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

generalized sector synthesis of output feedback control with anti windup structure this paper considers the control synthesis problem for linear time invariant lti continuous time systems with actuator saturation nonlinearities the control architecture is standard a nominal lti output feedback with an anti windup aw mechanism traditional approaches have focused on aw synthesis where the nominal controller is first designed to achieve a local performance in the linear region of the actuators and then the aw compensator is added to guarantee stability performance properties when the actuators saturate in contrast this paper proposes a synthesis method to design both the nominal controller and the aw compensator simultaneously using the generalized sector gs approach in the multiplier setting a sufficient condition is given in terms of linear matrix inequalities so that a given ellipsoid is synthesized as a domain of attraction for the closed loop system our result is shown to include the existing results on the simultaneous circle synthesis and the anti windup gs synthesis as special cases this paper also shows that the direct feed through term in the aw compensator does not contribute to enlarging the achievable domain of attraction within the multiplier gs framework

well posedness of feedback systems insights into exact robustness analysis and approximate computations this paper establishes a framework for robust stability analysis of linear time invariant uncertain systems the uncertainty is assumed to belong to an arbitrary subset of complex matrices the concept used here is well posedness of feedback systems leading to necessary and sufficient conditions for robust stability based on this concept some insights into exact robust stability conditions are given in particular frequency domain and state space conditions for well posedness are provided in terms of hermitian form inequalities it is shown that these inequalities can be interpreted as small gain conditions with a generalized class of scalings given by linear fractional transformations lft using the lft scaled small gain condition in the state space setting the duality is established between the h sub spl infin norm condition with frequency dependent scalings and the parameter dependent lyapunov condition connections to the existing results including the structured singular value and the integral quadratic constraints are also discussed finally we show that our well posedness conditions can be used to give a less conservative yet computable bound on the real structured singular value this result is illustrated by numerical examples

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

stabilization of uncertain systems via linear control this note considers the problem of stabilizing a linear dynamical system whose state equation includes a time varying uncertain parameter vector q cdot given the dynamics dot x t a q t x t b q t u t and a bounding set q for the values q t the objective is to choose a control law u t p x t guaranteeing uniform asymptotic stability for all admissible variations of q cdot our results differ from previous work in one fundamental way that is we show that when working with linear controllers it is possible to dispense with all assumptions on b cdot which have been made by previous authors e g see 1 9 this elimination of hypotheses on b cdot is accomplished roughly as follows the system sigma underline underline delta a q b q is shown to be equivalent to another system sigma underline underline delta a q b as far as stabilization is concerned since b is a constant matrix independent of q the desired result is readily obtained

robust performance analysis of uncertain lti systems dual lmi approach and verifications for exactness this paper addresses robust performance analysis problems of linear time invariant lti systems affected by real parametric uncertainties these problems known also as a special class of structured singular value computation problems are inherently intractable np hard problems as such intensive research effort has been made to obtain computationally tractable and less conservative analysis conditions where linear matrix inequality lmi plays an important nevertheless since lmi based conditions are expected to be conservative in general it is often the case that we cannot conclude anything if the lmi at hand turns out to be infeasible this motivates us to consider the dual of the lmi and examine the structure of the dual solution by pursuing this direction in this paper we provide rank conditions on the dual solution matrix under which we can conclude that the underlying robust performance is never attained in particular a set of uncertain parameters that violates the specified performance can be computed these results come from block moment matrix structure of the dual variable which is consistent with the recent results on polynomial optimization this particular structure enables us to make good use of simultaneous diagonalizability property of commuting diagonalizable matrices so that the sound rank conditions for the exactness verification can be obtained

brief paper fault tolerant control for singular systems with actuator saturation and nonlinear perturbation in this paper the problem of robust fault tolerant control for a class of singular systems subject to both time varying state dependent nonlinear perturbation and actuator saturation is investigated a sufficient condition for the existence of a fixed gain controller is first proposed which guarantees the regularity impulse free and stability of the closed loop system under all possible faults an optimization problem with lmi constraints is formulated to determine the largest contractively invariant ellipsoid an adaptive fault tolerant controller is then developed to compensate for the failure effects on the system by estimating the fault and updating the design parameter matrices online both of these two controllers are in the form of a saturation avoidance feedback with the advantage of relatively small actuator capacities compared with the high gain counterpart an example is included to illustrate the proposed procedures and their effectiveness

qualocation for a singularly perturbed boundary value problem a singularly perturbed one dimensional two point boundary value problem of reaction convection diffusion type is considered we generate a c 0 collocation like method by combining galerkin with an adapted quadrature rule using lobatto quadrature and splines of degree r we prove on a shishkin mesh for the qualocation method the same error estimate as for the galerkin technique the result is also important for the practical realization of finite element methods on shishkin meshes using quadrature formulas we report the results of numerical experiments that support the theoretical findings

robustness in the presence of mixed parametric uncertainty and unmodeled dynamics continuing the development of the structured singular value approach to robust control design the authors investigate the problem of computing mu in the case of mixed real parametric and complex uncertainty the problem is shown to be equivalent to a smooth constrained finite dimensional optimization problem in view of the fact that the functional to be maximized may have several local extrema an upper bound on mu whose computation is numerically tractable is established this leads to a sufficient condition of robust stability and performance a historical perspective on the development of the mu theory is included

