explicit characterization of decentralized coprime factors this paper is concerned with parametrization of all decentralized stabilizing controllers the auxiliary diagonal system which is defined by the diagonal elements of bezout factors plays an important role in the parametrization of decentralized controllers this paper gives an explicit characterization of the auxiliary system

a connection between state space and doubly coprime fractional representations explicit formulas for a doubly coprime fractional representation of the transfer function of a lumped lti system are given in terms of a stabilizable and detectable state space realization of the transfer function these formulas allow existing computational algorithms to be utilized for the purpose of computing doubly coprime factorizations several additional implications of this result are briefly discussed as are some immediate extensions and future research directions

frequency weighted controller reduction methods and loop transfer recovery abstract this paper shows that if one designs an lqg controller using the conventional technique of loop transfer recovery ltr then two frequency weighted controller reduction methods the enns frequency weighted balanced truncation enns 1984a ph d thesis stanford university ca 1984b proc 23rd cdc las vegas nv 127 132 and the bezout identity induced frequency weighted reduction method anderson and liu 1987 proc amer control conf mn will be equivalent under the condition that the plant transfer function is square nonsingular and minimum phase we also show that enns method is equivalent to the bezout identity induced frequency weighted reduction method if the controller itself is stable and a particular representation for the controller is assumed

matrix fraction construction of linear compensators a construction is given using polynomial matrix fractions of the compensator resulting from a combination of a linear feedback control law and estimator design

a revisit to inverse optimality of linear systems in this article we revisit the problem of inverse optimality for linear systems by applying certain explicit formulae for coprime matrix fraction descriptions cmfd of linear systems we propose a necessary and sufficient condition for a given stable state feedback law to be optimal for some quadratic performance index compared to existing results in the literature the proposed condition is simpler to check and interpret moreover it reduces the redundancy in the solutions of the associated algebraic riccati equation are as a direct application of the proposed results we consider the problem of inverse optimality of observer based state feedback to be specific for the case where the state is not fully known we consider the inverse optimality problem of an observer based state feedback for the closed loop system augmented by an observer more precisely it is shown that the observer based state feedback is inverse optimal for the closed loop system with some general forms of cost functions on

an lp newton method nonsmooth equations kkt systems and nonisolated solutions we define a new newton type method for the solution of constrained systems of equations and analyze in detail its properties under suitable conditions that do not include differentiability or local uniqueness of solutions the method converges locally quadratically to a solution of the system thus filling an important gap in the existing theory the new algorithm improves on known methods and when particularized to kkt systems derived from optimality conditions for constrained optimization or variational inequalities it has theoretical advantages even over methods specifically designed to solve such systems

doubly coprime factorization revisited in this note we derive a state space formula for all proper linear time invariant lti doubly coprime factorizations of a given lti plant as an application we present a new derivation of the state space formula for parameterization of all proper lti stabilizing controllers

annihilator structure of a principal ideal relation to optimal compensators abstract given a linear time invariant discrete time plant we consider the optimal control problem of minimizing by choice of a stabilizing compensator the seminorm of a selected closed loop map in a basic feedback system the seminorm can be selected to reflect any chosen performance feature and must satisfy only a mild condition concerning finite impulse responses we show that if the plant has no poles or zeros on the unit circle then the calculation of the minimum achievable seminorm is equivalent to the maximization of a linear objective over a convex set in a low dimensional euclidean space hence for a wide variety of optimal control problems one can compute the answer to an infinite dimensional optimization by a finite dimensional procedure this allows the use of effective numerical methods for computation

parametrization of ω stabilizing controllers and closed loop transfer matrices of a singular system abstract for a completely general singular system whose transfer matrix may be improper or even polynomial and an arbitrary set in the closed complex plane we give a state space parametrization of all controllers that achieve closed loop transfer matrices having all poles in and of the respective closed loop transfer matrices the main result is expressed in terms of a special type of state space realization called centered that exhibits the same attractive features and allows for formulas that bear essentially the same elegant simplicity of the standard proper case in particular the formulas can be specialized to get the parametrization of all controllers that achieve proper polynomial or proper and stable closed loop transfer matrices for a singular system

a chain scattering matrix description approach to h sup infinity control chain scattering matrix descriptions and j lossless coprime factorizations are employed to develop a relatively simple method for the synthesis of the continuous time h sup infinity suboptimal control analogously to the youla parameterization of all stabilizing controllers the authors derive an identity to generate all suboptimal controllers the derivation is carried out in terms of transfer function matrices although the final formulas are in state space the approach provides a clear connection at the transfer function level between coprime factorizations and algebraic riccati equations that are associated with the solutions of the problem

