Swing-up Control of Inverted Pendulum Using Pseudo-State Feedback

Abstract

Swing-up control, that is the transfer of a pendulum from a pendant state to the inverted one, is a good laboratory experiment of optimal control theory for non-linear control systems. The optimal control can be determined by the maximum principle and obtained as a function of time. Since the control is, however, determined in a feedforward fashion, the control is not robust to disturbances and uncertainties of the system, and the transfer of the state of the pendulum is not assured. In the paper, a robust swing-up control using a subspace projected from the whole state space is proposed. Based on the projected state space or pseudo-state, the control input is determined depending on the partitioning of the state as a bang-bang type control. The control algorithm is applied for a new type of pendulum (TI Tech pendulum), and the effectiveness and robustness of the proposed control are examined by experiments.

Stability and chaotic behavior of a PID controlled inverted pendulum subjected to harmonic base excitations by using the normal form theory

“…In accordance with the results of the previous section, we shall determine the stability conditions for the weak focus as a function of the parameters of the control law given by Eq (8). The stability analysis is carried out by using the normal form theory proposed by Bruno [27][28][29][30][31], since it provides a direct connection between the original and transformed equations of the system.…”

“…From Eqs (4) and (8), the mathematical model of the pendulum with harmonic base excitation and PID control can be written as follows:…”

“…Now we are going to analyze the advantages and appropriateness of the simple control law defined by Eq (8). It is well known that the unstable pointing-up position of the pendulum can be transformed into a stable one when the suspension point is excited in the vertical direction at high frequencies [21][22][23][24][25][26].…”

“…On the other hand, the problem of swinging up and controlling a pendulum has been considered in the classical Refs [7][8]. Other more complex control strategies reveal the great interest of the inverted pendulum in the field of control, as it is the case of control strategies based on space-state methods [9][10][11], control stabilization around homoclinic orbits [12], energy methods [13], passivity control [14] and bounded control [15] among others.…”

“…1 shows the layout of the pendulum system as well as the notation used to deduce the Lagrangian of the system. The pendulum is modeled by a mass m hanging at the end of a rod of negligible mass and length l, which is fixed to a support O [4][5][6], [7][8][9][10][11][12][13][14][15][16]. Let O'XY be an inertial frame and …”