on the complexity of computing the volume of a polyhedron we show that computing the volume of a polyhedron given either as a list of facets or as a list of vertices is as hard as computing the permanent of a matrix

polytope volume computation a combinatorial form of gram s relation for convex polytopes can be adapted for use in computing polytope volume we present an algorithm for volume computation based on this observation this algorithm is useful in finding the volume of a polytope given as the solution set of a system of linear inequalities p x e r ax b as an illustration we compute a formula for the volume of a projective image of the n cube from this formula we deduce that when a and b have rational entries so that the volume of p is also a rational number the number of binary digits in the denominator of the volume cannot be bounded by a polynomial in the total number of digits in the numerators and denominators of entries of a and b this settles a question posed by dyer and frieze

an inner loop controller guaranteeing robust transient performance for uncertain mimo nonlinear systems an output feedback controller has been recently proposed that has the following features 1 it is an inner loop controller so that it can be added on the existing closed loop system working in harmony with a pre designed possibly non robust outer loop controller 2 it robustifies the closed loop system in a way that the uncertain plant under external disturbance behaves like a disturbance free nominal plant and 3 it recovers the trajectory of the nominal closed loop system in time domain however it is restricted to the single input single output nonlinear systems in this technical note we extend this result for a class of multi input multi output mimo nonlinear systems having the same number of inputs and outputs the tools used in this synthesis are the singular perturbation theory and the multi variable circle criterion an example shows the effectiveness of the proposed method

global stabilization of a class of invertible mimo nonlinear systems in this paper we consider the class of input affine mimo nonlinear systems which are assumed to be invertible and input output linearizable this class includes as a trivial special case the class of mimo systems possessing a well defined vector relative degree it is shown that if a system in this class is strongly minimum phase in a sense specified in the paper it is globally stabilizable via partial state feedback or semiglobally stabilizable via dynamic output feedback the result substantially broadens the class of nonlinear mimo systems for which global semiglobal stabilization via limited state information is known to be possible

fuzzy rule based combination of linear and switching state feedback controllers this paper presents a fuzzy rule base combined controller which is a fuzzy rule based combination of linear and switching state feedback controllers for nonlinear systems subject to parameter uncertainties the switching state feedback controller is employed to drive the system states toward the origin when the system state approaches the origin the linear state feedback controller will gradually replace the switching state feedback controller the smooth transition between the linear and switching state feedback controllers is governed by the fuzzy rules by using the fuzzy rule based combination technique the proposed fuzzy rule base combined controller integrates the advantages of both the linear and switching state feedback controllers but eliminates their disadvantages as a result the proposed fuzzy controller provides good performance during the transient period and the chattering effect is removed when the system state approaches the origin stability conditions will be derived to guarantee the system stability furthermore a saturation function is employed to replace the switching component to alleviate the chattering during the transient period by using the proposed fuzzy rule based combination technique the steady state error introduced by the saturation function can be eliminated application examples will be given to show the merits of the proposed approach

local stabilization of minimum phase nonlinear systems abstract in this note we briefly review the notion of a nonlinear minimum phase system i e a system which when constrained in such a way as to produce zero output evolves with an asymptotically stable dynamics the main purpose of the paper is to show that any minimum phase nonlinear system can always be locally stabilized by smooth state feedback

integral inequalities for some special functions several integral inequalities for the classical hypergeometric confluent hypergeometric and confluent hypergeometric limit functions are given the related results for bessel and whittaker functions as well as for laguerre hermite and jacobi polynomials are discussed

a sweep plane algorithm for generating random tuples in simple polytopes a sweep plane algorithm by lawrence for convex polytope computation is adapted to generate random tuples on simple polytopes in our method an affine hyperplane is swept through the given polytope until a random fraction sampled from a proper univariate distribution of the volume of the polytope is covered then the intersection of the plane with the polytope is a simple polytope with smaller dimension in the second part we apply this method to construct a black box algorithm for log concave and t concave multivariate distributions by means of transformed density rejection author s abstract

