on 2 d pole placement the problem of characteristic polynomial assignment by dynamic output feedback in 2 d systems is solved via 2 d polynomial methods a simple necessary and sufficient condition is given the class of all assignable characteristic polynomials is described in a parametric form stable and causal solutions are treated

parametrization of all strictly causal stabilizing controllers within the framework of the factorization approach we present parametrizations of all strictly causal stabilizing controllers

dynamic feedback over commutative rings abstract we call a commutative ring r a ca n ring if for each n dimensional reachable system f g over r the system augmented by rank n projective modules with f augmented by a zero map and g augmented by an identity map is coefficient assignable we show that if r is a dedekind domain then r is a ca n 1 ring in particular a principal ideal domain is a ca n 1 ring we also show that if r is a ring with the gcs property and the 2 generator property then r is a ca 2 n 2 ring

2 d exact model matching the exact model matching problem in single input single output 2 d systems is solved by applying polyaomial techniques all solutions are described in a parametric form natural solvability conditions for causal and or stable solutions are given and a simple design procedure is proposed it consists in solving a linear 2 d polynomial equation which directly yields a suitable compensator

separability assignment problem for q dimensional linear discrete time systems separability assignment problem for q dimensional system is formulated sufficient conditions for the existence of a solution to the problem are presented an algorithm for finding the feedback gain matrix is given and illustrated by an example of a 4 d system

amalgamated free products of inverse semigroups abstract we study amalgamated free products in the category of inverse semigroups our approach is combinatorial graphical techniques are used to relate the structures of the inverse semigroups in a pushout square and we then examine amalgamated free products we show that an amalgam of inverse semigroups strongly embeds in the amalgamated free product thus providing an alternative proof of the strong amalgamation property for inverse semigroups a result due to t e hall we provide sufficient conditions for an amalgamated free product to bee unitary and we give necessary and sufficient conditions for the amalgamated free product of a special amalgam to bee unitary

regulation of split linear systems over rings coefficient assignment and observers a theory of regulators is developed for finite free split linear systems over a commutative ring k this is achieved by developing a theory of coefficient assignment and observers it is shown that the problem of coefficient assignment can be solved for reachable systems by using dynamic state feedback for strongly observable systems it is shown that observers with arbitrary dynamics can be constructed internal stability of observer and state feedback configuration is considered explicitly some examples are given to illustrate these techniques

the mixed initial boundary value problem for reducible quasilinear hyperbolic systems with linearly degenerate characteristics in this paper we prove that the c0 boundedness of solution implies the global existence and uniqueness of c1 solution to the mixed initial boundary value problem for linearly degenerate reducible quasilinear hyperbolic systems with nonlinear boundary conditions and we show by an example that the c0 norm of solution may blow up in a finite time this gives the mechanism of the formation of singularities caused by the interaction of boundary conditions with nonlinear hyperbolic waves the same result is still valid for the quasilinear hyperbolic system of rich type

feedback stabilization of multivariable two dimensional linear systems a constructive algorithm is presented for solving a two dimensional 2 d polynomial matrix equation that plays an important role in the output feedback stabilization of multivariable 2 d linear systems it is shown that this algorithm can have some computational advantages over the previous known ones based on the grobner basis approach as an application of the algorithm the class of all stabilizing strictly causal 2 d compensators is parametrized for a given stabilizable causal not necessarily strictly causal 2 d plant

the mixed initial boundary value problem for diagonalizable quasilinear hyperbolic systems in the first quadrant abstract in this paper we investigate the mixed initial boundary value problem for diagonalizable quasilinear hyperbolic systems with nonlinear boundary conditions on a half unbounded domain t x t 0 x 0 under the assumptions that system is strictly hyperbolic and linearly degenerate we obtain the global existence and uniqueness of c 1 solutions with the bounded l 1 l norm of the initial data as well as their derivatives and appropriate boundary condition based on the existence results of global classical solutions we also prove that when t tends to infinity the solutions approach a combination of c 1 travelling wave solutions under the appropriate assumptions of initial and boundary data the results can be applied to the equation of time like extremal surface in minkowski space r 1 1 n

stabilization of linear systems with non commensurate delays a stabilization method for linear systems with non commensurate delays in both state and control variables is presented this method is obtained by combining the algebraic stabilization method with the finite spectrum assignment method the present method can stabilize systems that algebraic methods cannot

