Order Infinite Dimensional Systems Research Articles

An approximation method for exact controls of vibrating systems with numerical viscosity

We analyze a method for the approximation of exact controls of a second order infinite dimensional system with bounded input operator. The algorithm combines Russell’s “stabilizability implies controllability” principle and a finite elements method of order θ with vanishing numerical viscosity. We show that the algorithm is convergent for any initial data in the energy space and that the error is of order θ for sufficiently smooth initial data. Both results are consequences of the uniform exponential decay of the discrete solutions guaranteed by the added viscosity and improve previous estimates obtained in the literature. Several numerical examples for the wave and the beam equations are presented to illustrate the method analyzed in this article.

Approximate controllability of second order infinite dimensional systems

Approximate controllability of second order infinite dimensional systems

Approximate controllability of infinite dimensional system with internal damping dependent on fractional powers of system operator

In this study, a class of models governed by a second-order (in time) evolution differential equation with damping and positive definite, self-adjoint system operator is studied. Damping term has a special, but practically important, form of a finite sum of fractional powers of the system operator. This type of damping operator can be used to model a variety of physical phenomena related to dissipation of energy empirically observed in physical systems. It may be used to model dissipation mechanisms resulting from air damping, internal structural damping, internal viscous damping, etc. Using spectral theory of linear unbounded operators and semi-group theory necessary and sufficient conditions of approximate controllability for second order infinite dimensional system with damping are formulated and proved. Some important, from the practical point of view, remarks and comments are also provided. In particular, necessary condition for approximate controllability is formulated and discussed. Finally, an illustrative example related to approximate controllability of distributed parameter system described by the partial differential equation with higher order spatial differential operators is presented. The example refers to a system describing dynamical behaviour of damped Kirchhoff–Love plate. The study extends earlier results on approximate controllability of damped second-order abstract evolution dynamical systems.

Optimization of second order spatially distributed systems with multiple interconnected actuator/sensor pairs

This work considers the economic aspects in control implementation of second order spatially distributed systems. By opting to use multiple actuator/sensor pairs with limited capacity, instead of a single actuator/sensor pair with higher price and larger capacity, this work provides a first view into the economic, computational and communication aspects. Considering a large number of actuator/sensor pairs and the a priori defined controller architecture of static output feedback along with the collocation assumption of the actuators and sensors, the resulting closed-loop system is expressed in terms of a static output feedback gain matrix that defines the communication amongst the actuator/sensor pairs. The design problem is then to find the values of the static feedback gain matrix in order to optimize a performance measure. Using the energy of the second order infinite dimensional system as the performance measure, the resulting optimization is expressed in terms of the solution to a parameterized operator Lyapunov equation. Simulation studies of the wave PDE in one spatial dimension for different communication topologies, reveal the effects of communication on the performance of the closed loop system.

On Stability Equivalence between Dynamic Output Feedback and Static Output Feedback for a Class of Second Order Infinite-Dimensional Systems

We consider stabilization for a class of abstract second order infinite-dimensional systems with collocated control and observation. We show that the closed-loop system under a proportional direct output feedback control is asymptotically stable if and only if the closed-loop system under some dynamic output feedback control is asymptotically stable. A Hautus test is developed to ensure the asymptotic stability. Two types of dynamic output feedback controls are investigated. The results are applied to some coupled wave-heat equations where the heat system is considered as a controller of the wave system. This study provides a different view in the study of the coupled systems described by partial differential equations.

Infinite-Dimensional Negative Imaginary Systems

We consider second order infinite-dimensional systems with force control and collocated position measurement interconnected with finite-dimensional controllers of the same type. We show that under assumptions that generalize those in the finite-dimensional case (the theory of negative imaginary systems), asymptotic stability of the closed-loop system can be concluded, but that the closed-loop system may not be exponentially stable nor input-output stable.

An Approximation Method for Exact Controls of Vibrating Systems

We propose a new method for the approximation of exact controls of a second order infinite dimensional system with bounded input operator. The algorithm combines Russell's “stabilizability implies controllability” principle with the Galerkin method. The main new feature of this work consists of giving precise error estimates. In order to test the efficiency of the method, we consider two illustrative examples (with the finite element approximations of the wave and the beam equations) and describe the corresponding simulations.

