a relative error bound for balanced stochastic truncation recently several methods have appeared for the approximation of power spectra notably balanced stochastic truncation bst it is shown that bst satisfies a relative error bound approximately twice the bound for the relative error method rem proper this offers a quantitative basis for the observation that bst and rem produce similar reduced order models balanced stochastic truncation can therefore be interpreted as providing a computationally simple algorithm for relative error approximation

an h minimax approach to the design of robust control systems abstract a realistic feedback design problem is posed based on the minimization of a weighted combination of the sensitivity and complementary sensitivity matrices a solution is obtained which makes use of the recently proposed methods for minimizing the sensitivity function alone

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

comments on robust fragile or optimal with reply the issue of fragile controllers produced by using popular robust and optimal control synthesis methods was raised by keel bhattacharyya in the above mentioned paper ibid vol 42 1997 the commenter points out that the controller fragility problem described in the paper is 1 well known in the literature 2 due to the poor choice of examples 3 can easily be fixed by changing the optimization criteria and 4 can be fixed by changing the implementation in reply keel bhattacharyya disagree with him on all counts and explain their argument

time varying versus time invariant compensation for rejection of persistent bounded disturbances and robust stabilization it is shown that time varying compensation does not improve the optimal rejection of persistent bounded disturbances this result is obtained by exploiting a key observation that any time varying compensator which yields a given degree of disturbance rejection must do so uniformly over time thereby removing any advantage of time variation this key observation is exploited to show that time varying compensation does not improve the optimal rejection of disturbances regardless of the norm used to measure the disturbances thus absolutely summable finite energy or persistent bounded disturbances may be treated in the same manner it is shown that time varying compensation does not help in the bounded input bounded output robust stabilization of time invariant plants with unstructured uncertainty in doing so it is also shown that the small gain theorem is both necessary and sufficient for the bounded input bounded output stability of certain linear time varying plants subject to unstructured linear time varying perturbations

generic buckling curves for specially orthotropic rectangular plates using a double affine transformation the classical buckling equation for specially orthotropic plates and the corresponding virtual work theorem are presented in a particularly simple fashion these dual representations are characterized by a single material constant called the generalized rigidity ratio whose range is predicted to be the closed interval from 0 to 1 if this prediction is correct then the numerical results using a ratio greater than 1 in the specially orthotropic plate literature are incorrect when natural boundary conditions are considered a generalized poisson s ratio is introduced thus the buckling results are valid for any specially orthotropic material hence the curves presented in the text are generic rather than specific the solution trends are twofold the buckling coefficients decrease with decreasing generalized rigidity ratio and when applicable they decrease with increasing generalized poisson s ratio since the isotropic plate is one limiting case of the above analysis it is also true that isotropic buckling coefficients decrease with increasing poission s ratio

spectral analysis of non linear systems involving square law operations abstract this paper contains a number of useful theoretical formulas to analyze the frequency domain properties of gaussian input data passing through non linear square law systems special bispectral density functions are defined and applied that are functions of a single variable from measurements of input data and output data only results are obtained to identify the separate frequency response functions for two models of linear systems in parallel with non linear square law systems non linear coherence functions are defined from these models which determine the proportion of the output spectrum due to the non linear operations together with ordinary coherence functions a measured output spectrum for these models can be decomposed into three components representing the linear operations the non linear operations and the remaining uncorrelated noise effects this material indicates also how to analyze other types of non linear models by employing similar techniques

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

on the robust stability of linear time invariant plants with unstructured uncertainty we consider general families of linear time invariant plants described by a nominal plant model with unstructured uncertainty we show under a rather weak assumption that when a family of systems is not robustly stabilizable by any linear time invariant feedback controller then no nonlinear time varying controller can robustly stabilize the given family

frequency weighted spl lscr sub spl infin norm and optimal hankel norm model reduction a new relative error model reduction method is proposed using frequency weighted balanced realization and explicit spl lscr sub spl infin norm error bounds are also derived for the relative error and multiplicative error the method only needs to solve two lyapunov equations it is further shown that this method is equivalent to the balanced stochastic truncation if the plant is square and minimum phase this paper also gives a complete solution to the frequency weighted hankel norm approximation with antistable weighting these results are then applied to spl lscr sub spl infin norm model reduction and several numerically effective algorithms are proposed it is shown through many numerical examples that these algorithms work very well and in many cases produce almost optimal solutions