youla like parametrizations subject to qi subspace constraints consider that a linear time invariant lti plant is given and that we wish to design a stabilizing controller for it admissible controllers are lti and must belong to a pre selected subspace that may impose structural restrictions such as sparsity constraints the subspace is assumed to be quadratically invariant qi with respect to the plant which from prior results guarantees that there is a convex parametrization of all admissible stabilizing controllers provided that an initial admissible stable stabilizing controller is provided this paper addresses the previously unsolved problem of extending youla s classical parametrization so that it admits qi subspace constraints on the controller in contrast with prior parametrizations the one proposed here does not require initialization and it does not require the existence of a stable stabilizing controller the main idea is to cast the stabilizability constraint as an exact model matching problem with stability restrictions which can be tackled using existing methods furthermore we show that when it exists the solution of the exact model matching problem can be used to compute an admissible stabilizing controller applications of the proposed parametrization on the design of norm optimal controllers via convex methods are also explored an illustrative example is provided and a special case is discussed for which the exact model matching problem has a unique and easily computable solution

partial eigenstructure assignment and its application to large scale systems the notion of partial eigenstructure assignment pea via linear state feedback control in linear multivariable systems is introduced this notion is a natural extension of eigenstructure assignment and partial eigenvalue assignment some theoretical basis for pea is provided and a parametric expression for feedback gain matrices achieving pea is derived an effective numerical algorithm for pea tailored to large scale systems is presented as an extension of the algorithm a recursive algorithm for eigenstructure assignment is presented these algorithms possess the following desired properties 1 compared to existing methods the presented algorithms significantly reduce the required computation time via transforming high dimensional matrix computations into low dimensional matrix computations 2 they can be implemented in a parallel fashion the proposed algorithm for pea is applied to modal control of large flexible space structure systems

a parametrization of stabilizing controllers for multirate sampled data systems all linear multirate controllers for a given multirate sampled data system plant are parameterized in terms of a single parameter q z that is allowed to be any stable transfer matrix provided that certain causality conditions are met the result parallels a well known parameterization result for single rate sampled data plants the parameterization suggests architectures for multirate controllers

a polynomial matrix method for computing stable rational doubly coprime factorizations this paper proposes a new method for computing stable rational doubly coprime factorizations from a given transfer matrix in contrast to the well known method which requires a state space representation the proposed method makes full use of polynomial matrices and the whole operation is carried out directly in the frequency domain furthermore the paper clarifies the meaning of the obtained factorization as a controller by showing that this factorization coincides with the one by the state space method for a suitable choice of feedback and observer gains

a new approach for computing the state feedback gains of multivariable systems this note presents some new results in linear systems first the relationship between the polynomial matrix description and the state space representation of multivariable systems is clarified then we show that once such a relationship is determined the coprime matrix fraction description can be easily computed and we can further develop a closed form formula to solve the pole assignment problem of multivariable systems such formula can be thought of as an extension of the ackermann s formula for mimo systems thus this note potentially gives us a clearer insight into linear systems from the theoretical viewpoint

a numerical algorithm for inner outer factorizations of real rational matrices abstract in this paper reliable algorithms are developed to perform the inner outer factorization of a strictly wide real rational matrix the derivations are based on a right coprime factorization of a certain right inverse of the given transfer matrix where the denominator factor is inner the implementation of the algorithms can be carried out efficiently using real matrix operations

on the design of linear multivariable feedback systems via constrained nondifferentiable optimization in h sup infinity spaces the design discussed is of linear lumped time invariant multivariable feedback systems subject to various frequency and time domain performance specifications the approach is based on the use of stabilizing controller parametrizations which result in the formulation of feedback system design problems as convex nondifferentiable optimization problems these problems are solvable by recently developed nondifferentiable optimization algorithms for the constrained minimization of regular uniformly locally lipschitz continuous functions in r sup n

normalized coprime factorizations and balancing for unstable nonlinear systems in this paper we first study the normalized right and left coprime factorizations of a nonlinear system from a state space point of view in order to do so we introduce the notion of inner and co inner nonlinear systems secondly we deal with balancing for unstable nonlinear systems for linear systems there are several ways to approach this problem and we generalize the ideas of balancing the normalized coprime factorization and of lqg balancing to the nonlinear case lqg balancing can be seen as measuring the influence of a state component on a certain future and past energy function we extend this interpretation to the nonlinear case furthermore we use the normalized coprime factorization to give another way of balancing the unstable nonlinear system we give the relation between these two methods of balancing which is similar to the relation in the linear case

robust stabilization of linear multivariable systems relations to approximation the stabilization of a linear system modelled as g a where g is a known rational transfer function and a is a perturbation satisfying w a2 g1 or equivalently there does not exist g with fewer poles in c than g such that w1 g g w2 a simple characterization of all robustly stabilizing controllers is then derived and state space formulae for maximally robust controllers are given finally reduced order controllers are considered

loop gain optimization of unstable non minimum phase feedback systems the paper presents a design method for optimizing the loop gain of unstable non minimum phase feedback systems because obvious gain bandwidth limitations exist in such systems the proposed best loop gain optimization goal is maximization of the stability gain and phase margins which are inherently low in such feedback systems the design procedure is based on coprime factorization of the plant and controller and the bezout identity an optimization process varies iteratively the observer based controller parameters until maximization of the gain and phase margins is achieved the same design technique also allows one to solve the mixed sensitivity h standard control problem for linear time invariant feedback systems

on doubly coprime factorizations abstract this paper proposes a new doubly coprime factorization technique which forms a basis of algebraic control theory this enables us to generalize a previous result on doubly coprime factorization furthermore it is shown that all stabilizing compensators possess a simple model reference structure

