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  1. 14 International Journal of Applied Mathematics and Computer Science
  2. 3 Zeszyty Naukowe. Automatyka / Politechnika Śląska
  3. 1 Archives of Electrical Engineering
  4. 1 Bulletin of the Polish Academy of Sciences. Mathematics
  5. 1 Bulletin of the Polish Academy of Sciences. Technical Sciences
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  1. 2 Starosolski Z.
  2. 2 Witkowska A.
  3. 2 Śmierzchalski R.
  4. 1 Abdelmalek I.
  5. 1 Abrishamifar A.
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1
Content available Hybrid synchronization and parameter estimation of a complex chaotic network of permanent magnet synchronous motors using adaptive integral sliding mode control
Siddique Nazam, Rehman Fazal U.
Bulletin of the Polish Academy of Sciences. Technical Sciences
|
2021
|
Vol. 69, nr 3
art. no. e137056
EN
The synchronisation of a complex chaotic network of permanent magnet synchronous motor systems has increasing practical importance in the field of electrical engineering. This article presents the control design method for the hybrid synchronization and parameter estimation of ring-connected complex chaotic network of permanent magnet synchronous motor systems. The design of the desired control law is a challenging task for control engineers due to parametric uncertainties and chaotic responses to some specific parameter values. Controllers are designed based on the adaptive integral sliding mode control to ensure hybrid synchronization and estimation of uncertain terms. To apply the adaptive ISMC, firstly the error system is converted to a unique system consisting of a nominal part along with the unknown terms which are computed adaptively. The stabilizing controller incorporating nominal control and compensator control is designed for the error system. The compensator controller, as well as the adopted laws, are designed to get the first derivative of the Lyapunov equation strictly negative. To give an illustration, the proposed technique is applied to 4-coupled motor systems yielding the convergence of error dynamics to zero, estimation of uncertain parameters, and hybrid synchronization of system states. The usefulness of the proposed method has also been tested through computer simulations and found to be valid.
2
Content available T–S fuzzy BIBO stabilisation of non-linear systems under persistent perturbations using fuzzy Lyapunov functions and non-PDC control laws
Salcedo José V., Martínez Miguel, García-Nieto Sergio, Hilario Adolfo
International Journal of Applied Mathematics and Computer Science
|
2020
|
Vol. 30, no. 3
529--550
EN
This paper develops an innovative approach for designing non-parallel distributed fuzzy controllers for continuous-time non-linear systems under persistent perturbations. Non-linear systems are represented using Takagi–Sugeno fuzzy models. These non-PDC controllers guarantee bounded input bounded output stabilisation in closed-loop throughout the computation of generalised inescapable ellipsoids. These controllers are computed with linear matrix inequalities using fuzzy Lyapunov functions and integral delayed Lyapunov functions. LMI conditions developed in this paper provide non-PDC controllers with a minimum ⋆-norm (upper bound of the 1-norm) for the T–S fuzzy system under persistent perturbations. The results presented in this paper can be classified into two categories: local methods based on fuzzy Lyapunov functions with guaranteed bounds on the first derivatives of membership functions and global methods based on integral-delayed Lyapunov functions which are independent of the first derivatives of membership functions. The benefits of the proposed results are shown through some illustrative examples.
3
Content available On three methods for bounding the rate of convergence for some continuous-time Markov chains
Zeifman Alexander, Satin Yacov, Kryukova Anastasia, Razumchik Rostislav, Kiseleva Ksenia, Shilova Galina
International Journal of Applied Mathematics and Computer Science
|
2020
|
Vol. 30, no. 2
251--266
EN
Consideration is given to three different analytical methods for the computation of upper bounds for the rate of convergence to the limiting regime of one specific class of (in)homogeneous continuous-time Markov chains. This class is particularly well suited to describe evolutions of the total number of customers in (in)homogeneous M/M/S queueing systems with possibly state-dependent arrival and service intensities, batch arrivals and services. One of the methods is based on the logarithmic norm of a linear operator function; the other two rely on Lyapunov functions and differential inequalities, respectively. Less restrictive conditions (compared with those known from the literature) under which the methods are applicable are being formulated. Two numerical examples are given. It is also shown that, for homogeneous birth-death Markov processes defined on a finite state space with all transition rates being positive, all methods yield the same sharp upper bound.
