brief multiplier based robust h design with time varying uncertainties a novel multiplier based approach for robust controller design with real parametric time varying uncertainty is proposed sufficient conditions for robustness analysis are derived in terms of noncausal dynamic multipliers that depend on the the rate of uncertainty variation an affine parametrization of such stability multipliers is introduced and the search for suitable multipliers is reduced to a convex minimization formulated as a linear matrix inequalities lmi problem stable dynamic scalings derived from this multiplier are employed to formulate an equivalent scaled h norm condition for robust stability based on which a m k m synthesis procedure is proposed to design robust controllers for the time varying parametric uncertainty considered

linear matrix inequalities in robustness analysis with multipliers abstract we show that a number of standard robustness tests can be reinterpreted as special cases of the application of the passivity theorem with the appropriate choice of multipliers we show how these tests can be performed using convex optimization over linear matrix inequalities

h controller synthesis for pendulum like systems abstract this paper focus on a stabilization problem for a class of nonlinear systems with periodic nonlinearities called pendulum like systems a notion of lagrange stabilizability is introduced which extends the concept of lagrange stability to the case of controller synthesis based on this concept we address the problem of designing a linear dynamic output controller which stabilizes in the lagrange sense a pendulum like system within the framework of the h control theory lagrange stabilizability conditions for uncertainty free systems and systems with norm bounded uncertainty in the linear part are derived respectively when these conditions are satisfied the desired stabilization output feedback controller can be constructed via feasible solutions of a certain set of linear matrix inequalities lmis

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

iterative methods for discrete nonlinear riemann hilbert problems abstract let d denote the complex unit disk t d and let f t r 2 r be a given function the paper presents iterative methods for the computation of functions w u iv which are holomorphic in d and satisfy the boundary relation f t u t v t 0 for all t t it is shown that an appropriate discretization of this boundary value problem can be obtained by replacing the class of holomorphic functions with the set of polynomials of degree not exceeding n and imposing the boundary condition at a grid of 2n uniformly distributed points on the circle t the proposed iterative methods for the solution of the discrete as well as the nondiscrete problems converge quadratically in a neighborhood of the solution error estimates for the iterative solutions are given in the scale of the sobolev spaces wk2

duality bounds in robustness analysis abstract multipliers are used in stability theory to reduce conservatism and exploit structural information about system components for an important class of stability problems which gives a resulting infinite dimensional multiplier optimization a corresponding dual problem is stated the dual gives valuable information about the original problem in particular error bounds for its finite dimensional approximations 1997 elsevier science ltd

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

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

lagrange stabilisation for uncertain phase controlled systems in this article we are concerned with the lagrange stabilisation problem for phase controlled systems with parameter uncertainties by using the kalman yakubovich popov lemma the frequency domain conditions for lagrange stability of nominal phase controlled systems are converted into linear matrix inequality lmi conditions the allowable parameter uncertainty bounds for controllability and observability as well as lmi conditions for lagrange stability of uncertain phase controlled systems are presented a controller design strategy based on the lmi method is proposed such that the uncertain phase controlled systems are lagrange stabilised a numerical example is provided to demonstrate the effectiveness of the proposed results

stability analysis and state feedback stabilization of pendulum like systems with multiple nonlinearities this paper addresses the lagrange stability analysis problem and the state feedback lagrange stabilization problem for pendulum like systems with multiple nonlinearities an existing method for analysing the lagrange stability of pendulum like systems with a single nonlinearity is generalized to pendulum like systems with multiple nonlinearities also a non degeneracy condition of the existing lagrange stability criterion is removed and a strict frequency domain inequality is used instead to study the state feedback lagrange stabilization problem this paper develops an extended strict bounded real lemma for linear systems which are not stable but stabilizable a sufficient condition for state feedback lagrange stabilization is proposed in terms of an algebraic riccati equation with a sign indefinite solution

brief paper factorization of multipliers in passivity and iqc analysis multipliers are often used to find conditions for the absolute stability of lur e systems they can be used either in conjunction with passivity theory or within the more recent framework of integral quadratic constraints iqcs we compare the use of multipliers in both approaches passivity theory requires that the multipliers have a canonical factorization and it has been suggested in the literature that this represents an advantage of the iqc theory we consider sufficient conditions on the nonlinearity class for the associated multipliers to have a canonical factorization

parametric robust structured control design we present a new approach to parametric robust controller design where we compute controllers of arbitrary order and structure which minimize the worst case h infty norm over a pre specified set of uncertain parameters at the core of our method is a nonsmooth minimization method tailored to functions which are semi infinite minima of smooth functions a rich test bench and a more detailed example illustrate the potential of the technique which can deal with complex problems involving multiple possibly repeated uncertain parameters

