A Ball-Balancing System for Demonstration of Basic Concepts in the State-Space Control Theory

1974 ◽  
Vol 11 (4) ◽  
pp. 367-376 ◽  
Author(s):  
V. Jørgensen
Keyword(s):  
Mathematical Model ◽  
Control Theory ◽  
Physical Properties ◽  
Graduate Education ◽  
State Space ◽  
The State ◽  
Basic Concepts ◽  
Conditions Of Stability ◽  
The Mathematical Model

The Ball-balancing Systems is intended to demonstrate the basic concepts in the state-space control theory in the graduate education. The physical properties of the system are stated and the mathematical model is evaluated. Conditions of stability are discussed.

Design and Experimental Validation of a LQG Control System for the Stabilization of a 2DOF Commercial Gimbal Using Physical Modelling Techniques

2020 ◽  
Author(s):  
Julián Andres Gómez Gómez ◽  
Camilo E. Moncada Guayazán ◽  
Sebastián Roa Prada ◽  
Hernando Gonzalez Acevedo
Keyword(s):  
Mathematical Model ◽  
Control System ◽  
State Space ◽  
Transient Response ◽  
Space Form ◽  
Angular Position ◽  
Physical Modelling ◽  
Motor Shaft ◽  
Brushless Dc ◽  
The Mathematical Model

Abstract Gimbals are mechatronic systems well known for their use in the stabilization of cameras which are under the effect of sudden movements. Gimbals help keeping cameras at previously defined fixed orientations, so that the captured images have the highest quality. This paper focuses on the design of a Linear Quadratic Gaussian, LQG, controller, based on the physical modeling of a commercial Gimbal with two degrees of freedom (2DOF), which is used for first-person applications in unmanned aerial vehicle (UAV). This approach is proposed to make a more realistic representation of the system under study, since it guarantees high accuracy in the simulation of the dynamic response, as compared to the prediction of the mathematical model of the same system. The development of the model starts by sectioning the Gimbal into a series of interconnected links. Subsequently, a fixed reference system is assigned to each link body and the corresponding homogeneous transformation matrices are established, which will allow the calculation of the orientation of each link and the displacement of their centers of mass. Once the total kinetic and potential energy of the mechanical components are obtained, Lagrange’s method is utilized to establish the mathematical model of the mechanical structure of the Gimbal. The equations of motion of the system are then expressed in state space form, with two inputs, two outputs and four states, where the inputs are the torques produced by each one of the motors, the outputs are the orientation of the first two links, and the states are the aforementioned orientations along with their time derivatives. The state space model was implemented in MATLAB’s Simulink environment to compare its prediction of the transient response with the prediction obtained with the representation of the same system using MATLAB’s SimMechanics physical modelling interface. The mathematical model of each one of the three-phase Brushless DC motors is also expressed in state space form, where the three inputs of each motor model are the voltages of the corresponding motor phases, its two outputs are the angular position and angular velocity, and its four states are the currents in two of the phases, the orientation of the motor shaft and its rate of change. This model is experimentally validated by performing a switching sequence in both the simulation model and the physical system and observing that the transient response of the angular position of the motor shaft is in accordance with the theoretical model. The control system design process starts with the interconnection of the models of the mechanical components and the models of the Brushless DC Motor, using their corresponding state space representations. The resulting model features six inputs, two outputs and eight states. The inputs are the voltages in each phase of the two motors in the Gimbal, the outputs are the angular positions of the first two links, and the states are the currents in two of the phases for each motor and the orientations of the first two links, along with their corresponding time derivatives. An optimal LQG control system is designed using MATLAB’s dlqr and Kalman functions, which calculate the gains for the control system and the gains for the states estimated by the observer. The external excitation in each of the phases is carried out by pulse width modulation. Finally, the transient response of the overall system is evaluated for different reference points. The simulation results show very good agreement with the experimental measurements.