design of fuzzy controllers for uncertain nonlinear systems using stability and robustness analyses this paper analyzes the stability and robustness of uncertain nonlinear systems and shows that the analysis results provide an efficient technique for the design of fuzzy controllers based on a fuzzy plant model describing an uncertain nonlinear plant this design involves the derivation of a stability criterion and a robust area in the uncertain parameter space in terms of some measures of the closed loop control system matrices an application example on balancing an inverted pendulum is given to illustrate the simple design methodology the stability and the robustness of the feedback system incorporated with the proposed fuzzy controller

input output to state stability this work explores lyapunov characterizations of the input output to state stability ioss property for nonlinear systems the notion of ioss is a natural generalization of the standard zero detectability property used in the linear case the main contribution of this work is to establish a complete equivalence between the ioss property and the existence of a certain type of smooth lyapunov function as corollaries one shows the existence of norm estimators and obtains characterizations of nonlinear detectability in terms of relative stability and of finite energy estimates

robust output regulation for invertible nonlinear mimo systems abstract the robust output regulation problem for a class of invertible nonlinear mimo systems is addressed in this paper under a mild assumption on the high frequency gain matrix and strongly minimum phase assumption a robust regulator is proposed with the combination of nonlinear internal model and a new extended high gain observer more explicitly the robust output regulation is achieved by employing a novel extended high gain observer based stabilizer such that the regulated system augmented with the nonlinear internal model is asymptotically stabilized the novelty of this approach consists in the use of an extended observer to estimate under mild assumptions all unmeasured terms

an approach to fuzzy control of nonlinear systems stability and design issues presents a design methodology for stabilization of a class of nonlinear systems first the authors represent a nonlinear plant with a takagi sugeno fuzzy model then a model based fuzzy controller design utilizing the concept of the so called parallel distributed compensation is employed the main idea of the controller design is to derive each control rule so as to compensate each rule of a fuzzy system the design procedure is conceptually simple and natural moreover the stability analysis and control design problems can be reduced to linear matrix inequality lmi problems therefore they can be solved efficiently in practice by convex programming techniques for lmis the design methodology is illustrated by application to the problem of balancing and swing up of an inverted pendulum on a cart

a gain matrix decomposition and some of its applications abstract any real square matrix m can be written as m u i l s where u is a matrix of 0 s 1 s and 1 s having exactly one nonzero element in each row and column l is a strictly lower triangular matrix and s is a symmetric positive semidefinite matrix the aim of this paper is to demonstrate the utility of this easily derived fact this is done in two ways first the decomposition is used to develop an identifier based solution to a simplified multivariable adaptive stabilization problem solved previously using nonidentifier based methods second it is briefly explained how to use the decomposition together with hysteresis switching and a certain lifted discrete time system representation to obtain an excitation free identifier based adaptive stabilizer for the entire class of n dimensional siso controllable observable discrete time linear process models this is accomplished by exploiting a new method of discrete time parameter adjustment called pseudo continuous tuning

comments on robust stabilization of a class of uncertain nonlinear systems via fuzzy control quadratic stabilizability h sup spl infin control theory and linear matrix inequalities this paper presents stability analysis for a class of uncertain nonlinear systems and a method for designing robust fuzzy controllers to stabilize the uncertain nonlinear systems first a stability condition for takagi and sugeno s fuzzy model is given in terms of lyapunov stability theory next new stability conditions for a generalized class of uncertain systems are derived from robust control techniques such as quadratic stabilization h sup spl infin control theory and linear matrix inequalities the derived stability conditions are used to analyze the stability of takagi and sugeno s fuzzy control systems with uncertainty which can be regarded as a generalized class of uncertain nonlinear systems the design method employs the so called parallel distributed compensation important issues for the stability analysis and design are remarked finally three design examples of fuzzy controllers for stabilizing nonlinear systems and uncertain nonlinear systems are presented

stability analysis and design of fuzzy control systems abstract the stability analysis and the design technique of fuzzy control systems using fuzzy block diagrams are discussed first we show the concept of fuzzy blocks and consider the connection problems of fuzzy blocks diagrams we derive some theorems and corollaries with respect to two basic types of connections of fuzzy blocks in order to preserve some properties in a connection of fuzzy blocks continuous piecewise polynomial membership functions are defined secondly a sufficient condition which guarantees the stability of fuzzy systems is obtained in terms of lyapunov s direct method we give an important fact based on this condition thirdly we propose a new design technique of a fuzzy controller the fuzzy block diagrams and the stability analysis are applied to the design problems of a model based fuzzy controller

