causal eigenvector function approximations and the problem of scaling an algorithm abstract eigenvector scaling implies the existence of degrees of freedom in the power series approximations of the eigenvector and dual eigenvector functions of the discrete time transfer function matrix an earlier paper gave consideration to the anticlockwise winding associated with anti causality and proposed an algorithm which scaled eigenvectors with a view to reducing their anti causal component the present paper proposes a different algorithm which although still formulated in the frequency domain places a direct penalty on the time domain anti causal component of the eigenvector sequences and avoids some of the aliasing difficulties that could arise in connection with the earlier algorithm the new algorithm also provides the systematic means of reducing the length of the causal component of the associated vector sequences and results in better eigenvector approximations as illustrated by two numerical examples

a differential algebraic riccati equation for applications in flow control we investigate existence and structure of solutions to quadratic control problems with semiexplicit differential algebraic constraints as they appear in the modeling of flows we introduce a decoupled representation of the state and the formal adjoint equations to identify the conditions for the existence of solutions we derive a differential algebraic riccati equation formulated in the original coefficients for the efficient numerical computation of the optimal solution

numerical algorithms and existence results on lq control of descriptor systems with conditions on x 0 and stability we present a numerical algorithm based on using orthogonal matrices and sufficient existence conditions for an lq control problem of descriptor systems which generalise the existing results the associated hamiltonian pencil may be singular and may have infinite generalised eigenvalues iges of multiplicity 1 besides on x 0 we present optimality conditions on x 0 we present sufficient conditions for stability of the optimal closed loop descriptor system as well as a stabilisation algorithm

direct transcription solution of high index optimal control problems and regular euler lagrange equations direct transcription methods are used to solve optimal control problems in many industrial settings models for physical systems often take the form of differential algebraic equations daes the index of the dae traditionally is viewed as an important factor in deciding whether a particular numerical approach should be used recently it has been observed that what the user thinks is the index of the dae may not be the same as the index available to the optimization software an investigation of this fact is underway in order to develop guidelines to assist users of various numerical optimal control packages this paper develops some theoretical results that will be needed for this development

discounted cost linear quadratic gaussian control for descriptor systems we consider the linear quadratic gaussian control problem with a discounted cost functional for descriptor systems on the infinite time horizon based on recent results from the deterministic frame

spectral factorization via hermitian pencils abstract the spectral factorization problem is solved for state space systems via results on the canonical forms and inertia of hermitian matrix pencils these algebraic results then give a deflation method for spectral factorization

the robustness properties of the linear quadratic regulators for singular systems the lqr linear quadratic regulator robustness properties of singular systems are investigated it is shown that the optimal regulators for singular systems with an appropriate selection of the feedback gain have the same robustness properties as those in regular systems namely at least 60 degrees guaranteed phase margin infinite gain margin and 50 gain reduction margin

impulse controllability and observability of rectangular descriptor systems the presence of impulsive responses in descriptor systems and how it relates to impulse controllability and impulse observability is considered it is shown that the equivalence between impulse controllability observability and the existence of an impulse eliminating semistate feedback output injection gain although true for square descriptor systems does not hold for rectangular descriptor systems in general hence necessary and sufficient conditions for the existence of an impulse eliminating semistate feedback output injection gain are presented

lq control of descriptor systems by cancelling structure at infinity this paper presents results concerning lq control of arbitrary descriptor systems and a numerical algorithm for achieving it the algorithm consists of two steps the first step is using the silverman algorithm orthogonal version transformation of the problem of lq control of descriptor system into a problem of lq control of a strictly proper system with singular or non singular new matrix 22 if new 22 is singular we perform the second step i e solving the lq singular control problem using again the silverman algorithm regarding the existing results on the lq singular control problem we have new results a counter example that this problem may not have a stable solution most general solvability conditions with or without stability and a new numerical algorithm based on using orthogonal matrices when stabilizability conditions are satisfied the algorithm places simultaneously stable poles in the closed loop system we show that the traditional riccati equation associated to this proble

lq control of descriptor systems a spectral factorisation approach a spectral factorisation algorithm of improper matrices and existence results under controllability conditions are presented based on these results an algorithm for linear quadratic lq control of rectangular descriptor systems without the detectability and infinite observability conditions is presented the matrix pencil associated with the euler lagrange differential equations can be singular can have finite generalised eigenvalues on the imaginary axis and infinite generalised eigenvalues of any multiplicity in the class of optimal state feedback controls we find a control that renders the closed loop system marginally stable and impulse free also we find conditions on x 0 that guarantee the optimality it is shown how the disturbance attenuation problem of descriptor systems can be re formulated as an optimal state feedback feedforward control problem for which the results of this article can be applied when applied to state space systems the algorithm of this article which is based on

