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Dynamical distributed control and synchronization

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Abstract

It is well known that most of real-world phenomena are described by partial differential equations. Nevertheless, for control design purposes it is very common to approximate them with a set of ordinary differential equations, since conventional design methods, such as calculus of variations or differential geometry, turn out to be very complex for this class of systems. However, by doing this, valuable properties are lost. In this work, we present a dynamical distributed control for nonlinear partial differential equation systems and we focus on solving the Generalized Synchronization problem, since this topic has multiple applications in the disciplines of engineering, biology, physics, etc. For the design of the control, we utilize a differential algebraic approach. The key ingredient of our design method is to find a canonical form of the given systems by means of the so-called partial differential primitive element. This representation is known as Generalized Observability Canonical Form and allows us to design a dynamical distributed control in a natural way. Additionally, to avoid a functional analysis for the stability of the resultant synchronization error, we propose to utilize tools from semi-group and spectral theory of infinite dimensional systems in a Hilbert space. As a result, we present a design approach less complex and, therefore, more accessible than most common design methods. Besides, with the proposed stability analysis, we obtain an easy criterion to select the control gains; hence, we can solve the generalized synchronization problem of partial differential equation systems in a simple way. To validate the effectiveness of the proposed control, we present two examples of generalized synchronization for reaction-diffusion systems and show their respective numerical results.

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Abbreviations

\(\Vert * \Vert \) :

Euclidean norm

\(*_{z}\) :

First-order derivative with respect to spatial coordinate z

\(\dot{*}\) :

First-order derivative with respect to time

\({\mathscr {A}}\) :

Linear operator

\(\partial ^{(n)}\) :

Aleatory differential operator of order n

\(\partial _i^{K}\) :

ith aleatory differential operator in K

\(\varPhi _\mathrm{m}\) :

Master system’s coordinate transformation

\(\varPhi _\mathrm{s}\) :

Slave system’s coordinate transformation

\(F_\mathrm{m}\) :

Master system’s state function

\(F_\mathrm{s}\) :

Slave system’s nonlinear state function

\(h_\mathrm{m}\) :

Master system’s output function

\(h_\mathrm{s}\) :

Slave system’s output function

\(H_\mathrm{{ms}}\) :

Coordinate transformation from slave to master state space

\({*}^{(n)}\) :

Derivative with respect to time of order n

\({\mathbb {R}}\) :

Real numbers

\({\mathbb {R}}^{n_\mathrm{m}}\) :

Master system’s n-dimensional state space

\({\mathbb {R}}^{n_\mathrm{s}}\) :

Slave system’s n-dimensional state space

\({\mathbb {Z}}\) :

Integer numbers

\(\sigma (*)\) :

Spectrum

\(\varSigma _\mathrm{m}\) :

Master system

\(\varSigma _\mathrm{s}\) :

Slave system

B :

Compact set

E :

Hilbert space

K, L :

Partial differential fields

\(K \langle u \rangle \) :

Partial differential field generated by K, u and the time partial derivatives of u

L/K :

Partial differential field extension

M :

Algebraic Manifold

\(r(*)\) :

Spectral radius

S(t):

Semi-group

\(\alpha _0\) :

Growth bound

\({\bar{y}}\) :

Partial differential primitive element

\(\delta \) :

Master system’s state vector in transformed coordinates

\(\eta \) :

Slave system’s state vector in transformed coordinates

\(\kappa _i\) :

Control gains

\(\mu \) :

Master system’s state vector

\(\nu \) :

Slave system’s state vector

\(\xi \) :

Aleatory system’s state vector in transformed coordinates

A :

Feed rate

B :

Reaction speed

\(D_X,D_Y\) :

Diffusion constants

e :

Synchronization error

\(u_\mathrm{m}\) :

Master system’s input

\(u_\mathrm{s}\) :

Slave system’s input

X, Y :

Substances concentrations

\(y_\mathrm{m}\) :

Master system’s output

\(y_\mathrm{s}\) :

Slave system’s output

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Authors and Affiliations

  1. Departamento de Control Automático, Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional, Av Instituto Politécnico Nacional 2508, San Pedro Zacatenco, Gustavo A. Madero, 07360, Mexico City, Mexico

    Juan Pablo Flores-Flores & Rafael Martínez-Guerra

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  1. Juan Pablo Flores-Flores
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  2. Rafael Martínez-Guerra
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Correspondence to Rafael Martínez-Guerra.

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Flores-Flores, J.P., Martínez-Guerra, R. Dynamical distributed control and synchronization. Nonlinear Dyn 103, 1663–1679 (2021). https://doi.org/10.1007/s11071-020-06191-4

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