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1

Optimal state feedback controller for balancing cube

Kowalczyk Adam, Piotrowski Robert

Journal of Automation Mobile Robotics and Intelligent Systems

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2022

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Vol. 16, No. 3

3--12

EN

In this paper, a nonlinear balancing cube system is considered, the concept for which is based on an inverted pendulum. The main purpose of this work was the modelling and construction of a balancing cube with the synthesis of the control system. The control objectives included swing-up and stabilization of the cube on its vertex at an unstable equilibrium. Execution of the intended purpose required, first, deriving a cognitive mathematical model. It was based on the Lagrange method. Next, a mathematical model for control purposes was derived. The project of the physical model of the balancing cube was presented. A stabilization system based on a linear quadratic regulator (LQR) was developed. Moreover, a swing-up mechanism was used to bring the cube close to the upper equilibrium point. The algorithm switching condition was important to enable the correct functioning of the system. The developed control system was verified in the Matlab environment. Finally, verifying experiments and comparisons among models (mathematical and physical) were performed.

2

System vibration control using linear quadratic regulator

Abdelrahman M., Aryassov G., Tamre M., Penkov I.

International Journal of Applied Mechanics and Engineering

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2022

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Vol. 27, no. 3

1--8

EN

Balancing a bipedal robot movement against external perturbations is considered a challenging and complex topic. This paper discusses how the vibration caused by external disturbance has been tackled by a Linear Quadratic Regulator, which aims to provide optimal control to the system. A simulation was conducted on MATLAB in order to prove the concept. Results have shown that the linear quadratic regulator was successful in stabilizing the system efficiently.

3

Shunt Active Power Filter Employing Robust Extended Complex Kalman Filter based Linear Quadratic Regulator Control Strategy for Power Quality Enhancement

Patjoshi Rajesh Kumar, Panigrahi Rakhee

Journal of Power Technologies

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2020

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Vol. 100, nr 2

120--128

EN

In this paper, a new reference current generation method is proposed for effective harmonics mitigation and reactive power compensation of three-phase shunt active power filter (SAPF) under grid perturbations. The proposed reference technique is specified as a self-regulator of dc-capacitor voltage. In operation, the proposed algorithm estimates the source reference current rapidly and adaptively through power system disruptions arising at source as well as load sides. The proposed technique employs the Robust Extended Complex Kalman filter (RECKF) algorithm to generate reference current, which confirms in phase action of SAPF with the functional power system, without being reliant upon any phase-locked loop (PLL) elements or proportional integral (PI) controller loop. As a consequence, an economical SAPF system can be designed. More-over, a Linear Quadratic Regulator (LQR) is formulated using RECKF methodology for delivering stability and robustness in the SAPF system. The design model and efficacy of the proposed algorithm are fully studied and assessed in a laboratory prototype employing dSPACE1104 to justify feasibility. The encouraging outcomes obtained experimentally demonstrate the efficiency of the proposed approach under both steady and dynamic conditions of the power system.

4

Projekt systemu sterowania dwukołowym robotem balansującym

Laddach Krzysztof, Łangowski Rafał

Zeszyty Naukowe Wydziału Elektrotechniki i Automatyki Politechniki Gdańskiej

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2019

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Nr 68

27--32

PL

W niniejszym artykule rozważony został problem sterowania dwukołowym robotem balansującym. Celem zaprezentowanych prac było zaprojektowanie systemu sterowania tego typu obiektem z wykorzystaniem metod nowoczesnej teorii sterowania. W ramach syntezy, bazując na wyprowadzonym w oparciu o zasady dynamiki Newtona modelu, dobrana została struktura systemu sterowania w postaci regulatora wykorzystującego sprzężenie zwrotne od stanu oraz sprzężenie w przód od zadanej wartości prędkości liniowej robota. Dobór wzmocnień regulatora w sprzężeniu od stanu wykonano z wykorzystaniem technologii sterowania optymalnego z kwadratowym wskaźnikiem jakości. Ponadto, z uwagi na niedostateczną jakość pomiarów używanych w sterowaniu, opracowano dyskretny filtr Kalmana. Jakość opracowanego rozwiązania została przetestowana, przede wszystkim symulacyjnie, w środowisku Matlab/Simulink. Zawarte rozważania agregują wiedzę umożliwiającą zaprojektowanie systemu sterowania obiektami tej klasy, co może być użyteczne szczególnie w różnych aplikacjach inżynierskich.

