structured singular values and stability analysis of uncertain polynomials part 1 the generalized m abstract one of the primary goals in this two part series is to establish a link between the structured singular value and results in stability analysis of uncertain polynomials another primary goal is to develop an improved method for accurately and efficiently computing the structured singular value for an important class of problems in robust stability analysis to achieve these goal we first introduce a generalized framework of structured singular values and next show how stability problems for uncertain polynomials may be studied in this framework part 1 of this series is devoted entirely to the generalized structured singular values specifically for the case when a certain matrix representing the nominal system is of rank one an analytical expression for the generalized notion is derived in this case which involves solving a convex optimization problem in one real variable and renders the structured singular value readily solvable in particular when the general framework is specialized to that of the standard structured singular value the expression is solved explicitly also for several additional important cases explicit solutions are obtained the framework as well as results will then be used in part 2 of this series to study stability problems for a class of polynomials whose coefficients are affine functions of real or complex uncertainties we demonstrate that the generalized structured singular value is a suitable notion for these problems and its solution unifies a number of results obtained previously via alternative approaches for several problems of interest we further demonstrate that stability conditions based upon the structured singular value and those in the spirit of kharitonov theorem can be derive from one another and hence establish a link between two drastically different type of results

extremal robustness properties of multilinear interval systems this paper deals with robustness properties of a control system subject to parameter uncertainty when the parameters appear affine multilinearly in the coefficients of the characteristic polynomial the robust hurwitz stability of such a multilinear interval polynomial family reduces to that of the stability of a set of manifolds derived from the kharitonov polynomials several extremal properties of these stability manifolds are derived these properties are useful in determining the worst case parametric h and nonlinear sector bounded stability margins for control systems containing interval parametric uncertainty they generalize previous results obtained by the authors for the linear case

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 output tracking control of siso plants with multiple operating points and with parametric and unstructured uncertainties the robust output tracking control problem is considered for single input single output plants with multiple operating points and with both parametric and unstructured uncertainties i e interval parameter uncertainties and norm bounded unmodelled dynamics the parametric uncertainties can involve parameter perturbations as well as degree variations the plant with uncertainties is assumed to be of minimum phase but is not required to have a fixed number of unstable poles a new method is presented by which a linear time invariant robust controller is designed in two steps first a nominal controller is designed to get exact output tracking for the nominal plant then a robust compensator is added to achieve robust properties it is shown that robust stability robust static state property robust transient property and robust tracking property can be achieved simultaneously a computationally tractable design procedure is presented a salient feature of our results shown in the present paper is th

estimation based approach for real time optimisation of uncertain nonlinear systems abstractin this paper the minimisation of an unknown but measurable cost function with uncertain dynamics is considered the drift term of the uncertain dynamical system and the gradient of the objective function are treated as unknown time varying parameters a novel estimation scheme based on almost invariant manifolds is proposed to estimate the time varying parameters a direct gradient based adaptive extremum seeking controller is designed to solve the uncertain optimisation problem this approach is shown to improve the transient performance of real time optimisation control systems

some new results for mckean s graphs with applications to kac s equation the main goal of the present paper is to sharpen some results about the error made when the wild sums used to represent the solution of the kac analog of boltzmann s equation are truncated at the n th stage more precisely in carlen carvalho and gabetta j funct anal 220 362 387 2005 one finds a bound for the above mentioned error which depends on an e on the one hand it is shown that the least negative eigenvalue of the linearized collision operator is the best possible exponent on the other hand e is an extra strictly positive number and a a positive coefficient which depends on e too thus it is interesting to check whether e can be removed from the above bound according to the aforesaid reference this problem is studied here by means of the probability distribution of the depth of a leaf in a mckean random tree in fact an accurate study of the probability generating function of such a depth leads to conclude that the above bound can be replaced with a n

robustness with real parametric and structured complex uncertainty the problem of robust stability and performance in feedback control systems with n real possibly repeated uncertain parameters and structured unmodelled dynamics the latter taking the form of m complex blocks dynamic uncertainties is considered a robustness margin rm is defined with respect to such model uncertainty structure and an algorithm is developed to compute it robust stability and performance is characterized in terms of rm in much the same way as with the multivariate stability margin km or the structured singular value an iterative procedure employing the above algorithm is given to compute tight bounds on km and these bounds are exact for the case m 3

robust stability and performance of interval plants abstract in this paper we consider the problem of robust performance for systems with unstructured and parametric perturbations the parametric uncertainty is modeled through a single input single output interval system description and the unstructured uncertainty through a norm bounded perturbation for this type of parametric robust performance we show that in order to synthesize a controller that achieves robust performance for a family of interval systems it is sufficient to synthesize that controller for a much smaller subset of the interval family regardless of the number of uncertain parameters in the original problem this reduced set is composed of thirty two one parameter families of systems a special class of controllers with stronger results is also presented