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

dynamic stabilization of regular linear systems we consider a general class of infinite dimensional linear systems called regular linear systems for which convenient representations are known to exist both in time and in frequency domain for this class of systems we investigate the concepts of stabilizability and detectability in particular their invariance under feedback and their relationship to exponential stability we introduce two concepts of dynamic stabilization the first formulated as usual with the plant and the controller connected in feedback and the second with two feedback loops even for finite dimensional systems the second concept stabilization with an internal loop in the controller is more general we argue that the more general concept is the natural one and we derive sufficient conditions under which an observer based stabilizing controller with an internal loop can be constructed

a treatment of jw axis model matching transformation zeros in the optimal h sup 2 and h sup infinity control designs a model matching transformation mmt zero is defined as a rank deficiency condition which prevents an h sup 2 or h sup infinity optimal control problem from being transformed into an equivalent model matching problem by imposing saturation constraints and accounting for additive instrument noise in the sensor and actuator signals all mmt zeros can be eliminated

optimal control on the doubly infinite continuous time axis and coprime factorizations we study the problem of existence of weak right or left or strong coprime factorizations in h infinity over the right half plane of an analytic function defined in some subset of the right half plane we give necessary and sufficient conditions for the existence of such coprime factorizations in terms of an optimal control problem over the doubly infinite continuous time axis in particular we show that an equivalent condition for the existence of a strong coprime factorization is that both the control and the filter algebraic riccati equation of an arbitrary realization that need not be well posed have a solution in general unbounded and not even densely defined and that a coupling condition involving these two solutions is satisfied the proofs that we give are partly based on corresponding discrete time results which we have recently obtained

towards a link between the factorization approach and mimo robust adaptive control based on a doubly coprime factorization theorem nett et al 1984 an adaptive controller for discrete time mimo systems consisting of a not necessarily minimum phase modelled part and fraction type unmodelled dynamics which are not necessarily stable non linear and time varying is developed then by virtue of a stability result which is itself of interest it is shown that the adaptive closed loop system is bibo stable even in the presence of small but not necessarily stable unmodelled dynamics and arbitrarily bounded initial conditions consequently an initial but important step for applying the factorization approach to mimo robust adaptive control is completely established

the construction of special coprime factorizations in discrete time given a rational discrete time transfer matrix it is shown how to construct two special kinds of coprime factorizations for it normalized and all pass denominator this is done in two ways first by carrying the existing continuous time construction results to discrete time via a bilinear transformation this is shown to be messy and computationally burdensome second direct constructions are derived in discrete time the derivations point out interesting differences between discrete and continuous time for these constructions

distributed system transfer functions of exponential order o is a real number we construct a transfer function algebra of fractions viz f o for modelling possibly unstable distributed systems such that i fcirc in f o is holomorphic in re s o i e is o stable iff fcirc is o exponentially stable and ii we allow delay in the direct input output transmission of the system this algebra is a a restriction of the algebra b o developed by callier and desoer 1978 1980 a b an extension of the algebra of proper rational functions such that the exponential order properties of the latter transfer functions of lumped systems are maintained the algebra f o can be used for modelling and feedback system design it is shown that standard semigroup systems are better modelled by a transfer function in f o rather than b o

an algorithm for obtaining the redundancy equations of lti systems in the context of model based failure detection and diagnosis there is a residual generation which can be obtained through a generalized parity space concept this paper presents a new algorithm involving the generation of redundancy or parity equations for dynamic linear time invariant systems the proposed scheme involves the successive elimination of state sub vectors and their replacement by output values as well as the processing of an output done one at a time to avoid matrix inversion some examples point out the numerical efficiency of the proposed procedure

strictly doubly coprime factorizations and all stabilizing compensators related to reduced order observers abstract doubly coprime factorizations dcf play an important role in the design of compensators for lumped linear time invariant systems they are defined as stable proper rational matrices and they not only constitute a starting point for the fractional approach to all stabilizing compensators but they are also closely related to the frequency domain design of optimal state feedback and estimation and of the resulting observer based compensators when defining dcfs related to reduced order observers some rational matrices become improper this can be prevented by the introduction of artificial stable dynamics the identity elements which cancel in the system and in the compensator transfer matrices such pole zero cancellations however are contrary to the usual notion of coprimeness it is shown here that these additional dynamics which have no meaning in an observer based compensator scheme are superfluous and that the parametrization of all stabilizing compensators is feasible on the basis of possibly nonproper factorizations not containing identity elements for such factorizations the new definition of strictly doubly coprime factorizations sdcf is proposed

