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

two cocycle of a kac moody group the structure of the group corresponding to a kac moody algebra is analyzed the two cocycle for the group is constructed in terms of local coordinates and the transition functions between different coordinate systems are computed the results are extended to the group mathrm map s 3 mathfrak g which arises in a 3 1 dimensional yang mills theory

the optimal projection equations for reduced order state estimation the singular measurement noise case the optimal projection equations for reduced order state estimation are generalized to allow for singular i e colored measurement noise the noisy and noise free measurements serve as inputs to dynamic and static estimators respectively the optimal solution is characterized by necessary conditions which involve a pair of oblique projections corresponding to reduced estimator order and singular measurement noise intensity

brief minimal and non minimal optimal fixed order compensators for time varying discrete time systems using the minimality property of finite horizon time varying compensators established in this paper and the moore penrose pseudo inverse instead of the standard inverse strengthened discrete time optimal projection equations sdope and associated boundary conditions are derived for finite horizon fixed order lqg compensation they constitute a two point boundary value problem explicit in the lqg problem parameters which is equivalent to first order necessary optimality conditions and which is suitable for numerical solution the minimality property implies that minimal compensators have time varying dimensions and that the finite horizon optimal full order compensator is not minimal the use of the moore penrose pseudo inverse is further exploited to reveal that the optimal projection approach can be generalised but only to partially include non minimal compensators furthermore the structure of the space of optimal compensators with arbitrary dimensions is revealed to a large extent max min compensator dimensions are introduced and their significance in solving numerically the two point boundary value problem is explained the numerical solution is presented in a recently published companion paper which relies on the results of this paper

homotopy methods for solving the optimal projection equations for the h2 reduced order model problem the optimal projection approach to solving the h2 reduced order model problem produces two coupled highly nonlinear matrix equations with rank conditions as constraints due to the resemblance of these equations to standard matrix lyapunov equations they are called modified lyapunov equations the algorithms proposed herein utilize probability one homotopy theory as the main tool it is shown that there is a family of systems the homotopy that make a continuous transformation from some initial system to the final system with a carefully chosen initial problem a theorem guarantees that all the systems along the homotopy path will be asymptotically stable controllable and observable one method which solves the equations in their original form requires a decomposition of the projection matrix using the drazin inverse of a matrix it is shown that the appropriate inverse is a differentiable function an effective algorithm for computing the derivative of the projection matrix that involves solving a set of sylvester equations is given another class of methods considers the equations in a modified form using a decomposition of the pseudogramians based on a contragredient transformation some freedom is left in making an exact match between the number of equations and the number of unknowns thus effectively generating a family of methods

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

a relation between group and cech cohomology in principal fiber bundles and anomalies a natural map between group cohomology of the structure group of a principal fiber bundle with coefficients in the space of functions from the total space into an abelian group and cech cohomology of the base space is defined a differential complex of local group cochains is constructed and an analog of the poincare lemma for group cohomology is proven by using the machinery of spectral sequences the cohomology of this complex is calculated and the connection between group cohomology and cech cohomology of the given principal fiber bundle is elucidated finally the non abelian and witten anomaly in this context is reviewed and the relevance of our results for lifting principal group actions is discussed

estimation of states and parameters for linear systems with nonlinearly parameterized perturbations we consider systems that can be described by a linear part with a nonlinear perturbation where the perturbation is parameterized by a vector of unknown constant parameters under a set of technical assumptions about the perturbation and its relationship to the outputs we present a modular design technique for estimating the system states and the unknown parameters the design consists of a high gain observer that estimates the states of the system together with the full perturbation and a parameter estimator constructed by the designer to invert a nonlinear equation we illustrate the technique on a simulated dc motor with friction

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

the optimal projection equations for reduced order discrete time state estimation for linear systems with multiplicative white noise the optimal projection equations obtained in 2 3 for reduced order discrete time state estimation are generalized to include the effects of state and measurement dependent noise to provide a model of parameter uncertainty in contrast to the single matrix riccati equation arising in the full order kalman filter case the optimal steady state reduced order discrete time estimator is characterized by three matrix equations one modified riccati equation and two modified lyapunov equations coupled by both an oblique projection and stochastic effects