stabilization of polynomially parametrized families of linear systems the single input case abstract given a continuous time family of finite dimensional single input linear systems parametrized polynomially such that each of the systems in the family is controllable there exists a polynomially parametrized control law making each of the systems in the family stable

transfer matrices realization and control of continuous time linear time varying systems via polynomial fractional representations abstract an algebraic theory of transfer matrices fractional representations and control for linear continuous time time varying systems based on the realization theory of input output maps is given it is shown for the first time that the realization of such systems specified by an abstract input output map as a module homomorphism over noncommutative polynomial rings can be established using an abstract kalman input output map defined over a ring of skew polynomials with time varying coefficients it is shown that in fact transfer matrices can be defined as formal power

bass serre theory for groupoids and the structure of full regular semigroup amalgams t e hall proved in 1978 that if s1 s2 u is an amalgam of regular semigroups in whichs1 s2 uis a full regular subsemigroup ofs1ands2 i e s1 s2 anduhave the same set of idempotents then the amalgam is strongly embeddable in a regular semigroupsthat containss1 s2 anduas full regular subsemigroups in this case the inductive structure of the amalgamated free produces1 us2was studied by nambooripad and pastijn in 1989 using ordman s results from 1971 on amalgams of groupoids in the present paper we show how these results may be combined with techniques from bass serre theory to elucidate the structure of the maximal subgroups ofs1 us2 this is accomplished by first studying the appropriate analogue of the bass serre theory for groupoids and applying this to the study of the maximal subgroups ofs1 us2 the resulting graphs of groups are arbitrary bipartite graphs of groups this has several interesting consequences for example ifs1ands2are combinatorial then the maximal subgroups ofs1 us2are free groups finite inverse semigroups may be decomposed in non trivial ways as amalgams of inverse semigroups

a geometric approach to the theory of 2 d systems the authors develop a geometric approach to the theory of 2 d two dimensional systems defining in a suitable way the notion of a sub 1 2 b sub 1 2 invariant subspaces and of controlled invariant subspaces such subspaces are shown to have good computational and feedback properties which make them useful in application in particular sufficient conditions and constructive procedures are obtained for the solutions of disturbance decoupling and model matching problems moreover it is shown that certain structural properties of a 2 d system can be described by means of a set of indexes defined in geometric terms and that such structural indexes can be used to reformulate the sufficient condition for model matching

pole assignment by dynamic feedback it is shown that by dynamic feedback the closed loop transfer matrix of a linear system can be made equal to a proper rational matrix of the form v s t 1 s t s t k s here v s is the numerator polynomial matrix of the open loop transfer matrix t s is a polynomial matrix which can be chosen arbitrarily up to some degree constraints t 2 s is a polynomial matrix whose invariant factors can be chosen arbitrarily up to some degree inequalities t k s is a polynomial matrix which is the denominator matrix in a matrix fraction description of the feedback matrix

algebraic aspects of linear control system stability after formulating the problem of linear control system stability in very general terms the problem is resolved by 1 establishing a necessary and sufficient condition for stability 2 exploring in some detail the nature of this condition and 3 developing a parametric characterization of all controllers which stabilize a given plant the problem formulation is based on a ring theoretic setting for linear systems and is distinguished by the use of a novel control system configuration any interconnection of two distinct systems can be represented in terms of this configuration while preserving the distinction between the two systems furthermore no other configuration has this property several benefits and extensions are accrued as a result of using this configuration first a completely general yet physically meaningful problem formulation results consequently state space and operator theoretic approaches to linear control system stability are thoroughly reconciled this reconciliation embraces continuous time as well as discrete time systems lumped as well as distributed systems and 1 d as well as n d systems second the controller parametrization affords four degrees of freedom two more than any existing parametrization the additional degrees of freedom make explicit the design opportunity associated with controller implementation and thus determine several important control system characteristics this result is expected to provide a theoretical basis for the development of comprehensive control system design methodologies wherein design considerations related to controller implementation are addressed

on the coefficient assignment of two dimensional systems in this note we prove that in two dimensional 2 d systems the arbitrary coefficient assignment cannot be achieved by using causal output feedbacks the problem however can be transformed into the design of systems over rings in order to find conditions guaranteeing the existence of a solution some related problems are also discussed

