Elements of mathematical phenomenology of self-organization nonlinear dynamical systems: Synergetics and fractional calculus approach

Improving the efficiency of the heat and power system, due to the synthesis of control based on the synergistic concept

“…At the end of the 20th century, a new approach to the synthesis of control systems emerged, which from its author A. Kolesnikov. called the method of Analytical Design of Aggregated Regulators (AKAR) [1][2][3][4][5][6][7][8][9][10]. The basis of this approach is the concept of synthesis of nonlinear feedbacks ensuring the asymptotic stability of the control system with respect to the required (from a practical point of view) motion (attractor) in the state space of the system [1].…”

Central collision of two rolling balls: theory and examples

“…The theory of the central collision of two bodies in translator moron (mass particles), as known theory, is starting point in our paper, and using Petrović's theory of Elements of mathematical phenomenology [5], [6] and Phenomenological mappings [7] are basic theories for obtaining kinetic parameters of the central collision of two rolling rigid, homogeneous and heavy, smooth balls along horizontal straight trace, as well as along curvilinear circle trace in vertical plane.. (Also, see papers from Special Issue of IJNLM [8], especially References [9][10][11][12][13][14]).…”

“…Listed expressions and relations (7)- (11) and conclusions a*-b*-c* of kinetic energy decreasing in comparison kinetic energy in pre-collision of kinetic state and post-collision kinetic state of the bodies in collision present Carnot's theorem (Lazare Carnot 1753-1824., Principes fondamenteaux de l'équilibre et de movement -1803) [25] of kinetic energy of bodies in translator kinetic states pre-and post collision (in arrival and outgoing kinetic states): " In the collision of two bodies in translator motion for arbitrary coefficient of the restitution, 1 0   k , lost of kinetic energy in decreasing during collision, is proportional to lost of the velocities.…”

“…. Using Petroviċ's theory of elements of mathematical phenomenology and phenomenological mappings [5][6][7][8][9][10][11][12][13][14] in parts of qualitative and mathematical analogies, we can indicate a qualitative and mathematical analogy between system of the translator dynamics and central collision (impact) dynamics of two bodies in translator motion pre-impact and post impact dynamics phenomena and system of the rolling two ball dynamics and central collision (impact) dynamics of two rolling balls in rolling motion pre-impact and post impact dynamics phenomena.…”

Unscented Kalman filter for continuous‐time nonlinear fractional‐order systems with process and measurement noises

“…The fractional-order G-L definition was the differential approximation recursion of the integer-order differential, and it is more suitable for signal processing [2]. Recently, the fractional-order calculus has been widely used in the physical mechanics [3], the biomedical science [4] and the automatic control [5] and so on. The development in the field of control is particularly rapid, such as fractional-order PID control [6], fractional-order sliding mode control [7] and iterative learning control [8].…”