a general class of sliding surface for sliding mode control in the conventional sliding mode control the relative degree of the chosen sliding surface is usually one this paper addresses a general class of sliding surface whose relative degree is no longer restricted to one we show that when the relative degree is more than two it is inevitable that there will exist a limit cycle for the nonlinear sign function and for the case of relative degree two the system possesses poor phase margin however these disadvantages can be remedied remarkably by smoothing out the control nonlinearity in a suitable boundary layer with a saturation function that ensures asymptotic stability of the overall system

design of variable structure controllers with continuous switching control a modified variable structure controller is proposed to deal with the conirol chattering problem which is one of the major difficulties for implementing the variable structure control vsc the new control strategy is effective for a class of uncertain nonlinear dynamical systems based on a suitably designed sliding sector a continuous switching controller whose control structure is switched whereas the control signal at switching remains continuous is constructed the complete attractiveness and the stability properties of the sliding sector in the sense of lyapunov stability ensure the retention of benefits achieved with the conventional vsc some typical examples are also investigated which show the improvement of system performance

approximation based control of nonlinear mimo time delay systems approximation based control is presented for a class of multi input multi output mimo nonlinear systems in block triangular form with unknown state delays neural networks nns are utilized to approximate and compensate for unknown functions in the system dynamics including the unknown bounds of the functions of delayed states the use of a separation technique removes the need for any assumption on the function of delayed states and allows the handling of multiple delays in each function of delayed states by combining the use of lyapunov krasovskii functionals and adaptive nn backstepping the proposed control guarantees that all closed loop signals remain bounded while the outputs converge to a neighborhood of the desired trajectories simulation results demonstrate the effectiveness of the proposed scheme

learning variable structure control approaches for repeatable tracking control tasks for the finite period tracking control task repeatable over iterations uniformly bounded learning control is synthesized into variable structure control to approximate the equivalent control and to realize perfect tracking

trees with minimal laplacian coefficients let g be a simple undirected graph with the characteristic polynomial of its laplacian matrix l g p g m email protected k 0 n 1 kc email protected n k it is well known that for trees the laplacian coefficient c n 2 is equal to the wiener index of g while c n 3 is equal to the modified hyper wiener index of the graph in this paper we characterize n vertex trees with given matching number m which simultaneously minimize all laplacian coefficients the extremal tree a n m is a spur obtained from the star graph s n m 1 with n m 1 vertices by attaching a pendant edge to each of certain m 1 non central vertices of s n m 1 in particular a n m minimizes the wiener index the modified hyper wiener index and the recently introduced incidence energy of trees defined as ie g email protected k 0 email protected k where m k are the eigenvalues of signless laplacian matrix q g d g a g we introduced a general r transformation which decreases all laplacian coefficients simultaneously in conclusion we illustrate on examples of wiener index and incidence energy that the opposite problem of simultaneously maximizing all laplacian coefficients has no solution

dynamic parallel distributed compensation for takagi sugeno fuzzy systems an lmi approach abstract this paper presents a unified systematic framework for designing dynamic feedback controllers for nonlinear systems described by takagi sugeno t s models both stabilization and multi objective control synthesis problems have been addressed the control laws are in the form of so called dynamic parallel distributed compensation dpdc which are essentially nonlinear dynamic feedback controllers the associated control synthesis problems are formulated as linear matrix inequality lmi problems i e the parameters of the dpdc controllers are obtained from a set of lmi conditions the approach in this paper can also be applied to hybrid or switching systems

design of decentralised adaptive sliding mode controllers for large scale systems with mismatched perturbations based on the lyapunov stability theorem a methodology for designing a decentralised adaptive sliding mode control scheme is proposed in this paper this scheme is implemented for a class of large scale systems with both matched and mismatched perturbations the perturbations and the interconnection terms are assumed to be norm bounded under certain mild conditions the decentralised sliding surfaces with adaptive mechanisms embedded are specially designed for each subsystem so that when each subsystem enters the sliding mode the mismatched perturbations and the effects of interconnections can be effectively overcome and achieve asymptotic stability the decentralised controller with embedded adaptive mechanisms is capable of driving the controlled state trajectories into the designated sliding surface in finite time this is also achieved without the knowledge of upper bounds of the perturbations except those of the uncertainties in the input channels a numerical example is included to demonstrate the feasibility of the proposed control scheme