a duality theory for robust stabilization and model reduction fuhrmann 1991 1993 developed a duality theory in the context of hanel norm approximation and nehari complementation problems the class of functions involved were the scalar antistable transfer functions this work was extended using normalized coprime factorizations by fuhrmann and ober 1993a to the class of all minimal transfer functions in this paper we extend the scope of the duality theory significantly the paper presents a unified approach to problems of hankel norm approximation model reduction and robust control of rational multivariable transfer functions the unification is achieved by considering two classes of transfer functions and corresponding normalized coprime factorization using the youla kucera parametrization of all stabilizing controllers we single out a unique controller by imposing a mcmillan degree minimization restriction on the doubly coprime factorizations with this controller we construct an associated stable transfer function which we call the characteristic function many problems on the original system can be reduced to the study of the characteristic function

on the stability of projections of balanced realizations abstract this paper generalizes a stability property concerning the state matrix of a balanced realization established by pernebo and silverman it is shown that stability is preserved under a general projection of the state matrix provided that the hankel singular values of the realization are distinct a necessary and sufficient condition for the case with nondistinct hankel singular values is also established

weighted hankel norm approximation calculation of bounds abstract frequency domain bounds are computed for the approximation error involved in the optimum hankel norm approximation procedure the bounds differ from those arising in the unweighted optimal hankel norm approximation procedure by a multiplicative factor depending solely on the weighting function this factor becomes unity when the weighting function is constant

state space solutions to hadamard weighted ℋ and ℋ 2 control problems variations of the standard h and h 2 control problems are presented state space solutions are given for the problems of weighting the individual elements of a closed loop transfer function while i satisfying an h frobenius norm bound or ii minimizing the h 2 norm of the closed loop the optimal h frobenius case is analysed and it is shown that the optimal controller is often unique in contrast to the standard h control problem the solutions of an algebraic riccati equation are examined in the h 2 case to reveal the structure of the h 2 optimal controller

reduced order controller design using coprime factor model reduction two procedures for reduced order controller design that incorporate coprime factor model reduction techniques and a related robust stabilization problem are proposed it is shown that robust stability can be achieved with these controllers the results obtained show how to reduce the controller degree systematically they clearly indicate how the maximum stability margin influences allowable model reduction error

the robust regulation problem with robust stability a subspace valued function approach among the most common purposes are the tracking of reference signals and the rejection of disturbance signals in the face of uncertainties the related design problem is called the robust regulation problem here we investigate the tradeoff between the robust regulation constraint and the requirement of robust stability with respect to perturbations in the normalized coprime factors of the plant we use an approach in terms of subspace valued functions associated to plant and controller in this way necessary and sufficient conditions for robust regulation with robust stability can be related to a simple geometric property

spectral factorization via hermitian pencils abstract the spectral factorization problem is solved for state space systems via results on the canonical forms and inertia of hermitian matrix pencils these algebraic results then give a deflation method for spectral factorization

divergence and flutter instability of elastic specially orthotropic plates subject to follower forces abstract divergence and flutter instabilities of a rectangular specially orthotropic plate simply supported on a pair of opposite sides subject to follower forces are computationally analyzed the effects of the tangential follower parameter aspect ratio boundary condition on the side opposite the loaded edge and material orthotropy on the magnitude of the critical load and the mode divergence or flutter of instability are detailed for the three boundary conditions considered built in simply supported and free free for the side opposite the loaded edge and given material orthotropy the plane is divided into regions corresponding to different governing instabilities both the number and size of these regions depend strongly upon the boundary condition considered as does the governing instability for a given boundary condition the configuration of the stability regions is independent of material orthotropy material orthotropy affects only the buckling loads corresponding to the instabilities and the size of the regions

the four block adamjan arov krein problem abstract this paper is concerned with the mixed sensitivity h design in the most general i e the four block case this problem involves in a crucial manner the so called four block operator the norm of which is the achievable feedback tolerance our objective in this paper is to provide an interpretation for all singular values of these singular values are given an adamjan arov krein interpretation in terms of the l distance of an l function to h l intuitively the singular values of are the various tolerance levels that can be achieved if we allow a various number of unstable poles in the closed loop we finally provide an upper bound on the number of singular values

optimal hankel norm approximation for stable systems with first order stable weighting functions abstract given a stable rational transfer function g s and weighting function w s the problem of finding g s of macmillan degree k so as to minimise w g g h is considered this problem is solved for w s s s with no assumptions on the signs of and this gives rise to approximations where w g g can be accurately bounded from above and below in terms of the hankel singular values of wg when 0

reduced order compensators via balancing and central control design for a structural control problem real time control of a physical system necessitates controllers that are low order in this article we compare two balanced truncation methods as a means of designing low order controllers for a nonlinear cable mass system the first is the standard technique of balanced truncation the second linear quadratic gaussian lqg balanced truncation can be thought of as balancing based on the controller and states that are important from the perspective of control and filter design are retained the control design applied to each reduced order model is the central controller we provide an overview of the central controller and devote attention to the design of this controller in the presence of balancing also described in this article is a method for reducing computational time in solving algebraic riccati equations for the design of low order lqg balanced controllers