EN

A two-wheeled balancing robot is one type of structures based on the principle of a mobile inverted pendulum. The main aim of controlling a balancing robot is to stabilise it at the upper equilibrium point and to enable its movement with the set speed. In this paper, a synthesis of a balancing robot control system has been given. Within its framework, based on the model using Newton's laws, a feedforward state-feedback control system structure has been designed. The controller parameter values have been computed using optimisation tools. More specifically, the linear-quadratic regulator (LQR) has been designed. Moreover, due to the insufficient quality of measurements used in the control system, a linear discrete Kalman filter has been developed. The complete control system has been implemented and tested in the Matlab/Simulink environment.

5

Metody strojenia regulatorów typu PID z wykorzystaniem technologii syntezy regulatorów od stanu

Czyżniewski Mateusz, Łangowski Rafał, Zubowicz Tomasz

Zeszyty Naukowe Wydziału Elektrotechniki i Automatyki Politechniki Gdańskiej

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2019

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Nr 68

9--12

PL

Układy sterujące typu PID są jednymi z najbardziej popularnych regulatorów wykorzystywanych w układach regulacji. W związku z tym znanych jest szereg metod doboru wartości ich parametrów (nastaw). Obok różnych metod inżynierskich czy analitycznych strojenia tego typu regulatorów, dostępne są również podejścia bazujące na optymalizacji. Wskaźnikiem jakości znajdującym w nich zastosowanie jest np. całka z kwadratu uchybu. W artykule przedstawiono alternatywny sposób wyznaczania nastaw regulatora z rodziny PID bazujący na rozwiązaniu liniowokwadratowego zadania optymalizacji wykorzystującego kryterium energetyczne. Uszczegóławiając, pokazano możliwość transformacji optymalnych nastaw regulatora bazującego na sprzężeniu zwrotnym od stanu obiektu w nastawy regulatora w klasycznym układzie regulacji. Jako aplikację wykorzystano nieliniowy matematyczny model dynamiki podwójnego odwróconego wahadła.

EN

The PID-type regulators are one of the most commonly used in control systems. Therefore, there are many different methods of selecting values of their parameters. Apart from various engineering or analytical methods of tuning such controllers, there are also approaches based on optimisation. In the latter case, a typical performance index is integral square error (ISE). In this paper, an alternative method of PID parameters tuning has been presented. This approach is based on a solution of the linear-square optimisation task with energy-based performance index. In the result, a state-feedback controller is obtained. Subsequently, the state-feedback gains are translated into PID regulator parameters. The proposed approach has been illustrated using a non-linear mathematical model of a double inverted pendulum.

6

A new efficient adaptive control of torsional vibrations induced by switched nonlinear disturbances

Wasilewski Maciej, Pisarski Dominik, Konowrocki Robert, Bajer Czesław I.

International Journal of Applied Mathematics and Computer Science

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2019

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Vol. 29, no. 2

285--303

EN

Torsional vibrations induced in drilling systems are detrimental to the condition of the machine and to the effectiveness of the engineering process. The cause of vibrations is a nonlinear and unknown friction between a drill string and the environment, containing jumps in its characteristics. Nonlinear behaviour of the friction coefficient results in self-excited vibration and causes undesirable stick-slip oscillations. The aim of this paper is to present a novel adaptive technique of controlling vibrating systems. The scheme is based on the linear quadratic regulator and uses direct measurements of the friction torque to synthesize its linear dynamic approximation. This approach allows generating a control law that takes into account the impact of the friction on the system dynamics and optimally steers the system to the desired trajectory. The controller’s performance is examined via numerical simulations of the stabilization of the drilling system. The proposed solution outperforms the comparative LQG regulator in terms of the minimization of the assumed cost functional and the overall stability of the control system under the nonlinear disturbance.

