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

an algebraic riccati equation approach to h optimization abstract this paper considers the general so called four block h optimal control problem with the assumption that system states are available for feedback it is shown that infimization of the h norm of the closed loop transfer function over all linear constant i e nondynamic stabilizing state feedback laws can be completely characterized via an algebraic riccati equation it is further shown that the optimal norm is not improved by allowing feedback to be dynamic thus the general state feedback h optimal control problem can be solved by iteratively solving one are and the controller can be chosen to be static gain

understanding the qr algorithm the qr algorithm is currently the most popular method for finding all eigenvalues of a full matrix while qr is now well understood by specialists in eigenvalue computations this understanding is not being conveyed effectively to the mathematical public many accounts present wilkinson s 1965 convergence proof others establish some of the connections between the qr algorithm the power method and inverse iteration usually much emphasis is rightly placed on the refinements such as shifts of origin which are required to make the algorithm competitive but practically all accounts fail to explain the basic meaning of qr iterations as a consequence the qr algorithm is widely thought to be difficult to understand the purpose of this paper is to try to convince the reader that the opposite is true in fact the qr algorithm is neither more nor less than a clever implementation of simultaneous iteration which is itself a natural easily understood extension of the power method this point

the constrained newton method on a lie group and the symmetric eigenvalue problem the newton method for estimating a critical point of a real function is formulated in a coordinate free manner on an arbitrary lie group convergence proofs for the numerical method are given an application of the general approach to computing the eigenvalues of a symmetric matrix is given and the resultant algorithm is compared with the classical shifted qr algorithm properties of the method described suggest that it is of interest for certain computations in online and adaptive environments

an algebraic approach to h output feedback control problems abstract we derive an output feedback controller which stabilizes a system and satisfies a prescribed h norm bound of the closed loop transfer function the proposed design method is available for any system i e there are no restrictions on d12 and d21 this approach utilizes only algebraic operations thus proofs are simple and clear

covariance controllers a new parametrization of the class of all stabilizing controllers abstract the class of all stabilizing controllers is parametrized in terms of a system matrix called the one at a time covariance and an arbitrary skew hermitian matrix the new parametrization is in closed form for any specified controller order and provides a deterministic version of the stochastic problem of covariance assignment

differential equations and the qr algorithm in this paper we consider a variety of isospectral flows on the set of n times n matrices these flows arise from lax pairs and can all be interpreted in terms of the qr decomposition for nonsingular matrices the asymptotics of these differential equations are considered in detail and for symmetric matrices these asymptotics provide a new method of solving the eigenvalue problem

robust stability via sign definite decomposition this paper considers the problem of robust stability of a polynomial family whose coefficients are polynomial functions of the parameters of interests the problem occurs in the design of a fixed order or fixed structure multivariable feedback controller parametrized by a real design parameter vector for a plant containing a vector of uncertain parameters the characteristic polynomials of such systems often contain coefficients which depend polynomially on and using results on sign definite decomposition a new stability test is developed that gives a sufficient condition for hurwitz stability of the family of closed loop systems that result when and vary over prescribed boxes this test is reminiscent of kharitonov s theorem even though the family of polynomials considered here is certainly not restricted to be interval or even convex moreover the test does reduce to kharitonov s theorem for the special case of interval polynomials using this criterion recursively and modularly sets of controllers that stabilize the family of uncertain plants are determined

stability of front solutions in inhomogeneous media abstract we use the fitzhugh nagumo f n model to study front solutions in inhomogeneous media inhomogeneity is modeled by adding a space dependent term z x in the equations changes in z shift the cubic nullcline up and down in the phase plane of the f n model this is used to mimic shifts in the cubic shaped nullcline of a bidomain model of calcium waves in egg cells physica d in press caused by changes in the total calcium concentration conditions for the existence and stability of stationary front solutions for a general class of z functions are obtained analytically when the dynamics are piecewise linear we show that spatial inhomogeneities cause pinning and oscillations of front solutions this is best demonstrated when z x is a linear ramp when the slope is large a linear ramp breaks the translational invariance of the stationary front and stabilizes the front the front becomes less stable as the slope decreases at a critical slope the front becomes unstable through a hopf bifurcation beyond which oscillations in the front occur this also occurs for nonlinear spatial inhomogeneities including a step increase with varying magnitude and steepness this bifurcation has been found previously physica d 106 1997 270 phys rev e 55 1997 366 in a local analysis of the f n model near a co dimension two point sometimes called a non equilibrium ising bloch nib bifurcation and with inhomogeneities of small magnitude the analysis presented here applies globally in parameter space and is valid for inhomogeneities of arbitrary magnitude we show that this bifurcation need not occur in conjunction with the nib point it is caused exclusively by spatial inhomogeneity a rich variety of wave phenomena including front oscillation front pinning and front reflection are shown to occur in inhomogeneous media

