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

monodromy of a class of analytic generalized nilpotent systems through their newton diagram newton diagram of a planar vector field allows to determine whether a singular point of an analytic system is a monodromic singular point we solve the monodromy problem for the nilpotent systems and we apply our method to a wide family of systems with a degenerate singular point so called generalized nilpotent cubic systems

on multipliers for bounded and monotone nonlinearities abstract recent results in equivalence between classes of multipliers for slope restricted nonlinearities are extended to multipliers for bounded and monotone nonlinearities this extension requires a slightly modified version of the zames falb theorem and a more general definition of phase substitution the results in this paper resolve apparent contradictions in the literature on classes of multipliers for bounded and monotone nonlinearities

an iqc approach to robust stability of aperiodic sampled data systems conditions for robust stability of sampled data systems with non uniform sampling patterns and structural uncertainties are derived the problem is tackled under the integral quadratic constraint iqc framework where the aperiodic sampling operation is modelled by an delay integration operator characterization based on integral quadratic constrains iqc is identified for this operator and the iqc theory is applied to derive convex stability criteria compared to the dominating lyapunov approach where the candidate lyapunov krasovskii functionals or looped functionals need to be tailored for the systems under consideration and therefore the stability conditions need to be re derived whenever additional uncertainties are considered the proposed approach has the advantage of avoiding such endeavor numerical examples are given to illustrate this main point and effectiveness of the proposed approach

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

incremental positivity nonpreservation by stability multipliers it is proved that stability multipliers such as zames falb 1968 multipliers popov 1961 multipliers and rl rc multipliers known to preserve the positivity of monotone memoryless nonlinearities do not in general preserve their incremental positivity our result implies that stability results based on these multipliers should be interpreted with caution since without incremental positivity the continuity of the closed loop system s input output relation may not be assured

optimization of lyapunov invariants in verification of software systems the paper proposes a control theoretic framework for verification of numerical software systems and puts forward software verification as an important application of control and systems theory the idea is to transfer lyapunov functions and the associated computational techniques from control systems analysis and convex optimization to verification of various software safety and performance specifications these include but are not limited to absence of overflow absence of division by zero termination in finite time absence of dead code and certain user specified assertions central to this framework are lyapunov invariants these are properly constructed functions of the program variables and satisfy certain properties analogous to those of lyapunov functions along the execution trace the search for the invariants can be formulated as a convex optimization problem if the associated optimization problem is feasible the result is a certificate for the specification

the zames falb iqc for systems with integrators a feedback interconnection of a neutrally stable linear time invariant system and a nonlinearity with 0 spl les x spl phi x spl les kx sup 2 is called critical because the worst case linearization is at best neutrally stable this characteristic makes the stability analysis of such systems particularly hard it is shown that an integrator and a sector bounded nonlinearity can be encapsulated in a bounded operator that satisfies several useful integral quadratic constraints which gives powerful tools for stability analysis of a general class of critically stable systems

equivalence between classes of multipliers for slope restricted nonlinearities different classes of multipliers have been proposed in the literature for obtaining stability criteria using passivity theory integral quadratic constraint iqc theory or lyapunov theory some of these classes of multipliers can be applied with slope restricted nonlinearities in this paper the concept of phase containment is defined and it is shown that several classes are phase contained within the class of zames falb multipliers there are two main consequences firstly it follows that the class of zames falb multipliers remains to date the widest class of available multipliers for slope restricted nonlinearities secondly further restrictions may be avoided when exploiting the parametrization of the other classes of multipliers

on the equivalent relationship between generalized performance robust stability and quadratic stability this technical note addresses the equivalent relationship between notions of generalized performance robust stability and quadratic stability for the feedback connection where is the transfer matrix of a nominal system and describes the set of uncertainty by defining the three notions in a more general setting the conventional equivalent relationship between robust stability and quadratic stability with respect to the norm bound uncertainty respectively the positive real uncertainty and the corresponding performance respectively the extended strict positive realness has been proved only special case of the results derived in the technical note a version of the kalman yakubovich popov lemma which plays a crucial role in establishing the equivalence between the generalized performance and the quadratic stability is also presented

stability analysis for time varying systems via quadratic constraints abstract in this paper we introduce the concept of quadratic constraint qc to deal with various stability problems for infinite dimensional linear discrete time varying ltv systems first we derive a necessary and sufficient condition for the close loop stability based on quadratic constraints qcs an equivalent condition of this stability criterion presents the relationship between the stabilization of each finite dimensional truncation system and that of the whole system moreover applying this stability criterion and the complete finiteness of the nest algebra we show that the plant is stabilizable if and only if it has a single strong representation finally we use qcs to give a necessary and sufficient condition for the robust stabilization of connected uncertainty set defined by gap metric