Time Domain Hydroelastic Analysis of a Flexible Marine Structure Using State-Space Models

2007 ◽  
Author(s):  
Reza Taghipour ◽  
Tristan Perez ◽  
Torgeir Moan
Keyword(s):  
Mathematical Model ◽  
State Space ◽  
Time Domain ◽  
State Space Model ◽  
Realization Theory ◽  
Regular Waves ◽  
Alternative State ◽  
Marine Structure ◽  
Hydroelastic Analysis ◽  
The Mathematical Model

This article deals with time-domain hydroelastic analysis of a marine structure. The convolution terms in the mathematical model are replaced by their alternative state-space representations whose parameters are obtained by using the realization theory. The mathematical model is validated by comparison to experimental results of a very flexible barge. Two types of time-domain simulations are performed: dynamic response of the initially inert structure to incident regular waves and transient response of the structure after it is released from a displaced condition in still water. The accuracy and the efficiency of the simulations based on the state-space model representations are compared to those that integrate the convolutions.

Optimization Control of the Bullwhip Effect in Supply Chain Inventory

2014 ◽  
Vol 945-949 ◽  
pp. 3187-3190
Author(s):  
Hai Dong ◽  
Jin Hua Liu ◽  
Liang Yu Liu
Keyword(s):  
Mathematical Model ◽  
Supply Chain ◽  
Control Theory ◽  
Inventory Control ◽  
Pid Controller ◽  
Bullwhip Effect ◽  
Control Gain ◽  
Optimization Control ◽  
Suitable Combination ◽  
The Mathematical Model

The bullwhip effect was caused by fuzzy demand among the enterprises. In order to reduce this effect, control theory was applied to solve the inventory in supply chain. Firstly, inventory control in supply chain and the bullwhip effect was researched. Secondly, a kind of proportional integral differential (PID) controller was developed for inventory control in a three-level supply chain, and the mathematical model of the PID controller for inventory control was presented. Finally, the results show that the PID controller can evidently alleviate the bullwhip effect and inventory fluctuations under the suitable combination of control gain.

Design of Observer for Direct Current Drive

2009 ◽  
Vol 147-149 ◽  
pp. 119-124
Author(s):  
Andrius Petrovas ◽  
Roma Rinkeviciene ◽  
Saulius Lisauskas
Keyword(s):  
Mathematical Model ◽  
Control Theory ◽  
State Space ◽  
Characteristic Equation ◽  
Direct Current ◽  
Angular Speed ◽  
Current Drive

Control theory proposes to apply observers in systems with uncertain plant parameters. Observers can be used for calculation of feedback gains and getting desired, even strictly determined response requirements. The paper considers design of controller for direct current drive. As state space variables are chosen armature current and angular speed of motor. Mathematical model of the motor, operating with load which is assumed as disturbance, is elaborated. Full order Luenberg observer is designed according to desired characteristic equation of system. The results of simulation are presented and discussed.

scholarly journals Avaliação de modelos matemáticos para estimativa da erosividade da chuva na região de São Mateus-ES

2016 ◽  
Vol 11 (3) ◽  
pp. 98
Author(s):  
Dione Pereira Cardoso ◽  
Fábio Ribeiro Pires ◽  
Robson Bonomo
Keyword(s):  
Mathematical Model ◽  
Mathematical Models ◽  
Holy Spirit ◽  
Climatic Conditions ◽  
The State ◽  
Rainfall Erosivity ◽  
Climatological Station ◽  
Average Rainfall ◽  
The Holy Spirit ◽  
The Mathematical Model