on the continuous realization of iterative processes many important mathematical problems are solved by iterative methods many of these iterative schemes may be regarded as the discrete realization of certain continuous dynamical systems this paper summarizes some of the recent developments in the continuous realization of several popular basic iterative methods

dynamical systems and non hermitian iterative eigensolvers simple preconditioned iterations can provide an efficient alternative to more elaborate eigenvalue algorithms we observe that these simple methods can be viewed as forward euler discretizations of well known autonomous differential equations that enjoy appealing geometric properties this connection facilitates novel results describing convergence of a class of preconditioned eigensolvers to the leftmost eigenvalue provides insight into the role of orthogonality and biorthogonality and suggests the development of new methods and analyses based on more sophisticated discretizations these results also highlight the effect of preconditioning on the convergence and stability of the continuous time system and its discretization

h control by state feedback an iterative algorithm and characterization of high gain occurrence we present an iterative algorithm to compute the achievable h norm by state feedbak for a standard regular problem and give a characterization when high gain feedback is necessary to approach the optimal value if there is no need for high gain components to approximate the optimal value we can reduce the problem to the computation of the h norm of a certain stable transfer matrix

all fixed order h sub spl infin controllers observer based structure and covariance bounds this note obtains a parameterization of the set of all stabilizing controllers of order less than or equal to the plant which yields for the closed loop transfer matrix a specified h sub spl infin norm bound the algebraic results of covariance control are applied to the h sub spl infin control problem to yield a parameterization in terms of the lyapunov matrix which carries many system properties such as h sub 2 performance covariance bounds system entropy at infinity etc all low order h sub spl infin controllers are shown to have observer based structure for reduced order models of the plant and are characterized by two riccati equations with a coupling condition

the dynamics of rayleigh quotient iteration this paper analyzes the rayleigh quotient iteration algorithm for symmetric matrices dynamical systems techniques are employed to characterize the sets of points for which the algorithm will converge to an eigenvector it is shown that these sets have full measure and that their geometric nature is related to the spacing of the eigenvalues

a grassmann rayleigh quotient iteration for computing invariant subspaces the classical rayleigh quotient iteration rqi allows one to compute a one dimensional invariant subspace of a symmetric matrix a here we propose a generalization of the rqi which computes a p dimensional invariant subspace of a cubic convergence is preserved and the cost per iteration is low compared to other methods proposed in the literature

stabilization of discrete time systems by first order controllers in this note we consider the problem of stabilizing a given but arbitrary linear time invariant discrete time system with transfer function p z by a first order discrete time feedback controller c z zx sub 1 x sub 2 z x sub 3 the complete set of stabilizing controllers is determined in the controller parameter space x sub 1 x sub 2 x sub 3 the solution involves the chebyshev representation of the characteristic equation on the unit circle the set is shown to be computable explicitly for fixed x sub 3 by solving linear equations involving the chebyshev polynomials in closed form and the three dimensional set is recovered by sweeping over the scalar parameter x sub 3 this result gives a constructive solution of 1 the problem of first order stabilizability of a given plant 2 simultaneous stabilization of a set of plants p sub i z and 3 stable or minimum phase first order stabilization of a plant the solution is facilitated by the fact that it is based on linear equations and the intersection of sets can be found by adding more equations in each case the complete set of solutions is found and this feature is essential and important for imposing further design requirements illustrative examples are included

least squares matching problems abstract a large class of matching problems can be thought of in the following way one is given a group g which acts on a riemannian manifold and one is given two points x 1 and x 2 in the manifold the problem is to find g g such that d g x 1 x 2 is as small as possible if the group has finite cardinality this is a combinatorial optimization problem whereas if g is a lie group one can use the calculus to obtain necessary conditions our main results establish circumstances under which one can relax the combinatorial problem to an appropriate lie group problem and then solve it by the method of steepest descent an interesting class of applications involves embedding the permutation group on n letters in the orthogonal group o n with the latter acting on r n n 1 2 via a symmetricized tensor product the convergence properties of the descent equation which arises in this way are analyzed in some detail

