almost invariant subspaces an approach to high gain feedback design part ii almost conditionally invariant subspaces in part i of this paper almost controlled invariant subspaces were studied in this part we will consider their duals the almost conditionally invariant subspaces these concepts give immediately by dualization of the almost disturbance decoupling control by state feedback the solution of the almost disturbance decoupled estimation problem finally we consider the problem of approximate disturbance decoupling by measurement feedback and it is shown that this problem is solvable to any arbitrary degree of accuracy if and only if 1 almost disturbance decoupling by state feedback and 2 almost disturbance decoupled estimation of the to be controlled output are both possible

regulator problem with internal stability a frequency domain solution this paper considers the general regulator problem with internal stability where the measured outputs are not necessarily the same as the regulated outputs using polynomial matrix techniques necessary and sufficient conditions are obtained in terms of skew primeness of two polynomial matrices one of these polynomial matrices represents the disturbance modes whereas the other is the polynomial system matrix representing the system zeros various special cases considered in the literature are also analyzed in terms of these necessary and sufficient conditions

equivalence between stability of partial realizations and feedback stabilization applications to reduced order stabilization abstract given a transfer function b z a z a classical feedback problem is to find a dynamic feedback d z c z such that the closed loop characteristic polynomial ca db z becomes stable antoulas has proved that this is equivalent to finding a stable partial realization of a certain sequence of real numbers the converse is also true every partial realization problem can be transformed into a feedback problem this paper gives a new formulation of the equivalence between feedback and realization of scalar systems as an application algebraic conditions for existence of reduced order stabilizing feedback are derived

input output stabilization in the general two sided model matching setup the problem of input output stabilization in a general two sided model matching setup is studied as a first step the problem is reduced to a pair of uncoupled bilateral diophantine equations over rh infinity then recent results on bilateral diophantine equations are exploited to obtain a numerically tractable solution given in terms of explicit state space formulae the resulting solvability conditions rely on two uncoupled sylvester equations accompanied by algebraic constraints this is in contrast to the corresponding one sided stabilization where no sylvester equations are required it is shown that imposing a mild simplifying assumption is instrumental in obtaining convenient parameterization of all stabilizing solutions which is affine in a single rh infinity parameter this demonstrates that if the aforementioned assumption is imposed the general two sided stabilization problem is similar to its one sided counterpart in the sense that the constraints imposed by a stability requirement can be resolved without increasing problem complexity less

design of multivariable feedback systems with stable plant this paper considers in a general algebraic framework the design of a unity feedback multivariable system with a stable plant the method is based on a simple parameterization of the four closed loop transfer functions in terms of p the plant transfer function and q h y 1 u 1 in particular the i o transfer function q h y 2 u 1 pq using the framework of rational transfer functions we show that the closed loop system will be exponentially stable if and only if q is exponentially stable furthermore if both p and q are strictly proper then the controller is also strictly proper the basic result is design theorem 2 an algorithm is given for obtaining strictly proper controllers such that the resulting i o map is decoupled all its poles can be chosen by the designer and the same holds for zeros except of course for the c zeros prescribed by the c zeros of the plant a discussion is included to temper these results by the constraints imposed by noise and plant saturation

new results on the algebraic theory of linear systems the solution of the cover problems abstract we give the complete solution of a large class of problems in linear system theory the so called cover problems these problems are formulated and solved both in the state space and in the input output frameworks the key concept which allows the effective parametrization of all solutions of the cover problems is that of the partial realizations of a sequence of matrices it is shown that the solutions of the state space cover problems can be expressed as state spaces of the partial realizations of appropriately defined sequences of matrices and the solution of the input output cover problems can be expressed as a simple function of the functions of the partial realizations of the sequences of mentioned above

optimal disturbance reduction in linear multivariable systems this paper presents a computational solution to an important optimization problem arising in optimal sensitivity theory the approach is to treat the multivariable problem exactly as the scalar problem in that stability constraints are handled via interpolation the resulting computations are easily implemented using existing methods

a new approach to synthesis problems in linear system theory we consider a rational nonlinear equation which is involved in the general synthesis design problem of linear multivariable systems the complete solution of this equation subject to causality stability and minimality constraints is provided formal power series and in particular the theory of partial realizations prove of central importance for the design problem

a polynomial characterization of mathcal a mathcal b invariant and reachability subspaces based on the state space model of p fuhrmann a link is laid between the geometric approach to linear system theory as developed by w m wonham and a s morse and the approach based on polynomial matrices in particular polynomial characterizations of a b invariant and reachability subspaces are given these characterizations are used to prove the equivalence of the disturbance decoupling problem and the exact model matching problem and also to connect the polynomial matrix and the geometric approach to the construction of observers finally constructive procedures and conditions are given for computing the supremal a b invariant subspace and reachability space and for checking the solvability of the exact model matching problem