multiobjective output feedback control via lmi optimization this paper presents an overview of a linear matrix inequality lmi approach to the multiobjective synthesis of linear output feedback controllers the design objectives can be a mix of h sub spl infin performance h sub 2 performance passivity asymptotic disturbance rejection time domain constraints and constraints on the closed loop pole location in addition these objectives can be specified on different channels of the closed loop system when all objectives are formulated in terms of a common lyapunov function controller design amounts to solving a system of linear matrix inequalities the validity of this approach is illustrated by a realistic design example

signless laplacians of finite graphs we survey properties of spectra of signless laplacians of graphs and discuss possibilities for developing a spectral theory of graphs based on this matrix for regular graphs the whole existing theory of spectra of the adjacency matrix and of the laplacian matrix transfers directly to the signless laplacian and so we consider arbitrary graphs with special emphasis on the non regular case the results which we survey old and new are of two types a results obtained by applying to the signless laplacian the same reasoning as for corresponding results concerning the adjacency matrix b results obtained indirectly via line graphs among other things we present eigenvalue bounds for several graph invariants an interpretation of the coefficients of the characteristic polynomial a theorem on powers of the signless laplacian and some remarks on star complements

adaptive statistic tracking control based on two step neural networks with time delays this paper presents a new type of control framework for dynamical stochastic systems called statistic tracking control stc the system considered is general and non gaussian and the tracking objective is the statistical information of a given target probability density function pdf rather than a deterministic signal the control aims at making the statistical information of the output pdfs to follow those of a target pdf for such a control framework a variable structure adaptive tracking control strategy is first established using two step neural network models following the b spline neural network approximation to the integrated performance function the concerned problem is transferred into the tracking of given weights the dynamic neural network dnn is employed to identify the unknown nonlinear dynamics between the control input and the weights related to the integrated function to achieve the required control objective an adaptive controller based on the proposed dnn is developed so as to track a reference trajectory stability analysis for both the identification and tracking errors is developed via the use of lyapunov stability criterion simulations are given to demonstrate the efficiency of the proposed approach

analysis and design of variable structure systems using a geometric approach multivariable variable structure systems in the sliding mode are studies using a geometric approach the properties of system order reduction and disturbance rejection are proved using projector theory new design methods for the choice of switching hyperplanes are derived for the closed loop eigenvalue eigenvector assignment problem

a decentralized model reference adaptive variable structure controller for large scale time varying delay systems in this note the problem of decentralized model reference adaptive variable structure control for a class of perturbed large scale systems with varying time delay interconnections is investigated based on the lyapunov stability theorem an adaptive variable structure control strategy for solving the robust tracking problem without the knowledge of upper bound of perturbations is developed the use of adaptive technique is to adapt the unknown upper bound of perturbations so that the objective of globally asymptotical stability is achieved once the system enters the sliding manifold the dynamics of controlled systems are insensitive to matching perturbations finally an example is given to demonstrate the feasibility of the proposed control scheme

continuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamic systems we consider a dynamic system containing uncertain elements only the set of possible values of these uncertainties is known based on this information a class of state feedback controls is proposed in order to guarantee uniform ultimate boundedness of every system response within an arbitrarily small neighborhood of the zero state these feedback controls are continuous functions of the state

laplacian coefficients of trees with a given bipartition let g be a graph of order n and g k 0n 1 kck n k the laplacian characteristic polynomial of g zhou and gutman 19 proved that among all trees of order n the kth coefficient ck is largest when the tree is a path and is smallest for a star in this paper for two given positive integers p and q p q we characterize the trees with a given bipartition p q which have the minimal and second minimal laplacian coefficients

robust adaptive sliding mode control for fuzzy systems with mismatched uncertainties this paper is devoted to design adaptive sliding mode controllers for the takagi sugeno t s fuzzy system with mismatched uncertainties and exogenous disturbances the uncertainties in state matrices are mismatched and norm bounded while the exogenous disturbances are assumed to be bounded with an unknown bound which is estimated by a simple and effective adaptive approach both state and static output feedback sliding mode control problems are considered in terms of linear matrix inequalities lmis both sliding surfaces and sliding mode controllers can be easily obtained via a convex optimization technique finally two simulation examples and a real experiment are utilized to illustrate the applicability and effectiveness of the design procedures proposed in this paper

