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

relaxation methods for monotropic programs we propose a dual descent method for the problem of minimizing a convex possibly nondifferentiable separable cost subject to linear constraints the method has properties reminiscent of the gauss seidel method in numerical analysis and uses the e complementary slackness mechanism introduced in bertsekas hosein and tseng 1987 to ensure finite convergence to near optimality as special cases we obtain the methods in bertsekas hosein and tseng 1987 for network flow programs and the methods in tseng and bertsekas 1987 for linear programs

robust adaptive l2 gain control of polytopic mimo lti systems lmi results passification based direct adaptive control is considered for polytopic uncertain linear time invariant multi input multi output systems linear matrix inequality based results are provided to guarantee that the adaptive algorithm passifies the system whatever the uncertain parameters in some given set the contributions are based on the introduction of a parallel feed forward shunt that liberates from the strong equality constraint pb c t g t often used for strict passification the shunt combined with the introduction of slack variables allows to conclude with easy to test conditions without restrictions on which data is uncertain in the process model the resulting adaptive control is such that the control gain is bounded in a chosen set whatever bounded disturbances moreover it proves to be not worse than computable parameter dependent static output feedback controls with respect to l2 gain attenuation a simple academic example illustrates the results keywords passivity robustness adaptive control lmi output feedback polytopic systems l2 gain

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

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

multivariable mrac design without gain symmetry conditions using a stabilizing multiplier a new multivariable mrac design for plants of arbitrary relative degree which does not require a stringent symmetry assumption related with the plant high frequency gain matrix is presented in contrast to previous results the new solution does not involve additional parametrization and filtering being thus close in structure and complexity to conventional solutions instead of the fragile symmetry assumption a less restrictive and robust condition is required which can be achieved using a multiplier

lmi characterization of structural and robust stability the discrete time case abstract this paper extends to the discrete time case some robust stability conditions recently obtained for continuous time systems those conditions are expressed in terms of linear matrix inequalities lmi being thus simply and efficiently computable as in the continuous time case parameter dependent lyapunov functions can be constructed and consequently the new approach can yield much sharper and less conservative results than the simultaneous stability approach in particular well known stability problems namely d stability and robust stability in the presence of diagonally structured uncertainty can be more efficiently addressed numerical examples are included to illustrate the advantages of the new stability conditions

on a convex parameter space method for linear control design of uncertain systems this paper presents a new procedure for continuous and discrete time linear control systems design it consists of the definition of a convex programming problem in the parameter space that when solved provides the feedback gain one of the most important features of the procedure is that additional design constraints are easily incorporated in the original formulation yielding solutions to problems that have raised a great deal of interest within the last few years this is precisely the case of the decentralized control problem and the quadratic stabilizability problem of uncertain systems with both dynamic and input uncertain matrices in this last case necessary and sufficient conditions for the existence of a linear stabilizing gain are provided and to the authors knowledge this is one of the first numerical procedures able to handle and solve this interesting design problem for high order continuous time or discrete time linear models the theory is illustrated by examples

design of strictly positive real systems using constant output feedback the authors present a linear matrix inequality lmi approach to the strictly positive real spr synthesis problem find an output feedback k such that the closed loop system t s is spr the authors establish that if no such constant output feedback k exists then no dynamic output feedback with a proper transfer matrix exists to make the closed loop system spr the existence of k to guarantee the spr property of the closed loop system is used to develop an adaptive control scheme that can stabilize any system of arbitrary unknown order and unknown parameters

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

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

lmi based algorithm for strictly positive real systems with static output feedback abstract an algorithm based on linear matrix inequalities lmis is proposed to find a constant output feedback matrix k o and a constant output tandem matrix f such that the controlled system a b k o c b f c is strictly positive real spr the number of output variables of the plant a b c is greater than or equal to the number of its input variables considering that spr systems with static output feedback are related to spr systems with static state feedback as shown in this manuscript the first step of the algorithm is to find a matrix f such that all transmission zeros of the system a b f c have negative real parts after finding this matrix f an output feedback matrix k o such that the system a b k o c b f c is spr is found another algorithm is proposed to specify a decay rate the results are applied to the simulation of electrical stimulation for paraplegic patients to vary knee joint angle from 0 to 60

robust discrete time simple adaptive regulation robust simple adaptive controllers are proposed for regulation of linear discrete time systems with parameter uncertainties sufficient conditions for closed loop stability of the proposed control scheme are derived these lead to the requirement of almost strictly positive realness of the plant which is shown to be equivalent under certain conditions to a minimum phase requirement a numerical example is given that demonstrates the applicability of the proposed method and the simplicity of its implementation

