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This paper proposes a robust controller for the optimal trajectory tracking problem of linear systems with an input delay and subjected to an external disturbance. We introduce a new integral transformation to convert the original error system with time delay into a delay-free one, and observe a key but simple relationship between the original state variables and the new state variables. We design the robust controller in two parts: one is a delayed optimal trajectory tracking controller used for implementing the control task, and the other is a disturbance observer used for compensating the external disturbance. Due to the delay effect, the current state in the controller is replaced with a predictor state. For demonstration, a robust controller to control the Back-and-Forth motion of an inverted pendulum on a cart is designed, where the pitch angle of the inverted pendulum is forced to be small by choosing a large weight of the pitch angle error in the quadratic performance criterion. Simulation results show that the proposed controller not only implements the control task well but also has a strong robustness against the disturbance.

This paper presents a control design method for implementing planar turning motion of a two-wheeled inverted pendulum with an input delay. The control task requires that the inverted pendulum is kept stabilized during the whole turning motion process along a pre-settled curve. Firstly, by using the theory of planar curve, key observations about the motion law of the two-wheeled mobile chassis are made and they are used to set up a dynamical trajectory tracking target. Then, by adjusting the parameters in the tracking target and the weights in the quadratic performance criterion, the optimal integral sliding mode controller based on a linear quadratic regulator is designed for keeping the vehicle body stabilized and tracking a circular path for the two-wheeled inverted pendulum. An illustrative example is given to demonstrate the validity of the theory with numerical simulation. Keywords Two-wheeled inverted pendulum • Input delay • Curvature tracking • Optimal control • Dynamical tracking target

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