an input normal form homotopy for the l sup 2 optimal model order reduction problem in control system analysis and design finding a reduced order model optimal in the l sup 2 sense to a given system model is a fundamental problem the problem is very difficult without the global convergence of homotopy methods and a homotopy based approach has been proposed the issues are the number of degrees of freedom the well posedness of the finite dimensional optimization problem and the numerical robustness of the resulting homotopy algorithm a homotopy algorithm based on the input normal form characterization of the reduced order model is developed here and is compared with the homotopy algorithms based on hyland and bernstein s optimal projection equations the main conclusions are that the input normal form algorithm can be very efficient but can also be very ill conditioned or even fail

low order integral action controller synthesis abstract a simple controller synthesis method is developed for certain classes of linear time invariant multi input multi output plants the number of poles in each entry of these controllers depends on the number of right half plane plant zeros and is independent of the number of poles of the plant to be stabilized furthermore these controllers have integral action so that they achieve asymptotic tracking of step input references with zero steady state error the designed controller s poles and zeros are all in the stable region with the exception of one pole at the origin for the integral action design requirement the freedom available in the design parameters may be used for additional performance objectives although the only goal here is stabilization and tracking of constant references

lqg control with an h sup infinity performance bound a riccati equation approach an lqg linear quadratic gaussian control design problem involving a constraint on h sup infinity disturbance attenuation is considered the h sup infinity performance constraint is embedded within the optimization process by replacing the covariance lyapunov equation by a riccati equation whose solution leads to an upper bound on l sub 2 performance in contrast to the pair of separated riccati equations of standard lqg theory the h sup infinity constrained gains are given by a coupled system of three modified riccati equations the coupling illustrates the breakdown of the separation principle for the h sup infinity constrained problem both full and reduced order design problems are considered with an h sup infinity attenuation constraint involving both state and control variables an algorithm is developed for the full order design problem and illustrative numerical results are given

on the deficiency indices and self adjointness of symmetric hamiltonian systems abstract the main purpose of this paper is to investigate the formal deficiency indices n of a symmetric first order system jf bf h f on an interval i where i r or i r here j b h are n n matrix valued functions and the hamiltonian h 0 may be singular even everywhere we obtain two results for such a system to have minimal numbers n 0 if i r resp n n if i r and a criterion for their maximality n 2n for i r as well as the quasi regularity this covers the kac krein and de branges trans amer math soc 99 1961 118 theorems on 2 2 canonical systems and some results from kogan and rofe beketov proc roy soc edinburgh sect a 74 1974 75 5 some conditions for a canonical system to have intermediate formal deficiency indices are presented too we also obtain a generalization of the well known titchmarsh sears theorem for second order sturm liouville type equations this contains results due to lidskii and krein as special cases we present two approaches to the above problems one dealing with formal deficiency indices and one dealing with ordinary deficiency indices our main non formal approach is based on the investigation of a symmetric linear relation smin which is naturally associated to a first order system this approach works in the framework of extension theory and therefore we investigate in detail the domain d s min of s min in particular we prove the so called regularity theorem for d s min as a byproduct of the regularity result we obtain very short proofs of generalizations of the main results of the paper by kogan and rofe beketov 1974 75

a linear programming approach to the synthesis of fixed structure controllers this paper describes a new approach to the synthesis of fixed structure and fixed order controllers such controllers are required in many practical applications a broad class of fixed structure controller synthesis problems can be reduced to the determination of a real controller parameter vector or simply a controller k k 1 k 2 k t so that a given set of real or complex polynomials of the form p s k po s k 1 p 1 s k t p t s is hurwitz the stability of the closed loop system requires a real characteristic polynomial to be hurwitz while several performance criteria can be satisfied by ensuring that a family of complex polynomials is hurwitz a novel feature of this paper is the exploitation of the interlacing property ip of hurwitz polynomials to construct arbitrarily tight approximations of the set of stabilizing controllers this is done by systematically constructing sets of linear inequalities in k the union of the feasible sets of linear inequalities provides an approximation of the set of all controllers k which render p s k hurwitz as the number of sets of linear inequalities increases and approaches infinity we show that the union of the feasible sets approaches the set of all stabilizing controllers of the desired structure the main tools that are used in the construction of the sets of linear inequalities are the hermite biehler theorem descartes rule of signs and its generalization we provide examples of the applicability of the proposed methodology to the synthesis of fixed order stabilizing controllers