a fuzzy lyapunov function approach to estimating the domain of attraction for continuous time takagi sugeno fuzzy systems this paper deals with stability analysis and control design problems for continuous time takagi sugeno t s fuzzy systems the first aim is to present less conservative linear matrix inequality lmi conditions to design controllers and assess the stability the second relevant contribution is to present a new strategy to find an inner estimate of the domain of attraction da via lmis the results are based on the fuzzy lyapunov functions flfs and non parallel distributed compensation non pdc approaches finally examples illustrate the effectiveness and merits of the proposed methods

global stability and performance of a simplified adaptive algorithm abstract most self tuning and adaptive control algorithms usually use reference models controllers or identifiers of about the same order as the controlled plant since the dimension of the plants in the real world may be very large or unknown implementation of adaptive control procedures may be difficult or sometimes impossible in this paper we prove global stability for a simple adaptive algorithm that can use low order model reference and controllers since no observers or identifiers are used in the adaptation process the algorithm is basically fitted for systems that are denominated as almost positive real it is shown that at the price of bounded rather than vanishing output tracking errors the simple algorithm can be applied in systems that can be stabilized via constant output feedback these procedures are believed to reduce considerably the effort required for implementation of adaptive control in practical applications especially in multivariable large scale systems

static output feedback control synthesis for linear systems with time invariant parametric uncertainties this technical note is concerned with the problem of designing robust static output feedback controllers for linear discrete and continuous time systems with time invariant polytopic uncertainties sufficient conditions for static output feedback stabilizing controller designs are given in terms of solutions to a set of linear matrix inequalities furthermore the results are also extended to h2 static output feedback controller designs numerical examples are given to illustrate the effectiveness of the proposed design methods

stabilizability conditions of multivariable uncertain systems via output feedback control a stabilizability condition is established for stabilizing multivariable uncertain dynamical systems with matched uncertainties by linear static output feedback control it is shown that under the assumptions of minimum phase and nonsingular high frequency gain for the nominal plant stabilization with linear static output feedback control is achievable a necessary and sufficient condition is also obtained for the existence of constant output feedback which makes the closed loop system strictly positive real

a unifying polyhedral approximation framework for convex optimization we propose a unifying framework for polyhedral approximation in convex optimization it subsumes classical methods such as cutting plane and simplicial decomposition but also includes new methods and new versions extensions of old methods such as a simplicial decomposition method for nondifferentiable optimization and a new piecewise linear approximation method for convex single commodity network flow problems our framework is based on an extended form of monotropic programming a broadly applicable model which includes as special cases fenchel duality and rockafellar s monotropic programming and is characterized by an elegant and symmetric duality theory our algorithm combines flexibly outer and inner linearization of the cost function the linearization is progressively refined by using primal and dual differentiation and the roles of outer and inner linearization are reversed in a mathematically equivalent dual algorithm we provide convergence results for the general case where outer and inner linearization are combined in the same algorithm

an epsilon relaxation method for separable convex cost network flow problems we propose a new method for the solution of the single commodity separable convex cost network flow problem the method generalizes the epsilon relaxation method developed for linear cost problems and reduces to that method when applied to linear cost problems we show that the method terminates with a near optimal solution and we provide an associated complexity analysis we also present computational results showing that the method is much faster than earlier relaxation methods particularly for ill conditioned problems

output feedback stabilization of uncertain dynamical systems given an uncertain dynamical system with a linear nominal part matched uncertainties and a measured output a sufficient condition is given for the existence of an output feedback stabilizing controller for the general multiinput multioutput case checking the sufficient condition involves the selection of a matrix f for the special class of single input single output systems f is a positive or negative scalar and it is shown that if the nominal system transfer function is strictly positive real then an output feedback stabilizing controller does indeed exist

simplified adaptive control via improved robust positive real conditions a new method is derived for embedding plants in a robust state feedback scheme to achieve strictly positive realness of the resulting augmented plants a state feedback gain is derived that guarantees the strictly positive realness of the closed loop in presence of polytopic type possibly time varying parameter uncertainties in the model that describes the plant this is achieved by assigning different lyapunov functions to each of the vertices of the uncertainty polytope the obtained feedback gain is used to apply existing methods for robust simplified adaptive control on systems with possibly time varying polytopic uncertainties

an lmi approach to stability analysis of coupled parabolic systems we analyze the exponential stability of a class of distributed parameter systems the system we consider is described by a coupled parabolic partial differential equation with spatially varying coefficients we approximate the coefficients by splitting space domains but take into account approximation errors during stability analysis using a quadratic lyapunov function we obtain sufficient conditions for exponential stability in terms of linear matrix inequalities

