brief multiplier based robust h design with time varying uncertainties a novel multiplier based approach for robust controller design with real parametric time varying uncertainty is proposed sufficient conditions for robustness analysis are derived in terms of noncausal dynamic multipliers that depend on the the rate of uncertainty variation an affine parametrization of such stability multipliers is introduced and the search for suitable multipliers is reduced to a convex minimization formulated as a linear matrix inequalities lmi problem stable dynamic scalings derived from this multiplier are employed to formulate an equivalent scaled h norm condition for robust stability based on which a m k m synthesis procedure is proposed to design robust controllers for the time varying parametric uncertainty considered

nonlinear maps preserving the reduced minimum modulus of operators abstract let x and y be infinite dimensional complex banach spaces and let b x resp b y denote the algebra of all bounded linear operators on x resp on y we describe bijective bicontinuous maps from b x to b y satisfying s t s t for all s t b x where t is the reduced minimum modulus of an operator t an analogue result for the finite dimensional case is obtained

maps preserving the local spectrum of jordan product of matrices abstract let m n c be the algebra of all n n complex matrices and fix a nonzero vector x 0 c n we show that a map from m n c into itself satisfies t s s t x 0 t s s t x 0 t s m n c if and only if there exists an invertible matrix a m n c such that a x 0 x 0 and t a t a 1 for all t m n c

spectrum nonincreasing maps on matrices abstract maps which do not increase the spectrum on complex matrices in a sense that sp a b sp a b are classified

existence of periodic solutions in a linear higher order system of difference equations in this paper we investigate the existence of nontrivial periodic solutions of a higher order system of difference equations our framework is based on an earlier periodicity condition of us we also use the theory of circulant matrices combined by a theorem of silvester on the computation the determinant of block matrices an illustrative application is given to show the effectiveness of our framework and to point out the connection between our periodicity results and some known stability conditions

nonlinear preservers of pseudospectral radius on banach spaces abstract given a complex banach space x we denote by l x the algebra of all linear bounded operators on it for e 0 let r e t denote the e pseudospectral radius of t l x for an infinite dimensional space x we characterize surjective maps l x l x such that r e t 1 t 2 r e t 1 t 2 for all t 1 t 2 l x an analogous result is obtained in the finite dimensional case with no surjectivity assumption on the map

nonlinear maps preserving the minimum and surjectivity moduli abstract let x and y be infinite dimensional complex banach spaces and let b x resp b y denote the algebra of all bounded linear operators on x resp on y we describe maps from b x onto b y satisfying c s t c s t for all s t b x where c stands either for the minimum modulus or the surjectivity modulus or the maximum modulus we also obtain analog results for the finite dimensional case

linear preserver problems a brief introduction and some special techniques abstract linear preserver problems concern the characterization of linear operators on matrix spaces that leave certain functions subsets relations etc invariant the earliest papers on linear preserver problems date back to 1897 and a great deal of effort has been devoted to the study of this type of question since then we present a brief picture of the subject aiming at giving a gentle introduction to the reader then we describe some techniques used in our recent papers on this type of problem

robustness in the presence of mixed parametric uncertainty and unmodeled dynamics continuing the development of the structured singular value approach to robust control design the authors investigate the problem of computing mu in the case of mixed real parametric and complex uncertainty the problem is shown to be equivalent to a smooth constrained finite dimensional optimization problem in view of the fact that the functional to be maximized may have several local extrema an upper bound on mu whose computation is numerically tractable is established this leads to a sufficient condition of robust stability and performance a historical perspective on the development of the mu theory is included

a multiloop generalization of the circle criterion for stability margin analysis in order to provide a theoretical tool well suited for use in characterizing the stability margins e g gain and phase margins of multiloop feedback systems multiloop input output stability results generalizing the circle stability criterion are considered generalized conic sectors with centers and radii determined by linear dynamical operators are employed to enable an engineer to specify the stability margins which he desires as a frequency dependent convex set of modeling errors including nonlinearities gain variations and phase variations which the system must be able to tolerate in each feedback loop without instability the resulting stability criterion gives sufficient conditions for closed loop stability in the presence of such frequency dependent modeling errors even when the modeling errors occur simultaneously in all loops so for example stability is assured as loop gains and phases vary throughout a set of nonzero measure whose boundaries are frequency dependent the stability conditions yield an easily interpreted scalar measure of the amount by which a muitiloop system exceeds or falls short of its stability margin specifications