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

system identification with generalized orthonormal basis functions a least squares identification method is studied that estimates a finite number of expansion coefficients in the series expansion of a transfer function where the expansion is in terms of recently introduced generalized basis functions the basis functions are orthogonal in h2 and generalize the pulse laguerre and kautz bases one of their important properties is that when chosen properly they can substantially increase the speed of convergence of the series expansion this leads to accurate approximate models with only a few coefficients to be estimated explicit bounds are derived for the bias and variance errors that occur in parameter estimates as well as in the resulting transfer function estimates

remarks on order reduction for a robustly suboptimal controller via singular perturbations abstract it is shown that under a generic assumption the suboptimally robust controller constructed according to glover and mcfarlane 1989 leads to a singularly perturbed system which may be reduced in the usual way avoiding thus the difficulties mentioned by habets 1991 the reduced controller obtained by this procedure is in fact optimal

identification of normalized coprime factors through constrained curve fitting this paper deals with identifying finite dimensional normalized right coprime factors nrcfs under a stochastic framework from a sequence of their frequency response estimates a two step procedure is suggested for the estimation in which two cost functions are sequentially minimized these cost functions are constructed on the basis of weighted curve fitting and the conditions on nrcfs a numerical procedure is proposed for computing the estimate moreover it is proved that under some conditions the estimate converges with probability 1 w p 1 to its actual value with increasing the data length and in general the optimal estimate is asymptotically approximately gaussian the efficiency of the proposed method in nrcf identification is illustrated by numerical simulations

hankel norm of sampled data systems generalizes the concept of hankel norm for purely continuous time systems to sampled data systems the hankel norm of a sampled data system is defined by taking into account intersample behaviors the formula of the hankel norm of a sampled data system is given by a hybrid state space model approach the authors show that the hankel norm is equal to that of an equivalent discrete time system two simple examples including sampled data controller reduction are demonstrated

performance preserving controller reduction via additive perturbation of the closed loop transfer function an h sub spl infin performance preserving controller order reduction method is proposed here performance preservation indicates that the h sub spl infin norm bound of the closed loop transfer function with reduced order controller is not greater than the h sub spl infin norm bound of the closed loop transfer function with full order controller we assume additive perturbations to the closed loop transfer function and obtain a sufficient condition for performance preservation two kinds of useful weightings are derived and the controller reduction problem is solved via a frequency weighted model reduction problem

robust stabilization in the gap metric controller design for distributed plants the problem of robustness in the gap metric for infinite dimensional systems is considered the problem of computing the optimal controller and the optimal robustness radius for a class of systems whose normalized coprime factors have elements which are h sub infinity functions with continuous boundary values is studied the underlying hankel and related operators which are important in the gap optimization problem are studied and relations between their singular values and vectors are established a computational approach to the optimal robustness problem is developed for single input single output systems whose transfer function is an inner function in h sub infinity times a rational function the procedure is applied to a general first order delay system and a closed form formula is obtained for the optimal controller the frequency response plots of the compensated system for various values of time delay are examined

nonlinear forced vibration of orthotropic rectangular plates using the method of multiple scales the method of multiple scales mms in conjunction with the galerkin method is used to analyze the nonlinear forced and damped response of a rectangular orthotropic plate subjected to a uniformly distributed harmonic transverse loading the effects of damping and in plane loads are considered the analysis considers simply supported as well as clamped plates for each case both movable and immovable edge conditions are considered by using mms all possible resonances such as primary subharmonic and superharmonic reso nances are studied for the undamped response without in plane loading comparisons of the mms results with those obtained by the finite element method show excellent agreement evelopment of composite materials comprising lam inates of orthotropic or multilayered anisotropic materi als recently has been receiving substantially growing research efforts due to the increasing demands for energy efficient high strength minimum weight aircraft designs many re searchers believe that the use of composite materials offers promising alternatives for aircraft designs thin laminated composite plates subjected to transverse periodic loadings could encounter deflections of the order of plate thickness or even higher responses of this kind cannot be predicted by using the linear theory consequently the need to study large amplitude deflection vibrations of composite structures is of paramount importance the literature survey shows that the equations of motion for the large deflection analysis of heterogeneous anisotropic plates using the von karman geometrical nonlinearity were first considered by whitney and leissa 1 based on these equa tions different methods of analysis have been developed by several researchers an excellent number of collections on nonlinear free and forced vibrations of composite plates cov ering the work through 1979 can be found in the comprehen sive book by chia 2 bert3 has conducted a survey on the dynamics of composite plates for the period 1979 81 a re view of the literature on nonlinear vibrations of plates can be found in the review paper by sathyamoorthy4 and the book by nayfeh and mook 5 large deflection analysis of symmetrically laminated rec equations of motion are presented in terms of the lateral displacement and stress function the equations are nondi mensionalized following the transformation introduced by brunelle and oyibo 9 though multimode analysis can be treated the present study is focused on single mode analysis a deflection function representing the first mode and satisfy ing the boundary conditions is assumed and subsequently the stress function is found next the galerkin method is applied to obtain the modal equation which is solved analytically by using the method of multiple scales mms 10 the mms also provides solutions for subharmonic and superharmonic reso nances the effects of damping ratio plate aspect ratio and in plane loading then are studied