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

parameter and controller dependent lyapunov functions for robust d stability and robust performance controller design this paper addresses the design of output controllers of chosen order in a given set for ensuring a number of desired properties such as robust d stability and robust performance it is shown that a sufficient condition for determining such controllers can be obtained by solving convex optimization problems with linear matrix inequality constraints through the introduction of a lyapunov function depending polynomially on both uncertainty and controller the proposed condition is also necessary for sufficiently large degrees of the polynomials involved under some assumptions

frequency domain conditions for strictly positive real functions frequency domain conditions for strictly positive real spr functions which appear in literature are often only necessary or only sufficient this point is raised in 1 2 where necessary and sufficient conditions in the s domain are given for a transfer function to be spr in this note the points raised in 1 i2 are clarified further by giving necessary and sufficient conditions in the frequency domain for transfer functions to be spr these frequency domain conditions are easier to test than those given in the s domain or time domain 1 2

a new decentralised controller design method for a class of strongly interconnected systems abstractin this paper two interconnected structures are first discussed under which some closed loop subsystems must be unstable to make the whole interconnected system stable which can be viewed as a kind of strongly interconnected systems then comparisons with small gain theorem are discussed and large gain interconnected characteristics are shown a new approach for the design of decentralised controllers is presented by determining the lyapunov function structure previously which allows the existence of unstable subsystems by fully utilising the orthogonal space information of input matrix some new understandings are presented for the construction of lyapunov matrix this new method can deal with decentralised state feedback static output feedback and dynamic output feedback controllers in a unified framework furthermore in order to reduce the design conservativeness and deal with robustness a new robust decentralised controller design method is given by combining with the parameter depend

a survey of extreme point results for robustness of control systems abstract this paper surveys a subset of the body of research which was sparked by kharitonov s theorem the focal point is extreme point results for robust stability and robust performance that is we give conditions under which satisfaction of a performance specification is ascertained for a family of systems by only checking a finite subset of the extreme members of this family the results which are surveyed apply mainly to systems with structured real parametric uncertainty in addition a number of counterexamples are given to illustrate cases for which an extreme point result does not hold for such cases a solution via the so called edge theorem is often possible

an e out of kilter method for monotropic programming the out of kilter method was first proposed by fulkerson 1961 for linear cost network flow and independently by minty 1961 for piecewise linear cost network flow and was then extended by rockafellar 1984 section 11k to piecewise linear cost monotropic programming we propose an extension of this method based on rockafellar s work and on ideas from e relaxation methods for convex cost network flow bertsekas et al 1997a de leone et al 1999 to monotropic programming in general the proposed method is amenable to a complexity analysis and its convergence is based on essentially a monotone decrease in vertical distance to the characteristic curve for each index

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

relaxation methods for network flow problems with convex arc costs we consider the standard single commodity network flow problem with both linear and strictly convex possibly nondifferentiable arc costs for the case where all arc costs are strictly convex we study the convergence of a dual gauss seide type relaxation method that is well suited for parallel computation we then extend this method to the case where some of the arc costs are linear as a special case we recover a relaxation method for the linear minimum cost network flow problem proposed in bertsekas 1 and bertsekas and tseng 2

a general theory for matrix root clustering in subregions of the complex plane we consider the general problem of root clustering of a matrix in the complex plane let a in c n times n and s subset c find the largest class of s and an algebraic criterion which is necessary and sufficient for lambda i a in s i 1 2 n we introduce two types of regions which constitute the largest class of s known to date the criterion is presented both for open regions and closed ones the results are used to define a design methodology for control systems moreover all classical results are shown to be special cases of the present theory

a new discrete time robust stability condition a new robust stability condition for uncertain discrete time systems with convex polytopic uncertainty is given it enables to check stability using parameter dependent lyapunov functions which are derived from lmi conditions it is shown that this new condition provides better results than the classical quadratic stability besides the use of a parameter dependent lyapunov function this condition exhibits a kind of decoupling between the lyapunov and the system matrices which may be explored for control synthesis purposes a numerical example illustrates the results

convex duality in stochastic optimization and mathematical finance this paper proposes a general duality framework for the problem of minimizing a convex integral functional over a space of stochastic processes adapted to a given filtration the framework unifies many well known duality frameworks from operations research and mathematical finance the unification allows the extension of some useful techniques from these two fields to a much wider class of problems in particular combining certain finite dimensional techniques from convex analysis with measure theoretic techniques from mathematical finance we are able to close the duality gap in some situations where traditional topological arguments fail

