nonconcavity of the spectral radius in levinger s theorem abstract let a r n n be a nonnegative irreducible square matrix and let r a be its spectral radius and perron frobenius eigenvalue levinger asserted and several have proven that r t r 1 t a t a increases over t 0 1 2 and decreases over t 1 2 1 it has further been stated that r t is concave over t 0 1 here we show that the latter claim is false in general through a number of counterexamples but prove it is true for a r 2 2 weighted shift matrices but not cyclic weighted shift matrices tridiagonal toeplitz matrices and the 3 parameter toeplitz matrices from fiedler but not toeplitz matrices in general a general characterization of the range of t or the class of matrices for which the spectral radius is concave in levinger s homotopy remains an open problem

wave propagation characterization and design of two dimensional elastic chiral metacomposite in this work a chiral metacomposite is proposed by integrating two dimensional periodic chiral lattice with elastic metamaterial inclusions for low frequency wave applications the plane harmonic wave propagation in the proposed metacomposite is investigated through the finite element technique and bloch s theorem band diagrams are obtained to illustrate wave properties of the chiral metacomposite effective dynamic properties of the chiral metacomposite are numerically calculated to explain low frequency bandgap behavior in the chiral metacomposite interestingly doubly negative effective density and modulus of the chiral metacomposite are found in a specific frequency range where a pass band with negative group velocity is observed tuning of the resulting low frequency bandgaps is then discussed by adjusting microstructure parameters of the metamaterial inclusion and lattice geometry specifically design of a metacomposite beam structure for the broadband low frequency vibration suppression is demonstrated

lqp dominant output feedbacks an output feedback matrix f will be said to dominate another bar f if the cost for f is not higher for any initial condition than the cost for bar f and if the cost is lower for some perhaps all initial conditions the output feedback obtained by for example the levine athans approach does not necessarily dominate the open loop feedback bar f 0 which could be unfortunate algorithms are given for finding feedbacks which dominate any given stabilizing feedback

design of digital pi dynamic controllers using projective controls extension of the projective controls approach to the design of discrete multivariable pi dynamic controllers for systems with external disturbances and controls feeding directly to the outputs is presented the procedure is applied to a ship boiler control problem to provide comparisons with the usual observer based approach

well posedness of feedback systems insights into exact robustness analysis and approximate computations this paper establishes a framework for robust stability analysis of linear time invariant uncertain systems the uncertainty is assumed to belong to an arbitrary subset of complex matrices the concept used here is well posedness of feedback systems leading to necessary and sufficient conditions for robust stability based on this concept some insights into exact robust stability conditions are given in particular frequency domain and state space conditions for well posedness are provided in terms of hermitian form inequalities it is shown that these inequalities can be interpreted as small gain conditions with a generalized class of scalings given by linear fractional transformations lft using the lft scaled small gain condition in the state space setting the duality is established between the h sub spl infin norm condition with frequency dependent scalings and the parameter dependent lyapunov condition connections to the existing results including the structured singular value and the integral quadratic constraints are also discussed finally we show that our well posedness conditions can be used to give a less conservative yet computable bound on the real structured singular value this result is illustrated by numerical examples

analysis of in plane wave propagation in periodic structures with sierpinski carpet unit cells abstract in this study the dispersion relationships and characteristics of directional periodic structure waveguides with sierpinski carpet unit cells were investigated finite element method was used to analyze the sierpinski carpet periodic structures with square pb cylinders inserted in the rubber background the dispersion curves of unit cells with different stages were investigated first then group velocities of periodic structures at given frequencies were calculated and used to analyze the directional waveguide characteristics in the sierpinski carpet periodic structures the dynamic responses of finite periodic structures were investigated to observe the directional features and the results were used to support the fact that sierpinski carpet unit cells with different stages can control the propagation behavior of elastic waves in specific ways the dispersion curves grew compressed as the stage increased multiple band characteristics were observed in the case of the second stage fractal configuration the directional propagation characteristics in sierpinski carpet periodic structures can be utilized to further extend the range of vibration reduction zones the results of this study have significant potential in terms of the application of sierpinski carpet periodic structures for vibration isolation

a new methodology for observer design and implementation observers are usually formulated as explicit systems of differential equations and implemented using standard ode solvers in this paper we show that there can be advantages in formulating the observer as a differential algebraic equation dae we propose a canonical dae observer and show that it can be implemented using standard dae solvers the dae implementation of the observer can be considered regardless of the design philosophy used we present a simple design strategy for this canonical dae observer based on the extended linearization method

extremal problems on the set of nonnegative definite matrices abstract an optimization problem of minimizing a real valued function of certain elements of a symmetric matrix subject to this matrix being nonnegative definite is considered optimality conditions are proposed the duality result of olkin and pukelsheim 1982 is extended to a wide class of such problems applications are discussed