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

a multiloop generalization of the circle criterion for stability margin analysis in order to provide a theoretical tool well suited for use in characterizing the stability margins e g gain and phase margins of multiloop feedback systems multiloop input output stability results generalizing the circle stability criterion are considered generalized conic sectors with centers and radii determined by linear dynamical operators are employed to enable an engineer to specify the stability margins which he desires as a frequency dependent convex set of modeling errors including nonlinearities gain variations and phase variations which the system must be able to tolerate in each feedback loop without instability the resulting stability criterion gives sufficient conditions for closed loop stability in the presence of such frequency dependent modeling errors even when the modeling errors occur simultaneously in all loops so for example stability is assured as loop gains and phases vary throughout a set of nonzero measure whose boundaries are frequency dependent the stability conditions yield an easily interpreted scalar measure of the amount by which a muitiloop system exceeds or falls short of its stability margin specifications

a unifying framework for robust synchronization of heterogeneous networks via integral quadratic constraints a general framework for analysing robust synchronization in large scale heterogenous networks is proposed based on the theory of integral quadratic constraints iqcs dynamic agents are represented as linear time invariant single input single output systems the agents exchange information according to a sparse dynamical interconnection operator in order to achieve synchronization where their outputs are steered to the same possibly time varying signal the main technical hindrance to applying iqcs in this context lies with the presence of the marginally stable dynamics which define the trajectory to which the agents outputs synchronize it is shown that by working with conditions defined on modified signal spaces of interest and exploiting the graph structure underlying the connections between the dynamic systems iqc methods can be applied directly to synchronization analysis without recourse to loop transformations which may obscure the inherent structural properties of the multi agent networked systems decentralized and scalable conditions for synchronization are proposed within this setting the iqc framework is demonstrated to unify and generalize some of the existing results in the literature including certain nyquist type consensus certificates for time delay systems moreover it allows the role of feedback in robustness against uncertainty to be better manifested within the context of synchronization

recent developments on the stability of systems with aperiodic sampling an overview this article presents basic concepts and recent research directions about the stability of sampled data systems with aperiodic sampling we focus mainly on the stability problem for systems with arbitrary time varying sampling intervals which has been addressed in several areas of research in control theory systems with aperiodic sampling can be seen as time delay systems hybrid systems input output interconnections discrete time systems with time varying parameters etc the goal of the article is to provide a structural overview of the progress made on the stability analysis problem without being exhaustive which would be neither possible nor useful we try to bring together results from diverse communities and present them in a unified manner for each of the existing approaches the basic concepts fundamental results converse stability theorems when available and relations with the other approaches are discussed in detail results concerning extensions of lyapunov and frequency domain methods for systems with aperiodic sampling are recalled as they allow to derive constructive stability conditions furthermore numerical criteria are presented while indicating the sources of conservatism the problems that remain open and the possible directions of improvement at last some emerging research directions such as the design of stabilizing sampling sequences are briefly discussed

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

frequency dependent error bounds for uncertain linear models we study frequency dependent error bounds for approximation and truncation of linear dynamic models with uncertainty a possible application is in model simplification relevant for control design the uncertainty is described by quadratic constraints and the error bounds are calculated based on solutions to linear matrix inequalities

duality in multiplier based robustness analysis frequency domain conditions involving multipliers are useful for robustness analysis the resulting analysis problem is generally convex but infinite dimensional and numerical solutions restricted to finite dimensional subspaces need to be considered the resulting finite dimensional problem can be transformed into a linear matrix inequality which can be solved using efficient algorithms the paper presents a format for the dual of the infinite dimensional problem the dual problem can be used to estimate the conservatism of particular finite dimensional subspaces

