Some remarks on adaptive stabilization of infinite-dimensional systems

It is the purpose of this note to show that a first-order adaptive controller stabilizes a large class of infinite-dimensional systems described by strongly continous semigroups. It is assumed that the plant is minimum-phase and has invertible high-frequency gain. Knowledge of the sign of the high-frequency gain is not required.     

Some remarks on static-feedback linearization for time-varying systems

This work summarizes some results about static state feedback linearization for time-varying systems. Three different necessary and sufficient conditions are stated in this paper. The first condition is the one by [Sluis, W. M. (1993). A necessary condition for dynamic feedback linearization. Systems & Control Letters, 21, 277–283]. The second and the third are the generalizations of known results due respectively to [Aranda-Bricaire, E., Moog, C. H., Pomet, J. B. (1995). A linear algebraic framework for dynamic feedback linearization. IEEE Transactions on Automatic Control, 40, 127–132] and to [Jakubczyk, B., Respondek, W. (1980). On linearization of control systems. Bulletin del’Academie Polonaise des Sciences. Serie des Sciences Mathematiques, 28, 517–522]. The proofs of the second and third conditions are established by showing the equivalence between these three conditions. The results are re-stated in the infinite dimensional geometric approach of [Fliess, M., Lévine J., Martin, P., Rouchon, P. (1999). A Lie–Bäcklund approach to equivalence and flatness of nonlinear systems. IEEE Transactions on Automatic Control, 44(5), 922–937].     

Markov chain approach to identifying Wiener systems

Identification of the Wiener system composed of an infinite impulse response (IIR) linear subsystem followed by a static nonlinearity is considered.The recursive estimates for unknown coefficients of the linear subsystem and for the values of the nonlinear function at any fixed points are given by the stochastic approx-imation algorithms with expanding truncations (SAAWET).With the help of properties of the Markov chain connected with the linear subsystem,all estimates derived in the paper are proved to be strongly consistent.In comparison with the existing results on the topic,the method presented in the paper simplifies the convergence analysis and requires weaker conditions.A numerical example is given,and the simulation results are consistent with the theoretical analysis.     

Some remarks on the stability of Mamdani fuzzy control systems

This paper presents a passivity approach to the stability analysis of Mamdani fuzzy control systems. The main idea of the paper is that the fuzzy controller can be considered from an input-output viewpoint as a nonlinear dissipative operator. An input-output approach is provided by considering a dynamical dependency of inputs of the fuzzy controller. Then the passivity theory is applied in order to obtain conditions for absolute stability of the closed-loop control system     

Some remarks about moment controllability for linear stochastic systems

The moment controllability of linear systems containing white multiplicative noise processes is considered. We introduce the definition ofpth moment controllability and we present a technique for obtaining sufficient conditions for this controllability. An example is given to illustrate the technique.     

Consensus in multi-agent systems with random delays governed by a Markov chain

This paper investigates the consensus problem in a multi-agent system with random delays governed by a Markov chain. The communication topology is assumed to be directed and fixed. With first-order dynamics under the sampled-data setting, we first convert the original system into a reduced-order one featuring the error dynamics. Accordingly, the consensus problem is transformed into the stabilization of the error dynamic system. Thereafter, based on the theory in stochastic stability for time-delay systems, a sufficient condition is established in terms of a set of linear matrix inequalities (LMIs). The mean square stability of the error dynamics is shown to guarantee consensus of the multi-agent system. By explicitly incorporating the transition probability of the random delay into consideration, the conservativeness in control design is reduced. A delay-dependent switching control scheme is developed by redesigning the adjacency matrix. Finally, simulation results are provided to verify the effectiveness of the proposed approach.     

Dwell-time controllers for stochastic systems with switching Markov chain

We study the problem of feedback stabilization of a family of nonlinear stochastic systems with switching mechanism modeled by a Markov chain. We introduce a novel notion of stability under switching, which guarantees a given probability that the trajectories of the system hit some target set in finite time and remain thereinafter. Our main contribution is to prove that if the expectation of the time between two consecutive switching (dwell time) is “sufficiently large”, then the system is stable under switching with guaranteed probability. We illustrate this methodology by constructing measurement feedback controllers for a wide class of stochastic nonlinear systems.     

Some remarks on duality over a commutative ring

This paper addresses the duality problem for dynamical, linear, time-invariant systems defined over a ring. The duality principle is at the core of theoretical results for systems defined over a field, but this principle can not be applied for systems over a ring. From the definition of controlled and conditioned invariant submodules in the geometric approach, we analyze the relationships among various notions of invariance, using the concept of orthogonal submodule. These logical relations are summarized in two non-equivalent schemes.     