second order directional derivatives of spectral functions a spectral function of a symmetric matrix x is a function which depends only on the eigenvalues of x l 1 x l 2 x l n x and may be written as l 1 x l 2 x l n x for some symmetric function in this paper we assume that is a c 1 1 function and discuss second order directional derivatives of such a spectral function we obtain an explicit expression of second order directional derivative for the spectral function

parametric compensator design a parametrization of dynamical output feedback compensators that will assign a complete set of distinct eigenvalues to a given state space system is presented the main result is a compact parametric expression for the gain matrices of an output feedback compensator of dynamical order i n m r explicitly characterized by two sets of free parameters the set of n l closed loop system eigenvalues and a set of n mr n m r 1 effective free parameters for n state r input m output systems these latter n effective free parameters are determined by a sequential design procedure the principal benefits of the explicit characterization of a parametric class of dynamical output feedback compensators lie in the ability to directly accommodate various different design criteria

an approximation scheme for distributionally robust nonlinear optimization we consider distributionally robust optimization problems drops with nonlinear and nonconcave dependence on uncertain parameters the drop can be written as a nonsmooth nonlinear program with a

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

solving maximum entropy sampling problems using factored masks we present a practical approach to anstreicher and lee s masked spectral bound for maximum entropy sampling and we describe favorable results that we have obtained with a branch and bound algorithm based on our approach by representing masks in factored form we are able to easily satisfy a semidefiniteness constraint moreover this representation allows us to restrict the rank of the mask as a means for attempting to practically incorporate second order information

the rate of convergence of the augmented lagrangian method for nonlinear semidefinite programming we analyze the rate of local convergence of the augmented lagrangian method in nonlinear semidefinite optimization the presence of the positive semidefinite cone constraint requires extensive tools such as the singular value decomposition of matrices an implicit function theorem for semismooth functions and variational analysis on the projection operator in the symmetric matrix space without requiring strict complementarity we prove that under the constraint nondegeneracy condition and the strong second order sufficient condition the rate of convergence is linear and the ratio constant is proportional to 1 c where c is the penalty parameter that exceeds a threshold overline c 0

a generalization of the structured singular value and its application to model validation the structured singular value spl mu gives a means of assessing the stability and performance of a system under a class of norm bounded structured perturbations this paper introduces a generalization of spl mu which guarantees stability with respect to perturbations where some elements of the perturbation structure are bounded from above and some are bounded from below this generalization is used to solve the frequency domain robust control model validation problem model validation determines whether or not there exist a model in the model set and a bounded noise signal that are consistent with the experimental input output measurements a computable convex upper bound to the generalized spl mu problem is presented and the relationship between the generalized spl mu problem and the standard spl mu and model validation problems is discussed

lowner s operator and spectral functions in euclidean jordan algebras we study analyticity differentiability and semismoothness of lowner s operator and spectral functions under the framework of euclidean jordan algebras in particular we show that many optimization related classical results in the symmetric matrix space can be generalized within this framework for example the metric projection operator over any symmetric cone defined in a euclidean jordan algebra is shown to be strongly semismooth the research also raises several open questions whose answers would be of strong interest for optimization research

floquet bloch decomposition for the computation of dispersion of two dimensional periodic damped mechanical systems floquet bloch theorem is widely applied for computing the dispersion properties of periodic structures and for estimating their wave modes and group velocities the theorem allows reducing computational costs through modeling of a representative cell while providing a rigorous and well posed spectral problem representing wave dispersion in undamped media most studies employ the floquet bloch approach for the analysis of undamped systems or for systems with simple damping models such as viscous or proportional damping in this paper an alternative formulation is proposed whereby wave heading and frequency are used to scan the k space and estimate the dispersion properties the considered approach lends itself to the analysis of periodic structures with complex damping configurations resulting for example from active control schemes the presence of damping materials or the use of shunted piezoelectric patches examples on waveguides with various levels of damping illustrate the performance and the characteristics of the proposed approach and provide insights into the properties of the obtained eigensolutions

derivatives of spectral functions a spectral function of a hermitian matrix x is a function which depends only on the eigenvalues of x 1x 2x nx and hence may be written f 1x 2x nx for some symmetric function f such functions appear in a wide variety of matrix optimization problems we give a simple proof that this spectral function is differentiable at x if and only if the function f is differentiable at the vector x and we give a concise formula for the derivative we then apply this formula to deduce an analogous expression for the clarke generalized gradient of the spectral function a similar result holds for real symmetric matrices

a quadratically convergent local algorithm on minimizing the largest eigenvalue of a symmetric matrix abstract optimization involving eigenvalues arise in many engineering problems we propose a new algorithm on minimizing the largest eigenvalue over an affine family of symmetric matrices under certain assumptions it is shown that if started close enough to the minimizer x the proposed algorithm converges to x quadratically the proposed algorithm can be readily extended to minimizing the largest eigenvalue over an affine family of hermitian matrices also it has been extended to minimizing sums of the largest eigenvalues of a symmetric or hermitian matrix

