an elementary proof of kharitonov s stability theorem with extensions gives an elementary proof of kharitonov s theorem using simple complex plane geometry without invoking the hermite bieler theorem kharitonov s theorem is a stability result for classes of polynomials defined by letting each coefficient vary independently in an arbitrary interval the result states that the whole class is hurwitz if and only if four special well defined polynomials are hurwitz the paper also gives elementary proofs of two previously known extensions for polynomials of degree less than six the requirement is reduced to fewer than four polynomials and the theorem is generalized to polynomials with complex coefficients

high order nonlinear initial value problems countably determined we give a numerical approximation of the solution of a high order nonlinear initial value problem by making use of certain properties of an adequate schauder basis

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

frequency domain criterion for robust stability of interval time delay systems abstract in this paper we characterize the boundary f b of the image f b of a box b in r m under a nonlinear mapping f r m c we generalize results recently reported by polyak and kogan 1993 necessary and sufficient conditions for robust stability of multiaffine systems mathematics research report 93 06 university of maryland baltimore county for multiaffine mappings and provide computationally tractable necessary and sufficient robust stability conditions for quasipolynomials with interval coefficients and interval delays a numerical stability verification for a quasipolynomial family with two interval delays is presented

characterization of zero locations of polytopes of real polynomials given a stability region of a polytope of polynomials the edge theorem can be used to verify certain root clustering properties of the polytope the authors consider the inverse problem starting with a polytope of real polynomials they present a result that allows the construction of the smallest set of regions in the complex plane which characterizes its zero location

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

robust stabilization of interval plants using lead or lag compensators abstract a lead or lag compensator robustly stabilizes a family of interval plants if and only if it simultaneously stabilizes the vertex plants

extended kronecker summation for cluster treatment of lti systems with multiple delays a new procedure is presented for determining the kernel and the offspring hypersurfaces for general linear time invariant lti dynamics with multiple delays these hypersurfaces as they have very recently been introduced in a concept paper r sipahi and n olgac automatica 41 2005 pp 1413 1422 form the basis of the overriding paradigm which is called the cluster treatment of characteristic roots ctcr in fact these two sets of hypersurfaces exhaustively represent the locations in the domain of the delays where the system possesses at least one pair of imaginary characteristic roots to determine the kernel and offspring we use the extraordinary features of the extended kronecker summation operation in this paper the end result is that the infinite dimensional problem reduces to a finite dimensional one and preferably into an eigenvalue problem following the procedure described in this paper we are able to shorten the computational time considerably in determining these hypersurfaces we demonstrate these concepts via some example case studies one of the examples treats a 3 delay system for this case another interesting perspective called the building block is also utilized to display the kernel in three dimensional space in the domain of spectral delays

algebraic design techniques for reliable stabilization in this paper we study two problems in feedback stabilization the first is the simultaneous stabilization problem which can be stated as follows given plants g 0 g 1 g l does there exist a single compensator c that stabilizes all of them the second is that of stabilization by a stable compensator or more generally a least unstable compensator given a plant g we would like to know whether or not there exists a stable compensator c that stabilizes g if not what is the smallest number of right half place poles counted according to their mcmillan degree that any stabilizing compensator must have we show that the two problems are equivalent in the following sense the problem of simultaneously stabilizing l 1 plants can be reduced to the problem of simultaneously stabilizing l plants using a stable compensator which in turn can be stated as the following purely algebraic problem given 2l matrices a 1 a l b 1 b l where a i b i are right coprime for all i does there exist a matrix m such that a i mb i is unimodular for all i conversely the problem of simultaneously stabilizing l plants using a stable compensator can be formulated as one of simultaneously stabilizing l 1 plants the problem of determining whether or not there exists an m such that a bm is unimodular given a right coprime pair a b turns out to be a special case of a question concerning a matrix division algorithm in a proper euclidean domain we give an answer to this question and we believe this result might be of some independent interest we show that given two n times m plants g 0 and g 1 we can generically stabilize them simultaneously provided either n or m is greater than one in contrast simultaneous stabilizability of two single input single output plants g 0 and g 1 is not generic

fixed point techniques and schauder bases to approximate the solution of the first order nonlinear mixed fredholm volterra integro differential equation abstract in this work we approximate the solution of the first order nonlinear fredholm volterra integro differential equation by means of a new method for its numerical resolution the main tools used to establish it are the properties of a biorthogonal system in a banach space and the banach fixed point theorem

a necessary and sufficient condition for the stability of nonnegative interval discrete systems the stability of nonnegative discrete systems with interval uncertainty is discussed for this class of system a necessary and sufficient condition for the stability is provided when all elements of the system matrix are assumed to be specified by intervals an illustrative example is used as a guideline to demonstrate the effectiveness of the result simple computational tests are suggested and their usefulness in this framework is discussed the possible extension of this result to more general classes of systems is addressed

