Robust H ∞ Filtering for a Class of Nonlinear Discrete-Time Markovian Jump Systems

This paper considers the problem of the robust H filtering for a class of nonlinear discrete-time Markovian jump systems with real time-varying norm-bounded parameter uncertainty. For each mode, the nonlinearity is assumed to satisfy the global Lipschitz conditions and appears in both the state and measured output equations. The problem that we address is the design of a nonlinear filter which ensures robust stochastic stability and a prescribed H performance level of the filtering error system for all admissible uncertainties. A sufficient condition for the solvability of this problem is obtained in terms of a set of linear matrix inequalities; an explicit expression of a desired nonlinear H filter is also given. Finally, an example is provided to demonstrate the effectiveness of the proposed approach.     

Robust disturbance attenuation of fluid dynamic systems with norm-bounded uncertainties

In this paper, we consider the problem of robust control of fluid dynamic systems in an Hsense. The system under studyis nonlinear in state and linear in inputs with parameter uncertaintiesin all entries. The controller is designed, via Hamilton–Jacobiequation and Riccati equation approaches, such that the resultingclosed-loop systems is robust stable and has a robust Hperformance.In particular, it is shown that the robust Hcontrol problemis solvable for a nonlinear system (governed by Navier–Stokes equation) in the neighbourhood of the origin if a robust Hcontrolproblem is solvable for a corresponding linearized system.     

Topological existence and stability for stackelberg problems

The aim of this paper is to study, in a topological framework, existence and stability for the solutions to a parametrized Stackelberg problem. To this end, approximate solutions are used, more precisely, -solutions and strict -solutions. The results given are of minimal character and the standard types of constraints are considered, that is, constant constraints, constraints defined by a finite number of inequalities, and more generally constraints defined by an arbitrary multifunction.     

The ε c -Product of a Schwartz b-Space by a Quotient Banach Space and Applications

We define the -product of a -space by a quotient Banach space. We give conditions under which this -product will be monic. Finally, we define the _c -product of a Schwartz b-space by a quotient Banach space and we give some examples of applications.     

On Some Isomorphism on the Category of b-Spaces

Given a nuclear b-space N, we show that if is a finite or -finite measure space and 1p, then the functors L _loc ^p (,N.) and NL ^p (,.) are isomorphic on the category of b-spaces of L. Waelbroeck.     

Asymptotic Analysis and Solution of a Finite-Horizon H ∞ Control Problem for Singularly-Perturbed Linear Systems with Small State Delay

A finite-horizon H state-feedback control problem for singularly-perturbed linear time-dependent systems with a small state delay is considered. Two approaches to the asymptotic analysis and solution of this problem are proposed. In the first approach, an asymptotic solution of the singularly-perturbed system of functional-differential equations of Riccati type, associated with the original H problem by the sufficient conditions of the existence of its solution, is constructed. Based on this asymptotic solution, conditions for the existence of a solution of the original H problem, independent of the small parameter of singular perturbations, are derived. A simplified controller with parameter-independent gain matrices, solving the original H problem for all sufficiently small values of this parameter, is obtained. In the second approach, the original H problem is decomposed into two lower-dimensional parameter-independent H subproblems, the reduced-order (slow) and the boundary-layer (fast) subproblems; controllers solving these subproblems are constructed. Based on these controllers, a composite controller is derived, which solves the original H problem for all sufficiently small values of the singular perturbation parameter. An illustrative example is presented.     

ε-optimality criteria for convex programming problems via exact penalty functions

In this paper, we present-optimality criteria for convex programming problems associated with exact penalty functions. Several authors have given various criteria under the assumption that such convex problems and the associated dual problems can be solved. We assume the solvability of neither the convex problem nor the dual problem. To derive our criteria, we estimate the size of the penalty parameter in terms of an-solution for the dual problem.     