generalized mixed μ bounds for real and complex multiple block uncertainty with internal matrix structure new absolute stability results that unify and extend existing structured singular value bounds for mixed uncertainty are developed using frequency domain arguments these bounds generalize prior upper bounds for mixed by permitting the treatment of non diagonal real uncertain blocks as well as accounting for internal matrix structure in the uncertainty several specializations to well known absolute stability criteria from the classical literature are given which allow for a systematic comparison of these results in addressing the problem of robust stability for real parameter uncertainty

a measure of worst case h performance and of largest acceptable uncertainty abstract the structured singular value ssv or is known to be an effective tool for assessing robust performance of linear time invariant models subject to structured uncertainty yet all single analysis provides is a bound on the uncertainty under which stability as well as h performance level of are guaranteed where is preselectable in this paper we introduce a related quantity which provides answers for the following questions i given determine the smallest with the property that for any uncertainty bounded by an h performance level of is guaranteed ii conversely given determined the largest with the property that again for any uncertainty bounded by an h performance level of is guaranteed properties of this quantity are established and approaches to its computation are investigated both unstructured uncertainty and structured uncertainty are considered

output feedback pole placement in the design of suboptimal linear quadratic regulators a design oriented methodology is presented for the construction of low order suboptimal output feedback compensators the design is suboptimal in the sense that it retains an l dimensional eigenspace from a reference optimal state feedback linear quadratic regulator problem if r is the number of outputs then for fixed l r it is known that the design requires a dynamic compensator of dimension l r the parameters of the compensator are determined by the solution of an associated output feedback pole placement problem using iterative dyadic pole placement a procedure is given which determines the solution of this problem without a priori assumptions on the dimension of the compensator the design is also extended to the class of stabilizable systems and the resulting compensator is shown to possess a separation property

structured and simultaneous lyapunov functions for system stability problems abstract it is shown that many system stability and robustness problems can be reduced to the question of when there is a quadratic lyapunov function of a certain structure which establishes stability of x ax for some appropriate a the existence of such a lyapunov function can be determined by solving a convex program we present several numerical methods for these optimization problems a simple numerical example is given

the u lagrangian of the maximum eigenvalue function in this paper we apply the u lagrangian theory to the maximum eigenvalue function la and to its precomposition with affine matrix valued mappings we first give geometrical interpretations of the u objects that we introduce we also show that the u lagrangian of la has a hessian which can be explicitly computed the second order development of the u lagrangian provides a second order development of la along a characteristic smooth manifold the set of symmetric matrices whose maximal eigenvalues have a fixed multiplicity the same results can be obtained when we precompose la with an affine matrix valued mapping a provided that this mapping satisfies a regularity condition em transversality condition we show that the hessian of the u lagrangian of la circ a coincides with the reduced hessian encountered in sequential quadratic programming finally we use the u lagrangian to derive second order algorithms for minimizing la circ a

duality in multiplier based robustness analysis frequency domain conditions involving multipliers are useful for robustness analysis the resulting analysis problem is generally convex but infinite dimensional and numerical solutions restricted to finite dimensional subspaces need to be considered the resulting finite dimensional problem can be transformed into a linear matrix inequality which can be solved using efficient algorithms the paper presents a format for the dual of the infinite dimensional problem the dual problem can be used to estimate the conservatism of particular finite dimensional subspaces

discrete time extensions of mixed μ bounds to monotonic and odd monotonic nonlinearities using the parameter dependent lyapunov function framework of haddad and bernstein this paper extends and connects some of the classical absolute stability results on discrete time systems with monotonic and odd monotonic feedback nonlinearities to modern robust stability and performance problems for constant real parameter uncertainty as in the continuous time case it is shown that an immediate application of this framework provides a generalization and reformulation of discrete time mixed theory in terms of lyapunov functions and riccati equations while blending h 2 robust performance within the framework

on analyticity of functions involving eigenvalues abstract let a z be an n n complex matrix whose elements depend analytically on z c m it is well known that any individual eigenvalue of a z may be nondifferentiable when it coalesces with others in this paper we investigate the analycity property of functions on the eigenvalues z 1 z n z of a z we first introduce the notion of functions that are symmetric with respect to partitions it is then shown that if a function c n c is analytic at a where a c m and is symmetric with respect to a certain partition induced by a then the composite function g z 1 z n z is analytic at a when z is real a z is symmetric or hermitian and the aforementioned assumptions hold so that g z is analytic at a we also derive formulae for its first and second order partial derivatives we apply the results to several problems involving eigenvalues

parameterization of frequency weighting for a two stage linear quadratic regulator based design abstract the purpose of this paper is to parametrize a class of lq frequency weightings and the associated dynamic controllers and to utilize the available freedom in a controller design methodology a two stage methodology is described in the first stage an acceptable solution to the conventional lq problem is obtained utilizing established guidelines for the selection of constant cost matrices to achieve satisfactory closed loop performance and an acceptable reference spectrum for the design model in the second stage frequency weighting is introduced to produce controller parameterizations which retain a part or all of the reference spectrum to preserve the desired dynamics associated with the design model and make the controller more robust with respect to the differences between the design model and the actual system