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

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

stacking sequence design of flat composite panel for flutter and thermal buckling in this paper we formulate the stacking sequence design of laminated composite flat panels for maximum supersonic flutter speed and maximum thermal buckling capacity the design is constrained to have stable limit cycle oscillation in the vicinity of the flutter boundary for the purpose of comparison we consider both linear and nonlinear panel flutter analyses because of manufacturing constraints the stacking sequence of the laminate is selected from a discrete set of orientation angles the plate is modeled using von karman plate equations and assumed to be subjected to uniform temperature differential the aerodynamic load on the plate is computed using third order piston theory the plate equations of motion are derived using hamilton s principle and discretized using the rayleigh ritz method the nonlinear panel flutter analysis is performed using an approximate perturbation approach the pareto fronts describing the trade off between thermal buckling and flutter are generated for different panel aspect ratios it is observed that optimal designs obtained for both linear and nonlinear flutter analyses are similar with only the predicted flutter speeds different this suggests that for the purpose of preliminary stacking sequence design linear flutter analysis is sufficient

robust controllers for uncertain linear multivariable systems abstract this paper is addressed to three distinct yet related topics in the design of controllers for imprecisely known linear multivariable systems in the first part it is supposed that the plant to be stabilized is subject to additive or multiplicative uncertainties and necessary and sufficient conditions are derived for the existence of a controller that stabilizes all plants within this band of uncertainty in the second part in contrast with the first part it is supposed that the number of unstable poles of the plant to be stabilized is not precisely known the type of plant uncertainty is the so called stable factor uncertainty and necessary and sufficient conditions are given for robust stabilization in the third part the model of uncertainty is a ball in the space of rational matrices metrized by the so called graph metric and sufficient conditions for robust stabilization are derived

optimal h multivariable robust controllers and the relationship to lqg design problems a new solution for the multivariable minimax optimal robust design problem is presented the relationship between lqg and h design methods is considered and the h solution is obtained from the solution of a related lqg problem the approach provides insights into the relationship between the h and lqg design philosophies the h solution involves the same type of steps which are involved in the polynomial based solutions of the lqg problem namely spectral factorization and the solution of linear diophantine equations a simplified design procedure is described which requires only the solution of a standard eigenvalue eigenvector problem

controller reduction concepts and approaches the problem of passing from a linear time invariant high order controller designed for a linear time invariant plant of presumably high order to a low order approximation of the controller is discussed the approximation problem is often best posed as a frequency weighted l sub infinity approximation problem many different controller representations are possible giving different performances of the various reduction algorithms

flutter of orthotropic panels in supersonic flow using affine transformations affine transformations are used in analyzing the flutter problem of rectangular simply supported orthotropic panels subjected to supersonic flow over one surface with the help of certain defined characteristic and bounded quantities a comprehensive solution which has the isotropic panels solution as a subset is found to this problem the physics of this very important aeroelastic problem which has been so obscure for a long time because of the presence of so many parameters is thus clearly exposed by showing how the aerodynamic and the elastic forces interact to produce the panel flutter phenomenon both the aerodynamic strip ackeret theory and lifting surface theory are compared and found to agree very well in the analysis hence complete stability boundaries are determined for both flat and buckled panels using the aerodynamic strip theory the simpler of the two theories

robust performance assessment of feedback control systems the proper measure of closed loop performance variation in the presence of model plant mismatch is discussed in this paper a generalized closed loop error transfer function which is a special representation of the dual youla parameter and has a close relationship with the pointwise n gap metric is proposed as the suitable means of representing closed loop performance variation in case of plant perturbation and the closed loop performance variation measure is accordingly defined as its maximum singular value frequency by frequency it is shown that this measure is essential and informative in characterizing closed loop performance variation this measure is also shown to be readily applicable to on line closed loop performance assessment or monitoring even without the explicit model of the plant its variant defined as the h function which features the relative performance variation as well as generalized stability margin variation with respect to the nominal plant is also discussed

