necessary and sufficient conditions for robust stability of discrete systems with coefficients depending continuously on two interval parameters abstract let a real polynomial in a complex variable whose coefficients are any given continuous functions of two real interval parameters be given necessary and sufficient conditions are derived for the polynomial to have all its zeros outside or inside the unit circle of the complex variable plane in a multi polynomial dependence of the coefficients on parameters the conditions involve in addition to a finite number of algebraic steps only resolving single variable fixed coefficients real polynomials

robust stability of a class of polynomials with coefficients depending multilinearly on perturbations necessary and sufficient conditions are given for robust stability of a family of polynomials each polynomial is obtained by a multilinearity perturbation structure restrictions on the multilinearity are involved but in contrast to existing literature these restrictions are derived from physical considerations stemming from analysis of a closed loop interval feedback system the main result indicates that all polynomials in the family of polynomials have their zeros in the strict left half plane if and only if two requirements are satisfied at each frequency the first requirement is the zero exclusion condition involving four kharitonov rectangles the second requirement is that a specially constructed theta 0 parameterized set of 16 intervals must cover the positive reals for each theta epsilon 0 2 pi

stability of polynomials under coefficient perturbation let the real polynomial a 0 x n a 1 x n 1 a n 1 x a n be stable and let the real numbers b k c k geq 0 0 leq k leq n be given we present a simple determinant criterion for finding the largest t 0 geq 0 such that the polynomial alpha 0 x n alpha 1 x n 1 alpha n 1 x alpha n is stable for all alpha k in a k b k t 0 a k c k t 0 cup a k 0 leq k leq n several further observations allow us to reduce the computational cost considerably

equivalent classes of critical circles we present numerical evidence that the fractal properties of the critical invariant circles of a typical area preserving twist map as summarized by the spectrum and the generalized dimensions d q depend only on the tails in the continued fraction expansion of the corresponding rotation numbers and d q are numerically the same for all critical invariant circles of the standard and sine maps which have the rotation numbers with the same periodic tail

robust stability of polynomials with multilinearly dependent coefficient perturbations analysis of the robust stability of a polynomial with multilinearly dependent coefficient perturbations is presented in this note some sufficient conditions for forming a convex polygon with the value set of the polynomials with multilinearly dependent coefficient perturbations are obtained a zero exclusion algorithm is then given to determine the d stability of such polynomials the well known kharitonov s theorem and the edge theorem for stability analysis can be included as special eases of the authors conclusions

structural analysis of robust control systems in this paper a necessary and sufficient condition for well posedness is presented using the inclusion condition of uncertainty and a set of all the systems corresponding to the uncertainty with which a closed loop system is well posed is given furthermore we analyze the topology of the set of systems based on the well posedness property we give the number of connected components in the set of uncertainty in the cases of complex matrix uncertainty and 2 spl times 2 real matrix uncertainty

method of robust stability analysis with highly structured uncertainties this paper presents a method of analyzing the stability of linear time invariant interconnected systems with uncertainties each subsystem is single input single output and its vector locus lies inside a polygon at each frequency it is shown that the convex closure of the image of the cartesian product of these polygons under the mapping phi det i fh agrees with the convex closure of the image of the vertices of these polygons under the mapping from this result the image can be estimated by a finite dumber of calculations a sufficient stability condition is obtained by applying this result to the multivariable nyquist stability criterion

some counterexamples in robust stability theory this note deals with the problem of determining if a linear system whose characteristics polynomial depends multilinearly on n independent uncertain real parameters i i 1 n is robustly stable it is shown by example that a polynomial in n variables may have a unique real root and that this observation disposes of several natural conjectures in robust stability theory in particular we show that in a certain sense there are no edge or m dimensional face kharitonov like theorems for the general multilinear case the result holds even when restricted to that subset of multilinear functions which can be written in the form f 1 n det i diag 1 n m for some complex matrix m

stability analysis of interval matrices another sufficient condition a sufficient condition for the stability of interval matrices is presented based on a lyapunov approach the condition while requiring the solution of a lyapunov matrix equation removes the restrictions imposed by heinen 1984 examples given illustrate the improvement of the proposed condition over the ones given by heinen and bialas 1983 and daoyi 1985

