system analysis via integral quadratic constraints this paper introduces a unified approach to robustness analysis with respect to nonlinearities time variations and uncertain parameters from an original idea by yakubovich 1967 the approach has been developed under a combination of influences from the western and russian traditions of control theory it is shown how a complex system can be described using integral quadratic constraints iqc for its elementary components a stability theorem for systems described by iqcs is presented that covers classical passivity dissipativity arguments but simplifies the use of multipliers and the treatment of causality a systematic computational approach is described and relations to other methods of stability analysis are discussed last but not least the paper contains a summarizing list of iqcs for important types of system components

affine parameter dependent lyapunov functions and real parametric uncertainty this paper presents new tests to analyze the robust stability and or performance of linear systems with uncertain real parameters these tests are extensions of the notions of quadratic stability and performance where the fixed quadratic lyapunov function is replaced by a lyapunov function with affine dependence on the uncertain parameters admittedly with some conservatism the construction of such parameter dependent lyapunov functions can be reduced to a linear matrix inequality lmi problem and hence is numerically tractable these lmi based tests are applicable to constant or time varying uncertain parameters and are less conservative than quadratic stability in the case of slow parametric variations they also avoid the frequency sweep needed in real spl mu analysis and numerical experiments indicate that they often compare favorably with spl mu analysis for time invariant parameter uncertainty

lpv system analysis via quadratic separator for uncertain implicit systems this paper considers a class of linear systems containing time varying parameters whose behavior is not known exactly we assume that the parameters vary within known intervals and there are known bounds on their rates of variation our objective is to give a computationally verifiable condition that guarantees stability of the system for all possible parameter variations we first point out that the information on the rate bounds can be exploited by considering an augmented system described by an implicit model we then develop a tool called the quadratic separator for stability analysis of uncertain implicit systems using this tool a sufficient stability condition is obtained for the original linear parameter varying system moreover we show that the stability condition obtained is equivalent to the existence of a lyapunov function that depends on the parameters in a linear fractional manner finally the computational aspects of the proposed stability conditions are addressed

p sv wave propagation in heterogeneous media velocity stress finite difference method i present a finite difference method for modeling p sv wave propagation in heterogeneous media this is an extension of the method i previously proposed for modeling sh wave propagation by using velocity and stress in a discrete grid the two components of the velocity cannot be defined at the same node for a complete staggered grid the stability condition and the p wave phase velocity dispersion curve do not depend on the poisson s ratio while the s wave phase velocity dispersion curve behavior is rather insensitive to the poisson s ratio therefore the same code used for elastic media can be used for liquid media where s wave velocity goes to zero and no special treatment is needed for a liquid solid interface typical physical phenomena arising with p sv modeling such as surface waves are in agreement with analytical results the weathered layer and corner edge models show in seismograms the same converted phases obtained by previous authors this method gives stable results for step discontinuities as shown for a liquid layer above an elastic half space the head wave preserves the correct amplitude finally the corner edge model illustrates a more complex geometry for the liquid solid interface as the poisson s ratio v increases from 0 25 to 0 5 the shear converted phases are removed from seismograms and from the time section of the wave field

solving quadratic distance problems an lmi based approach the computation of the minimum distance of a point to a surface in a finite dimensional space is a key issue in several system analysis and control problems the paper presents a general framework in which some classes of minimum distance problems are tackled via linear matrix inequality lmi techniques exploiting a suitable representation of homogeneous forms a lower bound to the solution of a canonical quadratic distance problem is obtained by solving a one parameter family of lmi optimization problems several properties of the proposed technique are discussed in particular tightness of the lower bound is investigated providing both a simple algorithmic procedure for a posteriori optimality testing and a structural condition on the related homogeneous form that ensures optimality a priori extensive numerical simulations are reported showing promising performances of the proposed method

kk theory for banach algebras ii equivariance and green julg type theorems abstract we extend the definition of the bivariant k theory kk ban from plain banach algebras to banach algebras equipped with an action of a locally compact hausdorff group g we also define a natural transformation from lafforgue s theory kk g ban into the new equivariant theory overcoming some technical difficulties that are particular to the equivariant case the categorical framework allows us to systematically define a descent homomorphism and to prove a green julg theorem a dual version of it and a generalised version that involves the action of a proper g space we also include a naive poincare duality theorem

