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1

Effective nonlinear predictive and CTC-PID control of rigid manipulators

Tatjewski Piotr

Journal of Automation Mobile Robotics and Intelligent Systems

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2024

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Vol. 18, No. 2

1--16

EN

Effective nonlinear control of manipulators with dynamically coupled arms, like those with direct drives, is the subject of the paper. Model-based predictive control (MPC) algorithms with nonlinear state-space models and most recent disturbance attenuation technique are proposed. This technique makes controller design and online calculations simpler, avoiding necessity of dynamic modeling of disturbances or resorting to additional techniques like SMC. The core of the paper are computationally effective MPC-NPL (Nonlinear Prediction and Linearization) algorithms, where computations at every sample are divided into two parts: prediction of initial trajectories using nonlinear model, then optimization using simplified linearized model. For a comparison, a known CTC-PID algorithm, which is also model-based, is considered. It is applied in standard form and also proposed in more advanced CTC-PID2dof version. For all algorithms a comprehensive comparative simulation study is performed, for a direct drive manipulator under disturbances. Additional contribution of the paper is investigation of influence of sampling period and of computational delay time on performance of the algorithms, which is practically important when using model-based algorithms with fast sampling.

2

Odporność nieliniowych, złożonych algorytmów sterowania serwonapędami z silnikami z magnesami trwałymi na czynniki związane z implementacją w układach rzeczywistych

Jastrzębski Marcin, Zawiślak Rafał

Przegląd Elektrotechniczny

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2023

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R. 99, nr 4

63--69

PL

Współczesne serwonapędy bezpośrednie z silnikami z magnesami trwałymi są sterowane przez nieliniowe, złożone algorytmy sterowania. Algorytmy te są wyprowadzane przy upraszczających założeniach, których nie można dotrzymać przy implementacji w układach rzeczywistych. Liczne czynniki dotyczące realizacji sterowania, pomiarów sygnałów, niepełnego modelowania silnika i maszyny roboczej pojawiają się na etapie implementacji i mogą wpływać destrukcyjnie na jakość regulacji osiąganą w układzie rzeczywistym. W artykule dokonano analizy takich czynników i oceniono istotność ich wpływu na pracę układu, w którym sterowanie wyznaczono metodami adaptacyjnego sterowania nieliniowego ze zmiennymi w czasie barierowymi funkcjami Lapunowa. Badania dotyczą serwonapędu z silnikiem liniowym z magnesami trwałymi. Wyniki przedstawiono w licznych symulacjach i poprzez eksperymenty przeprowadzone w układzie rzeczywistym.

EN

In the article, using the example of nonlinear adaptive control with the limitation of state variables in a finite time, the authors show the influence of factors considered as "deteriorating" the control on the accuracy of tracking a given trajectory. Based on an accurate model of a linear drive with a permanent magnet synchronous motor, the destructive effect of unmodulated additional dynamics and non-linearities in the control algorithm was analysed. The analysis of the obtained results made it possible to apply remedies and evaluate their effectiveness. The article presents simulations, and tests carried out on a real system.

3

Adaptacyjne sterowanie położeniem ramienia w układzie napędu elektrycznego z nieliniowym połączeniem sprężystym

Jastrzębski Marcin, Mosiołek Przemysław

Przegląd Elektrotechniczny

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2023

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R. 99, nr 4

51--58

PL

W pracy przedstawiono algorytm adaptacyjnego sterowania położeniem ramienia w układzie napędowym z nieliniowym połączeniem sprężystym. Do zaprojektowania algorytmu sterowania użyto metody „całkowania wstecz”. Zaproponowane w algorytmie prawa adaptacji realizują funkcję samostrojenia układu regulacji pozwalając projektantowi na nieznajomość parametrów układu napędowego. Działanie algorytmu zostało sprawdzone symulacyjnie oraz w układzie rzeczywistym.

EN

The paper presents an algorithm of adaptive control of the arm position in the drive with a non-linear elastic joint. The backstepping method was used to design the control algorithm. The laws of adaptation proposed in the algorithm implement the self-tuning function of the control system, and allow to avoid a process of identification parameters of the drive system. The operation of the algorithm has been verified in a simulation and in a real system.