7

Numerical error bound of optimal control for homogeneous linear systems

Daraghmeh Adnan, Qatanani Naji

Archives of Control Sciences

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2019

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Vol. 29, No. 2

323--337

EN

In this article we focus on the balanced truncation linear quadratic regulator (LQR) with constrained states and inputs. For closed-loop, we want to use the LQR to find an optimal control that minimizes the objective function which called "the quadratic cost function” with respect to the constraints on the states and the control input. In order to do that we have used formal asymptotes for the Pontryagin maximum principle (PMP) and we introduce an approach using the so called The Hamiltonian Function and the underlying algebraic Riccati equation. The theoretical results are validated numerically to show that the model order reduction based on open-loop balancing can also give good closed-loop performance.

8

Sterowanie bazujące na sprzężeniu od wektora zmiennych stanu z ograniczeniami serwonapędem z silnikiem PMSM

Tarczewski T., Skiwski M., Grzesiak L M., Zieliński M.

Przegląd Elektrotechniczny

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2018

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R. 94, nr 3

99--105

PL

W artykule przedstawiono syntezę regulatora bazującego na sprzężeniu od wektora zmiennych stanu z ograniczeniami sterującego serwonapędem. Do wyznaczenia współczynników wzmocnień regulatora zastosowano algorytm optymalizacyjny sztucznej kolonii pszczół. Przedstawiono metodę wprowadzenia ograniczeń do regulatora bazującą na dyskretnych równaniach predykcyjnych. Opracowany algorytm regulacji zaimplementowano w prototypowym serwonapędzie z tranzystorami mocy SiC MOSFET i przeprowadzono testy na stanowisku laboratoryjnym.

EN

In this paper, synthesis of constrained state feedback controller for servo-drive is presented. Gain coefficients of the controller were obtained by using an artificial bee colony based optimization algorithm. Constraints handling method that is based on discrete predictive equations has been proposed. Designed control algorithm has been implemented in a prototype servo-drive with SiC MOSFET power transistors and experimentally verified.

9

Application of artificial bee colony algorithm to auto-tuning of linear-quadratic regulator for PMSM position control

Tarczewski T., Grzesiak L M.

Przegląd Elektrotechniczny

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2016

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R. 92, nr 6

57--62

EN

In this paper auto-tuning process of linear-quadratic regulator (LQR) for position control of permanent magnet synchronous motor (PMSM) is presented. The novelty of the proposed solution lies in use of artificial bee colony (ABC) optimization algorithm to calculate gain coefficients of controller. In order to maintain selected state and control variables of drive in a permissible level, the ABC algorithm has been extend for solving constrained optimization problem. Proper operation of auto-tuning procedure was investigated by numerical simulations.

PL

W artykule przedstawiono proces samostrojenia regulatora liniowo-kwadratowego przeznaczonego do regulacji położenia silnika synchronicznego o magnesach trwałych. Oryginalność proponowanego rozwiązania polega na zastosowaniu algorytmu optymalizacyjnego sztucznej kolonii pszczół do wyznaczenia współczynników wzmocnień regulatora. W celu utrzymywania wartości wybranych zmiennych stanu i sygnałów sterujących w dopuszczalnym zakresie, algorytm optymalizacyjny rozszerzono o możliwość rozwiązywania problemów optymalizacyjnych z ograniczeniami. Poprawne działanie algorytmu automatycznego strojenia zostało zbadane w testach symulacyjnych.

10

Control Methods Design for a Model of Asymmetrical Quadrocopter

Beniak R., Gudzenko O.

Journal of Automation Mobile Robotics and Intelligent Systems

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2016

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Vol. 10, No. 2

40--49

EN

The paper describes the results of quadrocopters motion properties for the control based on the inverse dynamics method and optimal control method with synthesis linear-quadratic regulator (LQR). Motion of quadrocopters is tested for composite trajectories. The new model of asymmetrical quadrocopters, taking into account the rotation and shift of one arm relative to the other, was developed. A few criteria for evaluation of the effectiveness of control methods of quadrocopters are presented in this paper. An analysis of the results allows selecting a method for solving the problem of quadrocopters control and making recommendations for the formation of trajectories.