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

on the ordering of trees by the laplacian coefficients abstract we generalize the results from x d zhang x p lv y h chen ordering trees by the laplacian coefficients linear algebra appl 2009 doi 10 1016 j laa 2009 04 018 on the partial ordering of trees with given diameter for two n vertex trees t 1 and t 2 if c k t 1 c k t 2 holds for all laplacian coefficients c k k 0 1 n we say that t 1 is dominated by t 2 and write t 1 c t 2 we proved that among n vertex trees with fixed diameter d the caterpillar c n d has minimal laplacian coefficients c k k 0 1 n the number of incomparable pairs of trees on 18 vertices is presented as well as infinite families of examples for two other partial orderings of trees recently proposed by mohar for every integer n we construct a chain t i i 0 m of n vertex trees of length n 2 4 such that t 0 s n t m p n and t i c t i 1 for all i 0 1 m 1 in addition the characterization of the partial ordering of starlike trees is established by the majorization inequalities of the pendent path lengths we determine the relations among the extremal trees with fixed maximum degree and with perfect matching and further support the laplacian coefficients as a measure of branching

multiobjective h sub 2 h sub spl infin control for a linear time invariant system with several disturbance inputs and controlled outputs we show how to minimize the nominal h sub 2 norm performance in one channel while keeping bounds on the h sub 2 norm or h sub spl infin norm performance implying robust stability in the other channels this multiobjective h sub 2 h sub spl infin problem in an infinite dimensional space is reduced to sequences of finite dimensional convex optimization problems we show how to compute the optimal value and how to numerically detect the existence of a rational optimal controller if it exists we reveal how the novel trick of optimizing the trace norm of the youla parameter over certain convex constraints allows one to design a nearly optimal controller whose youla parameter is of the same order as the optimal one

on the laplacian coefficients of trees with a perfect matching let t be a tree with n vertices and let t k 0n 1 kck t n k be the characteristic polynomial of laplacian matrix of t it is well known that cn 2 t is equal to the wiener index of t while cn 3 t is equal to the modified hyper wiener index of t for two n vertex trees t1 and t2 write t1 t2 if ck t1 ck t2 for all 0 k n let n be the set of all trees with 2n vertices and perfect matchings except p2n and an 1 1 where ps is a path with s vertices and as t is a tree obtained from a star st 1 with t 1 vertices by attaching s pendent paths of length 2 to the center of st 1 in this paper we first give a graphic transformation that increases all laplacian coefficients of an arbitrary tree we then determine the minimum element and the second minimum element in n under the partial order and we finally identify the maximum element and second maximum element in n under the partial order furthermore we characterize some trees with extremal wiener indices and laplacian like energies

a unifying design of sliding mode and classical controllers a theory is given to unify output sliding mode control and classical control the idea is based on defining the sliding variable in such a way that once the system begins to slide the classical controller transfer function is realized this idea leads to the development of a hybrid sliding and classical controller which retains the merits of both types of controllers but eliminates their respective limitations the proposed method is robust and applies to non minimum phase single input single output siso systems no state measurement is required

on the laplacian coefficients of acyclic graphs abstract let g be a graph of order n and let g k 0 n 1 k c k n k be the characteristic polynomial of its laplacian matrix zhou and gutman recently proved that among all trees of order n the kth coefficient ck is largest when the tree is a path and is smallest for stars a new proof and a strengthening of this result is provided a relation to the wiener index is discussed

robust pole placement in lmi regions discusses analysis and synthesis techniques for robust pole placement in linear matrix inequality lmi regions a class of convex regions of the complex plane that embraces most practically useful stability regions the focus is on linear systems with static uncertainty on the state matrix for this class of uncertain systems the notion of quadratic stability and the related robustness analysis tests are generalized to arbitrary lmi regions the resulting tests for robust pole clustering are all numerically tractable because they involve solving linear matrix inequalities lmis and cover both unstructured and parameter uncertainty these analysis results are then applied to the synthesis of dynamic output feedback controllers that robustly assign the closed loop poles in a prescribed lmi region with some conservatism this problem is again tractable via lmi optimization in addition robust pole placement can be combined with other control objectives such as h sub 2 or h sub spl infin performance to capture realistic sets of design specifications physically motivated examples demonstrate the effectiveness of this robust pole clustering technique

