colored group field theory random matrix models generalize to group field theories gft whose feynman graphs are dual to higher dimensional topological spaces the perturbative development of the usual gft s is rather involved combinatorially and plagued by topological singularities which we discuss in great detail in this paper thus very difficult to control and unsatisfactory

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

the 1 n expansion of tensor models beyond perturbation theory we analyze in full mathematical rigor the most general quartically perturbed invariant probability measure for a random tensor using a version of the loop vertex expansion which we call the mixed expansion we show that the cumulants write as explicit series in 1 n plus bounded rest terms the mixed expansion recasts the problem of determining the subleading corrections in 1 n into a simple combinatorial problem of counting trees decorated by a finite number of loop edges

closed formulae for a parametric mixed sensitivity problem for pritchard salamon systems the parametric mixed sensitivity problem is posed for the class of irrational transfer matrices with a realization as a pritchard salamon system an exact solution is obtained in terms of two uncoupled standard riccati equations

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

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

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

admissibility of unbounded control operators for linear systems described by dot x t ax t bu t where a generates a semigroup on the state space x and b is an unbounded operator some necessary as well as some sufficient conditions are given for b to be admissible i e for any t the state x t should be in x and should depend continuously on the input u in l p this approach begins with an axiomatic description of such a system in terms of a functional equation the results are applied to the wave equation on a bounded interval

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

formulas for hankel singular values and vectors for a class of input delay systems this paper gives explicit formulas for hankel singular values and vectors for a class of multiple input delay systems the class consists of systems whose gramian is reducible to an integral operator with semi separable kernel this structure is exploited in deriving a transcendental equation for singular values

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

robust controller design for delay systems in the gap metric a robust stabilization problem in the gap metric is discussed for a system with delays in control the authors derive a design procedure of robust controllers based on finite dimensional ricatti equations and it is shown that the resulting controller has a structure of observer based predictive action by employing completing the square argument a game theoretic interpretation is also provided on the trade off between the initial uncertainties and attenuating the disturbance caused by plant uncertainties

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

robust control with respect to coprime factors of infinite dimensional positive real systems it is shown that infinite dimensional positive real systems can be robustly stabilized with respect to coprime factor perturbations with a robustness margin of at least 1 square root 2 this result is applied to dissipative colocated systems which arise in pde models of flexible structures

closed form boundary state feedbacks for a class of 1 d partial integro differential equations in this paper a problem of boundary stabilization of a class of linear parabolic partial integro differential equations p i des in one dimension is considered using the method of backstepping avoiding spatial discretization required in previous efforts the problem is formulated as a design of an integral operator whose kernel is required to satisfy a hyperbolic p i de the kernel p i de is then converted into an equivalent integral equation and by applying the method of successive approximations the equation s well posedness and the kernel s smoothness are established it is shown how to extend this approach to design optimally stabilizing controllers an adaptation mechanism is developed to reduce the conservativeness of the inverse optimal controller and the performance bounds are derived for a broad range of physically motivated special cases feedback laws are constructed explicitly and the closed loop solutions are found in closed form a numerical scheme for the kernel p i de is proposed its numerical effort compares favorably with that associated with operator riccati equations

robust stabilization bibo stability distance notions and robustness optimization abstract this paper studies robust stabilization of both linear shift invariant causal systems in an lp setting and linear time invariant causal continuous time systems in an lp p 1 or setting two key technical results in the paper establish the existence of lp and lp stable normalized coprime factorizations for discrete time and continuous time systems respectively which have coprime factorizations as lp and lp stable operators several distance measures for systems are then introduced including the graph metric the function the gap between the graphs of the systems and the projection gap it is shown that these distance measures lead to the weakest convergence notions for systems for which closed loop stability is a robust property the function can be computed using the dahleh pearson theory for l1 l1 optimal control robustness optimization in a directed p function is shown to be closely related to robustness optimization for bibo stable normalized coprime factor perturbations this result connects the stability margin of dahleh for coprime factor perturbations to the function these considerations are further supported by a robustness result in terms of the projection gap