11

Algebraic Riccati equation based Q and R matrices selection algorithm for optimal LQR applied to tracking control of 3rd order magnetic levitation system

Kumar E V., Jerome J.

Archives of Electrical Engineering

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2016

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Vol. 65, nr 1

151-168

EN

This paper presents an analytical approach for solving the weighting matrices selection problem of a linear quadratic regulator (LQR) for the trajectory tracking application of a magnetic levitation system. One of the challenging problems in the design of LQR for tracking applications is the choice of Q and R matrices. Conventionally, the weights of a LQR controller are chosen based on a trial and error approach to determine the optimum state feedback controller gains. However, it is often time consuming and tedious to tune the controller gains via a trial and error method. To address this problem, by utilizing the relation between the algebraic Riccati equation (ARE) and the Lagrangian optimization principle, an analytical methodology for selecting the elements of Q and R matrices has been formulated. The novelty of the methodology is the emphasis on the synthesis of time domain design specifications for the formulation of the cost function of LQR, which directly translates the system requirement into a cost function so that the optimal performance can be obtained via a systematic approach. The efficacy of the proposed methodology is tested on the benchmark Quanser magnetic levitation system and a detailed simulation and experimental results are presented. Experimental results prove that the proposed methodology not only provides a systematic way of selecting the weighting matrices but also significantly improves the tracking performance of the system.

12

Implementation of PID on PIC24F series microcontroller for speed control of a DC motor using MPLAB and Proteus

Aslam S., Hannan S., Sajjad U., Zafar W.

Advances in Science and Technology. Research Journal

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2016

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Vol. 10, nr 31

40--50

EN

Speed control of a DC motor is critical in most industrial systems where accuracy and protection are of essence. This paper presents simulations of Proportional Integral Derivative Controller (PID) on a 16-bit PIC 24F series microcontroller for speed control of a DC motor in the presence of load torque. The PID gains have been tuned by Linear Quadratic Regulator (LQR) technique and then it is implemented on a microcontroller using MPLAB, and finally simulated for speed control of a DC motor in Proteus Virtual System Modeling (VSM) software. Proteus has built in feature to add load torque to a DC motor so simulation results have been presented in three cases speed of DC motor is controlled without load torque, with 25% load torque and with 50% load torque. In all three cases PID effectively controls the speed of DC motor with minimum steady state error.

13

Control system of the grid-connected converter based on a state current regulator with oscillatory terms

Gałecki A., Kaszewski A., Grzesiak L.M., Ufnalski B.

Przegląd Elektrotechniczny

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2015

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R. 91, nr 1

65-69

EN

This paper presents a state-feedback current controller with oscillatory terms for three-phase grid-connected PWM converters. Use of the oscillatory terms allows for shaping the sinusoidal input currents of the converter under distorted grid voltage conditions. Linear-quadratic optimization method is used to calculate the current controller gains. In the converter control structure the state current controller and PI DC-link voltage controller are used. Mathematical analysis and simulation results of the proposed control system are presented and discussed.

XX

W artykule przedstawiono regulator prądu ze sprzężeniem od wektora stanu z rozszerzeniem o człony oscylacyjne opracowany dla trójfazowych przekształtników sieciowych PWM.Wykorzystanie członów oscylacyjnych pozwala na kształtowanie sinusoidalnego prądu wejściowego prostownika przy odkształconym napięciu sieci. Współczynniki wzmocnień regulatora prądu obliczono z wykorzystaniem optymalizacji liniowo-kwadratowej. W strukturze sterowania przekształtnikiem zastosowano opracowany regulator prądu oraz regulator PI napięcia DC-linku. Przedstawiono analizę matematyczną i omówiono wyniki badań symulacyjnych.

14

State-space current controller for the four-leg two-level grid-connected converter

Gałecki A., Kaszewski A., Grzesiak L. M., Ufnalski B.

