algebraic theory for robust stability of interconnected systems necessary and sufficient conditions we consider an interconnected system s o made of linear mulrivariable subsystems which are specified by matrix fractions with elements in a ring of stable scalar transfer functions h given that the k th subsystem is perturbed from g k n rk d k 1 to tilde g k n rk delta n rk d k delta d k 1 and that the system s o is h stable we derive a computationally efficient necessary and sufficient condition for the h stability of the perturbed system these fractional perturbations are more general than the conventional additive and multiplicative perturbations the result is generalized to handle simultaneous perturbations of two or more subsystems

a condition for robust stabilizability a family of linear time invariant finite dimensional plants is called robustly stabilizable if there exists a single controller that stabilizes each plant in the family a condition for robust stabilizability is given that requires the plant transmission zeros to remain in the left half complex plane and moreover imposes a restriction on the high frequency behavior of the plant transfer matrix

an iterative design procedure for simultaneous stabilization of mimo systems this paper considers the problem of finding one compensator which simultaneously stabilizes a family of mimo plants the family of plants bdp s q q q represented by their transfer function matrices may be uncountable thus allowing for the possibility of a continuum of variations in plant parameters a hermite invariant factorization of the plant is introduced and shown to be quite useful as far as stabilization is concerned to obtain the desired results a number of assumptions describing the set of allowable plants are imposed these assumptions include some regularity conditions on the plant and mimo versions of minimum phase and one sign high frequency gain requirements the satisfaction of these assumptions guarantees the existence of a strictly proper stable compensator c s guaranteeing simultaneous stabilization the algorithm used for controller design has one most attractive feature namely it is recursive in nature and allows the designer to select one compensator coefficient at a time

simultaneous controller design for time varying linear systems abstract in this paper we establish the parametrization of all simultaneously stabilizing controllers for two time varying linear systems a new approach to the simultaneous stabilization of three time varying linear systems is also provided

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

coherent states for the harmonic series irreducible representations of u p q i unitary operator coherent states unitary operator coherent states as defined by klauder 1963 perelomov 1972 and gilmore 1974 are considered for the harmonic series irreducible representations of u p q various properties of these states are reviewed including their reproducing kernel and measure explicit forms as well as the representation they lead to for the u p q generators starting from a contraction of the su p q lie algebra both dyson and holstein primakoff boson realisations of u p q are obtained in terms of pq pairs of boson creation annihilation operators and of the generators of a u p u q intrinsic group such boson realisations are applied to determine the matrix elements of the u p q generators between discrete bases classified according to the chain u p q contains implies u p u q finally the isomorphism between so 4 2 and su 2 2 is employed to derive the corresponding properties of the so 4 2 unitary operator coherent states for application in atomic physics

simultaneous stabilization and robust control of polynomial nonlinear systems using sos techniques this technical note is concerned with the simultaneous stabilization and robust performance control for a class of polynomial nonlinear systems building on the tenets of state dependent polynomial lyapunov functions we present sufficient conditions for simultaneous stabilization with and without hinfin performance these conditions can be verified by the recently developed sum of squares technique which essentially solves a linear matrix inequality feasibility problem

simultaneous stabilization using stable system inversion abstract this paper investigates the use of stable system inversion to design controllers for the simultaneous stabilization of multi input multi output systems a parametrization of all simultaneously stabilizing controllers for systems admitting a stable inverse is developed the controller is shown to have the structure of a generalized observer for strictly proper systems an intermediate step of the design requires solving a sequence of strong stabilization subproblems for systems with components having identical stable minimum phase transfer functions the design requires only a single model of the component in the controller in systems where a stable inverse fails to exist an interpolation scheme is proposed to develop a stabilizing controller

brief on simultaneous h control and strong h stabilization in this note the simultaneous h control problem is addressed it is first shown that the existence of a single controller which simultaneously stabilizes two systems with a given h norm level is equivalent to that of a strong h stabilization controller of an associated system moreover the result is extended to the case involving r 2 systems then a sufficient condition for the strong h stabilization is proposed based on the bounded real lemma finally a simple numerical example is presented to show the effectiveness of the result

simultaneous stabilization via linear state feedback control the problem of simultaneously stabilizing a finite collection of plants via a single compensator utilizing full state information is considered the authors assume that the complete state can be measured and seek a static linear state feedback control law sufficient conditions for such full state gains to exist are given and they show via several examples how stabilizing compensators can be designed