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

memoryless state feedback controller design for time delay systems with matched uncertain nonlinearities the robust control problem for a class of uncertain time delay systems is investigated the systems include multiple time delays and uncertain nonlinear functions to our best knowledge no controller design method for the systems is reported in the literature except for sliding mode control smc however the sliding mode controller must use the delayed state of the system so that the precise time delays have to be known and also a large memory is needed for controller implementation to overcome the shortcomings we employ the backstepping method to design a memoryless state feedback controller by constructing a novel lyapunov krasovskii functional we show that the resulting closed loop system is asymptotically stable simulations are done to verify the effectiveness of the proposed method

mixed h sub 2 h sub spl infin fuzzy output feedback control design for nonlinear dynamic systems an lmi approach this study introduces a mixed h sub 2 h sub spl infin fuzzy output feedback control design method for nonlinear systems with guaranteed control performance first the takagi sugeno fuzzy model is employed to approximate a nonlinear system next based on the fuzzy model a fuzzy observer based mixed h sub 2 h sub spl infin controller is developed to achieve the suboptimal h sub 2 control performance with a desired h sub spl infin disturbance rejection constraint a robust stabilization technique is also proposed to override the effect of approximation error in the fuzzy approximation procedure by the proposed decoupling technique and two stage procedure the outcome of the fuzzy observer based mixed h sub 2 h sub spl infin control problem is parametrized in terms of the two eigenvalue problems evps one for observer and the other for controller the evps can be solved very efficiently using the linear matrix inequality lmi optimization techniques a simulation example is given to illustrate the design procedures and performances of the proposed method

stability analysis of the simplest takagi sugeno fuzzy control system using circle criterion in this paper we present two sufficient conditions in the frequency domain for the global stability analysis of the simplest takagi sugeno t s fuzzy control system based on the circle criterion and its graphical interpretation these conditions are significant in graphically designing the simplest t s fuzzy controller in the frequency domain four numerical examples are provided to demonstrate the efficiency of our two frequency domain based conditions furthermore this t s fuzzy controller and the linear proportional controllers are also compared it is concluded that the simplest t s fuzzy controller can outperform the linear proportional controllers in a noisy environment with external disturbances

a robust multi objective dpdc for uncertain t s fuzzy systems this paper offers a design procedure for a robust pole placement problem with h objective via dynamic output feedback control for a class of uncertain fuzzy systems takagi sugeno t s fuzzy system is used for modeling and dynamic parallel distributed compensation dpdc technique is employed in the controller design solutions to the problems are derived in terms of a family of linear matrix inequalities lmis and uncertainties are directly considered in the conditions that determine the controller effectiveness of the proposed design technique is illustrated by simulation for an inverted pendulum system

simple method for model reference adaptive control abstract a simple method is presented for combined signal synthesis and parameter adaptation within the framework of model reference adaptive control theory the results are obtained using a simple derivation based on an improved lyapunov function

on the second largest laplacian eigenvalues of graphs the second largest laplacian eigenvalue of a graph is the second largest eigenvalue of the associated laplacian matrix in this paper we study extremal graphs for the extremal values of the second largest laplacian eigenvalue and the laplacian separator of a connected graph respectively all simple connected graphs with second largest laplacian eigenvalue at most 3 are characterized it is also shown that graphs with second largest laplacian eigenvalue at most 3 are determined by their laplacian spectrum moreover the graphs with maximum and the second maximum laplacian separators among all connected graphs are determined

