2020 (vol. 30) - Number 3

*M.A. Hammami, N.H. Rettab:*

On the region of attraction of dynamical systems: Application to Lorenz equations

*J. Rudy, J. Pempera, C. Smutnicki:*

Improving the TSAB algorithm through parallel computing

*Chiranjibe Jana, Madhumangal Pal , Guiwu Wei:*

Multiple Attribute Decision Making method based on intuitionistic Dombi operators and its application in~mutual fund evaluation

*S. Ogonowski, D. Bismor, Z. Ogonowski:*

Control of complex dynamic nonlinear loading process for electromagnetic mill

*N.A. Baleghi, M.H. Shafiei:*

An observer-based controller design for nonlinear discrete-time switched systems with~time-delay and affine parametric uncertainty

*Amine El Bhih, Youssef Benfatah, Mostafa Rachik:*

Exact determinantions of maximal output admissible set for a class of~semilinear discrete systems

*Archana Tiwari, Debanjana Bhattacharyya, K.C. Pati:*

Controllabilty and stability analysis on a group associated with Black-Scholes equation

*A. Sambas, I.M. Moroz, S. Vaidyanathan:*

A new 4-D hyperchaotic system with no equilibrium, its multistability, offset boosting and circuit simulation

*A.S.S. Abadi, P.A. Hosseinabadi, S. Mekhilef, A. Ordys:*

A new strongly predefined time sliding mode controller for a class of cascade high-order nonlinear systems

ACS Abstract:

**2019 (Volume 29)**

Number 1

A new D-stability arrea for linear discrete-time~systems

D. Krokavec, A. Filasova(Technical University of Kosice, Slovakia) |

The paper addresses the problem of constrained pole placement in discrete-time linear systems. The design conditions are outlined in terms of linear matrix inequalities for the D-stable ellipse region in the complex Z plain. In addition, it is demonstrated that the D-stable circle region formulation is the special case of by this way formulated and solved pole placement problem. The proposed principle is enhanced for discrete-lime linear systems with polytopic uncertainties.

**keywords:** discrete-time linear systems, state control, pole placement constraints, D-stability region, linear matrix inequalities, polytopic uncertainties

Forecasting the distribution of methane concentration levels in mine headings by means of model-based tests and in-situ measurements

J. Brodny, M. Tutak(Silesian University of Technology, Poland) |

The methane hazard is one of the most dangerous phenomena in hard coal mining. In a certain range of concentrations, methane is flammable and explosive. Therefore, in order to maintain the continuity of the production process and the safety of work for the crew, various measures are taken to prevent these concentration levels from being exceeded. A significant role in this process is played by the forecasting of methane concentrations in mine headings. This very problem has been the focus of the present article. Based on discrete measurements of methane concentration in mine headings and ventilation parameters, the distribution of methane concentration levels in these headings was forecasted. This process was performed on the basis of model-based tests using the Computational Fluid Dynamics (CFD). The methodology adopted was used to develop a structural model of the region under analysis, for which boundary conditions were adopted on the basis of the measurements results in real-world conditions. The analyses conducted helped to specify the distributions of methane concentrations in the region at hand and determine the anticipated future values of these concentrations. The results obtained from model-based tests were compared with the results of the measurements in real-world conditions. The methodology using the CFD and the results of the tests offer extensive possibilities of their application for effective diagnosis and forecasting of the methane hazard in mine headings.

**keywords:** CFD, forecasting the distribution of methane, mining heading, in-situ measurements, model tests

Model reduction problem of linear discrete systems: Admissibles initial states

A. Abdelhak(University Ibn Tofail, Morocco) | M. Rachik( University Hassan II, Morocco) |

Given a linear discrete system with initial state *x*_{0} and output function *y _{i}* we investigate a low dimensional linear system that produces, with a tolerance index

*ϵ*, the same output function when the initial state belongs to a specified set, called

*ϵ*-admissible set, that we characterize by a finite number of inequalities. We also give an algorithm which allows us to determine an

*ϵ*-admissible set.

**keywords:** linear discrete systems, model order reduction

Distributed controllability of one-dimensional heat equation in unbounded domains: The~Green's function approach

Asatur Zh. Khurshudyan( National Academy of Sciences of Armenia) |

We derive exact and approximate controllability conditions for the linear one-dimensional heat equation in an infinite and a semi-infinite domains. The control is carried out by means of the time-dependent intensity of a point heat source localized at an internal (finite) point of the domain. By the Green's function approach and the method of heuristic determination of resolving controls, exact controllability analysis is reduced to an infinite system of linear algebraic equations, the regularity of which is sufficient for the existence of exactly resolvable controls. In the case of a semi-infinite domain, as the source approaches the boundary, a lack of *L*^{2}-null-controllability occurs, which is observed earlier by Micu and Zuazua. On the other hand, in the case of infinite domain, sufficient conditions for the regularity of the reduced infinite system of equations are derived in terms of control time, initial and terminal temperatures. A sufficient condition on the control time, heat source concentration point and initial and terminal temperatures is derived for the existence of approximately resolving controls. In the particular case of a semi-infinite domain when the heat source approaches the boundary, a sufficient condition on the control time and initial temperature providing approximate controllability with required precision is derived

**keywords:** lack of controllability, exact controllability, approximate controllability, null-controllability, Green's function, heuristic method, infinite system of algebraic equations, regularity, fully regularity

Dynamic Contraction Method approach to digital longitudinal aircraft flight controller design

R. Czyba, Lukasz Stajer(Silesian University of Technology, Poland) |

This paper presents the design of digital controller for longitudinal aircraft model based on the Dynamic Contraction Method. The control task is formulated as a tracking problem of velocity and flight path angle, where decoupled output transients are accomplished in spite of incomplete information about varying parameters of the system and external disturbances. The design of digital controller based on the pseudo-continuous approach is presented, where the digital controller is the result of continuous-time controller discretization. A resulting output feedback controller has a simple form of a combination of low-order linear dynamical systems and a matrix whose entries depend nonlinearly on certain known process variables. Simulation results for an aircraft model confirm theoretical expectations.

