a generalization of myhill nerode theorem for fuzzy languages the well known myhill nerode theorem provides a necessary and sufficient condition for a language to be regular in the context of fuzzy languages and automata theory myhill nerode type theorems have been proved for fuzzy languages with finite range this paper introduces a new right equivalence relation on the free monoid of an alphabet based on the notion of factorization of fuzzy languages the index of this relation for a fuzzy language with infinite range can be finite this fact allows us to generalize the myhill nerode theorem for any kind of fuzzy languages in this paper is proved that the following two conditions are mutually equivalent for a given fuzzy language x i there exists a factorization such that the right equivalence relation of x defined via the factorization has a finite index ii the fuzzy language x is recognized by a fuzzy deterministic finite automaton

simultaneous h control of a finite collection of linear plants with a single nonlinear digital controller abstract this paper considers the problem of simultaneous h control of a finite collection of linear time invariant systems via a nonlinear digital output feedback controller the main result is given in terms of the existence of suitable solutions to riccati algebraic equations and a dynamic programming equation our main result shows that if the simultaneous h control problem for k linear time invariant plants of orders n 1 n 2 n k can be solved then this problem can be solved via a nonlinear time invariant controller of order n n 1 n 2 n k

asymptotic feedback stabilization a sufficient condition for the existence of control lyapunov functions abstract in this paper we study the feedback stabilization problem for a wide class of nonlinear systems that are affine in the control we offer sufficient conditions for the existence of control lyapunov functions that according to 3 23 and 28 30 guarantee stabilization at a specified equilibrium by means of a feedback law which is smooth except possibly at the equilibrium we note that the results of the paper present a local nature

a lyapunov like characterization of asymptotic controllability it is shown that a control system in bf r n is asymptotically controllable to the origin if and only if there exists a positive definite continuous functional of the states whose derivative can be made negative by appropriate choices of controls

stabilization with relaxed controls cannot in general be stabilized using a continuous closed loop control u x even if each state separately can be driven asymptotically to the origin an example is analyzed in section 2 in this paper we examine the possibility of stabilizing such systems with a continuous closed loop relaxed control we find indeed that the family of systems stabilizable with relaxed controls is larger than the family of those stabilizable with ordinary controls an even larger class is obtained if the continuity of the closed loop at the origin is not required the latter class includes all one dimensional systems for which states can be driven asymptotically to the origin this result does not hold in two dimensional systems and we provide a counter example it should be pointed that relaxed control type stabilization is used both in theory and in practice the method is known as dither we shall comment on the similarities lyapunov functions for the system 1 help us in the construction of the continuous closed loop stabilizers in fact we find that the existence of a smooth lyapunov function is equivalent to the existence of a stabilizing closed loop which is continuous except possibly at the origin an additional condition on the lyapunov function implies the continuity at the origin as well we present these results in section 4 after a brief introduction of closed loop relaxed controls notations and terminology in section 3 prior to that in section 2 we discuss an example illustrating the power of relaxed controls in the particular case of systems linear in the controls relaxed controls can be replaced by ordinary controls this is discussed in section 5 the role of lyapunov functions in the stability and stabilization theories is of course well known examples of systems with lyapunov functions are available in the literature we display some in section 6 along with general comments on the construction applications and counterexamples including one which cannot be continuously stabilized yet possesses a nonsmooth lyapunov function closed loop stabilization with ordinary controls is analyzed extensively in the literature see sontag 8 sussmann ll and references therein lyapunov functions techniques in stabil

a partial differential inequality for dissipative nonlinear systems abstract in this paper we show that storage functions for dissipative systems can be characterised by a partial differential inequality in the weak or viscosity sense the storage functions need not be differentiable this characterisation is stable with respect to perturbations we formulate a version of the positive real lemma for passive systems using the weak sense framework

brief on simultaneous h control and strong h stabilization in this note the simultaneous h control problem is addressed it is first shown that the existence of a single controller which simultaneously stabilizes two systems with a given h norm level is equivalent to that of a strong h stabilization controller of an associated system moreover the result is extended to the case involving r 2 systems then a sufficient condition for the strong h stabilization is proposed based on the bounded real lemma finally a simple numerical example is presented to show the effectiveness of the result

simultaneous stabilization of a set of nonlinear port controlled hamiltonian systems this paper investigates simultaneous stabilization of a set of nonlinear port controlled hamiltonian pch systems and proposes a number of results on the design of simultaneous stabilization controllers for the pch systems firstly the case of two pch systems is studied using the dissipative hamiltonian structural properties the two systems are combined to generate an augmented pch system with which some results on the control design are then obtained for the case that there exist parametric uncertainties in the two systems hamiltonian structures an adaptive simultaneous stabilization controller is proposed when there are external disturbances and parametric uncertainties in the two systems two simultaneous stabilization controllers are designed for the systems one is a robust controller and the other is a robust adaptive one secondly the case of more than two pch systems is investigated and a new result is proposed for the simultaneous stabilization of the systems finally two illustrative examples are studied by using the results proposed in this paper simulations show that the simultaneous stabilization controllers obtained in this paper work very well

simultaneous h infty control for nonlinear systems this note develops a novel method for designing simultaneous h infin state feedback controllers for a collection of single input nonlinear systems based on the kalman yakubovich popov lemma necessary and sufficient conditions for the existence of simultaneous h infin controllers are derived by the control storage function approach a universal formula for constructing continuous time invariant simultaneous h infin state feedback controllers is presented