Algebras almost commuting with Clifford algebras in a II∞ factor

We are concerned here with certain Banach algebras of operators contained within a fixed II factor N. These algebras may be thought of as noncommutative classifying spaces for the functor Ext _* ^N The basic objects of study are the algebras A _kN (for n=1, 2,...). Here, we are given an essentially unique representation of the complex Clifford algebra C _k N and the elements of A _k are those operators in N which exactly commute with the first k–1 generators of C _k and also commute with the kth generator modulo a symmetric ideal N. Up to isomorphism, these algebras are periodic with period 2.We determine completely the homotopy types of A ₁ ^–1 and A ₂ ^–1 Here, A ₁ ^–1 is homotopy equivalent to the space of (Breuer) Fredholm operators in N, while A ₂ ^–1 is homotopy equivalent to the group K _N ^–1 ={x N^–1¦ x=1+k, k K_N}. We use these results to compute the K-theory of A ₁ and A ₂.For a fixed C ^*-algebra A, we define abelian groups G _k,N(A) of equivalence classes of homomorphisms AA _k. Letting N = M (H) for a II₁ factor M we define similar abelian groups G _k(A, M) where we replace N by L(E) for countably generated right Hilbert M-modules E with (left) actions C _k L(E). Using ideas of Skandalis, we show that G _k,NG_k(A, M) so that the G _k,N are stable half-exact homotopy functors because the G _k(·, M) are such.In general, we show that G _k(A, M)KK ^k(A, M) and so our theory fits neatly into Kasparov KK-theory. We investigate many interesting examples from our point of view.     

Robust ℋ∞-Filtering Design with Pole Placement Constraint via Linear Matrix Inequalities

The problem of robust filtering design for continuous-time systems with convex bounded uncertainties is addressed in this paper. The aim is to determine a stable linear filter such that the filtering error system remains quadratically stable within a prespecified -attenuation level. Necessary and sufficient conditions for the existence of such robust filter are provided in terms of linear matrix inequalities, which can be solved efficiently through standard convex optimization procedures guaranteeing global convergence. Furthermore, as an improvement of the strategy, the filter dynamics can be constrained to some specific regions inside the left-half complex plane.     

Well-conditioned computation for H ∞ controller near the optimum

The present paper gives a procedure for determining a H optimal controller in the assumption that the game Riccati equations have stabilizing positive definite solutions at the optimum value. A specific feature of the construction is its first step consisting in balancing with respect to the positive definite stabilizing solutions of the Riccati equations. The justification is based on singular perturbations reduction.     

Controlled dual perturbations for l p -programming

l _p -programming is a common generalization of linear programming, quadratically constrained quadratic programming,l _p -constrainedl _p -approximation, and multiple criteria compromise programming. It is a type of convex programming with objective function and inequality constraints expressed by means ofl _p -norms. The dual program established by Peterson and Ecker is a maximization problem with a concave, upper-semicontinuous objective function over a set of constraints that are essentially linear. In developing a dual method for this problem, we face two major difficulties. One is the non-differentiability of the dual objective function and the other one is an efficient dual-to-primal conversion.In this paper, we introduce a mechanism to construct a suitably perturbed dual program with a differentiable concave objective function over linear constraints. Solving this well-constructed perturbed dual program, we can obtain an-optimal dual solution for an arbitrarily small number. Moreover, we show a way of constructing a linear program based on this dual solution. Then an-optimal primal solution can be obtained by solving the dual of this simple linear program.

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Zusammenfassung Diel _p -Optimierung ist eine Verallgemeinerung, die die lineare Optimierung, quadratische Optimierung mit quadratischen Restriktionen,l _p -Approximation mitl _p -Restriktionen, wie auch Vektoroptimierung umfaßt. Es handelt sich dabei um konvexe Optimierungsaufgaben, bei denen Zielfunktions- und Ungleichungsrestriktionen mittelsl _p -Normen ausgedrückt werden. Das duale Problem nach Peterson and Eckert ist ein Maximierungsproblem mit einer konkaven oberhalb-halbstetigen Zielfunktion über einer Menge von im wesentlichen linearen Restrictionen. Bei der Entwicklung einer dualen Lösungsmethode treten zwei Hauptschwierigkeiten auf: Die eine ist die Nicht-Differenzierbarkeit der dualen Zielfunktion, die andere besteht darin, eine effiziente Übertragung der dualen Lösung in eine primale zu finden.In dieser Arbeit führen wir eine Methode ein, die es gestattet, ein entsprechendes gestörtes duales Programm mit differenzierbarer konkaver Zielfunktion und linearen Restriktion aufzustellen. Bei der Lösung dieses wohl-strukturierten, gestörten dualen Problems erhalten wir eine-optimale Duallösung für beliebig kleines. Ferner zeigen wir einen Weg auf, wie, basierend auf dieser Duallösung, ein lineares Programm formuliert werden kann. Löst man das Dualproblem dieses einfachen linearen Programms, so erhält man eine-optimale Lösung für das Ausgangsproblem.

