# LQR kontrola linearnih dinamičkih sistema

Kešinović, Katarina (2014) LQR kontrola linearnih dinamičkih sistema. Diploma thesis, Faculty of Science > Department of Mathematics.

 Preview
PDF
Language: Croatian

The subject of this paper is a systematic overview of numerical techniques in engineer applications of the control theory. The problem of achieving the optimal control of a system of linear equations- $\dot{x}(t) = A(t)x(t) + B(t)u(t)$ is elaborated. Examples include linearquadratic regulation of the before mentioned system, which is achieved by minimization of the related functional: $I = \int t_{o}^{T}[x^{T}(t)Q(t)x(t) + u^{T}(t)R(t)u(t)]dt + x^{T}(t)Mx(t)$. Riccati matrix equations were used, and in this context, the mentioned functional is noted. Along with the results of Riccati theory, sufficient results of optimality for solving the problem of LQR control are also provided. It is also illustrated how these concepts appear e.g. in constructing active suspension of cars (in order to dampen the oscillations while driving on bumpy roads) or in appliances that need to regulate their own weight (to maintain an upright position on its two wheels with minute oscillations allowed). Possible buzzes on the sensors were removed by Kalman’s filter. The choice of Q and R matrices was varied and depending on it, system behavior on the same or different input data was monitored. The comparison with the Pole-placement method, which the LQR method is based on, is provided as well.