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This paper presents the modelling and simulation for optimal control design of nonlinear inverted pendulum-cart dynamic system using Proportional-Integral-Derivative (PID) controller and Linear Quadratic Regulator (LQR). LQR, an optimal control technique, and PID control method, both of which are generally used for control of the linear dynamical systems have been used in this paper to control the nonlinear dynamical system. The nonlinear system states are fed to LQR which is designed using linear state-space model. Inverted pendulum, a highly nonlinear unstable system is used as a benchmark for implementing the control methods. Here the control objective is to control the system such that the cart reaches at a desired position and the inverted pendulum stabilizes in upright position. The MATLAB-SIMULINK models have been developed for simulation of control schemes. The simulation results justify the comparative advantages of LQR control methods.