LQG control of an inverted pendulum

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LQG control of an inverted pendulum 1 Variables Cart position Pendulum angle Cart mass Pendulum mass Pendulum length Motor voltage Motor current Motor inductance Motor resistance Motor torque constant Disturbance force Internal gear ratio Motor gear radius Maximum voltage x θ M m 2l V i L R km Fd kg r Vmax 2 Energies Kinetic coenergy Magnetic coenergy Dissipation function External virtual work T ∗ = 1 2 M ˙x 2 + 1 2 m ( ˙x 2 G +˙ y 2 G ) + 1 2 ml 2 3 ˙θ 2 W ∗ m = 1 2 Li 2 + km θm i D = 1 2 Ri 2 δL = V δq − mgδy G + Fdδx 3 Kinematics xG = x + lsin θ yG = lcos θ ˙xG =˙ x + lcos θ˙θ ˙yG = −lsin θ˙θ ˙x 2 G =˙ x 2 + l 2 (cos θ) 2 ˙θ 2 + 2 ˙ xl cos θ˙θ ˙y 2 G = l 2 (sin θ) 2 ˙θ 2 δy G = −lsin θδθ θm = kgθr = kg x r 4 Energies (2) Kinetic coenergy Magnetic coenergy Dissipation function External virtual work T ∗ = 1 2 (M + m )˙x 2 + 1 2 4 3 ml 2 ˙θ 2 + ml cos θ˙x ˙θ W ∗ m = 1 2 L ˙q 2 + km kg r x ˙q D = 1 2 R ˙q 2 δL = V δq + mgl sin θδθ + Fdδx 5