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

g2 structures and octonion bundles abstract we use a g 2 structure on a 7 dimensional riemannian manifold with a fixed metric to define an octonion bundle with a fiberwise non associative product we then define a metric compatible octonionic covariant derivative on this bundle that is compatible with the octonion product the torsion of the g 2 structure is then shown to be an octonionic connection for this covariant derivative with curvature given by the component of the riemann curvature that lies in the 7 dimensional representation of g 2 we also interpret the choice of a particular g 2 structure within the same metric class as a choice of gauge and show that under a change of this gauge the torsion does transform as an octonion valued connection 1 form finally we also show an explicit relationship between the octonion bundle and the spinor bundle define an octonionic dirac operator and explore an energy functional for octonion sections we then prove that critical points correspond to divergence free torsion which is shown to be an octonionic analog of the coulomb gauge

controller design with regional pole constraints a design procedure is developed that combines linear quadratic optimal control with regional pole placement specifically a static and dynamic output feedback control problem is addressed in which the poles of the closed loop system are constrained to lie in specified regions of the complex plane these regional pole constraints are embedded within the optimization process by replacing the covariance lyapunov equation by a modified lyapunov equation whose solution in certain cases leads to an upper bound on the quadratic cost functional the results include necessary and sufficient conditions for characterizing static output feedback controllers with bounded performance and regional pole constraints sufficient conditions are also presented for the fixed order i e full and reduced order dynamic output feedback problem with regional pole constraints circular elliptical vertical strip parabolic and section regions are considered

l sub 2 optimal filter reduction a closed loop approach this paper is concerned with the problem of order reduction of a full order kalman filter for a stable linear signal model so that the steady state filtering error variance associated with the reduced order filter is minimized by an orthogonal parameterization the above problem is formulated to minimize the filtering error variance over a set of orthogonal matrices both continuous and iterative algorithms are derived to compute an optimal reduced order filter the algorithms are shown to possess good properties including the desirable convergence property the proposed algorithms are simple and effective numerical examples are presented to demonstrate the effectiveness and the significant advantages of the proposed algorithms over the existing open loop methods such as the well known balanced truncation method

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

probability one homotopy algorithms for solving the coupled lyapunov equations arising in reduced order h2 h modeling estimation and control optimal reduced order modeling estimation and control with respect to combined h 2 h criteria give rise to coupled lyapunov and riccati equations to develop reliable numerical algorithms for these problems this paper focuses on the coupled lyapunov equations which appear as a subset of the synthesis equations in particular this paper systematically examines the requirements of probability one homotopy algorithms to guarantee global convergence homotopy algorithms for nonlinear systems of equations construct a continuous family of systems and solve the given system by tracking the continuous curve of solutions to the family the main emphasis is on guaranteeing transversality for several homotopy maps based upon the pseudogramian formulation of the coupled lyapunov equations and variations based upon canonical forms these results are essential to the probability one homotopy approach by guaranteeing good numerical properties in the computational implementation of the homotopy algorithms

an implicit deflation method for ill conditioned sylvester and lyapunov equations this paper describes an effective algorithm for solving ill conditioned sylvester or lyapunov equations these equations can be solved by conventional methods for ill conditioned linear systems however such methods are not efficient since they require on the order of n 6 arithmetic operations on order n 2 data for these ill conditioned matrix equations an implicit deflation algorithm is proposed to implement a certain svd based minimum norm least squares approximate solution the method is practical since it relies only on the ability to solve a sylvester or lyapunov equation certain practical details are also discussed