wave propagation in two dimensional periodic lattices plane wave propagation in infinite two dimensional periodic lattices is investigated using floquet bloch principles frequency bandgaps and spatial filtering phenomena are examined in four representative planar lattice topologies hexagonal honeycomb kagome lattice triangular honeycomb and the square honeycomb these topologies exhibit dramatic differences in their long wavelength deformation properties long wavelength asymptotes to the dispersion curves based on homogenization theory are in good agreement with the numerical results for each of the four lattices the slenderness ratio of the constituent beams of the lattice or relative density has a significant influence on the band structure the techniques developed in this work can be used to design lattices with a desired band structure the observed spatial filtering effects due to anisotropy at high frequencies short wavelengths of wave propagation are consistent with the lattice symmetries

a unified approach to operator monotone functions abstract the notion of operator monotonicity dates back to a work by lowner in 1934 a map f s n s m is called operator monotone if a b implies f a f b here s n is the space of symmetric matrices with the semidefinite partial order often the function f is defined in terms of an underlying simpler function f of main interest is to find the properties of f that characterize operator monotonicity of f in that case it is said that f is also operator monotone classical examples are the lowner operators and the spectral scalar valued isotropic functions operator monotonicity for these two classes of functions is characterized in seemingly very different ways this work extends the notion of operator monotonicity to symmetric functions f on k arguments the latter is used to define generated k isotropic maps f s n s n k for any n k necessary and sufficient conditions are given for f to generate an operator monotone k isotropic map f when k 1 the k isotropic map becomes a lowner operator and when k n it becomes a spectral function this allows us to reconcile and explain the differences between the conditions for monotonicity for the lowner operators and the spectral functions

m form numerical range and the computation of the structured singular value an equivalent expressions for the structures singular value is proposed leading to an alternative algorithm for its computation the approach is based to the geometric properties of the m form numerical range of certain tuples of matrices similar to previously considered schemes the algorithm proposed is proved to give the correct value for block structures of size up to three for larger sizes insight is gained into the question of the possible gap between the structured singular value and its known upper bound

control of lpv systems with partly measured parameters the control of linear parameter varying lpv systems with time varying real uncertain parameters where only some of the parameters are measured and available for feedback is considered the control objectives are internal stability and disturbance attenuation in the sense of a bounded induced l sub 2 norm from the disturbance to the controlled output using parametric lyapunov functions the solvability conditions for dynamic output feedback controllers that depend on the measured parameters are expressed in terms of a set of linear matrix inequalities lmis and an additional coupling constraint which destroys the convexity of the overall problem by transforming the coupling constraint into a rank condition the problem is recast into a rank minimization problem with lmi constraints an example is included that demonstrates the application of the results

computational complexity of real structured singular value in l sub p setting this paper studies a generalized real structured singular value spl mu problem where uncertain parameters are bounded by an l sub p norm two results are presented the first one shows that this generalized spl mu problem is np hard for any given rational number p spl isin 1 spl infin the np hardness holds as long as le the size of the largest repeated block exceeds one this result generalizes the known np hardness result for the conventional spl mu problem with p spl infin the second result which strengthens the first one considers the approximability problem of the generalized spl mu we show that the problem of obtaining an estimate for the generalized spl mu with some guaranteed bounds on the relative error remains to be np hard regardless how large this bound is

the higher order derivatives of spectral functions in this paper we are interested in the higher order derivatives of functions of the eigenvalues of symmetric matrices with respect to the matrix argument we describe the formula for the k th derivative of such functions in two general cases the flrst case concerns the derivatives of the composition of an arbitrary not necessarily symmetric k times difierentiable function with the eigenvalues of symmetric matrices at a symmetric matrix with distinct eigenvalues the second case describes the derivatives of the composition of a k times difierentiable separable symmetric function with the eigenvalues of symmetric matrices at an arbitrary symmetric matrix we show that the formula signiflcantly simplifles when the separable symmetric function is k times continuously difierentiable as an application of the developed techniques we re derive the formula for the hessian of a general spectral function at an arbitrary symmetric matrix the new tools lead to a shorter cleaner derivation than the original one in 16 to make the exposition as self contained as possible we have included the necessary background results and deflnitions the proofs of the intermediate technical results are collected in the appendices

brief paper robust finite word length controller design a novel finite word length fwl controller design is proposed in the framework of a mixed m theory a robust fwl controller performance measure is first developed which takes into account the standard robust control requirements as well as the fwl implementation considerations and the corresponding controller design problem is naturally reformulated as a mixed m problem which can be treated effectively with the results of the mixed m theory

