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EN
This article deals with modelling and a flatness-based robust trajectory tracking scheme for a two degrees of freedom helicopter, which is subject to four types of tail rotor disturbances to validate the control scheme robustness. A mathematical model of the system, its differential flatness and a differential parametrization are obtained. The flat filtering control is designed for the system control with a partially known model, assuming the non-modelled dynamics and the external disturbances (specially the tail rotor ones) to be rejected by means of an extended state model (ultra-local model). Numerical and experimental assessments are carried out on a characterized prototype whose yaw angle (ψ), given by the z axis, is in free form, while the pitch angle (θ), which results from rotation about the y axis, is mechanically restricted. The proposed controller performance is tested through a set of experiments in trajectory tracking tasks with different disturbances in the tail rotor, showing robust behaviour for the different disturbances. Besides, a comparison study against a widely used controller of LQR type is carried out, in which the proposed controller achieves better results, as illustrated by a performance index.
EN
The global stability of discrete-time nonlinear systems with descriptor positive linear parts, positive scalar feedbacks and interval state matrices is addressed. Sufficient conditions for the global stability of this class of nonlinear systems are established. The effectiveness of these conditions is illustrated using numerical examples.
EN
A novel fault-tolerant tracking control scheme based on an adaptive robust observer for non-linear systems is proposed. Additionally, it is presumed that the non-linear system may be faulty, i.e., affected by actuator and sensor faults along with the disturbances, simultaneously. Accordingly, the stability of the robust observer as well as the fault-tolerant tracking controller is achieved by using the ℋ∞ approach. Furthermore, unknown actuator and sensor faults and states are bounded by the uncertainty intervals for estimation quality assessment as well as reliable fault diagnosis. This means that narrow intervals accompany better estimation quality. Thus, to cope with the above difficulty, it is assumed that the disturbances are over-bounded by an ellipsoid. Consequently, the performance and correctness of the proposed fault-tolerant tracking control scheme are verified by using a non-linear twin-rotor aerodynamical laboratory system.
EN
The diagnosis of systems is one of the major steps in their control and its purpose is to determine the possible presence of dysfunctions, which affect the sensors and actuators associated with a system but also the internal components of the system itself. On the one hand, the diagnosis must therefore focus on the detection of a dysfunction and, on the other hand, on the physical localization of the dysfunction by specifying the component in a faulty situation, and then on its temporal localization. In this contribution, the emphasis is on the use of software redundancy applied to the detection of anomalies within the measurements collected in the system. The systems considered here are characterized by non-linear behaviours whose model is not known a priori. The proposed strategy therefore focuses on processing the data acquired on the system for which it is assumed that a healthy operating regime is known. Diagnostic procedures usually use this data corresponding to good operating regimes by comparing them with new situations that may contain faults. Our approach is fundamentally different in that the good functioning data allow us, by means of a non-linear prediction technique, to generate a lot of data that reflect all the faults under different excitation situations of the system. The database thus created characterizes the dysfunctions and then serves as a reference to be compared with real situations. This comparison, which then makes it possible to recognize the faulty situation, is based on a technique for evaluating the main angle between subspaces of system dysfunction situations. An important point of the discussion concerns the robustness and sensitivity of fault indicators. In particular, it is shown how, by non-linear combinations, it is possible to increase the size of these indicators in such a way as to facilitate the location of faults.
5
Content available remote Highly nonlinear systems estimation using extended and unscented kalman filters
EN
The main idea of this study is to evaluate the estimation performance of extended and unscented Kalman filters (EKF and UKF). So, these latter are introduced to estimate the dynamic states of a similar model operating with identical covariance matrices in the same situation. The mean square error (MSE) criterion is used to quantify the estimation error between the actual and the estimated values. The simulation results obtained with Matlab/ Simulink software confirm the superiority and efficiency of UKF over EKF, especially when the system is highly non-linear under process and measurement noises, such is the case of the inverted double pendulum mounted on a cart (DIPC).
PL
Główną ideą tego badania jest ocena wydajności estymacji rozszerzonych filtrów Kalmana (EKF i UKF). Te ostatnie zostały wprowadzone w celu oszacowania stanów dynamicznych podobnego modelu działającego z identycznymi macierzami kowariancji. Kryterium błędu średniokwadratowego (MSE) służy do ilościowego określenia błędu oszacowania między wartościami rzeczywistymi i szacunkowymi. Wyniki symulacji uzyskane za pomocą oprogramowania Matlab i Simulink potwierdzają wyższość i wydajność UKF nad EKF, zwłaszcza gdy system jest wysoce nieliniowy
EN
The global stability of positive discrete-time time-varying nonlinear systems with time-varying scalar feedbacks is investigated. Sufficient conditions for the asymptotic stability of discrete-time positive time-varying linear systems are given. The new conditions are applied to discrete-time positive time-varying nonlinear systems with time-varying feedbacks. Sufficient conditions are established for the global stability of the discrete-time positive time-varying nonlinear systems with feedbacks.