Some remarks on Boolean sums

Summary Neciporuk, Lamagna/Savage and Tarjan determined the monotone network complexity of a set of Boolean sums if any two sums have at most one variable in common. Wegener then solved the case that any two sums have at most k variables in common. We extend his methods and results and consider the case that any set of h +1 distinct sums have at most k variables in common. We use our general results to explicitly construct a set of n Boolean sums over n variables whose monotone complexity is of order n ^5/3. The best previously known bound was of order n ^3/2. Related results were obtained independently by Pippenger.This paper was presented at the MFCS 79 Symposium, Olomouc, Sept. 79     

Some remarks on fuzzy graphs

We show that a fuzzy group can be associated with a fuzzy graph in a natural way. Some properties of fuzzy graphs are considered and we introduce the notions of eccentricity and center. Our examples indicate that results from (crisp) graph theory do not always have analogs for fuzzy graphs.     

Some remarks on optimal interpolation

We give in this note very simple new formula for the optimal solution of a generalized interpolation problem which arises in the weighted sensitivity H^∞-minimization for distributed systems. This is based on our work on skew Toeplitz operators from [2].     

Some remarks on estimation algebras

A necessary condition for the finite dimensionality of the estimation algebra of a single input, single output nonlinear system is derived and then extended to cover the multi-input case. A result of Hijab concerning the finite computability of the output function is also generalised.     

Some remarks on recognizable treeseries

Let K be a commutative ring, Σ a finite ranked alphabet and T _Σ the set of trees over Σ. We show that for any recognizable treeseries S:T → K having finite range and any subset A of K, the forest of all trees t ? T _Σ such that the corresponding coefficient (s, t) belongs to Ais recognizable.

In the case _Σ collapses to a monadic alphabet, we refind an analogous result concerning rational wordseries due to Schutzenberger [11] and Sontag [12]. Some applications are also given.     

Some remarks on non-algebraic adherences

We present some properties of non-algebraic adherences of languages, which can be consistently called context-sensitive adherences, because it is proved that in the Chomsky hierarchy of adherences, context-sensitive and recursive-enumerable adherences coincide. Concerning the center-mapping the traversing from algebraic to non-algebraic takes simple recursive to not necessarily recursive-enumerable. Except for the last equality, the Chomsky hierarchy of adherences is proper: R_Adh ? CF_Adh ? CS_Adh = RE_Adh and moreover, it can be refined in the non-algebraic case by considering the McNaughton-Nivat adherences.Regarding the programming context, adherences occur when we consider the set of divergent computations ‘shapes’ of a program or a program scheme. In this respect, the ‘interpretation’ is powerful enough to translate R_Adh to CS_Adh.     

Some remarks on a paper of G. Feichtinger

Summary The purpose of this note is to give some corrections and supplements to the paper [1] ofG. Feichtinger; especially we show that theorem 1 is wrong, prove the correct version of this theorem and make some comments on the notion of normal automorphism.

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Zusammenfassung In dieser Note werden einige Korrekturen und Ergänzungen zur der Arbeit [1] vonG. Feichtinger gebracht; insbesondere wird gezeigt, daß Theorem 1 falsch ist, es wird die richtige Version dieses Theorems bewiesen, und es werden einige Bemerkungen zum Begriff des normalen Automorphismus' gemacht.

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See the paper ofG. Feichtinger: Some Results on the Relation Between Automata and Their Automorphism Groups. Comp.1, 4, 327 (1966).     

Some remarks on a conjecture in parameter adaptive control

A.S. Morse has raised the following question: Do there exist differentiable functions

$\text{f:R}{}^2\text{→ R}\text{and}\text{g:R}{}^2\text{→ R}$

with the property that for every nonzero real number λ and every (x₀, y₀) ∈ $\text{R}$² the solution (x(t),y(t)) of

$\text{x}\text{?}\text{(t) = x(t) + λf(x(t),y(t))}$

,

$\text{y}\text{?}\text{(t) = g(x(t),y(t))}$

,

$\text{x(0) = x}{}_0\text{, y(0) = y}{}_0$

, is defined for all t ? 0 and satisfies

$\text{lim}\text{t → + ∞}$

and y(t) is bounded on [0,∞)? We prove that the answer is yes, and we give explicit real analytic functions f and g which work. However, we prove that if f and g are restricted to be rational functions, the answer is no.