time domain and frequency domain conditions for strict positive realness the author states various time domain and frequency domain conditions pertaining to multivariate strict positive real systems and establishes the relationship between these conditions their relationship is clarified by means of a three tier diagram each tier represents a set of equivalent conditions successively stronger than the lower tier the main theorem in the present study can be considered as the positive realness lemma for strictly positive real systems

h consensus control with spectrum constraints for stochastic multi agent systems subject to x u v dependent noises abstract this paper is concerned with the problem of h consensus control with spectrum constraints for continuous time multi agent systems mass with x u v dependent noises by means of the spectrum technique the definition of stability is introduced and the relationship between the convergence rate of state responses and the stability is discussed for the closed loop system the aim of the addressed problem is to design a static output feedback controller such that the closed loop system is stable and the prescribed h performance requirement is satisfied in virtue of the stochastic analysis method and kronecker product sufficient conditions are obtained for the existence of h consensus controller two examples are provided to illustrate the availability of the proposed techniques

comments on design of strictly positive real systems using constant output feedback a recent publication states and proves the conditions under which a linear time invariant system with state space realization a b c can be made strictly positive real via constant output feedback this note is intended to briefly present the development of the proof and to give due credit to the first proofs of this statement

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

exponential feedback passivity and stabilizability of nonlinear systems abstract motivated by n krasovskii s characterisation of exponential stability the concept of exponential passivity is introduced it is shown that to make a nonlinear system with factorisable high frequency gain matrix exponentially passive via either state or output feedback exponential minimum phaseness and invertibility conditions are necessary and sufficient these conditions also guarantee exponential output feedback stabilisability this result extends previous results concerning linear systems

epi convergent discretizations of stochastic programs via integration quadratures the simplest and the best known method for numerical approximation of high dimensional integrals is the monte carlo method mc i e random sampling mc has also become the most popular method for constructing numerically solvable approximations of stochastic programs however certain modern integration quadratures are often superior to crude mc in high dimensional integration so it seems natural to try to use them also in discretization of stochastic programs this paper derives conditions that guarantee the epi convergence of the resulting objectives to the original one our epi convergence result is closely related to some of the existing ones but it is easier to apply to discretizations and it allows the feasible set to depend on the probability measure as examples we prove epi convergence of quadrature based discretizations of three different models of portfolio management and we study their behavior numerically besides mc our discretizations are the only existing ones with guaranteed epi convergence for these problem classes in our tests modern quadratures seem to result in faster convergence of optimal values than mc

an e relaxation method for separable convex cost generalized network flow problems cover title the extended abstract of this article appeared in the proceedings of the 5th international ipco conference vancouver june 1996 p 1

technical communique mitigation of symmetry condition in positive realness for adaptive control feasibility of nonlinear and adaptive control methodologies in multivariable linear time invariant systems with state space realization a b c is apparently limited by the standard strictly positive realness conditions that imply that the product cb must be positive definite symmetric this paper expands the applicability of the strictly positive realness conditions used for the proofs of stability of adaptive control or control with uncertainty by showing that the not necessarily symmetric cb is only required to have a diagonal jordan form and positive eigenvalues the paper also shows that under the new condition any minimum phase systems can be made strictly positive real via constant output feedback the paper illustrates the usefulness of these extended properties with an adaptive control example

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

stochastic programming duality multipliers for unbounded constraints with an application to mathematical finance a new duality theory is developed for a class of stochastic programs in which the probability distribution is not necessarily discrete this provides a new framework for problems which are not necessarily bounded are not required to have relatively complete recourse and do not satisfy the typical slater condition of strict feasibility these problems instead satisfy a different constraint qualification called direction free feasibility to deal with possibly unbounded constraint sets and calmness of a certain finite dimensional value function to serve as a weaker condition than strict feasibility to obtain the existence of dual multipliers in this way strong duality results are established in which the dual variables are finite dimensional despite the possible infinite dimensional character of the second stage constraints from this infinite dimensional dual problems are obtained in the space of essentially bounded functions it is then shown how this framework could be used to obtain duality results in the setting of mathematical finance

some convex programs without a duality gap an important issue in convex programming concerns duality gap various conditions have been developed over the years that guarantee no duality gap including one developed by rockafellar network flows and monotropic programming wiley interscience new york 1984 involving separable objective function and affine constraints we show that this sufficient condition can be further relaxed to allow the constraint functions to be separable we also refine a sufficient condition involving weakly analytic functions by allowing them to be extended real valued

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