analysis of in plane wave propagation in hexagonal and re entrant lattices plane wave propagation in hexagonal and re entrant lattices is investigated through the application of bloch s theorem two dimensional dispersion relations are estimated and analyzed to illustrate peculiar properties of re entrant configurations and investigate the directional behavior of hexagonal lattices for varying construction angles special attention is devoted to the determination of phase and group wave velocities velocity plots are obtained for varying frequency and direction of wave propagation to analyze the dispersion characteristics of the lattices and further investigate their directional characteristics

fast computation of the multivariable stability margin for real interrelated uncertain parameters an alternative implementation of an algorithm for the computation of the multivariable stability margin to check the robust stability of feedback systems with real parametric uncertainty is proposed this method eliminates the need for the frequency search by reducing it to the testing of a finite number of conditions these conditions have a special structure that allows a significant improvement in the speed of computation

solving log determinant optimization problems by a newton cg primal proximal point algorithm we propose a newton cg primal proximal point algorithm ppa for solving large scale log determinant optimization problems our algorithm employs the essential ideas of ppa the newton method and the preconditioned cg solver when applying the newton method to solve the inner subproblem we find that the log determinant term plays the role of a smoothing term as in the traditional smoothing newton technique focusing on the problem of maximum likelihood sparse estimation of a gaussian graphical model we demonstrate that our algorithm performs favorably compared to existing state of the art algorithms and is much preferred when a high quality solution is required for problems with many equality constraints

characterization and efficient computation of the structured singular value the concept of structured singular value was recently introduced by doyle 1 as a tool for the analysis and synthesis of feedback systems with structured uncertainties it is a key to the design of control systems under joint robustness and performance specifications and it very nicely complements the h infty approach to control system design in this paper it is shown that the structured singular value can be obtained as the solution of several smooth optimization problems properties of these optimization problems are exhibited leading to a fast algorithm that always yields the structured singular value for block structures of size no larger than 3 and often does for block structures of larger size

strictly proper projective controllers for disturbance attenuation a design methodology is presented which applies projective controls to the disturbance attenuation problem for large flexible structures through this method low order strictly proper controllers are designed which improve the disturbance attenuation properties of the system a new parameterization of the closed loop system in terms of free parameters in the projective controller leads to an auxiliary minimization problem which determines the parameters of the controller a flexible structure example demonstrates the use of this methodology

rank preservation of matrices with structured uncertainties and its applications in robust control theory matrix rank is determined by the nonsingularity of its submatrices for matrices in which entries are quadratic functions of some uncertain parameters this paper derives sufficient conditions on parameters to that ensure the matrices preserve to some degrees the ranks they have when the parameters are all zero the rank preservation problem is converted to the nonsingularity analysis problem of the minors of the matrix in discussion and suitable tools such as the spl mu analysis method are used to solve the problem applications in robust control theory including tests for robust controllability observability minimum phaseness coprimeness and schur stability are given together with illustrative examples

bloch theorem with revised boundary conditions applied to glide screw and rotational symmetric structures abstract wave propagation in complex periodic systems is often addressed with the bloch theorem and consists in applying periodic boundary conditions to a discretized unit cell while this method has been developed for structures periodic by translation in a recent work for quasi one dimensional wave propagation it has been shown that screw translation plus rotation and glide translation plus reflection periodicities can be accounted for as well keeping the cartesian coordinate system but revisiting the periodic boundary conditions the goal of the present paper is to generalize this concept to quasi two dimensional wave propagation two dimensional waves propagating in three dimensional structures dispersion relations for a set of reduced problems are then compared to results from the classical method when available by considering a smaller periodicity the computational cost is decreased and the number of folding curves and non interacting intersecting curves is reduced improving their interpretability while the size of a unit cell is divided by a factor two when glide symmetries are considered this ratio is significantly increased for screw or rotational symmetries moreover the proposed revisited bloch method is applicable to screw symmetric structures that do not possess purely translational symmetries and for which the classical method cannot be used e g chiral nanotubes longitudinally wrinkled helicoids

nondifferentiable optimization algorithm for designing control systems having singular value inequalities it has been known for some time that many control system design requirements can be expressed as differentiable inequalities more recently it has been shown that important structural properties such as robustness and low noise sensitivity can be expressed as nondifferentiable inequalities involving the singular values of a system or return difference transfer function matrix this paper presents an optimization algorithm which permits all these constraints to be considered

method of centers for minimizing generalized eigenvalues abstract we consider the problem of minimizing the largest generalized eigenvalue of a pair of symmetric matrices each of which depends affinely on the decision variables although this problem may appear specialized it is in fact quite general and includes for example all linear quadratic and linear fractional programs many problems arising in control theory can be cast in this form the problem is nondifferentiable but quasiconvex so methods such as kelley s cutting plane algorithm or the ellipsoid algorithm of shor nemirovsky and yudin are guaranteed to minimize it in this paper we describe relevant background material and a simple interior point method that solves such problems more efficiently the algorithm is a variation on huard s method of centers using a self concordant barrier for matrix inequalities developed by nesterov and nemirovsky nesterov and nemirovsky have also extended their potential reduction methods to handle the same problem since the problem is quasiconvex but not convex devising a nonheuristic stopping criterion i e one that guarantees a given accuracy is more difficult than in the convex case we describe several nonheuristic stopping criteria that are based on the dual of a related convex problem and a new ellipsoidal approximation that is slightly sharper in some cases than a more general result to nesterov and nemirovsky the algorithm is demonstrated on an example determining the quadratic lyapunov function that optimizes a decay rate estimate for a differential inclusion

