Factional calculus-based modeling of mechanical systems: A case study on inverted pendulum dynamics

Abstract

Obtaining the mathematical model of a system's behavior and solving the equations of these models using appropriate solution methods is highly important for engineers. Modeling allows the system to be tested in a simulation environment before it is even produced, and it can be checked whether the system meets the requirements. At the same time, performance improvements can be made by changing the parameters of the modeled system. Models created for an airplane, car, or helicopter can provide a training environment for users. For these reasons, modeling the mathematical model of the system to give the closest results to real values is an important requirement. fractional calculus emerges as a different method for obtaining mathematical models for this purpose. Fractional calculus is a branch of mathematics that deals with fractional, complex order derivatives and integrals, in addition to integer-order derivatives. Using this mathematics, mathematical models containing fractional-order derivatives are obtained instead of integer-order derivatives obtained by classical methods. Fractional derivatives have several types such as Riemann-Liouville, Caputo, etc. Equations of motion should be obtained by choosing the appropriate one for modeling. The concepts related to fractional calculus are presented in Chapter 2. In this chapter, the emergence of fractional calculus and the relationship between classical integer-order derivatives and fractional derivatives are explained. In addition, the types of fractional derivatives are briefly given, and special functions used in the calculation of derivatives and integrals using this derivative are described. Chapter 3 provides information about system modeling approaches. The most commonly used classical methods for obtaining the motion equations of a system are Newtonian and Lagrangian mechanics. Newtonian mechanics analyzes the forces and moments that affect the system in order to obtain a mathematical model. These external forces and moments are determined using a free-body diagram. This method is quite easy for simple systems. However, Lagrangian mechanics is preferred for complex systems where determining the forces and moments acting on them is quite difficult. This is because Lagrangian mechanics uses the total kinetic and potential energy of the system to obtain the motion equations. The Lagrangian expression is obtained by subtracting the total potential energy from the total kinetic energy. Then this expression is substituted into the Euler-Lagrange equation to derive the motion equations. Equations obtained using these two methods contain integer derivatives. Different methods are available for obtaining fractional mathematical models. One of these methods is using the fractional Euler-Lagrange equation (FELE). With this equation, calculations are made using the Lagrangian expression in the classical method without obtaining the energy equations of the system in a fractional manner. The Fractional Lagrangian is obtained by replacing the derivatives in the classical Lagrangian with their "Fractional Derivative" counterparts. Fractional motion equations are then obtained using FELE. In Chapter 3, firstly, the equations for the two classical methods used for classical modeling, Newtonian and Lagrangian mechanics, are provided. Then, FELE is explained and its formulas are given. To demonstrate the application of these methods, the modeling of a 4-rotor aerial vehicle (Quadrotor System) is presented. First, the motion equations are obtained using Newtonian mechanics, and then they are obtained using Lagrangian mechanics. In addition, errors made in the open literature when obtaining the rotational motion equations have been corrected in this chapter. Finally, fractional motion equations for the quadrotor system are obtained using FELE. The important issue is the selection of the type of fractional derivative to be used in fractional modeling. The most important priority in modeling a mechanical system is adding initial and boundary conditions. For this reason, Caputo and Caputo-Fabrizio fractional derivatives can be preferred to add initial conditions into the equations in a physically interpretable way. The use of Caputo-Fabrizio fractional derivative in modeling is preferred to avoid non-singular kernel problems in solving the equations. After obtaining the equations, they need to be solved to acquire high accuracy results. Therefore, an appropriate solution method must be determined. These solution methods can be analytical or numerical. Analytical solution methods provide the exact results of the equations and express the system's solution with time-dependent functions. This allows for the derivation and integration of the obtained functions. However, a disadvantage of analytical solutions is that they become more complex and time-consuming as the system becomes more complex. Therefore, numerical solution methods can be preferred for nonlinear and complex systems. Numerical methods, on the other hand, provide approximate solutions to the equations and generally give results faster than analytical methods. Therefore, if the solutions are made with sufficient sensitivity, suitable results can be obtained for engineering purposes. In addition, there are also semi-analytical numerical solution methods. In these methods, the time interval for which the system is studied is divided into a certain time step, and analytical solutions are obtained for each time step to obtain results. In Chapter 4, analytical, numerical, and semi-analytical numerical methods are discussed. First, the differential transform method (DTM), which is a powerful analytical method for solving the classical model, is given, and it is applied to the double pendulum system as an example. It is seen that this method is not suitable for time-dependent systems because the results deviate from the Runge-Kutta (RK) solutions. Then, the improved version of this method, the multi-step differential transform method (MsDTM), is applied to the same system. At the same time, the system is solved using a powerful numerical solution method called RK, and it is seen that the two results are consistent with each other. However, it is also an important output that MsDTM's computation time is very long. In Chapter 4, a numerical method that uses matrix approach for solving the fractional model is also discussed. Since the solution of fractional differential equations is quite complex, numerical solution is preferred instead of analytical solution. At the same time, the fact that multi-step methods cannot be used in fractional differential equations is an important factor that restricts the analytical solution of these equations. Finally, in Chapter 5, all the methods mentioned have been applied to a nonlinear inverted pendulum system, which is a system modeled by two differential equations and exhibits oscillatory behavior. First, the energy equations of this system have been obtained, and then classical motion equations have been derived using Lagrangian mechanics. These equations were then solved using DTM, MsDTM, and RK methods, and the results were compared. It was observed that the results of DTM were quite deviant, but MsDTM gave the same results as RK. Then, the fractional motion equations of this system were obtained, and numerical equations were reached using matrix approach. The obtained equations were solved for different fractional orders, and the results were presented in the thesis. The important issue is that it is not known which fractional derivative order will give the actual model of the system. Therefore, graphs were obtained by trying different orders. Experimental data is required to determine which value is correct.