robust stability and performance of linear time varying systems in polytopic domains a sufficient condition to assess the stability of linear time varying systems in polytopic domains is given in this paper this condition is written as a feasibility test of linear matrix inequalities expressed at the polytope vertices and taking bounds on the time derivatives of the system parameters into account differently from other techniques in the literature there is no need of gridding procedures on the parameter space neither restrictive assumptions on the uncertainty structure the proposed test can be solved in polynomial time yielding as solution a parameter dependent lyapunov function that assures the system stability moreover the results can be applied to systems with affine parameter uncertainty and can be easily extended to deal with guaranteed cost computation numerical examples show that the proposed condition can lead to less conservative evaluations of the stability domain and of the performance index for this class of systems when compared to the results based on the conventional

method of power transformations for analysis of stability of nonlinear control systems abstract new frequency domain criteria for absolute stability are obtained these criteria essentially improve the estimates of stability regions in the parameter space compared with well known results of 7 8 and the circle criterion of j j bongiorno 1964

lmi characterization of structural and robust stability abstract this paper introduces several stability conditions for a given class of matrices expressed in terms of linear matrix inequalities lmi being thus simply and efficiently computable diagonal and simultaneous stability both characterized by polytopes of matrices are addressed using this approach a method particularly attractive to test a given matrix for d stability is proposed lyapunov parameter dependent functions are built in order to reduce conservativeness of the stability conditions the key idea is to relate hurwitz stability with a positive realness condition

an lmi condition for the robust stability of uncertain continuous time linear systems a new sufficient condition for the robust stability of continuous time uncertain linear systems with convex bounded uncertainties is proposed in this note the results are based on linear matrix inequalities lmis formulated at the vertices of the uncertainty polytope which provide a parameter dependent lyapunov function that assures the stability of any matrix inside the uncertainty domain with the aid of numerical procedures based on unidimensional search and the lmis feasibility tests a simple and constructive way to compute robust stability domains can be established

nonquadratic lyapunov functions for robust stability analysis of linear uncertain systems in this note we derive conditions of existence of a homogeneous polynomial lyapunov function of an arbitrary even degree establishing global asymptotic stability of linear system with box bounded uncertainty verification of these conditions is reduced to solving a convex minimization problem we produce numerical examples that demonstrate significant improvement in estimates of admissible uncertainty bounds compared with estimates obtained via the most commonly used quadratic lyapunov functions

an improved lmi condition for robust d stability of uncertain polytopic systems a sufficient condition for robust d stability of linear systems with polytope type uncertainties is proposed the result is based on a linear parameter dependent lyapunov function obtained from the feasibility test of a set of linear matrix inequalities lmis defined at the vertices of the polytope this improved lmi condition encompasses previous results based on additional inequalities as well as results based on extra variables

brief paper time varying discrete time linear systems with bounded rates of variation stability analysis and control design this paper investigates the problems of robust stability analysis and state feedback control design for discrete time linear systems with time varying parameters it is assumed that the time varying parameters lie inside a polytopic domain and have known bounds on their rate of variation a convex model is proposed to represent the parameters and their variations as a polytope and linear matrix inequality relaxations that take into account the bounds on the rates of parameter variations are proposed a feasible solution provides a parameter dependent lyapunov function with polynomial dependence on the parameters assuring the robust stability of this class of systems extensions to deal with robust control design as well as gain scheduling by state feedback are also provided in terms of linear matrix inequalities numerical examples illustrate the results

a nonreflecting boundary condition for discrete acoustic and elastic wave equations one of the nagging problems which arises in application of discrete solution methods for wave propagation calculations is the presence of reflections or wraparound from the boundaries of the numerical mesh these undesired events eventually override the actual seismic signals which propagate in the modeled region the solution to avoiding boundary effects used to be to enlarge the numerical mesh thus delaying the side reflections and wraparound longer than the range of times involved in the modeling obviously this solution considerably increases the expense of computation more recently nonreflecting boundary conditions were introduced for the finite difference method clayton and enquist 1977 reynolds 1978 these boundary conditions are based on replacing the wave equation in the boundary region by one way wave equations which do not permit energy to propagate from the boundaries into the numerical mesh this approach has been relatively successful except that its effectiveness degrades for event