free products of inverse semigroups a structure theorem is provided for the free product s inv t of inverse semigroups s and t each element of s inv t is uniquely expressible in the form r a a where a is a certain finite set of left reduced words and either a i i or a a a is a reduced word with aa e a the word a1 l am in s sgp t is called reduced if no letter is idempotent and left reduced if exactly a is idempotent the notation a stands for fl aa a e a under a product remarkably similar to scheiblich s product for free inverse semigroups the corresponding pairs a a form an inverse semigroup isomorphic with s inv t this description enables various properties of s inv t to be determined for example s inv t s u t is always completely semisimple and each of its subgroups is isomorphic with a finite subgroup of s or t if neither s nor t has a zero then s inv t is fundamental but in general fundamentality itself is not preserved

finite spectrum assignment and observer for multivariable systems with commensurate delays this paper is concerned with control of multivariable systems with commensurate delays the purpose of this paper is to enlarge the class of finite spectrum assignable systems to spectrally controllable multivariable systems with commensurate delays by overcoming an infinite number of eigenvalues which are contained in the system and make control very complex it is proved that if the system is spectrally controllable there is a delayed feedback matrix such that the closed loop system is spectrally controllable through a single input combining this result with previous ones about single input systems it is verified that spectral controllability is equivalent to finite spectrum assignability for multivariable systems with commensurate delays an observer for the system with delayed outputs is presented by using these results the multivariable system can be regulated as desired without preliminary knowledge of open loop eigenvalues and a stability test of transcendental functions whenever the system is spectrally controllable and spectrally observable

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

linear systems with commensurate time delays stability and stabilization independent of delay notions of exponential stability independent of delay and stabilizability independent of delay are developed for the class of delay differential systems of the retarded type with commensurate time delays various criteria for exponential stability independent of delay with a given order are specified in terms of matrices whose entries are functions of a single real parameter and polynomials in one variable whose coefficients are functions of a single real parameter sufficient conditions and a necessary condition based on local stabilizability are given for stabilizability independent of delay using state feedback with commensurate time delays constructive methods for determining a stabilizing feedback are also presented the last part of the paper deals with a standard type of observer and regulator with the requirement that the closed loop system be stable independent of delay

comments on 2 d exact model matching two examples are given to indicate that in 2 d coefficient assignment model matching some cancellations are unavoidable in the author s belief and the split condition not only the zero coprimeness is necessary

an asymptotic modal observer for linear autonomous time lag systems a full state vector observer is derived for a class of linear autonomous time lag systems represented by differential equations the major features of the approach are 1 an observer is constructed just for the set of unstable or poorly damped modes of the delayed system once they have been determined and 2 the design is accomplished by using a partial fraction expansion separating the unstable part from the system and then applying well known finite dimensional state vector techniques a procedure for the construction is given along with some numerical results to illustrate the simplicity of the observer

inverse monoids trees and context free languages this paper is concerned with a study of inverse monoids presented by a set x subject to relations of the form e i f i i i where e i and f i are dyck words i e idempotents of the free inverse monoid on x some general results of stephen are used to reduce the word problem for such a presentation to the membership problem for a certain subtree of the cayley graph of the free group on x in the finitely presented case the word problem is solved by using rabin s theorem on the second order monadic logic of the infinite binary tree some connections with the theory of rational subsets of the free group and the theory of context free languages are explored

an eigenvalue based approach for the stabilization of linear time delay systems of neutral type an eigenvalue based approach for the stabilization of linear neutral functional differential equations is presented which extends the recently developed continuous pole placement method for delay equations of retarded type the approach consists of two steps first the stability of the associated difference equation is determined and a procedure is applied to compute the supremum of the real parts of its characteristic roots which corresponds to computing the radius of the essential spectrum of the solution operator of the neutral equation no restrictions are made on the dimension of the system and the number of delays also the effect of small delay perturbations is explicitly taken into account as a result of this first step the stabilization problem of the neutral equation is reduced to a problem involving only a finite number of characteristic roots as a second step stabilization is achieved by shifting the rightmost or unstable characteristic roots to the left half plane in a quasi continuous way by applying small changes to the controller parameters and meanwhile monitoring other characteristic roots with a large real part a numerical example is presented

