The regulator equation is the fundamental equation whose solution must be found in order to solve the output regulation problem. It is a system of first-order partial differential equations (PDE) combined with an algebraic equation. The classical approach to its solution is to use the Taylor series with undetermined coefficients. In this contribution, another path is followed: the equation is solved using the finite-element method which is, nevertheless, suitable to solve PDE part only. This paper presents two methods to handle the algebraic condition: the first one is based on iterative minimization of a cost functional defined as the integral of the square of the algebraic expression to be equal to zero. The second method converts the algebraic-differential equation into a singularly perturbed system of partial differential equations only. Both methods are compared and the simulation results are presented including on-line control implementation to some practically motivated laboratory models.
A mathematical model of the microalgal growth under various light regimes is required for the optimization of design parameters and operating conditions in a photobioreactor. As its modelling framework, bilinear system with single input is chosen in this paper. The earlier theoretical results on bilinear systems are adapted and applied to the special class of the so-called intermittent controls which are characterized by rapid switching of light and dark cycles. Based on such approach, the following important result is obtained in the present paper: as the light/dark cycle frequency is going to infinity, the value of resulting production rate in the microalgal culture goes to a certain limit value, which depends on average irradiance in the culture only. As a case study, the so-called three-state model of photosynthetic factory, being a simple four-parameter model, is analyzed. The present paper shows various numerical simulations for the model parameters previously published and analyzed experimentally in the biotechnological literature. These simulation results are in a very good qualitative compliance with the well-known flashing light experiments, thereby confirming viability of the approach presented here.
In this paper, a modified version of the Chaos Shift Keying (CSK) scheme for secure encryption and decryption of data will be discussed. The classical CSK method determines the correct value of binary signal through checking which initially unsynchronized system is getting synchronized. On the contrary, the new anti-synchronization CSK (ACSK) scheme determines the wrong value of binary signal through checking which already synchronized system is loosing synchronization. The ACSK scheme is implemented and tested using the so-called \emph{generalized Lorenz system} (GLS) family making advantage of its special parametrization. Such an implementation relies on the parameter dependent synchronization of several identical copies of the GLS obtained through the observer-based design for nonlinear systems. The purpose of this paper is to study and compare two different methods for the anti-synchronization detection, including further underlying theoretical study of the GLS. Resulting encryption schemes are also compared and analyzed with respect to both the encryption redundancy and the encryption security. Numerical experiments illustrate the results.