large scale optimization of eigenvalues optimization problems involving eigenvalues arise in many applications let x be a vector of real parameters and let a x be a continuously differentiable symmetric matrix function of x we consi

an implicit small gain condition and an upper bound for the real structured singular value in this paper we develop an upper bound for the real structured singular value that has the form of an implicit small gain theorem the implicit small gain condition involves a shifted plant whose dynamics depend upon the uncertainty set bound and unlike prior bounds does not require frequency dependent scales or multipliers numerical results show that the implicit small gain bound compares favorably with real bounds that involve frequency dependent scales and multipliers

an interior point method for minimizing the maximum eigenvalue of a linear combination of matrices we consider the problem of finding lambda opt mathop inf limits x in r m lambda max a 0 sum limits i 1 m x i a i 1 and an optimal solution xopt if it exists here the a i are given symmetric matrices in ir n n for 0 i m and max a is the largest eigenvalue of a matrix a problem 1 can be rewritten as a convex differentiable problem with a positive definite constraint positive definiteness can easily be verified numerically via cholesky decomposition moreover the cone of positive definite matrices a is characterized by the domain of the smooth convex barrier function log det a

the design of output regulators for discrete time linear systems by projective controls the method of projective controls is applied to the design of static and low order dynamic output regulators for discrete time systems basic properties of the projective controls approach are obtained and the relation of the associated regulator to the state regulator as well as to the standard output regulator is determined an explicit relationship of dynamic regulators design by projective controls approach with observers is also developed and fundamental differences are highlighted

on occurrence of spectral edges for periodic operators inside the brillouin zone the paper discusses the following frequently arising question on the spectral structure of periodic operators of mathematical physics e g schrodinger maxwell waveguide operators etc is it true that one can obtain the correct spectrum by using the values of the quasimomentum running over the boundary of the reduced brillouin zone only rather than the whole zone or do the edges of the spectrum occur necessarily at the set of corner high symmetry points this is known to be true in 1d while no apparent reasons exist for this to be happening in higher dimensions in many practical cases though this appears to be correct which sometimes leads to the claims that this is always true there seems to be no definite answer in the literature and one encounters different opinions about this problem in the community in this paper starting with simple discrete graph operators we construct a variety of convincing multiply periodic examples showing that the spectral edges might occur deeply inside the brillouin zone on the other hand it is also shown that in a generic case the situation of spectral edges appearing at high symmetry points is stable under small perturbations this explains to some degree why in many maybe even most practical cases the statement still holds

efficient computation of a guaranteed lower bound on the robust stability margin for a class of uncertain systems sufficient conditions for the robust stability of a class of uncertain systems with several different assumptions on the structure and nature of the uncertainties can be derived in a unified manner in the framework of integral quadratic constraints these sufficient conditions in turn can be used to derive lower bounds on the robust stability margin for such systems the lower bound is typically computed with a bisection scheme with each iteration requiring the solution of a linear matrix inequality feasibility problem we show how this bisection can be avoided altogether by reformulating the lower bound computation problem as a single generalized eigenvalue minimization problem which can be solved very efficiently using standard algorithms we illustrate this with several important commonly encountered special cases diagonal nonlinear uncertainties diagonal memoryless time invariant sector bounded popov uncertainties structured dynamic uncertainties and structured parametric uncertainties we also present a numerical example that demonstrates the computational savings that can be obtained with our approach

quadratic expansions of spectral functions a function f on the space of n n real symmetric matrices is called spectral if it depends only on the eigenvalues of its argument that is f a f uau t for every orthogonal u and symmetric a in its domain spectral functions are in one to one correspondence with the symmetric functions on r n those that are invariant under arbitrary swapping of their arguments in this paper we show that a spectral function has a quadratic expansion around a point a if and only if its corresponding symmetric function has quadratic expansion around a the vector of eigenvalues we also give a concise and easy to use formula for the hessian of the spectral function in the case of convex functions we show that a positive definite hessian of f implies positive definiteness of the hessian of f 2002 elsevier science inc all rights reserved

representation of uncertain time delays in the h framework abstract how to represent an uncertain time delay in a form suitable for robust control in terms of the h norm and the structured singular value is discussed to use synthesis the uncertain delay has to be approximated to yield a proper rational interconnection matrix m s and a purely complex perturbation matrix s the parametric average of the delay is usually included in the nominal model while the uncertain perturbation covers variation around the average delay it is proposed to model the nominal time delay as uncertainty i e using a larger uncertainty set this yields a delay free nominal model which simplifies the controller synthesis for the cases studied the delay free nominal model does not yield a more conservative design than the average delay model despite its larger uncertainty set