proper stable transfer matrix factorizations and internal system descriptions the exact relations between coprime proper stable factorizations of p s and coprime polynomial matrix factorizations are derived and they directly lead to relations with internal descriptions of the plant in differential operator or state space form it is shown that obtaining any right or left proper stable coprime factorization is equivalent to state feedback stabilization or to designing a full order full state observer respectively solving the diophantine equation is shown to be equivalent to designing a full or reduced order observer of a linear functional of the state and to designing a stable inverse system and this suggests new computational methods to solve the diophantine

low order stabilizing controllers a method for designing low order stabilizing controllers is described it is shown that for some plants the order of stabilizing controllers may be less than or equal to l respectively m where l respectively m is the number of plant outputs respectively inputs an explicit characterization of a set of low order stabilizing controllers is given and numerical examples are provided to illustrate the results of the note

design of observer based compensators in the frequency domain the discrete time case for nth order systems with m outputs the design of compensators using state observers of order n x with 0 x m is presented in the z domain the design is carried out in equivalence with the usual time domain approach and the relations between the time and the frequency domain results are presented the state feedback and the observer are characterized by polynomial matrices in the frequency domain and the corresponding compensator matrix fraction descriptions are obtained from the solution of a linear diophantine equation in view of the optimal state estimation problem the use of an updated state estimate is of interest it is shown how this problem can also be solved in the frequency domain

numerically stable algorithm for transfer function matrix evaluation a numerically stable algorithm for the evaluation of the transfer function matrix corresponding to a state space description is proposed the algorithm performs orthogonal similarity transformations on the system matrices eigenvalue computations are used for pole and zero determinations

rational matrix gcds and the design of squaring down compensators a state space theory a state space construction for rational matrix greatest common divisors gcds of rational transfer matrices is given it is shown how the gcd results can be used to solve the problem of designing stable minimum phase squaring down compensators for multi variable plants one application is a direct state space construction for such compensators and a state space solution to fat plant h infinity control problems the results make use of the concepts of strongly observable systems and maximally observable systems and build upon the concepts introduced in the state space gcd extraction results for polynomial matrices of l m silverman and p van dooren 1979

pole placement and the observer based controller parameterization two properties related to observer based controller parameterization and pole placement are discussed it is shown that the poles of the closed loop system with observer based controller parameterization can be classified into three groups and each group of poles can be independently determined these three groups of poles are the regulator poles the observer poles and the poles of the added dynamics if the realization of the controller is not minimal then the uncontrollable and or unobservable controller poles can be removed and the order of the controller is minimized the set of these removable controller poles is a subset of the regulator and the observer poles the poles of the closed loop system with the minimal order controller will include all the poles of a parameter matrix and some of the regulator and the observer poles which are not the removable controller poles

state space formulae for discrete time h optimization abstract state space formulae and proofs are given for ail the important steps in discrete time h 8 optimization the steps closely follow the algorithms for continuous time systems but there are some technically involved differences in the detail that make their derivation non trivial

the design of linear controllers with symbolic algebra abstract modern transfer function methods use quite a lot of abstract algebra in the design of control systems in computing the final results symbolic algebra turns out to be an important tool in this paper symbolic computation algorithms will be developed to enable computations directly in the theoretically useful transfer function rings a symbolic software package called hinf will be developed and used both for educational and theoretical purposes

robust stabilization of singularly perturbed systems abstract we discuss the problem of designing stabilizing controllers for singularly perturbed systems on the basis of simplified models in 1 it was shown that a constant gain output feedback controller designed on the basis of the simplified model need not stabilize the true system containing both fast and slow modes this phenomenon was then expanded to include the case where the simplified system is strictly proper in 2 the objectives of this note are threefold i to show that given any proper system and any stabilizing controller for it that is proper but not strictly proper there exists a singular perturbation of the system that is destabilized by that controller ii to show that any strictly proper controller for a singularly perturbed system designed on the basis of a reduced order model will stabilize the true system for sufficiently small values of the fast dynamics parameter and iii to provide a characterization in the same spirit as 3 4 of the set of all strictly proper controllers that stabilize a given proper plant by combining these results it is possible to generate the class of all robustly stabilizing controllers for a given singularly perturbed system

a connection between normalized coprime factorizations and linear quadratic regulator theory given a transfer matrix described by a minimal state space triple a method is given for computing state space realizations for the numerator and denominator of a normalized stable right coprime factorization for the transfer matrix the method involves the solution of an algebraic riccati equation it allows the use of existing computational state space algorithms in finding normalized stable right coprime factorizations and avoids explicit calculations of spectral factors

parameterizations of stabilizing compensators by using reduced order observers reduced order stabilizing compensators are parameterized in terms of doubly coprime factorizations having state space representations based on a minimal order observer and its dual observer a free parameter of the compensators is restricted within strictly proper stable rational functions the order of the compensators is lower than that of nett s by the number of outputs or inputs