robust stabilization of normalized coprime factor plant descriptions with h sub infinity bounded uncertainty the problem of robustly stabilizing a family of linear systems is explicitly solved in the case where the family is characterized by h sub infinity bounded perturbations to the numerator and denominator of the normalized left coprime factorization of a nominal system this problem can be reduced to a nehari extension problem directly and gives an optimal stability margin all controllers satisfying a suboptimal stability margin are characterized and explicit state space formulas are given

frequency weighted model reduction with l error bounds abstract this paper considers frequency weighted model reduction the explicit lower and upper approximation error bounds are derived for certain classes of weighted model reduction problems the approximation is based on some unweighted approximations and the error bounds are given in terms of the hankel singular values of the weighted model

an efficient mixed methodology for free vibration and buckling analysis of orthotropic rectangular plates abstract in this paper a simple and efficient mixed ritz differential quadrature dq method is presented for free vibration and buckling analysis of orthotropic rectangular plates the mixed scheme combines the simplicity of the ritz method and high accuracy and efficiency of the dq method the accuracy of the proposed method is demonstrated by comparing the calculated results with those available in the literature it is shown that highly accurate results can be obtained using a small number of ritz terms and dq sample points the proposed method is suitable for the problem considered due to its simplicity and potential for further development

frequency weighted optimal hankel norm approximation of stable transfer functions abstract this paper shows how to introduce arbitrary frequency weighting into the optimal hankel norm approximation problem for scalar finite dimensional linear time invariant systems the procedure has applications to the closed form design of reduced order controllers

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

the four block adamjan arov krein problem for discrete time systems abstract we consider the problem of approximating in h norm a given discrete inverse time stable system possibly improper t t 11 t 12 t 21 t 22 by a discrete time system s with no more than r poles outside the unit disk possibly at infinity such that t 11 s t 12 t 21 t 22 here r 0 is an integer and 0 is a prescribed tolerance if r 0 this is the four block nehari problem while if we consider the one block case where t t11 we obtain the well known hankel norm approximation problem the theoretical developments are based on a frequency domain signature condition while the class of solutions is constructed in state space in terms of the solutions to two riccati equations

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

unified panel flutter theory with viscous damping effects the idea of a unified panel flutter theory investigated previously by the author is amplified to make the inclusion of damping effects possible the paper attempts to study the combined effects of the aerodynamic and structural types of damping on the flutter boundaries for simply supported panels in high supersonic mach number flows classical plate theory and two dimensional first order aerodynamics are used a closed form rather than modal approach is used in the solution of the partial differential equation in the analysis to avoid convergence problems the standing and traveling wave theories of panel flutter are compared the dual nature of damping stabilizing and destabilizing is exposed while the trends are similar to those already established for isotropic panels the variables compared are generically defined to account for both the isotropic as well as the orthotropic properties of the panels

unified aeroelastic flutter theory for very low aspect ratio panels a theory unifying the flutter analyses of orthotropic and isotropic panels having very low aspect ratios and exposed to an inviscid potential flow on the upper surfaces is developed the analysis considers an infinitely long panel of finite width simply supported on the side edges and resting on a spring foundation three dimensional linearized aerodynamic and classical plate theories are used traveling wave modes are utilized as solutions to the panel s aeroelastic equation of motion the resulting aerodynamic integral is evaluated approximately using both numerical and analytical methods the structural and aerodynamic generic variables which are instrumental in establishing this unified theory are derived using affine transformations subsonic as well as supersonic flutter and divergence boundaries are determined the effects of air to panel mass ratios and midplane forces are examined viscous structural damping is found to be destabilizing there is a general agreement between the results of this analysis and those obtained for isotropic panels by previous investigators

computation of the singular values of toeplitz operators and the gap metric it is widely recognized that the computation of gap metric is equivalent to a certain two block h problem i e the gap is equal to the norm of a certain two block operator however it can also be characterized as the smallest singular value of a certain toeplitz operator this paper derives a simple computational method for finding such singular values and the gap between two plants by using a state space approach

model reduction via balanced realizations an extension and frequency weighting techniques two model reduction methods for discrete systems related to balanced realizations are described the first is a technique which utilizes the least controllable and observable subsystem in deriving a balanced discrete reduced order model for this technique as l sup infinity norm bound on the reduction error is given the second method is a frequency weighting technique for discrete and continuous time systems where the input normal or output normal realizations are modified to include a simple frequency weighting for this technique l sup infinity norm bounds on the weighted reduction errors are obtained