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

a functional approach to lqg balancing the paper has as its theme a circle of problems related to lqg balancing with a special emphasis on the related problems of model reduction and robust stabilization the aim is to present a unified point of view to both previously described problems the unification is achieved by focusing on the study of three functions and the relationships between them and the various operators that are associated with these functions with an arbitrary transfer function g one can associate canonically two objects the conjugate inner function which is based on the nrcf and the r which is associated with the lqg controller of the function g the approach that is taken is functional emphasizing operators a balanced realization of a stable g arises as a matrix representation of the shift realization with respect to a basis made out of suitably normalized hankel singular vectors a similar result holds for lqg balanced realizations here the underlying hankel operator we study is h r where r u m v n and u v s

disturbance localization and pole shifting by dynamic compensation this paper concerns the problem of disturbance localization and pole shifting by dynamic compensation in multivariable linear systems with inaccessible state vector a necessary and sufficient condition for the solvability of the problem is given two synthesis procedures of the compensator are presented one is based on an observer and the other is based on a dual observer the observer designed by the proposed procedure acts as as unknown input observer for a linear function an improved procedure of designing a reduced order unknown input observer is also given the result may be applicable to reducing the compensator order

ddp and model following problem revisited abstract for linear multivariable systems with two input two output plant and compensator the paper solves the disturbance decoupling perfect tracking problems with stability results are obtained for the not necessarily robust as well as for the robust problem

regulator synthesis using c a b pairs the concept of a c a b pair of subspaces was introduced in an earlier paper by the author to study problems of compensator design here it is shown that the same concept plays a basic role in the theory of tracking and regulation from a few fundamental properties the main results of regulator theory are derived in a relatively simple and straightforward manner another new feature is that we are able to allow the presence of an unknown input thereby unifying the fields of tracking and regulation and disturbance decoupling all results are fully constructive and an example is given to illustrate the theory

disturbance decoupling by measurement feedback with stability or pole placement in this paper we solve the disturbance decoupling problem by measurement feedback and requiring stability or pole placement on the closed loop system the problem is attacked using the geometric approach through the concepts of a bmod mathcal b invariant and controllability subspaces and their duals a mid k invariant and complementary observability subspaces the solution of this problem has an interesting structure consisting of a feedback processor which decomposes into i a disturbance decoupling loop ii a disturbance input stabilization or pole placement loop and iii a controlled output stabilization or pole placement loop

causal factorization and linear feedback an algebraic framework for the investigation of linear dynamic output feedback is introduced pivotal in the present theory is the problem of causal factorization i e the problem of factoring two systems over each other through a causal factor the basic issues are resolved with the aid of the new concept of latency kernels

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

hierarchical error estimates for the energy functional in obstacle problems we present a hierarchical a posteriori error analysis for the minimum value of the energy functional in symmetric obstacle problems the main result is that the error in the energy minimum is up to oscillation terms equivalent to an appropriate hierarchical estimator the proof does not invoke any saturation assumption we even show that small oscillation implies a related saturation assumption in addition we prove efficiency and reliability of an a posteriori estimate of the discretization error and thereby cast some light on the theoretical understanding of previous hierarchical estimators finally we illustrate our theoretical results by numerical computations

a systematic approach to improve multiple lyapunov function stability and stabilization conditions for fuzzy systems this paper presents a systematic approach for decreasing conservativeness in stability analysis and control design for takagi sugeno ts systems this approach is based on the idea of multiple lyapunov functions together with simple techniques for introducing slack matrices unlike some previous approaches based on multiple lyapunov functions both the stability and the stabilization conditions are written as linear matrix inequality lmi problems the proposed approach reduces the number of inequalities and guarantees extra degrees of freedom to the lmi problems numeric examples illustrate the effectiveness of this method

on the class of all stabilizing regulators an alternative characterization is given of the class of stabilizing regulators which is convenient to achieve online or offline h sup 2 h sup infinity optimization for performance and robustness the characterization uses disturbance signal reconstruction and nulling filters the theory allows a reinterpretation of the necessary and sufficient conditions for regulation also connections between structurally stable regulation and the internal model principle are exposed

on recursiveness and related topics in linear systems a decomposition of matrix formal power series linear systems is developed the interconnections between formal power series realization pade approximation recursiveness berlekamp massey algorithm feedback interconnections matrix linear fractional representations matrix continued fractions and the matrix euclidean algorithm are thereby revealed

on the solvability of the general feedback control problem a new necessary and sufficient condition for the solvability of the general feedback control problem in linear systems is presented its advantage lies in the fact that it allows the complexity of the feedback controller to be incorporated in the theory of feedback design

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