maps preserving peripheral spectrum of jordan semi triple products of operators abstract let a and b be not necessarily unital or closed standard operator algebras on complex banach spaces x and y respectively for a bounded linear operator a on x the peripheral spectrum a of a is the set a z a z max a where a denotes the spectrum of a assume that a b is a map the range of which contains all operators of rank at most two it is shown that the map satisfies the condition that bab b a b for all a b a if and only if there exists a scalar c with 3 1 and either there exists an invertible operator t b x y such that a tat 1 for every a a or there exists an invertible operator t b x y such that a ta t 1 for every a a if x h and y k are complex hilbert spaces the maps preserving the peripheral spectrum of the jordan skew semi triple product ba b are also characterized such maps are of the form a uau or a ua t u where u b h k is a unitary operator a t denotes the transpose of a in an arbitrary but fixed orthonormal basis of h

state space approach to discrete time h control abstract state space solutions to a discrete time problem are given for a given number 0we give a characterization of all controllers such that the norm of the closed loop transfer function is less than the approach taken is based on the solution of two riccati equations using the stable deflating subspace of a symplectic pencil this ensures that no unnecessary assumptions on the a matrix on the realization of the transfer function are needed while the results are the direct analogues of those in the continuous time the state space derivation is considerably more difficult

orthogonality preserving transformations on indefinite inner product spaces generalization of uhlhorn s version of wigner s theorem abstract we present an analogue of uhlhorn s version of wigner s theorem on symmetry transformations for the case of indefinite inner product spaces this significantly generalizes a result of van den broek the proof is based on our main theorem which describes the form of all bijective transformations on the set of all rank one idempotents of a banach space which preserve zero products in both directions

condition spectra of special operators and condition spectra preservers abstract let h be a complex hilbert space and let l h be the algebra of all bounded linear operators on h descriptions of the condition spectra of various special classes of operators are established as application characterization is obtained for maps on l h leaving invariant the condition spectrum or the condition spectral radius of the product of operators analogous results are obtained for jordan triple product of operators

preservers of unitary similarity functions on lie products of matrices abstract denote by m n the set of n n complex matrices let f m n 0 be a continuous map such that f u a u f a for any complex unit a m n and unitary u m n f x 0 if and only if x 0 and the induced map t f t x is monotonically increasing on 0 for any rank one nilpotent x m n characterization is given for surjective maps on m n satisfying f a b b a f a b b a the general theorem is then used to deduce results on special cases when the function is the pseudo spectrum and the pseudo spectral radius

additive maps on standard operator algebras preserving parts of the spectrum let a and b be standard operator algebras on infinite dimensional complex banach spaces x and y respectively and let be a unital additive surjection from a onto b we introduce thirteen parts of the spectrum for elements in a and b and prove that if preserves any one of these parts of the spectrum then it is either an isomorphism or an anti isomorphism

sharp algebraic periodicity conditions for linear higher order difference equations in this paper we give easily verifiable but sharp in most cases necessary and sufficient algebraic conditions for the solutions of systems of higher order linear difference equations to be periodic the main tool in our investigation is a transformation recently introduced by the authors which formulates a given higher order recursion as a first order difference equation in the phase space the periodicity conditions are formulated in terms of the so called companion matrices and the coefficients of the given higher order equation as well

properties and preservers of the pseudospectrum abstract the interplay between the algebraic and analytic properties of a matrix and the geometric properties of its pseudospectrum is investigated it is shown that one can characterize hermitian matrices positive semi definite matrices orthogonal projections unitary matrices etc in terms of the pseudospectrum also characterizations are given to maps on matrices leaving invariant the pseudospectrum of the sum difference or product of matrix pairs it is shown that such a map is always a unitary similarity transform followed by some simple operations such as adding a constant matrix taking the matrix transpose or multiplying by a scalar in 1 1

a new view of the lp theory for a system of higher order difference equations it is proved under appropriate assumptions that the solutions of a linear system of higher order difference equations belong to l p that is the pth powers of the solutions are summable for some p 1 our results are based on a new transformation of the higher order system into a first step recursion where the companion matrices are well treatable from our point of view our theory is illustrated by examples including an interesting exponential stability result and a class of linear delay difference equations

a multiplier approach for the robust design of discrete time control systems with real complex uncertainties a novel approach for the design of robust controllers for discrete time uncertain systems using a multiplier approach is given our algorithm is based on a k m m synthesis type iteration involving a search for suitable multipliers and solving scaled h optimization problems the latter problems involve only the original plant data and the scalings are computed via standard spectral factorizations from the multiplier the design is performed entirely in the discrete time domain and the results can be used for multiobjective control design tchernychev and sideris 1997

pseudospectra of special operators and pseudospectrum preservers abstract denote by b h the banach algebra of all bounded linear operators on a complex hilbert space h let a b h and denote by a the spectrum of a for e 0 define the e pseudospectrum e a of a as e a z a e e b h e e in this paper the pseudospectra of several special classes of operators are characterized as an application complete descriptions are given of the maps of b h leaving invariant the pseudospectra of a b for different kinds of binary operations on operators such as the difference a b the operator product ab and the jordan product a b b a