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

to stabilize an interval plant family it suffices to simultaneously stabilize sixty four polynomials it is shown that stability of three specific polynomial families can be deduced from the stability of a finite number of polynomials these polynomial families are the characteristic polynomials of unity feedback loops with the controller in the forward path and where the plant includes a specific form of parameter uncertainty for the first polynomial family the plant has parameter uncertainty in the even or odd terms of the numerator or denominator polynomial for the second polynomial family the plant has a numerator or denominator which is an interval polynomial for the third polynomial family the plant is interval because of the structure of these results it is shown that they lead to robust stabilization results two examples are included the approach employed here was developed for plants with affine uncertainty it is demonstrated that considerable simplification results if the plants under investigation are interval

robust controller synthesis using kharitonov s theorem the authors present a method based upon kharitonov s theorem for synthesizing robustly stabilizing feedback controllers the approach is based upon a multiple plant model formulation with a quadratic cost functional sufficient conditions are obtained for characterizing robustly stabilizing static output feedback proportional controllers for mimo multiple input multiple output plants with denominator polynomial uncertainty

least squares approximation method for the solution of hammerstein volterra delay integral equations in this paper an efficient numerical method is developed for solving the hammerstein volterra delay integral equations by least squares ls approximation method which is based on a polynomial of degree n to compute an approximation to the solution of hammerstein volterra delay integral equations the convergence analysis of the approximation solution relative to the exact solution of the integral equation is proved and its accuracy is illustrated on two numerical examples the study of this integral equation is important because it has as a particular case the variant of a mathematical model from epidemiology

simultaneous stabilization of single input single output discrete time systems the problem of finding one compensator which simultaneously stabilizes a family of single input single output siso discrete time plants is considered the family of plants is described by the transfer functions pz q q in q which are generated via a z transformation a number of assumptions describing the set of allowable plants are then given these assumptions include some regularity conditions on the plants and a minimum phase requirement the satisfaction of these assumptions guarantees the existence of a strictly proper stable compensator c z for simultaneous stabilization an iterative computation method is provided for control design

on the nyquist envelope of an interval plant family in this note we study the envelope of the nyquist plots generated by an interval plant family and show that this boundary is not always contained in the nyquist plots of the kharitonov plants with this motivation in mind we give a sufficient condition for an envelope point to be contained in the nyquist plot of a kharitonov plant and use it to generate large and critical portions of the nyquist envelope e g we show that the outer nyquist envelope of a stable interval plant is generated by the nyquist plots of the kharitonov plants another by product of this sufficient condition is a framework for developing new extreme point results for interval feedback systems this framework is useful in computing the phase margin and the maximal peaking in the sensitivity and complementary sensitivity functions and in stating a robust version of the circle criterion we also use this framework to easily explain existing extreme point results for the gain margin the h sub spl infin norm and the positive realness of interval plants one conclusion which emerges is this seemingly all important properties of an interval feedback system are deducible from the kharitonov plants

mas consensus and delay limits under delayed output feedback in this technical note we study the consensus problem for discrete time multi agent systems over an undirected fixed network communication graph focusing on the robustness of consensus with respect to communication delay we assume that the agents input is subject to a constant albeit possibly unknown time delay and is generated by a distributed dynamic output feedback control protocol drawing upon concepts and techniques from robust control notably those concerning gain margin optimization and analytic interpolation we derive explicit closed form conditions for general linear agents to achieve consensus our results display an explicit dependence of the consensus condition on the delay value as well as on the agent s unstable poles and non minimum phase zeros showing that delayed communication between agents will generally hinder consensus and impose restrictions on the network topology we also show that a lower bound on the maximal delay allowable for consensus can be computed by a simple line search method