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

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

nonlinear filtering for discrete time linear systems with non gaussian initial data the optimal minimum mean square error filtering problem is discussed for a linear system with non gaussian initial distribution an input output realization is derived using an absolutely continuous change of probability measure a representation result is presented which facilitates the derivation of optimal filter realization

on adaptive control of nonlinearly parameterized nonlinear systems towards a constructive procedure abstract a new framework to design immersion and invariance adaptive controllers for nonlinearly parameterized nonlinear systems was recently proposed by the authors the key step is the construction of a monotone mapping via a suitable selection of a controller tuning function which has to satisfy some integrability conditions this translates into the need to solve a partial differential equation pde in this paper this result is extended providing some answers to the questions of characterization of monotonizable systems and solvability of the pde first adding to the design a nonlinear dynamic scaling we obviate the need to solve the pde second for the case of factorizable nonlinearities the following results are established i it is shown that the monotonicity condition is satisfied if a linear matrix inequality is feasible ii directly verifiable involutivity conditions that ensure the solution of the pde are presented iii an explicit formula for the required tuning function is given provided the regressor matrix satisfies some rank conditions hence adding a dynamic scaling this yields a constructive solution to the problem

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

a new reduced order adaptive filter for state estimation in high dimensional systems a simple reduced order adaptive filter optimal in the sense of minimum prediction error is proposed for estimating the state of high dimensional systems in which the process and observation noise statistics are unknown it is shown that implementation of this adaptive filter requires the solution of only two linear difference equations the dimensions of which are the dimensions of the full and reduced states respectively and that no solution of either an algebraic riccati equation or a lyapunov equation is needed in addition substantial gain in computer memory and cpu time is obtained by parametrization of the filter gain in the form of the product of two matrices one of which is a prescribed projection from the reduced space onto the full space a twin experiment on data assimilation with a quasi geostrophic ocean model shows the efficiency of the proposed approach

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

steady state kalman filtering with an h error bound abstract an estimator design problem is considered which involves both l 2 least squares and h worst case frequency domain aspects specifically the goal of the problem is to minimize an l 2 state estimation error criterion subject to a prespecified h constraint on the state estimation error the h estimation error 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 the l 2 state estimation error the principal result is a sufficient condition for characterizing fixed order i e full and reduced order estimators with bounded l 2 and h estimation error the sufficient condition involves a system of modified riccati equations coupled by an oblique projection i e idempotent matrix when the h constraint is absent the sufficient condition specializes to the l 2 state estimation result given in 2

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

nonlinear pi control of uncertain systems an alternative to parameter adaptation a novel approach to stabilization and trajectory tracking for nonlinear systems with unknown parameters and uncertain disturbances is developed we take a drastic departure from the classical adaptive control approach consisting of a parameterized feedback law and an identifier which tries to minimize a tracking or prediction error instead we propose a simple nonlinear pi structure that generates a stable error equation with a perturbation function that exhibits at least one root trajectories are forced to converge to this root by suitably adjusting the nonlinear pi gains we consider the two basic problems of i matched uncertainties when the uncertain terms are in the image of the input matrix and ii unknown control directions when the control signal is multiplied by a gain of unknown sign we show that without knowing the system parameters and with only basic information on the uncertainties we can achieve global asymptotic stability and global tracking without injecting high gains into the loop interestingly we prove that we can take as our nonlinear pi structure an activation function reminiscent of that used in neural networks although most of the results are derived assuming full state measurement we also present an observer based solution for a chain of integrators with unknown control direction the procedure is shown to provide simple solutions to the classical problems of neural network function approximation as well as eccentricity control and friction compensation of mechanical systems

a riccati equation approach to the singular lqg problem abstract the problem of optimal fixed order dynamic compensation for the singular lqg problem is considered necessary conditions characterizing the optimal compensator for the case involving both singular measurement noise and singular control weighting are given the solution consists of a set of two algebraic riccati equations and two lyapunov equations coupled by three projection matrices one projection is the standard order reduction projection while the other two projections reflect the two types of singularity that exist in the system the three projections are shown to satisfy disjointness conditions in addition to order reduction an advantage of the fixed structure approach is that differentiation which is often undersirable from a practical point of view and which may exist in the unconstrained optimal control can be avoided it is shown that the fixed order compensator agrees with the unconstrained solution when the latter possesses the same number of differentiations as are included in the prespecified controller structure and when the order is selected appropriately

reduced order lqg controllers for linear time varying plants abstract the problem of reduced order lqg optimization is addressed in a finite horizon linear time varying ltv system setting first order necessary conditions for local optimality in the parameter space is provided in terms of four coupled matrix differential equations this result provides a transparent generalization of the optimal projection equations of hyland and bernstein for the optimal steady state compensation of linear time invariant lti plants