Natural consensus filters for second order infinite dimensional systems

This work proposes consensus filters for a class of second order infinite dimensional systems. The proposed structure of the consensus filters is that of a local filter written in the natural setting of a second order formulation with the additional coupling that enforces consensus by minimizing the disagreement between all local filters. An advantage of the second order formulation imposed on the local filters is the natural interpretation that they retain, namely that the time derivative of the estimated position is equal to the estimated velocity. Stability analysis of the collective dynamics is achieved via the use of a parameter-dependent Lyapunov functional, and which guarantees that asymptotically, all filters agree with each other and that they all converge to the true state of the second order system.

Stabilization of an abstract second order system with application to wave equations under non-collocated control and observations

The objective of this paper is to deal with the stabilization of multi-dimensional wave equations under non-collocated control and observation with the following cases: (a) locally internal distributed control and observation; (b) internal distributed control and boundary observation; (c) boundary control and internal distributed observation; (d) locally internal distributed control and boundary observation. We first discuss the stability of an abstract second order infinite-dimensional system under the estimated state feedback control. The well-posedness, in the sense of D.Salamon, is specially emphasized. The result is then applied to the multi-dimensional wave equations with the aforementioned three different non-collocated control and observations (a)–(c). Finally, we prove the stabilizability of a wave equation with non-collocated control and observation (d) by using the same idea, although it cannot be cast into the abstract setting.

Approximate controllability of the n -th order infinite dimensional systems with controls delayed by the control devices

The main aim of this article is to derive general conditions for a few types of controllability at once for an arbitrary order abstract differential equation and arbitrary eigenvalues multiplicities, instead of conditions for fixed order equation and single eigenvalues. Another innovation of this article is taking into account delays caused by electronic control microcontrollers. This was possible thanks to analysis of the n-th order linear system in the Frobenius form, generating Jordan transition matrix of the confluent Vandermonde form. Using the explicit analytical form of the inverse confluent Vandermonde matrix enabled us to receive general conditions of different types of controllability for the infinite dimensional systems. We derived this analytical form of the inverse confluent Vandermonde matrix using new results from the linear algebra, presented in the paper by S. Hou and W. Pang, “Inversion of confluent Vandermonde matrices”, Int. J. Comput. Math. Appl., 43, pp. 1539–1547, 2002. † Author is a stipendiary of the Foundation for Polish Science.

Controllability of second order infinite dimensional dynamical systems with delays

The article is devoted to analysing the approximate absolute and approximate relative controllability of a given type second order infinite dimensional system. The considered dynamical system is governed by the evolution equation with three damping terms and three terms without derivatives. Following this aim, spectral theory for linear unbounded operators is involved. At first the representation of considered infinite dimensional dynamical system by the infinite series of finite dimensional systems is given. Next, two theorems on necessary and sufficient conditions of approximate absolute and approximate relative controllability of the considered system are formulated and proved. Finally, proven theorems are applied to the analysis of the elastic beam.

Numerical approach to the non-linear diofantic equations with applications to the controllability of infinite dimensional dynamical systems

The article is devoted to the analysis of the approximate controllability of a given type of second order infinite dimensional system, defined by the hyperbolic type partial differential equation with Dirichlet-like type zero boundary conditions defined in the n-dimensional rectangular prism. Following this aim, spectral theory for linear unbounded operators is involved. New numerical algorithms for solving some kind of non-linear diofantic equations, corresponding with the spectral properties of the unbounded operators, are presented and proved. An algorithm is optimized for eliminating the symmetrical solutions, not interesting in the scope of verification of the controllability. Next the limit which shows that the direct application of the analytical methods of verifying the controllability in the numerical approach is impossible is proved. Finally so-called partial approximate controllability is defined and the numerical algorithm for its verification is presented. Finally, proven theorems are applied to one particular infinite dimensional dynamical system.

Controllability of dynamical systems with constraints

The article is devoted to analysing the approximate controllability without constrains and constrained controllability with non-negative controls of a given type second order infinite dimensional system. The considered dynamical system is governed by evolution equation with three damping terms and three terms without derivatives. Following this aim spectral theory for linear unbounded operators is involved. At first, remind the representation of considered infinite dimensional dynamical system by the infinite series of finite dimensional systems. Next, two theorems on necessary and sufficient conditions of unconstrained and constrained approximate controllability of considered system are formulated and proved. Finally, proven theorems are applied to two particular dynamical systems, including one infinite dimensional.