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

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

stabilization of a linear plant via a stable compensator having no real unstable zeros abstract this paper studies the following problem under what conditions can a linear plant be stabilized by a stable compensator having no real zeros in the closed right half plane solving this problem is an important step towards to solving the problems of stabilization of linear systems via stable minimum phase compensation and simultaneous stabilization of n n 2 plants in the paper necessary and sufficient conditions are derived these conditions are completely general and therefore encompass unstable nonminimum phase plants as well it is shown that the stabilizability solely depends on the locations of real zeros and real poles of a given plant for a plant satisfying the stabilizability conditions a computational design procedure for constructing a desired compensator is also provided and illustrated via an example

the reproducing kernel method for solving the system of the linear volterra integral equations with variable coefficients in this paper we apply the reproducing kernel method to give the exact solution and approximate solution for the system of the linear volterra integral equations with variable coefficients some examples are given showing its effectiveness and convenience finally the numerical results obtained by the reproducing kernel method are superior to those obtained by other methods in farshid mirzaee 2010 4 tahmasbi and fard 2008 5 saeed and ahmed 2008 8

robust stabilizability for a class of transfer functions this paper is concerned with the robust stabilizability for single input single output plants robust stabilizability means that a fixed controller can stabilize simultaneously all the plants in a given class which is characterized by a frequency dependent uncertainty band function around the transfer function of a nominal model a necessary and sufficient condition for robust stabilizability is derived based on the well known nevanlinna pick theory in classical analysis it is shown that the values of the uncertainty band function should be restricted within a certain range at the unstable poles of the nominal model in order for the class to be robustly stabilizable a procedure of synthesizing a robust stabilizer is given and the parametrization of all the robust stabilizers is also shown

numerical computational solution of the volterra integral equations system of the second kind by using an expansion method an expansion method is used for treatment of second kind volterra integral equations system this method gives an analytic solution for the system the method reduces the system of integral equations to a linear system of ordinary differential equations after constructing boundary conditions this system reduces to a system of equations that can be solved easily with any of the usual methods finally for showing the efficiency of the method we use some numerical examples

on the order of simultaneously stabilizing compensators in this paper simultaneous strong stabilization problem is considered and it is shown that there is no upper bound for the minimal order of a simultaneously strongly stabilizing compensator in terms of the plant orders a similar problem was also considered by smith et al 1986 where it was shown that such a bound does not exist for the strong stabilization problem of a single plant but the examples given in the article were forcing an approximate unstable pole zero cancellation or forcing the distance between two distinct unstable zeros to go to zero in this paper it is shown that 1 if approximate unstable pole zero cancellation does not occur and the distances between distinct unstable zeros are bounded below by a positive constant then it is possible to find an upper bound for the minimal order of a strongly stabilizing compensator and 2 for the simultaneous strong stabilization problem even for the two plant case such a bound cannot be found

non euclidian metrics and the robust stabilization of systems with parameter uncertainty this paper considers from a complex function theoretic point of view certain kinds of robust synthesis problems in particular we use a certain kind of metric on the disk the hyperbolic metric which allows us to reduce the problem of robust stabilization of systems with many types of real and complex parameter variations to an easily solvable problem in non euclidian geometry it is shown that several apparently different problems can be treated in a unified general framework a new result on the gain margin problem for multivariable plants is also given finally we apply our methods to systems with real zero or pole variations

comments on extreme point results for robust stabilization of interval plants with first order compensators the authors point out a simple alternative proof of the main result theorem 1 of the paper by barmish hollot kraus and tempo 1992 this proof follows immediately on applying the vertex result of hollot and yang 1990 to the generalization of kharitonov s theorem given by chapellat and bhattacharyya 1988 1989 the authors also point out that more general vertex results are known

galerkin method for constrained variational equations and a collage based approach to related inverse problems for a general variational equation with constraints we present both the stability of the corresponding galerkin method and a collage theorem for a related inverse problem in addition we demonstrate the applicability of the results by considering forward and inverse boundary value problems of some linear impulsive differential equations

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

numerical approaches for systems of volterra fredholm integral equations in this work we introduce two numerical methods for solving systems of volterra fredholm integral equations in the nonlinear case we suggest a fixed point method where the iterations are perturbed in a suitable way according to a schauder basis in the banach space of continuous functions c a b 2 in the linear case we propose a collocation method based on a particular class of approximating functions in both methods convergence analysis and or low computational cost are analysed taking into account the properties of the basis under consideration numerical results confirm the theoretical study

linear volterra integro differential equation and schauder bases in this paper we present a numerical method to approximate the solution of the linear volterra integro differential equation by making use of the expression of a function of the banach space c 0 1 in terms of a basis schauder we are able to define a sequence of functions which approximate in the uniform sense the solution of such equation likewise we study the error committed in each approximation some advantages that this method possess are that it is not necessary to solve linear equation systems and moreover the involved integrals are immediate