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

the optimal projection equations for model reduction and the relationships among the methods of wilson skelton and moore first order necessary conditions for quadratically optimal reduced order modeling of linear time invariant systems are derived in the form of a pair of modified lyapunov equations coupled by an oblique projection which determines the optimal reduced order model this form of the necessary conditions considerably simplifies previous results of wilson 1 and clearly demonstrates the quadratic extremality and nonoptimality of the balancing method of moore 2 the possible existence of multiple solutions of the optimal projection equations is demonstrated and a relaxation type algorithm is proposed for computing these local extrema a component cost analysis of the model error criterion similar to the approach of skelton 3 is utilized at each iteration to direct the algorithm to the global minimum

brs cohomology and topological anomalies the occurrence of non abelian anomalies in gauge theories and gravitation first discovered via perturbative techniques is now completely explained from the mathematical point of view by means of the family index theorem of atiyah and singer here we make contact between this approach and brs cohomology by showing that they yield the same non abelian anomalies provided a certain restriction to local functionals is not introduced from the very beginning in particular this solves the unicity problem for this kind of anomalies local brs cohomology is still relevant for the abelian case

group cocycles line bundles and anomalies the relation between complex line bundles and certain group cocycles is explored in general to obtain explicit formulae for the transition functions and curvature of the determinant line bundle det of a family of dirac operators coupled to yang mills fields a covariant derivative on sections of det is constructed which realizes the curvature and minimally couples to the integrated anomaly which thus appears as a functional magnetic field on gauge orbit space the transcription of group cohomological cocycles into geometrical line bundles information is refined in such a way that the relevant cohomology groups can be computed in many cases giving insight into the classification of lifts of principal group actions

optimal reduced order state estimation for unstable plants the problem of optimal reduced order steady state state estimation is considered for the case in which the plant has unstable poles in contrast to the standard full order estimation problem involving a single algebraic riccati equation the solution to the reduced order problem involves one modified riccati equation and one lyapunov equation coupled by a projection matrix this projection is completely distinct from the projection obtained by bernstein and hyland 1985 for stable plants

robust reduced order nonstrictly proper state estimation via the optimal projection equations with petersen hollot bounds a state estimation design problem involving parametric plant uncertainties is considered an error bound suggested by recent work of petersen and hollot is utilized for guaranteeing robust estimation necessary conditions which generalize the optimal projection equations for reduced order state estimation are used to characterize the estimator which minimizes the error bound the design equations thus effectively serve as sufficient conditions for synthesizing robust estimators an additional feature is the presence of a static estimation gain in conjunction with the dynamic kalman estimator i e a nonstrictly proper estimator

contragredient transformations applied to the optimal projection equations the optimal projection approach to solving the h2 reduced order model problem produces two coupled highly nonlinear matrix equations with rank conditions as constraints it is not obvious from their original form how they can be differentiated and how some algorithm for solving nonlinear equations can be applied to them a contragredient transformation a transformation which simultaneously diagonalizes two symmetric positive semi definite matrices is used to transform the equations into forms suitable for algorithms for solving nonlinear problems three different forms of the equations obtained using contragredient transformations are given an svd based algorithm for the contragredient transformation and a homotopy algorithm for the transformed equations are given together with a numerical example

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

technical communique reduced order kalman filter with unknown inputs this paper presents an optimal reduced order kalman filter for discrete time dynamic stochastic linear systems with unknown inputs the problem is to estimate a part of the state vector in the case where none of the observations are assumed to be noise free the proposed filter is obtained by minimizing the trace of the estimation error covariance matrix with respect to the remaining degrees of freedom after noninteresting state and unknown inputs decoupling the necessary and sufficient conditions for stability and convergence of the filter are established