output to state stability and detectability of nonlinear systems the notion of input to state stability iss has proved to be useful in nonlinear systems analysis this paper discusses a dual notion output to state stability oss a characterization is provided in terms of a dissipation inequality involving storage lyapunov functions combining iss and oss there results the notion of input output to state stability ioss which is also studied and related to the notion of detectability the existence of observers and output injection

output input stability implies feedback stabilization abstract we study the recently introduced notion of output input stability which is a robust variant of the minimum phase property for general smooth nonlinear control systems this paper develops the theory of output input stability in the multi input multi output setting we show that output input stability is a combination of two system properties one related to detectability and the other to left invertibility for systems affine in controls we derive a necessary and sufficient condition for output input stability which relies on a global version of the nonlinear structure algorithm this condition leads naturally to a globally asymptotically stabilizing state feedback strategy for affine output input stable systems

on the construction of static stabilizers and static output trackers for dynamically linearizable systems related results and applications it is shown that the property of dynamic linearizability to be understood as linearizability by means of the dynamic extension algorithm implies the existence of static possibly time varying control laws yielding asymptotic output tracking with arbitrary speed of convergence and asymptotic stabilization with a computable bound on the region of attraction similar results hold for systems which are only input output linearizable by means of dynamic state feedback provided that the inverse dynamics possess certain stability properties applications of these results to the problem of regional stabilization of a vtol aircraft is considered together with the tracking problem for a class of flexible joints robots moreover a novel parameterization for flexible joint robots is also proposed

on normal forms of nonlinear systems affine in control the nonlinear equivalences of both finite and infinite zero structures of linear systems have been well understood for single input single output systems and have found many applications in nonlinear control theory the extensions of these notions to multiple input multiple output systems have proven to be highly sophisticated in this paper we propose constructive algorithms for decomposing a nonlinear system that is affine in control these algorithms require modest assumptions on the system and apply to general multiple input multiple output systems that do not necessarily have the same number of inputs and outputs they lead to various normal form representations and reveal the structure at infinity the zero dynamics and the invertibility properties all of which represent nonlinear equivalences of relevant linear system structural properties

on the convergence of an extended state observer for nonlinear systems with uncertainty abstract the extended state observer first proposed by jingqing han in j q han a class of extended state observers for uncertain systems control decis 10 1 1995 85 88 in chinese is the key link toward the active disturbance rejection control that is taking off as a technology after numerous successful applications in engineering unfortunately there is no rigorous proof of convergence to date in this paper we attempt to tackle this long unsolved extraordinary problem the main idea is to transform the error equation of objective system with its extended state observer into a asymptotical stable system with a small disturbance for which the effect of total disturbance error is eliminated by the high gain

stabilization by output feedback of multivariable invertible nonlinear systems in this paper the problem of global stabilization of a rather general class of mimo nonlinear systems is addressed the systems considered in the paper are invertible have a trivial zero dynamics and possess a normal form in which certain multipliers are functions of the state vector of a special kind while special structural dependence of such multipliers on the components state vector has been exploited before in the context of achieving stabilization via full state feedback though the novelty in the approach of this paper is that a peculiar structure is identified which happens to be intimately related to the property of uniform complete observability thus making the design of observers possible and to the property of uniform invertibility relations never established before in the literature as a result for this class of mimo nonlinear systems a dynamic output feedback law can be designed yielding semiglobal and even global under appropriate assumptions asymptotic stability

output feedback stabilization of nonlinear mimo systems having uncertain high frequency gain matrix abstract the purpose of this paper is to provide a method for semi global asymptotic stabilization of a nonlinear minimum phase mimo system under a mild hypothesis of the so called high frequency gain matrix this result is based on a non trivial extension to the mimo setting of the approach based on the use of extended observers as a byproduct a dynamic output feedback control is obtained that asymptotically stabilizes the equilibrium of the closed loop system in spite of uncertainties in the high frequency gain matrix

