a state space formulation for optimal hankel norm approximations the optimal hankel norm approximation problem studied in 1 is reformulated in a state space setting the crucial extension theorem is reestablished in this framework the minimal degree optimal approximation is then derived in terms of state space parameters

an optimal controller reduction by covariance equivalent realizations based upon the theory of covariance equivalent realizations 1 this note presents a technique to obtain reduced order controllers via an optimal projection of full order lqg controllers this projection is optimal in the sense that it is a weighted least square approximation to a covariance equivalent projection of the full order optimal closed loop system the technique which is capable of extracting optimal minimal lqg controllers whenever they exist is also shown to yield controllers that are equivalent to those obtained by component cost analysis 12

on direct versus indirect methods for reduced order controller design comparisons are obtained between direct parameter optimization and indirect controller reduction methods in reduced order controller design in particular the authors compare the controller canonical correlation coefficients method of c de villemagne and r e skelton 1988 to the optimal projection theory of d c hyland and d s bernstein 1984 it is noted that suboptimal methods may in certain cases be simpler to implement than optimal methods thus the choice of direct versus indirect methods can be viewed as a tradeoff between design simplicity and actual control system performance

balancing of lossless and passive systems different balancing techniques are applied to lossless nonlinear systems with open loop balancing applied to their scattering representation it is shown that they all lead to the same result the pair of to be balanced functions is given by two copies of the physical energy function yielding thus no information about the relative importance of the state components in a balanced realization in particular in the linear lossless case all balancing singular values and similarity invariants are equal to one this result is extended to general passive systems in which case the to be balanced functions are ordered into a single sequence of inequalities and the similarity invariants are all less than or equal to one

balanced forms canonicity and parameterization we show that balanced forms are canonical and continuously parameterize the set of systems of a given order which are asymptotically stable minimal and have distinct singular values the advantage of the balanced canonical form over other classical canonical forms lies precisely in the guaranteed stability and minimality of the parameterization we also derive formulas allowing the synthesis of systems using gradient methods in balanced form

a projection approach to covariance equivalent realizations of discrete systems covariance equivalent realization theory has been used recently in continuous systems for model reduction 1 4 and controller reduction 2 5 in model reduction this technique produces a reduced order model that matches q 1 output covariances and q markov parameters of the full order model in controller reduction it produces a reduced controller that is close to matching q 1 output covariances of the full order controller and q markov parameters of the closed loop system for discrete systems a method was devised to produce a reduced order model that matches the q 1 covariances 6 but not any markov parameters this method requires a factorization to obtain the input matrix and since the dimension of this matrix factor depends upon rank properties not known a priori this method may not maintain the original dimension of the input vector hence this method 6 is obviously not suitable for controller reduction a new projection method is described that matches q covariances and q markov parameters of the original system while maintaining the correct dimension of the input vector

reduction of state variables based on regulation and filtering performances abstract this paper provides a principal component analysis of linear discrete time systems on the basis of optimal control and estimation the analysis is to reveal the important state components which remain necessary for reducing performance degradation under dimensional constraints on control and estimation laws the trade off relations between the dimension and performance degradation are expressed as system invariants representing the importance of each principal component which are characterized as the eigenvalues of matrices depending on the solutions of both lyapunov and riccati equations based on the analysis the paper also provides model reduction techniques for the systems generating the optimal input and estimate with the desirable properties of stability reachability and observability being preserved in the reduced systems

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

on dual algebraic riccati equations a method is presented to solve dual algebraic riccati equations it is shown that only one schur decomposition of the hamiltonian matrix is necessary to solve the dual equations