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

robust reduced order nonstrictly proper state estimation via the optimal projection equations with guaranteed cost bounds a state estimation design problem involving parametric plant uncertainties is considered an estimation error bound suggested by multiplicative white noise modeling is utilized for guaranteeing robust estimation over a specified range of parameter uncertainties necessary conditions that generalize the optimal projection equations for reduced order state estimation are used to characterize the estimator that minimizes the error bound the design equations thus effectively serve as sufficient conditions for synthesizing robust estimators additional features include the presence of a static estimation gain in conjunction with the dynamic kalman estimator to obtain a nonstrictly proper estimator

immersion and invariance adaptive control of nonlinearly parameterized nonlinear systems a new framework to design adaptive controllers for nonlinearly parameterized systems is proposed in this technical note the key step is the construction of a monotone mapping which explicitly depends on some of the estimator tuning parameters convexity that is related to monotonicity has been explored by several authors the property is assumed a priori and is useful for the design only in some region of state space which is completed switching the controller in our approach monotonicity can be used in the whole state space and moreover is enforced by the designer effectively becoming a synthesis tool one consequence of these features is that the controller does not rely on state dependent switching in order to dispose of degrees of freedom to render the function monotone we depart from standard adaptive control and adopt instead the recently introduced immersion and invariance approach

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

order reduction in linear state estimation under performance constraints the design and analysis of minimal order state estimators for possibly time varying linear systems under constraints on the maximal allowable mean square error are considered a global lower bound on the optimal error is derived along with a lower bound on the minimal estimator order needed for meeting the performance constraint the ideal reduced order estimator which satisfies the lower bound is derived along with conditions for its realizability when the ideal estimator is not realizable its structure forms a suboptimal estimator which maintains in some sense a local optimality property and is called the pseudoideal estimator the mean square error of the pseudoideal estimator defines upper bounds on the optimal error and on the estimator order needed for meeting the performance constraint the lower and the upper bounds on the order define a reduced search set for the design problem when the distance between the ideal and the pseudoideal estimators is sufficiently small in a certain numerical sense the pseudoideal estimator may be considered optimal for practical purposes

reduced order h filtering for singular systems abstract this paper solves the problem of reduced order h filtering for singular systems the purpose is to design linear filters with a specified order lower than the given system such that the filtering error dynamic system is regular impulse free or causal stable and satisfies a prescribed h performance level one major contribution of the present work is that necessary and sufficient conditions for the solvability of this problem are obtained for both continuous and discrete singular systems these conditions are characterized in terms of linear matrix inequalities lmis and a coupling non convex rank constraint moreover an explicit parametrization of all desired reduced order filters is presented when these inequalities are feasible in particular when a static or zeroth order h filter is desired it is shown that the h filtering problem reduces to a convex lmi problem all these results are expressed in terms of the original system matrices without decomposition which makes the design procedure simple and directly last but not least the results have generalized previous works on h filtering for state space systems an illustrative example is given to demonstrate the effectiveness of the proposed approach

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

structurally balanced controller order reduction with guaranteed closed loop performance abstract this paper considers controller order reduction with the consideration of closed loop stability and performance a closed loop structurally balancing method is proposed to reduce the controller orders for a class of systems and controllers the resulting reduced order controllers if they exist are guaranteed to stabilize the corresponding systems error bounds are also derived for the closed loop performance degradation

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

robust controller design based on reduced order plants two dual controller design methods are proposed for linear time invariant multi input multi output systems where designs based on a reduced order plant robustly stabilizer higher order plants with additional poles or zeros in the stable region the additional poles or zeros are considered as multiplicative perturbations of the reduced plant the methods are tailored towards closed loop stability and performance and they yield estimates for the stability robustness and performance of the final design they can be considered as formalizations of two classical heuristic model reduction techniques one method neglects a plant pole sufficiently far to the left of dominant poles and the other cancels a sufficiently small stable plant zero with a pole at the origin

parametrization and optimization of reduced order filters this paper deals with the design of reduced order filters for linear signal models to ensure that the transfer function from the noise to the filtering error is not only stable but also has a minimum h sub 2 norm one of the major results in the paper is the discovery and parametrization of a set of filters of fixed order which lead to stable filtering error transfer functions such a parametrization is given in terms of an arbitrary orthogonal projection matrix and a given full order filter as a consequence the problem of minimizing the h sub 2 norm of the filtering error transfer function over the set of reduced order filters can be formulated as an equivalent unconstrained parametric optimization problem over a compact manifold a gradient flow based algorithm is proposed to find an optimal solution to the problem nice properties of the algorithm including the convergence property are established theoretically as well as demonstrated numerically with an example