preservers of local spectrum of matrix jordan triple products abstract let m n be the space of n n complex matrices and for a nonzero vector e c n and t m n let t e denote the local spectrum of t at e maps on m n which preserve the local spectrum of jordan triple product of matrices at e in a sense that t s t e t s t e t s m n are characterized with no surjectivity assumption on them

preservers of radial unitary similarity functions on products of operators abstract let s be a subset of the algebra l h of all bounded linear operators on an infinite dimensional complex hilbert space h and let f s 0 be a radial unitary similarity invariant function under some assumption on s and f characterizations are obtained for surjective maps on s satisfying f t s f t s t s s where the binary operation stands for the product or the jordan semi triple product on operators analogous descriptions are obtained for the finite dimensional case without the surjectivity assumption on

stability switches in linear delay difference equations the paper discusses necessary and sufficient conditions for the asymptotic stability of the zero solution of the linear delay difference equation y n 1 email protected n email protected n k where a b are complex numbers and k is a positive integer compared to the case when a b are real numbers the stability behavior of this equation turns out to be much richer in particular if a b 1 then as k monotonously increases the equation may switch finite times from asymptotic stability to instability and vice versa we describe an interesting structure of the set of these stability switches their explicit values and apply the obtained results to some important delay difference equations and their systems

spectrum preserving linear maps abstract let x and y be banach spaces we show that a spectrum preserving surjective linear map from b x to b y is either of the form t ata 1 for an isomorphism a of x onto y or the form t bt b 1 for an isomorphism b of x onto y

maps leaving functional values of operator products invariant abstract let a be a standard operator algebra on a complex hilbert space h of dimension greater than 2 by invariants of certain functional values of operator products we characterize some surjective maps on a furthermore several kinds of general preserver problems on standard operator algebras are solved when we take respectively the functional as for example k numerical radius k 1 operator norm ky fan k norm schatten p norm 1 p and so on

controller design with real parametric uncertainty a number of techniques have been developed in recent years for the analysis and design of controllers which are robust with respect to structured complex uncertainty in particular the complex synthesis procedure has been successfully applied to a number of engineering problems however the presence of real parametric uncertainty in the problem description substantially complicates matters so that standard complex synthesis techniques are no longer adequate in this paper we develop a procedure to tackle the mixed real and complex synthesis problem this procedure involves a d g k iteration between computing the mixed upper bound and solving an h optimal control problem and has guaranteed convergence to a local minimum of the nonconvex problem the procedure has been implemented in software and several controller designs are compared with the corresponding complex synthesis designs

the complex structured singular value a tutorial introduction to the complex structured singular value is presented with an emphasis on the mathematical aspects of the based methods discussed here have been useful for analysing the performance and robustness properties of linear feedback systems several tests for robust stability and performance with computable bounds for transfer functions and their state space realizations are compared and a simple synthesis problem is studied uncertain systems are represented using linear fractional transformations lfts which naturally unify the frequency domain and state space methods

preservers of pseudo spectral radius of operator products abstract let h be an infinite dimensional complex hilbert space and let l h be the algebra of all bounded linear operators on h for e 0 and t l h let r e t denote the e pseudo spectral radius of t we characterize surjective maps on l h which satisfy r e t s r e t s for all t s l h we also obtain analogous result for the finite dimensional case without the surjectivity assumption on

jordan triple multiplicative surjective maps on b h abstract let h be a hilbert space and b h be the algebra of all bounded linear operators on h in this paper we obtain a structure theorem of maps preserving the jordan triple zero product on b h then this is applied to give a full characterization of a surjective map b h b h satisfying a b a a b a

preservers of local spectra of matrix sums abstract let m n c be the set of n n complex matrices and for a nonzero vector e c n and t m n c let t e denote the local spectrum of t at e we characterize surjective maps on m n c which satisfy t s e t s e for all matrices t and s we arrive at the same conclusion by supposing that t s e t s e for all matrices t and s without the surjectivity assumption on continuous maps from m n c onto itself that preserve the local spectral radius of the sum of matrices at a nonzero fixed vector are also characterized

maps preserving the spectrum of generalized jordan product of operators abstract let a 1 a 2 be standard operator algebras on complex banach spaces x 1 x 2 respectively for k 2 let i 1 i m be a sequence with terms chosen from 1 k and define the generalized jordan product t 1 t k t i 1 t i m t i m t i 1 on elements in a i this includes the usual jordan product a 1 a 2 a 1 a 2 a 2 a 1 and the triple a 1 a 2 a 3 a 1 a 2 a 3 a 3 a 2 a 1 assume that at least one of the terms in i 1 i m appears exactly once let a map a 1 a 2 satisfy that a 1 a k a 1 a k whenever any one of a 1 a k has rank at most one it is shown in this paper that if the range of contains all operators of rank at most three then must be a jordan isomorphism multiplied by an m th root of unity similar results for maps between self adjoint operators acting on hilbert spaces are also obtained