balancing and model reduction for second order form linear systems model reduction of second order form linear systems is considered where a second order form reduced model is desired the focus is on reduction methods that employ or mimic moore s balance and truncate 1981 first we examine second order form model reduction by conversion to first order form and obtain a complete solution for this problem then new gramians and input output i o invariants for second order systems are motivated and defined based on these direct second order balancing methods are developed this leads naturally to direct second order form analogs for the well known first order form balance and truncate model reduction method explicit algorithms are given throughout the paper

on prediction type reduced order observers for discrete time systems we define the combination of the standard reduced order observer with one step ahead state predictor as the prediction type reduced order observer we first obtain a simple and significant relation that ensures the equivalence of the reduced and the full order observers in the steady state we show the fact that a prediction type reduced order does not always provide a reduced order controller we also discuss the implications of the equivalence of the observers on state space representation of the doubly coprime factorization and yjb parametrization we use the equivalence relation to clarify the feedback properties achieved by the loop transfer recovery technique using the prediction type reduced order observer at both the input and output of a square and minimum phase plant in addition we discuss the computational requirements for the observer based controllers

linear control theory with an h 0e optimality criterion this expository paper sets out the principal results in bf h infty control theory in the context of continuous time linear systems the focus is on the mathematical theory rather than computational methods

brief bezout identity related to reduced order observer based controllers for singular systems a design method of causal reduced order observer based controllers for singular systems is presented a new state space representation of the bezout identity for singular systems is developed a parameterization of all causal reduced order controllers is also given the general results given here can be extended to various cases including minimal and full order observer based controllers

an interpolation theory approach to h controller degree bounds abstract we derive upper bounds for the mcmillan degree of all h optimal controllers associated with design problems which may be embedded in a certain generalized regular configuration our analysis is confined to problems of the first kind which are characterized by the assumption that both p12 s and p21 s are square but not necessarily of the same size this paper which uses interpolation theory complements a previous paper which addresses the same problem through an approach based on approximation theory we demonstrate that the interpolation theory approach is more direct and circumvents a number of the technical difficulties in the previous method the final outcome is a much shorter proof as a by product we achieve a new result on the degree of an optimal solution of the matrix nevanlinna pick problem

controller reduction via stable factorization and balancing abstract a controller reduction procedure based on a representation of a controller as a matrix function defined using stable proper transfer functions and employing a balancing technique is studied in this paper for a certain right coprime factorization of an lqg designed controller k s n s d 1 s we approximate using a balancing technique the pair d s n s t by a low order pair d1 s n1 s t defining a factorization of the reduced order controller k 1 s n1 s d1 1 s we show that reducing the controller order in this way is motivated in a natural way which leads to the expectation of both good stability properties and good accuracy of approximation of closed loop behaviour this is also demonstrated in some examples

technical communique doubly coprime factorizations related to any stabilizing controllers in state space given a plant and an arbitrary stabilizing controller we present state space formulas for doubly coprime factorizations related to the given controller then we determine parametrizations of all stabilizing controllers centered on that controller

parameterization of reduced order compensator in the frequency domain after a brief survey of the relations between time and frequency domain representations of the observer based linear state feedback loop the parameterization of the compensator is discussed it is shown that the free parameters specified by the state feedback matrix k and the output error feedback matrix l in the time domain appear as coefficients in the polynomial matrices dtilde s and s characterizing the closed loop dynamics in the frequency domain these polynomial matrices can either be chosen arbitrarily pole placement problem or they may be derived as solutions from the corresponding optimal control or the optimal estimation problems respectively starting from the polynomial matrices dtilde s and s the doubly coprime fractional representations of the system and of the compensator transfer matrices can be computed directly without recurrence to the time domain results from these representations the right and left coprime compensator matrix fraction descriptions can easily be evalu

foundations of a bicoprime factorization theory bicoprime factorizations bcfs are a generalization of the well known coprime factorizations commonly used in control theory however they have received negligible attention from the academic community so far this technical note lays the foundations of a bcf theory the theory is built from the ground up starting with the basic characteristics of such factorizations before moving on to state space parameterizations of bcfs and internal stability some advantages of bcfs are outlined including the possibility of reduced dimension internal stability tests an uncertainty structure induced by bcfs is also examined and the associated robust stability analysis tests provided in multiple instances it is shown how coprime factor results have their roots in the more abstract and more general bcfs

quantum linear coherent controller synthesis a linear fractional representation approach abstract this paper is concerned with a linear fractional representation approach to the synthesis of linear coherent quantum controllers for a given linear quantum plant the plant and controller represent open quantum harmonic oscillators and are modelled by linear quantum stochastic differential equations the feedback interconnections between the plant and the controller are assumed to be established through quantum bosonic fields in this framework conditions for the stabilization of a given linear quantum plant via linear coherent quantum feedback are addressed using a stable factorization approach the class of all stabilizing quantum controllers is parameterized in the frequency domain coherent quantum weighted 2 and control problems for linear quantum systems are formulated in the frequency domain finally a projected gradient descent scheme is outlined for the coherent quantum weighted 2 control problem