4
Content available One DOF robot manipulator control through type-2 fuzzy robust adaptive controller
Naderolasli Amir, Chatraei Abbas
Journal of Automation Mobile Robotics and Intelligent Systems
|
2019
|
Vol. 13, No. 1
65--70
EN
In this article, the one DOF robot manipulator control is assessed through second type robust fuzzy-adaptive controller. The objective is to obtain a tracking path with appropriate accuracy. The stability of the closed loop system is verified through Lyapunov stability theory and the efficiency of tracking is analyzed subject to the constraints and uncertainty. In order to design the fuzzy controller a set of if-then fuzzy rules are considered which describe the system input-output behavior. Simulation and the results of the experiments on the one DOF robots indicate the effectiveness of the proposed methods.
5
Content available Fundamental limitations of the decay of generalized energy in controlled (discrete-time) nonlinear systems subject to state and input constraints
Selek István, Ikonen Enso
International Journal of Applied Mathematics and Computer Science
|
2019
|
Vol. 29, no. 4
629--639
EN
This paper is devoted to the analysis of fundamental limitations regarding closed-loop control performance of discrete-time nonlinear systems subject to hard constraints (which are nonlinear in state and manipulated input variables). The control performance for the problem of interest is quantified by the decline (decay) of the generalized energy of the controlled system. The paper develops (upper and lower) barriers bounding the decay of the system’s generalized energy, which can be achieved over a set of asymptotically stabilizing feedback laws. The corresponding problem is treated without the loss of generality, resulting in a theoretical framework that provides a solid basis for practical implementations. To enhance understanding, the main results are illustrated in a simple example.
6
Content available Adaptive backstepping tracking control for an over-actuated DP marine vessel with inertia uncertainties
Witkowska A., Śmierzchalski R.
International Journal of Applied Mathematics and Computer Science
|
2018
|
Vol. 28, no. 4
679--693
EN
Designing a tracking control system for an over-actuated dynamic positioning marine vessel in the case of insufficient information on environmental disturbances, hydrodynamic damping, Coriolis forces and vessel inertia characteristics is considered. The designed adaptive MIMO backstepping control law with control allocation is based on Lyapunov control theory for cascaded systems to guarantee stabilization of the marine vessel position and heading. Forces and torque computed from the adaptive control law are allocated to individual thrusters by employing the quadratic programming method in combination with the cascaded generalized inverse algorithm, the weighted least squares algorithm and the minimal least squares algorithm. The effectiveness of the proposed control scheme is demonstrated by simulations involving a redundant set of actuators. The evaluation criteria include energy consumption, robustness, as well accuracy of tracking during typical vessel operation.
7
Content available The dynamics of a parametrically driven damped pendulum
Das A., Kumar K.
International Journal of Applied Mechanics and Engineering
|
2015
|
Vol. 20, no. 2
257--266
EN
Ordered and chaotic states of a parametrically driven planar pendulum with viscous damping are numerically investigated. The damping makes the number of chaotic windows fewer but with larger width. Stroboscopic maps of the chaotic motion of the pendulum, driven either subharmonically or harmonically, show strange attractors with inversion symmetry in the phase plane.
8
Content available Nonlinear control for a diesel engine: A CLF-based approach
Kuzmych O., Aitouche A., El-Hajjaji A., Bosche J.
International Journal of Applied Mathematics and Computer Science
|
2014
|
Vol. 24, no. 4
821--835
EN
In this paper, we propose a control Lyapunov function based on a nonlinear controller for a turbocharged diesel engine. A model-based approach is used which predicts the experimentally observed engine performance for a biodiesel. The basic idea is to develop an inverse optimal control and to employ a Lyapunov function in order to achieve good performances. The obtained controller gain guarantees the global convergence of the system and regulates the flows for the variable geometry turbocharger as well as exhaust gas recirculation systems in order to minimize the NOx emission and the smoke of a biodiesel engine. Simulation of the control performances based on professional software and experimental results show the effectiveness of this approach.
9
Content available remote Speed Sensorless Control of Induction Motor Based on a Non-Linear Flux and Speed Observer with Parameters Estimation and Considering Iron Loss
Alemi-Rostami M., Abrishamifar A.
