and substituting them into the partial differential equations (PDEs) results in a model consisting of a set of pure hyperbolic PDEs. The resulting PDE system involves a hyperbolic operator in which the velocity matrix is spatially varying, non-symmetric, and its eigenvalues are not necessarily negative through of the domain. The C₀-semigroup generation property of such an operator is proven and it is shown that the generated C₀-semigroup is exponentially stable and, consequently, the ORE has a unique and non-negative solution. Conversion of the ORE into a matrix Riccati differential equation allows the use of a numerical scheme to solve the control problem.     

Optimal fault detection for linear discrete time-varying systems

This paper deals with the problem of observer-based fault detection for linear discrete time-varying (LDTV) systems. A problem formulation is first proposed to address the optimization of the fault detection filter (FDF) design, which is expressed in terms of maximizing a finite horizon H_∞/H_∞ or H₋/H_∞ performance index. This formulation can be applied to FDF design of LDTV systems subject to l₂-norm bounded unknown inputs or stochastic noise sequences. It is shown that a unified optimal solution to the FDF can be obtained by solving the discrete time Riccati equation and the optimal FDF is not unique. A numerical example is given to illustrate the proposed method.     

Optimal control of stable weakly regular linear systems

The paper extends quadratic optimal control theory to weakly regular linear systems, a rather broad class of infinite-dimensional systems with unbounded control and observation operators. We assume that the system is stable (in a sense to be defined) and that the associated Popov function is bounded from below. We study the properties of the optimally controlled system, of the optimal cost operatorX, and the various Riccati equations which are satisfied byX. We introduce the concept of an optimal state feedback operator, which is an observation operator for the open-loop system, and which produces the optimal feedback system when its output is connected to the input of the system. We show that if the spectral factors of the Popov function are regular, then a (unique) optimal state feedback operator exists, and we give its formula in terms ofX. Most of the formulas are quite reminiscent of the classical formulas from the finite-dimensional theory. However, an unexpected factor appears both in the formula of the optimal state feedback operator as well as in the main Riccati equation. We apply our theory to an extensive example.Part of the results reported here were obtained while the second author was visiting FUNDP Namur, under the Belgian Program on Inter-University Poles of Attraction initiated by the Belgian state, Prime Minister's Office, Science Policy Programming. The scientific responsibility is assumed by the authors.     

Weak and strong stabilization of linear systems

We study the problem of stabilizing a distributed linear system on a subregion of its geometrical domain. We are concerned with two methods： the first approach enables us to characterize a stabilizing control via the steady state Riccati equation, and the second one is based on decomposing the state space into two suitable subspaces and studying the projections of the initial system onto such subspaces. The obtained results are performed through various examples.     

Structural stability of linear difference equations in Hilbert space

In this note, we prove that linear difference equations in Hilbert space are structurally stable if and only if they have an exponential dichotomy.     

A new approach to static output feedback stabilization of linear dynamic systems

The topic of this study is output feedback control of linear control system with output. Use of condensed forms of the linear system under static output feedback (SOF) control is made to derive sufficient conditions for stabilization. A minimal-order control-free observer is presented.