lyapunov functionals in complex spl mu analysis conditions for robust stability of linear time invariant systems subject to structured linear time invariant uncertainties can be derived in the complex spl mu framework or equivalently in the framework of integral quadratic constraints these conditions can be checked numerically with linear matrix inequality lmi based convex optimization using the kalman yakubovich popov lemma we show how lmi tests also yield a convex parametrization of a subset of lyapunov functionals that prove robust stability of such uncertain systems we show that for uncertainties that are pure delays the lyapunov functionals reduce to the standard lyapunov krasovksii functionals that are encountered in the stability analysis of delay systems we demonstrate the practical utility of the lyapunov functional parametrization by deriving bounds for a number of measures of robust performance beyond the usual h sub spl infin performance these bounds can be efficiently computed using convex optimization over linear matrix inequalities

a multiplier approach for the robust design of discrete time control systems with real complex uncertainties a novel approach for the design of robust controllers for discrete time uncertain systems using a multiplier approach is given our algorithm is based on a k m m synthesis type iteration involving a search for suitable multipliers and solving scaled h optimization problems the latter problems involve only the original plant data and the scalings are computed via standard spectral factorizations from the multiplier the design is performed entirely in the discrete time domain and the results can be used for multiobjective control design tchernychev and sideris 1997

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

robust performance against time varying structured perturbations in this paper we consider the problem of robust performance analysis for some nominal system m z against bounded linear time varying structured feedback perturbations we introduce a natural input output notion of rate of variation for a linear time varying operator we then exhibit upper and lower bounds on the maximum rate of variation of these perturbations against which robust performance is achievable using these bounds we show that the existence of frequency dependent d scales that render the norm of m z less than one is necessary and sufficient for robust performance against arbitrarily slowly varying structured linear perturbations of norm less than one this result suggests that it is natural and well justified to deal with the frequency dependent upper bound for the spl mu norm rather than spl mu itself this is reassuring given that the upper bound is easily and reliably computable while computation of complex spl mu appears difficult

approximate solution of periodic riemann boundary value problem for analytic functions a direct method for the approximate solution of periodic riemann boundary value problems for analytic functions is given through the approximation by complex splines by the cardinal splines of the first degree we get the approximation of the canonical function based on the approximate result of singular integrals with hilbert kernel furthermore we obtain the approximate solution which may be sufficiently close to the exact solution to any degree when the partition is sufficiently fine

stability multipliers and spl mu upper bounds connections and implications for numerical verification of frequency domain conditions the paper shows the equivalence between the following two approaches for obtaining sufficient conditions for the robust stability of systems with structured uncertainties 1 apply the classical absolute stability theory with multipliers and 2 use modern spl mu theory specifically the spl mu upper bound obtained by fan et al 1991 in particular the relationship between the stability multipliers used in absolute stability theory and the scaling matrices used in the cited reference is explicitly characterized the development hinges on the derivation of certain properties of a parametrized family of complex linear matrix inequalities a result of independent interest the derivation also suggests a general computational framework for checking the feasibility of a broad class of frequency dependent conditions based on which bisection schemes can be devised to reliably compute several quantities of interest for robust control

dichotomy property and lagrange stability for uncertain pendulum like feedback systems abstract in this note property of dichotomy and lagrange stability of the uncertain pendulum like systems with additional multiplicative and h uncertainty are considered respectively by using some results of h theory the property of dichotomy and lagrange stability of uncertain pendulum like systems with multiple equilibria are transformed into l or h norm condition of some transfer functions the convergence problem of all bounded solutions for some uncertain pendulum like systems can be converted into the internal stability of another system under unstructured perturbations

lagrange stabilization of pendulum like systems a pseudo h_ infty control approach this paper studies the lagrange stabilization of a class of nonlinear systems whose linear part has a singular system matrix and which have multiple periodic in state nonlinearities both state and output feedback lagrange stabilization problems are considered the paper develops a pseudo h control theory to solve these stabilization problems in a similar fashion to the strict bounded real lemma in classic h control theory a pseudo strict bounded real lemma is established for systems with a single unstable pole sufficient conditions for the synthesis of state feedback and output feedback controllers are given to ensure that the closed loop system is pseudo strict bounded real the pseudo h control approach is applied to solve state feedback and output feedback lagrange stabilization problems for nonlinear systems with multiple nonlinearities an example is given to illustrate the proposed method