kharitonov like theorems for robust performance of interval systems abstract the sensitivity function of a control system is an important concept in performance analysis classical filter design as well as modern robust h control for an interval system we prove that the maximal h norm of its sensitivity function is achieved at twelve out of sixteen kharitonov vertices similar kharitonov like results are established for the complementary sensitivity function and strict positive realness of interval systems these results are useful in robust performance analysis and h control design for dynamic systems under parametric perturbations

an overview of extremal properties for robust control of interval plants abstract this paper provides an overview of recent results on the problem of robust stability and performance of feedback control systems in the presence of plant perturbations the widely studied class of interval plants is considered and an effort is made to cover different types of plant perturbations by incorporating in addition to the interval plant description unstructured norm bounded perturbations extremality results for these classes of uncertain systems are provided in a unifying framework robust performance of control systems is addressed in the paper and a number of extremal results are given for the computation of the structured singular value function for feedback systems with interval plants also classical frequency response analysis for interval plant controller families of open loop transfer functions is surveyed with the aim of providing extremal results for the computation of specification parameters such as phase or gain margins and sensitivity and complementary sensitivity peaks

a generalization of kharitonov s theorem robust stability of interval plants the robust stability problem is considered for interval plants in the case of single input multioutput or single output multi input systems a necessary and sufficient condition for the robust stabilization of such plants is developed using a generalization of v l kharitonov s theorem 1978 the generalization given provides necessary and sufficient conditions for the stability of a family of polynomials delta s q sub 1 s p sub 1 s q sub m s p sub m s where the q sub i are fixed and the p sub i are interval polynomials the coefficients of which are regarded as a point in parameter space which varies within a prescribed box this generalization called the box theorem reduces the question of the stability of the box in parameter space to the equivalent problem of the stability of a prescribed set of line segments it is shown that for special classes of polynomials q sub i s the set of line segments collapses to a set of points and this version of the box theorem in turn reduces to kharitonov s original theorem

frequency response of uncertain systems with interval plants frequency domain properties of uncertain rational functions depending on two independent interval polynomials are investigated several simplifications for efficient computation of the envelope of the nyquist plots of the uncertain family are reported stronger computational reduction results are given for the bode magnitude and phase plots of an interval plant controller family of transfer functions these results allow for a clear understanding of the extremality properties of several frequency domain performance specifications commonly used in control system design

application of the finite inclusions theorem to robust analysis and synthesis in a recent publication the finite nyquist theorem and its corollary the finite inclusions theorem fit were stated and proved fit provides a simple condition for a family of polynomials with parameter uncertainty to have all its roots in a given convex region of the complex plane since fit requires knowledge of the family s value set at only finitely many points along the region s boundary it provides a powerful tool for robust analysis and synthesis of parametrically uncertain systems in this paper we demonstrate this fact by 1 employing fit in robust stability analysis 2 using fit in the computation of the weighted h norm of a transfer function with parameter uncertainty and 3 carrying out fit synthesis for an automatic bus steering problem

on robust stability of interval control systems the authors develop results on the robust stability of a nonlinear control system containing both parametric as well as unstructured uncertainty the basic system considered is that of the classical lur e problem of nonlinear control theory a robust version of the lur e problem consisting of a family of linear time invariant systems subjected simultaneously to bounded parameter variations and feedback perturbations from a family of sector bounded nonlinear gains is presently treated by using the kharitonov theorem to develop some extremal results on positive realness of interval transfer functions i e a family of rational transfer functions with bounded independent coefficient perturbations the authors determine the size of a sector of nonlinear feedback gains for which absolute stability can be guaranteed these calculations amount to the determination of the stability margin of the system under joint parametric and nonlinear feedback perturbations

absolute stability of systems with parametric uncertainty and nonlinear feedback this paper gives an extension of the absolute stability analysis of systems whose forward path contains a linear time invariant subsystem with uncertain coefficients and whose feedback path contains a conic nonlinearity the results provide a simple frequency domain interpretation of stability that can be of importance in compensator design and robust stability analysis in nonlinear systems

frequency response of interval plant controller families abstract in this paper a general frequency domain result for families of interval plant with a fixed linear controller is given it is shown that the locus of the polar diagrams of frequency responses of the transfer functions of an interval plant controller family is bounded by the polar plots relative to the 32 segments of transfer functions of the interval plant family easy proofs of several important results such as the generalization of the theorem of kharitonov for feedback systems with interval plants or the robust version of the small gain theorem for the same class of systems are constructed by using the general result more importantly an immediate consequence of the main theorem is that extremal phase gain margins or sensitivity complementary sensitivity peaks for systems of the family can be deduced from those of the 32 segments of the interval plant family

robust stability of multivariable systems with both real parametric and norm bounded uncertainties the robust stability of a multivariable system with both structured real parametric and unstructured norm bounded uncertainties is considered it is shown that if the real parametric uncertainty enters only linearly and independently into the numerator matrix and the denominator characteristic polynomial then the robust stability or stability margin can be tested or calculated exactly it is also shown that this in fact gives a way to compute a class of real structured singular values

