Mathematic modeling and optimal control of unstable electromechanical system

U ovom radu proveden je postupak matematičkog modeliranja nestabilnog elektromehaničkog sustava kolica na klackalici i sinteza procjenitelja stanja i optimalnog regulatora koji navedeni sustav stabilizira u ravnotežnom položaju. Model je izveden korištenjem Euler-Lagrangeovih jednadžbi za mehanički dio sustava, te korištenjem Kirchhoffovih zakona za električni dio. Svođenjem modela koji je isprva opisan običnim nelinearnim diferencijalnim jednadžbama drugog reda na sustav diferencijalnih jednadžbi prvog reda, omogućena je analiza sustava u prostoru stanja, a nakon linearizacije nelinearnog modela sustav se analizirao u vidu upravljivosti, osmotrivosti i stabilnosti. Nakon analize sustava, provedena je sinteza Luenbergerovog procjenitelja stanja primjenom empirijskog pravila za smještanje vlastitih vrijednosti sustava i optimalnog linearno kvadratnog regulatora po varijablama stanja (eng. Linear Quadratic Regulator, LQR), koji je proširen s integralnim djelovanjem (eng. Linear Quadratic Integral control, LQI), a čija su pojačanja dobivena rješavanjem algebarske Riccatijeve jednadžbe. Na kraju je provedena simulacija reguliranog sustava u alatu Simulink, tako da se u sustav dovodio poremećaj poznatog iznosa. Rezultati simulacije uspoređeni su s eksperimentom provedenim na laboratorijskom sustavu te su predložena eventualna poboljšanja sustava vođenja.This thesis shows the mathematical modelling process of an unstable electromechanical system of a cart on a seesaw and the synthesis of a state observer and an optimal regulator which stabilizes the system at an equilibrium point. The model was derived using the Euler-Lagrange equations for the mechanical subsystem and Kirchhoff's laws for the electrical subsystem. Rewriting the model which was at first described by ordinary differential equations of second order as a system of first order differential equations, allowed the use of a state space approach in linearizing and then analyzing the system in terms of controllability, observability and stability. After the analysis of the system, a full state Luenberger estimator and an optimal linear quadratic regulator with integral action (LQI) were designed using an empirical rule for eigenvalue placement for the state estimator, whereas the gain matrix for the LQI was calculated by solving the algebraic Riccati equation. Finally, a simulation was conducted in Simulink, where a known disturbance was introduced to the system. The results of the simulation were compared to those of an experiment made using the laboratory setup and improvements of the control system were proposed