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 3

A new fuzzy iterative learning control algorithm for single joint manipulator

Meng Wang, Guangrong Bian, Hongsheng Li(Wuhan University of Technology, China) |

This paper present a new fuzzy iterative learning control design to solve the trajectory tracking problem and performing repetitive tasks for rigid robot manipulators. Several times' iterations are needed to make the system tracking error converge, especially in the first iteration without experience. In order to solve that problem, fuzzy control and iterative learning control are combined, where fuzzy control is used to tracking trajectory at the first learning period, and the output of fuzzy control is recorded as the initial control inputs of ILC. The new algorithm also adopts gain self-tuning by fuzzy control, in order to improve the convergence rate. Simulations illustrate the effectiveness and convergence of the new algorithm and advantages compared to traditional method.

**keywords:** iterative learning control, fuzzy control, fuzzy gain adjustment, single joint manipulator

Analysis, adaptive control and synchronization of a novel 4-D hyperchaotic hyperjerk system via backstepping control method

Sundarapandian Vaidyanathan(Vel Tech University, India) |

A hyperjerk system is a dynamical system, which is modelled by an n-th order ordinary differential equation with n ≥ 4 describing the time evolution of a single scalar variable. Equivalently, using a chain of integrators, a hyperjerk system can be modelled as a system of n first order ordinary differential equations with n ≥ 4. In this research work, a 4-D novel hyperchaotic hyperjerk system with two nonlinearities has been proposed, and its qualitative properties have been detailed. The novel hyperjerk system has a unique equilibrium at the origin, which is a saddle-focus and unstable. The Lyapunov exponents of the novel hyperjerk system are obtained as L_1 = 0.14219, L_2 = 0.04605, L_3 = 0 and L_4 = -1.39267. The Kaplan-Yorke dimension of the novel hyperjerk system is obtained as D_{KY} = 3.1348. Next, an adaptive controller is designed via backstepping control method to stabilize the novel hyperjerk chaotic system with three unknown parameters. Moreover, an adaptive controller is designed via backstepping control method to achieve global synchronization of the identical novel hyperjerk systems with three unknown parameters. MATLAB simulations are shown to illustrate all the main results derived in this research work on a novel hyperjerk system.

**keywords:** hyperchaos, hyperjerk system, adaptive control, backstepping control, synchronization

Tools and methods for RTCP-nets modeling and verification

M. Szpyrka, J. Biernacki, A. Biernacka(AGH University of Science and Technology, Poland) |

RTCP-nets are high level Petri nets similar to timed colored Petri nets, but with different time model and some structural restrictions. The paper deals with practical aspects of using RTCP-nets for modeling and verification of real-time systems. It contains a survey of software tools developed to support RTCP-nets. Verification of RTCP-nets is based on coverability graphs which represent the set of reachable states in the form of directed graph. Two approaches to verification of RTCP-nets are considered in the paper. The former one is oriented towards states and is based on translation of a coverability graph into nuXmv (NuSMV) finite state model. The later approach is oriented towards transitions and uses the CADP toolkit to check whether requirements given as µ-calculus formulae hold for a given coverability graph. All presented concepts are discussed using illustrative examples.

**keywords:** RTCP-nets, Petri nets, model checking, coverability graphs, nuXmv, CADP

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

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

In the paper a Lyapunov matrices approach to the parametric optimization problem of time-delay systems with two commensurate delays and a P-controller is presented. The value of integral quadratic performance index of quality is equal to the value of the Lyapunov functional for the initial function of time-delay system. The Lyapunov functional is determined by means of the Lyapunov matrix.

**keywords:** time-delay system, Lyapunov matrix, Lyapunov functional

Optimal and suboptimal control of a standard Brownian motion

M. Lefebvre(Polytechnique Montreal, Canada) |

The problem of optimally controlling a standard Brownian motion until a fixed final time is considered in the case when the final cost function is an even function. Two particular problems are solved explicitly. Moreover, the best constant control as well as the best linear control are also obtained in these two particular cases.

**keywords:** stochastic control, Wiener process, best linear control

Membership functions for fuzzy focal elements

S. Porebski, E. Straszecka(Silesian University of Technology, Poland) |

The paper presents a study on data-driven diagnostic rules, which are easy to interpret by human experts. To this end, the Dempster-Shafer theory extended for fuzzy focal elements is used. Premises of the rules (fuzzy focal elements) are provided by membership functions which shapes are changing according to input symptoms. The main aim of the present study is to evaluate common membership function shapes and to introduce a rule elimination algorithm. Proposed methods are first illustrated with the popular Iris data set. Next experiments with five medical benchmark databases are performed. Results of the experiments show that various membership function shapes provide different inference efficiency but the extracted rule sets are close to each other. Thus indications for determining rules with possible heuristic interpretation can be formulated.

**keywords:** diagnostic rule extraction, medical diagnosis support, fuzzy focal elements, membership functions, Dempster-Shafer theory

Explicit finite-difference scheme for the numerical solution of the model equation of nonlinear hereditary oscillator with variable-order fractional derivatives

R.I. Parovik(Vitus Bering Kamchatka State University, Russia) |

The paper deals with the model of variable-order nonlinear hereditary oscillator based on a numerical finite-difference scheme. Numerical experiments have been carried out to evaluate the stability and convergence of the difference scheme. It is argued that the approximation, stability and convergence are of the first order, while the scheme is stable and converges to the exact solution.

**keywords:** nonlinear hereditary oscillator, finite-difference scheme, Cauchy problem, fractional derivatives, numerical experiment

Equitable coloring of graphs. Recent theoretical results and new practical algorithms

H. Furmanczyk, A. Jastrzebski, M. Kubale(Gdansk University of Technology, Poland) |

In many applications in sequencing and scheduling it is desirable to have an underlaying graph as equitably colored as possible. In this paper we survey recent theoretical results concerning conditions for equitable colorability of some graphs and recent theoretical results concerning the complexity of equitable coloring problem. Next, since the general coloring problem is strongly NP-hard, we report on practical experiments with some efficient polynomial-time algorithms for approximate equitable coloring of general graphs.

**keywords:** computer experiments, corona graph, equitable chromatic number, equitable coloring conjectures, NP-hardness, polynomial heuristics

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