Linear Quadratic Regulator Control of an Inverted Pendulum with Friction
Linear Quadratic Regulator Control of an Inverted Pendulum with Friction
The linear quadratic regulator (LQR) method is used to generate a control force that brings an inverted pendulum from an initial condition back to the upright position in an optimal way. The state space is used to represent the dynamics of the system. Static and Coulomb friction forces act as external disturbances. Coulomb friction causes an oscillation of the cart position around the equilibrium position (). When only viscous friction is present, LQR brings the pendulum to the upright position since viscous friction is included in the state matrix, while Coulomb friction is included in neither the nor the matrix. A standard friction model is used and is described below. The eigenvalues of the closed-loop state matrix (where is the gain vector generated by LQR) are all located in the left side of the complex plane, showing that the resulting system is stable.
x'(t)=Ax(t)+Bu(t)
x=0
A
A
B
A-BK
K