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

robust l 1 performance analysis for linear systems with parametric uncertainties in this contribution a computational approach for analysing the robust l gain or the robust l1 performance of uncertain linear systems is developed in particular the system s state space matrices may have a rational dependence on structured parametric time invariant or time varying uncertainties the computation is based on robust semi definite programming and provides a trade off between accuracy and computational effort a novel matrix inequality condition to determine the star norm of discrete time systems is derived as an auxiliary result

robust control with structure perturbations the problem of making a given stabilizing controller robust so that the closed loop system remains stable for prescribed ranges of variations of a set of physical parameters in the plant the problem is treated in the state space and transfer function domains in the state space domain a stability hypersphere is determined in the parameter space using lyapunov theory the radius of this hypersphere is iteratively increased by adjusting the controller parameters until the prescribed perturbation ranges are contained in the stability hypersphere in the transfer function domain a corresponding stability margin is defined and optimized on the basis of the recently introduced concept of the largest stability hypersphere in the space of coefficients of the closed loop characteristic polynomial the design algorithms are illustrated by examples

on the kalman yakubovich popov lemma the purpose of this note is to present a new elementary proof for the multivariable k y p lemma a minimum of linear algebra and finite dimensional convexity theory is used

structured and simultaneous lyapunov functions for system stability problems abstract it is shown that many system stability and robustness problems can be reduced to the question of when there is a quadratic lyapunov function of a certain structure which establishes stability of x ax for some appropriate a the existence of such a lyapunov function can be determined by solving a convex program we present several numerical methods for these optimization problems a simple numerical example is given

stability analysis of systems with generalized frequency variables a class of large scale multi agent systems with decentralized information structures can be represented by a linear system with a generalized frequency variable in this paper we investigate fundamental properties of such systems stability and d stability exploiting the dynamical structure specifically we first show that such system is stable if and only if the eigenvalues of the connectivity matrix lie in a region of the complex plane specified by the generalized frequency variable the stability region is characterized in terms of polynomial inequalities leading to an algebraic stability condition we also show that the stability test can be reduced to a linear matrix inequality lmi feasibility problem involving generalized lyapunov inequalities and that the lmi result can be extended for robust stability analysis of systems subject to uncertainties in the interconnection matrix we then extend the result to d stability analysis to meet practical requirements and provide a unified treatment of d stability conditions for ease of implementation finally numerical examples illustrate utility of the stability conditions for the analysis of biological oscillators and for the design of cooperative stabilizers

brief paper comparisons among robust stability criteria for linear systems with affine parameter uncertainties this paper is concerned with the problem of comparisons among robust stability criteria for a class of uncertain linear systems where the system state matrices considered are affinely dependent on the uncertain parameters at first a robust stability criterion for the class of systems to be affinely quadratically stable aqs is derived based on the vertex separator approach where affine parameter dependent lyapunov functions are exploited to prove stability then comparison results between the robust stability criterion and the existing tests for aqs are given in terms of degree of conservatism a numerical example is given to illustrate the results

brief paper non fragile h filter design for linear continuous time systems this paper studies the problem of non fragile h filter design for linear continuous time systems the filter to be designed is assumed to include additive gain variations which result from filter implementations a notion of structured vertex separator is proposed to approach the problem and exploited to develop sufficient conditions for the non fragile h filter design in terms of solutions to a set of linear matrix inequalities lmis the designs guarantee the asymptotic stability of the estimation errors and the h performance of the system from the exogenous signals to the estimation errors below a prescribed level a numerical example is given to illustrate the effect of the proposed method

robust s regularity of matrix pencils applied to the analysis of descriptor models abstract this paper deals with the robust s regularity of an uncertain linear matrix pencil e a and its applicability to the analysis of uncertain linear descriptor models of the form e x ax for the continuous case or ex k 1 ax k for the discrete case lmi linear matrix inequality based conditions for the robust s regularity of e a are established when both a and e are affected by lft linear fractional transform based uncertainties these conditions though sufficient are also necessary in numerous cases of practical interest moreover the relevance of the conditions for the analysis of the d admissibility of descriptor models d stability and impulse freeness or causality is also discussed