state space formulae for all stabilizing controllers that satisfy and h norm bound and relations to risk sensitivity abstract given a linear system all stabilizing controllers such that a specified closed loop transfer function has h norm less than a given scalar are parametrized this characterization involves the solution to two algebraic riccati equations each with the same order as the system and further gives gives feasible controllers also with this order the relationship to the risk sensitive lqg stochastic control problem is established giving an equivalence between robust and stochastic control

the projected gradient methods for least squares matrix approximations with spectral constraints the problems of computing least squares approximations for various types of real and symmetric matrices subject to spectral constraints share a common structure this paper describes a general procedure in using the projected gradient method it is shown that the projected gradient of the objective function on the manifold of constraints usually can be formulated explicitly this gives rise to the construction of a descent flow that can be followed numerically the explicit form also facilitates the computation of the second order optimality conditions examples of applications are discussed with slight modifications the procedure can be extended to solve least squares problems for general matrices subject to singular value constraints

h sub spl infin design with first order controllers the problem of determining all first order controllers c s x 1s x2 s x3 which stabilize a given single input single output siso linear time invariant lti plant of arbitrary order has been recently solved in this note these results are extended to determine the subset of controllers which also satisfy various robustness and performance specifications which can be formulated as specific hinfin norm constraints the problem is solved by converting the hinfin problem into the simultaneous stabilization of the closed loop characteristic polynomial and a family of related complex polynomials the stability boundary of each of these polynomials can be computed explicitly for fixed x3 by solving linear equations the union of the resulting stability regions yields the set of all x1 and x2 which simultaneously satisfy the hinfin condition and closed loop stability for a fixed x3 the entire three dimensional set meeting specifications is obtained by sweeping x3 over the stabilizing range

liapunov and covariance controllers the early work of liapunov produced some of the most powerful tools for stability analysis that remain to this day to capture the class of all stabilizing controllers one would be well served by posing the problem in terms of the existence of a liapunov function since a liapunov function is known to exist for stable systems for linear systems this paper derives the set of all quadratic liapunov functions for output feedback control problems and in this way parametrizes the set of all stabilizing controllers of fixed order this is a unifying framework from which all other controllers can be produced by special choices of the free parameters in these controllers we will show how to choose the free parameters to produce all covariance controllers and all h controllers of fixed order these results also apply to robustness analysis and provide a closed form expression for the set of all stabilizing real structured perturbations due to the assignment of a matrix property to the system e g covaria

self similar flows abstract large classes of self similar isospectral flows can be viewed as continuous analogues of certain matrix eigenvalue algorithms in particular there exist families of flows associated with the qr lr and cholesky eigenvalue algorithms this paper uses lie theory to develop a general theory of self similar flows which includes the qr lr and cholesky flows as special cases also included are new families of flows associated with the sr and hr eigenvalue algorithms the basic theory produces analogues of unshifted single step eigenvalue algorithms but it is also shown how the theory can be extended to include flows which are continuous analogues of shifted and multiple step eigenvalue algorithms

the continuous time rayleigh quotient flow on the sphere a continuous time differential equation inspired by the rayleigh quotient iteration for a symmetric matrix a is studied the set of all continuous solutions termed the rayleigh quotient flow is shown to be a time scaled version of the newton flow for rayleigh quotient cost functional the scaling factor ensures that the rate of variation of the rayleigh quotient is constant and positive along solutions this interpretation leads to a precise phase portrait for the rayleigh quotient flow it is shown that a complete non degenerate solution of the rayleigh quotient flow visits each of the eigenvectors of a in ascending order

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

relations between maximum entropy h control and combined h lqg control abstract the maximum entropy h control problem and a combined h lqg control problem are presented in a unified way and shown to be equivalent in the case of a full order controller and equalized h lqg weights the result links two separate approaches which have distinctly different methods of solution only two decoupled algebraic riccati equations are needed to state the controller maximum value of the entropy and minimum value of the auxiliary cost

ordinary differential equations and the symmetric eigenvalue problem in this paper the authors develop a general framework for calculating the eigenvalues of a symmetric matrix using ordinary differential equations new algorithms are suggested and old algorithms including qr are interpreted

disturbance attenuation and h optimization a design method based on the algebraic riccati equation this note presents a method for designing a state feedback control law to reduce the effect of disturbances on the output of a given linear system the problem under consideration involves attenuating the disturbances to a prespecified level the construction of the state feedback control law requires the solution of a certain algebraic riccati equation by applying the procedure with successively smaller values of the prespecifled disturbance level the result also provides an approach to the h infty optimization problem for the state feedback case