bundle reduction and the alignment distance on spaces of state space lti systems this paper introduces a large class of differential geometric distances between finite dimensional linear dynamical systems collectively called the alignment distance contrary to the existing distances the alignment distance is based on the state space description of dynamical systems and is defined on the manifolds of systems of fixed order and fixed input output dimension under a matrix rank constraint e g minimality controllability or observability while the quotient topology and principal fiber bundle structure associated with such manifolds have been known since the early days of modern control theory distances natural to this structure have not been studied the starting point for defining such a distance is to identify a linear system of order n with its equivalence class of state space realizations all related by the so called similarity action i e state space change of basis under gl n the lie group of nonsingular n times n matrices the main idea of the alignment distance is to first find the best state space change of basis that brings a realization of a system as close as possible to a realization of another system the alignment step and then compare the aligned realizations a direct implementation of this idea due to noncompactness of gl n is complicated however using the notion of reduction of the structure group of a principal bundle we show that the change of basis can be restricted to an orthogonal change of basis provided one uses realizations in a reduced subbundle this key observation brings about significant computational benefits as a technical contribution possibly of independent interest we show that several forms of realization balancing available in the control literature have differential geometric significance and are indeed examples of reducing the structure group from gl n to its subgroup of orthogonal matrices o n the alignment distance can be defined for stable and unstable systems discrete or continuous time and stochastic systems

principal component analysis in linear systems controllability observability and model reduction kalman s minimal realization theory involves geometric objects controllable unobservable subspaces which are subject to structural instability specifically arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces this situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization structural instability associated with geometric theories is not unique to control it arises in the theory of linear equations as well in this setting the computational problems have been studied for decades and excellent tools have been developed for coping with the situation one of the main goals of this paper is to call attention to principal component analysis hotelling 1933 and an algorithm golub and reinsch 1970 for computing the singular value decompositon of a matrix together they form a powerful tool for coping with structural instability in dynamic systems as developed in this paper principal component analysis is a technique for analyzing signals singular value decomposition provides the computational machinery for this reason kalman s minimal realization theory is recast in terms of responses to injected signals application of the signal analysis to controllability and observability leads to a coordinate system in which the internally balanced model has special properties for asymptotically stable systems this yields working approximations of x c x bar o the controllable and unobservable subspaces it is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces

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

on the bilinear transformation of lqg balanced realizations abstract this contribution shows that some useful properties of lqg balanced realizations of continuous time systems are preserved by applying bilinear transformations this fact suggests that lqg balancing of discrete systems and its approximation can be performed by using the typical schemes successfully adopted for continuous time systems in the paper the approximation of discrete linear time invariant systems by using the singular perturbation approximation approach for lqg balanced realizations is analyzed showing its usefulness in approximating discrete models including unstable ones for closed loop aims

state space design of low order stabilizers an algorithm is presented for stabilizing a linear multivariable system with a controller of fixed dynamic order this is an output feedback stabilization problem the algorithm attempts to solve this by a sequence of approximate pole assignment problems the approximation is driven by the optimization of an objective function consisting of a weighted sum of the condition number of the closed loop eigenvectors and the norm of the difference between the computed and actual equivalent state feedback controls

functional models in linear algebra abstract this paper examines the role of functional models in linear algebra and some related areas of application notably linear systems theory this approach adopts an intermidiate level of abstraction somewhere between abstract module theory on the one hand and matrix theory on the other it is a powerful method which simplifies many proofs provides insights and illuminates the less obvious connections between linear algebra and operator theory approximation theory moment problems interpolation and control theory

minimal balanced realization of transfer function matrices using markov parameters a new relation between the controllability observability matrix and gramian for any stable realization is used to enable the development of an algorithm for obtaining a minimal balanced realization directly from the transfer function matrix the algorithm employs the markov parameters and a positive definite symmetric matrix which is composed solely from the coefficients of the least common denominator polynomial of the transfer function matrix

technical communique improvement of system order reduction via balancing using the method of singular perturbations in this paper we first present several alternative techniques for system order reduction based on system balancing by employing the method of singular perturbations the techniques obtained have the same robustness accuracy evaluated with respect to the h norm of the reduced order system as the two techniques existing in the control literature known as the direct truncation and balancing residualization methods the newly proposed techniques preserve the exact dc gain as the original system and produce very good to excellent accuracy at low and medium frequencies we also present a simple formula that fixes the dc gain for the direct truncation method in the second part of the paper we propose how to perform order reduction of a balanced system using theory of singular perturbations that can produce very good accuracy at high frequencies particularly for systems that have lightly damped highly oscillatory modes the result obtained for such kind of systems requires that the modes with higher hankel singular values should be neglected and those with smaller hankel singular values should be retained this is a counter intuitive result in the spirit of h theory but an intuitive result in the spirit of theory of singular perturbations the second part of the paper raises an important question about validity of the system robustness estimate based on the h norm for linear systems with lightly damped highly oscillatory modes