4

Nieliniowe sterowanie predykcyjne ramion manipulatorów

Tatjewski Piotr

Pomiary Automatyka Robotyka

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2023

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R. 27, nr 2

47--58

PL

Przedmiotem artykułu są algorytmy sterowania predykcyjnego (typu MPC - Model Predictive Control) ramion manipulatorów sztywnych. Zastosowano MPC z modelem w przestrzeni stanów i wykorzystano najnowszą technikę tłumienia zakłóceń i błędów modelowania, pozwalającą uniknąć dynamicznego modelowania zakłóceń lub uciekania się do dodatkowych technik ich eliminowania, takich jak SMC. Rozważane są przede wszystkim najbardziej efektywne obliczeniowo algorytmy MPC-NPL (NPL - Nonlinear Prediction and Linearization), w dwóch wersjach: z optymalizacją QP (Quadratic Programming) z ograniczeniami i z jawną optymalizacją bez ograniczeń i spełnieniem ograniczeń nierównościowych a posteriori. Dla wszystkich rozważanych algorytmów przeprowadzono kompleksową analizę symulacyjną sterowania manipulatorem z napędem bezpośrednim, przy dwóch rodzajach zakłócenia: zewnętrznym i parametrycznym. Wyniki porównano z uzyskanymi dla znanego algorytmu CTC-PID (CTC - Computer Torque Control), uzyskując lepszą jakość regulacji algorytmami MPC. Zbadano wpływ długości okresu próbkowania i obliczeniowego opóźnienia sterowania na jakość regulacji, co jest istotne dla algorytmów z szybkim próbkowaniem opartych na modelach.

EN

The subject of the article are predictive control algorithms (of MPC type - Model Predictive Control) for rigid manipulator arms. MPC with a state-space model and with the latest disturbance and modeling error suppression technique was applied, which avoids dynamic disturbance modeling or resorting to additional disturbance cancellation techniques, such as SMC. First of all, the most computationally efficient MPC-NPL (Nonlinear Prediction and Linearization) algorithms are considered, in two versions: the first with constrained QP (Quadratic Programming) optimization and the second with explicit (analytical) optimization without constraints and satisfying a posteriori inequality constraints. For all considered algorithms, a comprehensive simulation analysis was carried out for a direct drive manipulator, with two kinds of disturbances: external and parametric. The obtained results were compared with those for the well-known CTC-PID algorithm (CTC - Computer Torque Control), showing better control quality with MPC algorithms. In addition, the influence of the length of the sampling period and of the computational delay on control quality was investigated, which is important for model-based algorithms with fast sampling.

5

Controlling bifurcations in high-speed rotors utilizing active gas foil bearings

Papadopoulos Anastasios, Gavalas Ioannis, Chasalevris Athanasios

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2023

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Vol. 71, nr 6

art. no. e146796

EN

High-speed rotors on gas foil bearings (GFBs) are applications of increasing interest due to their potential to increase the power-toweight ratio in machines and also formulate oil-free design solutions. The gas lubrication principles render lower (compared to oil) power loss and increase the threshold speed of instability in rotating systems. However, self-excited oscillations may still occur at circumferential speeds similar to those in oil-lubricated journal bearings. These oscillations are usually triggered through Hopf bifurcation of a fixed-point equilibrium (balanced rotor) or secondary Hopf bifurcation of periodic limit cycles (unbalanced rotor). In this work, an active gas foil bearing (AGFB) is presented as a novel configuration including several piezoelectric actuators that shape the foil through feedback control. A finite element model for the thin foil mounted in some piezoelectric actuators (PZTs), is developed. Second, the gas-structure interaction is modelled through the Reynolds equation for compressible flow. A simple physical model of a rotating system consisting of a rigid rotor and two identical gas foil bearings is then defined, and the dynamic system is composed with its unique source of nonlinearity to be the impedance forces from the gas to the rotor and the foil. The third milestone includes a linear feedback control scheme to stabilize (pole placement) the dynamic system, linearized around a speed-dependent equilibrium (balanced rotor). Further to that, linear feedback control is applied in the dynamic system utilizing polynomial feedback functions in order to overcome the problem of instability.

