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

a procedure for the identification of linear and non linear multi degree of freedom systems abstract most of the modern techniques for vibration analysis are based on the modal approach however when the systems under study are not lightly damped so that there are significant cross damping terms in the governing equations when there are closely spaced natural frequencies or when non linearities exist then the modal approach is extremely limited a number of techniques have been proposed to deal with these problems but the methods tend to produce models which are subjective in this paper a generalized non iterative approach in which spectral techniques are used is presented for the identification of multi degree of freedom systems from input output records the method is suitable for the analysis not only of systems with significant cross damping but also of highly coupled non linear systems the implementation of the technique is illustrated with two numerically simulated examples

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

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

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

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

some general results on the seepage exclusion problem asymptotic roof boundary layer solutions are obtained for the quasilinear seepage exclusion problem for ellipsoidal cavities of arbitrary aspect ratio and arbitrary horizontal axial index k the maximum dimensionless potential at the cavity wall max is given 0 by the equation max k 2 f0 k f1 k f2 k 2 where 1 2 sorptive number apical curvature and the fn are polynomials in 2 and k 2 of degree n 1 there is thus a hierarchy in importance of the geometrical parameters determining max is dominant is next in importance and the effect of k is relatively minor the above equation includes as special cases previous results for circular elliptic and parabolic cylinders and for spheres spheroids and paraboloids for elliptic discs 0 max 1 k 4 12 f0 k f1 k 12 f2 k 14 where 1 1 characteristic horizontal length and the fn are polynomials in k 2 of degree n 1 this equation includes as special cases previous results for strips and circular discs the various results are relevant also to cavities with ellipsoidal and flat elliptic roofs but otherwise of arbitrary though not too irregular configuration the corresponding nonlinear analysis is straightforward

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

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

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

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

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