stabilization of nonlinear systems nonlinearly depending on fast time varying parameters an immersion and invariance approach the problem of stabilization of nonlinear systems which depend nonlinearly on fast time varying parameters is considered in the technical note it is assumed that if the plant parameters were known a static state feedback controller that achieves the stabilization objective with a lyapunov like function that is independent of the parameters is known a constructive procedure to update the unknown parameters of the controller based on the immersion and invariance approach is proposed the main contribution of the paper is to show that the proposed controller guarantees global convergence to zero of the systems state for arbitrary time variations of the plant parameters provided the controller parameters are bounded to ensure the latter condition an assumption that in the single parameter case is strictly weaker than the monotonicity condition invoked in previous studies is imposed stabilization is achieved via a rather unique combination of gradient like parameter estimation and the construction of a monotonic signal that counters the deleterious effect of the parameter variations several simulation examples illustrate the applicability of the suggested method

the optimal projection equations for fixed order sampled data dynamic compensation with computation delay for an lqg type sampled data regulator problem which accounts for computational delay and utilizes an averaging a d device the equivalent discrete time problem is shown to be of increased order due to the inclusion of delayed measurement states the optimal projection equations for reduced order discrete time compensation are applied to the augmented problem to characterize low order controllers the design results are illustrated on a tenth order flexible beam example

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

on partially augmented observers for systems with coloured noises the problem of optimal fixed order observers for continuous time linear systems with deterministic input and coloured process and measurement noises is considered optimal estimation of the state involves augmentation of the system and as a result the observer has the augmented dimension in this paper the order of the observer is prefixed to a value which ranges between the dimension of the original system and that of the augmented system the presence of a deterministic input imposes the structure of an observer on the estimator the optimal observer estimator under these dimensional and structural constraints is then derived solution is given for the general case where the intensity of the white noise component of the measurement noise may be singular necessary conditions fully characterizing the optimal observer estimate are given the solution consists of a modified riccati and three modified lyapunov equations coupled by three projection matrices the three projections correspond to the order

kalman filtering with constrained output injection in applications involving large scale systems such as discretized partial differential equations it is often of interest to use data to estimate state variables associated with a subregion of the spatial domain in this paper we derive an extension of the classical kalman filter in which data injection is confined to a subspace of the system states

the analysis of elliptic families i metrics and connections on determinant bundles in this paper we construct the quillen metric on the determinant bundle associated with a family of elliptic first order differential operators we also introduce a unitary connection on and calculate its curvature our results will be applied to the case of dirac operators in a forthcoming paper

reduced order h sub spl infin filtering for stochastic systems this paper deals with the reduced order h sub spl infin filtering problem for stochastic systems necessary and sufficient conditions are obtained for the existence of solutions to the continuous time and discrete time problems in terms of certain linear matrix inequalities lmis and a coupling nonconvex rank constraint condition furthermore when these conditions are feasible an explicit parametrization of all desired reduced order filters corresponding to a feasible solution is given in particular when the reduced order filter is restricted to be a static one then simple conditions expressed by lmis only without any rank constraints are derived and a parametrization of all solutions is also given finally an illustrative example is provided to show the effectiveness of the proposed approach

optimal observers for systems with colored noises the problem of optimal full order observers for continuous time linear systems with colored process and measurement noises is considered in such cases optimal estimation of the state involves augmenting the system thus a higher order observer is required the structure of a full order observer is assumed and necessary conditions for the optimal observer are derived the conditions are given for the general case where the intensity of the white noise component of the measurement noise may be singular the solution consists of a modified riccati equation and a lyapunov equation coupled by two projection matrices in the singular case and one projection matrix in the nonsingular case

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

quantal phase factors accompanying adiabatic changes a quantal system in an eigenstate slowly transported round a circuit c by varying parameters r in its hamiltonian r will acquire a geometrical phase factor exp i c in addition to the familiar dynamical phase factor an explicit general formula for c is derived in terms of the spectrum and eigenstates of r over a surface spanning c if c lies near a degeneracy of c takes a simple form which includes as a special case the sign change of eigenfunctions of real symmetric matrices round a degeneracy as an illustration c is calculated for spinning particles in slowly changing magnetic fields although the sign reversal of spinors on rotation is a special case the effect is predicted to occur for bosons as well as fermions and a method for observing it is proposed it is shown that the aharonov bohm effect can be interpreted as a geometrical phase factor