6

Trajectory planning and motion control schemes for 2DoF planar parallel manipulator Biglide type with elastic joints: A comparative study

Litim EL Mostafa, Elawady Iman, Belmokhtar Samira, Senouci Mohamed, Omari Abdelhafid

Przegląd Elektrotechniczny

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2021

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R. 97, nr 8

88--95

EN

This research paper develops a nonlinear backstepping sliding mode design scheme for the motion control of two-degree of freedom planar parallel robot. The main objective of this paper is to gain a strong control in trajectory tracking case. However, dynamical equations of motion for a 2DoF parallel manipulator, including structured and unstructured uncertainties, are taken into account. Furthermore, the hybrid control strategy is based on backstepping scheme and on a switching function is that are presented for high accuracy of a mixed space tracking trajectory of robot. Also, the application of control technique in presence of parameter uncertainties in a massive change is studied. In addition, the benefit of this method is that it imposes the intended stability properties by first fixing the Lyapunov candidate functions and then calculating the other functions in a recursive way. Therefore, simulation outcomes are shown so as to assess the tracking performance and to evaluate the total stability of the closed-loop system. Finally, the results accomplished in simulation show the efficiency of the controller proposed for a parallel robot with two degrees of freedom Biglide type with elastic joints.

PL

W artykule opracowano nieliniowy tryb ślizgowy backstepping do sterowania ruchem równoległego robota planarnego 2DoF z elastycznymi przegubami. Badane jest zastosowanie techniki sterowania w obecności niepewności parametrów przy masywnej zmianie. Zaletą proponowanej metody jest to, że narzuca ona zamierzone właściwości stabilności poprzez wcześniejsze ustalenie unkcji Lyapunova. Wyniki uzyskane w symulacji wskazują na skuteczność proponowanego regulatora.

7

A numerically efficient fuzzy MPC algorithm with fast generation of the control signal

Marusak Piotr M.

International Journal of Applied Mathematics and Computer Science

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2021

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Vol. 31, no. 1

59--71

EN

Model predictive control (MPC) algorithms are widely used in practical applications. They are usually formulated as optimization problems. If a model used for prediction is linear (or linearized on-line), then the optimization problem is a standard, i.e., quadratic, one. Otherwise, it is a nonlinear, in general, nonconvex optimization problem. In the latter case, numerical problems may occur during solving this problem, and the time needed to calculate control signals cannot be determined. Therefore, approaches based on linear or linearized models are preferred in practical applications. A novel, fuzzy, numerically efficient MPC algorithm is proposed in the paper. It can offer better performance than the algorithms based on linear models, and very close to that of the algorithms based on nonlinear optimization. Its main advantage is the short time needed to calculate the control value at each sampling instant compared with optimization-based numerical algorithms; it is a combination of analytical and numerical versions of MPC algorithms. The efficiency of the proposed approach is demonstrated using control systems of two nonlinear control plants: the first one is a chemical CSTR reactor with a van de Vusse reaction, and the second one is a pH reactor.

8

Adaptive, nonlinear state transformation-based control of motion in presence of hard constraints

Kabziński J., Mosiołek P.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2020

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Vol. 68, nr 5

963–-971

EN

The paper describes a nonlinear controller design technique applied to a servo drive in the presence of hard state constraints. The approach presented is based on nonlinear state-space transformation and adaptive backstepping. It allows us to impose hard constraints on the state variables directly and to achieve asymptotic tracking of any reference trajectory inside the constraints, despite unknown plant parameters. Two control schemes (with and without integral action) are derived, investigated and then compared. Several examples demonstrate the main features of the design procedure and prove that it may be applied in case of motion control problems in electric drive automation.

9

Efficient MPC algorithms with variable trajectories of parameters weighting predicted control errors

Nebeluk Robert, Marusak Piotr

Archives of Control Sciences

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2020

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Vol. 30, No. 2

325--363

EN

Model predictive control (MPC) algorithms brought increase of the control system performance in many applications thanks to relatively easily solving issues that are hard to solve without these algorithms. The paper is focused on investigating how to further improve the control system performance using a trajectory of parameters weighting predicted control errors in the performance function of the optimization problem. Different shapes of trajectories are proposed and their influence on control systems is tested. Additionally, experiments checking the influence of disturbances and of modeling uncertainty on control system performance are conducted. The case studies were done in control systems of three control plants: a linear non-minimumphase plant, a nonlinear polymerization reactor and a nonlinear thin film evaporator. Three types of MPC algorithms were used during research: linear DMC, nonlinear DMC with successive linearization (NDMC–SL), nonlinear DMC with nonlinear prediction and linearization (NDMC–NPL). Results of conducted experiments are presented in greater detail for the control system of the polymerization reactor, whereas for the other two control systems only the most interesting results are presented, for the sake of brevity. The experiments in the control system of the linear plant were done as preliminary experiments with the modified optimization problem. In the case of control system of the thin film evaporator the researched mechanisms were used in the control system of a MIMO plant showing possibilities of improving the control system performance.

