MODELING, SIMULATION AND STABILIZATION OF A DOUBLE QUAD INVERTED PENDULUM SYSTEM (DQIP) USING A LINEAR QUADRATIC REGULATOR (LQR) CONTROLLER
MODELING, SIMULATION AND STABILIZATION OF A DOUBLE QUAD INVERTED PENDULUM SYSTEM (DQIP) USING A LINEAR QUADRATIC REGULATOR (LQR) CONTROLLER
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Date
2014-08
Authors
MOHAMMAD, ABDULLAHI
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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.
Description
A Thesis Submitted to the Department of Electrical and Computer Engineering, Ahmadu
Bello University, Zaria, in Partial Fulfillment of the Requirements for the Award of
Master of Science (M.Sc) Degree in Electrical Engineering.
August, 2014
Keywords
MODELING,, SIMULATION,, STABILIZATION,, INVERTED,, PENDULUM SYSTEM,, LINEAR QUADRATIC REGULATOR