necessary and sufficient conditions for robust stability of linear systems with multiaffine uncertainty structure this correspondence is concerned with an image f b sup m of a box b sup m spl sub r sup m under a multiaffine transformation f r sup m spl rarr c the authors introduce a notion of a principal point of b sup m and prove that the boundary of f b sup m is covered by images of principal points the authors exploit this result to provide necessary and sufficient robust stability conditions for polynomials whose coefficients are multiaffine functions of parameters an application of the general criterion obtained in the correspondence to the particular case of systems with a cascade of first order uncertain blocks leads to a computationally tractable procedure that verifies stability of the systems

characterization of robust controllers abstract this paper presents a procedure for designing a robust controller which is able to stabilize a multivariable system with interval uncertainties in the system parameters the uncertain parameters are assumed to lie within known intervals the fundamental idea behind the procedure is to characterize the set of robust controllers as a convex set by the new matrix inequalities result a suitable controller can then be chosen by means of an optimization algorithm the controller thus chosen will be able to ensure that the closed loop matrix lies within desired intervals for all the prescribed interval uncertainties in the system parameters

a simple proof of kharitonov s theorem an alternative proof of kharitonov s theorem on the stability of linear time invariant continuous systems under parameter variations is presented

brief paper assigning closed loop invariant polynomials over polytopes for linear periodic systems this paper provides a parameterization of all output feedback periodic controllers providing closed loop invariant polynomials varying over some assigned polytopes for linear discrete time periodic plants using the notion of time invariant system associated to a periodic system the parameterization is provided in terms of the general solution of a suitably defined matrix diophantine equation

nonanalytic twist maps and frenkel kontorova models we give a review of our recent works on a class of twist maps and frenkel kontorova fk models in which the external potential function is nonanalytic c1 and endowed with a variable degree of inflection z when z 3 recurrence of invariant curves has been observed an inverse residue criterion for the reappearance of an invariant curve complementary to the residue criterion for the disappearance of an invariant curve is introduced to make a precise determination of the reappearance point we have also studied the local and global scaling behaviors of these curves the critical exponents the singularity spectrum and the generalized dimensions are found to vary with z when 2 z 3 but remain the same for z 3 the degree of inflection thus plays a role reminiscent of that of dimensionality in phase transitions with z 2 and 3 corresponding respectively to the lower and upper critical dimensions when the same nonanalytic function is substituted for the sinusoidal external potential in the standard fk model the same conclusions are reached there exists a sequence of pinning depinning transitions and the critical exponents of the phonon gap the coherence length and the peierls nabarro barrier all show the same dependence on z

assigning invariant polynomials over polytopes this paper provides a parameterization of all output feedback controllers yielding closed loop invariant polynomials varying over some assigned polytopes

an algorithm for robust pole assignment via polynomial approach a computational method for designing controllers which attempt to place the roots of the characteristic polynomial of an uncertain system inside some prescribed regions is presented the analysis is based on transfer functions of characteristic polynomials and the problem is formulated as one of semi infinite programming an example of an application is given to illustrate this approach

infinite systems of hyperbolic functional differential inequalities we consider initial value problems for infinite systems of first order partial functional differential equations the unknown function is the functional argument in equations and the partial derivations appear in the classical sense a theorem on the existence of a solution and its continuous dependence upon initial data is proved the cauchy problem is transformed into a system of functional integral equations the existence of a solution of this system is proved by using integral inequalities and the iterative method infinite differential systems with deviated argument and differential integral systems can be derived from the general model by specializing given operators

feedback minimax sensitivity and optimal robustness in this paper we look for feedbacks that minimize the sensitivity function of a linear single variable feedback system represented by its frequency responses sensitivity to disturbances and robustness under plant perturbations are measured in a weighted h infty norm in an earlier paper zames proposed an approach to feedback design involving the measurement of sensitivity by multiplicative seminorms which have certain advantages over the widely used quadratic norm in problems where there is plant uncertainty or where signal power spectra are not fixed but belong to sets the problem was studied in a general setting and some h infty examples were solved here a detailed study of the single variable case is undertaken the results are extended to unstable plants and explicit formulas for the general situation of a finite number of right half plane rhp plant zeros or poles are provided the q or approximate inverse parametrization of feedbacks that maintain closed loop stability is extended to the ease of unstable plants the h infty and wiener hopf approaches are compared

on the computation of solutions of systems of interval polynomial equations systems of algebraic equations with interval coefficients are very common in several areas of engineering sciences the computation of the solution of such systems is a central problem when the characterization of the variables related by such systems is desired in this paper we characterize the solution of systems of algebraic equations with real interval coefficients the characterization is obtained considering the approach introduced in j comput appl math 136 2001 271