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

low rank approximate solutions to large sylvester matrix equations in the present paper we propose an arnoldi based method for solving large and sparse sylvester matrix equations with low rank right hand sides we will show how to extract low rank approximations via a matrix krylov subspace method we give some theoretical results such an expression of the exact solution and upper bounds for the norm of the error and for the residual numerical experiments will also be given to show the effectiveness of the proposed method

brief paper optimal reduced order compensation of time varying discrete time systems with deterministic and white parameters the finite horizon optimal compensation problem is considered in the case of linear time varying discrete time systems with deterministic and white stochastic parameters and quadratic criteria the dimensions of the compensator are a priori fixed and may be time varying also the dimensions of the system may be time varying strengthened discrete time optimal projection equations sdope are developed which within the class of minimal compensators are equivalent to the first order necessary optimality conditions based on the sdope and their associated boundary conditions two numerical algorithms are presented to solve the two point boundary value problem one is a homotopy algorithm while the second iterates the sdope repeatedly forward and backward in time the latter algorithm is much more efficient and constitutes a generalization of the single iteration of the control and estimation riccati equations associated with the full order problem for systems with deterministic parameters the algorithms are illustrated with a numerical example the case of systems with deterministic parameters will be treated as a special case of systems with white parameters

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

the optimal projection equations for reduced order state estimation first order necessary conditions for optimal steady state reduced order state estimation for a linear time invariant plant in the presence of correlated disturbance and nonsingular measurement noise are derived in a new and highly simplified form in contrast to the lone matrix riccati equation arising in the full order kalman filter case the optimal steady state reduced order estimator is characterized by three matrix equations one modified riccati equation and two modified lyapunov equations coupled by a projection whose rank is precisely equal to the order of the estimator and which determines the optimal estimator gains this coupling is a graphic reminder of the suboptimality of proposed approaches involving either model reduction followed by full order estimator design or full order estimator design followed by estimator reduction techniques the results given here complement recently obtained results which characterize the optimal reduced order model by means of a pair of coupled modified lyapunov equations 7 and the optimal fixed order dynamic compensator by means of a coupled system of two modified riceati equations and two modified lyapunov equations 6

minimum entropy static output feedback control with an h sub spl infin norm performance bound the problem of designing static output feedback controllers is considered the controller is required to minimize the closed loop entropy at s spl infin and to satisfy a prescribed h sub spl infin norm bound the design equations consist of a modified riccati equation and a lyapunov equation that are coupled via a projection these equations are solved using a homotopy algorithm the design procedure is illustrated by two examples the first one is a classical example that was solved in the literature using convex optimization techniques the second example is one of designing a gain scheduled normal acceleration loop for an air launched unmanned air vehicle

some open problems in matrix theory arising in linear systems and control control theory has long provided a rich source of motivation for developments in matrix theory accordingly we discuss some open problems in matrix theory arising from theoretical and practical issues in linear systems theory and feedback control the problems discussed include robust stability matrix exponent induced norms stabilizability and pole assignability and nonstandard matrix equations a substantial number of references are included to acquaint matrix theorists with problems and trends in this application area feedback control theory has long provided a rich source of motivation for developments in matrix theory the purpose of this paper is to discuss several open problems in matrix theory that arise from theoretical and practical issues in feedback control theory and the associated area of linear systems theory many of these problems are remarkably simple to state are of intense interest in control theory and applications and yet remain unsolved besides leading to the resolution of these problems it is hoped that this paper will help to stimulate increased interaction between matrix and control theorists accord

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

observer based compensators for fixed order h control this paper present a two riccati inequality approach for the design of fixed order h suboptimal controllers with an observer based structure the two riccati inequalities are only one way coupled and can be solved consecutively the fixed order controller has the structure of a fixed order estimator of the state feedback control law in the presence of the worst case disturbances it retains the observed based compensator structure of the full order central controller of the standard h design but adds a fixed order dynamic and mappings between the full order and the fixed order state spaces in the full order special case this controller recovers to the minimal entropy central controller of the standard h design

minimal residual methods augmented with eigenvectors for solving sylvester equations and generalized sylvester equations in this paper we propose minimal residual methods augmented with eigenvectors for solving large sylvester matrix equations ax xb c and generalized sylvester matrix equations axb x c the subspace from which the approximate solution is extracted is the kronecker product subspace of two augmented krylov subspaces numerical experiments report the effectiveness of these methods