a parameter dependent approach to robust h_ infty filtering for time delay systems this paper is concerned with the problem of robust h infin filtering for linear continuous time systems with polytopic parameter uncertainties and time varying delay in the state we utilize the polynomially parameter dependent idea to solve the robust h infin filtering problem with new linear matrix inequality conditions obtained for the existence of admissible filters these conditions are developed based on homogeneous polynomially parameter dependent matrices of arbitrary degree the delay dependence and polynomial parameter dependence guarantee the proposed approach to be potentially less conservative which is shown via a numerical example

an lmi approach to constrained optimization with homogeneous forms abstract this paper considers the problem of determining the minimum euclidean distance of a point from a polynomial surface in r n it is well known that this problem is in general non convex the main purpose of the paper is to investigate to what extent linear matrix inequality lmi techniques can be exploited for solving this problem the first result of the paper shows that a lower bound to the global minimum can be achieved via the solution of a one parameter family of linear matrix inequalities lmis it is also pointed out that for some classes of problems the solution of a single lmi problem provides the lower bound the second result concerns the tightness of the bound it is shown that optimality of the lower bound amounts to solving a system of linear equations an application example is finally presented to show the features of the approach

bivariant algebraic k theory we show how methods from k theory of operator algebras can be applied in a completely algebraic setting to define a bivariant m1 stable homotopy invariant excisive k theory of algebras over a fixed unital ground ring h a b 7 kk a b which is universal in the sense that it maps uniquely to any other such theory it turns out kk is related to c weibel s homotopy algebraic k theory kh we prove that if h is commutative and a is central as an h bimodule then kk h a kh a we show further that some calculations from operator algebra kk theory such as the exact sequence of pimsner voiculescu carry over to algebraic kk

construction of optimal lyapunov function for systems with structured uncertainties the robust stability problem for nominally linear system with nonlinear time varying structured perturbations p sub j j 1 q is considered the system is of the form x spl dot a sub n x sup q spl sigma sub j 1 p sub j a sub j x when lyapunov direct method is utilized to solve the problem usually some quadratic form is chosen as a lyapunov function the note presents a procedure of selection of optimal lyapunov function from the class of quadratic forms under some weak conditions the procedure is effective in solving the robust stability problem the procedure is simple requires only such numerical routines as inverting positive definite symmetric matrices and determining the eigenvalues and eigenvectors of symmetric matrices it is expected that when optimal lyapunov function is used for a robust linear feedback controller design the resulting controller will have the smallest in some sense gains the examples demonstrate the effectiveness of the method the method is effective for large scale systems it worked well for 24 dimensional system

parameterized lmis in control theory a wide variety of problems in control system theory fall within the class of parameterized linear matrix inequalities lmis that is lmis whose coefficients are functions of a parameter confined to a compact set such problems though convex involve an infinite set of lmi constraints and hence are inherently difficult to solve numerically this paper investigates relaxations of parameterized lmi problems into standard lmi problems using techniques relying on directional convexity concepts an in depth discussion of the impact of the proposed techniques in quadratic programming lyapunov based stability and performance analysis mu analysis and linear parameter varying control is provided illustrative examples are given to demonstrate the usefulness and practicality of the approach

piecewise lyapunov functions for robust stability of linear time varying systems in this paper we investigate the use of two term piecewise quadratic lyapunov functions for robust stability of linear time varying systems by using the so called s procedure and a special variable reduction method we provide numerically efficient conditions for the robust asymptotic stability of the linear time varying systems involving the convex combinations of two matrices an example is included to demonstrate the usefulness of our results

computation of nonconservative stability perturbation bounds for systems with nonlinearly correlated uncertainties consideration is given to the problem of robust stability analysis of linear dynamic systems with uncertain physical parameters entering as polynomials in the state equation matrices a method is proposed giving necessary and sufficient conditions for computing the uncertain system stability margin in parameter space which provides a measure of maximal parameter perturbations preserving stability of the perturbed system around a known stable nominal system a globally convergent optimization algorithm that enables solutions to the problem to be obtained is presented using the polynomial structure of the problem the algorithm generates a convergent sequence of interval estimates of the global extremum these intervals provide a measure of the accuracy of the approximating solution achieved at each step of the iterative procedure some numerical examples are reported showing attractive features of the algorithm from the point of view of computational burden and convergence behavior