strong stabilization over polytopes the problem considered consists of finding a stable controller assigning a closed loop characteristic polynomial over a stability domain specified as a polytopic region of the space of characteristic polynomial coefficients the controller is required to be stable with respect to the same polytopic stability region specified for the closed loop system

simultaneous partial pole placement a new approach to multimode system design simultaneous partial pole placement of a family of single input single output plants is proposed as a generalization of the classical pole placement and stabilization problems this problem finds application in the design of a compensator for a family of linear dynamical systems in this note we show that the proposed problem is equivalent to a new class of transcendental problem using stable minimum phase rational functions with real coefficients a necessary condition for the solvability of the associated transcendental problem is obtained finally a counterexample to the following conjecture is obtained pairs of simultaneously stabilizable plants of bounded mcmillan degree have simultaneously stabilizing compensators of bounded mcmillan degree

a class of multilinear uncertain polynomials to which the edge theorem is applicable this paper gives a class of multilinear uncertain polynomials to which the edge theorem is applicable in the paper a notion of symmetric uncertainty structure is introduced the main result is as follows if a multilinear uncertain polynomial has a symmetric uncertainty structure and the uncertain parameters are nonoverlapping then the corresponding value sets will be convex polygons this means that the use of the zero exclusion condition to test the robust stability of such uncertain polynomials is feasible since we need only to calculate the extreme points of the convex polygons furthermore a version of the edge theorem can be used to test the robust stability of such uncertain polynomials

easily testable sufficient conditions for the robust stability of systems with multilinear parameter dependence abstract a number of robust stability problems take the following form a polynomial has real coefficients which are multilinear in real parameters that are confined to a rectangular axis parallel box in parameter space an efficient method is required for checking the stability of this set of polynomials we present two sufficient conditions in this paper which can be applied in many cases they involve checking certain properties at the corners and edges of the parameter space box

robust stability of polynomials with multilinear parameter dependence abstract the problem is studied of testing for stability a class of real polynomials in which the coefficients depend on a number of variable parameters in a multilinear way we show that the testing for real unstable roots can be achieved by examining the stability of a finite number of corner polynomials obtained by setting parameters at their extreme values while checking for unstable complex roots normally involves examining the real solutions of up to m 1 simultaneous polynomial equations where m is the number of parameters when m 2 this is an especially simple task

perturbation bounds for robust stability in linear state space models abstract this paper addresses the aspect of stability robustness of linear systems in the time domain upper bounds on the linear perturbation of an asymptotically stable linear system are obtained to maintain stability both for structured as well as unstructured perturbations using the lyapunov approach for structured perturbation the resulting bound is such that it garners the structural information about the nominal as well as the perturbation matrix into a single unified expression in the case of unstructured perturbations special features of the nominally stable matrix are exploited resulting in simpler expressions for the bound without the need to solve the lyapunov equation improvement of the proposed measures is illustrated with the help of examples

a note on robust pole placement pole placement algorithm in single input single output siso systems is discussed with respect to the corresponding real stability radius of the resulting closed loop polynomial

robust schur stability of a polytope of polynomials the main objective of the authors is to provide a necessary and sufficient condition for a polytope of polynomials to have all its zeros inside the unit circle the criterion obtained serves as a discrete time counterpart for results in s bialas 1985 and f fu and b r barmish 1987 for the continuous case also the results are reduced to operations on n 1 n 1 matrices it is concluded that by the edge result of a c bartlett et al 1987 it suffices to check the exposed edges in order to determine whether a polytope of polynomials has all its zeros in a simply connected region d

characterization of allowable perturbations for robust stability the problem of determining stability for a closed loop linear multivariable process in the presence of ill defined process perturbations is considered it is shown that m matrices may be used to characterize a class of allowable perturbations

characterization of allowable perturbations of a closed loop system with pole placement controller the problem of determining stability of a closed loop transfer function in the presence of coefficient perturbations is considered the results of interval polynomial theory are used to characterize the allowable interval of variation

a new approach to the optimization of envelope constrained filters with uncertain input in envelope constrained filtering the filter is optimized subject to the constraint that the filter response to a given signal lies within a specified envelope or mask in a number of signal processing applications the envelope constrained filtering problem is more directly relevant than least squares approximation based approaches the present authors develop an efficient method for solving an extended version of the envelope constrained filtering problem in which the input pulse is not known exactly but is known to lie within a specified mask this envelope constrained problem with uncertain input ecui has been examined elsewhere but the algorithm proposed there for its solution has in general inferior convergence characteristics