10

Wpływanie kształtem trajektorii referencyjnej na działanie układów regulacji predykcyjnej

Nebeluk Robert, Marusak Piotr

Pomiary Automatyka Robotyka

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2019

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R. 23, nr 1

21--30

PL

Zastosowanie algorytmów regulacji predykcyjnej MPC do regulacji wielu procesów nieliniowych, o różnym stopniu trudności, często umożliwia osiągniecie bardzo dobrej jakości regulacji. Jest to możliwe ze względu na odpowiednie uwzględnienie w strukturze algorytmu informacji uzyskanych z modelu procesu. Do formułowania zadania optymalizacji dla algorytmów regulacji predykcyjnej najczęściej zakładana jest stała trajektoria referencyjna dla całego horyzontu predykcji. W artykule przedstawiono możliwości poprawy jakości regulacji przez zastosowanie trajektorii referencyjnej zmiennej na horyzoncie predykcji. Podczas porównywania jakości regulacji z wykorzystaniem trajektorii referencyjnych o różnej postaci, wzięto pod uwagę dwie wielkości. Pierwszą z nich jest czas narastania wyjścia obiektu regulacji, a drugą - przeregulowanie. Badania prowadzono w układach regulacji dwóch obiektów: nieminimalnofazowego obiektu liniowego oraz nieliniowego reaktora polimeryzacji. Do regulacji w przypadku pierwszego obiektu, zastosowano algorytm DMC, a w przypadku drugiego - algorytm bazujący na modelu nieliniowym, z nieliniową predykcją i linearyzacją (NDMC-NPL). Przedstawione wyniki dobrze ilustrują możliwości wpływania, za pomocą trajektorii referencyjnych o różnych kształtach, na poprawę jakości regulacji oferowanej przez algorytmy predykcyjne.

EN

Applying model predictive control (MPC) algorithms to control many processes, of different difficulty level, often allows improving control quality. It is possible by including information received from a process model in the algorithm structure. When defining the optimization problem for the predictive control algorithms most often a time-constant reference trajectory is assumed. Possibilities of improving the control quality by applying a time reference trajectory variable on the prediction horizon are presented in the paper. Two quantities are considered when comparing control quality: the rise time of the control plant output, and the overshoot. The experiments were conducted in the control systems of two control plants: a linear nonminimumphase plant and a nonlinear polymerization reactor. In the control system of the first control plant the DMC predictive control algorithm was used. For the nonlinear reactor the NDMC-NPL algorithm based on a nonlinear model was applied. It is demonstrated that by using the reference trajectories of different shapes it is possible to improve control quality offered by the MPC control algorithms.

11

Geometry and flatness of m-crane systems

Nowicki M., Respondek W., Piasek J., Kozłowski K.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2019

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Vol. 67, nr 5

893--903

EN

We propose a class of m-crane control systems that generalizes two- and three-dimensional crane systems. We prove that each representant of the described class is feedback equivalent to the second order chained form with drift. In consequence, we prove that it is differentially flat. Then we investigate its control properties and derive a control law for tracking control problem.

12

Heuristic control of nonlinear power systems: Application to the infinite bus problem

Hossain Musabbir Md, Khurshudyan Asatur Zh.

Archives of Control Sciences

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2019

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Vol. 29, No. 2

279--288

EN

In this paper, we apply the heuristic method for determination of control functions for controllability analysis of nonlinear power systems. The problem of control of quasi-linear systems under proper assumptions on the nonlinear term is considered in the general statement. Making use of the Green’s function solution of nonlinear systems, the exact and approximate controllability conditions are expressed in terms of unknown controls in an explicit form. The way of resolving controls determination is discussed. As a particular application, a one-machine infinite-bus system is considered described by a coupled system of three first order ordinary differential equations. Two heuristic forms of admissible controls are considered providing approximate controllability within the same amount of time having different intensities. Results of numerical simulations are presented and discussed.

13

Practical hybrid solutions based on nonlinear controllers applied to PFC boost converter

Rosa A. H. R., Morais L. M. F., Seleme S. I.

Przegląd Elektrotechniczny

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2018

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R. 94, nr 6

8--14

EN

In this work, it has been investigated the combination of nonlinear controllers applied to PFC (Power Factor Corrector) boost converter. Advantages of each included technique are discussed and considered: SFL (State Feedback Linearization), PBC (Passivity based control), IDAPBC (InterconnectionandDampingAssignmentPassivityBasedControl)andthePLL(PhaseLockedLoop). Wealsoaddtheanalysesoftuninggainsofthe nonlinear control methods. In addition, we propose practical hybrid solutions to the PFC boost (αβ-IDAPBC), which carry out robustness advantages. To evaluate the feasibility of the proposed methods, a Hardware in The Loop (HIL) tests are developed.