EN
This paper addresses the issue of data-driven smoothing parameter (bandwidth) selection in the context of nonparametric system identification of dynamic systems. In particular, we examine the identification problem of the block-oriented Hammerstein cascade system. A class of kernel-type Generalized Regression Neural Networks (GRNN) is employed as the identification algorithm. The statistical accuracy of the kernel GRNN estimate is critically influenced by the choice of the bandwidth. Given the need of data-driven bandwidth specification we propose several automatic selection methods that are compared by means of simulation studies. Our experiments reveal that the method referred to as the partitioned cross-validation algorithm can be recommended as the practical procedure for the bandwidth choice for the kernel GRNN estimate in terms of its statistical accuracy and implementation aspects.
EN
A wide variety of approaches for set-valued simulation, parameter identification, state estimation as well as reachability, observability and stability analysis for nonlinear discrete-time systems involve the propagation of ellipsoids via nonlinear functions. It is well known that the corresponding image sets usually possess a complex shape and may even be nonconvex despite the convexity of the input data. For that reason, domain splitting procedures are often employed which help to reduce the phenomenon of overestimation that can be traced back to the well-known dependency and wrapping effects of interval analysis. In this paper, we propose a simple, yet efficient scheme for simultaneously determining outer and inner ellipsoidal range enclosures of the solution for the evaluation of multi-dimensional functions if the input domains are themselves described by ellipsoids. The Hausdorff distance between the computed enclosure and the exact solution set reduces at least linearly when decreasing the size of the input domains. In addition to algebraic function evaluations, the proposed technique is-for the first time, to our knowledge-employed for quantifying worst-case errors when extended Kalman filter-like, linearization-based techniques are used for forecasting confidence ellipsoids in a stochastic setting.
9
Content available remote Memristive devices In three-phase systems
EN
The aim of the research presented in a paper was to provide trustworthy simulation results for symmetrical three-phase systems with memristive load. The memristors in the system are combined with linear resistors in order to limit the current in the element. Linear drift model of the memristorwasconsideredinMatlabsimulations. ItisbasedonStrukovmodelwithBiolekwindow. Highnonlinearityofmemristorresultsindeformation of most of the signals in the system. Since the voltage of the neutral point is highly non-sinusoidal it affects on other signals like phase voltage, phase currents, delta voltages. A Fast Fourier Transform (FFT) is applied to chosen signals in order to provide a frequency spectrum. On this basis a Total Harmonic Distortion (THD) parameter was calculated.
PL
W pracy zaprezentowano wyniki badań symulacyjnych nad układem trójfazowym symetrycznym z obciążeniem elementami memrystorowymi. Memrystory w obwodzie odbiornika są połączone szeregowo z rezystorami liniowymi w celu ograniczenia prądu. W obliczeniach symulacyjnych przyjęto model memrystora "linear drift", bazujący na modelu Strukova z oknem Biolka. Wysoka nieliniowość elementów memrystorowych skutkuje odkształceniem większości sygnałów w obwodzie. Skoro napięcie punktu neutralnego odbiornika wykazuje wysoką nieliniowość, to skutkuje to odkształceniem pozostałych sygnałów, t.j. napięć fazowych, prądów fazowych czy napięć przewodowych. Do wybranych sygnałów zastosowano Szybką Transformatę Fouriera (FTT) w celu zaprezentowania widma częstotliwościowego. Na tej podstawie obliczono Współczynnik Zawartości Harmonicznych.
EN
In this paper, we prove the existence of a unique weak solution for a class of fractional systems of Schrodinger equations by using the Minty-Browder theorem in the Cartesian space. To this aim, we need to impose some growth conditions to control the source functions with respect to dependent variables.
EN
The global (absolute) stability of nonlinear systems with fractional positive and not necessarily asymptotically stable linear parts and feedbacks is addressed. The characteristics u = f(e) of the nonlinear parts satisfy the condition k1e ≤ f(e) ≤ k2e for some positive k1 and k2. It is shown that the fractional nonlinear systems are globally asymptotically stable if the Nyquist plots of the fractional positive linear parts are located on the right-hand side of the circles (−1/k1,−1/k2).
EN
The global stability of positive continuous-time standard and fractional order nonlinear feedback systems is investigated. New sufficient conditions for the global stability of these classes of of positive nonlinear systems are established. The effectiveness of these new stability conditions is demonstrated on simple examples of positive nonlinear systems.