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

a necessary and sufficient condition for the stability of interval matrices consider the linear dynamical system described by the equation xdot t ax t where a is a constant real matrix with dimensions n n and the elements of matrix a are not known precisely we know only the upper and lower bounds of the intervals in which the elements of matrix a are confined in this paper there is proved a necessary and sufficient condition for the stability of solutions of eqn 1 with incomplete information about the elements of matrix a

closed loop hyperstability of interval plants this note examines the conditions for robust bounded input bounded output closed loop stability of interval plants when there is a nonlinear element in the feedback path specifically the authors present the robust generalized circle criterion and popov s criterion

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

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

numerical solution of linear volterra integral equations system of the second kind abstract there are several numerical approaches for solving linear volterra integral equations system of the second kind in this paper we have presented a method for numerical solution of linear volterra integral equations system based on the power series method the major advantage of which is being derivative free also this method reproduces the analytical solution when the exact solutions are polynomial the numerical results prove that the presented method is very effective and simple the software used for the numerical calculations in this study was matlab 7 4

a correction to rationalized haar wavelet bases to approximate solution of nonlinear fredholm integral equations with error analysis applied mathematics and computation 265 2015 304 312 abstract in the recent paper rationalized haar wavelet bases to approximate solution of nonlinear fredholm integral equations with error analysis applied mathematics and computation 265 2015 304 312 the authors have approximated the solutions of nonlinear fredholm integral equations nfies of the second type by using the successive approximations method in any stage the rationalized haar wavelets rhws and the corresponding operational matrices were applied to approximate the integral operator in theorem 4 2 page 307 of the reference 4 the authors introduced an upper bound for the error and explicitly stated that the rate of convergence of ui to u is o qi in which i is the number of iterations q is the contraction constant u is the exact solution and ui is the approximate solution in ith iteration this statement is not true and we prove carefully that the rate of convergence will be o iqi

all constant gain stabilizing controllers for an interval delay system with uncertain parameters a computationally tractable design method is presented which explicitly yields the entire set of all constant gain controllers which robustly stabilize a family of systems with uncertainty a desired optimal controller may then be selected from the feasible set the method applies to the class of systems where a delay element e hs with interval finite or infinite uncertainty about h is followed by a plant characterized by a rational transfer function with independent interval coefficients illustrative numerical examples are provided

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

single term walsh series method for systems of linear volterra integral equations of the second kind this paper generalizes single term walsh series method for solving systems of linear volterra integral equations of the second kind using the properties of walsh functions this method reduces these types of equations to a linear system of algebraic equations in addition this method i does not require any operational matrices ii needs less computational effort iii gives both piecewise constant and discrete solution iv yields good agreement with exact solutions convergence results and error analysis are discussed numerical examples are presented to illustrate the computational efficiency of the method

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

about some numerical approaches for mixed integral equations abstract in this paper we tackle on mixed volterra fredholm integral equations as in linear as in non linear cases to the aim to obtain numerical solutions of these models the authors propose in the linear case the direct collocation method using a p order quasi interpolating spline class and in the nonlinear case the fixed point method based on polynomial approximation built by schauder tensor bases the advantages of both methods are outlined and their convergence is studied numerical results confirm the theoretical statements

shortest curves in planar regions with curved boundary abstract a general framework is presented for describing shortest curve algorithms and their time complexity in regions of the plane whose boundaries may be curved an algorithm that accepts curved boundary jordan regions along with given start and end points and produces the shortest curve between them is presented its time complexity is bounded by the product of the complexity of the region s boundary and that of the output shortest curve when the region is a simple polygon with n vertices the time bound is o nk where k is the number of vertices in the shortest curve a second algorithm produces shortest curves in multiply connected regions with possibly curved boundary

direct multiple shooting method for solving approximate shortest path problems we use the idea of the direct multiple shooting method presented by bock in proceedings of the 9th ifac world congress budapest pergamon press 1984 for solving optimal control problems to introduce an algorithm for solving some approximate shortest path problems in motion planning the algorithm is based on a direct multiple shooting discretization that includes a collinear condition a continuity condition type in the direct multiple shooting method multiple shooting structure and approximation conditions in the case of monotone polygons it is implemented by a c code and a numerical example shows that our algorithm significantly reduces the running time and memory usage of the system