a first order lyapunov robustness method for linear systems with uncertain parameters a method for stability robustness analysis based on a quadratic lyapunov function that varies linearly with uncertainty parameters is derived linear time invariant systems with structured uncertainties are discussed the lyapunov function is optimized numerically to maximize the robustness region in parameter space numerical results are given for four examples in which the first order method is compared to previous lyapunov methods for robustness analysis while the zero order method is slightly better than the first order method for one example the first order method is clearly superior in the other three more realistic examples the first order method is especially superior for applications to active control of flexible structures where robustness with respect to unmodeled coupling between modeled modes are important issues for such applications the first order method is much better at detecting the increased robustness associated with increased separation between frequencies

piecewise linear quadratic optimal control the use of piecewise quadratic cost functions is extended from stability analysis of piecewise linear systems to performance analysis and optimal control lower bounds on the optimal control cost are obtained by semidefinite programming based on the bellman inequality this also gives an approximation to the optimal control law an upper bound to the optimal cost is obtained by another convex optimization problem using the given control law a compact matrix notation is introduced to support the calculations and it is proved that the framework of piecewise linear systems can be used to analyze smooth nonlinear dynamics with arbitrary accuracy

equivariant algebraic kk theory and adjointness theorems abstract we introduce an equivariant algebraic kk theory for g algebras and g graded algebras we study some adjointness theorems related with crossed product trivial action induction and restriction in particular we obtain an algebraic version of the green julg theorem which gives us a computational tool

stability robustness bounds for linear state space models with structured uncertainty in this note we consider the robust stability analysis problem in linear state space models we consider systems with structured uncertainty some lower bounds on allowable perturbations which maintain the stability of a nominally stable system are derived these bounds are shown to be less conservative than the existing ones

robust performance analysis for structured linear time varying perturbations with bounded rates of variation the robust performance analysis problem is considered for linear time invariant lti systems subject to block diagonal structured and bounded linear time varying ltv perturbations with specified maximal rates of variation analysis methods are developed in terms of semidefinite programming for the computation of upper and lower bounds for the optimum robust performance level the upper bound computation is based on an integral quadratic constraint iqc test developed using a generalized version of the so called swapping lemma the lower bound computation method employs an extended version of the power distribution theorem together with a generalized version of the kalman yakubovich popov kyp lemma and serves as a means to assess the conservatism of the computed upper bounds in the case of dynamic ltv perturbations as corollaries of the underlying result for lower bound computation it is shown for general block diagonal uncertainty structures that thefrequency dependent constant d scaling tests are exact for robust performance analysis against arbitrarily slow fast dynamic ltv perturbations respectively

a convex approach to robust stability for linear systems with uncertain scalar parameters in this paper robust stability for linear systems with several uncertain complex and or real scalar parameters is studied a countable family of conditions sufficient for robust stability is given in terms of solvability of some simple linear matrix inequalities lmis these conditions are of increasing precision and it is shown conversely that robust stability implies solvability of these lmis from a certain rank and beyond this result constitutes an extension of the characterization by solvability of lyapunov inequality of the asymptotic stability for usual linear systems it is based on the search of parameter dependent quadratic lyapunov functions polynomial of increasing degree in the parameters

a parameter dependent lyapunov function for a polytope of matrices a new sufficient condition for a polytope of matrices to be hurwitz stable is presented the stability is a consequence of the existence of a parameter dependent quadratic lyapunov function which is assured by a certain linear constraint for generating extreme matrices of the polytope the condition can be regarded as a duality of the known extreme point result on quadratic stability of matrix polytopes where a fixed quadratic lyapunov function plays the role the obtained results are applied to a polytope of second degree polynomials for illustration

window controlled cmp crosscorrelation analysis for surface waves in laterally heterogeneous media cmp crosscorrelation cmpcc analysis of surface waves enhances lateral resolution of surface wave analyses we found the technique of window controlled cmpcc analysis which applies two kinds of spatial windows to further improve the lateral resolution of cmpcc analysis first a spatial weighting function given by the number of crosscorrelation pairs is applied to cmpcc gathers because the number of crosscorrelation pairs is concentrated near the cmp the lateral resolution in extracting dispersion curves on cmps can be improved second crosscorrelation pairs with longer receiver spacing are excluded to further improve lateral resolution although removing crosscorrelation pairs generally decreases the accuracy of phase velocity estimations the required accuracy to estimate phase velocities is maintained by considering the wavenumber resolution defined for given receiver configurations when applied to a synthetic data set simulating a laterally heterogeneous structure window controlled cmpcc analysis improved the retrieval of the lateral variation in local dispersion curves beneath each cmp we also applied the method to field seismic data across a major fault the window controlled cmpcc analysis improved lateral variations of the inverted s wave velocity structure without degrading the accuracy of s wave velocity estimations we discovered that window controlled cmpcc analysis is effective in improving lateral resolution of dispersion curve estimations with respect to the original cmpcc analysis

