Modeling of dynamic systems and nonlinear system identification

Abstract

One of the primary goals of science is to identify and describe the structures and physical laws of nature. When the data corresponding to the input and output of a physical system is available, but the underlying rules and the structure of the system are unknown, it is essential to employ various approaches to determine these rules and structures. Determination of the underlying rules and structure of a system, particularly in some operation regions, is a difficult task because of the existence of some nonlinearities in the structure of the model. Therefore, choosing a reliable approach to identify the structure of the model in the different working regions of the system is crucial. For this purpose, system identification has been established as a critical technique for assisting in the modeling of complex engineering systems. System identification includes all processes of establishing a mathematical model of the systems by measured input-output datasets. The developed mathematical models using system identification methods are commonly used for monitoring, controller design, fault detection, system response prediction, optimization, and other purposes. The procedure of system identification could be classified into three steps: First, the structure of the mathematical model has to be determined. The structure of the mathematical model could be represented with linear or nonlinear models. Second, the unknown coefficients of the mathematical model should be determined by simulation or experimental input-output datasets. Finally, the model with the identified parameters has to be validated with the new input-output datasets. The major aims of this research could be listed as: In the first step, it is planned to develop transparent nonlinear mathematical models of the mechanical systems in a way that each term of the model could be physically interpreted. These models are called "white-box" models, which are developed using physical rules like Kirchhoff's and Newton's rules. Second, the thesis aims to properly determine the nonlinear models of the physical systems utilizing an appropriate system identification methodology. Third, it aims to investigate the existence of the identified physical phenomena, like nonlinear frictional terms, and dead-zone using different statistical methods. To fulfill these purposes, the following steps are performed: First, the general mathematical models of some physical systems are developed. The mathematical models of the physical systems include linear and various nonlinear equations. The linear equations of the model are developed utilizing some physical rules like Kirchhoff's and Newton's rules, etc. For the nonlinear part of the models, the nonlinear equations of some physical phenomena, like nonlinear friction equations and dead-zone, along with time-delay, are compiled and added to the general mathematical model of the physical systems. Then, the appropriate input signals are generated to stimulate all the dynamics of the physical systems in their different working regions. This is performed to capture the effect of all the possible existing nonlinearities in the system's output. In the next step, the output of the mathematical models is collected, and input-output data sets are established. Then, the Particle Swarm Algorithm (PSO) algorithm is coded to determine the unknown parameters of the general mathematical model of the system using input-output datasets. The PSO algorithm's results are evaluated by utilizing the conventional Nonlinear Least Squared Errors (NLSE) estimation method. Afterward, various statistical tests, including the confidence interval test and the null hypothesis test, are executed to investigate the identification results' validity. Finally, using some model evaluation criteria such as Mean Squared Errors (MSE) and coefficient of determination (R2), the capability of the determined models in computing the output of the real systems is evaluated. The framework suggested in this thesis is implemented for four case studies as benchmark problems, ranging from simple to complex in two steps. Initially, two case studies, namely a Direct Current (DC) motor, and a solenoid actuator are chosen, and their mathematical models with various combinations of nonlinearities are constructed in the first stage. The simulation data for both the DC motor and solenoid actuator models are established by utilizing the nonlinear models. First, all kinds of friction nonlinearities are incorporated into the real mathematical models of these components, followed by adding some likely friction nonlinearities to check the effectiveness of the identification algorithms. After that, the identification and validation frameworks are utilized to ascertain the model parameters and verify the credibility of the outcomes. Furthermore, a PSO algorithm with multiple cost functions is used to optimize the design parameters of a solenoid actuator to improve its performance. The second stage involves obtaining actual experimental data from real mechanical systems, which is then utilized to examine the framework developed in the simulation studies. The initial benchmark problem involves collecting real data from the experimental apparatus of the ball and beam mechanism by providing appropriate input signals. Moreover, the identification algorithm's effectiveness is tested for various experimental conditions for the mechanism of the ball and beam. In the second benchmark problem, real data is acquired from a 6-degree-of-freedom (DOF) UR5 robotic manipulator by providing appropriate trajectories. Then, the model parameters are determined, and the reliability of the outcomes is examined using the identification and validation frameworks.