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

zames falb multipliers for convergence rate motivating example and convex searches the bound of convergence rates for discrete time lurye sometimes written lur e systems has recently attracted much attention the contributions of this technical note are twofold firstly we sho

stability robustness of a feedback interconnection of systems with negative imaginary frequency response a necessary and sufficient condition expressed simply as the dc loop gain i e the loop gain at zero frequency being less than unity is given in this note to guarantee the internal stability of a feedback interconnection of linear time invariant lti multiple input multiple output systems with negative imaginary frequency response systems with negative imaginary frequency response arise for example when considering transfer functions from force actuators to colocated position sensors and are commonly important in for example lightly damped structures the key result presented here has similar application to the small gain theorem which refers to the stability of feedback interconnections of contractive gain systems and the passivity theorem which refers to the stability of feedback interconnections of positive real or passive systems a complete state space characterization of systems with negative imaginary frequency response is also given in this note and also an example that demonstrates the application of the key result is provided

absolute stability with a generalized sector condition this paper generalizes the linear sector in the classical absolute stability theory to a sector bounded by concave convex functions this generalization allows more flexible or more specific description of the nonlinearity and will thus reduce the conservatism in the estimation of the domain of attraction we introduce the notions of generalized sector and absolute contractive invariance for estimating the domain of attraction of the origin necessary and sufficient conditions are identified under which an ellipsoid is absolutely contractively invariant in the case that the sector is bounded by piecewise linear concave convex functions these conditions can be exactly stated as linear matrix inequalities moreover if we have a set of absolutely contractively invariant aci ellipsoids then their convex hull is also aci and inside the domain of attraction we also present optimization technique to maximize the absolutely contractively invariant ellipsoids for the purpose of estimating the domain of attraction the effectiveness of the proposed method is illustrated with examples

continuous time analysis eigenstructure assignment and h sub 2 synthesis with enhanced linear matrix inequalities lmi characterizations this note describes a new framework for the analysis and synthesis of control systems which constitutes a genuine continuous time extension of results that are only available in discrete time in contrast to earlier results the proposed methods involve a specific transformation on the lyapunov variables and a reciprocal variant of the projection lemma in addition to the classical linearizing transformations on the controller data for a wide range of problems including robust analysis and synthesis multichannel h sub 2 stateand output feedback syntheses the approach leads to potentially less conservative linear matrix inequality lmi characterizations this comes from the fact that the technical restriction of using a single lyapunov function is to some extent ruled out in this new approach moreover the approach offers new potentials for problems that cannot be handled using earlier techniques an important instance is the eigenstructure assignment problem blended with lyapunov type constraints which is given a simple and tractable formulation

exponential decay rate conditions for uncertain linear systems using integral quadratic constraints this technical note develops linear matrix inequality lmi conditions to test whether an uncertain linear system is exponentially stable with a given decay rate alpha these new alpha exponential stability tests are derived for an uncertain system described by an interconnection of a nominal linear time invariant system and a troublesome perturbation the perturbation can contain uncertain parameters time delays or nonlinearities this technical note presents two key contributions first alpha exponential stability of the uncertain lti system is shown to be equivalent to internal linear stability of a related scaled system this enables derivation of alpha exponential stability tests from linear stability tests using integral quadratic constraints iqcs this connection requires iqcs to be constructed for a scaled perturbation operator the second contribution is a list of iqcs derived for the scaled perturbation using the detailed structure of the original perturbation finally connections between the proposed approach and related work are discussed

on stability of systems with aperiodic sampling devices this technical note is concerned with stability analysis of aperiodic sampled data systems the stability problem is tackled from a pure discrete time point of view where the at sampling behavior of the system is modelled as the response of a nominal discrete time lti system in feedback interconnection with a structured uncertainty conditions under which the uncertainty is positive real pr are identified based on the pr property a number of integral quadratic constraints iqc are derived and the iqc theory is applied to derive stability conditions numerical examples are given to illustrate the effectiveness of the proposed approach

systems with sector bound nonlinearities a behavioral approach the class of nonlinear dynamical systems obtained by interconnection of a linear time invariant dynamical system and a memoryless time invariant sector bound nonlinearity has been widely studied in literature in this paper it is shown that many important results for this class of systems can be elegantly unified using behavioral theory of dynamical systems systems with slope restricted nonlinearities are also considered

a nonlinear functional approach to lft model validation model validation provides a useful means of assessing the ability of a model to account for a specific experimental observation and has application to modeling identification and fault detection in this paper we consider a new approach to the model validation problem by deploying quadratic functionals and more generally nonlinear functionals to specify noise and dynamical perturbation sets specifically we consider a general linear fractional transformation framework for the model structure and use constraints involving nonlinear functional inequalities to specify model non linearities and unknown perturbations and characteristics of noise and disturbance signals sufficient conditions for invalidation of such models are provided in terms of semidefinite programming problems

