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

controller reduction with error performance continuous and discrete time cases this paper is concerned with the problem of controller reduction for linear systems both continuous and discrete time cases are considered with necessary and sufficient conditions obtained for the existence of desired reduced order controllers in solving this problem two approaches are presented the first approach is based on the projection lemma where the admissible controllers can be parameterized after a set of conditions are satisfied and the second one directly incorporates the controller matrices to be determined into a set of conditions by introducing new techniques and thus no parameterization procedure is needed these necessary and sufficient conditions are formulated in terms of linear matrix inequalities lmis plus some equality constraints since these conditions are not convex the cone complementarity linearization ccl idea is exploited to cast them into sequential minimization problems subject to lmi constraints which can be readily solved by standard numerical software a num

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

singular monge ampere foliations this paper generalizes results of lempert and szoke on the structure of the singular set of a solution of the homogeneous monge ampere equation on a stein manifold their a priori assumption that the singular set has maximum dimension is shown to be a consequence of regularity of the solution in addition their requirement that the square of the solution be c 3 everywhere is replaced by a smoothness condition on the blowup of the singular set under these conditions the singular set is shown to inherit a finsler metric which in the real analytic case uniquely determines the solution of the monge ampere equation these results are proved using techniques from contact geometry

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

controller reduction for linear parameter varying systems with a priori bounds in this paper we develop a controller reduction procedure for linear parameter varying lpv systems the method uses synthesis riccati inequalities for the normalized robust stabilization problem as a basis for the approximation the technique provides a priori error bounds which are used to obtain closed loop stability conditions and performance degradation level we also generalize the relative model reduction method to lpv systems and give an energy interpretation to the controller reduction procedure to illustrate the method a reduced order controller is synthesized and tested on a nonlinear missile model

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

reduction of controller fragility by pole sensitivity minimization presents a method for the reduction of controller fragility the method is based on the sensitivity of closed loop poles to perturbations in the controller parameters by means of a state space parameterization of the controller the closed loop pole sensitivity can be reduced a controller fragility measure based on the closed loop pole sensitivity is proposed conditions for the optimal state space realization of the controller are presented along with a numerical method for obtaining the solution

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

controller reduction by h sub infinity balanced truncation h sub infinity balanced truncation may be used to obtain reduced order plants or controllers the plant possibly unstable is compensated using a particular robustly stabilizing controller the two riccati equations involved are then used to define a set of closed loop input output invariants called the h sub infinity characteristic values that part of the plant or controller corresponding to small h sub infinity characteristic values is discarded to give a reduced order plant or controller by exploiting an intimate connection with coprime factorization a simple a priori test is derived for the ability of such a reduced order controller to stabilize the full order plant the performance of the resulting closed loop may also be bounded a priori i e in terms of the prespecified level of robustness and the discarded h sub infinity characteristic values

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

dissipativity preserving model reduction for large scale distributed control systems we propose a dissipativity preserving structured model reduction method for distributed control systems as a fundamental tool to develop structured model reduction we first establish dissipativity preserving model reduction for general linear systems on the basis of a singular perturbation approximation to this end by deriving a tractable expression of singular perturbation models we characterize dissipativity preservation in terms of a projection like transformation of storage functions and we show that the resultant approximation error is relevant to the sum of neglected eigenvalues of an index matrix next utilizing this dissipativity preserving model reduction we develop a structured controller reduction method for distributed control systems the major significance of this method is to preserve the spatial distribution of dissipative controllers and to provide an a priori bound for the performance degradation of closed loop systems in terms of the cal h 2 norm the efficiency of the proposed method is verified through a numerical example of vibration suppression control for interconnected second order systems

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

frequency domain uncertainty and the graph topology a new metric on linear time invariant systems is defined this metric is no greater than the gap metric and is in fact the smallest metric for which a certain robust stabilization result holds unlike other known metrics which induce the graph topology it has a clear frequency response interpretation this allows questions regarding robustness in the face of parametric uncertainty to be considered in terms of this metric

balanced performance preserving controller reduction two methods for controller reduction based on the closed loop system performance preserving criterion are proposed they are the closed loop block balanced and the closed loop structurally balanced truncating methods reduced order controllers are obtained directly by solving the closed loop system reduction problem with performance preserving weights