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

solution of systems of integro differential equations using numerical treatment of fixed point we approximate the solution of a system of nonlinear mixed fredholm volterra integro differential equations of the second kind using fixed point techniques and schauder bases in certain banach spaces the convergence and error are studied several numerical examples are given and the obtained numerical approximations are compared with the corresponding exact solutions

multiple shooting approach for computing approximately shortest paths on convex polytopes abstract in this paper we use a multiple shooting approach in solving boundary value problems for ode to introduce a novel iterative algorithm for computing an approximate shortest path between two points on the surface of a convex polytope in 3d namely the polytope is partitioned into subpolytopes shooting points and a straightness condition are established the algorithm specifies how to combine shortest paths between shooting points in subpolytopes to become the required shortest path by the straightness condition in particular the algorithm does not rely on steiner points and graph tools on the entire polytope it is implemented in c and a comparison with agarwal har peled and karia s algorithm on the accurate construction of the shortest path is presented

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

control of discrete time hybrid stochastic systems a realistic stochastic control problem for hybrid systems with markovian jump parameters can have switching parameters in both the state and the measurement equations furthermore both the base and jump states in general are not perfectly observed there are only two existing controllers for this problem both with complexity exponentially increasing with time the authors present another control algorithm for stochastic systems with markovian jump parameters this algorithm is derived through the use of stochastic dynamic programming and is designed to be used for realistic stochastic control problems i e with noisy state observations this scheme has fixed computational requirements at each stage and a natural parallel implementation simulation results are used to compare the algorithm with previous schemes

structured singular values and stability analysis of uncertain polynomials part 2 a missing link abstract in part 1 of this paper a generalized notion of structured singular value is introduced and studied specifically in the case when a certain matrix is of rank one in part 2 of this series we demonstrate that the framework developed in part 1 is particularly suitable for a class of stability problems and its unifies several frequency based stability conditions obtained via alternative approaches generally these problems correspond to a class of uncertain polynomials whose coefficients are perturbed in an affine fashion several stability conditions are derived readily from solution of the structured singular value presented in part 1 which either extend or coincide with previously known results in particular for family of interval and diamond polynomials we further show that the stability conditions based on the structured singular value and the corresponding results in the spirit of kharitonov theorem lead to one another thus establishing a link between the two types of drastically different results and accomplishing one of our main goals in this series a related robust stabilization problem is also investigated in this framework

a root distribution criterion for interval polynomials the problem of finding the conditions under which an interval polynomial has a given number of roots in the open left half plane and the other roots in the open right half plane irrespective of the values of its coefficients is considered a simple criterion is provided to test interval polynomials for the root distribution invariance viewed as an extension of kharitonov s theorem the goal is to provide an alternative theorem and then give an efficient means of checking the root distribution invariance

some new results on the simultaneous stabilizability of a family of single input single output systems abstract in this paper we obtain sufficiency conditions on the problem of simultaneous stabilization of a parameterized family of single input single output plants by a non switching compensator the sufficiency conditions include simultaneous stabilization and simultaneous pole assignment of a finite number of plants

a sufficient condition for simultaneous stabilization the condition under which it is possible to find a single controller that stabilizes k single input single output linear time invariant systems p sub i s i 1 k is investigated the concept of avoidance in the complex plane is introduced and used to derive a sufficient condition for k systems to be simultaneously stabilizable a method for constructing a simultaneous stabilizing controller is also provided and is illustrated by an example

on the thirty two virtual polynomials to stabilize an interval plant this paper studies the conservatism of the 32 virtual polynomials to stabilize an interval plant it is shown that working with the 32 virtual vertices is generally less conservative than with the kharitonov polynomials of the smallest interval polynomial containing the characteristic polynomial polytope by means of the former it is possible to find all the controllers such that the value set of the polytope of characteristic polynomials is applied in two quadrants as a maximum for each spl omega while using the latter only some of them can be found the cases in which both methods coincide are also analyzed and the conditions on the numerator and denominator of the controller are developed thus this coincidence can be known a priori from the characteristics of the coefficients of the numerator and denominator of the controller it is shown that these conditions are satisfied by the first order controllers

block bialternate sum and associated stability formulae abstract a new block bialternate sum of a partitioned square matrix with itself is defined and some basic properties established it has similar properties to the bialternate sum but it preserves the block structure of the summand it is used in place of the block kronecker sum and the block lyapunov sum to solve maximal stability problems of stable systems under low gain integral control and of singularly perturbed systems it is also used to find the critical gains for a first order controller with a scalar plant from which all stabilizing first order controllers can be constructed in each stability problem the internal dimensions of the resulting closed formulae are lower than those currently available