computation of approximate null vectors of sylvester and lyapunov operators this paper describes an effective algorithm for computing approximate null vectors of certain matrix operators associated with sylvester or lyapunov equations the singular value decomposition and rank revealing qr methods are two widely used stable algorithms for numerical determination of the rank and nullity of a matrix a these methods however are not readily applicable to sylvester and lyapunov operators since they require on the order of n sup 6 arithmetic operations on order n sup 2 data for these problems a variant of inverse power iteration is employed to compute orthonormal bases for singular subspaces associated with the small singular values the method is practical since it relies only on the ability to solve a sylvester or lyapunov equation certain practical aspects are considered and a direct refinement technique is proposed to enhance the convergence of the algorithm

the xy centring algorithm for the dual lmi problem a new approach to fixed order control design many fixed order suboptimal control problems with stability performance and robustness specifications can be reduced to a search for a matrix x 0 satisfying a linear matrix inequality lmi while x 1 satisfies another lmi this paper defines a certain class of these problems we shall call the dual lmi problem and a computational algorithm to solve our dual lmi problem is given properties and limitations of the algorithm are discussed in comparison with the existing algorithm the min max algorithm an extension to optimal control problems is provided numerical examples for the fixed order stabilization problem and the static output feedback linear quadratic optimal control problem demonstrate the applicability of the proposed algorithm

parameter dependent lyapunov functions and the popov criterion in robust analysis and synthesis many practical applications of robust feedback control involve constant real parameter uncertainty whereas small gain or norm bounding techniques guarantee robust stability against complex frequency dependent uncertainty thus entailing undue conservatism since conventional lyapunov bounding techniques guarantee stability with respect to time varying perturbations they possess a similar drawback in this paper we develop a framework for parameter dependent lyapunov functions a less conservative refinement of fixed lyapunov functions an immediate application of this framework is a reinterpretation of the classical popov criterion as a parameter dependent lyapunov function this result is then used for robust controller synthesis with full order and reduced order controllers

on the robustness of meop design versus asymptotic lqg synthesis an analysis is presented of generalized lqg linear quadratic gaussian type algorithms with regard to robustness characteristics comparisons are made of the maximum entropy optimal projection meop methodology versus the linear quadratic gaussian with loop transfer recovery lqg ltr technique and the parametric robust linear quadratic gaussian synthesis procedure the performance of a reduced order direct controller design is compared with the full order designs on a example representative of typical flexible structure dynamics

the optimal projection equations for fixed order dynamic compensation first order necessary conditions for quadratically optimal steady state fixed order dynamic compensation of a linear time invariant plant in the presence of disturbance and observation noise are derived in a new and highly simplified form in contrast to the pair of matrix riccati equations for the full order lqg case the optimal steady state fixed order dynamic compensator is characterized by four matrix equations two modified riccati equations and two modified lyapunov equations coupled by a projection whose rank is precisely equal to the order of the compensator and which determines the optimal compensator gains the coupling represents a graphic portrayal of the demise of the classical separation principle for the reduced order controller case

robust controller synthesis using the maximum entropy design equations this note presents an application of the optimality conditions obtained in 1 for dynamic compensation in the presence of state control and measurement dependent noise by solving these equations which represent a fundamental generalization of standard steady state lqg theory a series of increasingly robust control designs is obtained for the example considered in 2

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

limit point criterion for singular linear dirac differential systems the deficiency index problem of a singular linear dirac differential system is studied in this paper several criterion are presented under which the linear dirac systems are in the limit point case or the strong limit point case some of results extensively generalize and improve the previous results obtained by krall hinton and shaw moreover as a corollary of our results the levinson type criteria for dirac systems is obtained

controller reduction by component cost analysis component cost analysis 1 is used to develop a method for controller reduction the reduction of the controller is based upon the participation of the controller states in the value of a quadratic performance metric the controller states which have the smallest contribution to the performance metric are truncated to produce the reduced controllers an error index is defined to evaluate the reduced controller compared to the optimal lqg controller and bounds on this index are derived a numerical example is included to illustrate the procedure

on classification of second order differential equations with complex coefficients this paper is concerned with the deficiency index problem of second order differential equations with complex coefficients it is known that this class of equations is classified into cases i ii and iii according to the number of linearly independent solutions in suitable weighted square integrable spaces in this study the original equation is reformulated into a new formally self adjoint differential system by introducing a new spectral parameter and the relationship between the classifications of the equation and the system is obtained moreover the exact dependence of cases ii and iii on the corresponding half planes is given and some criteria of the three cases are established