Linear Constant Systems Research Articles

Continuous time stochastic adaptive control: non-explosion, ϵ-consistency and stability

For completely observed continuous time constant parameter stochastic linear systems, an indirect adaptive control law is presented which, subject principally to a weak location hypothesis concerning the true parameter, and a persistent excitation hypothesis, generates ϵ-consistent recursive least squares parameter estimates and ensures the system is mean square sample path stable. The adaptive control algorith mentails (i) recursively calculating the least squares estimate of the system parameter, and (ii) recursively generating the LQR feedback gain matrix lying in a set of matrix gains γ known to contain a stabilizing gain. The a.s. non-explosion of the system is a direct consequence of this construction.

Identification and rational L 2 approximation A gradient algorithm

This paper deals with the identification of linear constant dynamical systems when formalized as a rational approximation problem. The criterion is the l 2 norm of the transfer function, which is of interest in a stochastic context. The problem can be expressed as nonlinear optimization in a Hilbert space, but standard algorithms are usually not well adapted. We present a generic recursive procedure to find a local optimum of the criterion in the case of scalar systems. Our methods are borrowed from differential theory mixed with a bit of classical complex analysis. To our knowledge, the algorithm described in this paper is the first that ensures convergence to a local minimum.

Comments on ‘Driving a linear constant system by a piecewise constant control’

Earlier results on controllability by means of piecewise constant controls are reviewed. Implications of the resulting freedom in the choice of the sampling instants are also indicated as well as consequences for the corresponding observability problem, i.e. state estimation by means of output samples.

Analytical and numerical solutions to higher index linear variable coefficient DAE systems

Analytical and numerical solutions to the singular linear system of differential equations A( t) x′( t) + B( t) x( t) = f( t) are investigated. Closed-form analytical solutions of systems transformable to standard canonical form, SCF (see Campbell (1983)) are derived. Equivalent ways to compute the analytic solution are presented which are of interest in a symbolic/numeric computing environment. Many numerical methods for solving DAE systems are based on Gear's backward difference formula (BDF) method. It is proved that the BDF method and the modified BDF method, MBDF (see Clark (1986)) are stable and O( h k ) accurate after a maximum of ( s − 1) k + 1 steps for linear variable coefficient systems in SCF, where s is the dimension of the algebraic part. These new results are valid for arbitrarily higher index systems and generalize well-known results for linear constant coefficient singular systems to the variable coefficient case. Numerical experiments confirm the order and stability results.

Generalized disturbance localization using energy-based interaction measures

An energy-based, continuous-valued measure of interaction between the input and output channels of a linear constant system is formulated and used as the basis for a theory of near disturbance localization for discrete-time systems. The generalized disturbance localization problem is formulated as the minimization problem of finding the feedback gain matrix F for which this interaction measure is minimized. The solutions obtained are shown to correspond where appropriate with those predicted by the geometric state-space approach. Connections are made, too, to results on system invertibility and on ‘cheap’ linear-quadratic control problems.

Generalized controller canonical form for linear and nonlinear dynamics

A generalized controller canonical form for linear and nonlinear dynamics is proposed. It is obtained using the theorem of the primitive element from differential algebra. The derivation of the controller form does not apply to multivariable constant linear systems. With the transformations, all nonlinear dynamics can be exactly linearized by dynamic feedback. The main departure from standard theory is that transformations may depend on input derivatives. Once differential-algebraic tools are introduced, the proofs of the results are easy. >

On successive pole assignment by linear-quadratic optimal feedbacks

This paper studies a method of shifting poles of linear constant systems via LQ optimal feedback. By making use of a solution to the so-called inverse regulator problem, this method enables us to shift poles successively by pairs while maintaining the well-known advantages of LQ regulators. Polynomial fractional representation is effectively used both to characterize the maximal pole-assignable area, which clarifies the theoretical bound of the LQ optimal pole shifting, and to derive an algorithm. A numerical example is given to illustrate the results.

The Numerical Solution of Higher Iindex Differential/Algebraic Equations by Implicit Methods

This paper studies the order, stability, and convergence properties of implicit Runge–Kutta (IRK) methods applied to differential/algebraic systems with index greater than one. These methods do not in general attain the same order of accuracy for higher index differential/algebraic systems as they do for index 1 systems or for purely differential systems. Necessary and sufficient conditions on the method coefficients are derived to ensure that the local and global errors of the method attain a given order of accuracy for higher index linear constant coefficient systems. IRK methods applied to nonlinear semi-explicit index 2 systems are studied, and a sufficient set of conditions is derived which ensures that a method is accurate to a given order for these systems. Finally, some numerical experiments are presented that illustrate the theoretical results and demonstrate the effects of roundoff' errors on the solution.

Optimization of Interconnecting Parameters Regions of Stability for Large-scale Systems

In this paper, it has been proposed by the authors optimization of interconnecting parameters regions of stability for large-scale systems. Making use of the method of weighted sum type of Lyapunov function, the optimization of interconnecting parameters regions of stability has been realized on the 2-order linear constant composite system. At the same time, it has been obtained some interconnecting parameters regions of stability of n-order linear and nonlinear constant large-scale systems.