robustness of sampled data control systems having parametric uncertainty a conic sector approach the robustness of sampled data control systems when the open loop system is modeled by a family of plants having affine or interval uncertainty structure is considered using the conic sector approach as adapted to sampled data systems the authors give a sufficient condition for robust stability that depends only on edge parameters in the case of affine uncertainty and on vertex parameters when interval uncertainty is assumed

h sub 2 norm checking for systems with parameter uncertainties this paper studies the problem of h sub 2 norm checking for a mimo system with parameter uncertainties under the assumptions that the coefficients of the numerator polynomial of the system s transfer function are multi affine functions of the uncertain parameter that varies in a convex polytope and that the coefficients of the denominator polynomial of the system s transfer function are affine functions of the uncertain parameter which varies in a convex polytope a vertex checking result is obtained

robust parametric classical control design this paper extends well known graphical design tools such as the nyquist plot bode plot and nichols charts to the domain of control systems containing parameter uncertainty we have specifically considered linear interval control systems namely those where the uncertain parameters appear linearly in the characteristic polynomial and vary in intervals because in these cases exact construction of frequency envelopes is possible using these developments we calculate various extremal stability margins for such systems and these are used with standard classical control design techniques to deal with robustness with respect to parametric uncertainty in addition we give an effective technique to iteratively select values of the design parameters within given ranges so that the closed loop system produces improved gain or phase margin

robust stability under structured and unstructured perturbations the problem of robust stability for linear time invariant single output control systems subject to both structured parametric and unstructured h sub infinity perturbations is studied a generalization of the small gain theorem which yields necessary and sufficient conditions for robust stability of a linear time invariant dynamic system under perturbations of mixed type is presented the solution involves calculating the h sub infinity norm of a finite number of extremal plants the problem of calculating the exact structured and unstructured stability margins is then constructively solved a feedback control system containing a linear time invariant plant which is subject to both structured and unstructured perturbations is considered the case where the system to be controlled is interval is treated and a nonconservative easily verifiable necessary and sufficient condition for robust stability is given the solution is based on the extremal of a finite number of line segments in the plant parameter property of a finite number of line segments in the plant parameter space along which the points closest to instability are encountered

necessary and sufficient conditions on negative imaginariness for interval siso transfer functions and their interconnection this article is concerned with the study of the negative imaginariness of interval transfer functions necessary and sufficient conditions for strictly negative imaginary interval transfer functions subject to parameter uncertainty are first derived based on kharitonov s theorem then certain properties of the interconnection system with negative imaginary interval transfer functions are analyzed finally an example is provided to illustrate the main results of the article

stability conditions for polytopes of polynomials in considering the stability or performance of linear systems with uncertain parameters one is led to consider the simultaneous stability of families of polynomials for a general stability region in the complex plane the author gives a stability criterion for polytopes of polynomials in terms of the stability of a minimal number of corners and edges of the polytope the testing set of edges and corners depends entirely on the edge directions of the polytope hence the results are particularly applicable to problems with fixed directions but varying lengths and positions of the edges applications to robust pole placement of uncertain systems and computation of worst case h sub infinity norms are demonstrated

on the parametric h sub spl infin problems of weighted interval plants in this paper we consider the parametric h sub spl infin problems of weighted interval plants a sufficient condition will be given on weighting function w s such that for any interval plant p s q r the maximal h sub spl infin norm of w s p s q r is achieved at one of the 16 so called weighted kharitonov plants it is shown that for first order and some high order stable weighting functions the extremality property is enjoyed generally by the maximal h sub spl infin norm of weighted interval plants

vertices and segments of interval plants are not sufficient for step response analyses abstract interval plants are of interest in control theory as models of uncertain systems they are useful because many worst case analyses of these models are simple to carry out for example robust stability of an interval plant can be determined by investigating only the four kharitonov vertices of the denominator polynomial also the maximum peak of the bode magnitude plot can be found using just 16 special plant vertices these 16 vertices are connected by 32 special line segments most stability and frequency domain analyses that cannot be done using only the special vertices can be carried out using just the segments from these results it is tempting to conjecture that the 16 vertices or at least the 32 segments are adequate for step response analyses this paper presents examples showing that these conjectures are not true

a necessary and sufficient condition for robust bibo stability abstract this paper is concerned with obtaining necessary and sufficient conditions for robustly stabilizing a class of plants characterized by a known linear shift invariant plant with additive perturbations of the form of possibly nonlinear time varying l stable operators we will show that in some sense the small gain theorem is necessary and sufficient even when the perturbations are restiricted to a class of linear operators

global well posedness of the 2d critical dissipative quasi geostrophic equation in the triebel lizorkin spaces abstract we prove global well posedness of the 2d critical dissipative quasi geostrophic equation for small initial data in the triebel lizorkin spaces f p q s s 2 p p q 1 in particular our result implies the global existence for small initial data in the sobolev space h s p s 2 p p 1 since f p 2 s h s p the key ingredients of our proof are the pointwise exponential decay estimate of the semigroup e t and maximum principle