nonexistence of simple hyperbolic blow up for the quasi geostrophic equation the problem we are concerned with is whether singularities form in finite time in incompressible fluid flows it is well known that the answer is no in the case of euler and navier stokes equations in dimension two in dimension three it is still an open problem for these equations in this paper we focus on a two dimensional active scalar model for the 3d euler vorticity equation constantin majda and tabak 7 suggested by studying rigorous theorems and detailed numerical experiments a general principle if the level set topology in the temperature field for the 2d quasi geostrophic active scalar in the region of strong scalar gradients does not contain a hyperbolic saddle then no finite time singularity is possible numerical simulations showed evidence of singular behavior when the geometry of the level sets of the active scalar contain a hyperbolic saddle there is a naturally associated notion of simple hyperbolic saddle breakdown the main theorem we present in this paper shows that such breakdown cannot occur in finite time we also show that the angle of the saddle cannot close in finite time and it cannot be faster than a double exponential in time using the same techniques we see that analogous results hold for incompressible 2d and 3d euler these results were announced in 9 but with a slight difference in the definition of a simple hyperbolic saddle the definition given in section 4 generalizes the one given in the announcement see also constantin 4 discussed in section 7 remark 5 below the whole work described in this paper is basically part of the author s thesis i am particularly grateful to my thesis advisor charles fefferman for his attention support guidance and advice i am indebted to d christodoulou and p constantin for helpful corrections and suggestions i wish to thank a majda for suggesting the subject and e tabak for discussions and com

the optimal regulation of generalized state space systems with quadratic cost abstract in this paper the optimal feedback control for regulating generalized state space systems with quadratic cost is presented by using an algebraic method and the weighting matrix q in the cost is allowed to be positive semi definite

rational spectral factorization using state space methods abstract we describe a state space based algorithm to find the minimum phase spectral factor of any rational spectral density whether it be proper improper or polynomial

bi c eigenvector sequences and the design of causal commutative controllers abstract the use of laurent power series expansions of the eigenvector matrix of a linear multivariable transfer function matrix g z holds the key to the physical realization of commutative controllers in general however such controllers would be anti causal it is the purpose of this paper to show that there are enough degrees of freedom in the choice of the controller eigenfunctions both to effect gain phase compensation of the frequency response of the eigenfunctions of g z and to force the resulting control law to be causal the results of the paper are shown to be superior to those possible through the use of approximate commutative controllers proposed earlier

h2 and h approximations for eigenvalues vector functions of transfer matrices h2 and h weighting sequence approximations to rational and irrational functions trefethen 1981 helton et al 1989 are pertinent to characteristic locus design the most effective technique combines inverse discrete fourier transforms with what amount to h2 approximations in this paper we discuss the theoretical background and develop some new h2 and h techniques the resulting algorithms can be applied to general rational irrational function and vector function approximation problems they are shown to be particularly effective in the approximation of eigenvalue vector functions of transfer function matrices the derived techniques provide some very useful tools for the characteristic locus design method

the linear quadratic optimal regulator for descriptor systems in this paper we investigate the linear quadratic optimal regulator problem for the continuous time descriptor system e dot x ax bu where e is in general a singular matrix we solve first a general finite horizon problem by applying the calculus of variations to derive the optimal trajectory of the vector consisting of the concatenated descriptor codescriptor and control vectors from this trajectory the optimal feedback gain relating the control and descriptor variable can be computed by transforming to a coordinate system which can be computed by performing a singular value decomposition of e we derive several riccati differential equations all of which have the same solution this solution gives the optimal cost the steady state optimal feedback gain can be computed by solving an eigenvalue eigenvector problem formulated from the untransformed system parameters in general there does not exist a unique optimal feedback gain but rather the gain is constrained to lie in a linear variety whose dimension is equal to the number of inputs times the rank deficiency of e

a two dimensional model for quasigeostrophic flow comparison with the two dimensional euler flow abstract a simple two dimensional model for quasigeostrophic flow is contrasted with the two dimensional incompressible euler equations the model arises under the assumptions of fast rotation uniform stratification and uniform potential vorticity it is found that the more loca feed back of the quasigeostrophic model gives rise to strongly nonlinear front formation as opposed to two dimensional euler where the steepening process of mature fronts obeys a nonlocal nearly linear mechanism

on linear quadratic optimal control problems for time varying descriptor systems we deal with linear quadratic optimal control problems for time varying descriptor systems in a hilbert space setting a sufficient solvability condition is given by means of an appropriately stated linear boundary value problem bvp and by discussing the special structure of the regular differential system inherent in this bvp

commutative controllers revisited parallel computation a new lease of life this paper presents a design methodology for multivariable control that combines time and frequency domain descriptions of the plant to produce appropriate compensators the proposed methodology relies on a novel convolution implementation made possible by developments in vlsi computer technology and parallel processing algorithms the result is an algorithm that completely reduces the multivariable design problem to a set of siso problems to which classical frequency response techniques can be applied the paper also establishes conditions for the existence of the necessary controller representations and proposes methods to ensure that these conditions are satisfied simulation results are presented to demonstrate the utility of the methodology

singular lq problem for nonregular descriptor systems in this paper a singular linear quadratic lq problem for non regular descriptor systems is investigated under some general conditions the optimal control and the optimal state of the lq problem are given the optimal control is synthesized as state feedback all the finite eigenvalues of the closed loop system are located on the left half complex plane the state of the closed loop system has the least free entries

high frequency energy cascades in inviscid hydrodynamics abstract with the aim of gaining insight into the notoriously difficult problem of energy and vorticity cascades in high dimensional incompressible flows we take a simpler and very well understood low dimensional analog and approach it from a new perspective using the fourier transform specifically we study numerically and analytically how kinetic energy moves from one scale to another in solutions of the hyperbolic or inviscid burgers equation in one spatial dimension 1d we restrict our attention to initial conditions which go to zero as x the main result we report here is a fourier analytic way of describing the cascade process we find that the cascade proceeds by rapid growth of a crossover scale below which there is asymptotic power law decay of the magnitude of the fourier transform