Przegląd Elektrotechniczny
|
2013
|
R. 89, nr 3a
89--97
EN
In this paper, a control method is presented for induction motor which offers high efficiency and high dynamics even considering the influences of iron loss. Recently, research to consider the influences of iron loss has been made in the vector control of an induction motor. Vector control method is a quite complex task which demands precise information about the rotor speed and the position of the magnetic flux. The vector control method presented in this paper, estimates both rotor speed and amplitude magnetic flux rotor. However, there are also applications in which even speed sensors should be omitted. In this method, three-phase motor currents and DC link voltage are measured by means of a nonlinear observer and considering the Lyapunov function for current error, motor parameters, rotor shaft and position of flux are estimated. To stabilize the control system, the Lyapunov function for error, is used. Then, using the genetic algorithm, the value of controlling coefficients their and general effects on system's behavior are obtained. The simulation and experimental results confirm the resistant performance and the proper dynamic efficiency of this method.
PL
W artykule przedstawiono algorytm sterowania maszyną indukcyjną, uwzględniający straty w żelazie. Sterowanie opiera się na bezczujnikowym pomiarze prędkości obrotowej, przy pomocy obserwatora. Korzystając z pomiarów prądów fazowych i napięcia DC-link, z funkcji Lyapunov’a wyznaczana jest prędkość i strumień wirnika. Implementacja algorytmu genetycznego pozwoliła na ocenę wpływu współczynników regulacji na odpowiedź układu. Przedstawiono wyniki symulacyjne i eksperymentalne.
10
Content available Sharply-weak regular linear extensions of dynamical systems leading up to regular systems
Tkocz-Piszczek E.
Zeszyty Naukowe. Matematyka Stosowana / Politechnika Śląska
|
2011
|
z. 1
51--68
EN
In this paper we consider some sign–changing Lyapunov function in research on regularity and sharply-week regularity of sets of linear extensions of dynamical systems. By regularity we mean exponential dichotomy of linear differential systems.
PL
Wartykule podjęto tematykę regularności liniowych rozszerzeń układów dynamicznych, która równoważna jest z wykładniczą dychotomią liniowych układów różniczkowych. Przeprowadzono badanie regularności przy użyciu znakozmiennych funkcji Lapunowa. Ponadto, przedstawiono doprowadzenie słabo regularnych układów do regularnych.
11
Content available remote Stability of impulsive Hopfield neural networks with Markovian switching and time-varying delays
Raja R., Sakthivel R., Anthoni S. M., Kim H.
International Journal of Applied Mathematics and Computer Science
|
2011
|
Vol. 21, no 1
127-135
EN
The paper is concerned with stability analysis for a class of impulsive Hopfield neural networks with Markovian jumping parameters and time-varying delays. The jumping parameters considered here are generated from a continuous-time discrete-state homogenous Markov process. By employing a Lyapunov functional approach, new delay-dependent stochastic stability criteria are obtained in terms of linear matrix inequalities (LMIs). The proposed criteria can be easily checked by using some standard numerical packages such as theMatlab LMI Toolbox. A numerical example is provided to show that the proposed results significantly improve the allowable upper bounds of delays over some results existing in the literature.
12
Content available remote Convergence method, properties and computational complexity for Lyapunov games
Clempner J. B., Poznyak A. S.
International Journal of Applied Mathematics and Computer Science
|
2011
|
Vol. 21, no 2
349-361
EN
We introduce the concept of a Lyapunov game as a subclass of strictly dominated games and potential games. The advantage of this approach is that every ergodic system (repeated game) can be represented by a Lyapunov-like function. A direct acyclic graph is associated with a game. The graph structure represents the dependencies existing between the strategy profiles. By definition, a Lyapunov-like function monotonically decreases and converges to a single Lyapunov equilibrium point identified by the sink of the game graph. It is important to note that in previous works this convergence has not been guaranteed even if the Nash equilibrium point exists. The best reply dynamics result in a natural implementation of the behavior of a Lyapunov-like function. Therefore, a Lyapunov game has also the benefit that it is common knowledge of the players that only best replies are chosen. By the natural evolution of a Lyapunov-like function, no matter what, a strategy played once is not played again. As a construction example, we show that, for repeated games with bounded nonnegative cost functions within the class of differentiable vector functions whose derivatives satisfy the Lipschitz condition, a complex vector-function can be built, where each component is a function of the corresponding cost value and satisfies the condition of the Lyapunov-like function. The resulting vector Lyapunov-like function is a monotonic function which can only decrease over time. Then, a repeated game can be represented by a one-shot game. The functionality of the suggested method is successfully demonstrated by a simulated experiment.