inequalities for formal power series and entire functions we present several integral and exponential inequalities for formal power series and for both arbitrary entire functions of exponential type and generalized borel transforms they are obtained through certain limit procedures which involve the multiparameter binomial inequalities integral inequalities for continuous functions and weighted norm inequalities for analytic functions some applications to the confluent hypergeometric functions bessel functions laguerre polynomials and trigonometric functions are discussed also some generalizations are given

multivariable adaptive control using high frequency gain matrix factorization in this note we extend the application of a less restrictive condition about the high frequency gain matrix to design stable direct model reference adaptive control for a class of multivariable plants with relative degree greater than one the new approach is based on a control parametrization derived from a factorization of the high frequency gain matrix k sub p in the form of a product of three matrices one of them being diagonal three possible factorizations are presented only the signs of the diagonal factor or equivalently the signs of the leading principal minors of k sub p are assumed known

global stabilization of partially linear composite systems a linear stabilizable nonlinear asymptotically stable cascade system is globally stabilizable by smooth dynamic state feedback if a the linear subsystem is right invertible and weakly minimum phase and b the only linear variables entering the nonlinear subsystem are the output and the zero dynamics corresponding to this output both of these conditions are coordinate free and there is freedom of choice for the linear output variable this result generalizes several earlier sufficient conditions for stabilizability moreover the weak minimum phase condition for the linear subsystem cannot be relaxed unless a growth restriction is imposed on the nonlinear subsystem

asymptotic stabilization of minimum phase nonlinear systems how a class of multivariable nonlinear systems can be stabilized about an equilibrium via smooth state feedback is shown more precisely conditions under which for every compact set of initial states it is possible to design a feedback law which drives to the equilibrium all initial states in this compact set are described the theory includes the development of globally defined transformations of the system equations to their global normal form

applying young diagrams to 2 symmetric fuzzy measures with an application to general fuzzy measures abstract in this paper we apply young diagrams a well known object appearing in combinatorics and group representation theory to study some properties of the polytope of 2 symmetric fuzzy measures with respect to a given partition a subpolytope of the set of fuzzy measures the main result in the paper allows to build a simple and fast algorithm for generating points on this polytope in a random fashion besides we also study some other properties of this polytope as for example its volume in the last section we give an application of this result to the problem of identification of general fuzzy measures

weighted norm inequalities for analytic functions we present a weighted norm inequality involving convolutions of arbitrary analytic functions and certain confluent hypergeometric functions this result implies a family of weighted norm inequalities both for entire functions of exponential type and for generalized hypergeometric series the approach is based on author s general inequality for continuous functions and some hypergeometric transformations

on random generation of fuzzy measures in this paper we deal with the problem of obtaining a random procedure for generating fuzzy measures we use the fact that the polytope of fuzzy measures is an order polytope so that it has special properties that allow to build a uniform algorithm first we derive an exact procedure based on an existing procedure to generate random linear extensions then we study the applicability of this algorithm to the polytope of fuzzy measures showing that the complexity grows dramatically with the cardinality of the referential set next we study other heuristics appearing in the literature for the polytope of fuzzy measures our results seem to mean that these procedures cannot be applied to this case either finally we propose another heuristic that reduces the complexity and could be used instead of the other procedures we finish comparing the performance of this heuristic with the other possibilities showing that our alternative seems to work better for the polytope of fuzzy measures

a modified algorithm for invertibility in nonlinear systems the algorithm of hirschorn 1 for the construction of inverses of nonlinear systems is modified and new sufficient conditions for invertibility and inverse system for a larger class of systems are obtained for the systems which do not satisfy the invertibility conditions a characterization of input space on which the input output map is injective is derived

output input stability and minimum phase nonlinear systems this paper introduces and studies the notion of output input stability which represents a variant of the minimum phase property for general smooth nonlinear control systems the definition of output input stability does not rely on a particular choice of coordinates in which the system takes a normal form or on the computation of zero dynamics in the spirit of the input to state stability philosophy it requires the state and input of the system to be bounded by a suitable function of the output and derivatives of the output modulo a decaying term depending on initial conditions the class of output input stable systems thus defined includes all affine systems in global normal form whose internal dynamics are input to state stable and also all left invertible linear systems whose transmission zeros have negative real parts as an application we explain how the new concept enables one to develop a natural extension to nonlinear systems of a basic result from linear adaptive control