<p>Objetivou-se estimar a erosividade da chuva, mediante seis modelos matemáticos, de regressão linear avaliando entre estes, qual é mais indicado para as condições climáticas da região de São Mateus-ES. Os dados pluviométricos foram obtidos junto à Agência Nacional das Águas-ANA, sendo de 1947 a 2014 para Itauninhas, de 1971 a 2014 para Barra Nova, de 1981 a 2014 para São João da Cachoeira Grande e de 1993 a 2014 para Boca da Vala. Para estimar a erosividade da chuva, a partir da precipitação anual e do coeficiente de chuva, foram utilizadas diferentes equações utilizadas em outros estados com aplicação ao estado do Espírito Santo ou ajustadas para o próprio estado. Para os modelos matemáticos (II) e (I), os valores médios foram de 6.541,2 MJ ha<sup>-1</sup> mm h<sup>-1</sup> ano<sup>-1</sup> a 936,357 MJ ha<sup>-1</sup> mm h<sup>-1</sup> ano<sup>-1</sup> (Itauninhas), de 6.995,855 MJ ha<sup>-1</sup> mm h<sup>-1</sup> ano<sup>-1</sup> a 1.420,296 MJ ha<sup>-1</sup> mm h<sup>-1</sup> ano<sup>-1</sup> (Barra Nova), de 6.297,272 MJ ha<sup>-1</sup> mm h<sup>-1</sup> ano<sup>-1</sup> a 1.014,815 MJ ha<sup>-1</sup> mm h<sup>-1</sup> ano<sup>-1</sup> (São João da Cachoeira Grande) e de 5.427,659 MJ ha<sup>-1</sup> mm h<sup>-1</sup> ano<sup>-1</sup> a 1.626,489 MJ ha<sup>-1</sup> mm h<sup>-1</sup> ano<sup>-1</sup> (Boca da Vala). Para os municípios de Barra Nova e Boca da Vala a erosividade da chuva foi estimada pela equação EI<sub>30</sub> = 6,4492*pi – 391,63 com distribuição leptocúrtica. Para as outras duas localidades, a distribuição foi platicúrtica. A estação climatológica com o maior valor de erosividade média da chuva foi Barra Nova, enquanto Boca da Vala apresentou a menor erosividade, considerando apenas a estimativa da erosividade da chuva pelo modelo matemático II. Os maiores e menores valores de erosividade da chuva foram obtidos com os modelos matemáticos I e II. Para estimar a erosividade da chuva, nas condições climáticos da região de São Mateus-ES, o modelo matemático mais adequado é o II.</p><p align="center"><strong><em>Evaluation of mathematical models to estimate rainfall erosivity in the region of São Mateus-ES</em></strong></p><p><strong>Abstract</strong><strong>: </strong>This study aimed to estimate the rainfall erosivity by six mathematical models, linear regression, and evaluate these, which is more suitable for the climatic conditions of São Mateus-ES region. The rainfall data were obtained from the National Water Agency-ANA, and 1947-2014 for Itauninhas, 1971-2014 to Barra Nova, 1981-2014 for São João da Cachoeira Grande and 1993-2014 for Boca da Vala. To estimate the rainfall erosivity, from the annual precipitation and rainfall coefficient were used different equations used in other states with application to the state of the Holy Spirit or adjusted to the state itself. For mathematical models (II) and (I), the average values were 6541.2 MJ ha<sup>-1</sup> mm h<sup>-1</sup> year<sup>-1</sup> to 936.357 MJ ha<sup>-1</sup> mm h<sup>-1</sup> year<sup>-1</sup> (Itauninhas) of 6995.855 MJ mm ha<sup>-1</sup> h<sup>-1</sup> year<sup>-1</sup> to 1420.296 MJ ha<sup>-1</sup> mm h<sup>-1</sup> year<sup>-1</sup> (Barra nova), to 6297.272 MJ ha<sup>-1</sup> mm h<sup>-1</sup> year<sup>-1</sup> and 1014.815 MJ mm ha<sup>-1</sup> h<sup>-1</sup> year<sup>-1</sup> (São João da Cachoeira Grande) and 5427.659 MJ ha<sup>-1</sup> mm h<sup>-1</sup> year<sup>-1</sup> to 1626.489 MJ ha<sup>-1</sup> mm h<sup>-1</sup> year<sup>-1</sup> (Boca da Vala). For the municipalities of Barra Nova and Boca da Vala the rainfall erosivity was estimated by EI<sub>30</sub> = 6.4492*pi - 391.63 with leptokurtic distribution. For the other two locations, the distribution was platykurtic. The climatological station with the highest amount of average rainfall erosivity was Barra Nova, while Boca da Vala had the lowest erosivity, considering only an estimated rainfall erosivity by the mathematical model II. The highest and lowest values erosivity of the rain were obtained with the mathematical models I and II. To estimate the rainfall erosivity in the climatic conditions of São Mateus-ES region, the most suitable mathematical model is II.</p>

scholarly journals OPTIMIZING THE NATURAL ILLUMINATION OF RANDOM SHAPE PREMISES

2016 ◽  
Vol 2 (1) ◽  
pp. 181-184
Author(s):  
Чернышёва ◽  
Kseniya Chernysheva ◽  
Брусенцев ◽  
Aleksandr Brusentsev
Keyword(s):  
Mathematical Model ◽  
Mathematical Problem ◽  
Technical Problem ◽  
Main Task ◽  
Random Shape ◽  
Natural Illumination ◽  
Basic Concepts ◽  
The Mathematical Model