stabilization of linear distributed control systems with unbounded delay in this paper we study the asymptotic stabilization of linear distributed parameter control systems with unbounded delay assuming that the semigroup of operators associated with the uncontrolled and nondelayed equation is compact and that the phase space is a uniform fading memory space we characterize those systems that can be stabilized using a feedback control as consequence we conclude that every system of this type is stabilizable with an appropriated finite dimensional control

asymptotic tracking for 2 d and delay differential systems abstract a method for asymptotic tracking in 2 d systems is presented and is applied to discrete 2 d systems and delay differential systems the method consists of solving two linear equations in 2 d polynomials solvability conditions are derived and the questions of stability causality and properness are discussed

control of linear systems with fixed noncommensurate point delays the first solution is given to the fundamental open problem of stabilizability and detectability necessary sufficient conditions for internal stabilization by feedback of retarded and a large class of neutral delay differential systems with several fixed noncommensurate point delays using causal compensators observers and state feedback or dynamic output feedback which are also the same type of neutral or retarded delay differential systems with fixed point delays only our results are rank conditions on the system matrices zi f g and zif t h t evaluated at points in the complex plane and are the weakest possible generally applicable sufficient such rank conditions for stabilization of neutral systems in the light of what is known on the stability of such systems these conditions are necessary for most practical purposes the class of systems we consider includes all retarded delay differential systems with noncommensurate fixed point delays in the case of retarded systems these rank conditions are necessary and sufficient conditions for stabilization via compensators which are causal retarded delay differential systems with fixed point delays only these constitute the first full solution of these previously unsolved problems of stabilizability and detectability which together are necessary and sufficient conditions for internal stabilization by feedback for delay differential systems even in the retarded single fixed point delay case an application of our results to a problem of practical importance in control of linear systems with no delays provides a stabilization criterion interesting in itself

amalgamated free products of inverse semigroups abstract this paper is the first of two papers devoted to the study of amalgamated free products of inverse semigroups the subject of our second paper is the structure of amalgamated free products while this one is concerned with offering concrete descriptions of normal forms we define a lower bounded amalgam of inverse semigroups and present a procedure by which the schutzenberger automata of the amalgamated free product can be constructed those automata which can be such a schutzenberger automaton are characterized yielding a canonical form for the amalgamated free product our proof makes use of the graph theoretic ideas developed by jones margolis meakin and stephen for presentations of inverse semigroups

global existence and asymptotic behavior of classical solutions of quasilinear hyperbolic systems with linearly degenerate characteristic fields abstract in this paper we study the global existence and the asymptotic behavior of classical solution of the cauchy problem for quasilinear hyperbolic system with constant multiple and linearly degenerate characteristic fields we prove that the global c 1 solution exists uniquely if the bv norm of the initial data is sufficiently small based on the existence result on the global classical solution we show that when the time t tends to the infinity the solution approaches a combination of c 1 traveling wave solutions finally we give an application to the equation for time like extremal surfaces in the minkowski space time r 1 n

invertibility of linear systems described by differential difference equations abstract a complete algebraic theory of invertibility of linear systems described by differential difference equations with commensurable delays is developed introducing the notion of k integral l delay inverse considering the transfer function matrix the matrix fraction and rosenbrock s system matrix descriptions respectively in the forms q 1 s t s q 1 s p s and v s s i f g h j where q s is a polynomial t s q s and p s are p r p p and p r matrices over r s and f g h and j are n n n r p n and p r matrices over r of the system nascs for invertibility of the delay systems in the above sense are shown to depend on a new notion of weak monicity of a polynomial g s over r s constructed from the minors of t s p s or the system matrix v s while for a stable inverse with or without delay in addition to the property of stable weak monicity a rank condition at complex points of a row vector constructed from the same matrices is also to be satisfied this paper demonstrates that apart from the full rank property of the matrix t s p s or v s which constitutes the nasc for inversion of delay free systems the concept of weakly monic stable polynomials plays the major role in deciding the invertibility of delay systems these criteria of invertibility are then employed to establish nascs for regulability of systems with commensurable delays under feedback the paper is concluded with a detailed description of a constructive procedure establishing the pertinent existence theorems as to how the inherent integrations and the inherent delays of are determined and a stable inverse with minimum delays and minimum integrations is computed the novelty of this construction procedure is that the inverse system is generated via the theory of realization