on minimizing the largest eigenvalue of a symmetric matrix abstract optimization problems involving eigenvalues arise in many engineering problems in this paper we consider the problem of minimizing the largest eigenvalue over an affine family of symmetric matrices this problem has a variety of applications such as the stability analysis of dynamic systems or the computation of structured singular values given 0 we give an optimality condition which ensures that the largest eigenvalue is within error bound of the solution also a new line search rule is proposed and it is shown to have good descent properties when the multiplicity of the largest eigenvalue the solution is known a new algorithm for the optimization problem under consideration is proposed some numerical experiments on the proposed algorithm are presented

design of a class of multirate systems using a maximum relative spl lscr sup 2 error criterion a criterion for designing the class of multirate systems for rate changing is presented this criterion arises from a model matching perspective with maximum relative spl lscr sup 2 error over a general class of inputs and is a natural extension of the standard chebyshev method for filter design using multirate and convex analysis techniques the criterion is shown to lead to a convex matrix valued function approximation problem an algorithm using convex optimization is proposed to solve the problem an example illustrates the use of the algorithm and effectiveness over standard filter design techniques

semi definite matrix constraints in optimization positive semi definite matrix constraints arise in a number of optimization problems in which some or all of the elements of a matrix are variables such as the educational testing and matrix modification problems the structure of such constraints is developed including expressions for the normal cone feasible directions and their application to optimality conditions a computational framework is given within which these concepts can be exploited and which permits the quantification of second order effects the matrix of lagrange multipliers in this formulation is shown to have an important relationship to the characterization of the normal cone modifications of various iterative schemes in nonlinear programming are considered in order to develop an effective algorithm for the educational testing problem and comparative numerical experiments are described it is shown that a particular choice of the penalty parameter for an l 1 exact penalty function is appropriate for this type of problem the be

constrained maximum entropy sampling a fundamental experimental design problem is to select a most informative subset having prespecified size from a set of correlated random variables instances of this problem arise in many applied domains such as meteorology environmental statistics and statistical geology in these applications observations can be collected at different locations and possibly at different times information is measured by entropy practical situations have further restrictions on the design space for example budgetary limits geographical considerations as well as legislative and political considerations may restrict the design space in a complicated manner using techniques of linear algebra combinatorial optimization and convex optimization we develop upper and lower bounds on the optimal value for the gaussian case we describe how these bounds can be integrated into a branch and bound algorithm for the exact solution of these design problems finally we describe how we have implemented this algorithm and we present computational results for estimated covariance matrices corresponding to sets of environmental monitoring stations in the ohio valley of the united states

the design of optimal output regulators for linear multivariable systems with constant disturbances a method recently developed for solving the output regulator problem is extended to linear systems with constant disturbances the design of static and low order dynamic regulators for linear systems based on the use of projective controls is reviewed first the proposed method determines through a unified procedure the order and the parameters of the compensator together with the feedback parameters this methodology is then applied to systems with external disturbances and various control structures incorporating integral action are investigated in the light of the proposed methodology

on the norms used in computing the structured singular value different norms are considered to replace the euclidean norm in an algorithm given by m h k fan and a l tits ibid vol 33 p 284 9 1988 which is used for the computation of the structured singular value of any matrix the algorithm is explained and it is shown that the l sub 1 norm is the best possible norm in a certain sense

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

on problems involving eigenvalues for uncertain matrices by structured singular values this technical note considers problems that involve eigenvalues of uncertain matrices expressed in linear fractional form using the structured singular value mu and the skewed structured singular value nu in particular positive definiteness conditions maximum and minimum eigenvalues and generalized eigenvalues of uncertain matrices are expressed using mu and nu the obtained results allow us to acquire information of an uncertain matrix efficiently by the existing computational tools that provide practically useful approximations to the values of mu and nu

a convex parameterization of robustly stabilizing controllers this paper treats synthesis of robust controllers for linear time invariant systems uncertain real parameters are assumed to appear linearly in the closed loop characteristic polynomial the main contribution is to give a convex parameterization of all controllers that simultaneously stabilize the system for all possible parameter combinations with the new parameterization certain robust performance problems can be stated in terms of quasi convex optimization

state space solutions to standard h sub 2 and h sub infinity control problems simple state space formulas are derived for all controllers solving the following standard h sub infinity problem for a given number gamma 0 find all controllers such that the h sub infinity norm of the closed loop transfer function is strictly less than gamma it is known that a controller exists if and only if the unique stabilizing solutions to two algebraic riccati equations are positive definite and the spectral radius of their product is less than gamma sup 2 under these conditions a parameterization of all controllers solving the problem is given as a linear fractional transformation lft on a contractive stable free parameter the state dimension of the coefficient matrix for the lft constructed using the two riccati solutions equals that of the plant and has a separation structure reminiscent of classical lqg i e h sub 2 theory this paper is intended to be of tutorial value so a standard h sub 2 solution is developed in parallel