EN
A new two-stage approach to the identification of polynomial Wiener systems is proposed. It is assumed that the linear dynamic system is described by a transfer function model, the memoryless nonlinear element is invertible and the inverse nonlinear function is a polynomial. Based on these assumptions and by introducing a new extended parametrization, the Wiener model is transformed into a linear-in-parameters form. In Stage I, parameters of the transformed Wiener model are estimated using the least squares (LS) and instrumental variables (IV) methods. Although the obtained parameter estimates are consistent, the number of parameters of the transformed Wiener model is much greater than that of the original one. Moreover, there is no unique relationship between parameters of the inverse nonlinear function and those of the transformed Wiener model. In Stage II, based on the assumption that the linear dynamic model is already known, parameters of the inverse nonlinear function are estimated uniquely using the IV method. In this way, not only is the parameter redundancy removed but also the parameter estimation accuracy is increased. A numerical example is included to demonstrate the practical effectiveness of the proposed approach.
EN
This investigation is concerned with robust analysis and control of uncertain nonlinear systems with parametric uncertainties. In contrast to the methodologies from the field of linear parameter varying systems, which employ convex structures of the state space representation in order to perform analysis and design, the proposed approach makes use of a polytopic form of a generalisation of the characteristic polynomial, which proves to outperform former results on the subject. Moreover, the derived conditions have the advantage of being cast as linear matrix inequalities under mild assumptions.
EN
The positivity and absolute stability of a class of fractional nonlinear continuous-time and discrete-time systems are addressed. Necessary and sufficient conditions for the positivity of this class of nonlinear systems are established. Sufficient conditions for the absolute stability of this class of fractional positive nonlinear systems are also given.
EN
This paper presents a novel approach to the design of fuzzy state feedback controllers for continuous-time non-linear systems with input saturation under persistent perturbations. It is assumed that all the states of the Takagi–Sugeno (TS) fuzzy model representing a non-linear system are measurable. Such controllers achieve bounded input bounded output (BIBO) stabilisation in closed loop based on the computation of inescapable ellipsoids. These ellipsoids are computed with linear matrix inequalities (LMIs) that guarantee stabilisation with input saturation and persistent perturbations. In particular, two kinds of inescapable ellipsoids are computed when solving a multiobjective optimization problem: the maximum volume inescapable ellipsoids contained inside the validity domain of the TS fuzzy model and the smallest inescapable ellipsoids which guarantee a minimum *-norm (upper bound of the 1-norm) of the perturbed system. For every initial point contained in the maximum volume ellipsoid, the closed loop will enter the minimum *-norm ellipsoid after a finite time, and it will remain inside afterwards. Consequently, the designed controllers have a large domain of validity and ensure a small value for the 1-norm of closed loop.
17
Content available Investigation of the memristor nonlinear properties
EN
The study of nonlinear systems is an important research topic for scientists and researchers. Memristor, for a long time, it remained just as a theoretical element and rarely appeared in the literature because of having no simple and practical realization. In this paper, we reviewed the theoretical substantiation of the memristor and conducted a practical study of its nonlinear properties using the memristor company KNOWM of series BS-AF-W 16DIP. We also investigated the characteristics of the memristor via the LabView environment.
PL
Badanie systemów nieliniowych jest ważnym tematem dla badaczy i naukowców. Memrystor przez długi czas pozostawał elementem teoretycznym i rzadko pojawiał się w literaturze z powodu braku prostej i praktycznej realizacji. W tym artykule zostały przedstawione teoretyczne uzasadnienie memrystora i badania jego właściwości nieliniowych na przykładzie memrystora firmy KNOWM serii BS-AF-W 16DIP. Zostały przeprowadzone badania charakterystyk memrystora w środowisku LabView.
EN
Backstepping is a method designed to control nonlinear systems. Formulated for the first time and described by Petar Kokotovic around 1990. The main purpose of backstepping is stabilisation of each subsystem’s structure; the function needs to be of an adequate class. The present task consists in minimising the vehicle body displacement from the position of equilibrium after it has been knocked out of it with the force exerted on the wheel, which has the purpose of simulating the wheel encountering an obstacle. The model depicts ¼ of the vehicle suspension with a conical spring or classical spring, and the damper with magnetorheological fluid which modifies damping of the suspension. The Mathcad programme facilitated the calculations, the model was designed using the Matlab – Simulink environment. Simulations were carried out on the model with the linear characteristics of a spring, as well as on the model with the implemented nonlinear spring. All the investigated cases were accompanied by the control tests that comprised building the model without control, i.e. with the constant value of damping
EN
The purpose of this work is to propose and characterize fractional descriptor reduced-order perfect nonlinear observers for a class of fractional descriptor discrete-time nonlinear systems. Sufficient conditions for the existence of these observers are established. The design procedure of the observers is given and demonstrated on a numerical example.
EN
In this study the fixed-fixed column subjected to axial Euler’s load has been investigated. The load is placed between the fixed ends of the structure and its location can be changed along column’s length. The boundary problem of free vibrations of the mentioned system has been formulated on the basis of Bernoulli – Euler theory and taking into account non-linear axial deformation relationship. Due to non-linear expressions the solution of the problem was done with small parameter method. In the paper the change of the first vibration frequency in relation to location and magnitude of the loading force was obtained. The relationship between natural vibration frequency and the amplitude is also discussed.
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