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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
Although the explicit commutativitiy conditions for second-order linear time-varying systems have been appeared in some literature, these are all for initially relaxed systems. This paper presents explicit necessary and sufficient commutativity conditions for commutativity of second-order linear time-varying systems with non-zero initial conditions. It has appeared interesting that the second requirement for the commutativity of non-relaxed systems plays an important role on the commutativity conditions when non-zero initial conditions exist. Another highlight is that the commutativity of switched systems is considered and spoiling of commutativity at the switching instants is illustrated for the first time. The simulation results support the theory developed in the paper.
EN
This article presents the usage of a zero-sum differential game to control a nonlinear object, which, in the analysed problem, was a mathematical pendulum. The obtained control was optimal with regard to adopted quality indicator for the worst interference. The two-point boundary value problem was solved numerically by means of the Dircol software application. Numerical solutions, meeting all the necessary optimality conditions, were obtained for different values of the rough parameter and for different values of damping.
PL
W artykule wykorzystano grę różniczkową o sumie zerowej do sterowania obiektem nieliniowym, jakim w analizowanym problemie jest wahadło matematyczne. Uzyskano sterowanie optymalne ze względu na przyjęty wskaźnik jakości, przy najgorszym zakłóceniu. Dwupunktowy problem brzegowy został rozwiązany numerycznie przy wykorzystaniu programu Dircol. Rozwiązania numeryczne spełniające wszystkie warunki konieczne optymalności zostały otrzymane dla różnych wartości parametru szorstkości oraz przy różnych wartościach tłumienia.
EN
In the paper, a novel control structure based on the fuzzy logic and model predictive control methodologies for an elastic two-mass drive system is proposed. In order to reduce the computational requirements of the classical MPC methodology, the multi parametric programming (MPT) approach is used. The robustness of the system is ensured by implementation of three MPT controllers generated for different operation points and a supervisory fuzzy system. The main goal of the fuzzy system is suitable shaping of the control signal. The effectiveness of the proposed approach is checked in simulation and experimental tests. In order to show the properties of the proposed control structure, a critical comparison with an adaptive classical MPC controller is carried out. Both control structures are tested taking into account the performance and possibility of real-time implementation.
5
Content available remote Porównanie wybranych algorytmów sterowania napędem PMSM w sterowaniu nadążnym
PL
W niniejszym artykule przedstawiono porównanie czterech regulatorów podczas zadania nadążania za zadaną trajektorią. Obiektem symulacji jest napęd z silnikiem PMSM, na którego wale umieszczono dodatkowo masę bezwładną oraz masę skupioną, poruszającą się promieniście. Regulator w żaden sposób nie jest informowany o położeniu masy skupionej. Dodatkowo na układ oddziałuje tarcie, które również nie zostało zamodelowane. Porównano ze sobą regulator PID, regulator ślizgowy w postaci klasycznej, quasi-ślizgowy oraz supertwisting. Na końcu wyniki przedstawiono w tabeli.
EN
This paper presents comparison of four selected tracking control algorithms. The simulated plant is a PMSM drive with additional well-balanced inertia load and a point mass, which is moving radially. The controller has no information about position of the point mass. Moreover, friction is present in the plant, but it is unmodelled in the controller. PID, classical sliding mode, quasi-sliding mode and super-twisting controllers are compared. The paper ends with the table of advantages and disadvantages of investigated control algorithms.
EN
Bidirectional Inductive power transfer (IPT) systems behave as high order resonant networks and hence are highly sensitive to changes in system parameters. Traditional PID controllers often fail to maintain satisfactory power regulation in the presence of parametric uncertainties. To overcome these problems, this paper proposes a robust controller which is designed using linear matrix inequality (LMI) techniques. The output sensitivity to parametric uncertainty is explored and a linear fractional transformation of the nominal model and its uncertainty is discussed to generate a standard configuration for μ-synthesis and LMI analysis. An H∞ controller is designed based on the structured singular value and LMI feasibility analysis with regard to uncertainties in the primary tuning capacitance, the primary and pickup inductors and the mutual inductance. Robust stability and robust performance of the system is studied through μ-synthesis and LMI feasibility analysis. Simulations and experiments are conducted to verify the power regulation performance of the proposed controller.