the graph metric for unstable plants and robustness estimates for feedback stability in this paper a graph metric is defined that provides a measure of the distance between unstable multivariable plants the graph metric induces a graph topology on unstable plants which is the weakest possible topology in which feedback stability is robust using the graph metric it is possible to derive estimates for the robustness of feedback stability without assuming that the perturbed and unperturbed plants have the same number of rhp poles if the perturbed and unperturbed systems have the same rhp poles then it is possible to obtain necessary and sufficient conditions for robustness with respect to a given class of perturbations as an application of these results the design of stabilizing controllers for unstable singularly perturbed systems is studied finally the relationship of the graph metric to the gap metric introduced by zames and el sakkary is studied in detail in particular it is shown that the robustness results of zames and el sakkary do not enable one to conclude the causality of the perturbed system whereas the present results do

coprime factorization for regular linear systems mild sufficient conditions are given for the existence of a doubly coprime factorization of the transfer function of a regular linear system as well as formulae for such a factorization the results are illustrated by two examples of delay systems one of which has infinitely many unstable poles

fractional balanced reduction model reduction via fractional representation a method for the model reduction of finite dimensional linear time invariant fdlti plants is proposed which uses normalized fractional representations is proposed the method dubbed fractional balanced reduction applies balance and truncate to a special representation of the graph operator of the plant this operation yields the graph operator of a reduced order plant the method has such properties as existence of an a priori error bound in the graph metric and preservation of sector containment coupling fractional representations with principal component analysis gives a model reduction method that is capable of producing in a numerically stable way a good reduced order model using the whole full order model sector properties are also preserved these are useful for deciding stability when nonlinearities are in the loop

brief paper algebraic and topological aspects of the regulator problem for lumped linear systems the regulator problem with robustness is solved for systems modelled by rational transfer matrices a topology for possibly unstable plants is presented a necessary and sufficient condition is derived for the existence of a proper controller which provides internal stability and output regulation throughout an open neighbourhood of the plant a characterization of all such controllers is determined

a chain scattering matrix approach to the h output feedback control for state delayed systems a state space coprime factorization approach to the h controllers design for state delayed systems is developed the coprime factors that are generated by the present factorization possess the same delayed state as that of the original system this property together with the well known youla parameterization and the chain scattering matrix description provides a relatively simple method to obtain a family of the internally stabilizing h output feedback controllers a numerical example is given to demonstrate the validity and the applicability of the proposed approach

projection free approximate balanced truncation of large unstable systems in this article we show that the projection free snapshot based balanced truncation method can be applied directly to unstable systems we prove that even for unstable systems the unmodified balanced proper orthogonal decomposition algorithm theoretically yields a converged transformation that balances the gramians including the unstable subspace we then apply the method to a spatially developing unstable system and show that it results in reduced order models of similar quality to the ones obtained with existing methods due to the unbounded growth of unstable modes a practical restriction on the final impulse response simulation time appears which can be adjusted depending on the desired order of the reduced order model recommendations are given to further reduce the cost of the method if the system is large and to improve the performance of the method if it does not yield acceptable results in its unmodified form finally the method is applied to the linearized flow around a cylinder at re 100 to show that it actually is able to accurately reproduce impulse responses for more realistic unstable large scale systems in practice the well established approximate balanced truncation numerical framework therefore can be safely applied to unstable systems without any modifications additionally balanced reduced order models can readily be obtained even for large systems where the computational cost of existing methods is prohibitive

a new approach for calculating doubly coprime matrix fraction descriptions using the concepts of state feedback in generalized state space explicit formulas for calculating doubly coprime matrix fraction descriptions mfds of the transfer matrix of a regular state space system and the polynomial generalized bezout identity elements corresponding to those coprime mfds are presented they can be easily calculated with the help of existing computational algorithms and software packages

an analysis of the pole zero cancellations in h optimal control problems of the first kind the aim of this paper is to study the pole zero cancellations which occur in a class of h infty optimal control problems which may be embedded in the configuration of fig 1 h infty control problems are said to be of the first kind if both p 12 s and p 21 s are square but not necessarily of the same size it is primarily this class of problems which will concern us here a general bound on the mcmillan degree of all controllers which are stabilizing and lead to a closed loop which satisfies bf mathcal r s infty leqq rho rho need not be optimal in the l infty norm sense is derived as illustrated in fig 1 bf mathcal r s is the transfer function relating y 1 s to u 1 s if the mcmillan degree of p s in fig 1 is n we show that in the single loop siso case the corresponding unique h infty optimal controller never requires more than n 1 states in the multivariable case there is a continuum of optimal controllers whose mcmil

