Sergej Celikovsky

BRIDGE THE GAP BETWEEN THE LORENZ SYSTEM AND THE CHEN SYSTEM

This paper introduces a unified chaotic system that contains the Lorenz and the Chen systems as two dual systems at the two extremes of its parameter spectrum. The new system represents the continued transition from the Lorenz to the Chen system and is chaotic over the entire spectrum of the key system parameter. Dynamical behaviors of the unified system are investigated in somewhat detail.

Secure synchronization of a class of chaotic systems from a nonlinear observer approach

The aim of this note is two-fold. First, it discusses synchronization of chaotic systems from a control theoretic point of view and introduces the concept of secure synchronization, i.e., a communication scheme that resists possible intrusion based on either adaptive or robust control techniques. Second, for a large class of chaotic systems, i.e., the generalized Lorenz system, global exponential synchronization via a scalar communication signal is suggested and its security is analyzed from a control theoretic viewpoint. Both theoretical analysis and numerical simulations are provided, verifying the proposed chaos-synchronization-based secure communication design principle and methodology.

Constructive nonsmooth stabilization of triangular systems

The problem of local asymptotic continuous feedback stabilization of single-input nonlinear systems is considered here. The aim is to explicitly construct a continuous asymptotically stabilizing feedback for a class of nonlinear systems that is known to be continuous feedback asymptotically stabilizable, but for which the known results do not provide an effective method to compute the stabilizer. Computer simulations are included to show practical applicability of our approach.

CHAOS SYNTHESIS BY MEANS OF EVOLUTIONARY ALGORITHMS

This paper introduces the notion of chaos synthesis by means of evolutionary algorithms and develops a new method for chaotic systems synthesis. This method is similar to genetic programming and gr...

Equivalence of nonlinear systems to triangular form: the singular case

The problem of state equivalence of a given nonlinear system to a triangular form is considered here. The solution of this problem has been known for the regular case, i.e. when there exists a certain nested sequence of regular and involutive distributions. It is also known that in this case the corresponding system is linearizable using a smooth coordinate change and static state feedback. This paper deals with the singular case, i.e. when the nested sequence of involutive distributions of the system contains singular distributions. Special attention is paid to the so-called bijective triangular form. Geometric, coordinates-free criteria for the solution of the above problem as well as constructive, verifiable procedures are given. These results are used to obtain a result in the nonsmooth stabilization problem.

Singular perturbation based solution to optimal microalgal growth problem and its infinite time horizon analysis

The problem of the optimal microalgal growth is considered here. The objective is to maximize the specific growth rate of microalgae by manipulating the irradiance. The model describing the growth of microalgae is based on the mechanistic description in the form of the so called photosynthetic factory (PSF) resulting into the second order bilinear system which is, nevertheless, known in biotechnological literature to comprise many important features of microalgal growth. To obtain the solution of optimal control problem, the singular perturbation approach is used here to reduce fast components of system dynamics leading to a less dimensional system with more complex performance index which allows a nice analytical solution. Its infinite horizon analysis shows that the optimal solution on large time intervals tends to the optimal steady state of PSF thereby supporting the hypothesis often mentioned in the biotechnological literature. Finally, the numerical algorithm to compute optimal control is applied to the original non-reduced system giving very similar results as the reduction based approach.

Model for Photosynthesis and Photoinhibition: Parameter Identification Based on the Harmonic Irradiation

A method for parameter identification of a model describing the growth of the algae is presented. The method is based on the description in the form of the so-called photosynthetic factory. The experimental data are gained by measuring the steady-state photosynthetic production when the input of the photosynthetic factory (light intensity) is a harmonic signal. Estimation of parameters is based on a sufficient number of experiments compared with simulated data via the least-squares technique. As the input signal is harmonic and the dynamics of the unforced system is exponentially stable, the resulting asymptotical steady-state trajectory of the photosynthetic factory is also periodic and can be computed via determining an appropriate center manifold graph by solving the corresponding first-order partial differential equation. The latter is performed by the finite-element method. The application of the proposed method is demonstrated on an example using real experimental data.

O_{2}

This paper aims to further improve previously developed design for the Acrobot walking based on the partial exact feedback linearization of order 3. Namely, such an exact system transformation leads to an almost linear system where error dynamics along trajectory to be tracked is a 4 dimensional linear time varying system having 3 time varying entries only, the remaining entries are either zero or equal to one. Previously, the exponentially stable tracking was obtained by solving quadratic stability of a linear system with polytopic uncertainty applying LMI methods to solve this problem numerically. Here, the new approach is presented allowing to design the tracking feedback and to prove the corresponding stability completely analytically. Moreover, this approach gives even better results than the LMI based one in the sense of the convergence speed. The key idea of the novel approach presented here is that it manages to use part of the information about the mentioned time dependent entries, thereby reducing overly conservativeness of the previous LMI based design. Numerical simulations of the Acrobot walking based on the above mentioned new analytical design are demonstrated as well.

Response Measurement

A new control concept for a class of simple underactuated mechanical system, the so-called acrobot, is presented here. Despite being seemingly a simple system, the acrobot comprises many important difficulties when controlling the most challenging underactuated system - the walking robot. This paper presents the design of the asymptotical tracking of the prescribed trajectory generated by a suitable open-loop input of the acrobot. Such a design is based on the partial exact linearization of the third order combined with a certain robust stabilization technique. The proposed control is then demonstrated by the exponential tracking of the walking-like trajectory of the acrobot. Besides theoretical proofs, our approach is supported by numerical simulations and illustrated by acrobot movement animations.

Analytical design of the Acrobot exponential tracking with application to its walking

The problem of local stabilizability of locally controllable nonlinear systems is considered. It is well known that, contrary to the linear case, local controllability does not necessarily imply stabilizability. A class of nonlinear systems for which local controllability implies local asymptotic stabilizability using continuous static-state feedback is described, as for this class of systems the well-known Hermes controllability condition is necessary and sufficient for local controllability.