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

the general distance problem in h optimal control theory abstract the general distance problem that arises in h optimal control theory is considered transformations that reduce the general h problem to the general distance problem are reviewed and an iterative scheme called iteration is considered in which the general distance problem is reduced to a standard best approximation problem the iteration is viewed as a problem of finding the zero crossing of a function this function is shown to be continuous monotonically decreasing convex and bounded by some very simple functions these properties make it possible to obtain very rapid convergence of the iterative process the issues of model reduction and approximation in h synthesis will also be addressed

robust stabilization of delay systems by approximation of coprime factors given a delay system with transfer function g s h2 s h1 s where h1 s 01npi s e is and h2 s 02nqi s e is with 0 0 1 n1 0 0 n2 the pi being polynomials of degree i and i 0for i 0 and the qi polynomials of degree di 0 for each i the robust stabilization of a class of perturbed coprime factors of this system is considered asymptotic estimates are obtained based on recent results on the approximation and stabilization of normalized coprime factors an explicit formula is given for the normalized coprime factor stability margin for the case of a multivariable transfer function of the form g s e str s with r rational

performance bounds in cal h _ infty optimal control for stable siso plants with arbitrary relative degree this note deals with performance bounds for the h infin optimal control of discrete time lti plants the case studied corresponds to stable scalar plants with arbitrary relative degree but no finite non minimum phase zero by using nehari s theorem and a reformulation of the standard youla parameterization a closed form expression for the characteristic polynomial of the associated eigenvalue problem is obtained also we derive an analytic expression for the optimal h infin cost as a function of the plant relative degree

bibo stability robustness in the presence of coprime factor perturbations a necessary and sufficient condition for robustly stabilizing a family of plants described by perturbations of fixed coprime factors of a plant is given the computation of the largest stability margin is discussed via solving a nonsquare l sup 1 optimal control problem an algorithm for obtaining lower approximations of the minimum value of the optimization problem mu sub 0 is proposed this together with the standard algorithm which provides upper approximations allows mu sub 0 to be computed within any degree of accuracy

smooth weight optimization in ℋ loop shaping design abstract smooth variation in the magnitude response of weights facilitates loop shaping design as it prevents the cancellation of important modes of the system for example lightly damped poles zeros of flexible structures when the shaped plant is formed based on closed loop design specifications for accurate fitting of transfer functions to magnitude data smooth weights also allow low order transfer functions to be used when such smooth variations in their magnitude responses are computed point wise in frequency in this paper smoothness constraints for weights expressed as gradient constraints on a log scale in db decade which is intuitive from a design perspective are imposed in a weight optimization framework for loop shaping control this work builds on a lanzon weight optimization in loop shaping automatica 41 1 2005 1201 1208 where additional constraints are formulated in linear matrix inequality lmi form to cast a complete weight optimization framework the resulting algorithm thus maximizes the robust stability margin and simultaneously synthesizes smooth weights along with a stabilizing controller a numerical example is given to elucidate the efficacy of the smoothness constraints in loop shaping control

on hermitian solutions of the symmetric algebraic riccati equation the structure of the set of hermitian solutions of the matrix quadratic equation xdx xa a x c 0 is studied under the conditions that c c d is positive semidefinite and a d is stabilizable new features e g nonexistence of the minimal solution appear in contrast with the known case when a d is controllable

exact computation of infimum for a class of continuous time h optimal control problem with a nonzero direct feedthrough term from the disturbance input to the controlled output a noniterative method for the computation of infimum for a class of continuous time h optimal control problem is considered in this paper the problem formulation is fairly general and does not place any restrictions on any direct feedthrough terms of the given systems the method is applicable to systems where i the transfer function from the disturbance input to the measurement output is free of imaginary axis invariant zeros and left invertible and ii the transfer function from the control input to the controlled output of the given system is free of imaginary axis invariant zeros and right invertible the result presented in this paper is a continuation of the previous work of the author and his co workers chen et al 1992 in which the direct feedthrough term from the disturbance input to the measurement output of the given system are required to be zero

a mechanical model for the buckling of unilaterally constrained rectangular plates a mechanical model is described for the problem of buckling of unilaterally constrained finite rectangular plates due to the nature of the imposed constraint on the plate s lateral deflection w solving for the buckling load required the solution of a nonlinear partial differential equation in w while the plates were modeled along the lines of classical plate theory the nonlinearity arose from the fact that the plates were attached to nonlinear elastic foundations exhibiting a deformation sign dependent force displacement relationship this feature was introduced to model the unilateral constraint the influence of different boundary conditions material orthotropy and transverse load distributions was investigated for each case the weak form of the governing differential equation was solved via the galerkin s method investigations of the buckling loads of rectangular plates attached to such foundations and subjected to a uniform inplane stress field showed the validity of this approach for the cases investigated and compared to some previous exact results reported in the literature

coprime factorization and optimal control on the doubly infinite discrete time axis we study the problem of strongly coprime factorization over h infinity of the unit disc we give a necessary and sufficient condition for the existence of such a coprime factorization in terms of an optimal control problem over the doubly infinite discrete time axis in particular we show that an equivalent condition for the existence of such a coprime factorization is that both the control and filter algebraic riccati equation of an arbitrary realization have a solution in general unbounded and even nondensely defined and that a coupling condition involving these solutions is satisfied

