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

stability of polynomials under coefficient perturbation let the real polynomial a 0 x n a 1 x n 1 a n 1 x a n be stable and let the real numbers b k c k geq 0 0 leq k leq n be given we present a simple determinant criterion for finding the largest t 0 geq 0 such that the polynomial alpha 0 x n alpha 1 x n 1 alpha n 1 x alpha n is stable for all alpha k in a k b k t 0 a k c k t 0 cup a k 0 leq k leq n several further observations allow us to reduce the computational cost considerably

stabilization of uncertain systems via linear control this note considers the problem of stabilizing a linear dynamical system whose state equation includes a time varying uncertain parameter vector q cdot given the dynamics dot x t a q t x t b q t u t and a bounding set q for the values q t the objective is to choose a control law u t p x t guaranteeing uniform asymptotic stability for all admissible variations of q cdot our results differ from previous work in one fundamental way that is we show that when working with linear controllers it is possible to dispense with all assumptions on b cdot which have been made by previous authors e g see 1 9 this elimination of hypotheses on b cdot is accomplished roughly as follows the system sigma underline underline delta a q b q is shown to be equivalent to another system sigma underline underline delta a q b as far as stabilization is concerned since b is a constant matrix independent of q the desired result is readily obtained

analysis and control of the jump modes behavior of 2 d singular systems part i structural stability this paper considers the problem of structural stability of 2 d singular systems firstly some properties of structural stability of 2 d general singular systems are presented sufficient and necessary conditions for the structural stability of the 2 d singular systems are given then by extending the lyapunov approach for the structural stability of 1 d continuous singular systems to the discrete case a generalized lyapunov equation approach to the analysis of the structural stability of 2 d singular roesser models 2 d srm is proposed the existence of a solution to the generalized lyapunov equation gives a sufficient condition for the structural stability of the 2 d srm

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

linear stable unity feedback system necessary and sufficient conditions for stability under nonlinear plant perturbations the authors consider a linear not necessarily time invariant stable unity feedback system where the plant and the compensator have normalized right coprime factorizations they study two cases of nonlinear plant perturbations additive and feedback with four subcases resulting from 1 allowing exogenous input to delta p or not 2 allowing the observation of the output of delta p or not the plant perturbation delta p is not required to be stable using the factorization approach the authors obtain necessary and sufficient conditions for all cases in terms of two pairs of nonlinear pseudostate maps simple physical considerations explain the form of these necessary and sufficient conditions finally the authors obtain the characterization of all perturbations delta p for which the perturbed system remain stable

method of robust stability analysis with highly structured uncertainties this paper presents a method of analyzing the stability of linear time invariant interconnected systems with uncertainties each subsystem is single input single output and its vector locus lies inside a polygon at each frequency it is shown that the convex closure of the image of the cartesian product of these polygons under the mapping phi det i fh agrees with the convex closure of the image of the vertices of these polygons under the mapping from this result the image can be estimated by a finite dumber of calculations a sufficient stability condition is obtained by applying this result to the multivariable nyquist stability criterion

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

mixed finite elements for second order elliptic problems in three variables two families of mixed finite elements one based on simplices and the other on cubes are introduced as alternatives to the usual raviart thomas nedelec spaces these spaces are analogues of those introduced by brezzi douglas and marini in two space variables error estimates inl2 andh s are derived

on the solution of an inverse scattering problem in seismic while drilling technology abstract this paper concerns the solution of an inverse scattering problem arising in seismic exploration the problem to be analyzed is the prediction of the acoustic wave speeds ahead of a drill bit device located at the center of a well bore and regarded as a seismic source from measurements of the displacements at recording positions located near the surface of the exploration well the forward model is stated in terms of displacements assuming cylindrical symmetry with the symmetry axis located at the center of the borehole absorbing boundary conditions are employed at the artificial boundaries the inverse problem is formulated as the minimization of a quadratic cost functional which is solved using a nonlinear iterative procedure known as quasilinearization that uses the frechet derivatives of the displacements with respect to the wave speed parameters to update the parameter values during the iteration approximations to the solution of the differential equations associated with the forward problem and the frechet derivatives which are needed to discretize the optimization procedure were obtained using an explicit finite element procedure results on the existence uniqueness and regularity of the differential problem and the frechet derivatives are derived as well as on their continuity with respect to the parameter convergence results for the iterative estimation procedure are also given numerical examples illustrating the application of the algorithm to estimate the wave speeds in model problems within the context of the seismic while drilling technique are presented