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

robustness with dynamic iqcs an exact state space characterization of nominal stability with applications to robust estimation for robustness analysis with integral quadratic constraints we formulate a new positivity condition on the solution of the corresponding linear matrix inequality which is necessary and sufficient for nominal stability of the underlying system the application of this technical result is illustrated by a complete solution of the l 2 gain and robust h 2 estimator design problems if the uncertainties are characterized by dynamic integral quadratic constraints

stability analysis of some classes of input affine nonlinear systems with aperiodic sampled data control in this paper we investigate the stability analysis of nonlinear sampled data systems which are affine in the input we assume that a stabilizing controller is designed using the emulation technique we intend to provide sufficient stability conditions for the resulting sampled data system this allows to find an estimate of the upper bound on the asynchronous sampling intervals for which stability is ensured the main idea of the paper is to address the stability problem in a new framework inspired by the dissipativity theory furthermore the result is shown to be constructive numerically tractable criteria are derived using linear matrix inequality for polytopic systems and using sum of squares technique for the class of polynomial systems

modified nevanlinna pick interpolation and feedback stabilization of linear plants with uncertainty in the gain factor abstract this paper is a sequel to an earlier work in which we applied classical novnnlinna pick interpolation to the stabilization of a single input output plant with uncertainly in the gain factor by means of an asymptotically stable control and feedback sensor in this present work we show that for the proper design we need a modification of the classical interpolation procedure a point not clearly brought out in our previous paper

brief multiplier theory for stability analysis of anti windup control systems we apply the passivity theorem with appropriate choice of multipliers to develop a general framework for analyzing stability of anti windup bumpless transfer awbt schemes for appropriate choices of the multipliers we reduce the resulting sufficient conditions for stability to equivalent linear matrix inequalities lmis we show that a number of previously reported attempts to analyze awbt stability using seemingly diverse techniques such as the popov circle and off axis circle criteria the optimally scaled small gain theorem generalized m upper bound and describing functions can all be generalized within the framework of this paper

a note on bartlett adjustments to likelihood ratio tests summary the invariance properties of the likelihood ratio statistic are used to question the validity of certain terms arising in the expansion of its null distribution it is suggested that the null distribution of bartlett adjusted tests is the appropriate reference chi squared distribution with error of order n 3 2 where n is the sample size bartlett adjustment factors are a widely used method for improving the large sample chi squared approximation to the null distribution of maximum likelihood ratio statistics among the major developments into the calculation and effect of the correction factors are those due to lawley 1956 hayakawa 1977 and barndorff nielsen cox 1984 for regular problems lawley showed that all of the cumulants of the bartlett adjusted log likelihood ratio statistic for testing a composite hypothesis agree with those of the appropriate chi squared reference distribution with error of order n 3 2 where n is the sample size assumed large hayakawa obtained an asymptotic expansion to order n l for the null distribution of the test statistic considered by lawley and obtained a slightly different result following barndorff nielsen cox 1984 let hayakawa s expansion for the null distribution of the likelihood ratio statistic for testing a composite hypothesis be written as f x qd x 24n f a2qd 4 x 2a2 al qd 2 x a2 al qd x o n 3 2 1 here qd x denotes the density function of the chi squared distribution with d degrees of freedom d being the dimension of the vector parameter specified under the null hypothesis and the terms al and a2 given in 7 of hayakawa are functions of the moments of certain derivatives of the log likelihood function for the special case d 1 and a single nuisance parameter the term a2 expressed in a particularly convenient form given in the imperial college thesis of g m cordeiro is also given here in 2 note that p 1 12dn fal is the bartlett adjustment factor lawley s result corresponds to that in 1 with the term a2 taken to be zero and in many commonly occurring situations the term a2 as given by hayakawa when calculated explicitly is in fact zero our purpose here is to suggest that there is an error in the expansion due to hayakawa and that in 1 the term a2 should not appear making hayakawa s result explicitly agree with that of lawley the basis of the argument is presented in the next section