on the computation of proper feedforward internally stabilizing and pole placing compensators by combining two important results from the polynomial and fractional representation approaches we derive a state space formula for the direct computation of a proper feedforward internally stabilizing and pole placing compensator c s

stabilizing solutions of the h algebraic riccati equation the algebraic riccati equation studied in this paper is related to the suboptimal state feedback h control problem it is parametrized by the h norm bound we want to achieve the objective of this paper is to study the behavior of the solution to the riccati equation as a function of it turns out that a stabilizing solution exists for all but finitely many values of larger than some a priori determined bound on the other hand for values smaller than there does not exist a stabilizing solution the finite number of exception points can be characterized as switching points where eigenvalues of the stabilizing symmetric solution can switch from negative to positive with increasing after the final switching point the solution will be positive semidefinite we obtain the following interpretation the riccati equation has a stabilizing solution with k negative eigenvalues if and only if there exists a static feedback such that the closed loop transfer matrix has k unstable poles and an l norm strictly less than

a frequency domain approach to control of singular systems in this paper singular control systems are studied from the frequency domain point of view the proper stable factorization of a singular system is derived the stability of the closed loop system consisting of a singular system and controller is addressed most significantly all proper stabilizing controllers are parameterized for a given singular system

h optimal control part 2 solution for controllers abstract the problem is considered of bounding the h norm of the transfer function t y1u1 relating the output y 1 to the input u 1 for the feedback configuration shown in fig 1 subject to the closed loop system being internally stable a new criterion for internal stability is derived in terms of the q parameterization where q k i p22 k 1 by means of such a stability theory it is shown that the h control problem can be decomposed into two one sided model matching problems of the form considered in part 1 hung 1989 of the paper a closed form state space solution with a tight mcmillan degree bound is obtained for h optimal or suboptimal controllers an iterative algorithm for obtaining the h optimal controller is provided computationally the algorithm requires the solution of only two algebraic riccati equations in each iteration for which the upper bound on t y1u1 is adjusted

on improving control loop robustness of model matching controllers this note first characterizes the class of all stabilizing controllers for a two degree of freedom control system which achieve a prescribed achievable transfer function the char acterization is in terms of an arbitrary proper stable transfer function with this characterization robust model matching is formulated as a standard hm optimization problem this means that standard controller designs for a nominal plant such as linear quadratic gaussian ones can be enhanced to give improved robustness properties using hm design tech niques

bounded real and positive real balanced truncation using σ normalised coprime factors in this article we will extend the method of balanced truncation using normalised right coprime factors of the system transfer matrix by meyer 1990 3 to balanced truncation with preservation of half line dissipativity special cases are preservation of positive realness and bounded realness we consider a half line dissipative input output system with quadratic supply rate given by a nonsingular symmetric matrix sigma with the property that its positive signature is equal to the number of input components of the system the transfer matrix of such system allows a sigma normalised right coprime factorisation we associate with such factorisation two lyapunov equations one of which is nonstandard involving the indefinite matrix sigma balancing will be based on making the unique solutions of these two lyapunov equations equal and diagonal the diagonal elements will be called the hankel sigma singular values because their squares are the nonzero eigenvalues of the composition of the graph hankel operator multiplication by sigma and the adjoint graph hankel operator this method of balanced truncation will be shown to preserve stability minimality and half line dissipativeness we will characterize the classical positive real and bounded real characteristic values in terms of the new hankel sigma singular values finally we will derive one step error bounds for the special case of balanced truncation of bounded real systems c 2009 elsevier b v all rights reserved

quantitative domains via fuzzy sets part i continuity of fuzzy directed complete posets this paper deals with quantitative domain theory via fuzzy sets it examines the continuity of fuzzy directed complete posets dcpos for short based on complete residuated lattices first we show that a fuzzy partial order in the sense of fan and zhang and an l order in the sense of belohlavek are equivalent to each other then we redefine the concepts of fuzzy directed subsets and continuous fuzzy dcpos we also define and study fuzzy galois connections on fuzzy posets we investigate some properties of continuous fuzzy dcpos we show that a fuzzy dcpo is continuous if and only if the fuzzy double downward arrow operator has a right adjoint we define fuzzy auxiliary relations on fuzzy posets and approximating fuzzy auxiliary relations on fuzzy dcpos we show that a fuzzy dcpo is continuous if and only if the fuzzy way below relation is the smallest approximating fuzzy auxiliary relation

convergent ritz approximations of the set of stabilizing controllers abstract a necessary and sufficient condition for linear exponential sums to be dense in l p is derived under this condition a ritz approximation converges which has been proposed to solve linear control problems state space representations of orthonormal basis functions for multivariable systems are given

