Generalized linear dynamical systems and the classification problem

We consider the classification of generalized linear controllable systems over the field = ℂ or = ℝ under transformations defined by the action of the group GL _n ( ) × GL _n ( ). We review the recent results of Cobb, Helmke, Shayman, Zhou, Hinrichsen, O’Halloran, and others on the geometric structure of the set of orbits C _n,m ( ) of generalized linear controllable systems, which in particular prove smoothness, compactness, and projectivity of C _n,m ( ) and evaluate its dimension. We show that C _n,m ( ) is a natural compactification of the set of orbits of ordinary linear controllable systems ∑ _n,m ( ) and the boundary C _n,m ( ) − ∑ _n,m ( ) consists of the orbits of singular generalized systems. __________ Translated from Nelineinaya Dinamika i Upravlenie, No. 4, pp. 153–166, 2004.     

Solution of linear systems from an optimal control problem arising in wind simulation

Several solution strategies for a class of large, sparse linear systems with a block 2 × 2 structure arising from the finite element discretization of an optimal control problem in wind simulation are introduced and analyzed. Block preconditioners and a sparse direct solver on the original coupled system are compared with a preconditioned GMRES iteration applied to a reduced system (Schur complement). Theoretical and experimental results demonstrate the effectiveness of the reduced system approach. Copyright © 2009 John Wiley & Sons, Ltd.     

State estimation of linear dynamical systems under bounded control

The problem of estimation of all possible states that a linear system under bounded control may take (namely, the reachable or attainable set) is addressed. A number of previously developed Lyapunov techniques for estimating the reachable set of ann-dimensional linear system are extended and compared. The techniques produce over-estimates in the form ofn-dimensional ellipsoids. Illustrative examples are solved.This work was supported by the Natural Sciences and Engineering Research Council of Canada, Operating Grant Nos. A-0621 and A-4080, and by Research Personnel Support from the Dean of Science, Memorial University of Newfoundland, Canada.     

Asymptotically optimal estimation in a linear regression problem with random errors in coefficients

We consider the problem of estimating an unknown one-dimensional parameter in the linear regression problem in the case when the independent variables (called coefficients in the article) are measured with errors, and the variances of the principal observations can depend on the main parameter. We study the behavior of two-step estimators, previously introduced by the authors, which are asymptotically optimal in the case when the independent variables are measured without errors. Under sufficiently general assumptions we find necessary and sufficient conditions for the asymptotic normality and asymptotic optimality of these estimators in the new setup.     

Stabilization of linear dynamical systems by optimal controls of linear-quadratic problems

A method of obtaining a bounded feedback which stabilizes a linear dynamical system is described. The attainment of a positional solution of a special auxilliary linear-quadratic problem of optimal control using an optimal regulator is the basis of the method. The results are illustrated for the well-known problem of the stabilization of a pendulum in the upper unstable positions by means of inertial controls.     

A convex optimal control problem involving a class of linear hyperbolic systems

In this paper, we consider a convex optimal control problem involving a class of linear hyperbolic partial differential systems. A computational algorithm which generates minimizing sequences of controls is devised. The convergence properties of the algorithm are investigated.     

Reduction of dimension of optimal estimation problems for dynamical systems with singular perturbations

The possibility of applying the method of integral manifolds to the reduction of optimal filtering problems for systems with low energy dissipation is explored. For such systems, it is shown that the slow subsystem of matrix Riccati differential equations turns out to have a higher dimension than expected, which leads to an increase in the dimension of the reduced problems. An optimal filter is constructed for the Langevin equation and for a dynamic model of a single-link flexible manipulator.     

Asymptotic estimation of states of linear nonautonomous systems with scalar output

For a linear nonautonomous system with one output, the existence conditions for an identifier that estimates the current state with exponential rate are established. An algorithm for constructing such an identifier is described and its dependence on small perturbations of coefficients is studied.     

Non-singular locally optimal ellipsoidal approximation of the estimate of the states of linear systems

A problem which arises when estimating the attainability domains of linear dynamical systems by ellipsoids is investigated in a short time interval in the case when the initial position of the system in phase space is known precisely for some at least coordinates. A method is proposed which allows one to avoid problems associated with the degeneracy of the right-hand sides of the differential equations of the locally optimal ellipsoidal approximation. The mathematical meaning of these equations is made more precise in the case of the minimization of the phase volume. An example is given.     

Perturbed linear stochastic dynamical systems

This paper is concerned with a class of stochastic differential equations which arises by adding a nonlinear term involving a small parameter δ to the drift coefficient of a linear stochastic system. First, a stochastic representation of the solutions of a certain class of Cauchy problems is studied. Second, a finite time expansion in powers of δ of the expectations of functions is established. Third, the corresponding ergodic expansions are sought with additional assumptions regarding the existence of a unique ergodic measure.     

Example of a linear abnormal optimal control problem

We give an example of a linear optimal control problem for which the Pontryagin maximum principle holds on the optimal control only for λ ⁰ = 0.     

Asymptotically optimal estimation in the linear regression problem in the case of violation of some classical assumptions

We consider the problem of estimating the unknown parameters of linear regression in the case when the variances of observations depend on the unknown parameters of the model. A two-step method is suggested for constructing asymptotically linear estimators. Some general sufficient conditions for the asymptotic normality of the estimators are found, and an explicit form is established of the best asymptotically linear estimators. The behavior of the estimators is studied in detail in the case when the parameter of the regression model is one-dimensional.     

Inversion of linear delay dynamical systems

We consider the problem of inversion of a dynamical system, that is, the problem of real-time reconstruction of the unknown input of the system on the basis of measurements of its output. We consider linear systems of functional-differential equations in the case of commensurable delays. We obtain necessary conditions for the invertibility of this class of systems and suggest an inversion algorithm that permits one to obtain an estimate of the unknown input signal with given accuracy.     

On observing the states of uncertain dynamical systems

For the implementation of a stable asymptotic reduced-order Luenberger observer, which is not sensitive to unknown disturbances entering the system, it is shown that specific columns of the measurement matrixC must be different from zero.This research was supported by the Technion VPR Fund, L. Rogow Aeronautical Research Fund.The author wishes to thank E. P. Ryan, G. Leitmann, and S. Gutman for helpful discussions.