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

robust stabilization of infinite dimensional systems by finite dimensional controllers abstract the problem of robustly stabilizing an infinite dimensional system with transfer function g subject to an additive perturbation is considered it is assumed that g b 0 of systems introduced by callier and desoer 3 the perturbation satisfies w1 w2 c now write w1gw2 g1 g1 where g1 is rational and totally unstable and g2 is stable generalizing the finite dimensional results of glover 12 this family of perturbed systems is shown to be stabilizable if and only if min g 1 the smallest hankel singular value of g 1 a finite dimensional stabilizing controller is then given by k w 2 k 1 i g 2 k 1 1 w 1 where 2 is a rational approximation of g2 such that g 2 g 2 min g 1 and k1 robustly stabilizes g1 to margin the feedback system g k will then be stable if w1 w2

spline approximation for retarded systems and the riccati equation the purpose of this paper is to introduce a new spline approximation scheme for retarded functional differential equations the special feature of this approximation scheme is that it preserves the product space structure of retarded systems and approximates the adjoint semigroup in a strong sense these facts guarantee the convergence of the solution operators for the differential riccati equation in a strong sense numerical findings indicate a significant improvement in the convergence behaviour over both the averaging and the previous spline approximation scheme

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

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

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

spectral factorization and lq optimal regulation for multivariable distributed systems a necessary and sufficient condition is proved for the existence of a bistable spectral factor with entries in the distributed proper stable transfer function algebra x1d49c in the context of distributed multivariable convolution systems with no delays a by product is the existence of a normalized coprime fraction of the transfer function of such a possibly unstable system with entries in the algebra of fractions over x1d49c we next study semigroup state space systems sgb with bounded sensing and control having a transfer function with entries in and consider its standard lq optimal regulation problem having an optimal state feedback operator k0 for a system sgb a formula is given relating any spectral factor of a transfer function coprime fraction power spectral density to k0 a by product is the description of any normalized coprime fraction of the transfer function in terms of k0 finally we describe an alternative way of finding the solution operator k0 of the lq problem using spectral factor

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

diagrammatics of a colored syk model and of an syk like tensor model leading and next to leading orders the sachdev ye kitaev syk model is a model of q interacting fermions gross and rosenhaus have proposed a generalization of the syk model which involves fermions with different flavors in terms of feynman graphs those flavors are reminiscent of the colors used in random tensor theory this gives us the opportunity to apply some modern yet elementary tools developed in the context of random tensors to one particular instance of such colored syk models we illustrate our method by identifying all diagrams which contribute to the leading and next to leading orders of the 2 point and 4 point functions in the large n expansion and argue that our method can be further applied if necessary in the second part we focus on the recently introduced gurau witten tensor model and also extract the leading and next to leading orders of the 2 point and 4 point functions this analysis turns out to be remarkably more involved than in the colored syk model

dynamic stabilization of regular linear systems we consider a general class of infinite dimensional linear systems called regular linear systems for which convenient representations are known to exist both in time and in frequency domain for this class of systems we investigate the concepts of stabilizability and detectability in particular their invariance under feedback and their relationship to exponential stability we introduce two concepts of dynamic stabilization the first formulated as usual with the plant and the controller connected in feedback and the second with two feedback loops even for finite dimensional systems the second concept stabilization with an internal loop in the controller is more general we argue that the more general concept is the natural one and we derive sufficient conditions under which an observer based stabilizing controller with an internal loop can be constructed

observability of nontrivial small solutions for neutral systems abstract in this paper we introduce the concept of observability of nontrivial small solutions for neutral functional differential equations with a single point delay and prove a matrix type criterion

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

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

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

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

optimal robustness in the gap metric the application of the gap metric to robust stabilization of feedback systems is considered in particular a solution to the problem of robustness optimization in the gap metric is presented the problem of robust stabilization under simultaneous plant controller perturbations is addressed and the least amount of combined plant controller uncertainty measured by the gap metric that can cause instability of a nominally stable feedback system is determined included are a detailed summary of the main properties of the gap metric and the introduction of a dual metric called the t gap metric a key contribution of this study is to show that the problem of robustness optimization in the gap metric is equivalent to robustness optimization for normalized coprime factor perturbations this settles the question as to whether maximizing allowable coprime factor uncertainty corresponds to tolerating the largest ball of uncertainty in a well defined metric

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

the linear quadratic control problem for infinite dimensional systems with unbounded input and output operators this paper establishes a general semigroup framework for solving quadratic control problems with infinite dimensional state space and unbounded input and output operators

robust stabilizability of normalized coprime factors the infinite dimensional case the problem of robustly stabilizing a linear system subject to h bounded perturbations in the numerator and the denominator of its normalized left coprime factorization is considered for a class of infinite dimensional systems this class has possible unbounded finite rank input and output operators which include many delay and distributed systems the optimal stability margin is expressed in terms of the solutions of the control and filter algebraic riccati equations