**keywords:** nonlinear systems, MIMO systems, aircraft control, digital controller, singular perturbation

Preview Control applied for humanoid robot motion generation

M. Szumowski, M.S. Zurawska, T. Zielinska(Warsaw University of Technology, Poland) |

This paper presents a concept of humanoid robot motion generation using the dedicated simplified dynamic model of the robot (Extended Cart-Table model). Humanoid robot gait with equal steps length is considered. Motion pattern is obtained here with use of Preview Control method. Motion trajectories are first obtained in simulations (off-line) and then they are verified on a test-bed. Tests performed using the real robot confirmed the correctness of the method. Robot completed a set of steps without losing its balance.

**keywords:** motion generation, humanoid robot, Preview Control, Extended Cart-Table model, off-line method

Combined modified function projective synchronization of different systems through adaptive control

N.A. Almohammadi, E.O. Alzahrani( King Abdulaziz University, Saudi Arabia) | M.M. El-Dessoky(Mansoura University, Egypt) |

This work describes a new study to achieve a combination of modified function projective synchronization between three different chaotic systems through adaptive control. Using the Lyapunov function theory, the asymptotic stability of the error dynamics is obtained and discussed. Further, we set some appropriate initial conditions for the state variables and assigning specific values to the parameters and obtain the graphical results, which shows the efficiencies of the new method. Finally, we summarized our work with conclusion and references.

**keywords:** modified function projective synchronization, novel chaotic system, four-scroll chaotic system, adaptive control

On discounted LQR control problem for disturbanced singular system

L. Yulianti, A. Nazra, Zulakmal, Muhafzan( Universitas Andalas, Indonesia) | A. Bahar( Universiti Teknologi Malaysia) |

The LQR (linear quadratic regulator) control problem subject to singular system constitutes a optimization problem in which one must be find an optimal control that satisfy the singular system and simultaneously to optimize the quadratic objective functional. In this paper we establish a sufficient condition to obtain the optimal control of discounted LQR optimization problem subject to disturbed singular system where the disturbance is time varying. The considered problem is solved by transforming the discounted LQR control problem subject to disturbance singular system into the normal LQR control problem. Some available results in literatures of the normal LQR control problem be used to find the sufficient conditions for the existence of the optimal control for discounted LQR control problem subject to disturbed singular system. The final result of this paper is in the form a method to find the optimal control of discounted LQR optimization problem subject to disturbed singular system. The result shows that the disturbance is vanish with the passage of time.

**keywords:** discounted LQR control, singular system, disturbance

Absolute stability of a class of positive nonlinear continuous-time and discrete-time systems

T. Kaczorek(Bialystok University of Technology, Poland) |

The positivity and absolute stability of a class of nonlinear continuous-time and discrete-time systems are addressed. Necessary and sufficient conditions for the positivity of this class of nonlinear systems are established. Sufficient conditions for the absolute stability of this class of nonlinear systems are also given.

**keywords:** absolute stability, positive, nonlinear, discrete-time, continuous-time, systems

Sliding mode tracking control for unmanned helicopter using extended disturbance observer

I. Ullah, H-L. Pei(South China University of Technology, China) |

This paper presents a robust control technique for small-scale unmanned helicopters to track predefined trajectories (velocities and heading) in the presence of bounded external disturbances. The controller design is based on the linearized state-space model of the helicopter. The multivariable dynamics of the helicopter is divided into two subsystems, longitudinal-lateral and heading-heave dynamics respectively. There is no strong coupling between these two subsystems and independent controllers are designed for each subsystem. The external disturbances and model mismatch in the longitudinal-lateral subsystem are present in all (matched and mismatched) channels. This model mismatch and external disturbances are estimated as lumped disturbances using extended disturbance observer and an extended disturbance observer based sliding mode controller is designed for it to counter the effect of these disturbances. In the case of heading-heave subsystem, external disturbances and model mismatch only occur in matched channels so a second order sliding mode controller is designed for it as it is insensitive to matched uncertainties. The control performance is successfully tested in Simulink.

**keywords:** unmanned helicopter, external disturbances, sliding mode control, extended disturbance observer, mismatched uncertainty

Chaos in a simple snap system with only one nonlinearity, its adaptive control and real circuit design

V.-T. Pham(Ton Duc Thang University, Vietnam) | Sundarapandian Vaidyanathan(Vel Tech University, India) | Ch. Volos(Aristotle University of Thessaloniki, Greece) | S. Jafari(Amirkabir University of Technology, Iran) | F.E. Alsaadi, F.E. Alsaadi(King Abdulaziz University, Saudi Arabia) |

We study an elegant snap system with only one nonlinear term, which is a quadratic nonlinearity. The snap system displays chaotic attractors, which are controlled easily by changing a system parameter. By using analysis, simulations and a real circuit, the dynamics of such a snap system has been investigated. We also investigate backstepping based adaptive control schemes for the new snap system with unknown parameters.

**keywords:** chaos, chaotic systems, hyperjerk systems, adaptive control, backstepping control, circuit design

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