h sub spl infin design with pole placement constraints an lmi approach this paper addresses the design of state or output feedback h sub spl infin controllers that satisfy additional constraints on the closed loop pole location sufficient conditions for feasibility are derived for a general class of convex regions of the complex plane these conditions are expressed in terms of linear matrix inequalities lmis and the authors formulation is therefore numerically tractable via lmi optimization in the state feedback case mixed h sub 2 h sub spl infin synthesis with regional pole placement is also discussed finally the validity and applicability of this approach are illustrated by a benchmark example

a variable structure approach to feedback regulation of uncertain dynamical systems feedback control of uncertain dynamical systems is considered adopting a deterministic approach based on the theory of variable structure systems a feedback regulator is developed which guarantees global asymptotic stability or global ultimate boundedness under certain hypotheses on the bounded system uncertainties and disturbances

brief paper adaptive controller design for uncertain fuzzy systems using variable structure control approach based on the variable structure control vsc theory we develop an adaptive fuzzy control system design method for uncertain takagi sugeno fuzzy models with norm bounded uncertainties we relax the restrictive assumptions that each nominal local system model shares the same input channel and the norm bound of the uncertainty is known which are usually invoked in the traditional vsc based fuzzy control design methods as the local controller we use a vsc law with a switching feedback control term and an adaptation law to account for the norm bounded uncertainties in terms of lmis we derive a sufficient condition for the existence of linear sliding surfaces guaranteeing the asymptotic stability we present an lmi characterization of such sliding surfaces we also give an lmi based design algorithm together with a numerical design example

a robust h_ infty non pdc design scheme for singularly perturbed t s fuzzy systems with immeasurable state variables this paper addresses an observer based robust h infty fuzzy controller design method for singularly perturbed nonlinear systems with external disturbances which are represented by uncertain takagi sugeno t s fuzzy models for a practical output feedback design the premise variables of the fuzzy controller and the fuzzy observer are considered unknown in general a fuzzy lyapunov function is utilized for the synthesis of the nonparallel distributed compensation based controller the closed loop singularly perturbed t s fuzzy system is asymptotically robustly stable in the absence of disturbances and satisfies an h infty norm condition in the presence of disturbances for all positive values of the singular perturbation parameter within the given desired bound using the descriptor redundancy approach some strict linear matrix inequality lmi conditions are derived for the h infty robustness of the closed loop system when uncertainties are simultaneously considered in all state space matrices of each fuzzy subsystem the main disadvantage of a fuzzy lyapunov function based design is that the bounds on the derivatives of the membership functions must be known a priori inspired by a recent work that provides the bounds of the membership derivatives for a given arbitrary compact set on the state space a new method is proposed that guarantees that the closed loop observer based uncertain singularly perturbed t s fuzzy system remains robustly stable in the compact set using finsler s lemma the resulting design conditions are converted to lmis two numerical examples are provided to show the effectiveness of the proposed method

on the laplacian coefficients of graphs under some transformations abstract we generalize three approaches on graph transformations respectively from stevanovic and ilic 2009 16 and tan 2011 19 we also generalize an approach of graph transformations on the spectral radius of adjacency matrix into the laplacian coefficients of graphs from li and feng 1979 26 moreover we determine the unique tree having the third maximal laplacian coefficients among all n vertex trees

the energy of a graph abstract the energy e g of a simple graph g is defined to be the sum of the absolute values of the eigen values of g if g is a k regular graph on n vertices then e g k k n 1 n k b 2 and this bound is sharp it is shown that for each 0 there exist infinitely many n for each of which there exists a k regular graph g of order n with k e g b 2 two graphs with the same number of vertices are equienergetic if they have the same energy we show that for any positive integer n 3 there exist two equienergetic graphs of order 4n that are not cospectral

stabilization of uncertain dynamic systems including state delay the authors consider a class of time varying systems with time varying state delay and uncertain input and parameters they present sufficient conditions on the open loop to obtain an arbitrary degree closed loop stability they construct a min max controller from knowledge of the upper bound of the delay

starlike trees are determined by their laplacian spectrum in this paper we show that if g is a starlike tree then it is determined by its laplacian spectrum moreover we prove some facts about trees with the same adjacency spectrum as a starlike tree

