# solution

Consider an inverted pendulum system as shown in Figure 1. The system is driven by a DC motor. A belt-pulley mechanism is used for power transmission between the DC motor and the cart. You are supposed to analyze the stability of the system and design a controller for vertical positioning of the pendulum by using Root-Locus method. Please use Matlab/Simulink for your modeling and controller design applications. Figure 1. Schematic of Inverted Pendulum. The system contains mechanical and electromechanical components. For these sub-systems, system parameters are given below: 1. Mechanical System Parameters (Cart and Belt Pulley) â€¢ Mass of the cart (Mor): 0.9kg â€¢ Mass of the pendulum (mg): 0.1 kg â€¢ Friction of the cart (brat): 0.01 (Nm’s) Length of the pendulum to CG (I): 0.24 m Moment of inertia of the pendulum (J.):0.0053 kg m2 â€¢ Radius of pulley (Cres): 0.023 m 2. Electromechanical System Parameters (DC Motor) â€¢ Moment of inertia of the rotor (Jawa): 0.01 kg m2 â€¢ Motor viscous friction constant (best):0.1 Nm’s â€¢ Electromotive force constant (Kw): 0.01 Virad’s â€¢ Motor Torque Constant (Kr): 0.01 N. Amp â€¢ Electric Resistance (Break): 1 Ohm â€¢ Electric Inductance (La): 0.5 H 1. Obtain free-body diagrams of the mechanical and electromechanical systems 2. Mathematical modeling of the system, a. Derive the equations of motion of sub-systems. (Mechanical and electromechanical systems) 6. Obtain the required transfer functions of mechanical and electromechanical systems for pendulum angle control application. c. Obtain the open-loop transfer function of the system. (Hint: Connect the subsystems serially) 3. Perform the Routh Stability Analysis for the open-loop system.

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