     

Existence and construction of Δ1-splines of class Ck on a three-directional mesh

Let ^* be the equilateral triangulation of the plane and let ₁ ^* be the equilateral triangle formed by four triangles of ^*. We study the space of piecewise polynomial functions in C ^k (R ²) with support ₁ ^*, having a sufficiently high degree n and which are invariant with respect to the group of symmetries of ₁ ^*. Such splines are called ₁ ^*-splines. We first compute the dimension of this space in function of n and k. Then, for any fixed k0, we prove the existence of ₁ ^*-splines of class C ^k and minimal degree, but these splines are not unique. Finally, we describe an algorithm computing the Bernstein–Bézier coefficients of these splines.     

On the gi/m/∞ service system with batch arrivals and different types of service distributions

We study the infinite-server system with batch arrivals ands different types of customers. With probabilityp _i an arriving customer is of typei (i=1,..., s) and requires an exponentially distributed service time with parameter _i (G ^GI /M ₁ ...M _s /). For theGI ^GI /M ₁...M _s / system it is shown that the binomial moments of thes-variate distribution of the number of type-i customers in the system at batch arrival epochs are determined by a recurrence relation and, in steady state, can be computed recursively. Furthermore, forG ^GI /M ₁...M _s /, relations between the distributions (and their binomial moments) of the system size vector at batch arrival and random epochs are given. Thus, earlier results by Takács [14], Gastwirth [9], Holman et al. [11], Brandt et al. [3] and Franken [6] are generalized.     

A value iteration method for undiscounted multichain Markov decision processes

This paper proposes a value iteration method which finds an-optimal policy of an undiscounted multichain Markov decision process in a finite number of iterations. The undiscounted multichain Markov decision process is reduced to an aggregated Markov decision process, which utilizes maximal gains of undiscounted Markov decision sub-processes and is formulated as an optimal stopping problem. As a preliminary, sufficient conditions are presented under which a policy is-optimal.

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Zusammenfassung In dieser Arbeit wird eine Wertiterationsmethode vorgeschlagen, die eine-optimale Politik für einen undiskontierten nicht-irreduziblen Markovschen Entscheidungsprozeß (MEP) in endlichen vielen Schritten liefert. Der undiskontierte nicht-irreduzible MEP wird auf einen aggregierten MEP reduziert, der maximale Gewinn eines undiskontierten Sub-MEP verwendet und als optimales Stopp-Problem formuliert wird. Zu Beginn werden hinreichende Bedingungen für die-Optimalität einer Politik angegeben.

     

H ∞-Control for Markovian Jumping Linear Systems with Parametric Uncertainty

This paper studies the problem of H -control for linear systems with Markovian jumping parameters. The jumping parameters considered here are two separable continuous-time, discrete-state Markov processes, one appearing in the system matrices and one appearing in the control variable. Our attention is focused on the design of linear state feedback controllers such that both stochastic stability and a prescribed H -performance are achieved. We also deal with the robust H -control problem for linear systems with both Markovian jumping parameters and parameter uncertainties. The parameter uncertainties are assumed to be real, time-varying, norm-bounded, appearing in the state matrix. Both the finite-horizon and infinite-horizon cases are analyzed. We show that the control problems for linear Markovian jumping systems with and without parameter uncertainties can be solved in terms of the solutions to a set of coupled differential Riccati equations for the finite-horizon case or algebraic Riccati equations for the infinite-horizon case. Particularly, robust H -controllers are also designed when the jumping rates have parameter uncertainties.     

Almost π-Lattices

In this paper we establish some conditions for an almost -domain to be a -domain. Next -lattices satisfying the union condition on primes are characterized. Using these results, some new characterizations are given for -rings.     

Analysis of Error Growth Via Stability Regions in Numerical Initial Value Problems

This paper concerns the stability analysis of numerical methods for solving time dependent ordinary and partial differential equations. In the literature stability estimates for such methods were derived, under a condition which can be viewed as a transplantation of the Kreiss resolvent condition (from the unit disk to the stability region S of the numerical method). These estimates tell us that errors in the numerical time stepping process cannot grow faster than linearly with min{s,n}. Here n denotes the number of time steps, and s stands for the order of the (spatial discretization) matrices involved.In this paper we address the natural question of whether the above stability estimates can be improved so as to imply an error growth at a slower rate than min{s,n} (when n, s). Our results concerning this question are as follows: (a) for all (practical) Runge–Kutta and other one-step formulas, we show that the estimates from the literature are sharp in that error growth at the rate min{s,n} can actually occur, (b) for linear multistep formulas we find that, rather surprisingly, some of the stability estimates can substantially be improved and extended, whereas others are sharp.The results proved in this paper are also relevant to (suitably scaled spatial discretization) matrices whose -pseudo-eigenvalues lie at a distance not exceeding K from the stability region S of the time stepping method, for all >0 and fixed constant K.