2020 (vol. 30) - Number 4

*A.B. Malinowska, R. Kamocki, R. Almeida, T. Odzijewicz:*

On the existence of optimal consensus control for the fractional Cucker--Smale model

*R. Grymin, W. Bożejko, J. Pempera, M. Wodecki, Z. Chaczko:*

Algorithm for solving the Discrete-Continuous Inspection Problem

*T. Kaczorek, A. Ruszewski:*

Global stability of discrete-time nonlinear systems with descriptor standard and fractional positive linear parts and scalar feedbacks

*R. Arora, K. Gupta:*

Fuzzy goal programming technique for multi-objective indefinite quadratic bilevel programming problem

*P. Nowak, J. Czeczot, P. Grelewicz:*

Tuning rules for industrial use of the second-order Reduced Active Disturbance Rejection Controller

*M.I. Gomoyunov:*

Optimal control problems with a fixed terminal time in linear fractional-order systems

*Muhafzan, A. Nazra, L. Yulianti, Zulakmal, R. Revina:*

On LQ optimization problem subject to fractional order irregular singular systems

*M. Blizorukova, V. Maksimov:*

On one algorithm for reconstruction of an disturbance in a linear system of ordinary differential equations

ACS Abstract:

**2020 (Volume 30)**

Number 4

On the existence of optimal consensus control for the fractional Cucker--Smale model

R. Almeida(University of Aveiro, Portugal) | R. Kamocki(University of Lodz, Poland) | A.B. Malinowska(Bialystok University of Technology, Poland) | T. Odzijewicz( Warsaw School of Economics, Poland) |

This paper addresses the nonlinear Cucker-Smale optimal control problem under the interplay of memory effect. The aforementioned effect is included by employing the Caputo fractional derivative in the equation representing the velocity of agents. The sufficient conditions for the existence of solutions to the considered problem are proved and the analysis of some particular problems is illustrated by two numerical examples.

**keywords:** fractional calculus, fractional differential systems, flocking model, consensus, optimal control

Algorithm for solving the Discrete-Continuous Inspection Problem

R. Grymin, W. Bożejko, J. Pempera, M. Wodecki(Wrocław University of Science and Technology, Poland) | Z. Chaczko(University of Technology Sydney, Australia) |

The article introduces an innovative approach for the inspection challenge that represents a generalization of the classical Traveling Salesman Problem. Its principle idea is to visit continuous areas (circles) in a way, that minimizes travelled distance. In practice, the problem can be defined as an issue of scheduling unmanned aerial vehicle which has discrete-continuous nature. In order to solve this problem the use of local search algorithms is proposed.

**keywords:** discrete-continuous optimization, UAV, VTOL, autonomous drone, zero emission

Global stability of discrete-time nonlinear systems with descriptor standard and fractional positive linear parts and scalar feedbacks

T. Kaczorek, A. Ruszewski(Białystok University of Technology, Poland) |

The global stability of discrete-time nonlinear systems with descriptor positive linear parts and positive scalar feedbacks is addressed. Sufficient conditions for the global stability of standard and fractional nonlinear systems are established. The effectiveness of these conditions is illustrated on numerical examples.

**keywords:** stability, fractional, positive, nonlinear, discrete-time, feedback, system

Fuzzy goal programming technique for multi-objective indefinite quadratic bilevel programming problem

R. Arora, K. Gupta(University of Delhi, India) |

Bilevel programming problem is a non-convex two stage decision making process in which the constraint region of upper level is determined by the lower level problem. In this paper, a multi-objective indefinite quadratic bilevel programming problem (MOIQBP) is presented. The defined problem (MOIQBP) has multi-objective functions at both the levels. The followers are independent at the lower level. A fuzzy goal programming methodology is employed which minimizes the sum of the negative deviational variables of both the levels to obtain highest membership value of each of the fuzzy goal. The membership function for the objective functions at each level is defined. As these membership functions are quadratic they are linearized by Taylor series approximation. The membership function for the decision variables at both levels is also determined. The individual optimal solution of objective functions at each level is used for formulating an integrated pay-off matrix. The aspiration levels for the decision makers are ascertained from this matrix. An algorithm is developed to obtain a compromise optimal solution for (MOIQBP). A numerical example is exhibited to evince the algorithm. The computing software LINGO 17.0 has been used for solving this problem.