elementary proof of controller parametrization without coprime factorizability recently a controller parametrization method without coprime factorizability was presented by mori in this note we present this method using elementary matrix operations unlike in the earlier paper good knowledge of commutative algebra and module theory is not required

a new algorithm for finite spectrum assignment of single input systems with time delay finite spectrum assignment for time delay systems is the elimination of delay operators from the characteristic function of the closed loop system and the arbitrary assignment of poles the control consists of polynomials in the delay operator and finite laplace transforms an algorithm for computing the control matrix is presented in particular the control matrix over rational functions of a delay operator is computed and expanded to partial fractions partial fractions are systematically transformed to finite laplace transforms

a geometric approach to dynamic feedback cyclization over commutative rings a different geometric point of view is introduced in the study of dynamic feedback cyclization over rings this considerably simplifies the theory in the free case in the sense that the main results can now be obtained by simple proofs the non free case remains to be examined

on necessary and sufficient conditions for regulation of linear systems over rings necessary and sufficient conditions are given for regulation of linear systems over rings using observers and causal dynamic state feedback systems with the polynomial fractional representation property the results are then used to obtain stabilizability conditions for systems over integers delay differential systems systems over polynomial rings and to obtain conditions to make a 2 d system nonrecursive

on the control of linear systems whose coefficients are functions of parameters the stabilization and regulation of linear discrete time systems whose coefficients depend on one or more parameters is studied for linear systems whose coefficients are continuous functions of real or complex parameters respectively analytic or rational functions of real parameters it is shown that reachability of the system for all values of the parameters implies that the system can be stabilized using gains that are also continuous analytic rational functions of the parameters closed form expressions for a collection of stabilizing gains are given in terms of the reachability gramian for systems which are stabilizable for all values of the parameters it is shown that continuous analytic rational stabilizing gains can be computed from a finite time solution to a riccati difference equation whose coefficients are functions of the parameters these results are then applied to the problem of tracking and disturbance rejection in the case when both the plant and the exogenous signals contain parameters

block form control of linear time invariant discrete time systems defined over commutative rings abstract a new approach based on a block input block output state model is developed for the study of linear time invariant discrete time systems whose coefficients belong to a commutative ring with 1 it is well known that such systems arise in the study of various classes of complex linear systems including systems depending on parameters and multidimentional systems by time compressing the block input state representation new results are obtained on the construction of a memoryless block form state feedback control law that yields a type of assignability and or deadbeat control these results are then dualized to yield results on a new type of state observer based on a block of output measurements the observer and state feedback controller are then combined to yield new results on input output regulation and set point tracking for systems defined over an arbitrary commutative ring in the last part of the paper the block input form is utilized to study the stabilization of systems defined over a normed algebra

robust observers for systems with parameters abstract in many cases linear multivariable models which describe physical systems contain structured parametric type uncertainties specifically it may be that the state space model is expressed in a form which involves parameters we first consider circumstances in which a parameter independent observer can be constructed for the system and suggest conditions that guarantee its existence we also give conditions under which such a robust observer can be subsequently used with additional dynamics in a parameter free configuration for regulation

a constructive algorithm for the complete set of compensators for two dimensional feedback system design in the recent factorization approach to linear feedback system design it has been found that in order to completely solve a generalized stabilization problem a coprime condition in the ring of stable transfer functions must be solved in this note two dimensional 2 d systems are considered and this stable coprime condition is solved for stable rings a well known 2 d zero coprime condition is a special case of this condition this is the first constructive method for obtaining the complete set of solutions or compensators for 2 d feedback system design some examples of systems treated as 2 d systems are temporal spatial spatial spatial delay differential multipass etc

regulation of linear systems over rings by dynamic output feedback abstract a general algebraic solution to the problem of regulation of linear systems over arbitrary commutative rings by dynamic output feedback is given which extends the theory of regulation for such systems furthermore the notion of row column regularity for polynomial matrices over a class of rings bezout domains is introduced this concept is shown to simplify the regulation problem considerably for these rings compared to the general case

