Multidimensional Linear Systems Research Articles

Certain martingale methods of parameter estimation in dynamic system

PurposeTo find martingale theory and methods of parameter vector estimation of multidimensional linear control system in dynamic system.Design/methodology/approachIn parameter vector estimation of dynamic system, due to the limitation of conventional methods, the authors study the iterative formula and consistency of parameter vector estimation using the theory and methods of martingale, and extend its application field.FindingsSome conditions of strongly consistent estimation and iterative formula are obtained. And the consistency of parameter vector estimation in multidimensional linear control system is discussed.Practical implicationsSolved some practical problems of parameter vector estimation in dynamic system.Originality/valueThe paper solve the accuracy problem of parameter vector estimation of multidimensional linear control system in dynamic system and extend parameter vector estimation application field.

Reduction of the Rosenbrock matrix in analysis of invariant zeros of the linear MIMO-System

The problem is considered of reduction of the Rosenbrock matrix in analysis of invariant zeros of a linear multidimensional dynamical system with many inputs and many outputs. Approaches to the Rosenbrock matrix reduction are compared, which are carried out with the aid of the transformation of initial system equations to the Yokoyama canonical form and on the basis of zero divisors of numerical matrices. It is shown that the approach to the Rosenbrock matrix reduction on the basis of zero divisors is less arduous and offers the possibility of obtaining a well-posed problem in eigenvalues of the generalized linear bundle, while the reduction on the basis of subspaces of A.N. Krylov leads to the problem in eigenvalues of the nonlinear (polynomial) bundle of matrices, and its stipulation is generally rather poor. Various versions of the reduced matrices are presented, which are obtained in accordance with properties of the dynamical system. Properties of the Rosenbrock reduced matrices are analyzed.

Suboptimal minimax design of constrained parabolic systems with mixed boundary control

The paper concerns minimax control problems for linear multidimensional parabolic systems with distributed uncertain perturbations and control functions acting in mixed (Robin) boundary conditions. The main goal is to design a feedback control regulator that ensures the required state performance and robust stability under any feasible perturbations and minimize an energy-type functional under the worst perturbations from the given area. We design and justify an easily implemented suboptimal structure of the feedback boundary regulator and compute its optimal parameters ensuring the required state performance and robust stability of the nonlinear closed-loop control system on the infinite horizon.

Mean-square stability of second-order Runge–Kutta methods for multi-dimensional linear stochastic differential systems

In this paper, the mean-square stability of second-order Runge–Kutta schemes for multi-dimensional linear stochastic differential systems is studied. Motivated by the work of Tocino [Mean-square stability of second-order Runge–Kutta methods for stochastic differential equations, J. Comput. Appl. Math. 175 (2005) 355–367] and Saito and Mitsui [Mean-square stability of numerical schemes for stochastic differential systems, in: International Conference on SCIentific Computation and Differential Equations, July 29–August 3 2001, Vancouver, British Columbia, Canada] we investigate the mean-square stability of second-order Runge–Kutta schemes for multi-dimensional linear stochastic differential systems with one multiplicative noise. Stability criteria are established and numerical examples that confirm the theoretical results are also presented.

Ellipsoid-based parametric estimation in the linear multidimensional systems with uncertain model description

Parametric estimation under uncertainty of the plant model description was considered within the framework of the ellipsoidal approach to the problems of guaranteed estimation. The unknown multidimensional plant whose parameter vector should be estimated was assumed to be linear and static, and uncertainty of its "input-output" model, to have both additive and multiplicative components. The external ellipsoidal approximations of the nonconvex information sets guaranteeing that the vector of possible plant parameters is contained in them were constructed from the results of observations. The method of their construction comes to semidefinite programming, that is, to minimization of the linear function constrained by the linear matrix inequalities which are readily realized by the numerical methods.

Periodic, Strict, and Quasi-Invariance in Linear Systems

This brief deals with shift invariance in linear systems (LSs), using a group theoretical approach. The group approach is necessary to obtain general definitions, which can be applied to continuous-time, discrete-time, multirate and also multidimensional LSs. Thus, a unique definition is given for periodic invariance and strict invariance (SI) and also for quasi-invariance (QI), a new concept, intermediate between periodic and SI. The importance of QI relies upon the fact that it is comprehensive of practically all multirate LSs

Robust and efficient quantization and coding for control of multidimensional linear systems under data rate constraints