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

simultaneous stabilization for a set of boolean control networks abstract this paper investigates simultaneous stabilization of a collection of boolean control networks bcns by using the semi tensor product method and presents a number of new results first an algebraic expression of the bcns is obtained by the semi tensor product based on which some necessary and sufficient conditions are presented to solve the simultaneous stabilization problem by a free control sequence a state feedback control and an output feedback control respectively second using the column stacking form of matrices a new procedure is established to design both state feedback and output feedback controllers for the simultaneous stabilization problem the study of two illustrative examples shows that the new results obtained in this paper are very effective in solving simultaneous stabilization of a collection of bcns

feedback system design the tracking and disturbance rejection problems the problem of designing a compensator for a specified plant which simultaneously stabilizes the resultant feedback system and causes it to track a prescribed family of inputs and or reject prescribed disturbances is considered a set of linear design equations in the space of stable systems is formulated in a general linear systems setting and an explicit parameterization of the resultant solution space is obtained for a class of generalized multivariate systems the theory is illustrated with several single and multivariate examples

a generalization of the routh hurwitz stability criteria and an application to a problem in robust controller design the classical results of routh and hurwitz on the stability of polynomials are generalized from the open left half plane to certain other subregions g of the complex plane by our method arbitrary conic sections can be handled the results are interpreted as criteria for g stability of linear systems a numerical algorithm to compute the criteria is given an application to a problem in robust controller design is outlined

a state space interpretation of simultaneous stabilization a state space interpretation of some previously known results on simultaneous stabilization is given the results reviewed were obtained using the yuola lemma

observer parameterization for simultaneous observation the stable inverse approach is used to obtain the observers for the simultaneous observation of a given set of plants a parameterization in terms of a stable inverse and a stable null space is proposed for all simultaneous observers to verify the effectiveness of the proposed method a design example is also given

simultaneous stabilization of linear single input single output plants the problem of designing a single compensator that simultaneously stabilizes several distinct plants is considered here using the fractional representation approach owing to saeks and murray 1982 1983 necessary and sufficient conditions for the existence of such a compensator are given and a design algorithm is proposed

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

on bounds for the existence of a bistable controller abstract by s r we shall denote the set of polynomials nonzero on the closed disk z z r the problem to be studied is given as follows avoidance problem given polynomials a and b nonnegative integers n m and positive numbers r 1 r 2 r 3 determine if there exist p s r 1 q s r 2 with degrees n and m respectively such that ap bq is in s r 3 it is shown that the avoidance problem for r 1 r 2 r 3 1 is equivalent to the well known bistable stabilization problem for a large class of control systems also a general class of optimization problems for solving the avoidance problem is introduced the class of optimization problems includes problems with sets more general than disks s r in addition an algorithm is proposed for solving these optimization problems and numerical experience is reported as an application bistable controllers for the plant p z z z 2 are shown to exist for values of as low as 0 0246 thus improving on previously reported results by blondel et al for this difficult case furthermore it is shown how in some cases it is possible to find optimal controllers by using a combination of numerical and algebraic techniques and a variant of the algorithm already introduced as an example a low order controller is found for p for the case when 7 4 3 0 0717 this method could be applied to other problems where the plant depends on a parameter

finite size corrections for inclined interfaces in two dimensions exact results for ising and solid on solid models analysis of finite size corrections for the surface tension and surface stiffness coefficients in two dimensional models with inclined interfaces is presented we obtain a universal leading contribution proportional to lnl l for the 2d system of sizel by explicit calculations for restricted and unrestricted solid on solid models and the square lattice ising model we demonstrate the gaussian nature of rough interfaces with fixed ends and derive the leading 1 l type corrections for appropriate surface quantities

on an open problem in bistable stabilization we propose an equivalent representation of an open problem in bistable stabilization using this new formulation we derive two simple necessary condition tests for the solution of the problem for certain special cases we also obtain the sufficient conditions

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

brief condition and algorithm for simultaneous stabilization of linear plants this paper is devoted to the problem of simultaneous stabilization of linear plants it is shown that four given plants p i s i 1 2 3 4 are simultaneously stabilizable if and only if there exist four stable polynomials h i s i 1 2 3 4 which simultaneously satisfy two polynomial equations if we can only find four stable h i s i 1 2 3 4 to satisfy one of the two polynomial equations then two controllers c 1 s and c 2 s having a common denominator or numerator can be obtained such that c 1 s simultaneously stabilizes p 1 s and p 3 s and c 2 s simultaneously stabilizes p 2 s and p 4 s which is called the simultaneous stabilization in groups based on such two controllers a design algorithm for the simultaneous stabilization of three plants is induced

brief paper reliable stabilization using a multi controller configuration given a plant p we consider the problem of designing a pair of controllers c 1 and c 2 such that their sum stabilizes p and in addition each of them also stabilizes p should the other one fail this is referred to as the reliable stabilization problem it is shown that every strongly stabilizable plant can be reliably stabilized moreover one of the two controllers can be specified arbitrarily subject only to the constraint that it should be stable the stabilization technique is extended to reliable regulation

multivariable nyquist criteria root loci and pole placement a geometric viewpoint classical control theory is concerned with the topics in our title in the context of single input single output systems there is now a large and growing literature on the extension of these ideas to the multiinput multioutput case this development has posed certain difficulties some due to the intrinsic nature of the problem and some we would argue due to an inadequate reflection on what the multivariable problem calls for in this paper we describe what seems to us to be the natural multivariable analogs of these concepts from classical control theory a rather satisfactory generalization of the nyquist criterion will be described and a clear analog of the asymptotic properties of the root locus will be obtained in the multiparameter case however an example is given which illustrates the quite surprising fact that the root locus map is not always continuous at infinite gains this calls for a new ingredient a compactification of the space of gains and perhaps the most interesting new feature in this circle of ideas comes in the area of pole placement this problem is difficult in the multivariable case but by establishing a correspondence with a classical set of problems in geometry we are able to understand its main aspects and to derive results on pole placement by output feedback over the real field