ellipsoidal approximation of the stability domain of a polynomial the stability domain in the space of coefficients of a polynomial is a nonconvex set in general in this note we propose a new convex ellipsoidal inner approximation of this set derived via optimization over linear matrix inequalities as a by product we obtain new simple sufficient conditions for stability that may prove useful in robust control design

expansion of det a b and robustness analysis of uncertain state space systems in this note an expansion for the determinant of the sum of two matrices det a b is given a robustness analysis approach for state space systems with uncertain parameters is investigated by using kharitonov s theorem some useful results are obtained several illustrative examples are given

necessary and sufficient conditions for root clustering of a polytope of polynomials in a simply connected domain linear time invariant systems with uncertain parameters are considered a test is given which is both necessary and sufficient for root clustering of a family of polytopic polynomials in a simply connected domain the test does not require checking a continuum of polynomials special cases include stability testing of continuous time systems and of discrete time systems with uncertain parameters as well as relative stability tests of such systems interval polynomials can also be handled as a special case

singularity spectrum of a critical kam torus the authors numerically determine the f alpha spectrum of scaling indices for critical kam tori in the area preserving standard map similarities and differences between the corresponding orbits of circle maps are discussed

invariance of the strict hurwitz property for polynomials with perturbed coefficients given a strictly hurwitz polynomial f lambda lambda n a n 1 lambda n 1 a n 2 lambda n 2 a 1 lambda a 0 it is of interest to know how much the coefficients a i can be perturbed while simultaneously preserving the strict hurwitz property for systems with n leq 4 maximal intervals of the a i are given in a recent paper by guiver and bose 1 in this note a theorem of kharitonov is exploited to obtain a general result for polynomials of any degree

a generalization of kharitonov s four polynomial concept for robust stability problems with linearly dependent coefficient perturbations kharitonov s four polynomial concept is generalized to the case of linearly dependent coefficient perturbations and more general zero location regions to this end a specially constructed scalar function of a scalar variable is instrumental to the robustness analysis the present work is motivated by two fundamental limitations of kharitonov s theorem namely 1 the theorem only applies to polynomials with independent coefficient perturbations and 2 it only applies to zeros in the left hand plane

robust hurwitz stability via sign definite decomposition abstract in this paper we first consider the problem of determining the robust positivity of a real function f x as the real vector x varies over a box x r l we show that it is sufficient to check a finite number of specially constructed points this is accomplished by using some results on sign definite decomposition we then apply this result to determine the robust hurwitz stability of a family of complex polynomials whose coefficients are polynomial functions of the parameters of interest we develop an eight polynomial vertex stability test that is a sufficient condition of hurwitz stability of the family this test reduces to kharitonov s well known result for the special case where the parameters are just the polynomial coefficients in this case the result is tight this test can be recursively and modularly used to construct an approximation of arbitrary accuracy to the actual stabilizing set the result is illustrated by examples

sensitivities of stability constraints and their applications let the real polynomial a s a 0 a 1 s a n s n with the coefficients being known differentiable functions a k x be given and let the constraints g i x 0 determine the strictly hurwitz property of the polynomial a s a simple and efficient method to calculate the derivatives partial g i x partial x j is proposed then the application of the method to the problem of stability of polynomials under coefficient perturbation by gradient optimization is discussed also a theorem characterizing the stability region and the newly introduced regions of nondestabilizing perturbations is given

physically based parameterizations for designing adaptive algorithms abstract structural properties are examined of systems with physical component values as parameters both state variable realizations and transfer function descriptions are investigated the transfer functions in particular are shown to be the ratios of polynomials with coefficients multilinear in the parameters these structures prove useful in formulating adaptive algorithms

robust pole assignment in linear state feedback numerical methods are described for determining robust or well conditioned solutions to the problem of pole assignment by state feedback the solutions obtained are such that the sensitivity of the assigned poles to perturbations in the system and gain matrices is minimized it is shown that for these solutions upper bounds on the norm of the feedback matrix and on the transient response are also minimized and a lower bound on the stability margin is maximized a measure is derived which indicates the optimal conditioning that may be expected for a particular system with a given set of closed loop poles and hence the suitability of the given poles for assignment

robust pole assignment for synthesizing feedback control systems using recurrent neural networks this paper presents a neurodynamic optimization approach to robust pole assignment for synthesizing linear control systems via state and output feedback the problem is formulated as a pseudoconvex optimization problem with robustness measure i e the spectral condition number as the objective function and linear matrix equality constraints for exact pole assignment two coupled recurrent neural networks are applied for solving the formulated problem in real time in contrast to existing approaches the exponential convergence of the proposed neurodynamics to global optimal solutions can be guaranteed even with lower model complexity in terms of the number of variables simulation results of the proposed neurodynamic approach for 11 benchmark problems are reported to demonstrate its superiority