Przegląd Elektrotechniczny

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2014

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R. 90, nr 11

63--66

EN

In this paper a state-feedback current controller for four-leg grid-connected converter is presented. Two current control structures are proposed. The elementary one consists of the state regulator and a reference input matrix. The augmented one consists of the elementary structure with additional integral terms. Linear-quadratic optimization method is used to determine the controller coefficients. The developed current controller in the cascaded control structure with an outer PI DC-link voltage controller is used. Simulation results of the proposed control system are presented and discussed at the end of the paper.

PL

W artykule przedstawiono regulator prądu ze sprzężeniem od wektora stanu dla czterogałęziowego prostownika sieciowego. Zaproponowano dwie struktury regulacji prądu. Podstawowa zawiera regulator stanu i macierz wejścia. Rozszerzona składa się ze struktury podstawowej z dodanymi członami całkującymi. Dobór współczynników wzmocnień regulatora przeprowadzono z wykorzystaniem optymalizacji liniowo-kwadratowej. Opracowany regulator prądu współpracuje w kaskadowej strukturze regulacji z nadrzędnym regulatorem napięcia typu PI. W końcowej części artykułu przedstawiono i omówiono wyniki badań symulacyjnych.

15

A state-space approach for control of NPC type 3-level sine wave inverter used in FOC PMSM drive

Tarczewski T., Grzesiak L. M., Wawrzak A., Karwowski K., Erwiński K.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2014

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Vol. 62, nr 3

439--448

EN

This paper presents the design and analysis process of state feedback controllers for NPC type 3-level sine wave inverter. In order to achieve good dynamic features in a state feedback control an internal model of reference input and a feedforward path are introduced. During synthesis process of the state feedback controllers maximum permissible dynamics of voltage control in the linear range of modulation is taken into account. It is shown that initially gains of the controller and feedforward path are non-stationary and depend on the angular velocity. Stationary approximation of a non-stationary state feedback controller is presented in details. Proposed state feedback control structures for continuous voltage shaping NPC type 3-level voltage source inverter are examined in FOC PMSM drive. The novelty of the presented controller lays in a stationary approximated state feedback control structure designed in terms of maximum permissible dynamics of a voltage control system. Simulation and experimental results (at the level of 3 kW) of the designed control algorithms are included.

16

Controlling Chaotic Systems Using Aggregated Linear Quadratic Regulator

Wang D., Ha P.

Przegląd Elektrotechniczny

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2012

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R. 88, nr 7b

336--340

EN

A systematic design method for controlling chaotic systems is presented in this paper. The aggregated multiple local models are adopted to express chaotic systems. The Linear Quadratic Regulator (LQR) theory is proposed to design state feedback control system for each local model. Multi-model control strategy has come into being by combining of T-S fuzzy model and LQR. The global stability of closed loop control system can guarantee and it is illustrated with several chaotic systems as examples.

PL

Zaproponowano projekt sterowania systemem chaotycznym. Zaadaptowano wypadkowe (aggregated) połączenie wielu lokalnych modeli do opisu system chaotycznego. Każdy lokalny model jest sterowany z wykorzystaniem teorii LQR – linear quadratic regulator.

17

High precision permanent magnet synchronous servo-drive with lqr position controller

Tarczewski T., Grzesik L.M.

Przegląd Elektrotechniczny

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2009

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R. 85, nr 8

42-47

EN

This paper presents a design approach to a state controller for permanent magnet synchronous motor (PMSM) servo-drives. Firstly, the well known non-linear mathematical model of PMSM is evaluated to a linear decoupled form by introducing new variables. An internal input model was used to achieve zero steady state error for ramp input as well as disturbance rejection. The proposed control method was implemented in a microcomputer system (TMS320F2812-DSP) and tested in a setup system.

PL

Przedstawiono sposób projektowania regulatora stanu dla serwonapędu z silnikiem synchronicznym o magnesach trwałych. Zastosowano wewnętrzny model liniowo narastającego sygnału referencyjnego w celu wyeliminowania błędu pozycji w stanie ustalonym oraz wyeliminowania wpływu zakłóceń . Zaproponowaną metodę sterowania zaimplementowano w procesorze sygnałowym (TMS320F2812-DSP) i zweryfikowano na stanowisku laboratoryjnym.