on the relationship between the model order reduction problem and the simultaneous stabilization problem this note points out the fact that under certain conditions the model order reduction problem is very closely tied to the simultaneous stabilization problem these conditions occur when the reduced order model is to be used to design a suboptimal controller for a high order plant under these conditions the controller must not only stabilize the low order model but must also stabilize the high order plant the result concerning the existence of a controller that will simultaneously stabilize two plants is used to conclude that the approximate inclusion of any unstable real modes of the high order plant in the low order model will guarantee the existence of such a controller

an algebraic theory for design of controllers for linear multivariable systems part i structure matrices and feedforward design in this two part paper a theory is presented in which several control problems can be solved the theory is applicable to a general class of linear multivariable systems where the measured output does not have to be equal to the controlled output or the state the system may be affected by nonmeasurable disturbances only controllers which stabilize the system and have proper transfer functions are allowed i e the controllers have to be physically realizable it is shown that the solutions to control problems of servo type e g problems of model matching decoupling and invertibility are special cases of the solution to a more general problem analogously the solutions to problems of regulator type e g disturbance decoupling output regulation and pole placement are also speclal cases of the solution to a more general problem it is shown that problems of servo type and regulator type can be solved independently of each other in part i generalized polynomials are introduced as a mathematical framework structue matrices which describe how well a system can be controlled are defined finally problems of servo type are solved part ii mainly deals with problems of regulator type

simultaneous stabilization of nonlinear systems we study the problem of simultaneous stabilization of a given set of nonlinear plants by one nonlinear not necessarily stable compensator we obtain a necessary and sufficient condition under which there is a single compensator which stabilizes a given set of n nonlinear plants this note emphasizes the importance of the following problem when is a nonlinear unstable plant stabilizable by an incrementally stable compensator

robust control of the characteristic values of systems with possible parameter variations an analytical method to control by multiparameter output feedback the characteristic values of a closed loop multi input multi output system inside or outside general aubregions of the complex plane is presented the method pertains to analogue as well as discrete systems the design method is robust in the sense that the characteristic values are designed to cluster inside the desired subregion even though any number of the specifying parameters are only known to within a given interval the interval may possibly be disjoint and possibly infinite completely unknown examples are provided which illustrate the broadness and effectiveness of the method

simultaneous observation of linear systems this paper considers the problem of designing an observer which can observe the states for each of a given set of plants the concept of the simultaneous observation is introduced the simultaneous observation is achieved using coprime factorization techniques necessary and sufficient conditions for the existence of simultaneous observation are derived the observer design is discussed and an example is given

robust stabilizability for linear systems with both parameter variation and unstructured uncertainty the authors consider the problem of finding a single compensator which stabilizes a plant having both parameter variation and high frequency unstructured uncertainty an uncertain system model is proposed to characterize plants containing those two different uncertainties closed loop stability criteria are then proposed for the model robust stabilizability for single input single output siso systems is studied a number of assumptions describing the set of admissable siso plants are imposed the satisfaction of these assumptions guarantees the existence of a proper stable compensator c s achieving robust stabilization the algorithm used for controller design is recursive in nature and allows the designer to select one compensator coefficient at a time also the minimum order of the desired stabilizer c s depends only on the largest relative degree of the nominal plants which often leads to a lower order compensator

algebraic theory of linear multivariable feedback systems this paper presents an algebraic theory for analysis and design of linear multivariable feedback systems the theory is developed in an algebraic setting sufficiently general to include as special cases continuous and discrete time systems both lumped and distributed designs are implemented by construction of a controller with two vector inputs and one vector output use of controllers of this type is shown to generate convenient stability results and convenient global parametrizations of all i o maps and all disturbance to output maps achievable for a given plant by a stabilizing compensator these parametrizations are then used to show that any such i o map and any such disturbance to output map may be simultaneously realized by choice of an appropriate controller in the special case of lumped systems it is shown that the design theory can be reduced to manipulations involving polynomial matrices only the resulting design procedure is thus shown to be more efficient computationally finally the problem of asymptotically tracking a class of input signals is considered in the general algebraic setting it is shown that the classical results on asymptotic tracking can be generalized to this setting additionally sufficient conditions for robustness of asymptotic tracking and robustness of stability are developed

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

synthesis of linear multivariable regulators this paper generalizes the results obtained by the authors in 1 and presents new results on the construction of proper controllers for linear multivariable regulators these results may lead to efficient computation methods that do not involve smith forms or the order relations used in 1 to construct proper controllers the methods are based on the recently developed theory of generalized polynomials 2 we discuss several numerical algorithms and how they can be applied to the regulator problem we also point out an important computational problem in need of an efficient numerical algorithm