a functional approach to lqg balancing the paper has as its theme a circle of problems related to lqg balancing with a special emphasis on the related problems of model reduction and robust stabilization the aim is to present a unified point of view to both previously described problems the unification is achieved by focusing on the study of three functions and the relationships between them and the various operators that are associated with these functions with an arbitrary transfer function g one can associate canonically two objects the conjugate inner function which is based on the nrcf and the r which is associated with the lqg controller of the function g the approach that is taken is functional emphasizing operators a balanced realization of a stable g arises as a matrix representation of the shift realization with respect to a basis made out of suitably normalized hankel singular vectors a similar result holds for lqg balanced realizations here the underlying hankel operator we study is h r where r u m v n and u v s

lqg balancing for continuous time infinite dimensional systems in this paper we study the existence of linear quadratic gaussian lqg balanced realizations for continuous time infinite dimensional systems lqg balanced realizations are those for which the optimal cost operator for the system and its dual system are equal and diagonal the class of systems we consider is that of distributional resolvent linear systems which includes well posed linear systems as a subclass we prove the existence of lqg balanced realizations under a finite cost condition for both the system and its dual system we also show that an lqg balanced realization of a well posed transfer function is well posed we further show that approximately controllable and observable lqg balanced realizations are unique up to a unitary state space transformation finally we show that the spectrum of the product of the optimal cost operator of a system and its dual system is independent of the particular realization our method of proof shows the connections with coprime factorizations lyapunov balanced realizations and discrete time systems the main reason for studying lqg balanced realizations is that truncated lqg balanced realizations provide a good approximation of the original system we show that under certain conditions this is also true in the infinite dimensional case by proving an error bound in the gap metric

linear quadratic gaussian balancing for model reduction of differential algebraic systems we generalise the model reduction method of linear quadratic gaussian balanced truncation to linear time invariant systems governed by differential algebraic equations the presented method relies on the solution of generalised algebraic riccati equations a simple a priori error bound in the gap metric is given

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

a characterization of monotone unidimensional latent variable models recently the problem of characterizing monotone unidimensional latent variable models for binary repeated measures was studied by ellis and van den wollenberg and by junker we generalize their work with a de finetti like characterization of the distribution of repeated measures x x 1 x 2 that can be represented with mixtures of likelihoods of independent but not identically distributed random variables where the data satisfy a stochastic ordering property with respect to the mixing variable the random variables x j may be arbitrary real valued random variables we show that the distribution of x can be given a monotone unidimensional latent variable representation that is useful in the sense of junker if and only if this distribution satisfies conditional association ca and a vanishing conditional dependence vcd condition which asserts that finite subsets of the variables in x become independent as we condition on a larger and larger segment of the remaining variables in x it is also interesting that the mixture representation is in a certain ordinal sense unique when ca and vcd hold the characterization theorem extends and simplifies the main result of junker and generalizes methods of ellis and van den wollenberg to a much broader class of models exchangeable sequences of binary random variables also satisfy both ca and vcd as do exchangeable sequences arising as location mixtures in the same way that de finetti s theorem provides a path toward justifying standard i i d mixture components in hierarchical models on the basis of our intuitions about the exchangeability of observations this theorem justifies one dimensional latent variable components in hierarchical models in terms of our intuitions about positive association and redundancy between observations because these conditions are on the joint distribution of the observable data x they may also be used to construct asymptotically power 1 tests for unidimensional latent variable models

fractional balanced reduction model reduction via fractional representation a method for the model reduction of finite dimensional linear time invariant fdlti plants is proposed which uses normalized fractional representations is proposed the method dubbed fractional balanced reduction applies balance and truncate to a special representation of the graph operator of the plant this operation yields the graph operator of a reduced order plant the method has such properties as existence of an a priori error bound in the graph metric and preservation of sector containment coupling fractional representations with principal component analysis gives a model reduction method that is capable of producing in a numerically stable way a good reduced order model using the whole full order model sector properties are also preserved these are useful for deciding stability when nonlinearities are in the loop