Generalized linear systems with lumped or distributed parameters and differential vector spaces

Abstract By employing differential vector spaces, we obtain results for linear systems with constant or varying coefficients and with lumped or distributed parameters which were only known for constant linear systems described by rational transfer matrices. Questions such as the characterization of input and output, input-output equivalence, invertibility, feedback decoupling and realization are investigated.

Generalized linear systems with lumped or distributed parameters and differential vector spaces

Abstract By employing differential vector spaces, we obtain results for linear systems with constant or varying coefficients and with lumped or distributed parameters which were only known for constant linear systems described by rational transfer matrices. Questions such as the characterization of input and output, input-output equivalence, invertibility, feedback decoupling and realization are investigated.

Improved condition for the optimality off decentralised control for large-scale systems

For a linear constant large-scale system composed for interconnected subsystems, if a decentralised control law which minimises a quadratic cost function with respect to individual decoupled subsystems is used, this simple control strategy is shown to result in a globally optimal interconnected system, provided the condition given by M. Ikeda and D.D. Siljak (1982) is satisfied and the quadratic cost function is modified to include the additional terms arising from the subsystem interactions. An improved condition for the global optimality is given, which implies that, as long as a local eigenvalue condition given in the paper is satisfied for the decentralised local optimal control and the global detectability condition is satisfied, global optimality can also be ensured. It can be expected that, owing to this new simple condition, some of the difficulties in the application of decentralised optimal control approaches could be overcome. >

Sensitivities of frequency response functions of linear dynamic systems to variations in design parameter values

Expressions for the sensitivities of the frequency response functions of linear constant coefficient systems with distinct eigenvalues to perturbations in the values of their design parameters are derived. The expressions are of each of two forms, the one involving complex number (phasor) representation of the responses and the sensitivities, the other involving separate amplitude and phase representations. The relations derived contain eigenvalue, eigenvector and adjoint eigenvector sensitivities. Engineering applications of the theory are suggested.

New Tools in Rational L2 Approximation

This paper follows new trends in system identification and deals with the problem of finding a best approximant of a given (possibly non rational) linear constant discrete system. The criterion is the l2 norm of the transfer function, which is of interest in a stochastic context. Though this problem can be formalized as an optimization problem in a suitable hilbert space, standard algorithms don't fit the geometry of the situation. In this article, we present a generic algorithm to find a local optimum of the criterion in the special case when all dynamical systems involved have a single input and a single output. It is based on the recursive structure of the problem which appears in studying it from our viewpoint. It should be noticed that our methods are borrowed from the by now classical differential theory mixed up with a bit of classical complex analysis. To our knowledge, the procedure described in this paper is the first one that ensures convergence to a local minimum. About developments, we believe that the results described therein could give rise to an extension to the multi-input multi-output case, but this has not been done yet, and belongs to ongoing research.

Driving a linear constant system by a piecewise constant control

Recent studies have shown that complete controllability by means of all admissible controls is equivalent to complete controllability by means of piecewise constant controls with n preassigned switching times. This paper addresses the problem of determining the switching times and provides a simple algorithm for the derivation of a piecewise constant control function which accomplishes the transfer of the system from its current position to a desired reachable target. A discussion on the applications of the results obtained in this study to design problems, especially to computer-controlled systems, is given.

Continued-fraction decomposition of linear systems in the state space

Various results pertaining to a continued-fraction decomposition of matrix formal power series, and in particular of multi-input multi-output linear constant systems, are presented from a state-space point of view.

The relationship between the stability and the optimality of linear systems (II)

Different from the inverse problem put forward by R.E.Kalman, another kind of inverse problem of linear optimal control is proposed and discussed in [1] as follows: Given an asymptotically stable linear constant system and a nonnegative quadratic performance index, when can a state-feedback be separated from the stable system so that this state-feedback control law is optimal for the given index? In this paper this problem is extended. Similar conclusions are obtained for linear discrete systems and linear time-variable systems. According to these conclusions we can say that the correspondence between the asymptotically stable system and the optimal feedback system is the inherent character of all kinds of linear systems.

A generalization of Rosenbrock's theorem

This note presents the simple generalization of Rosenbroek's theorem, concerning the limits of linear state variable feedback in modifying the dynamics of constant linear multivariable systems.

Feedback equivalence of constant linear systems

Two constant linear systems are said to be feedback equivalent if one can be transformed into the other via an element of the “feedback group”, which acts by state space feedback and by change of basis in the state and input spaces. Let C n,m be the space of n-dimensional completely reachable systems with m-dimensional input (pairs of matrices, n × n and n × m). The action of the feedback group partitions the space C n,m into finitely many orbits (equivalence classes), and the closure of each orbit is a union of orbits. If one views orbit closure as ‘deformation’, then orbit closure may be considered in terms of perturbations or system failure. In this paper we determine: (1) a classification of the orbits, and (2) the orbits contained in the closure of a given orbit. Both of these problems have been solved previously (see [1,4,6,3]); here we present simple proofs and point out a connection between this problem and the analogous problem for nilpotent matrices.

A note on the invertibility of nonlinear input-output differential systems

A new conceptual framework employing differential algebra is proposed for solving the long-standing problem of inverting nonlinear input-output differential systems. Simple characterizations of left and right invertibility are given which rest upon an invariant integer which, for constant linear systems, turns out to be the rank of the transfer matrix.