coprime factor based anti windup synthesis for parameter dependent systems abstract this paper presents a novel anti windup compensator synthesis for linear parameter varying lpv systems that have linear fractional transformation lft representations the lpv anti windup synthesis is based on a right coprime factor description of the lpv plant to be controlled it is shown that the anti windup compensator synthesis reduces to solving a robust l 2 gain state feedback control problem for the lpv lft system subject to input saturations the synthesis takes advantage of unstructured multipliers that is multipliers that do not commute with the parameter structure and exploits a detailed description of the parameter trajectory polytope the anti windup compensator is given in an explicit lpv lft form sharing with the plant the same order and same parameter structure

simultaneous parametric uncertainty modeling and robust control synthesis by lft scaling given a set of experimental data points in the plant parameter space the objective of this paper is to give a systematic method for designing a controller to achieve a prescribed level of robust performance for the closed loop system a standard procedure would be to model the data first and then to design a controller our focus here instead is to complete these two tasks simultaneously eliminating the conservatism caused by the fact that the data is modeled independently of the closed loop objective to this end integral quadratic constraints iqcs are used as a tool for uncertainty modeling we propose the lft scaling to integrate the iqc modeling with the control synthesis it is shown that the lft scaling provides a condition dual to the standard iqc for linear time invariant systems and this duality result is essential for solving the simultaneous modeling and synthesis problem a solution is given in terms of matrix inequalities and its computational aspects are discussed the proposed method is applied to the control design for an ac linear servo motor and experimental results demonstrate that our method achieves the performance superior to that achieved by existing methods

gain scheduling via linear fractional transformations abstract a linear finite dimensional plant with rational state space parameter dependence is controlled using a parameter dependent controller the parameters are known to take on values in a unit ball and are known in real time the goal of control is to stabilize the parameter dependent closed loop system and provide disturbance error attenuation as measured in induced l 2 norms the approach taken uses the optimally scaled small gain theorem and solves the control synthesis problem by reformulating the existence conditions into a finite dimensional convex optimization

induced l2 gain computation for rational lpv systems using finsler s lemma and minimal generators this paper proposes a novel method to compute an upper bound on the induced l2 gain for a linear parameter varying lpv system with rational parameter dependence the proposed method relies on a standard dissipation inequality condition the storage function is a quadratic function of the state and a rational function of the parameters the specific parameter dependence is restricted to involve fixed rational functions and an affine term with free decision variables finsler s lemma and affine annihilators are used to formulate sufficient linear matrix inequality lmi conditions for the dissipativity relation the dimension and conservatism of the resulting lmi problem are reduced by the joint application of minimal generators and maximal annihilators an lpv model of a pendulum cart system is used to demonstrate the proposed method and compare it to existing techniques in the literature

low rank exploitation in semidefinite programming for control many control related problems can be cast as semidefinite programs even though there exist polynomial time algorithms and excellent publicly available solvers the time it takes to solve these problems can be excessive what many of these problems have in common in particular in control is that some of the variables enter as matrix valued variables this leads to a low rank structure in the basis matrices which can be exploited when forming the newton equations in this article we describe how this can be done and show how our code called strul can be used in conjunction with the semidefinite programming solver sdpt3 the idea behind the structure exploitation is classical and is implemented in lmi lab but we show that when using a modern semidefinite programming framework such as sdpt3 the computational time can be significantly reduced finally we describe how the modelling language yalmip has been changed in such a way that our code which can be freely downloaded can be interfaced using sta

characterization and efficient computation of the structured singular value the concept of structured singular value was recently introduced by doyle 1 as a tool for the analysis and synthesis of feedback systems with structured uncertainties it is a key to the design of control systems under joint robustness and performance specifications and it very nicely complements the h infty approach to control system design in this paper it is shown that the structured singular value can be obtained as the solution of several smooth optimization problems properties of these optimization problems are exhibited leading to a fast algorithm that always yields the structured singular value for block structures of size no larger than 3 and often does for block structures of larger size

a generalized state space for singular systems systems of the form e dot x ax bu y cx with e singular are studied of particular interest are the impulsive modes that may appear in the free response of such systems when arbitrary initial conditions are permitted modes that are associated with natural system frequencies at infinity a generalized definition of system order that incorporates these impulsive degrees of freedom is proposed and concepts of controllability and observability are defined for the impulsive modes allowable equivalence transformations of such singular systems are specified the present framework is shown to overcome several difficulties inherent in other treatments of singular systems and to extend in a natural and satisfying way many results previously known only for regular state space systems

augmented lyapunov function approach to l2 gain analysis for discrete time systems with saturation nonlinearities abstract this paper presents a method for l 2 gain analysis of discrete time systems with saturation nonlinearity by using an augmented lyapunov functional by using sector and slope properties of the saturation nonlinearity the l 2 gain performance criterion is obtained via lmi formulation the proposed lyapunov function have augmented quadratic form with states as well as the sector bounded nonlinear function numerical examples show that the proposed method is less conservative than the existing ones