**keywords:** bilevel programming, indefinite quadratic programming, multi-objective programming, pay-off matrix, Taylor series approximation, LINGO 17.0

Tuning rules for industrial use of the second-order Reduced Active Disturbance Rejection Controller

P. Nowak, J. Czeczot, P. Grelewicz(Silesian University of Technology, Poland) |

In the paper, problem of proper tuning of second-order Reduced Active Disturbance Rejection Controller (RADRC2) is considered in application for industrial processes with significant (but not dominant) delay time. For First-Order plus Delay Time (FOPDT) and Second-Order plus Delay Time (SOPDT) processes, tuning rules are derived to provide minimal Integral Absolute Error (IAE) assuming robustness defined by gain and phase margins. Derivation was made using optimization procedure based on D-partition method. The paper also shows results of comprehensive simulation validation based on exemplary benchmark processes of more complex dynamics as well as final practical validation. Comparison with PID controller shows that RADRC2 tuned by the proposed rules can be practical alternative for industrial control applications.

**keywords:** reduced-order ADRC, tuning, practical validation, industrial application

Optimal control problems with a fixed terminal time in linear fractional-order systems

M.I. Gomoyunov(Ural Branch of the Russian Academy of Sciences, and Ural Federal University, Russia) |

The paper deals with an optimal control problem in a dynamical system described by a linear differential equation with the Caputo fractional derivative. The goal of control is to minimize a Bolza-type cost functional, which consists of two terms: the first one evaluates the state of the system at a fixed terminal time, and the second one is an integral evaluation of the control on the whole time interval. In order to solve this problem, we propose to reduce it to some auxiliary optimal control problem in a dynamical system described by a first-order ordinary differential equation. The reduction is based on the representation formula for solutions to linear fractional differential equations and is performed by some linear transformation, which is called the informational image of a position of the original system and can be treated as a special prediction of a motion of this system at the terminal time. A connection between the original and auxiliary problems is established for both open-loop and feedback (closed-loop) controls. The results obtained in the paper are illustrated by examples.

**keywords:** optimal control, fractional derivatives, linear systems, open-loop control, feedback control, reduction

On LQ optimization problem subject to fractional order irregular singular systems

Muhafzan, A. Nazra, L. Yulianti, Zulakmal, R. Revina(Universitas Andalas, Indonesia) |

In this paper we discuss the linear quadratic (LQ) optimization problem subject to fractional order irregular singular systems. The aim of this paper is to find the control-state pairs satisfying the dynamic constraint of the form a fractional order irregular singular systems such that the LQ objective functional is minimized. The method of solving is to convert such LQ optimization into the standard fractional LQ optimization problem. Under some particularly conditions we find the solution of the problem under consideration.

**keywords:** linear quadratic optimization, fractional order, irregular singular system, Caputo fractional derivative, Mittag-Leffler function

On one algorithm for reconstruction of an disturbance in a linear system of ordinary differential equations

M. Blizorukova, V. Maksimov( Ural Branch of the Russian Academy of Sciences, Russia) |

The problem of reconstructing an unknown disturbance under measuring a part of phase coordinates of a system of linear differential equations is considered. Solving algorithm is designed. The algorithm is based on the combination of ideas from the theory of dynamical inversion and the theory of guaranteed control. The algorithm consists of two blocks: the block of dynamical reconstruction of unmeasured coordinates and the block of dynamical reconstruction of an input.

**keywords:** dynamical reconstruction, guaranteed control, stable algorithm

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