AbstractRecently, we reported results on coding strategies for scalar feedback systems with data‐rate‐limited feedback channels in which the data‐rate constraints are time varying. Such rate‐varying channels are typically encountered in communication networks in which links between nodes are subject to noise, congestion, and intermittent disruption.The present paper describes results of extending this research into the multidimensional domain. An important consideration is that for systems of dimension greater than one, many classical feedback designs cannot be realized for operation near the theoretical minimum possible data rate. A novel control coding scheme will be presented, and in terms of this, it will be shown that the advantages of coarse signal quantization that had been reported earlier for scalar systems remain in the multidimensional case. The key is to allocate the communication bandwidth efficiently among faster and slower modes. We discuss various strategies that allocate bandwidth by scheduling the time slots assigned to each mode. In particular, we propose a ‘robust attention varying’ technique, whose merit will be discussed in terms of its robustness with respect to time‐varying communication channel capacity and also in terms of how well it operates when the feedback channel capacity is near the theoretical minimum data rate. Copyright © 2006 John Wiley & Sons, Ltd.

Asymptotic Stability of Second-Order Linear Time-Varying Systems

M ANY mechanical systems belong to a special class of dynamical systems that are generally described by a secondorder linear matrix differential equation after linearization about an equilibrium condition. When the system characteristics change with time, the coefficient matrices of the equation are time-varying. Examples can be found in all fields of engineering, a space station carrying a mobile crane and a space structure extended in orbit being typical examples from space engineering. This study is motivated by the latter, where the stabilization of a spacecraft undergoing a configuration change in a gravity gradient torque was investigated [1]. In this paper, we investigate the stability of the following secondorder multidimensional linear time-varying system:

Control design for suppressing transversal patterns in reaction–(convection)–diffusion systems

The stabilization of planar stationary fronts solutions in a two-dimensional rectangular or cylinder domain, in which a diffusion–convection–reaction process occurs, is studied by reducing the original two-variable PDEs model to an approximate one-dimensional model that describes the behavior of the front line. We consider the control strategy based on sensors placed at the designed front line position and actuators that are spatially-uniform or space dependent. We present a systematic control design that determines the number of required sensors and actuators, their position and their form. The control used linear analysis of a lumped truncated model and concepts of finite and infinite zeros of linear multidimensional systems.

A Solvable Lie Algebra Condition for Stability of Linear Multidimensional Systems

This note analyzes exponential stability of a class of linear discrete multidimensional systems. Using a multidimensional comparison principle for estimating the system componentwise exponential convergence and a solvable Lie algebra condition, a sufficient condition for exponential stability of linear multidimensional systems is presented. The stability condition can be easily examined by computing the system matrices in finite steps. This is demonstrated by an example.

ROBUST CONTROL OF STATIONARY PLANAR FRONTS IN REACTION-DIFFUSION SYSTEMS

This paper considers a new approach to construct a feedback controller that stabilizes a front line solution of a nonlinear parabolic distributed (reaction- diffusion) system in a planar domain. The controller incorporates several space- dependent actuators that respond to sensors located at the front position. Sensor numbers and its locations are chosen by the multivariable root-locus technique for the finite-dimensional approximation of the original PDE model. The concept of finite and infinite zeros of linear multidimensional systems is used. The theoretical results are confirmed by computer simulations.

Iterative design of robust controllers under unknown bounded perturbation norms

A suboptimal robust controller for a multidimensional linear stationary nominal system with output- and control-operator perturbations and a bounded external perturbation is designed. Perturbation norms (weights) are assumed to be unknown and estimated from measurement data. The quality index of the closed-loop control system is taken to be the worst ??-norm for the output of the system in the class of admissible perturbations. The quality index for systems in the class of linear stationary stabilizing controllers is a fractional linear function of induced norms of transfer matrices of the closed-loop system and perturbation norms. The classical design of suboptimal robust controllers under known perturbation norms is reduced to a standard ?1-optimization problem, and optimal estimation of unknown perturbation norms is reduced to a linear programming problem. Therefore, the iterative sequential method of controller design and perturbation norm estimation is used as a heuristic method for designing suboptimal robust controllers under unknown perturbation norms. Modeling results corroborate the effectiveness of this method.

Bounded Inputs and the Representation of Linear System Maps

We give an expression for the most general input-output map associated with the members of a certain important large family of multidimensional linear shift-invariant systems with bounded Lebesgue-measurable inputs. The expression given is an iterated function-space limit of a convolution. We also give a necessary and sufficient condition under which the limit can be written as a convolution with an integrable impulse-response function. A key role is played by a certain family of weighting operators. It is observed that for the large family of inputs and maps addressed, the Dirac impulse-response concept is in fact not the key concept concerning the representation of H, and that instead the inputoutput properties of H are determined in general by a certain type of family of responses. Some related material concerning engineering education is also given.