robust tracking relation between the numerators mode readability and inverse internal model in the robust multivariable asymptotic tracking problem with two output plants it is shown that if the numerator related to the controlled output is fixed while the other numerator and the denominator are perturbed then there is no solution to the problem if however the whole part of the plant related to the controlled output is stable and fixed while the other part is arbitrarily perturbed the problem has a solution and the compensator that solves the problem incorporates an inverse internal model of the signal to be tracked

the existence and design of functional observers for two dimensional systems abstract this paper is concerned with functional observers for two dimensional 2d systems described by the fornasini marchesini local state space second model a necessary and sufficient condition is given for the existence of asymptotic functional observers a more tractable necessary condition for the existence of asymptotic functional observers is also presented as it helps exclude those 2d systems for which there is no asymptotic functional observer a constructive design method is then proposed for the design of asymptotic functional observers using a rank condition on the given system matrices and a linear matrix inequality lmi technique two illustrative examples are provided to demonstrate the feasibility and effectiveness of the proposed method

a new fast algorithm for robust stability analysis of linear control systems with linearly dependent parametric uncertainties abstract in this paper a fast algorithm is proposed for the computation of stability margins in parameter space for linear control systems subject to structured uncertainties the case in which plant transfer function coefficients are affine in a set of physical uncertain parameters is considered the paper shows also how the proposed algorithm can be used to solve another interesting problem in robust control analysis i e the determination of the region of pole location of a closed loop control system including an entire family of uncertain plants

on the structure of 2 d observers this note gives necessary and sufficient conditions for asymptotic reconstructibility of the state of multivariable 2d systems the existence and the structure of the observers are related to the matrices appearing in a bezout identity a counterexample shows that the state estimation cannot be performed in general by a static feedback on the estimation error this note includes also a technique for designing dynamic state observers

perfect observers for singular 2 d fornasini marchesini models a concept of the perfect observer for singular two dimensional 2 d fornasini marchesini models is proposed sufficient conditions are established for the existence of the perfect observer a procedure for design of the perfect observer is also derived and illustrated by a numerical example

some results on the problems of decentralized reliable stabilization concerns the problems of decentralized reliable control in linear time invariant multivariable control systems by the stable factorization approach decentralized reliable control is defined as the problem in which the closed loop system remains stable in spite of failure in the interconnection between the subplants based on the decentralized parametric characterization developed by desoer and gundes in 1988 a necessary and sufficient condition for the decentralized version of the strong stabilization problem is first proposed next it is shown that the decentralized reliable stabilization problem is solvable if and only if the augmented plant is decentralized strongly stabilizable moreover it is concluded that the problem of centralized reliable multiple control can be viewed as the problem of decentralized reliable stabilization of an augmented plant

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 necessary and sufficient condition for stability of a perturbed system the problem of determining the stability of a feedback system in the presence of perturbation is considered a necessary and sufficient condition under which the perturbed system remains stable is obtained this condition is very general in particular it does not require the perturbed plant to have the same number of poles as the nominal plant

linear system stability under parameter uncertainties a sufficient condition for linear system stability under parameter variations is presented it yields parameter bounds which are not conservative and the test is easily implemented on a digital computer

necessary and sufficient condition for stability robustness under known additive perturbation a necessary and sufficient condition is derived for the stability robustness of a unity feedback closed loop system involving a strictly proper plant p0 and a stabilizing compensator c0 under the assumption that the plant p0 is perturbed to p0 p0 where p0 is a known strictly proper rational matrix

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

global parametrization of feedback systems with nonlinear plants for unity feedback systems with a plant p and controller c we show that given a nonlinear stable p the class of all controller stabilizing the feedback system is globally parametrized by a stable map q we also show how q is useful in studying the effects of modeling errors on the stability

strictly hurwitz property invariance of quartics under coefficient perturbation in this note it is shown how the maximum interval centered around each of the coefficients of a prescribed fourth degree strictly hurwitz polynomial may be determined so that the strictly hurwitz property remains invariant when each coefficient might be perturbed to any arbitrary value within its permissible interval

finite element methods for a composite model in elastodynamics the propagation of waves through a composite isotropic nonhomogeneous elastic system omega consisting of an elastic solid omega s with an imbedded fluid saturated porous medium omega p is analyzed the linear elastodynamic equation for omega s and biot s low frequency dynamic equations for omega p are chosen to describe the propagation with appropriate boundary conditions at the interface between the media and along the artificial boundaries of the model first the existence and uniqueness of the solution are analyzed and then the continuous and discrete time galerkin procedures for obtaining approximate solutions are defined standard finite element subspaces of h 1 omega are used to approximate the displacement vector components for the solid part of omega while mixed finite element subspaces of h operatorname div omega p are employed to approximate the vector displacement for omega p convergence of optimal order is derived for the methods under certain sm