delay robustness of interconnected passive systems an integral quadratic constraint approach we consider networks of passive systems with time delays in the interconnections and present a stability analysis technique with the help of the integral quadratic constraint iqc framework unlike the classical passivity approach that fails to characterize delay robustness and the small gain approach that conservatively accounts for arbitrarily large delays the new technique gives sharp stability estimates that depend on the duration of delay since the effect of delay depends on its duration relative to the time scales of the system we make use of a roll off iqc that captures magnitude roll off at high frequencies thus providing the critical time scale information we then combine this roll off iqc with an output strict passivity iqc that incorporates gain and phase information and demonstrate the benefit of this combined iqc approach on a cyclic interconnection structure with delay finally we develop a technique to verify these iqcs for classes of nonlinear state space models and present an example from internet congestion control

a framework for worst case and stochastic safety verification using barrier certificates this paper presents a methodology for safety verification of continuous and hybrid systems in the worst case and stochastic settings in the worst case setting a function of state termed barrier certificate is used to certify that all trajectories of the system starting from a given initial set do not enter an unsafe region no explicit computation of reachable sets is required in the construction of barrier certificates which makes it possible to handle nonlinearity uncertainty and constraints directly within this framework in the stochastic setting our method computes an upper bound on the probability that a trajectory of the system reaches the unsafe set a bound whose validity is proven by the existence of a barrier certificate for polynomial systems barrier certificates can be constructed using convex optimization and hence the method is computationally tractable some examples are provided to illustrate the use of the method

distributed nonlinear control of mobile autonomous multi agents abstract this paper studies the distributed nonlinear control of mobile autonomous agents with variable and directed topology a new distributed nonlinear design scheme is presented for multi agent systems modeled by double integrators with the new design the outputs of the controlled agents asymptotically converge to each other as long as a mild connectivity condition is satisfied moreover the velocity derivative of the output of each agent can be restricted to be within any specified neighborhood of the origin which is of practical interest for systems under such physical constraint the new design is still valid if one of the agents is a leader and the control objective is to achieve leader following as an illustration of the generality and effectiveness of the presented methodology the formation control of a group of unicycle mobile robots with nonholonomic constraints is revisited instead of assuming the point robot model the unicycle model is transformed into two double integrators by dynamic feedback linearization and the proposed distributed nonlinear design method is used to overcome the singularity problem caused by the nonholonomic constraint by properly restricting the velocities simulation results are included to illustrate the theoretical results

a popov criterion for systems with slowly time varying parameters a popov criterion is derived for systems with slowly time varying parameters the parameters and their time derivatives are assumed to take values in convex polytopes

feedback and optimal sensitivity model reference transformations multiplicative seminorms and approximate inverses in this paper the problem of sensitivity reduction by feedback is formulated as an optimization problem and separated from the problem of stabilization stable feedback schemes obtainable from a given plant are parameterized salient properties of sensitivity reducing schemes are derived and it is shown that plant uncertainty reduces the ability of feedback to reduce sensitivity the theory is developed for input output systems in a general setting of banach algebras and then specialized to a class of multivariable time invariant systems characterized by n times n matrices of h infty frequency response functions either with or without zeros in the right half plane the approach is based on the use of a weighted seminorm on the algebra of operators to measure sensitivity and on the concept of an approximate inverse approximate invertibility of the plant is shown to be a necessary and sufficient condition for sensitivity reduction an indicator of approximate invertibility called a measure of singularity is introduced the measure of singularity of a linear time invariant plant is shown to be determined by the location of its right half plane zeros in the absence of plant uncertainty the sensitivity to output disturbances can be reduced to an optimal value approaching the singularity measure in particular if there are no right half plane zeros sensitivity can be made arbitrarily small the feedback schemes used in the optimization of sensitivity resemble the lead lag networks of classical control design some of their properties and methods of constructing them in special cases are presented

stability analysis of systems with uncertain time varying delays stability of systems in the presence of bounded uncertain time varying delays in the feedback loop is studied the delay parameter is assumed to be an unknown time varying function for which the upper bounds on the magnitude and the variation are given the stability problem is treated in the integral quadratic constraint iqc framework criteria for verifying robust stability are formulated as feasibility problems over a set of frequency dependent linear matrix inequalities the criteria can be equivalently formulated as semi definite programs sdp using kalman yakubovich popov lemma as such checking robust stability can be performed in a computationally efficient fashion