EN
In the frame structure of stacker cranes harmful mast vibrations may appear due to the inertial forces of acceleration or the braking movement phase. This effect may reduce the stability and positioning accuracy of these machines. Unfortunately, their dynamic properties also vary with the lifted load magnitude and position. The purpose of the paper is to present a controller design method which can handle the effect of a varying lifted load magnitude and position in a dynamic model and at the same time reveals good reference signal tracking and mast vibration reducing properties. A controller design case study is presented step by step from dynamic modeling through to the validation of the resulting controller. In the paper the dynamic modeling possibilities of single-mast stacker cranes are summarized. The handling of varying dynamical behavior is realized via the polytopic LPV modeling approach. Based on this modeling technique, a gain-scheduled controller design method is proposed, which is suitable for achieving the goals set. Finally, controller validation is presented by means of time domain simulations.
EN
Interrupt Timed Automata (ITA) are an expressive timed model, introduced to take into account interruptions according to levels. Due to this feature, this formalism is incomparable with Timed Automata. However several decidability results related to reachability and model checking have been obtained. We add auxiliary clocks to ITA, thereby extending its expressive power while preserving decidability of reachability. Moreover, we define a parametrized version of ITA, with polynomials of parameters appearing in guards and updates. While parametric reasoning is particularly relevant for timed models, it very often leads to undecidability results. We prove that various reachability problems, including robust reachability, are decidable for this model, and we give complexity upper bounds for a fixed or variable number of clocks, levels and parameters.
PL
Artykuł opisuje możliwości syntezy krzepkiego stabilizatora systemowego minimalizującego normę H∞, którego właściwości dynamiczne kształtowane są przez wybór odpowiedniej funkcji wagowej. Rozważana jest efektywność metody w procesie projektowania stabilizatora systemowego.
EN
The paper discusses robust PSS synthesis. The PSS is an H∞ controller, what means that minimises H∞ norm of transfer function between the exogenous signals such as reference inputs and disturbances, and the error signals which are to be minimised to meet the control objective. The dynamic properties of the plant are shaped by choosing appropriate weighting function applied to the plant output and input signals. Question appears: does such type of PSS design lead to effective PSS which can be applied in real power system?
EN
The paper deals with the problem of robust predictive fault-tolerant control for nonlinear discrete-time systems described by the Takagi-Sugeno models. The proposed approach is based on a triple stage procedure, i.e. it starts from fault estimation while the fault is compensated with a robust controller. The robust controller is designed without taking into account the input constraints related with the actuator saturation that may change due to its faulty behaviour. Thus, to check the compensation feasibility, the robust invariant set is developed, which takes into account the input constraints. If the current state does not belong to the robust invariant set, then suitable predictive control actions are performed in order to enhance the invariant set. This appealing phenomenon makes it possible to enlarge the domain of attraction, which makes the proposed approach an efficient solution for the fault-tolerant control. The final part of the paper shows an illustrative example regarding the application of the proposed approach to the twin-rotor system.
EN
The problem of output regulation deserves a special attention particularly when it comes to the regulation of nonlinear systems. It is well-known that the problem is not always solvable even for linear systems and the fact that some demanding applications require not only magnitude but also rate actuator constraints makes the problem even more challenging. In addition, real physical systems might have parameters whose values can be known only with a specified accuracy and these uncertainties must also be considered to ensure robustness and on the other hand because they can be crucial for the type of behaviour exhibited by the system as it happens with the celebrated chaotic systems. The present paper proposes a robust control method for output regulation of chaotic systems with parameter uncertainties and subjected to magnitude and rate actuator constraints. The method is an extension of a work recently addressed by the same authors and consists in decomposing the nonlinear system into a stabilizable linear part plus a nonlinear part and in finding a control law based on the small-gain principle. Numerical simulations are performed to validate the effectiveness and robustness of the method using an aeronautical application. The output regulation is successfully achieved without exceeding the input constraints and stability is assured when the parameters are within the specified intervals. Furthermore, the proposed method does not require much computational effort because all the control parameters are computed offline.