a note on hankel approximation of all pass functions abstract the paper extends the results obtained by fuhrmann and ober on hankel approximation with an arbitrary transfer function g we can associate using left and right coprime factorization two all pass functions u and u and also a function r called the characteristic function of g which is connected with the lqg controller of the function g we analyse the connections between the hankel operators h u h r and h u h r and also establish a one to one correspondence between the optimal hankel approximation of h r and that of h u with some additional property finally the duality results on nehari complements and hankel approximations are extended

the structure of characteristic values for classes of linear time invariant systems abstractin this paper a particular structure of the characteristic values found in some classes of systems is dealt with in particular for the linear time invariant systems in minimal form a b c d with transfer function matrix g s for which if i is a characteristic value then also 1 i is consequently when the system order is odd a singular value is equal to one the classes of systems for which this property holds are characterised by the existence of specific state space representations moreover the same structure of the characteristic values can be also retrieved for the class of even g s gt s and odd g s gt s systems several examples related to multi input multi output mimo systems are reported also referring to real physical systems

model reduction via balanced state space representations a model reduction procedure based on balanced state space representations is studied in this paper the reduced order model is examined from the point of view of stability controllability and observability both continuous time and discrete time systems are considered

optimal angle reduction a behavioral approach to linear system approximation we investigate the problem of optimal state reduction under minimization of the angle between system behaviors the angle is defined in a worst case sense as the largest angle that can occur between a system trajectory and its optimal approximation in the reduced order model this problem is analyzed for linear time invariant finite dimensional systems in a behavioral l2 setting without reference to input output decompositions and stability considerations the notion of a weakest past future link is introduced and it is shown how this concept is applied for the purpose of model reduction a method that reduces the state dimension by one is presented and shown to be optimal specific algorithms are provided for the numerical implementation of the approximation method the concepts and results are explicitly translated to an input output setting and related to balancing hankel norm reduction and normalized doubly coprime factorizations

parametric generalized singular perturbation approximation for model order reduction a new model reduction technique for the approximation of balanced realization is introduced the method involves a further generalization of the generalized singular perturbation approximation by adding several parameters that can be tuned according to a specified performance criterion an a priori bound can be computed guaranteeing the quality of the approximated model in the whole range of parameter variations two numerical examples conclude this paper

a gramian based approach to model reduction for uncertain systems the technical note considers a problem of model reduction for a class of uncertain systems with structured norm bounded uncertainty the technical note introduces controllability and observability gramians in terms of certain parameterized algebraic riccati inequalities based on these gramians three model reduction approaches are investigated for the underlying uncertain systems

acausal models and balanced realizations of stationary processes abstract we study acausal realizations of stationary or stationary increment processes in particular we characterize the family of models whose corresponding spectral factors have a fixed zero structure acausal models with a fixed zero structure are related to each other by a certain group of state feedback transformations which is naturally parametrized by the solution set of a homogeneous algebraic riccati equation each feedback transformation reflects some of the eigenvalues of the generator matrix a of the representation to a mirror image with respect to the imaginary axis dually acausal models with a fixed pole structure are parametrized by a dual riccati equation and by a corresponding family of output injection transformations from a general standpoint the results of this study clarify the role played by dual pairs of riccati equations in spectral factorization and may be relevant to other problem areas than stochastic modelling one natural application of the concepts discussed in the paper is to stochastic balancing balancing of models with an essentially arbitrary eigenvalue location can be accomodated very naturally in this framework a balancing algorithm involving the solution of a dual pair of riccati equations is discussed