love wave waveform inversion in time domain for shallow shear wave velocity abstracthigh frequency surface wave techniques are widely used to estimate s wave velocity of near surface materials surface wave methods based on inversions of dispersion curves are only suitable to laterally homogeneous or smoothly laterally varying heterogeneous earth models due to the layered model assumption during calculation of dispersion curves waveform inversion directly fits the waveform of observed data and it can be applied to any kinds of earth models we have used the love wave waveform inversion in the time domain to estimate near surface s wave velocity we used the finite difference method as the forward modeling method the source effect was removed by the deconvolution technique which made our method independent of the source wavelet we defined the difference between the deconvolved observed and calculated waveform as the misfit function we divided the model into different sizes of blocks depending on the resolution of the love waves and we updated the s wave velocity of each blo

a new class of universal lyapunov functions for the control of uncertain linear systems the authors analyze the problem of synthesizing a state feedback control for the class of uncertain continuous time linear systems affected by time varying memoryless parametric uncertainties they consider as candidate lyapunov functions the elements of the class spl sigma sub p sup z which is formed by special homogeneous positive definite functions they show that this class is universal in the sense that a lyapunov function exists if and only if there exists a lyapunov function in spl sigma sub p sup z they prove this result in a constructive way showing that such a lyapunov function can always be obtained by smoothing a polyhedral function for which construction algorithms are available the authors show that unlike the polyhedral lyapunov functions these functions allow us to derive explicit formulas for the stabilizing controller

nonquadratic lyapunov functions for robust control abstract this paper deals with the problem of controlling a linear continuous time system with structured time varying parameter uncertainties and input disturbances with a lyapunov function approach in contrast with most of the previous results in the literature we do not confine our attention to the class of quadratic lyapunov functions conversely the basic motivation of this paper is to determine whether there exist other functions that can be conveniently used as candidate lyapunov functions this question has a positive answer the proposed class is that of polyhedral norms or more generally of polyhedral minkowski functionals we show that the class of these functions is universal in the sense that if the problem of ultimately bounding the state in an assigned convex set via state feedback control can be solved via a lyapunov function and a continuous state feedback compensator then it can be solved via a polyhedral lyapunov function and a possibly different continuous control moreover we show that the control can be piecewise linear a numerical technique for constructing the controller is presented for the case in which the uncertainty constraint sets are polyhedral

a new robust d stability condition for real convex polytopic uncertainty abstract the problem of robust d stability analysis with respect to real convex polytopic uncertainties is tackled a new lmi based sufficient condition for the existence of parameter dependent lyapunov functions is proposed this condition generalises previously published conditions numerical comparisons with quadratic stability results as well as previous results based on parameter dependent lyapunov functions illustrate the relevance of this new condition finally this result appears to be promising for robust multi objective performance analysis and control synthesis purposes

h guaranteed cost computation by means of parameter dependent lyapunov functions a linear matrix inequality approach to compute h guaranteed costs by means of parameter dependent lyapunov functions is presented in this paper the uncertain linear time invariant systems are supposed to belong to convex bounded domains polytope type uncertainty the conditions proposed are less conservative than other parameter dependent based methods from the literature providing better estimates of worst case h norms in both continuous and discrete time systems as illustrated by means of numerical examples

brief homogeneous lyapunov functions for systems with structured uncertainties the problem addressed in this paper is the construction of homogeneous polynomial lyapunov functions hplfs for linear systems with time varying structured uncertainties a sufficient condition for the existence of an hplf of given degree is formulated in terms of a linear matrix inequalities lmi feasibility problem this condition turns out to be also necessary in some cases depending on the dimension of the system and the degree of the lyapunov function the maximum norm of the parametric uncertainty for which there exists a homogeneous polynomial lyapunov function is computed by solving a generalized eigenvalue problem the construction of such lyapunov functions is efficiently performed by means of popular convex optimization tools for the solution of problems in lmi form comparisons with other classes of lyapunov functions through numerical examples taken from the literature show that hplfs are a powerful tool for robustness analysis