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

realisation and approximation of linear infinite dimensional systems with error bounds the class of linear infinite dimensional systems with finite dimensional inputs and outputs whose impulse response h satisfies h in l 1 cap l 2 0 infty mathbb c p times m and induces a nuclear hankel operator is said to be of nuclear type for this class of systems it is shown that balanced or output normal realisations always exist and their truncations converge to the original system in various topologies furthermore explicit l infty bounds on the transfer function errors l 1 and l 2 bounds on the impulse response errors and hilbert schmidt and nuclear bounds on the hankel operator errors are obtained these truncations also generate an approximating sequence to the optimal hankel norm approximations to the original system and various error bounds of these approximants are deduced

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 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

coprime factorization for regular linear systems mild sufficient conditions are given for the existence of a doubly coprime factorization of the transfer function of a regular linear system as well as formulae for such a factorization the results are illustrated by two examples of delay systems one of which has infinitely many unstable poles

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

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

robustly stabilizing controllers for dissipative infinite dimensional systems with collocated actuators and sensors we derive formulas for normalized doubly coprime factorizations over h for a class of nonexponentially stabilizable systems using these results we solve the problem of robust stabilization with respect to normalized coprime factor perturbations for a class of infinite dimensional systems s a b b d with finite rank inputs and outputs and dissipative operator a these systems are not exponentially stabilizable we give a parameterization of controllers that achieve a given robustness margin

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

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

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

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

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

method for reducing the search complexity in analysis by synthesis coding a method of encoding speech includes a limited search of a tree code excitation codebook with a closed loop gain calculation for each test path under consideration the gain calculation occurs when minimizing an error distance measurement between a synthetic signal defined by each test path being considered and the appropriate speech signal by optimizing a scaling factor of the synthetic signal the encoding method achieves a significant reduction in computational complexity with minimal loss of performance

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

approximation of unstable infinite dimensional systems using coprime factors abstract the effectiveness of comprime factor techniques in l 2 and l model reduction of unstable linear systems is analysed asymptotic estimates are given of the achievable error in the stable and unstable parts of the approximate system measured in a number of different norms some involving the associated hankel operators the chordal metric is introduced as a measure of approximation and is shown to yield the graph topology finally it is deduced that asymptotically optimal l 2 and l convergence rates can be obtained for a large class of unstable systems

the 1 n expansion of tensor models with two symmetric tensors it is well known that tensor models for a tensor with no symmetry admit a 1 n expansion dominated by melonic graphs this result relies crucially on identifying jackets which are globally defined ribbon graphs embedded in the tensor graph in contrast no result of this kind has so far been established for symmetric tensors because global jackets do not exist in this paper we introduce a new approach to the 1 n expansion in tensor models adapted to symmetric tensors in particular we do not use any global structure like the jackets we prove that for any rank d a tensor model with two symmetric tensors and interactions the complete graph k d 1 admits a 1 n expansion dominated by melonic graphs

method of and device for quantizing excitation gains in speech coders based on analysis by synthesis techniques an optimum excitation signal for each subframe is determined in a speech coder based on analysis by synthesis techniques and operating on frames of samples divided into a number of subframes the excitation signal includes a shape contribution innovation and an amplitude contribution gain which are quantized separately a circuit it for gain quantization includes means qu for determining a gain index for each subframe a comparison logic network cfr for detecting the maximum value taken by the gain index in the frame and means for computing a normalized index for each subframe as a difference between the maximum index and the gain index relevant to that subframe the coded signal includes the coded values of the maximum index and of the normalized indexes as information on the gain relevant to a frame

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

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

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

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

speech coding apparatus and method using classification rules a speech coding apparatus and method uses classification rules to code an utterance while consuming fewer computing resources the value of at least one feature of an utterance is measured during each of a series of successive time intervals to produce a series of feature vector signals representing the feature values the classification rules comprise at least first and second sets of classification rules the first set of classification rules map each feature vector signal from a set of all possible feature vector signals to exactly one of at least two disjoint subsets of feature vector signals the second set of classification rules map each feature vector signal in a subset of feature vector signals to exactly one of at least two different classes of prototype vector signals each class contains a plurality of prototype vector signals according to the classification rules a first feature vector signal is mapped to a first class of prototype vector signals the closeness of the feature value of the first feature vector signal is compared to the parameter values of only the prototype vector signals in the first class of prototype vector signals to obtain prototype match scores for the first feature vector signal and each prototype vector signal in the first class at least the identification value of at least the prototype vector signal having the best prototype match score is output as a coded utterance representation signal of the first feature vector signal