adaptive robust h filter design for linear systems with time varying uncertainty this article studies the problem of designing robust h filters for linear uncertain systems the uncertainty parameters are assumed to be time varying unknown but bounded which appear affinely in the matrices of system models an adaptive mechanism is introduced to construct novel filters with variable gains which can reduce the conservativeness of the traditional robust h filters the proposed adaptive filter design conditions are given in terms of linear matrix inequalities a numerical example is presented to illustrate the effectiveness of the proposed strategy

application of free vibration analysis of membranes using the non dimensional dynamic influence function in this paper the multi domain method of subdividing the membranes of interest into several domains is presented for applications of free vibration analysis of arbitrarily shaped membranes the method is especially effective for concavely shaped membranes with high concavity and multi connected membranes with a hole compared with the single domain method the frequency equations of these membranes are given by the determinants of the system matrices obtained when the compatibility conditions are considered on the common boundaries along which the sub domains adjoin each other although many boundary nodes are distributed along the fixed boundaries on which transverse displacements are prescribed to be zero the number of boundary nodes has no effect on the order of the system matrix the order is determined to be equal to the value corresponding to twice the number of interior nodes distributed on the common boundaries thus eigenvalues the roots of the frequency equation may be obtained from the determinant of the system matrix of a small order the case studies presented in the paper reveal that the eigenvalues calculated by this method are in good agreement with those obtained by the fem program ansys

parameter dependent lyapunov functions for linear systems with constant uncertainties robust stability of linear time invariant systems with respect to structured uncertainties is considered the small gain condition is sufficient to prove robust stability and scalings are typically used to reduce the conservatism of this condition it is known that if the small gain condition is satisfied with constant scalings then there is a single quadratic lyapunov function which proves robust stability with respect to all allowable time varying perturbations in this technical note we show that if the small gain condition is satisfied with frequency varying scalings then an explicit parameter dependent lyapunov function can be constructed to prove robust stability with respect to constant uncertainties this lyapunov function has a rational quadratic dependence on the uncertainties

lyapunov based exact stability analysis and synthesis for linear single parameter dependent systems we propose a class of polynomially parameter dependent quadratic ppdq lyapunov functions for assessing the stability of single parameter dependent linear time invariant s pdlti systems in a non conservative manner it is shown that the stability of s pdlti systems is equivalent to the existence of a ppdq lyapunov function a bound on the degree of the polynomial dependence of the lyapunov function in the system parameter is given explicitly the resulting stability conditions are expressed in terms of a set of matrix inequalities whose feasibility over a compact and connected set can be cast as a convex finite dimensional optimisation problem extensions of the approach to state feedback controller synthesis are also provided

robust stabilization of uncertain linear systems quadratic stabilizability and h sup infinity control theory the problem of robustly stabilizing a linear uncertain system is considered with emphasis on the interplay between the time domain results on the quadratic stabilization of uncertain systems and the frequency domain results on h sup infinity optimization a complete solution to a certain quadratic stabilization problem in which uncertainty enters both the state and the input matrices of the system is given relations between these robust stabilization problems and h sup infinity control theory are explored it is also shown that in a number of cases if a robust stabilization problem can be solved via lyapunov methods then it can be also be solved via h sup infinity control theory based methods

exact calculation of the multiloop stability margin a mapping theorem by l a zadeh and c a desoer 1963 serves as the basis for an algorithm that computes the stability margin k sub m of diagonally perturbed multivariable feedback systems without conservatism the stability margin determination not only verifies system stability but also quantifies how much further the plan uncertainties can be extended before instability occurs the essence of the computational approach is the realization that the true image of a given domain can be approximated with arbitrary accuracy by first subdividing the domain into the requisite number of subdomains and then forming the union of the convex hulls of their images the technique is illustrated in an example with a plant model having three uncertainties in general the method is applicable to either siso or mimo lti systems whose uncertainties can be modeled with noninteracting coefficients in the open loop transfer function representations

affine parameter dependent lyapunov functions for lpv systems with affine dependence this paper deals with the certification problem for robust quadratic stability robust state convergence and robust quadratic performance of linear systems that exhibit bounded rates of variation in their parameters we consider both continuous time ct and discrete time dt parameter varying systems in this paper we provide a uniform method for this certification problem in both cases and we show that contrary to what was claimed previously the dt case requires a significantly different treatment compared to the existing ct results in the established uniform approach quadratic lyapunov functions which are affine in the parameter are used to certify robust stability robust convergence rates and robust performance in terms of linear matrix inequality feasibility tests to exemplify the procedure we solve the certification problem for mathscr l 2 gain performance both in the ct and the dt cases a numerical example is given to show that the proposed approach is less conservative than a method with slack variables