PL

W pracy zbadano możliwość zastosowania nieliniowego sterownika do przekształtnika typu boost stosowanego do poprawy współczynnika mocy. Wzięto pod uwagę: SFL – State Feedback Linearizaton, PBC - Passivity Based Control, IDAPBC – Interconection and Damping Assignement PBC oraz PLL - Phase Locked Loop. Przeanalizowano też strojenie wzmocnienia. Zaproponowano praktyczne hybrydowe rozwiązanie.

14

Adaptive control of two-mass drive system with nonlinear stiffness

Kabziński J., Mosiołek P.

Przegląd Elektrotechniczny

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2018

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R. 94, nr 3

49--54

EN

The paper describes a nonlinear controller design technique for a servo drive in the presence of nonlinear friction together with a flexible shaft connecting the motor and the load. The shaft is characterized by the nonlinear stiffness curve. Two different type of the nonlinear stiffness curve are considered. The proposed controller is based on adaptive backstepping, modified by the use of command filtering. The proposed approach allows to accomplish the rigorous proof of the closed-loop system stability. Several experiments prove the control effectiveness.

PL

Opisano problem sterowania prędkością układu napędowego z nieliniowym tarciem, połączeniem sprężystym i nieznanymi parametrami. Elastyczne połączenie jest opisane przy pomocy nieliniowej funkcji sztywności. Rozważane są dwa typy nieliniowej funkcji sztywności: wypukła i wklęsła. Układy regulacji są projektowane przy pomocy metod „adaptive backstepping” z filtracją wartości zadanych. Opisano szereg eksperymentów, które ilustrują charakterystyczne właściwości układu regulacji.

15

High performances of polynomial and nonlinear backstepping control strategies of an induction motor fed by matrix converter

Boumediene H., Hassaine S., Mazari B.

Przegląd Elektrotechniczny

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2017

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R. 93, nr 11

133--139

EN

The main objective of this paper is to present the performance analysis of the oriented flux control of an induction motor associated with a matrix converter. A polynomial technique of RST type is used for speed control. As for the control of internal current loops, the technique used is based on the nonlinear approach. Overall, the proposed feedback law is asymptotically stable, which is shown in the context of the Lyapunov theory. The design of the control laws by the backstepping technique has been detailed while taking account of the non-linearities in the design phase of the control system. The objective is to obtain a good transient response and a good capacity of rejection of charge disturbance. The induction motor incorporating the proposed control techniques (RST-Backstepping) has been successfully implemented in numerical simulation using Matlab/Simulink under different operating conditions where the static and dynamic responses of the system are evaluated. It can be seen that the proposed control technique provides good speed monitoring performance. For internal loops, overall stability is ensured and the proposed approach presents good robustness to the uncertainties of the system parameters.

PL

W artykule zaprezentowano analizę właściwości sterowania silnikiem indukcyjnym za pośrednictwem przetwornika macierzowego. Zastosowano wielomianową technikę RST do sterowania prędkością. Do sterowania pętlą prądową zastosowano metodę nieliniową. Zaproponowane sprzężenie zwrotne jest asymtotycznie stabilne w konteksście teorii Lapunova. Numeryczne symulacje wykazały skuteczność zaproponowanej metody.

16

A novel method for the design of switching surfaces for discretized MIMO nonlinear systems

Luis-Delgado J. D., Al-Hadithi B. M., Jiménez A.

International Journal of Applied Mathematics and Computer Science

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2017

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Vol. 27, no. 1

5--17

EN

Designing variable structure control with sliding mode (VSC-SM) control schemes needs a switching function or a sliding surface which guarantees the global stability of the closed-loop system. Despite the fact that a wide range of design approaches has been proposed for solving this mathematical problem, the number of proposed methodologies for nonlinear systems is not very extensive, especially for discrete time nonlinear MIMO systems, and most of them require some coordinate system transformation. Therefore, it is not an easy task to find a design scheme that can be applied to discrete time nonlinear MIMO systems. The proposed methodology introduces a mathematical tool: a switching surface equation for a class of MIMO nonlinear systems through an explicit equation without any coordinate transformation. This equation makes use of an implicit linearizing process via the Taylor expansion that allows the use of linear procedures for the design of switching surfaces and the forward Euler method to obtain a discrete time dynamics representation. An illustrative example is included to show the advantages of the proposed design methodology.