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

robustly stabilizing controllers with respect to left coprime factor perturbations for infinite dimensional linear systems we solve the problem of robust stabilization with respect to left coprime factor perturbations for a large class of irrational transfer functions with a realization in terms of unbounded operators the key condition is that the associated dynamical system and its dual should satisfy a finite cost condition so that two optimal cost operators exist we obtain an explicit parameterization of infinitely many robustly stabilizing controllers in terms of these optimal cost operators and the generating operators of the realization

spectral properties of finite dimensional perturbed linear semigroups abstract many interesting cauchy problems arising in applications can be treated as finite dimensional perturbations of relatively simple problems in most applications these perturbations take place via boundary conditions it is the aim of this paper to investigate such cauchy problems and especially the spectral properties of the corresponding generators some general results are obtained that unify several well known results for functional differential equations and age dependent population dynamics applications to control problems as well as some nonlinear generalizations will appear in forthcoming papers

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

infinite dimensional linear systems with unbounded control and observation a functional analytic approach abstract the object of these notes is to develop a unifying framework for the functional analytic representation of infinite dimensional linear systems with unbounded input and output operators on the basis of the general approach new results are derived on the wellposedness of feedback systems and on the linear quadratic control problem the implications of the theory for large classes of functional and partial differential equations are discussed in detail keywords representation of infinite dimensional systems semigroups boundary control feedback linear quadratic control

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

analytic system problems and j lossless coprime factorization for infinite dimensional linear systems abstract this paper extends the coprime factorization approach to the synthesis of internally stabilizing controllers satisfying an h norm bound to a class of systems with irrational transfer matrices using the coprime factorization description the h control problem can be reduced to two stable analytic system problems such problems have solutions if and only if a certain j lossless factorization exists the full h synthesis problem is shown to be equivalent to the solution of two nested j lossless factorizations if the irrational transfer matrix has a state space realization then the known state space formulas for the h control problem may be recovered using the relationship between j lossless factorizations and solutions of riccati equations however the results derived here are valid for a larger class of infinite dimensional systems

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

depth first algebraic codebook search for fast coding of speech a codebook is searched in view of encoding a sound signal this codebook consists of a set of codevectors each of 40 positions and comprising n non zero amplitude pulses assignable to predetermined valid positions to reduce the search complexity a depth first search is used which involves a tree structure with levels ordered from 1 through m a path building operation takes place at each level whereby a candidate path from the previous level is extended by choosing a predetermined number of new pulses and selecting valid positions for said new pulses in accordance with a given pulse order rule and a given selection criterion a path originated at the first level and extended by the path building operations of subsequent levels determines the respective positions of the n non zero amplitude pulse of a candidate codevector use of a signal based pulse position likelihood estimate during the first few levels enable initial pulse screening to start the search on favorable conditions a selection criterion based on maximizing a ratio is used to assess the progress and to choose the best one among competing candidate codevectors

backstepping observers for a class of parabolic pdes in this paper we design exponentially convergent observers for a class of parabolic partial integro differential equations p i des with only boundary sensing available the problem is posed as a problem of designing an invertible coordinate transformation of the observer error system into an exponentially stable target system observer gain output injection function is shown to satisfy a well posed hyperbolic pde that is closely related to the hyperbolic pde governing backstepping control gain for the state feedback problem for several physically relevant problems the observer gains are obtained in closed form the observer gains are then used for an output feedback design in both collocated and anti collocated setting of sensor and actuator the order of the resulting compensator can be substantially lowered without affecting stability explicit solutions of a closed loop system are found in particular cases

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

linear state space systems in infinite dimensional space the role and characterization of joint stabilizability detectability several fundamental results from the theory of linear state space systems in finite dimensional space are extended to encompass a class of linear state space systems in infinite dimensional space the results treated are those pertaining to the relationship between input output and internal stability the problem of dynamic output feedback stabilization and the concept of joint stabilizability detectability a complete structural characterization of jointly stabilizable detectable systems is obtained the generalized theory applies to a large class of linear state space systems assuming only that i the evolution of the state is governed by a strongly continuous semigroup of bounded linear operators ii the state space is hilbert space iii the input and output spaces are finite dimensional and iv the sensing and control operators are bounded general conclusions regarding the fundamental structure of control theoretic problems in infinite dimensional space can be drawn from these results