13
Content available remote The choice of the forms of Lyapunov functions for a positive 2D Roesser model
Kaczorek T.
International Journal of Applied Mathematics and Computer Science
|
2007
|
Vol. 17, no 4
471-475
EN
The appropriate choice of the forms of Lyapunov functions for a positive 2D Roesser model is addressed. It is shown that for the positive 2D Roesser model: (i) a linear form of the state vector can be chosen as a Lyapunov function, (ii) there exists a strictly positive diagonal matrix P such that the matrix ATPA-P is negative definite. The theoretical deliberations will be illustrated by numerical examples.
14
Content available remote Extended Lie algebraic stability analysis for switched systems with continuous-time and discrete–time subsystems
Zhai G., Xu X., Lin H., Liu D.
International Journal of Applied Mathematics and Computer Science
|
2007
|
Vol. 17, no 4
447-454
EN
We analyze stability for switched systems which are composed of both continuous-time and discrete-time subsystems. By considering a Lie algebra generated by all subsystem matrices, we show that if all subsystems are Hurwitz/Schur stable and this Lie algebra is solvable, then there is a common quadratic Lyapunov function for all subsystems and thus the switched system is exponentially stable under arbitrary switching. When not all subsystems are stable and the same Lie algebra is solvable, we show that there is a common quadratic Lyapunov-like function for all subsystems and the switched system is exponentially stable under a dwell time scheme. Two numerical examples are provided to demonstrate the result.
15
Content available remote A new fuzzy Lyapunov approach to non-quadratic stabilization of Takagi-Sugeno fuzzy models
Abdelmalek I., Goléa N., Hadjili M. L.
International Journal of Applied Mathematics and Computer Science
|
2007
|
Vol. 17, no 1
39-51
EN
In this paper, new non-quadratic stability conditions are derived based on the parallel distributed compensation scheme to stabilize Takagi-Sugeno (T-S) fuzzy systems. We use a non-quadratic Lyapunov function as a fuzzy mixture of multiple quadratic Lyapunov functions. The quadratic Lyapunov functions share the same membership functions with the T-S fuzzy model. The stability conditions we propose are less conservative and stabilize also fuzzy systems which do not admit a quadratic stabilization. The proposed approach is based on two assumptions. The first one relates to a proportional relation between multiple Lyapunov functions and the second one considers an upper bound to the time derivative of the premise membership functions. To illustrate the advantages of our proposal, four examples are given.
16
Content available remote A backstepping approach to ship course control
Witkowska A., Tomera M., Śmierzchalski R.
International Journal of Applied Mathematics and Computer Science
|
2007
|
Vol. 17, no 1
73-85
EN
As an object of course control, the ship is characterised by a nonlinear function describing static manoeuvring characteristics that reflect the steady-state relation between the rudder deflection and the rate of turn of the hull. One of the methods which can be used for designing a nonlinear ship course controller is the backstepping method. It is used here for designing two configurations of nonlinear controllers, which are then applied to ship course control. The parameters of the obtained nonlinear control structures are tuned to optimise the operation of the control system. The optimisation is performed using genetic algorithms. The quality of operation of the designed control algorithms is checked in simulation tests performed on the mathematical model of a tanker. In order to obtain reference results to be used for comparison with those recorded for nonlinear controllers designed using the backstepping method, a control system with the PD controller is examined as well.
17
Badanie stabilności układu hybrydowego z zastosowaniem wielokrotnej przedziałami liniowej funkcji Lapunowa
Starosolski Z.