normalised coprime factorizations for nonstrictly proper systems some earlier results of d g meyer and g f franklin see ibid vol ac 32 pp 227 8 march 1987 are extended and a procedure is derived for obtaining directly a normalized right coprime factorization of a system that may not be strictly proper

closed form solutions for nonlinear quasi unsteady transonic aerodynamics the existence of these solutions is established the full nonlinear unsteady velocity potential equations for an airfoil are considered a suitable mapping scheme is employed in transforming these full nonlinear equations into the hodograph plane the closed form solutions can be obtained for the nonlinear unsteady aerodynamic characteristics of an airfoil in potential flow dips were observed in the pressure distributions as the freestream mach number is varied it appears that there are finite optimum reduced frequencies for the pressure distributions

the parallel projection operators of a nonlinear feedback system abstract this paper defines and studies a pair of nonlinear parallel projection operators associated with a nonlinear feedback system these operators have been seen to play an important role in the robustness and design of linear systems especially in the theory of the gap metric the use of weighted gaps in control system design and glover mcfarlane loop shaping we show that the input output l 2 stability of a feedback system amounts to a coordinatization of the input and output spaces which is also equivalent to the existence of a pair of nonlinear parallel projection operators onto the graph of the plant and the inverse graph of the controller respectively these projections are shown to have equal norms whenever one of the feedback elements is linear a bound on this norm is given in the case of passive systems with unity negative feedback

controller approximation approaches for preserving h sub spl infin performance investigates the design of reduced order controllers using an h sub spl infin framework given a stabilizing controller which satisfies a prespecified level of closed loop h sub spl infin performance sufficient conditions are derived for another controller to be stabilizing and satisfy the same level of h sub spl infin performance such controllers are said to be p spl gamma admissible where p is the model of the plant under consideration and spl gamma is the required level of prespecified h sub spl infin performance the conditions are expressed as norm bounds on particular frequency weighted errors where the weights are selected to make a specific transfer function a contraction the design of reduced order p spl gamma admissible controllers is then formulated as a frequency weighted model reduction problem it is advantageous for the required weights to be large in some sense solutions which minimize either the trace or the determinant of the inverse weights are characterized we show that the procedure for minimizing the determinant of the inverse weights always gives a direction where the weights are the best possible to conclude we demonstrate by way of a numerical example that when used in conjunction with a combined model reduction convex optimization scheme the proposed design procedures are effective in substantially reducing controller complexity

a structured approximation problem with applications to frequency weighted model reduction the authors consider the problem of finding necessary and sufficient conditions for the existence of a q with no more than k poles in the closed left half plane such that the norm of a certain matrix

state space formulae for all stabilizing controllers that satisfy and h norm bound and relations to risk sensitivity abstract given a linear system all stabilizing controllers such that a specified closed loop transfer function has h norm less than a given scalar are parametrized this characterization involves the solution to two algebraic riccati equations each with the same order as the system and further gives gives feasible controllers also with this order the relationship to the risk sensitive lqg stochastic control problem is established giving an equivalence between robust and stochastic control

feedback stability under simultaneous gap metric uncertainties in plant and controller abstract the stability robustness of a feedback system is studied in this paper by assuming that the plant and the controller are subject to independent uncertainties and that the uncertainties are measured by the gap metric a fairly complete solution is obtained by exploring the trigonometric structure of the graphs of the plant and the controller

gap metric problem for mimo delay systems parametrization of all suboptimal controllers we consider the problem of robustness optimization in the gap metric for mimo systems with a scalar time delay we present an algorithm for the parametrization of all suboptimal controllers in our algorithm the aak theory plays the central role using this approach suboptimal controllers can be found by computing solutions to certain infinite rank operator equations we show that these equations can be solved numerically by modifying the two point boundary value problem approach reported earlier for the computation of optimal robustness radius we present a numerical example to illustrate the procedure

generic karman rostovstev plate equations in an affine space the karman rostovstev plate equations are affinely transformed into equations that depend on the generic plate constants generalized rigidity generalized extensional rigidity and generalized poisson ratio associated with the well known linearized orthotropic plate slab equations furthermore since the generalized extensional rigidity is a function of generalized rigidity and generalized poisson ratio only with generalized poisson ratio being a weak parameter the only fundamental constant is generalized rigidity this means that regardless of what analysis technique is used the results can be cast in terms of only one strong material parameter whose range is from zero to one the affine stretching is performed on the dependent variables as well as the independent variables so that the dependence of all quantities of interest on generalized rigidity and generalized poisson ratio are easily calculated e g even auxiliary quantities such as stress strain all weighted stress resultants etc some typical results static side load response collapse load limited by rib buckling are presented by modifying an approximate technique due to donnell 1976