global estimates for mixed methods for second order elliptic equations global error estimates in l2 92 l 9 and h s q 2 in r2 or a re derived for a mixed finite element method for the dirichlet problem for the elliptic operator lp div a grad p bp cp based on the raviart thomas nedelec space vh x wh c h div 92 x l2 g2 optimal order estimates are obtained for the approximation of p and the associated velocity field u a grad p bp in l2 g2 and h s 92 0 6 s k 1 and if q c r2 forp in lv2

a note on robust pole placement pole placement algorithm in single input single output siso systems is discussed with respect to the corresponding real stability radius of the resulting closed loop polynomial

brief paper jump modes analysis and observer design for discrete singular systems this paper is concerned with the reconstructability and detectability of the jump modes of discrete singular systems the notion of jump modes can be considered as the discrete counterpart of the impulsive modes of continuous singular systems which has been extensively studied the effect of jump modes on the existence of reduced order normal luenberger observers for discrete singular systems is investigated necessary and sufficient conditions for the existence of normal luenberger observers and the corresponding observer design procedure are obtained

stability analysis and stabilization of 2 d singular roesser models abstract this paper addresses the problem of controller design for 2 d singular roesser models sufficient stability conditions are formulated in terms of strict linear matrix inequalities lmis for 2 d singular systems described by the roesser models then a set of sufficient conditions for existence of desired a state feedback controller are obtained numerical examples are provided to demonstrate the effectiveness of the proposed method

algebraic theory of linear time invariant feedback systems with two input two output plant and compensator this paper present an algebraic theory for linear time invariant multi input multi output systems with two input two output plant and compensator using the factorization approach this system configuration considered by doyle and nett is most general in that any interconnection of two systems can be represented in terms of this scheme other system configurations encountered in compensator design problems are special cases of this system configuration the analysis and synthesis applies to continuous time as well as discrete time systems and to lumped as well as distributed systems since the algebraic setting is completely general the compensator parametrization has four degrees of freedom the paper is self contained and is tutorial in the sense that it develops the main results of nett without introducing the concept of containment

two families of mixed finite elements for second order elliptic problems two families of mixed finite elements one based on triangles and the other on rectangles are introduced as alternatives to the usual raviart thomas nedelec spaces error estimates inl 2 andh 5 are derived for these elements a hybrid version of the mixed method is also considered and some superconvergence phenomena are discussed

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

on implementable transfer matrices a straightforward proof is given of the sufficient condition for implementable transfer matrices a similar result is obtained also when the controlled output is not accessible for feedback

analysis and control of the jump modes behavior of 2 d singular systems part ii regular observer and compensator design this paper considers the problems of regular observer and compensator design for 2 d acceptable singular systems first the notions of jump mode reconstructability and jump mode reachability for 2 d acceptable singular systems are proposed necessary and sufficient conditions are presented for the jump mode reconstructability and reachability respectively then we show that there exists a reduced order regular observer for a 2 d acceptable singular system if the system is detectable and jump mode reconstructable while there exists a reduced order regular compensator for a 2 d acceptable singular system if the system is stabilizable and jump mode reachable

maximal polytopic stability domains in parameter space for uncertain systems abstract some results on the stability and pole location of families of uncertain systems with characteristic polynomial coefficients affected by structured perturbations are presented the case in which the coefficients are affine in a set of system physical parameters is considered assuming that the parameters are subject to polytopic perturbations of a given structure for these cases which often arise in system stability robustness analysis as for example in the robustification of controllers for multivariable closed loop systems a procedure is given to estimate maximal stability polytopic regions in the space of parameters this result is also used to derive simple necessary and sufficient conditions for the discrete time stability of convex combinations of two polynomials

admissible additive perturbations for the general feedback system with two input two output plant and compensator the class of all admissible additive perturbations of the plant which do not destabilize the closed loop is established

allowable coefficient perturbations with preserved stability of a hurwitz polynomial given a polynomial p s ansn an 11sn 1 a1s a0 which satisfies the routh hurwitz criterion a procedure is given for finding non conservative upper bounds on the allowable variations of the coefficients such that the perturbed polynomial maintains the stability property

robustness results in linear quadratic gaussian based multivariable control designs the robustness of control systems with respect to model uncertainty is considered using simple frequency domain criteria available and new results are derived under a common framework in which the minimum singular value of the return differences transfer matrix is the key quantity in particular robustness results associated with multivariable control systems designed on the basis of linear quadratic lq and the linear quadratic gaussian lqg design methodologies are presented