numerical methods in control we study classical control problems like pole assignment stabilization linear quadratic control and h control from a numerical analysis point of view we present several examples that show the difficulties with classical approaches and suggest reformulations of the problems in a more general framework we also discuss some new algorithmic approaches

robust pole assignment abstract this paper presents new theorems on the theory of interval matrix inequalities and the theory of polynomials with interval roots and applies them to the problem of robust pole placement we formulate optimization problems and derive convergent iterative algorithms which allow the designer to find controllers that place closed loop poles within desired intervals for plants with unknown but bounded parameter uncertainties the algorithms are computationally reasonable and provide a useful addition to currently existing control cad tools

when is the multiaffine image of a cube a convex polygon abstract we give two simple sufficient conditions under which the multiaffine image in the complex plane of an m dimensional cube is a convex polygon a third condition which in some generic sense is necessary and sufficient is then obtained our conditions involve checking the locations of the image of the vertices of the cube these results help determine whether a parameterized family of polynomials th stable

some discrete time counterparts to kharitonov s stability criterion for uncertain systems in 1 kharitonov gave an elegant and simple stability criterion for continuous time systems this note reports on similar results for discrete time systems

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

robust stability under a class of nonlinear parametric perturbations this paper considers the robust stability verification of linear time invariant systems admitting a class of nonlinear parametric perturbations the general setting is one of determining the closed loop stability of systems whose open loop transfer functions consist of powers products and ratios of polytopes of polynomials apart from this general setting two special cases of independent interest are also considered the first special case concerns uncertainties in the open loop gain and real poles and zeros while the second special case treats uncertainties in the open loop gain and complex poles and zeros in light of the zero exclusion principle robust stability is equivalent to zero exclusion of the value sets of the system characteristic function a value set consists of the values of the characteristic functions at a fixed frequency the main results of this paper are as follows 1 the value set of the characteristic function at each fixed frequency is determined by the edges and some frequency dependent internal line segments 2 consequently hurwitz invariance verification simplifies to that of checking certain continuous scalar functions for avoidance of the negative real axis 3 for the case of real zero pole gain variations the critical lines are all frequency independent and therefore the determination of the robust stability is even simpler 4 for the case of complex zero pole gain variations the critical internal lines are shown to be either frequency independent or to be confined in certain two dimensional planes or three dimensional boxes

technical communique robust stability and performance with fixed order controllers this paper deals with the problem of synthesizing or designing a feedback controller of fixed dynamic order for a linear time invariant plant for a fixed plant as well as for an uncertain family of plants containing parameter uncertainty so that stability robust stability and robust performance are attained the desired closed loop specifications considered here are given in terms of a target performance vector representing a desired closed loop design in general these point targets are unattainable with a fixed order controller by enlarging the target from a fixed point set to an interval set the solvability conditions are relaxed and a solution is enabled results from the parametric robust control literature can be used to design the interval target family so that the poles and zeros of the closed loop system are guaranteed to remain in a prescribed region of the complex plane and closed loop performance measured for instance in the h norm are attained even when plant uncertainty is present it is shown that it is possible to devise a computationally simple linear programming approach that attempts to meet the desired closed loop specifications the approach developed here gives the entire set of controllers attaining the specifications as a convex set and also can be recast to give the lowest order controller attaining specifications

fractal properties of critical invariant curves we examine the dimension of the invariant measure for some singular circle homeomorphisms for a variety of rotation numbers through both the thermodynamic formalism and numerical computation the maps we consider include those induced by the action of the standard map on an invariant curve at the critical parameter value beyond which the curve is destroyed our results indicate that the dimension is universal for a given type of singularity and rotation number and that among all rotation numbers the golden mean produces the largest dimension

universal scaling of critical quasiperiodic orbits in a class of twist maps recently we have shown that the fractal properties of the critical invariant circles of the standard map as summarized by the spectrum and the generalized dimensions d q depend only on the tails in the continued fraction expansion of the corresponding rotation numbers in buric n mudrinic m and todorovic k 1997 30 l161 in the present paper this result is extended on the whole class of sufficiently smooth area preserving twist maps of cylinders we present numerical evidence that the and d q are the same for all critical invariant circles of any such map which have the rotation numbers with the same tail