2018 (vol. 28) - Number 2

*E. Roszkowska:*

Hybrid motion control for multiple mobile robot systems

*C. Civelek:*

Stability analysis of engineering/physical dynamic systems using residual energy function

*M. Ziolko, M. Nowak:*

Design of transmultiplexer integer filters

*S. Vaidyanathan, S. Jafari, V.-T. Pham, A.T. Azar:*

A 4-D chaotic hyperjerk system with a hidden attractor, adaptive backstepping control and circuit design

*T. Kaczorek:*

An extension of Klamka's method to positive descriptor discrete-time linear systems with bounded inputs

*A. Ratajczak:*

Motion planning for nonholonomic systems with earlier destination reaching

*K.C. Patra, B.K. Dakua:*

Investigation of limit cycles and signal stabilization of two dimensional systems with memory type nonlinear elements

ACS Abstract:

**2018 (Volume 28)**

Number 2

Hybrid motion control for multiple mobile robot systems

E. Roszkowska(Wroclaw University of Science and Technology, Poland) |

This article presents a hybrid control system for a group of mobile robots. The components of this system are the supervisory controller(s), employing a discrete, event-driven model of concurrent robot processes, and robot motion controllers, employing a continuous time model with event-switched modes. The missions of the robots are specified by a sequence of to-be visited points, and the developed methodology ensures in a formal way their correct accomplishment.

**keywords:** multi-robot system, hybrid control

Stability analysis of engineering/physical dynamic systems using residual energy function

C. Civelek(Ege University, Turkey) |

In this article, an engineering/physical dynamic system including losses is analyzed in relation to the stability from an engineer's/physicist's point of view. Firstly, conditions for a Hamiltonian to be an energy function, time independent or not, is explained herein. To analyze stability of engineering system, Lyapunov-like energy function, called residual energy function is used. The residual function may contain, apart from external energies, negative losses as well. This function includes the sum of potential and kinetic energies, which are special forms and ready-made (weak) Lyapunov functions, and loss of energies (positive and/or negative) of a system described in different forms using tensorial variables. As the Lypunov function, residual energy function is defined as Hamiltonian energy function plus loss of energies and then associated weak and strong stability are proved through the first time-derivative of residual energy function. It is demonstrated how the stability analysis can be performed using the residual energy functions in different formulations and in generalized motion space when available. This novel approach is applied to RLC circuit, AC equivalent circuit of Gunn diode oscillator for autonomous, and a coupled (electromechanical) example for nonautonomous case. In the nonautonomous case, the stability criteria can not be proven for one type of formulation, however, it can be proven in the other type formulation.

**keywords:** stability analysis, residual energy function as Lyapunov function, physical dynamic systems, coupled engineering systems

Design of transmultiplexer integer filters

M. Ziolko, M. Nowak(AGH University of Science and Technology, Poland) |

The designing of transmultiplexer systems relies on determining filters for the transmitter and receiver sides of multicarrier communication system. The perfect reconstruction conditions lead to the bilinear equations for FIR filter coefficients. Generally there is no way of finding all possible solutions. This paper describes methods of finding a large family of solutions. Particular attention is devoted to obtaining algorithms useful in fixed-point arithmetic needed to design the integer filters. As a result, the systems perform perfect reconstruction of signals. Additionally, a simple method is presented to transform any transmultiplexer into an unlimited number of different transmultiplexers. Finally, two examples of integer filters that meet perfect reconstruction conditions are shown. The first illustrates a FIR filter which does not require multiplications. The frequency properties of filters and signals are discussed for the second example.

**keywords:** fixed point arithmetic, MIMO systems, integer digital filters

A 4-D chaotic hyperjerk system with a hidden attractor, adaptive backstepping control and circuit design

S. Vaidyanathan(Vel Tech University, India) | S. Jafari(Amirkabir University of Technology, Iran) | V.-T. Pham(Ton Duc Thang University, Vietnam) | A.T. Azar(Nile University, Egypt) |

A novel 4-D chaotic hyperjerk system with four quadratic nonlinearities is presented in this work. It is interesting that the hyperjerk system has no equilibrium. A chaotic attractor is said to be hidden attractor when its basin of attraction has no intersection with small neighborhoods of equilibrium points of the system. Thus, our new non-equilibrium hyperjerk system possesses a hidden attractor. Chaos in the system has been observed in phase portraits and verified by positive Lyapunov exponents. Adaptive backstepping controller is designed for the global chaos control of the non-equilibrium hyperjerk system with a hidden attractor. An electronic circuit for realizing the non-equilibrium hyperjerk system is also introduced, which validates the theoretical chaotic model of the hyperjerk system with a hidden chaotic attractor.

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

An extension of Klamka's method to positive descriptor discrete-time linear systems with bounded inputs

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

The minimum energy control problem for the positive descriptor discrete-time linear systems with bounded inputs by the use of Weierstrass-Kronecker decomposition is formulated and solved. Necessary and sufficient conditions for the positivity and reachability of descriptor discrete-time linear systems are given. Conditions for the existence of solution and procedure for computation of optimal input and the minimal value of the performance index is proposed and illustrated by a numerical example.

**keywords:** descriptor, positive, discrete-time, linear, system, Weierstrass-Kronecker decomposition, minimum energy control

Motion planning for nonholonomic systems with earlier destination reaching

A. Ratajczak(Wroclaw University of Science and Technology, Poland) |

The motion planning problem consists in finding a control function which drives the system to a desired point. The motion planning algorithm derived with an endogenous configuration space approach assumes that the motion takes place in an arbitrary chosen time horizon. This work introduces a modification to the motion planning algorithm which allows to reach the destination point in time, which is shorter than the assumed time horizon. The algorithm derivation relies on the endogenous configuration space approach and the continuation (homotopy) method. To achieve the earlier destination reaching a new formulation of the task map and the task Jacobian are introduced. The efficiency of the new algorithm is depicted with simulation results.

**keywords:** motion planning, nonholonomic, endogenous configuration space, homotopy, continuation, earlier destination reaching

Investigation of limit cycles and signal stabilization of two dimensional systems with memory type nonlinear elements

K.C. Patra, B.K. Dakua(C.V. Raman College of Engineering, India) |

The paper presents a simple, systematic and novel graphical method which uses computer graphics for prediction of limit cycles in two dimensional multivariable nonlinear system having rectangular hysteresis and backlash type nonlinearities. It also explores the avoidance of such self-sustained oscillations by determining the stability boundary of the system. The stability boundary is obtained using simple Routh Hurwitz criterion and the incremental input describing function, developed from harmonic balance concept. This may be useful in interconnected power system which utilizes governor control. If the avoidance of limit cycle or a safer operating zone is not possible, the quenching of such oscillations may be done by using the signal stabilization technique which is also described. The synchronization boundary is laid down in the forcing signal amplitudes plane using digital simulation. Results of digital simulations illustrate accuracy of the method for 2x2 systems.

**keywords:** describing function, dither signal, incremental input describing function, limit cycles, signal stabilization, stability boundary

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