global analysis of piecewise linear systems using impact maps and surface lyapunov functions this paper presents an entirely new constructive global analysis methodology for a class of hybrid systems known as piecewise linear systems pls this methodology infers global properties of pls solely by studying the behavior at switching surfaces associated with pls the main idea is to analyze impact maps i e maps from one switching surface to the next switching surface such maps are known to be unfriendly maps in the sense that they are highly nonlinear multivalued and not continuous we found however that an impact map induced by an linear time invariant flow between two switching surfaces can be represented as a linear transformation analytically parametrized by a scalar function of the state this representation of impact maps allows the search for surface lyapunov functions sulf to be done by simply solving a semidefinite program allowing global asymptotic stability robustness and performance of limit cycles and equilibrium points of pls to be efficiently checked this new analysis methodology has been applied to relay feedback on off and saturation systems where it has shown to be very successful in globally analyzing a large number of examples in fact it is still an open problem whether there exists an example with a globally stable limit cycle or equilibrium point that cannot be successfully analyzed with this new methodology examples analyzed include systems of relative degree larger than one and of high dimension for which no other analysis methodology could be applied this success in globally analyzing certain classes of pls has shown the power of this new methodology and suggests its potential toward the analysis of larger and more complex pls

a cutting plane algorithm for robustness analysis of periodically time varying systems an algorithm for robustness analysis of periodic systems is derived the system under consideration consists of a linear periodically time varying plant in feedback interconnection with a structured uncertainty conditions for robust stability and robust performance can be formulated in terms of periodic integral quadratic constraints iqcs in this way the robustness analysis becomes a problem of optimizing the parameters of the iqc a cutting plane algorithm is suggested for solving this infinite dimensional optimization problem

a structure exploiting preprocessor for semidefinite programs derived from the kalman yakubovich popov lemma semidefinite programs derived from the kalman yakubovich popov kyp lemma are quite common in control and signal processing applications the programs are often of high dimension which makes them hard or impossible to solve with general purpose solvers here we present a customized preprocessor kypd that utilizes the inherent structure of this particular optimization problem the key to an efficient implementation is to transform the optimization problem into an equivalent semidefinite program this equivalent problem has much fewer variables and the matrices in the linear matrix inequality constraints are of low rank kypd can use any primal dual solver for semidefinite programs as an underlying solver

iqc based l2 control of linear periodic systems abstract stability analysis of linear periodically time varying systems via integral quadratic constraints is extended to the problem of control design a full state feedback controller that satisfies exponential stability and l 2 gain disturbance attenuation from an external disturbance to a controlled output is designed for linear systems with periodically time dependent system matrices the main result relies on dual forms of certain integral quadratic constraints the solvability conditions for the problem are cast as a set of finite dimensional linear matrix inequalities and thus they are easily solvable moreover the best possible disturbance attenuation level can be obtained as a convex problem

interconnections of nonlinear systems with mixed small gain and passivity properties and associated input output stability results abstract a negative feedback interconnection consisting of two causal nonlinear systems is shown to be input output stable when a mixed small gain and passivity assumption is placed on each of the systems the mixed small gain and passivity property captures the well known notions of passivity and small gain associated with systems the property can be appropriately reduced to an input and output strictly passive system description or alternatively can be reduced to a description of a system with small gain more importantly the property captures the concept of a blending of the small gain and passivity ideas this concept of blending can be visualized for example by considering linear time invariant systems that exhibit passivity type properties at say low frequencies and lose these passivity type properties but have small gain at high frequencies

a cutting plane method for solving kyp sdps semidefinite programs originating from the kalman yakubovich popov lemma are convex optimization problems and there exist polynomial time algorithms that solve them however the number of variables is often very large making the computational time extremely long algorithms more efficient than general purpose solvers are thus needed to this end structure exploiting algorithms have been proposed based on the dual formulation in this paper a cutting plane algorithm is proposed in a comparison with a general purpose solver and a structure exploiting solver it is shown that the cutting plane based solver can handle optimization problems of much higher dimension