overlapping block balanced canonical forms for various classes of linear systems abstract through the use of balanced realizations it has been possible to derive parametrizations and canonical forms for various classes of minimal linear systems of given dimension a possible problem of these parametrizations is that they are not overlapping this could be a drawback for the application of balanced parametrizations in such areas as system identification model reduction and optimization it is the topic of this paper to derive overlapping parametrizations which are closely related to the existing balanced parametrizations we first introduce input normal canonical forms which are defined through a novel way of choosing nice selections of columns of the reachability matrix these canonical forms provide overlapping parametrizations in the sense that they form a real analytic atlas of the manifold of systems which are considered then we introduce so called block balanced input normal forms which use the previously constructed input normal forms as building blocks the classes of systems for which such parametrizations are given are the stable minimal systems positive real minimal systems bounded real minimal systems and the class of all minimal systems of given mcmillan degree the results include both the single input single output and the multivariable case in the single input single output case however the issue of choosing nice selections of columns does not occur therefore in this case the derivation and presentation of the results is considerably simplified

linear quadratic gaussian balancing for discrete time infinite dimensional linear systems in this paper we study the existence of linear quadratic gaussian lqg balanced realizations for discrete time infinite dimensional systems lqg balanced realizations are those for which the smallest nonnegative self adjoint solutions of the control and filter riccati equations are equal we show that the control filter riccati equation has a nonnegative self adjoint solution if and only if the system is output input stabilizable our main result is that the transfer function of a discrete time linear system has an approximately controllable and observable lqg balanced realization iff it has an input and output stabilizable realization the corresponding control and filter riccati equations have unique nonnegative self adjoint solutions moreover approximately controllable and observable lqg balanced realizations are unique up to a unitary state space transformation finally we show that the spectrum of the product of the smallest nonnegative self adjoint solutions of the control and filter riccati equations is independent of the particular realization

balanced gains and their significance for l 2 model reduction the singular values do not give enough information for model reduction in the l2sense other invariants are introduced which help describe the contribution of each state to the l2magnitude of the impulse response this leads to another suboptimal balanced truncation criterion

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

variance error interpolation and experiment design we investigate how the variance error associated with the prediction error identification is related to the power spectral densities of the input and the additive noise at the output let f e i w be the ratio of the input power spectral density psd to the output noise psd we characterize the set of all functions f for which the variance error remains constant this analysis results a minimal finite dimensional affine parameterization of the variance error this parameterization connects our analysis with the theory of nevanlinna pick interpolation it is shown that the set of all f for which the variance error remains constant can be characterized by the solutions of a nevanlinna pick interpolation problem this insight has interesting consequences in optimal input design where it is possible to use some recent tools in analytic interpolation theory to tune shape of the input psd to suit certain needs

some integral properties for balanced realizations of scalar systems certain structural properties for balanced minimal realizations of scalar transfer functions are used to develop closed forms for the semi infinite integral of the impulse response and of the square of the impulse response

controller reduction using canonical interactions a conceptual approach to controller reduction is proposed that is based on closed loop considerations the method assesses the interactions between the plant and a high order controller as described by canonical correlation coefficients a free parameter in the algorithm of the reduced controller makes it possible to trade stability robustness versus performance several examples demonstrate the efficacy of the method

reduced order control based on approximate inertial manifolds a reduced order method based on approximate inertial manifolds is applied to optimal control problems in infinite dimensional state spaces a detailed analysis of the method is given for the linear quadratic regulator problem the method can also be applied to higher order control systems with an appropriate decomposition of the state space in terms of slow and fast exponential decay

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

on the solution of a nonlinear matrix equation for mimo symmetric realizations abstract a result concerning a particular solution of a nonlinear solution matrix equation reported in a previous paper 2 and interesting for studying siso systems is extended to mimo symmetric realizations for such a class of systems other important siso results can be generalized

hankel singular values and lqg characteristic values of discrete time linear systems in cascade with inner systems recent results have shown that for continuous time systems obtained by cascading an asymptotically stable system with an inner system the first n hankel singular values are greater than or equal to those of the original system similarly cascading a system in minimal form with an inner system the same property holds for the linear quadratic gaussian lqg characteristic values in this article we consider the discrete time case and demonstrate that the property also holds for these systems a very important consequence stemming from these results is that the hankel singular values and the lqg characteristic values of input delayed discrete systems are greater than or equal to those of their zero delay counterpart