robust fuzzy h control for uncertain nonlinear systems via state feedback an lmi approach this paper presents a method for designing robust fuzzy h controllers which stabilize nonlinear systems and guarantee an induced l2 norm bound constraint on disturbance attenuation for all admissible uncertainties takagi and sugeno fuzzy models with uncertainties are used as the model for the uncertain nonlinear systems fuzzy control systems utilize the concept of the so called parallel distributed compensation pdc using a single quadratic lyapunov function the stability with decay rate and disturbance attenuation for takagi and sugeno fuzzy model with parameter uncertainties are discussed a sufficient condition for the presence of the robust fuzzy h controllers is then presented in terms of linear matrix inequalities lmis finally design examples of robust fuzzy h controllers for uncertain nonlinear systems are presented

the signless laplacian coefficients and incidence energy of bicyclic graphs abstract let q g x det x i q g i 1 n 1 i i x n i be the characteristic polynomial of the signless laplacian matrix of a graph g of order n this paper investigates how the signless laplacian coefficients i e coefficients of q g x change after some graph transformations these results are used to prove that the set b n of all bicyclic graphs of order n has exactly two minimal elements with respect to the partial ordering of their coefficients furthermore we present a sharp lower bound for the incidence energy of bicyclic graphs of order n and characterize all extremal graphs

laplacian coefficients of trees with given number of leaves or vertices of degree two abstract let g be a simple undirected n vertex graph with the characteristic polynomial of its laplacian matrix l g det i l g k 0 n 1 k c k n k it is well known that for trees the laplacian coefficient c n 2 is equal to the wiener index of g while c n 3 is equal to the modified hyper wiener index of graph using a result of zhou and gutman on the relation between the laplacian coefficients and the matching numbers in subdivided bipartite graphs we characterize the trees with k leaves pendent vertices which simultaneously minimize all laplacian coefficients in particular this extremal balanced starlike tree s n k minimizes the wiener index the modified hyper wiener index and recently introduced laplacian like energy we prove that graph s n n 1 p has minimal laplacian coefficients among n vertex trees with p vertices of degree two in conclusion we illustrate on examples of these spectrum based invariants that the opposite problem of simultaneously maximizing all laplacian coefficients has no solution and pose a conjecture on extremal unicyclic graphs with k leaves

ordering of trees with fixed matching number by the laplacian coefficients abstract let g be a simple undirected graph with the characteristic polynomial of its laplacian matrix l g p g x k 0 n 1 k c k x n k aleksandar ilic a ilic trees with minimal laplacian coefficients comput math appl 59 2010 2776 2783 identified n vertex trees with given matching number q which simultaneously minimize all laplacian coefficients in this paper we give another proof of this result generalizing the approach in the above paper we determine n vertex trees with given matching number q which have the second minimal laplacian coefficients we also identify the n vertex trees with a perfect matching having the largest and the second largest laplacian coefficients respectively extremal values on some indices such as wiener index modified hyper wiener index laplacian like energy incidence energy of n vertex trees with matching number q are obtained in this paper

a transformed lure problem for sliding mode control and chattering reduction the sliding mode control problem is investigated with the frequency domain approach the authors show that the sliding mode control can be transformed into a lure problem with this formulation it is shown that the conventional sliding mode control with a nonlinear sign function can only guarantee to make the overall system stable but not asymptotically stable the authors also consider a practical situation where the sign nonlinearity may contain a hysteresis loop for this situation they show that even if the hysteresis loop is very small there still exists a limit cycle however chattering phenomenon caused by the limit cycle behavior can be eliminated by a certain class of nonlinearities

a variable structure control with simple adaptation laws for upper bounds on the norm of the uncertainties in deterministic design of feedback controllers for uncertain dynamical systems upper bounds on the norm of the uncertainties are an important clue to guarantee the stability of the closed loop system sometimes however these upper bounds may not be easily obtained because of the complexity of structure of the uncertainties simple adaptation laws for upper bounds on the norm of the uncertainties are proposed and these adaptive upper bounds are used to design a variable structure control system that guarantees the asymptotic stability of uncertain dynamical systems