normalized h sub spl infin controller reduction with a priori error bounds this paper describes an approach to the reduction of controllers for the normalized coprime factor robustness problem as well as the normalized h sub spl infin problem it is shown that a relative error approximation of a coprime factor representation of any suboptimal controller leads to a stability guarantee and an upper bound on the performance degradation when the reduced order controller is implemented when the approximation is performed on the controller generator guaranteed a priori stability and performance bounds are obtained in terms of the synthesis riccati equation solutions of the normalized h sub spl infin control problems

fixed structure pid controller design for standard h control problem a design method of multivariable pid controllers that satisfy the performance index of the standard h control problem and linear constraints on the control structure is proposed nonconvex convex matrix inequalities obtained from the h norm condition for each frequency are approximated by linear matrix inequalities respectively a sequence of pid gains is obtained by solving a feasibility problem of the linear matrix inequalities starting from a stabilizing pid gain iteratively the pid gain converges to a local optimal solution and the sequence of the h norm bound is monotonically nonincreasing

the structure of legendre foliations the local and global structure of legendre foliations of contact manifolds is analysed the main invariant of a legendre foliation is shown to be a quadratic form on the tangent bundle to the foliation the fundamental quadratic form the equivalence problem is solved in the case when the fundamental quadratic form is nondegenerate and a generalization of chern s solution to the equivalence problem for finsler manifolds is obtained a normal form for legendre foliations is given which is closely related to weinstein s structure theorem for lagrangian foliations it is shown that every compact simply connected leaf of a legendre foliation is diffeomorphic to a sphere

model reduction for nonlinear systems with incremental gain or passivity properties in this paper model reduction techniques for a class of nonlinear systems are proposed specifically nonlinear systems are considered that can be decomposed as the feedback interconnection of a high order linear subsystem and a nonlinear subsystem of relatively low order allowing for the application of well developed reduction techniques for linear systems in this setting conditions are given under which internal stability as well as passivity or a bound on the l 2 gain are preserved for the reduced order nonlinear model additionally a priori error bounds are given in the derivation of the error bound an incremental gain or incremental passivity property of the nonlinear subsystem is shown to be instrumental additionally the techniques developed in this paper are applied in the scope of controller reduction as is illustrated by means of an industrial temperature control benchmark example

abstract cauchy problems for quasilinear operators whose domains are not necessarily dense or constant abstract the solvability of the abstract cauchy problem for the quasilinear evolution equation u t a u t u t for t 0 and u 0 u 0 d is discussed here a w w y is a family of closed linear operators in a real banach space x such that y d a w y for w y y is another banach space which is continuously embedded in x and d is a closed subset of y the existence and uniqueness of c 1 solutions to the cauchy problem is proved without assuming that y is dense in x or d a w is independent of w the abstract result is applied to obtain an l 1 valued c 1 solution to a size structured population model

normalized coprime robust stability and performance guarantees for reduced order controllers constructing a reduced order controller from a high dimensional plant is commonly necessary the reduce then design approach constructs the controller from a reduced order plant design then reduce reduces a full order controller in both cases we present sufficient conditions for the full order plant and reduced order controller to achieve closed loop stability or performance these conditions motivated primarily by the gap metric reveal model reduction orders that guarantee stability or performance the control of the linearized ginzburg landau system provides validation

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

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

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

upper and lower bounds for approximation in the gap metric upper and lower bounds for the closest approximant of degree k n in the gap metric to a plant of degree n are obtained the bounds are expressed in terms of the singular values of two hankel operators defined in terms of the symbol of the graph of the plant the question of robust stability and performance of feedback systems is examined in the context of approximation of plant and controller in the gap metric

feedback properties of multivariable systems the role and use of the return difference matrix for linear time invariant multivariable feedback systems the feedback properties of plant disturbance attenuation sensor noise response stability margins and sensitivity to plant and sensor variation are quantitatively related to the bode magnitude versus frequency plots of the singular values of the return difference matrix i l and of the associated inverse return difference matrix i l 1 implied fundamental limits of feedback performance are quantitatively described and design tradeoffs are discussed the penalty function in the stochastic linear quadratic gaussian lqg optimal control problem is found to be a weighted sum of the singular values with the weights determined by the quadratic cost and noise intensity matrices this enables systematic tuning of lqg cost and noise matrices so that the resulting optimal return difference and inversereturn difference meet inequality constraints derived from design specifications on feedback properties the theory has been used to synthesize a multivariable automatic controller for the longitudinal dynamics of an advanced fighter aircraft