17

Velocity controller for a class of vehicles

Herman P., Adamski W.

Foundations of Computing and Decision Sciences

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2017

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Vol. 42, No. 1

43--58

EN

This paper addresses the problem of velocity tracking control for various fully-actuated robotic vehicles. The presented method, which is based on transformation of equations of motion allows one to use, in the control gain matrix, the dynamical couplings existing in the system. Consequently, the dynamics of the vehicle is incorporated into the control process what leads to fast velocity error convergence. The stability of the system under the controller is derived based on Lyapunov argument. Moreover, the robustness of the proposed controller is shown too. The general approach is valid for 6 DOF models as well as other reduced models of vehicles. Simulation results on a 6 DOF indoor airship validate the described velocity tracking methodology.

18

Offset-free nonlinear Model Predictive Control with state-space process models

Tatjewski P.

Archives of Control Sciences

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2017

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Vol. 27, no. 4

595--615

EN

Offset-free model predictive control (MPC) algorithms for nonlinear state-space process models, with modeling errors and under asymptotically constant external disturbances, is the subject of the paper. The main result of the paper is the presentation of a novel technique based on constant state disturbance prediction. It was introduced originally by the author for linear state-space models and is generalized to the nonlinear case in the paper. First the case with measured state is considered, in this case the technique allows to avoid disturbance estimation at all. For the cases with process outputs measured only and thus the necessity of state estimation, the technique allows the process state estimation only - as opposed to conventional approach of extended process-and-disturbance state estimation. This leads to simpler design with state observer/filter of lower order and, moreover, without the need of a decision of disturbance placement in the model (under certain restrictions), as in the conventional approach. A theoretical analysis of the proposed algorithm is provided, under applicability conditions which are weaker than in the conventional approach. The presented theory is illustrated by simulation results of nonlinear processes, showing competitiveness of the proposed algorithms.

19

Nonlinear control of underwater robotic vehicle in plane motion

Garus J.

Zeszyty Naukowe Akademii Marynarki Wojennej

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2016

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R. 57 nr 3 (206)

15--23

EN

A control system supporting motion of an underwater robotic vehicle along a reference trajectory in the horizontal plane is presented in the paper. A waypoint line-of-sight scheme and nonlinear PD control law are applied to calculate command signals. Parameters of the proposed control law are tuned using genetic algorithms. The validity and advantages of the approach are illustrated through numerical simulation results.

PL

W artykule przedstawiono koncepcję system automatycznego sterowania ruchem płaskim robota podwodnego wzdłuż trajektorii odniesienia. Układem odpowiedzialnym za wyznaczanie sygnałów sterujących jest autopilot, w którym zaimplementowano nieliniowy regulator PD. Parametry regulatora dostrojono z wykorzystaniem algorytmów genetycznych. Zamieszczono wyniki badań symulacyjnych ruchu robota w płaszczyźnie poziomej z wykorzystaniem zaproponowanego algorytmu sterowania.

20

Adaptive motion control with state constraints using barrier Lyapunov functions

Jastrzębski M., Kabziński J., Mosiołek P.

Przegląd Elektrotechniczny

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2016

|

R. 92, nr 4

112--119

EN

A servo control with unknown system parameters and the constraints imposed on the maximal position and velocity is considered. The barrier Lyapunov functions approach is applied to assure the preservation of bounds in any conditions. The system performance is compared for three cases of the controller design: based on quadratic Lyapunov functions, based on barrier Lyapunov functions if only position constraints are imposed and based on barrier Lyapunov functions if both position and velocity bounds are present. The tuning rules are discussed and several numerical experiments demonstrating features of the proposed control and the influence of the parameters are presented.

PL

Opisano problem sterowania napędowym układem nadążnym z nieznanymi parametrami i ograniczeniami nałożonymi na maksymalne wartości położenia i prędkości. Porównano właściwości trzech układów regulacji: ze sterowaniem zaprojektowanym na podstawie kwadratowych funkcji Lapunowa, ze sterowaniem zaprojektowanym na podstawie barierowych funkcji Lapunowa i ograniczeniem na położenie, oraz ze sterowaniem zaprojektowanym na podstawie barierowych funkcji Lapunowa przy ograniczeniach na położenie i prędkość. Opisano szereg eksperymentów, które ilustrują charakterystyczne właściwości układu regulacji i dostarczają wniosków co do wyboru parametrów algorytmu sterowania.