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 2

Reachability of standard and fractional continuous-time systems with constant inputs

K. Rogowski(Bialystok University of Technology, Poland) |

The reachability of standard and fractional-order continuous-time systems with constant inputs is addressed. Positive and non-positive continuous-time linear systems are considered. Necessary and sufficient conditions for the existence of such constant inputs that steers the system from zero initial conditions to the given final state in desired time are derived and proved. As an example of such systems the electrical circuits with DC voltage sources are presented.

**keywords:** constant input, reachability, fractional system, positive system

Endogenous timing in a vertically differentiated mixed duopoly with Cournot competition

L. Feng(Jiaxing University, China) | M. Gu(Shanghai Jiao Tong University, China) |

This paper compares the equilibrium outcomes under simultaneous and sequential output setting in a mixed duopoly in a vertically differentiated market. When the timing of the output game is determined endogenously, it is shown that simultaneous play in the quality stage with the public firm acting as the high-quality producer and simultaneous play in the second period in the output stage turn out to be the unique subgame perfect Nash equilibrium, which contrasts with the endogenous timing in a purely private duopoly.

**keywords:** vertical differentiation, endogenous timing, public firm, private firm

Minimum energy control of descriptor discrete-time linear systems by the use of Weierstrass-Kronecker decomposition

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

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

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

Pointwise observation of the state given by parabolic system with boundary condition involving multiple time delays

A. Kowalewski(AGH University of Science and Technology, Poland) |

Various optimization problems for linear parabolic systems with multiple constant time delays are considered. In this paper, we consider an optimal distributed control problem for a linear parabolic system in which multiple constant time delays appear in the Neumann boundary condition. Sufficient conditions for the existence of a unique solution of the parabolic equation with the Neumann boundary condition involving multiple time delays are proved. The time horizon T is fixed. Making use of the Lions scheme, necessary and sufficient conditions of optimality for the Neumann problem with the quadratic cost function with pointwise observation of the state and constrained control are derived.

**keywords:** distributed control, parabolic system, time delays, pointwise observation

Relaxed formulation of the design conditions for Takagi-Sugeno fuzzy virtual actuators

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

The H-infinity norm approach to virtual actuators design, intended to Takagi-Sugeno fuzzy continuous-time systems, is presented in the paper. Using the second Lyapunov method, the design conditions are formulated in terms of linear matrix inequalities in adapted bounded real lemma structures. Related to the static output controller, and for systems under influence of single actuator faults, the design steps are revealed for a three-tank system plant.

**keywords:** nonlinear dynamic systems, Takagi-Sugeno fuzzy models, fault tolerant control, static output controllers, virtual actuators, linear matrix inequalities

Determinig of an object orientation in 3D space using direction cosine matrix and non-stationary Kalman filter

R. Bieda, K. Jaskot(Silesian University of Technology, Poland) |

This paper describes a method which determines the parameters of an object orientation in 3D space. The rotation angles calculation bases on the signals fusion obtained from the inertial measurement unit (IMU). The IMU measuring system provides information from a linear acceleration sensors (accelerometers), the Earth's magnetic field sensors (magnetometers) and the angular velocity sensors (gyroscopes). Information about the object orientation is presented in the form of direction cosine matrix whose elements are observed in the state vector of the non-stationary Kalman filter. The vector components allow to determine the rotation angles (roll, pitch and yaw) associated with the object. The resulting waveforms, for different rotation angles, have no negative attributes associated with the construction and operation of the IMU measuring system. The described solution enables simple, fast and effective implementation of the proposed method in the IMU measuring systems.

Adaptive observer-based fault estimation for a class of Lipschitz nonlinear systems

N. Oucief, S. Labiod(University of Jijel, Algeria) | M. Tadjine(The National Polytechnic School, Algeria) |

Fault input channels represent a major challenge for observer design for fault estimation. Most works in this field assume that faults enter in such a way that the transfer functions between these faults and a number of measured outputs are strictly positive real (SPR), that is, the observer matching condition is satisfied. This paper presents a systematic approach to adaptive observer design for joint estimation of the state and faults when the SPR requirement is not verified. The proposed method deals with a class of Lipschitz nonlinear systems subjected to piecewise constant multiplicative faults. The novelty of the proposed approach is that it uses a rank condition similar to the observer matching condition to construct the adaptation law used to obtain fault estimates. The problem of finding the adaptive observer matrices is formulated as a Linear Matrix Inequality (LMI) optimization problem. The proposed scheme is tested on the nonlinear model of a single link flexible joint robot system.

**keywords:** nonlinear adaptive observer, fault estimation, strictly positive real, Lipschitz systems, observer matching condition, LMI

A non integer order, state space model for one dimensional heat transfer process

K. Oprzedkiewicz(AGH University of Science and Technology, Poland) | E. Gawin(High Vocational School, Tarnow, Poland) |

In the paper a new, state space, non integer order model for one dimensional heat transfer process is presented. The model is based on known semigroup model. The derivative with respect to time is described by the non integer order Caputo operator, the spatial derivative is described by integer order operator. The elementary properties of the state operator are proven. The solution of state equation is calculated with the use of Laplace transform. Results of experiments show, that the proposed model is more accurate than analogical integer order model in the sense of square cost function.

**keywords:** non-integer order systems, heat transfer process, diffusion equation, infinite dimensional systems, Feller semigroups

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