PL
W artykule przedstawiono nowy odporny na zmianę mechanicznej stałej czasowej maszyny roboczej predykcyjny regulator prędkości z miękko przełączanym sterowaniem. Proponowana struktura sterowania bazuje na predykcyjnym regulatorze obliczanym off-line dla różnych wartości nieznanego parametru, następnie w zależności od estymaty tego parametru następuje miękkie przełączanie pomiędzy poszczególnymi prawami sterowania MPC. We wstępie omówiono problematykę sterowania napędami z połączeniem sprężystym i zmiennym momentem bezwładności. W kolejnych rozdziałach przedstawiono model napędu i proponowaną strukturę sterowania. Rozważania teoretyczne zostały poparte badaniami symulacyjnymi.
EN
The paper presents a new robust predictive speed controller with soft-switched controls. The proposed control structure is based on the predictive controller off-line calculated (based on mpQP) for different values of the unknown parameter. Then, depending on the parameter estimates soft switching approach follows between the MPC control laws. A preliminary point discusses the problems of motion control of the drive with elastic coupling and variable parameters. In the following chapters presents a model of the drive and the proposed control structure. Theoretical considerations are supported by simulation studies.
13
EN
Control system for a variable inertia drive based on a hybrid P–PI sliding mode controller and reference trajectory generator is described in this paper. Sliding mode block with boundary layer (active for large error values) and the P–PI block (active for small error values) are combined in one controller. Simulation results show that this control approach can decrease tracking error, enhance system robustness and attenuate high frequency chattering in the control signal. The results of simulation are confirmed by experimental studies.
PL
W artykule przedstawiono układ sterowania dla napędu o zmiennej bezwładności. W układzie tym zaproponowano generator trajektorii wzorcowej oraz hybrydowy regulator położenia i prędkości. W regulatorze połączono blok regulatora ślizgowego z warstwą graniczną (aktywny dla dużych wartości uchybu) z blokiem P–PI (aktywnym dla małych wartości uchybu). Badania symulacyjne wykazały, że takie podejście zmniejsza błędy śledzenia, poprawia odporność i eliminuje szybkie oscylacje sygnału sterującego. Badania eksperymentalne potwierdziły wyniki.
PL
W artykule przedstawiono wyniki symulacyjne adaptacyjnego i odpornego regulatora prędkości z zastosowaniem sztucznej sieci neuronowej dla napędu z silnikiem PMSM w dwu-pętlowej strukturze MFC. Zostały porównane dwa regulatory neuronowe, jeden działający w trybie offline, drugi w trybie online. Przedstawione wyniki badań symulacyjnych ilustrują poprawne działanie adaptacyjnej i odpornej regulacji prędkości na zmianę parametrów układu napędowego.
EN
This paper presents the results of simulation of adaptive and robust speed controller for PMSM motor drive by structure of the MFC. Both controllers use the technique of artificial neural networks. An adaptive speed control is trained online, and robust is trained offline method. The simulation results confirm the correct operation of the robust and adaptive speed control to change the parameters of the drive system.
EN
This paper examines the problem of designing a robust H∞ fuzzy controller with D-stability constraints for a class of nonlinear dynamic systems which is described by a Takagi–Sugeno (TS) fuzzy model. Fuzzy modelling is a multi-model approach in which simple sub-models are combined to determine the global behavior of the system. Based on a linear matrix inequality (LMI) approach, we develop a robust H∞ fuzzy controller that guarantees (i) the L2-gain of the mapping from the exogenous input noise to the regulated output to be less than some prescribed value, and (ii) the closed-loop poles of each local system to be within a specified stability region. Sufficient conditions for the controller are given in terms of LMIs. Finally, to show the effectiveness of the designed approach, an example is provided to illustrate the use of the proposed methodology.
EN
An Active Queue Management (AQM) robust control strategy for Traffic Control Protocol (TCP) data transfer is proposed. To this purpose, the TCP behaviour is first approximated by a second–order model with delayed input obtained from the linearization of an efficient and commonly used nonlinear fluid–based model. The adopted feedback control structure uses a fractional– order PI controller. To ensure the desired robustness, the parameter regions where such a controller guarantees a given modulus margin (inverse of the H1 norm of the sensitivity function) are derived. An example commonly used in the literature is worked out to show that the suggested graphically–based design technique is simple to apply while it limits the effects of disturbances and of the unmodelled dynamics.