The problem of providing the illumination in diverse industrial premises, despite the numerous methods and ways to solve it is still relevant. In this article, the example of a specific technical problem considers the possibility of optimizing the natural illumination of random shape premises. The main task &amp; basic concepts and definitions from the area of metering lighting which are necessary for solving this task are formulated. The mathematical model of natural illumination and the mathematical problem of optimization of natural illumination are described here. A brief description of the algorithm for solving the problem is given also.

scholarly journals Study and Development of Robust Control Systems for Educational Drones

2021 ◽  
pp. 301-308
Author(s):  
Maria Letizia Corradini ◽  
Gianluca Ippoliti ◽  
Giuseppe Orlando ◽  
Simone Terramani
Keyword(s):  
Mathematical Model ◽  
Robust Control ◽  
Control Theory ◽  
Sliding Mode Control ◽  
Informal Education ◽  
Sliding Mode ◽  
Educational Robotics ◽  
Altitude Control ◽  
The Mathematical Model ◽  
Robust Control Systems

AbstractThis paper considers the problem of attitude and altitude control of quadrotors using the sliding mode control theory. The mathematical model of the quadrotor is derived using the Euler-Newton formalism. The sliding-mode is applied to the Parrot Mambo minidrone, which is a strong example of bringing educational robotics to formal (MATLAB, Python, JavaScript), non-formal (Tynker, Blockly, Swift Playground) and informal education. The control considered shows good performance and enhanced robustness.

scholarly journals After Strict Comparison, the Conclusion of Double Slit Test Is in Conflict With the Mathematical Model

2021 ◽  
Author(s):  
Xia Ruhuai
Keyword(s):  
Mathematical Model ◽  
Physical Properties ◽  
Wave Length ◽  
Optical Fiber Communication ◽  
Single Frequency ◽  
Particle Model ◽  
Causality Test ◽  
Light Waves ◽  
The Mathematical Model ◽  
Double Slit

Abstract The double slit test, as it is known, results in a series of alternating streaks of light and dark, presumed to be caused by interfering light waves. Careful comparison of experimental results and mathematical models reveals that the plausible theory of light-wave interference contains many fatal flaws. For example, the mathematical model does not have a mechanism for regularly producing multiple dark streaks. In practice, the spot where the dark streaks should appear is the brightest spot. Since only the electron radiates photons outward as it lowers its energy and returns to its ground state, the photon waves generated in the slit (possibly a vacuum) are of unknown origin. When light waves interfere with each other, photons can be infinitely subdivided and multiplied; After the head of the photon reaches the screen, the rest of the photon can still participate in interference; The effects of reflected light waves that most satisfy the interference conditions are not shown in the fringe. The important influence of polarization direction is not considered in the interference condition. In the causality test, the photon that has collapsed into a particle should be retested with a double slit test. In the causality test, the photon that has collapsed into a particle should be retested with a double slit test. The phenomenon of "observation determines outcome" has also been observed. When the phase is shifted π or the signal is reversed, it becomes a negative wave that cancels itself out. The transmittance of light waves through a double-slit device should not be so weak. As a fundamental physical property, waves must have bandwidth, but light and quantum only have a single frequency, which does not conform to the Fourier transform principle that waves must follow. Light waves in optical fiber communication must produce modulation effect, which has not been shown in reality. If the light waves from different slits interfere with each other, a mask similar to a two-slit device will not allow the lithographer to work properly. A photon whose size is much smaller than the wave length does not represent the whole wave but only represents a sample of the wave, but the corresponding physical properties are not shown. A pair of stars orbiting each other without changing color only proves that redshift is impossible but not that the speed of light is constant. Due to the late birth of radio technology, many physical properties of electromagnetic waves were not included in the light and matter waves proposed earlier. The results of the double slit test only prove that the simple quantum particle model is incorrect but do not provide evidence that the wave model is correct.

