MODELING, SIMULATION AND STABILIZATION OF A DOUBLE QUAD INVERTED PENDULUM SYSTEM (DQIP) USING A LINEAR QUADRATIC REGULATOR (LQR) CONTROLLER

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MODELING, SIMULATION AND STABILIZATION OF A DOUBLE QUAD INVERTED PENDULUM SYSTEM (DQIP) USING A LINEAR QUADRATIC REGULATOR (LQR) CONTROLLER

Abstract:

Inverted Pendulum phenomenon has been critical in the modeling and stabilization of space-ships, rockets, humanoid robots, etc and also in understanding classical control systems. This research is aimed at extending the modeling and complexity of the inverted pendulum from single-link or double-link to a multilink type, with particular emphasis on the double-quad inverted pendulum (DQIP) i.e. 8-link system. The nonlinear model of the DQIP was obtained using Eular-Lagrange formulation. To ease the analysis and control of the DQIP, the model was linearized using the Jacobi Matrix based on the Taylor’s series expansion approach. In order to validate the linear model, a comprehensive Simulink model of the DQIP was developed from which the Quad Inverted Pendulum (QIP) model was extracted. The response of the extracted QIP model was then compared with the responses reported in literatures for other QIP models and was found to outperform them. LQR was introduced after it was confirmed that the system was controllable and observable but not stable. This validated the DQIP model developed. The performance of the DQIP was then tested against the performance specifications of ≤ 2% maximum overshoot, ≤ 20 seconds settling time and zero steadystate error. With the following values of weighting matrices, which were used to penalize the states (Q) and input (R) respectively: Q=dig[500, 5000, 5000, 5000, 5000, 5000, 5000, 5000, 5000] and R=[0.8] , maximum overshoots of 2.05%, 1.7%, 1.575%, 1.52%, 1.45%, 1.26%, 1.21% and 0.106% for the first, second, third, fourth, fifth, sixth, seventh and eighth pendulum were obtained with settling time of 17.5 seconds and zero steady/state error. The LQR designed has therefore demonstrated its ability to stabilize all the pendulum links about a vertical position.

MODELING, SIMULATION AND STABILIZATION OF A DOUBLE QUAD INVERTED PENDULUM SYSTEM (DQIP) USING A LINEAR QUADRATIC REGULATOR (LQR) CONTROLLER

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