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

analysis of finite element methods and domain decomposition algorithms for a fluid solid interaction problem this paper is concerned with the finite element galerkin approximations for a fluid solid interaction model proposed in x feng p lee and y wei appl anal submitted both continuous time and discrete time approximations are formulated and analyzed optimal order a priori estimates for the errors in l infty h 1 and l infty l 2 are derived the main difficulty for the error estimates is caused by the interface conditions which describe the interaction between a fluid and a solid on their contact surface and it is overcome by using a boundary duality argument of douglas and dupont numer math 20 1973 pp 213 237 to handle the terms involving the interface conditions parallelizable domain decomposition algorithms are also proposed and analyzed for efficiently solving the finite element systems

bicoprime factorizations of the plant and their relation to right and left coprime factorizations in a general algebraic framework starting with a bicoprime factorization p n sub pr d sup 1 n sub pl a right coprime factorization n sub p d sub p sup 1 a left coprime factorization d sup 1 sub p n sub p and the generalized bezout identities associated with the pairs n sub p d sub p and d sub p n sub p are obtained the set of all h stabilizing compensators for p in the unity feedback configuration s p c are expressed in terms of n sub pr d n sub pt and the elements of the bezout identity the state space representation p c si a sup 1 b is included as an example

invariance of the strict hurwitz property for polynomials with perturbed coefficients given a strictly hurwitz polynomial f lambda lambda n a n 1 lambda n 1 a n 2 lambda n 2 a 1 lambda a 0 it is of interest to know how much the coefficients a i can be perturbed while simultaneously preserving the strict hurwitz property for systems with n leq 4 maximal intervals of the a i are given in a recent paper by guiver and bose 1 in this note a theorem of kharitonov is exploited to obtain a general result for polynomials of any degree

a pole placement design approach for systems with multiple operating conditions the author proposes design procedures based on state space pole placement techniques for systems with multiple operating conditions this is the so called simultaneous pole placement problem first the full state feedback problem is studied in which a nonlinear local pole placement solution is proposed the design condition is formulated in terms of the rank condition of a multimode controllability matrix then the output feedback problem is approached using a multimodel controller design which is an extension of the observer design to multimode systems the design is decomposed into separated global pole placement subproblems and a local pole placement subproblem for a system with some operating conditions having modes on the j omega axis but no modes at the origin in the open right half of the complex plane stabilizability and detectability conditions for the design of an asymptotically stabilizing control are established without any restriction on the number of inputs or outputs relations of this approach to other simultaneous control design approaches are pointed out

on the generic dimension of controllable subspaces it is proved that the generic dimension of the controllable subspace remains unchanged even if the free parameters in a linear system are allowed to vary with time more precisely it is proved that the generic rank of the controllability matrix equals the generic rank of a controllability matrix for a discrete time varying system as a by product a new proof of s hosoe s 1980 characterization of the generic dimension of the controllable subspace is obtained

sensitivities of stability constraints and their applications let the real polynomial a s a 0 a 1 s a n s n with the coefficients being known differentiable functions a k x be given and let the constraints g i x 0 determine the strictly hurwitz property of the polynomial a s a simple and efficient method to calculate the derivatives partial g i x partial x j is proposed then the application of the method to the problem of stability of polynomials under coefficient perturbation by gradient optimization is discussed also a theorem characterizing the stability region and the newly introduced regions of nondestabilizing perturbations is given

robustness of pole location in perturbed systems abstract in this paper we present some results on robustness of location of roots of polynomials in given regions of the complex plane for unknown but bounded perturbations on the polynomial coefficients a geometric approach in coefficient space is exploited to derive maximal deviations in a given class of admissible perturbations of characteristic polynomial coefficients of an uncertain linear system from their nominal values preserving system poles in a given region of the complex plane it is also shown that the solution of this problem can be used to give computationally feasible necessary and sufficient conditions such that all the roots of the members of a family of polynomials lie in a given open region of the complex plane this last result can be considered an extension of the result of the well known theorem of kharitonov it is also outlined how the proposed technique can be used to deal with families of polynomials with linearly correlated coefficient perturbations