Notes on linear systems and frequency responses

We show that not all continuous multidimensional linear shift-invariant systems are characterized by their frequency responses (even when they exist), but that the members of a certain important large family of multidimensional linear systems are completely characterized by their suitably defined frequency responses. Copyright © 2005 John Wiley & Sons, Ltd.

Structured Noncommutative Multidimensional Linear Systems

We introduce a class of multidimensional linear systems with evolution along a free semigroup. The transfer function for such a system is a formal power series in noncommuting indeterminates. Standard system-theoretic properties (the operations of cascade/parallel connection and inversion, controllability, observability, Kalman decomposition, state-space similarity theorem, minimal state-space realizations, Hankel operators, realization theory) are developed for this class of systems. We also draw out the connections with the much earlier studied theory of rational and recognizable formal power series. Applications include linear-fractional models for classical discrete-time systems with structured, time-varying uncertainty, dimensionless formulas in robust control, multiscale systems and automata theory, and the theory of formal languages.

Multidimensional Schur coefficients and BIBO stability

In the framework of BIBO stability tests for one-dimensional (1-D) linear systems, the Schur-Cohn stability test has the appealing property of being a recursive algorithm. This is a consequence of the simultaneously algebric and analytic aspect of the Schur coefficients, which can be also regarded as reflection coefficients. In the multidimensional setting, this dual aspect gives rise to two extension of the Schur coefficients that are no longer equivalent. This paper presents the two extensions of the Schur-Cohn stability test that derive from these extended Schur coefficients. The reflection-coefficient approach was recently proposed in the 2-D case as a necessary but non sufficient condition of stability. The Schur-type multidimensional approach provides a stronger condition of stability, which is necessary and sufficient condition of stability for multidimensional linear system. This extension is based on so-called slice function associated to n-variable analytic functions. Several examples are given to illustrate this approach.

PARAMETRIZING ALL SOLUTIONS OF UNCONTROLLABLE MULTIDIMENSIONAL LINEAR SYSTEMS

Using an algebraic analysis approach, we derive a necessary and sufficient condition so that we can parametrize all solutions of a multidimensional linear system by glueing the controllable sub-behaviour with the autonomous elements. Effective algorithms checking this condition are obtained. This result generalizes a result of 1-D linear systems for a class of multidimensional linear systems.

Notes on Linear Systems and Impulse Responses

We show that the members of a certain important large family of multidimensional linear systems are completely characterized by their (suitably defined) impulse responses—which we show exist as certain limits. Not all linear shift-invariant systems are characterized by their impulse responses. Our results contribute to the reconstruction of an important part of the engineering theory of linear systems that had been based on the flawed concept of the Dirac impulse function.

Conservative linear systems, unitary colligations and Lax–Phillips scattering: multidimensional generalizations

Connections between conservative linear systems, Lax–Phillips scattering, and operator model theory are well known. A common thread in all the theories is a contractive, analytic, operator-valued function on the unit disc T(z) having a representation of the form T(z) = D + z C (I − zA)−1 B, known as the transfer or frequency-response function in the system-theory community, the scattering function in the mathematical physics community, and the characteristic operator function in the operator theory community. Here we consider analogues of this circle of ideas in the more general setting of multidimensional systems/multi-evolution scattering systems/multivariable function-theoretic operator theory. Three particular extensions are discussed; from the point of view of system theory, these involve (1) a multidimensional linear system with transfer function a contractive analytic operator function on the unit polydisc in complex Euclidean space, (2) a non-commutative multidimensional linear system with evolution along a free semigroup and with transfer function equal to a formal power series in non-commuting indeterminants, and (3) an overdetermined multidimensional linear system with transfer function identified as a bundle mapping between two Hermitian vector bundles over an algebraic curve embedded in complex projective space. This survey updates an earlier survey by the first author appearing in 2000.

A differential operator approach to multidimensional optimal control

We use recent improvements in the parametrizations of controllable linear multidimensional systems to show how to transform the study of a linear quadratic optimal problem into that of a variational problem without constraints. We give formal conditions on the differential module defined by the linear control system to pass from the Pontryagin approach to a purely Euler–Lagrange variational problem. This formal approach uses the cost function in order to link the locally exact sequence formed by the controllable system and its parametrizations with the sequence formed by their formal adjoint operators. In the case of partial differential equations, this scheme is typical for any problem of linear elasticity theory and electromagnetism.