parameter dependent lyapunov functions and the popov criterion in robust analysis and synthesis many practical applications of robust feedback control involve constant real parameter uncertainty whereas small gain or norm bounding techniques guarantee robust stability against complex frequency dependent uncertainty thus entailing undue conservatism since conventional lyapunov bounding techniques guarantee stability with respect to time varying perturbations they possess a similar drawback in this paper we develop a framework for parameter dependent lyapunov functions a less conservative refinement of fixed lyapunov functions an immediate application of this framework is a reinterpretation of the classical popov criterion as a parameter dependent lyapunov function this result is then used for robust controller synthesis with full order and reduced order controllers

numerical methods of lyapunov function construction and their application to the absolute stability problem the minimax approach is used to study numerical selection of a lyapunov function from a multiparameter class of functions a numerical method is suggested for plotting an absolute stability region in the space of system gains for a control system with several nonstationary nonlinearities

a converse lyapunov theorem for a class of dynamical systems which undergo switching the authors investigate the stability of a system in which the dynamics at any instant in time will follow one of a fixed set of vector fields they allow switching between members of the family of vector fields to be completely random the focus of this paper is to prove a converse lyapunov theorem for this class of systems

fundamental connections among the stability conditions using higher order time derivatives of lyapunov functions for the case of linear time invariant systems abstract it has already been recognized that looking for a positive definite lyapunov function such that a high order linear differential inequality with respect to the lyapunov function holds along the trajectories of a nonlinear system can be utilized to assess asymptotic stability when the standard lyapunov approach examining only the first derivative fails in this context the main purpose of this paper is on one hand to theoretically unveil deeper connections among existing stability conditions especially for linear time invariant lti systems and from the other hand to examine the effect of the higher order time derivatives approach on the stability results for uncertain polytopic lti systems in terms of conservativeness to this end new linear matrix inequality lmi stability conditions are derived by generalizing the concept mentioned above and through the development relations among some existing stability conditions are revealed examples illustrate the improvement over the quadratic approach

lpv control and full block multipliers a novel construction of linear parameter varying controllers is presented for arbitrarily fast parameter variations it is shown how full block multipliers allow to considerably reduce conservatism various auxiliary results of independent interest are included

analysis and synthesis of robust control systems via parameter dependent lyapunov functions in this paper the problem of robust stability of systems subject to parametric uncertainties is considered sufficient conditions for the existence of parameter dependent lyapunov functions are given in terms of a criterion which is reminiscent of but less conservative than popov s stability criterion an equivalent frequency domain criterion is demonstrated the relative sharpness of the proposed test and existing stability criteria is then discussed the use of parameter dependent lyapunov functions for robust controller synthesis is then considered it is shown that the search for robustly stabilizing controllers may be limited to controllers with the same order as the original plant a possible synthesis procedure and a numerical example are then discussed

stability analysis and gain scheduled state feedback control for continuous time systems with bounded parameter variations the problems of robust stability analysis and state feedback control based on gain scheduling for continuous time systems with time varying parameters that have bounded rates of variation and lie inside a polytope are addressed in this article with respect to previous results in the literature two main contributions of the article are i the robust stability analysis conditions are less conservative and demand less computational effort than the existing ones ii the conditions can be extended to cope with the problem of control design for this class of system parameter dependent linear matrix inequality lmi conditions are given for the existence of a parameter dependent lyapunov function quadratic in the state and homogeneous polynomially of arbitrary degree in the parameter assuring robust stability two convex procedures based on lmis exhibiting distinct complexities are proposed to solve the problem of robust stability an extension to deal with the computation of a stabilising parameter dependent state feedback gain for this class of time varying systems is also provided as a sequence of lmis of increasing precision examples illustrate the results including comparisons with other techniques from the literature

robust performance analysis with lmi based methods for real parametric uncertainty via parameter dependent lyapunov functions robust performance analysis for linear time invariant systems with linear fractional transformation real parametric uncertainty is considered new conditions of robust stability performance based on parameter dependent lyapunov functions are proposed the robust stability performance measures are robust pole location robust h sub spl infin performance and robust h sub 2 performance linear matrix inequality lmi based sufficient conditions for the existence of parameter dependent lyapunov functions are derived by using the quadratic separation concept the performances of the proposed conditions are compared with existing tests