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

local controllability reachability and reconstructability of the general singular model of 2d systems necessary and sufficient conditions for the local controllability reachability and reconstructability of the general singular model of 2d linear systems are established

four input four output feedback systems robust asymptotic behavior the author solves the asymptotic tracking disturbance rejection problem for feedback systems with two input two output plant and compensator a factorization method recently developed by c n nett 1986 and c a desoer and a n gundes 1987 is used

properties of min max controllers in uncertain dynamical systems we discuss properties of min max controllers previously obtained for uncertain dynamical systems in particular we present a new model for switching action and re examine asymptotic stability we demonstrate the attractiveness of switching surfaces and comment on the insensitivity of solutions in those surfaces to the uncertainty

complete characterization of the spherical spectral set the problem of computing the spectral set of any given spherical family of polynomials is considered an algorithm based on the decomposition of the spectral set in three disjoint subsets with relate points having the same dimension of the back image in the parameter space is devised moreover the structure of the polynomial function which characterizes the main subset is analyzed such an analysis leads to significant simplifications for boundary generation and bounds computation of the spectral set

a relationship between sensitivity and stability of multivariable feedback systems both comparison sensitivity and robust stability analysis attempt to extend concepts which are well understood in scalar feedback systems to multivariable control system design although the two bodies of work have developed about a decade apart the basic philosophies are similar both attempt to maintain a system property sensitivity reduction versus stability in the presence of plant modeling errors this paper demonstrates how the similarity in philosophy translates into a similarity in results and interpretations

robust stability with structured real parameter perturbations this paper considers the problem of robust stabilization of a linear time invariant system subject to variations of a real parameter vector for a given controller the radius of the largest stability hypersphere in this parameter space is calculated this radius is a measure of the stability margin of the closed loop system the results developed are applicable to all systems where the closed loop characteristic polynomial coefficients are linear functions of the parameters of interest in particular this always occurs for single input multioutput or single output multiinput systems where the transfer function coefficients are linear or affine functions of the parameters many problems with transfer function coefficients which are nonlinear functions of physical parameters can be cast into this mathematical framework by suitable weighting and redefinition of functions of physical parameters as new parameters the largest stability hyperellipsoid for the case of weighted perturbations and a stability polytope in parameter space are also determined based on these calculations a design procedure is proposed to robustify a given stabilizing controller this algorithm iteratively enlarges the stability hypersphere or hyperellipsoid in parameter space and can be used to design a controller io stabilize a plant subject to given ranges of parameter excursions these results are illustrated by an example

parametrization of all stabilizing imc controllers for unstable plants a complete parametrization of all stabilizing imc controllers for open loop unstable plants is presented within this general setting the class of all steady state offset free imc controllers for asymptotically constant inputs is given

an algorithm for the computation of the structured complex stability radius abstract perturbed state space models of the form x a bdc x are considered where a b c are given matrices a is stable and d is an unknown complex perturbation matrix a reliable and efficient algorithm for the computation of the norm of the minimal norm destabilizing matrix d is presented the performance of the algorithm and possible applications are illustrated

exact calculation of the multiloop stability margin a mapping theorem by l a zadeh and c a desoer 1963 serves as the basis for an algorithm that computes the stability margin k sub m of diagonally perturbed multivariable feedback systems without conservatism the stability margin determination not only verifies system stability but also quantifies how much further the plan uncertainties can be extended before instability occurs the essence of the computational approach is the realization that the true image of a given domain can be approximated with arbitrary accuracy by first subdividing the domain into the requisite number of subdomains and then forming the union of the convex hulls of their images the technique is illustrated in an example with a plant model having three uncertainties in general the method is applicable to either siso or mimo lti systems whose uncertainties can be modeled with noninteracting coefficients in the open loop transfer function representations

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

on sufficient conditions for the stability of interval matrices using gershgorin s diagonal dominance theorem new sufficient conditions for the stability of interval matrices are obtained the new conditions remove a restriction in heinen s 1983 analysis and are applicable to a wider class of problems than previously possible

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

decoupling linear multiinput multioutput plants by dynamic output feedback an algebraic theory this paper presents an algebraic theory for the design of a decoupling compensator for linear time invariant multiinput multioutput systems the design method uses a two input one output compensator which gives a convenient parametrization of all diagonal input output i o maps and all disturbance to output d o maps achievable by a stabilizing compensator for a given plant it is shown that this method has two degrees of freedom any achievable diagonal i o map and any achievable d o map can be realized simultaneously by a choice of an appropriate compensator the difference between all achievable diagonal and nondiagonal i o maps and the cost of decoupling is discussed for some particular algebraic settings

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

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

on ultimate boundedness control of uncertain systems in the absence of matching assumptions this paper describes an approach to ultimate boundedness control under hypotheses less stringent than heretofore required instead of imposing the so called matching conditions of previous authors a certain decomposition of the dynamics is performed the resulting decomposed system has two parts a matched portion and a mismatched portion a certain measure of mismatch m is defined and it is shown that effective control is possible as long as m does not exceed some critical mismatch threshold m it is seen that this threshold depends on the choice of feedback hence it becomes possible to investigate the tradeoffs between the size of the mismatch threshold and the sizes of the feedback gains which achieve the desired ultimate boundedness of solutions