Zeszyty Naukowe. Automatyka / Politechnika Śląska
|
2006
|
z. 146
85-104
PL
W niniejszej pracy badano stabilność układu hybrydowego z zastosowaniem wielokrotnej funkcji Lapunowa. Do parametryzacji wielokrotnej funkcji Lapunowa użyto funkcji przedziałami liniowej. Rozważano układ hybrydowy złożony z dwóch dynamik przełączanych w zależności od stanu dyskretnego i liniowej funkcji przełączającej. Cechą charakterystyczną badanego układu jest poruszanie się po trajektoriach dwóch różnych dynamik w tym samym regionie płaszczyzny stanu. Stabilność danego układu jest analizowana przez rozwiązanie problemu programowania liniowego (PPL). W wyniku rozwiązania PPL otrzymuje się wartości przedziałami liniowej wielokrotnej funkcji Lapunowa (PLWFL). W pracy zostały przedstawione warunki konieczne, jakie musi spełniać PLWFL, aby badany układ hybrydowy był asymptotycznie stabilny.
EN
In the paper a study of application of piecewise linear Lyapunov function as a Multiple Lyapunov function for analyzing the stability of hybrid system is presented. An example of hybrid system composed of two linear dynamics with switching is analyzed. The switching is controlled by a discrete control function. Both dynamics are asymptocally stable and both are defined under common region of the state space. The stability of hybrid system is proved by solving the linear programming problem. The conditions for asymptotic stability of hybrid linear system based on Piecewise Linear Multiple Lyapunov Function are presented.
18
Content available remote Modeling shortest path games with Petri nets: A Lyapunov based theory
Clempner J.
International Journal of Applied Mathematics and Computer Science
|
2006
|
Vol. 16, no 3
387-397
EN
In this paper we introduce a new modeling paradigm for shortest path games representation with Petri nets. Whereas previous works have restricted attention to tracking the net using Bellman’s equation as a utility function, this work uses a Lyapunov-like function. In this sense, we change the traditional cost function by a trajectory-tracking function which is also an optimal cost-to-target function. This makes a significant difference in the conceptualization of the problem domain, allowing the replacement of the Nash equilibrium point by the Lyapunov equilibrium point in game theory. We show that the Lyapunov equilibrium point coincides with the Nash equilibrium point. As a consequence, all properties of equilibrium and stability are preserved in game theory. This is the most important contribution of this work. The potential of this approach remains in its formal proof simplicity for the existence of an equilibrium point.
19
Content available remote A new robust approach to MCS algorithms. Application to an AC machine
Naceri F., Belkacem S.
Archives of Electrical Engineering
|
2006
|
Vol. 55, nr 2
163-178
EN
The robustness of MCS algorithms against rapidly varying system parameters, external disturbances and system non-linearity is investigated. The MCS algorithm is a signifiant developpment of MRAC and is similary based on the hyperstability theory of Popov. The hyperstability theory guarantees the global asymptotic stability of the error vector (i.e. the difference between the reference model and system states). Finally a new approach has been successfully implemented to Field-Oriented Controlled drive. Discussion on theoretical aspects, such as, selection of a reference model, stability analysis, gain adaptive and steady state error are included. Results of simulations are also presented.
PL
Przeanalizowano odporność algorytmów syntezy minimalnego sterownika przy szybkozmiennych wartościach parametrów układu, zakłóceniach zewnętrznych i przy nieliniowości. Algorytm syntezy minimalnego sterownika jest istotnym krokiem w rozwoju sterowania adaptacyjnego z modelem odniesienia, opartym na teorii hyperstabilności Popowa. Zapewnia ona szeroką stabilność asymetryczną wektora błędu (różnicy między stanami modelu odniesienia i układu). Nowe podejście zastosowano pomyślnie w sterowaniu polowo-zorientowanym. Przedyskutowano takie zagadnienia teoretyczne jak wybór modelu odniesienia, analizę stabilności, błąd doboru wzmocnienia i błąd w stanie ustalonym. Przedstawiono rezultaty badań symulacyjnych.
20
Content available remote Non-quadratic performance design for Takagi-Sugeno fuzzy systems
Bernal M., Husek P.
International Journal of Applied Mathematics and Computer Science
|
2005
|
Vol. 15, no 3
383-391
EN
This paper improves controller synthesis of discrete Takagi-Sugeno fuzzy systems based on non-quadratic Lyapunov functions, making it possible to accomplish various kinds of control performance specifications such as decay rate conditions, requirements on control input and output and disturbance rejection. These extensions can be implemented via linear matrix inequalities, which are numerically solvable with commercially available software. The controller design is illustrated with an example.
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