Nonlinear optimal control and its application to a two-wheeled robot

This research studies two advanced nonlinear optimal control techniques, i.e., the freezing control and the iteration scheme, and their associated applications, such as a single inverted pendulum (IP) on a cart system and a two-wheeled robot (TWR) system. These techniques are applied to stabilise the highly unstable nonlinear systems in the vertical upright position when facing different initial pitch angles. Different linear optimal controllers (linear quadratic regulator and linear quadratic Gaussian) and nonlinear optimal controllers are designed and applied to the models for concurrent control of all state variables. The controlled systems are tested in simulation and the best performing control design is eventually implemented on a robot prototype built with an educational kit – the LEGO EV3, after practical factors such as motor voltage limitation, gyro sensor drift and model uncertainties have been considered, analysed and dealt with. Simulations and experiments on the TWR robot prototype demonstrate the superiority of the nonlinear freezing optimal control technique, showing larger operation ranges of the robot pitch angle and better response performances (i.e., shorter rise time, less overshoot and reduced settling time) than the linear optimal control methods. In particular, a novel mixing method to create a new nonlinear model (Model AB) from two different models on the same physical prototype with an increased controllable region of the TWR system is introduced, for the first time, for the calculations of optimal feedback gains for the system. Significantly, the utilisation of this mixed model, combined with the nonlinear freezing controller, achieves true global control of the TWR, even from an initial pitch angle of 90° (i.e., the horizontal position), when a motor with a saturated voltage of 48V and nominal torque of 298 mNm is adopted in simulation tests. This is wider than the angle achievable from the primary model (Model A) and any other single feedback control method on TWR reported in the literature. Robustness tests when introducing model uncertainties by adding mass and height on the TWR also illustrate excellent control performances from the nonlinear optimal control in both simulations and hardware implementations.