EN
An off-line methodology was proposed for enhancing the robustness of an initial Generalized Predictive Control (GPC) by convex optimization of the Youla parameter. However, this procedure of robustification is restricted with the case of the systems affected only by unstructured uncertainties. This paper proposes an extension of this method to the systems subjected to both unstructured and structured polytopic uncertainties. The main idea consists in adding supplementary constraints to the optimization problem which validates the Lipatov stability condition at each vertex of the polytope. These polytopic uncertainties impose a set of non convex quadratic constraints. The globally optimal solution is found by means of the GloptiPoly3 software. Therefore, this robustification provides stability robustness towards unstructured uncertainties for the nominal system, while guaranteeing stability properties over a specified polytopic domain of uncertainties. Finally, an illustrative example is given.
PL
Problemy poruszane w artykule koncentrują się na ważnych, współczesnych kwestiach dotyczących projektowania mechatronicznego odpornego sterowania rozmytego ruchem nadążnym mobilnych robotów kołowych. W artykule została przeprowadzona synteza algorytmów sterowania na podstawie teorii stabilności Lapunova wraz z analizą symulacji trzech algorytmów sterowania rozmytego ruchem nadążnym mobilnego robota kołowego z uwzględnieniem niedokładności parametrycznej: analitycznego sterowania ślizgowego, sterowania ślizgowego z rozmytym sterowaniem równoważnym, rozmytego sterowania ślizgowego.
EN
The problems addressed in the article concern important and up-to-date issue of the mechatronic design of the robust control systems of follow-up motion of the mobile wheeled robots. In the article the synthesis of control algorithms based on stability theory of Lapunov was carried out along with simulation analysis of the three control algorithms of the follow-up motion of the mobile wheeled robot with consideration of the parametric inaccuracy: analytic robust sliding control algorithm, robust sliding control algorithm with fuzzy compensation control, fuzzy algorithm of sliding control.
19
Content available Evolutionary method of robust controller computation
PL
Matematyczne metody doboru współczynników regulatora odpornego w przestrzeniach H∞ są bardzo skomplikowane. Projektant układu regulacji musi wykazywać się znajomością technik analizy funkcjonalnej. Do rozwiązywania problemów optymalizacji tego rodzaju doskonale nadają się algorytmy ewolucyjne. W artykule przedstawiono metodę oraz wyniki symulacji podczas doboru współczynników równania regulatora odpornego. Do doboru użyte są tylko dwa kryteria: sprawdzenie stabilności i zależność geometryczna - minimalizacja największej odległości między krzywymi Nyquista operacji G(jω) i 1/F(jω), gdzie G(jω) i F(jω) są transmitancjami regulatora oraz obiektu regulacji w układzie sprzężenia zwrotnego.
EN
Mathematical methods of robust controller coefficients selection in H∞ spaces are very complicated. A control system integrator has to know functional analysis methods. To solve this kind of problem, evolutionary algorithms can be used. The paper presents both the method and simulation results of evolutionary algorithms application for a robust controller coefficients selection. To select robust controller, only two requirements are used: stability check and geometric dependency - minimizing the maximum distance between Nyquist diagrams of operations - G(jω) and 1/F(jω). Where G(jω) and F(jω) are controller and plant transfer functions in a feedback control system.
EN
In the paper the problem of ship autopilot design based on feedback linearization method combined with the robust control approach, is considered. At first the nonlinear ship model (of Norrbin type) is linearized with the use of the simple system nonlinearity cancellation. Next, bearing in mind that exact values of the model parameters are not known, the ensuing inaccuracies are taken as disturbances acting on the system. Thereby is obtained a linear system with an extra term representing the uncertainty which can be treated by using robust, H∞optimal control techniques. The performed simulations of ship course-changing process confirmed a high performance of the proposed controller despite the assumed significant errors of its parameters.
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