simultaneous stabilization of linear and nonlinear systems by means of nonlinear state feedback the simultaneous stabilization resp asymptotic stabilization of a countable family of control systems consists of finding a control which stabilizes resp asymptotically stabilizes all the systems in the family in this paper we introduce a new method which enables us to show that given any countable family of stabilizable nonlinear systems there exists a continuous state feedback law which simultaneously stabilizes not asymptotically the family then by enriching this method we prove that any finite family of stabilizable linear time invariant lti systems can be simultaneously exponentially stabilized by means of nonlinear time varying state feedback we also derive sufficient conditions for the simultaneous asymptotic stabilizability of countably infinite families of lti systems finally sufficient conditions for the simultaneous asymptotic stabilizability of finite families of nonlinear systems are provided and used for the simultaneous asymptotic stabilization of certain pairs of nonlinear homogeneous systems

simultaneous stabilization with near optimal lqr performance we consider the use of linear time varying controllers for simultaneous stabilization and performance we prove that for every finite set of plants we can design a linear time varying controller which provides not only closed loop stability but also near optimal lqr type performance by adding a mild nonlinearity in the form of a nonlinear delay we end up with an improved controller which provides near optimal performance for each possible model as well as for the case when the plant changes occasionally

a universal construction of artstein s theorem on nonlinear stabilization abstract this note presents an explicit proof of the theorem due to artstein which states that the existence of a smooth control lyapunov function implies smooth stabilizability moreover the result is extended to the real analytic and rational cases as well the proof uses a universal formula given by an algebraic function of lie derivatives this formula originates in the solution of a simple riccati equation

simultaneous quadratic stabilization for discrete time nonlinear systems a constructive method is developed for designing simultaneously quadratically stabilizing state feedback controllers for a collection of single input discrete time nonlinear systems in the context of quadratic control lyapunov functions a necessary and sufficient condition for the existence of simultaneously quadratically stabilizing state feedback laws is derived an explicit formula for constructing continuous simultaneously stabilizing controllers is presented the obtained results are verified by a numerical example

brief practical l2 disturbance attenuation for nonlinear systems in this work a notion of generalized l 2 gain for nonlinear systems where the gain is considered as a function of the state instead of a global constant is presented this new notion seems to be adequate to characterize the gain properties of several nonlinear systems which do not possess a uniform l 2 gain property i e the l 2 gain depends on the operating point moreover a notion of practical l 2 gain attenuation which extends the standard definition and parallels mutatae mutandis the concepts of practical stability is also proposed

fuzzy language on free monoid in this paper we shall introduce the notion of fuzzy regular language adjunctive language and dense language and discuss their algebraic character and algebraic properties

robust h control for polytopic nonlinear control systems this technical note develops a robust control storage function method for solving the h control problem of single input polytopic nonlinear control systems necessary and sufficient conditions for verifying robust control storage functions are provided moreover conditions for the existence of continuous robust h state feedback controllers are derived an explicit formula for constructing continuous robust h controllers is presented finally a numerical example is given for illustration

fuzzy automata with e moves compute fuzzy measures between strings this paper introduces fuzzy automata with transitions by empty string moves and shows their relationship with other classes of classical fuzzy automata the move represents a state change of the automaton without consuming any symbol of the input string in approximate string matching moves allow to model the effect of the insertion of a symbol one of the possible edit operations we provide a fuzzy measure between strings based on the concepts of string alignments and fuzzy edit operations the main contribution of this paper is to prove that a particular class of fuzzy automata with moves computes those fuzzy measures without restricting the number of errors between the strings given a fuzzy measure a building method for constructing the fuzzy automaton with moves that computes it is also proposed

on the recognizability of fuzzy languages ii it is shown that a fuzzy language is recognized by a finite monoid iff it has finitely many right derivatives iff it has finitely many left derivatives finally we show that the behaviors of lukasiewicz automata have the above properties

a sufficient condition for simultaneous stabilization the condition under which it is possible to find a single controller that stabilizes k single input single output linear time invariant systems p sub i s i 1 k is investigated the concept of avoidance in the complex plane is introduced and used to derive a sufficient condition for k systems to be simultaneously stabilizable a method for constructing a simultaneous stabilizing controller is also provided and is illustrated by an example

simultaneous stabilization for a collection of single input nonlinear systems this paper presents a novel method for designing a controller that simultaneously stabilizes a collection of single input nonlinear systems the control lyapunov function approach is used to derive necessary and sufficient conditions for the existence of time invariant simultaneously stabilizing state feedback controllers additionally a universal formula for constructing a continuous simultaneously stabilizing controller when the provided sufficient condition is satisfied is presented for any collection of second order and third order feedback linearizable systems in canonical form global simultaneous stabilization via a single state feedback controller is shown to be always possible two examples are included for illustration

non three space properties and szlenk index abstract we prove that for any non zero countable ordinal which is not additively indecomposable the property of having szlenk index not exceeding is not a three space property this complements a result of brooker and lancien which states that if is additively indecomposable then having szlenk index not exceeding is a three space property

fractional representation algebraic geometry and the simultaneous stabilization problem an explicit relationship between the fractional representation approach feedback system design and the algebro geometric approach to system theory is formulated and used to derive a global solution to the feedback system problem these techniques are then applied to the simultaneous stabilization problem yielding a natural geometric criterion for a set of plants to be simultaneously stabilized by a single compensator