scholarly journals After Strict Comparison, the Conclusion of the Double Slit Test Is in Conflict with the Mathematical Model

2021 ◽  
Author(s):  
Xia Ruhuai
Keyword(s):  
Mathematical Model ◽  
Physical Properties ◽  
Light Wave ◽  
Optical Fiber Communication ◽  
Particle Model ◽  
Matter Waves ◽  
General Law ◽  
Light Waves ◽  
The Mathematical Model ◽  
Double Slit

Abstract Since they were born before physical achievements in question, light and matter waves, which have almost no physical properties, are abstract purely mathematical concepts rather than objects. The result of the double-slit test, a series of dark and bright streaks, was determined to be caused by interfering light waves. However, this plausible principle does not stand up to scientific scrutiny, and it is not difficult to find many flaws in the conclusion upon further study. For example, (1) the principle of generating multiple regular dark stripes cannot be found in the mathematical model. (2) In the mathematical model, the location where the dark stripe should appear most is actually the spot with the highest brightness. (3) Since the quantum property must be expressed in the process of interference, interference will not be triggered if all conditions are not met at the same time, and therefore, the principle of interference will be rejected. (4) Since the light comes from the radiation of the electron transition, it is impossible to generate a light source in a slit that may be a vacuum. (5) Interfering light waves can be subdivided into multiple parts and proliferate greatly. (6) A headless wave of light left outside the screen after the head disappears can still induce much interference. (7) There is no interference effect caused by reflected light waves. (8) The effect of polarization direction on interference is neglected, and the interference condition is incomplete. (9) In the causality test, to verify the effect of collapse, the photon that has collapsed into a particle should be tested again with a double slit test. (10) Whether the phenomenon of "observation determines outcome" is tampered with by observation. (11) When the phase is shifted π or the signal is reversed, waves become negative waves capable of annihilating themselves, which is typical of antimatter. In this case, matter and antimatter are conjoined. (12) If the general law of waves is followed, not only the targeted light wave passes through the slit. (13) The original version of the double-slit test cannot be reproduced, and the test results are different from those presented by contemporary technology. (14) Waves must obey Fourier's principle, but light waves, quantum waves and matter waves do not. (15) The modulation effect that waves must produce is not present in optical fiber communication. (16) If the light from different slits must interfere with each other, a mask full of slits will cause the lithographer to fail. (17) Since the size is much smaller than the wavelength, the photon is only a sampling of the light wave, but the corresponding physical properties are not presented. (18) The fact that the two stars orbiting each other do not change color only proves that the phenomenon of redshift is impossible but does not support the inference that the speed of light is constant. (19) The Michelson-Morley test process is not open and transparent enough, and pinhole diffraction and mechanical processing using broken lines instead of curves cannot be ruled out. In conclusion, Newton's particle model was wrong, but neither was the light wave theory.

On waiting times to populate an environment and a question of statistical inference

1999 ◽  
Vol 36 (1) ◽  
pp. 261-267 ◽  
Author(s):  
F. Thomas Bruss ◽  
M. Slavtchova-Bojkova
Keyword(s):  
Mathematical Model ◽  
Statistical Inference ◽  
Waiting Time ◽  
Branching Process ◽  
Waiting Times ◽  
The State ◽  
Higher Moments ◽  
Total Progeny ◽  
Cycle Lengths ◽  
The Mathematical Model

The mathematical model we consider here is the classical Bienaymé–Galton–Watson branching process modified with immigration in the state zero.We study properties of the waiting time to explosion of the supercritical modified process, i.e. that time until all beginning cycles which die out have disappeared. We then derive the expected total progeny of a cycle and show how higher moments can be computed. With a view to applications the main goal is to show that any statistical inference from observed cycle lengths or estimates of total progeny on the fertility rate of the process must be treated with care. As an example we discuss population experiments with trout.

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