the complex structured singular value a tutorial introduction to the complex structured singular value is presented with an emphasis on the mathematical aspects of the based methods discussed here have been useful for analysing the performance and robustness properties of linear feedback systems several tests for robust stability and performance with computable bounds for transfer functions and their state space realizations are compared and a simple synthesis problem is studied uncertain systems are represented using linear fractional transformations lfts which naturally unify the frequency domain and state space methods

efficient computation of a guaranteed lower bound on the robust stability margin for a class of uncertain systems sufficient conditions for the robust stability of a class of uncertain systems with several different assumptions on the structure and nature of the uncertainties can be derived in a unified manner in the framework of integral quadratic constraints these sufficient conditions in turn can be used to derive lower bounds on the robust stability margin for such systems the lower bound is typically computed with a bisection scheme with each iteration requiring the solution of a linear matrix inequality feasibility problem we show how this bisection can be avoided altogether by reformulating the lower bound computation problem as a single generalized eigenvalue minimization problem which can be solved very efficiently using standard algorithms we illustrate this with several important commonly encountered special cases diagonal nonlinear uncertainties diagonal memoryless time invariant sector bounded popov uncertainties structured dynamic uncertainties and structured parametric uncertainties we also present a numerical example that demonstrates the computational savings that can be obtained with our approach

frequency domain criteria of robust stability for slowly time varying systems the problem of stability of feedback systems with structured slowly time varying uncertain gains is considered for the case when the pair uncertain gain derivative belongs to a given convex set a sufficient frequency domain condition of stability is obtained this condition is an mimo generalization of the siso results derived in the 60s in the context of the positivity theory

analysis of systems with slope restricted nonlinearities using externally positive zames falb multipliers this paper proposes an approach for assessing the stability of feedback interconnections where one element is a static slope restricted nonlinearity and the other element is a linear system the approach is based on the use of zames falb multipliers where the dynamic portion of the multiplier is chosen as an externally positive noncausal transfer function by restricting attention to a subset of these multipliers a set of pure linear matrix inequality conditions is obtained which requires no initial parameterization by the user a useful byproduct of using externally positive systems is that the results are applicable to nonodd slope restricted nonlinearities which is not the case for all classes of zames falb multipliers

integral quadratic constraint approach vs multiplier approach integral quadratic constraints iqc arise in many optimal and or robust control problems the iqc approach can be viewed as a generalization of the classical multiplier approach in the absolute stability theory in this paper we study the relationship between the two approaches for robust stability analysis using a slightly modified multiplier approach we show that the existence of an iqc is equivalent to the existence of a multiplier in most known cases it is hoped that this result provides some new insight into both approaches and makes them more useful in robust control applications

gain scheduling control with dynamic multipliers by convex optimization in this paper we provide a solution to the gain scheduling synthesis problem gsp for structured parametric and dynamic passive uncertainties which encompasses an approach for norm bounded uncertainties as considered in classical structured singular value theory if compared to known approaches the distinguishing feature is the use of frequency dependent multipliers to guarantee stability and performance of the controlled system on the basis of suitable parameterizations of these multipliers we develop necessary and sufficient conditions for the existence of gain scheduling controllers and discuss the benefits of this new design methodology

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

computer aided stability analysis renders popov criterion obsolete a class of multipliers that can be used for producing alternative stability tests for systems having a single incrementally conic nonlinearity f x has been nearly parameterized by zames and falb 1 the present note demonstrates how linear programming can be used to compute optimal multipliers leading to a frequency domain stability criterion that is less conservative than the standard graphical criteria

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

on stability of discrete time lti systems with varying time delays this technical note concerns robust stability analysis of discrete time linear time invariant lti systems with varying time delays the stability problem is tackled under the integral quadratic constraint iqc framework the time delay system in question is viewed as feedback interconnection of an lti stable operator and the so called delay difference operator for which iqc characterizations are required in order to apply iqc analysis one of the main contributions of the technical note is to derive novel iqc characterizations for the delay difference operator based on the assumption that the time varying delay parameter and its variation are bounded a set of new stability criteria emerges as the result numerical examples are given to illustrate the effectiveness of the new stability criteria another distinct contribution is the idea of introducing bounds on the so called m step total variation and m step average variation of the time varying delay parameter which is proven to be useful in stability analysis