technical communique wiener hopf approach to derivation of rational doubly coprime factorizations this paper presents a method to construct a stable rational doubly coprime factorization dcf of a transfer matrix proper or improper from a given polynomial dcf wiener hopf factor polynomials are used to rationalize the polynomial dcf matrices it is shown that the suggested construction procedure results in the stable rational dcf description that includes the normalized matrix fractional description mfd and inner coprime mfd as special cases

output feedback control of linear two time scale systems output feedback control of linear time invariant singularly perturbed systems is studied the set of all compensators that stabilize a singularly perturbed system while preserving its two time scale structure is parameterized the parameterization is used to show that any two frequency scale stabilizing compensator can be asymptotically approximated by a compensator designed via a sequential procedure in this procedure a fast high frequency compensator is designed first to stabilize the fast model of the system then a strictly proper slow low frequency compensator is designed to stabilize a modified slow model the parallel connection of the two compensators forms a two frequency scale stabilizing compensator for the singularly perturbed system

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

discrete time lqg ltr technique for systems with time delays loop transfer recovery ltr is applied for lqg controllers with prediction estimators for a class of square minimum phase discrete time linear systems with time delays we consider the class of plants described by the triple a b c whose time delay structure is characterized by the following hypothesis on the plant impulse response cb cab cad 2b 0 det cad 1b 0 when the estimator gain is fixed by recovery technique it is shown that the loop transfer matrix at the plant input is equal to the transfer matrix obtained with the optimal digital regulator having d sample controller delays described in 6 the dual result corresponding to the situation where the feedback gain is fixed by recovery technique is also analysed

a state space parametrization of discrete time two degree of freedom integral controllers abstract using the observer based integral controller proposed by ishihara h j guo and h takeda 1992 automatica 28 599 603 as a basic controller we propose a state space parametrization of discrete time two degree of freedom integral controllers accounting computation delay the proposed parametrization can easily be obtained by using the solution of an observer based regulator problem for a plant clear interpretation of the parametrization as an observer based integral controller is given as application we discuss the loop transfer recovery and the dead beat control using the free parameters

a state space approach to parameterization of stabilizing controllers for nonlinear systems a state space approach to the youla parameterization of stabilizing controllers for linear and nonlinear systems is suggested the stabilizing controllers or a class of stabilizing controllers for nonlinear systems are characterized as fractional transformations of stable parameters the main idea behind this approach is to decompose the output feedback stabilization problem into state feedback and state estimation problems the parameterized output feedback controllers have separation structures this machinery allows the parameterization of stabilizing controllers to be conducted directly in state space without using coprime factorization

doubly coprime matrix fraction representations using proportional and derivative feedback concepts in generalized state space systems the concepts of proportional and derivative state feedback are used to derive some explicit formulas for doubly coprime matrix fraction representations of generalized state space systems specifically proportional and derivative state feedback is used to determine the doubly coprime matrix fraction representations and to solve the corresponding generalized bezout identity in polynomial matrix form

a nonminimal representation of reduced order observers abstract a new nonminimal representation of reduced order observers of order n k in the time and in the frequency domain is introduced this reduced order observer is based on a full order plant model where the order reduction to n k results from an infinite gain feedback of k outputs this new representation is crucial for establishing the links between time and frequency domain design methods incorporating reduced order observers the main result are the doubly coprime fractional representations of a transfer matrix related to an observer of order n k

survey paper from youla kucera to identification adaptive and nonlinear control some 20 years ago formulae were presented for the set of all linear time invariant controllers stabilizing a linear time invariant plant this paper traces the development of many ideas from these formulae covering linear h 2 and h control identification adaptive control and nonlinear systems

stabilizability and stable proper factorizations for linear time varying systems the objective of this paper is to study in detail stabilizability and stable proper factorizations for linear time varying discrete time systems our main results are i if a linear time varying system can be stabilized by dynamic state feedback then it can also be stabilized by memoryless state feedback ii a complete characterization of the existence of stable proper factorizations for linear time varying input output operators this characterization is nontrivial there exist input output operators that do not admit stable proper factorizations

nyquist type stability test for undamped elastic structures a transfer function with possibly infinitely many imaginary axis poles is used to represent a linear multiresonant elastic structure with negligible damping a stable coprime factorization of this transfer function is presented this factorization is the basis for a graphical nyquist type feedback stability test that does not require counting encirclements of the critical point the test can be used to assess the input output stability of a damped or undamped structure with stable feedback compensators using displacement and velocity sensors the test is applied to a continuous slewing beam example and it is shown that contrary to the finite dimensional case this feedback system is unstable whenever either the velocity feedback compensator transfer function is strictly proper or a certain feedback delay which can be arbitrarily small is present

2dof controller parametrization for systems with a single i o delay this note puts forward a parametrization of all stabilizing two degrees of freedom controllers for possibly unstable processes with dead time the proposed parametrization is based on a doubly coprime factorization of the plant and takes the form of a generalized smith predictor dead time compensator feedback part and a finite dimensional feedforward part prefilter some alternative dead time compensation schemes and disturbance attenuation limitations are also discussed