galerkin schemes and inverse boundary value problems in reflexive banach spaces we develop the galerkin method for a recent version of the lax milgram theorem the generation of the corresponding finite dimensional subspaces for concrete boundary value problems leads us to consider certain biorthogonal systems in the reflexive banach spaces in question in addition we present an application to the numerical solution of inverse problems involving certain elliptic boundary value problems

comments on a necessary and sufficient condition for the stability of interval matrices in a recent paper by bialas a new proof of a known result concerning interval polynomials was given this result is summarized here and a much stronger result is mentioned bialas main theorem on the stability of interval matrices is shown to be false

shortest paths help solve geometric optimization problems in planar regions the goal of this paper is to show that the concept of the shortest path inside a polygonal region contributes to the design of efficient algorithms for certain geometric optimization problems involving simple polygons computing optimum separators maximum area or perimeter inscribed triangles a minimum area circumscribed concave quadrilateral or a maximum area contained triangle the structure for the algorithms presented is as follows a decompose the initial problem into a low degree polynomial number of optimization problems b solve each individual subproblem in constant time using standard methods of calculus basic methods of numerical analysis or linear programming these same optimization techniques can be applied to splinegons curved polygons first a decomposition technique for curved polygons is developed this technique is substituted for triangulation in creating equally efficient curved versions of the algorithms for the shortest path tree ray shooting and two point shortest path

a perspective on the numerical treatment of volterra equations abstract we discuss the properties and numerical treatment of various types of volterra and abel volterra integral and integro differential equations

on the sixty four polynomials of djaferis to stabilize an interval plant this paper shows that 32 of the 64 polynomials developed by djaferis to stabilize an interval plant with a controller are superfluous thus it is sufficient to stabilize 32 polynomials simultaneously moreover the construction developed in this note allows the implementation of these polynomials in a simple way

the solution of a transcendental problem and its applications in simultaneous stabilization problems a complete solution for a special transcendental problem in system theory is given one of the interpretations of this transcendental problem is the following simultaneous stabilization problem under what conditions can two linear plants be simultaneously stabilized by a single compensator having no real poles in the closed right half of the complex plane these conditions are completely general and therefore encompass unstable non minimum phase plants as well it is shown that the simultaneous stabilizability of two given plants via such a restricted compensator depends solely on the locations of the real zeros of the two plants and their difference plant in the closed right half of the complex plane for two plants satisfying the stabilizability conditions a computational design procedure for constructing a desired compensator is provided and illustrated a computationally checkable necessary condition for simultaneous stabilization of three plants is also given

controller synthesis via the 4 k polynomial approach abstract in some recent work it was shown that to stabilize systems with real parameter uncertainty it suffices to find a controller that simultaneously stabilizes a finite number of polynomials these polynomials include those generated from the vertex plants as well as some generated by some fictitious vertex plants that involve the controller this paper deals with the issues of existence of such a controller controller synthesis and conservativeness of the design it is shown how this approach can enhance the stability robustness of an h design

root clustering for convex combination of complex polynomials in some important applications such as the edge theorem it is required that a polynomial is alpha stable along a parameter segment using the critical constraint the necessary and sufficient conditions for root clustering alpha stability of convex combinations of complex polynomials are presented the approach is general and requires that a certain real polynomial has no zeros in the open interval

on the stability properties of polynomials with perturbed coefficients given a polynomial p c s s n t 1 s n 1 t n 0 which is hurwitz or p d z z n t 1 z n 1 t n 0 which has zeros only within or on the unit circle it is of interest to know how much the coefficients t j can be perturbed while preserving the stability properties in this note a method is presented for obtaining the largest hypersphere centered at t t t 1 t n containing only polynomials which are stable

fractional representation algebraic geometry and the simultaneous stabilization problem an explicit relationship between the fractional representation approach feedback system design and the algebro geometric approach to system theory is formulated and used to derive a global solution to the feedback system problem these techniques are then applied to the simultaneous stabilization problem yielding a natural geometric criterion for a set of plants to be simultaneously stabilized by a single compensator

biorthogonal systems for solving volterra integral equation systems of the second kind with the aid of biorthogonal systems in adequate banach spaces the problem of approximating the solution of a system of nonlinear volterra integral equations of the second kind is turned into a numerical method that allows it to be solved numerically

to stabilize a k real parameter affine family of plants it suffices to simultaneously stabilize 4 4 polynomials a controller stabilizes an entire family of plants with affine uncertainty if it simultaneously stabilizes a finite number of polynomials an upper bound for this number is 4k