robust stability analysis of sparsely interconnected uncertain systems in this paper we consider robust stability analysis of large scale sparsely interconnected uncertain systems by modeling the interconnections among the subsystems with integral quadratic constraints we show that robust stability analysis of such systems can be performed by solving a set of sparse linear matrix inequalities we also show that a sparse formulation of the analysis problem is equivalent to the classical formulation of the robustness analysis problem and hence does not introduce any additional conservativeness the sparse formulation of the analysis problem allows us to apply methods that rely on efficient sparse factorization techniques and our numerical results illustrate the effectiveness of this approach compared to methods that are based on the standard formulation of the analysis problem

reconciling nu gap metric and iqc based robust stability analysis this technical note elaborates on the flexibility of an approach that combines gap metric and integral quadratic constraint iqc based analysis in the study of uncertain feedback interconnections of distributed parameter transfer functions it is established that a standard gap ball robust stability result can be recovered within the blended iqc gap framework which only requires the existence of gap continuous paths within the uncertainty set of interest this is achieved in part by showing that sufficiently small gap balls are pathwise connected in the graph topology a linear fractional characterisation of the gap is a key ingredient this characterisation is underpinned by a certain j spectral factorisation also shown to exist herein

the proximal augmented lagrangian method for nonsmooth composite optimization we study a class of optimization problems in which the objective function is given by the sum of a differentiable but possibly nonconvex component and a nondifferentiable convex regularization term we introduce an auxiliary variable to separate the objective function components and utilize the moreau envelope of the regularization term to derive the proximal augmented lagrangian a continuously differentiable function obtained by constraining the augmented lagrangian to the manifold that corresponds to the explicit minimization over the variable in the nonsmooth term the continuous differentiability of this function with respect to both primal and dual variables allows us to leverage the method of multipliers mm to compute optimal primal dual pairs by solving a sequence of differentiable problems the mm algorithm is applicable to a broader class of problems than proximal gradient methods and it has stronger convergence guarantees and a more refined step size update rules than the alternating direction method of multipliers admm these features make it an attractive option for solving structured optimal control problems we also develop an algorithm based on the primal descent dual ascent gradient method and prove global exponential asymptotic stability when the differentiable component of the objective function is strongly convex and the regularization term is convex finally we identify classes of problems for which the primal dual gradient flow dynamics are convenient for distributed implementation and compare contrast our framework to the existing approaches

zames falb multipliers for quadratic programming in constrained linear model predictive control a quadratic program must be solved on line at each control step and this constitutes a nonlinearity if zero is a feasible point for this quadratic program then the resultant nonlinearity is sector bounded we show that if the nonlinearity is static then it is also monotone and slope restricted hence we show the existence of zames falb multipliers for such a nonlinearity the multipliers may be used in a general and versatile analysis of the robust stability of input constrained model predictive control

nonsmooth h_infty synthesis we develop nonsmooth optimization techniques to solve h infty synthesis problems under additional structural constraints on the controller our approach avoids the use of lyapunov variables and therefore leads to moderate size optimization programs even for very large systems the proposed framework is versatile and can accommodate a number of challenging design problems including static fixed order fixed structure decentralized control design of pid controllers and simultaneous design and stabilization problems our algorithmic strategy uses generalized gradients and bundling techniques suited for the h infty norm and other nonsmooth performance criteria we compute descent directions by solving quadratic programs and generate steps via line search convergence to a critical point from an arbitrary starting point is proved and numerical tests are included to validate our methods the proposed approach proves to be efficient even for systems with several hundreds of states

a less conservative lmi condition for stability of discrete time systems with slope restricted nonlinearities many conditions have been found for the absolute stability of discrete time lur e systems in the literature it is advantageous to find convex searches via lmis where possible in this technical note we construct two less conservative lmi conditions for discrete time systems with slope restricted nonlinearities the first condition is derived via lyapunov theory while the second is derived via the theory of integral quadratic constraints iqcs and noncausal zames falb multipliers both conditions are related to the jury lee criterion most appropriate for systems with such nonlinearities and the second generalizes it numerical examples demonstrate a significant reduction in conservatism over competing approaches

brief stability and l2 gain properties of lpv systems stability and l 2 gain properties of linear parameter varying systems are obtained under assumed bounds on either the maximum or average value of the parameter rate