design of coupled decentralized output regulators decentralized static output regulators are designed via projective controls in cases where the controllers are directly coupled i e when the controls of other decision makers appear in the local outputs for two decision makers the solution is reduced to the problem of solving a general non square algebraic riccati equation the existence and uniqueness of the solution is proved and then determined analytically these results are then generalized to an arbitrary number of decision makers finally the possibilities of the sequential design of static and dynamic regulators for this type of decentralized information structure are investigated a numerical example illustrates the design procedure

computational methods for parametric lq problems a survey iterative methods for finding the optimal constant feedback gains for parametric lq problems notably for optimal constant output feedback problems are surveyed the connections of several methods to loss function expansions are discussed with important implications to the understanding of their convergence properties especially the descent anderson moore method levine athans like methods and the newton method are considered convergence results are also included the initialization problem and the output feedback stabilization problem are also discussed furthermore it is shown that the concepts and methods surveyed in this paper are useful in solving many realistic generalized parametric lq problems as well notably so called robust parametric lq problems

asymptotic expansions of the ordered spectrum of symmetric matrices abstract in this work we build on ideas of torki 2001 6 and show that if a symmetric matrix valued map t a t has a one sided asymptotic expansion at t 0 of order k then so does t m a t where m is the m th largest eigenvalue we derive formulas for computing the coefficients a 0 a 1 a k in the asymptotic expansion as an application of the approach we give a new proof of a classical result due to kato 1976 3 about the one sided analyticity of the ordered spectrum under analytic perturbations finally as a demonstration of the derived formulas we compute the first three terms in the asymptotic expansion of m a t e for any fixed symmetric matrices a and e

an introduction to a class of matrix cone programming in this paper we define a class of linear conic programming which we call matrix cone programming or mcp involving the epigraphs of five commonly used matrix norms and the well studied symmetric cone mcp has recently been found to have many important applications for example in nuclear norm relaxations of affine rank minimization problems in order to make the defined mcp tractable and meaningful we must first understand the structure of these epigraphs so far only the epigraph of the frobenius matrix norm which can be regarded as a second order cone has been well studied here we take an initial step to study several important properties including its closed form solution calm bouligand differentiability and strong semismoothness of the metric projection operator over the epigraph of the l 1 l infty spectral or operator and nuclear matrix norm respectively these properties make it possible to apply augmented lagrangian methods which have recently received a great deal of interests due to their high efficiency in solving large scale semidefinite programming to this class of mcp problems the work done in this paper is far from comprehensive rather it is intended as a starting point to call for more insightful research on mcp so that it can serve as a basic tool to solve more challenging convex matrix optimization problems in years to come

a µ based optimal finite word length controller design a novel optimal finite word length fwl controller design is proposed in the framework of m theory a computationally tractable close loop stability measure with fwl implementation considerations of the controller is derived based on the m theory and the optimal fwl controller realizations are obtained by solving the resulting optimal fwl realization problem using linear matrix inequality techniques

probability that a band gap extremum is located on the irreducible brillouin zone contour for the 17 different plane crystallographic lattices abstract after recalling the first brillouin zone bz and the first irreducible brillouin zone ibz of a lattice in terms of its plane crystallographic group we investigate the danger of restricting a band gap detection to the contour of the ibz instead of its full ibz based on hundreds of porous phononic crystal simulations we provide for the 17 plane crystallographic groups i statistics of the band gap localizations ii probabilities to get non full band gaps and iii averages of the bandwidth error made when only the ibz contour is considered it is found that for phononic crystals the ibz contour provides accurate results only for highly symmetric lattices

derivatives of compound matrix valued functions abstract primary matrix functions and spectral functions are two classes of orthogonally invariant functions on a symmetric matrix argument many of their properties have been investigated thoroughly and find numerous applications both theoretical and applied in areas ranging from engineering image processing optimization and physics we propose a family of maps that provide a natural connection and generalization of these two classes of functions the family of maps also contains the well known multiplicative and additive compound matrices we explain when each member of this family is a differentiable function and exhibit a formula for its derivative

real m bounds based on fixed shapes in the nyquist plane parabolas hyperbolas cissoids nephroids and octomorphs in this paper we introduce new bounds for the real structured singular value the approach is based on absolute stability criteria with plant dependent multipliers that exclude the nyquist plot from fixed plane curve shapes containing the critical point jo unlike half plane and circle based bounds the critical feature of the fixed curve bounds is their ability to differentiate between the real and imaginary components of the uncertainty since the plant dependent multipliers have the same functional form at all frequencies the resulting graphical interpretation of the absolute stability criteria are frequency independent in contrast to the frequency dependent off axis circles that arise in standard real bounds