2018 (vol. 28) - Number 4

*T. Kaczorek:*

A new method for computation of positive realizations of fractional linear continuous-time systems

*A. Biswas, A. Sarkar:*

Development of dual hesitant fuzzy prioritized operators based on Einstein operations with their application to multi-criteria group decision making

*L. Li, R. Zhang, J. Wang, X. Shang:*

Some q-orthopair linguistic Heronian mean operators with their application to multi-attribute group decision making

*J. Duda:*

Lyapunov matrices approach to the parametric optimization of a system with two delays and a PD controller

*M. Lhous, M. Rachik, El M. Magri:*

Ideal observability for bilinear discrete-time systems with and without delays in observation

*P.T. Anh, A. Babiarz, A. Czornik, M. Niezabitowski:*

Variation of constant formulas for fractional difference equations

*K. Bingi, R. Ibrahim, M. Noh Karsiti, S. Miya Hassan, V. Rajah Harindran:*

A comparative study of 2DOF PID and 2DOF fractional order PID controllers on a class of unstable systems

ACS Abstract:

**2016 (Volume 26)**

Number 1

PID control of FOPDT plants with dominant dead time based on the modulus optimum criterion

J. Cvejn(University of Pardubice, Czech Republic) |

The modulus optimum (MO) criterion can be used for analytical design of the PID controller for linear systems with dominant dead time. However, although the method usually gives fast and non-oscillating closed-loop responses, in the case of large dead time the stability margin gets reduced and even non-stable behavior can be observed. In this case a correction of the settings is needed to preserve the stability margin. We describe and compare two methods of design of the PID controller based on the MO criterion that for the stable first-order systems with dead time preserve the stability margin, trying to keep maximum of the performance of the original MO settings.

**keywords:** PID controller, process control, dead time, modulus optimum, magnitude optimum

A novel 3-D jerk chaotic system with three quadratic nonlinearities and its adaptive control

Sundarapandian Vaidyanathan(Vel Tech University, India) |

This paper announces an eight-term novel 3-D jerk chaotic system with three quadratic nonlinearities. The phase portraits of the novel jerk chaotic system are displayed and the qualitative properties of the jerk system are described. The novel jerk chaotic system has two equilibrium points, which are saddle-foci and unstable. The Lyapunov exponents of the novel jerk chaotic system are obtained as L1 = 0.20572, L2 = 0 and L3 = -1.20824. Since the sum of the Lyapunov exponents of the jerk chaotic system is negative, we conclude that the chaotic system is dissipative. The Kaplan-Yorke dimension of the novel jerk chaotic system is derived as DKY = 2.17026. Next, an adaptive controller is designed via backstepping control method to globally stabilize the novel jerk chaotic system with unknown parameters.cheap led light bulbs Moreover, an adaptive controller is also designed via backstepping control method to achieve global chaos synchronization of the identical jerk chaotic systems with unknown parameters. The backstepping control method is a recursive procedure that links the choice of a Lyapunov function with the design of a controller and guarantees global asymptotic stability of strict feedback systems. MATLAB simulations have been depicted to illustrate the phase portraits of the novel jerk chaotic system and also the adaptive backstepping control results.

**keywords:** chaos, chaotic system, dissipative chaotic system, adaptive control, backstepping control, synchronization

Decomposition method and its application to the extremal problems

H. Gorecki, M. Zaczyk(AGH University of Science and Technology, Poland) |

In the article solution of the problem of extremal value of x(t) is presented, for the n-th order linear systems. The extremum of x(t) is considered as a function of the roots s1, s2, ... sn of the characteristic equation. The obtained results give a possibility of decomposition of the whole n-th order system into a set of 2-nd order systems.

**keywords:** extremal problems, decomposition, characteristic equation, transmittance

Coexistence of synchronization and anti-synchronization in chaotic systems

U.E. Vincent(Lancaster University, UK) | L. Ren, R. Guo(Qilu University of Technology, China) |

The coexistence of anti-synchronization and synchronization in chaotic systems is investigated. A novel algorithm is proposed to determine the variables of the master system that should anti-synchronize with corresponding variables of the slave system. Control strategies that guarantee the coexistence of synchronization and anti-synchronization in the unified chaotic system are presented; while numerical simulations are employed to validate and illustrate the effectiveness of the proposed method.

**keywords:** coexistence synchronization, complete synchronization, anti-synchronization, unified chaotic system

Lyapunov matrices approach to the parametric optimization of a neutral system

J. Duda(AGH University of Science and Technology, Poland) |

The Lyapunov functionals are used to test the stability of systems, in calculation of the robustness bounds for uncertain time delay systems, in computation of the exponential estimates for the solutions of time delay systems. The method of determination of a Lyapunov functional for a time delay system with one delay was presented by Repin. Duda used the Repin's method to determination of a Lyapunov functional for a system with two delays, for a system with both lumped and distributed delay, led light bulbs for home for a system with time-varying delay. In last years a method of determination of a Lyapunov functional by means of Lyapunov matrices is very popular, see for example. The Lyapunov quadratic functionals are also used to calculation of a value of a quadratic performance index of quality in the process of the parametric optimization for time delay systems. One constructs a functional for a system with a time delay with a given time derivative whose is equal to the negatively defined quadratic form of a system state. The value of that functional at the initial state of a time delay system is equal to the value of a quadratic performance index of quality. In the paper a Lyapunov matrices approach to the parametric optimization problem of a neutral system is presented. This paper extends the results of Duda’s works onto a neutral system.

**keywords:** neutral system, Lyapunov matrix, Lyapunov functional

Analysis, adaptive control and circuit simulation of a novel finance system with dissaving

O.I. Tacha, Ch.K. Volos, I.N. Stouboulos, I.M. Kyprianidis(Aristotle University of Thessaloniki, Greece) |

In this paper a novel 3-D nonlinear finance chaotic system consisting of two nonlinearities with negative saving term, which is called `dissaving' is presented. The dynamical analysis of the proposed system confirms its complex dynamic behavior, which is studied by using well-known simulation tools of nonlinear theory, such as the bifurcation diagram, Lyapunov exponents and phase portraits. Also, some interesting phenomena related with nonlinear theory are observed, such as route to chaos through a period doubling sequence and crisis phenomena. In addition, an interesting scheme of adaptive control of finance system's behavior is presented. Furthermore, the novel nonlinear finance system is emulated by an electronic circuit and its dynamical behavior is studied by using the electronic simulation package Cadence OrCAD in order to confirm the feasibility of the theoretical model.

**keywords:** chaotic finance system, dissaving, bifurcation diagram, Lyapunov exponent, phase portrait, adaptive control, nonlinear circuit

Elman neural network for modeling and predictive control of delayed dynamic systems

A. Wysocki, M. Lawrynczuk(Warsaw University of Technology, Poland) |

The objective of this paper is to present a modified structure and a training algorithm of the recurrent Elman neural network which makes it possible to explicitly take into account the time-delay of the process and a Model Predictive Control (MPC) algorithm for such a network. In MPC the predicted output trajectory is repeatedly linearized on-line along the future input trajectory, which leads to a quadratic optimization problem, nonlinear optimization is not necessary. A strongly nonlinear benchmark process (a simulated neutralization reactor) is considered to show advantages of the modified Elman neural network and the discussed MPC algorithm. The modified neural model is more precise and has a lower number of parameters in comparison with the classical Elman structure. The discussed MPC algorithm with on-line linearization gives similar trajectories as MPC with nonlinear optimization repeated at each sampling instant.

**keywords:** dynamic models, process control, model predictive control, neural networks, Elman neural network, delayed systems

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