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

design of optimal reduced order h2 filters abstract this paper shows that the design of a reduced order optimal h 2 filter is equivalent to a frequency weighted l 2 norm model reduction problem

robust strong stabilization via modified popov controller synthesis in this paper the authors merge the parameter dependent lyapunov function framework used to construct robust popov controllers with the a priori and a posteriori approaches for obtaining stable compensators specifically the authors derive constructive sufficient conditions that yield robust lyapunov and asymptotically stable stabilizing popov dynamic compensators

kac moody groups topology of the dirac determinant bundle and fermionization the relation between kac moody groups and algebras and the determinant line bundle of the massless dirac operator in two dimensions is clarified analogous objects are studied in four space time dimensions and a generalization of witten s fermionization mechanism is presented in terms of the topology of the dirac determinant bundle

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

parameter convergence in nonlinearly parameterized systems a large class of problems in parameter estimation concerns nonlinearly parametrized systems over the past few years a stability framework for estimation and control of such systems has been established we address the issue of parameter convergence in such systems in this paper systems with both convex concave and general parameterizations are considered in the former case sufficient conditions are derived under which parameter estimates converge to their true values using a min max algorithm in the latter case to achieve parameter convergence a hierarchical min max algorithm is proposed where the lower level consists of a min max algorithm and the higher level component updates the bounds on the parameter region within which the unknown parameter is known to lie using this hierarchical algorithm a necessary and sufficient condition is established for global parameter convergence in systems with a general nonlinear parameterization in both cases the conditions needed are shown to be stronger than linear persistent excitation conditions that guarantee parameter convergence in linearly parametrized systems explanations and examples of these conditions and simulation results are included to illustrate the nature of these conditions a general definition of nonlinear persistent excitation that leads to parameter convergence is proposed at the end of this paper

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

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

brief h controller reduction for nonlinear systems sufficient conditions are proposed for the existence of reduced order fixed order controllers solving the standard nonlinear h output feedback control problem state space formulas for such reduced order h controllers are also derived in terms of the solutions of two hamilton jacobi inequalities the development uses only elementary concepts of dissipativity and differential game thus the proofs given are simple and clear

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

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

adaptive control of a class of non linearly parametrized systems using convexification in this paper we deal with the problem of adaptive control of a class of non linearly parametrized plants the main contribution is to show that through a well chosen reparametrization it is possible to transform the non convex nonlinear parametrization into a convex one a property that is used to design a globally stable adaptive controller

computational methods for parametric lq problems a survey iterative methods for finding the optimal constant feedback gains for parametric lq problems notably for optimal constant output feedback problems are surveyed the connections of several methods to loss function expansions are discussed with important implications to the understanding of their convergence properties especially the descent anderson moore method levine athans like methods and the newton method are considered convergence results are also included the initialization problem and the output feedback stabilization problem are also discussed furthermore it is shown that the concepts and methods surveyed in this paper are useful in solving many realistic generalized parametric lq problems as well notably so called robust parametric lq problems

robust static and dynamic output feedback stabilization deterministic and stochastic perspectives three parallel gaps in robust feedback control theory are examined sufficiency versus necessity deterministic versus stochastic uncertainty modeling and stability versus performance deterministic and stochastic output feedback control problems are considered with both static and dynamic controllers the static and dynamic robust stabilization problems involve deterministically modeled bounded but unknown measurable time varying parameter variations while the static and dynamic stochastic optimal control problems feature state control and measurement dependent white noise general sufficiency conditions for the deterministic problems are obtained using lyapunov s direct method while necessary conditions for the stochastic problems are derived as a consequence of minimizing a quadratic performance criterion the sufficiency tests are then applied to the necessary conditions to determine when solutions of the stochastic optimization problems also solve the deterministic robust stability problems as an additional application of the deterministic result the modified riccati equation approach of petersen and hollot is generalized in the static case and extended to dynamic compensation

robust reduced order modeling via the optimal projection equations with peterson hollot bounds an optimal reduced order modeling problem with parametric plant uncertainty is considered a model reduction bound suggested by recent work of i r petersen and c v hollot is utilized for guaranteeing robust reduced order modeling over a specified range of uncertain parameters necessary conditions which generalize the optimal projection equations for model reduction are used to characterize the reduced order model which minimizes the model reduction bound the optimality equations thus effectively serve as sufficient conditions for characterizing robust reduced order models