stabilization of hereditary distributed parameter control systems abstract in this paper we study the asymptotic stabilization of semilinear distributed parameter control systems with delay assuming that the semigroup of operators associated with the uncontrolled and undelayed equation is compact we prove that a well known rank condition which implies the feedback stabilization for lumped systems can be extended to this type of systems

a note on dynamic output feedback for linear systems over rings it is shown that two previous regulation techniques one from systems over rings the other from systems over fields can be combined with a slight modification to obtain a dynamic output feedback compensator for regulating stabilizable and detectable systems over rings the advantage of the resulting technique is that it is a more direct and explicit output feedback technique than the available alternative technique of using observers and dynamic state feedback configuration and as such it brings more insight into regulation of linear systems over rings

amalgams of finite inverse semigroups abstract we show that the word problem is decidable for an amalgamated free product of finite inverse semigroups in the category of inverse semigroups this is in contrast to a recent result of m sapir that shows that the word problem for amalgamated free products of finite semigroups in the category of semigroups is in general undecidable

initial boundary value problem for the equation of timelike extremal surfaces in minkowski space this paper investigate the mixed initial boundary value problem for the equation of timelike extremal surfaces in minkowski space r1 1 n in the first quadrant under the assumptions that the initial data are bounded and the boundary data are small we prove the global existence and uniqueness of the c2 solutions of the initial boundary value problem for this kind of equation based on the existence results on global classical solutions we also show that as t tends to infinity the first order derivatives of the solutions approach c1 traveling wave under the appropriate conditions on the initial and boundary data geometrically this means the extremal surface approaches a generalized cylinder which is an exact solution

pole assignment of two dimensional systems for separable characteristic equations abstract this paper considers the problem of pole assignment of linear time invariant multivariable two dimensional systems the desirable closed loop characteristic function is considered to be separable an algorithm is presented for the solution of the problem using state feedback the necessary condition for the problem to have a solution is established and an illustrative example is given

output feedback stabilization of delay systems via generalization of the transformation method the left characteristic matrix equation l c m e of a delay system has been employed to develop a state feedback stabilization theory for delay systems analogously the right characteristic matrix equation r c m e has led to the development of an observer theory for this class of systems the characteristic matrix equation approach however only permits the relocation of integral multiples of n poles where n represents the system order this restriction is removed in this work through a generalization of the l c m e and r c m e in a manner which permits the stabilization of an arbitrary but finite number of unstable modes the resulting state feedback controller and state estimator permit the output feedback stabilization of the most general linear autonomous delay system under the minimal assumptions of spectral stabilizability and detectability

the dynamic feedback equivalence over principal ideal domains we show that over a principal ideal domain the dynamic feedback equivalence for not necessarily reachable linear systems is reduced to the feedback equivalence for one augmented systems we also obtain the dynamic feedback classification for two dimensional linear systems

a necessary and sufficient condition for feedback stabilization in a factorial ring the necessity for a coprime fractional representation of a plant in order to completely parametrize all stabilizing compensators by the youla theory is examined it is shown that the coprimeness is necessary for systems in a factorial ring a class inclusive of multidimensional scalar systems some observations on partial state transfer functions in the feedback configuration are found to be useful in showing the necessity

on matrix fraction representations for linear systems over commutative rings this paper deals with the problem of existence of polynomial matrix fraction representations for transfer matrices of linear systems over rings as well as the related realization theory these representations are then used in establishing new results for various classes of systems including split systems the relevance of these results to the regulation of various nonclassical classes of linear systems for example delay differential systems two dimensional systems etc is also discussed

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

robust controllers for multivariable model matching systems abstract the robustness properties of multivariable perfect model matching systems are presented in order to stabilize the nominal system a general parametrized 2 d controller is introduced a necessary and sufficient condition of robust stabilization is derived for the perfect model matching system under external disturbances and non linear time varying perturbations the results presented could be viewed as the extension of the work by astrom 1980

parameterization of stabilizing controllers with either right or left coprime factorization a parameterization method of stabilizing controllers that requires only one of right left coprime factorizations is presented as an illustrative model we will consider the multidimensional system with structural stability for which it is not yet known in general whether or not a doubly coprime factorization always exists a parameterization of stabilizing controllers for the model is presented