a transformation approach to stochastic model reduction a new and direct approach to stochastic model reduction is developed the order reduction algorithm is obtained by establishing an equivalence between canonical correlation analysis and solutions to algebraic riccati equations also the concept of balanced stochastic realization bsr plays a fundamental role asymptotic stability of the reduced order realization is established and spectral domain interpretations for the bsr are given

distance measures for uncertain linear systems a general theory in this paper we propose a generic notion of distance between systems that can be used to measure discrepancy between open loop systems in a feedback sense under several uncertainty structures when the uncertainty structure is chosen to be four block or equivalently normalized coprime factor uncertainty then this generic distance measure reduces to the well known nu gap metric associated with this generic distance notion we also define a generic stability margin notion that allows us to give the distance measure a feedback interpretation by deriving generic robust stability and robust performance results the proposed distance notion and the corresponding results exploit a powerful generalization of the small gain theorem which handles perturbations in rfrl infin rather than only in rfrh infin when the uncertainty structure is fixed to one of the standard structures e g additive multiplicative inverse multiplicative coprime factor four block or any mixtures of the above we give a step by step procedure based on model validation ideas that shows how the generic notion of distance and the correspondingly generic winding number conditions can be reduced to simple formulae this work provides a unified framework that captures and embeds previous results in this area and also completes the picture by showing how other results of a similar nature can be obtained from the same framework the techniques used involve only basic linear algebra so they also provide a simplification of previous advanced proofs furthermore the various distance measures so created can be used for non conservative model embedding into the smallest uncertain family an illustrative example is also given that demonstrates the superior qualities above the nu gap metric of a particular distance measure obtained from this work in situations where the plant is lightly damped all systems considered in this paper are linear time invariant

a new set of invariants for linear systems application to reduced order compensator design a new set of invariants for linear systems weighting the contribution of each state component to the inherent closed loop lqg behavior of the system is presented together with applications to model order reduction and reduced order compensator design

two riccati formulae for the discrete time h control problem a two riccati state space approach to the discrete time h control problem is presented the main theoretical tool is the kalman szego popov yakubovich system written in adequate forms in order to reduce the problems to a standard form for which the solutions are easily obtained the necessity part of our main results uses a signed counterpart of the positivity theorem

supersonic flutter analysis of composite plates and shells a high precision doubly curved quadrilateral thin shell finite element is used the composite material property is included using classical lamination theory and the supersonic aerodynamic effect is included using linearized piston theory the normal modes approach is adopted results are presented for composite plates and curved panels and composite cylindrical and conical shells parametric studies concerning the effects of boundary conditions fiber orientation degree of orthotropy and flow angle on the flutter characteristics are presented

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a controller degree bound for h 0e control problems of the second kind this paper is a continuation of our work on mathcal h infty optimal control problems which may be embedded in the linear fractional configuration of fig 1 in two previous articles 19 20 a controller degree bound was established for problems in which both p 12 s and p 21 s are square problems of the first kind if the mcmillan degree of p s is n it was shown that there exist mathcal h infty optimal controllers with mcmillan degree no greater than n 1 here we switch our attention to problems of the second kind that is we allow p 12 s to have more rows than columns with p 21 s square or alternatively we allow p 21 s to have more columns than rows with p 12 s square our main result shows that the degree bound derived previously for problems of the first kind carries over to problems of the second kind without change in addition to the controller degree bound our analysis suggests a number of modifications which are easily made to cur

a characterization of all solutions to the four block general distance problem all solutions to the four block general distance problem which arises in h infty optimal control are characterized the procedure is to embed the original problem in an all pass matrix which is constructed it is then shown that part of this all pass matrix acts as a generator of all solutions special attention is given to the characterization of all optimal solutions by invoking a new descriptor characterization of all pass transfer functions as an application necessary and sufficient conditions are found for the existence of an h infty optimal controller following that a descriptor representation of all solutions is derived

eigenvalue similarity rules for symmetric cross ply laminated plates as recently demonstrated the stability and vibration problems of orthotropic plates can be efficaciously treated in suitable affine spaces and similarity rules independent of individual problem boundary conditions can be formulated and very effectively used in analysis and design the present work extends these concepts to symmetric cross ply laminated plates by employing the affine stretching process utilizing some previous material constant definitions due to tsai and generalizing the existing similarity rules as examples similarity rules are employed to find the frequency spectrum of a cross ply symmetric plate from the corresponding isotropic solution and a similarity solution is presented for the vibration of an axially loaded simply supported cross ply symmetric plate buckling similarity rules are also presented