brief circle and popov criteria as tools for nonlinear feedback design we convert classical absolute stability criteria into nonlinear design procedures which employ efficient numerical tools such as lmi s we first analyze an earlier circle criterion design and show that its feasibility is limited by conditions on the unstable part of the zero dynamics and the relative degree then we develop an extended circle criterion design which eliminates the relative degree obstacle the restrictions on the zero dynamics are relaxed by using the popov multiplier which also reduces controller complexity the results are illustrated with several design examples

robust performance analysis for structured linear time varying perturbations with bounded rates of variation the robust performance analysis problem is considered for linear time invariant lti systems subject to block diagonal structured and bounded linear time varying ltv perturbations with specified maximal rates of variation analysis methods are developed in terms of semidefinite programming for the computation of upper and lower bounds for the optimum robust performance level the upper bound computation is based on an integral quadratic constraint iqc test developed using a generalized version of the so called swapping lemma the lower bound computation method employs an extended version of the power distribution theorem together with a generalized version of the kalman yakubovich popov kyp lemma and serves as a means to assess the conservatism of the computed upper bounds in the case of dynamic ltv perturbations as corollaries of the underlying result for lower bound computation it is shown for general block diagonal uncertainty structures that thefrequency dependent constant d scaling tests are exact for robust performance analysis against arbitrarily slow fast dynamic ltv perturbations respectively

robustness under bounded uncertainty with phase information the authors consider uncertain linear systems where the uncertainties in addition to being bounded also satisfy constraints on their phase in this context the authors define the phase sensitive structured singular value ps ssv of a matrix and show that sufficient and sometimes necessary conditions for stability of such uncertain linear systems can be rewritten as conditions involving ps ssv they then derive upper bounds for ps ssv computable via convex optimization they extend these results to the case where the uncertainties are structured diagonal or block diagonal for instance

robust stability with time varying structured uncertainty considers the problem of assessing robust stability in the presence of block diagonally structured time varying dynamic uncertainty it is shown that robust stability holds only if there exist constant scalings which lead to a small gain condition the notion of stability here is finite gain stability over finite energy signals in sharp contrast to the case of time invariant dynamic uncertainty this result is not limited by the number uncertainty blocks these results parallel previous results regarding finite gain stability over persistent bounded signals

control of a class of pendulum like systems with lagrange stability in this paper the control problem of linear perturbed systems is investigated some strategies of controller design guaranteeing the closed loop systems with property of pendulum like and lagrange stability are proposed the given method is applied to the system of synchronous single phase motor with pulsing vibrating moment

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

discrete riemann hilbert problems interpolation of simply closed curves and numerical conformal mapping abstract let g be a simply connected region with 0 g and with a twice lipschitz continuously differentiable boundary curve and let z 1 n be an even number of n 2 n equidistant grid points on the unit circle sfnc z sfnc 1 with z 1 1 then there exists for all sufficiently large n a polynomial p n of degree n 1 normalized by the condition that the coefficient p 0 0 and the coefficients p 1 and p n 1 are real such that p n satisfies the interpolation condition p n z for all 1 n in a neighbourhood of the normalized conformal mapping function there is exactly one such interpolating polynomial the sequence of these p n converges to the conformal mapping function as n if is three times lipschitz continuously differentiable then the p n are also conformal mappings of the unit circle onto regions which approximate g an important tool for the theoretical investigation is a discrete analogon of the riemann hilbert problem we present two fast procedures for the numerical solution of the discrete riemann hilbert problem a conjugate gradient method with computational cost o n log n and a toeplitz matrix method with cost o n log 2 n using this one can calculate the interpolating polynomials by a newton method and in this way obtains very effective methods for the numerical approximation of the conformal mapping function

sampling method for semidefinite programmes with non negative popov function constraints an important class of optimisation problems in control and signal processing involves the constraint that a popov function is non negative on the unit circle or the imaginary axis such a constrain

duality and basis functions for robust h 2 performance analysis the robust h2 performance of a plant with a prescribed uncertain component refers to its white noise rejection level in the worst case with respect to its uncertainty set in this paper we derive a dual formulation for the computation of the robust h2 performance of a plant with unit gain structured uncertainty the dual framework gives a tight lower bound on robust h2 performance with no duality gap with respect to the primal approach paganini 1995a we also show how the upper and lower bounds for the robust h2 performance of a plant may be computed using semidefinite programming linear matrix inequality type formulae based on basis functions and provide an example which enables us to demonstrate the relative merits and the complementary nature of the basis function and the gridding approaches to robust h2 performance analysis for both the primal and the dual formulations