## 1 Introduction  
Control design for nonlinear systems has always been a challenging task. Generally, the control approaches can be classified into linear and nonlinear catalogs. In the linear approaches, by simplifying or even neglecting partial nonlinearities, the systems are linearized around equilibrium points such that the classical and modern linear control theories [1, 2] are applicable. However, in order to give consideration of the nonlinearities as much as possible to meet the increasing demand of high-quality control performances, various nonlinear control approaches such as the feedback linearization [3], the sliding mode control [4], the adaptive methodology [5], the finite time control [6], the backstepping approach [7], and so forth have been proposed and applied. As a promising and viable nonlinear control approach, the continuous-time optimal predictive control (CTOPC) methodology [8] extends a novel prospect for the control design of the nonlinear systems. In the prototype CTOPC (PCTOPC), state of the dynamical system or the prescribed reference is predicted by a linear combination of its high-order derivatives by means of the Taylor series expansion. Then, the optimal predictive control law is obtained by minimizing a quadratic receding horizon performance index (RHPI), which is the function of the predictive response errors. The control law has the merit of giving full consideration of all the system nonlinearities due to the complete inclusion of the system dynamics, which makes the PCTOPC more attractive and have many successful applications in various fields during the past decades such as the flight control [9, 10], the permanent magnet synchronous motor control [11], angle tracking control of a steer-by-wire system [12], the DC-DC converter control [13, 14, 15], the photovoltaic inverter control [16, 17],the wind energy system design [18; 19], the water tank system design [20; 21], the active suspension system design [22], stabilization of an artificial gas-lift process [23], path following system design of autonomous mobile vehicles [24; 25], and solar power plant control [26]. Though the PCTOPC has drawn much attention in the aforementioned areas, its inadequacy is also obvious. That is, the closed-loop system is susceptible to the unknown external uncertainties since the PCTOPC only takes into account of the current tracking errors between the prescribed reference and the system output, making poor robustness and large steady-state tracking errors easy to occur. Thus far, only few previous works have been dedicated to address such problems [9; 19; 21; 23; 24; 25; 26]. Therefore, targeting the inadequacy and extending an alternative methodology to enhance the robustness of the PCTOPC are meaningful and of engineering significance. As one of the most widely used approaches around the world, the linear proportional-integral-differential (PID), which not only has the merits of simple controller structure and easy implementation procedures, but also has the strong capability on dealing with the unknown disturbances, has many successful application examples in various fields [27; 28; 29]. In the PID, considering the terms of accumulative tracking errors can help the closed-loop systems reduce stead-state errors. By assigning appropriate proportional, integral, and differential gain coefficients, satisfactory system response time and input performance can be acquired. Hence, the PID approach has been successfully applied to enhance the system robustness for the existing approaches [30; 31]. Due to those advantages, this paper combines the PCTOPC with the linear PID approach for closed-loop system robustness enhancement. Next, the proposed approach is named as PID-CTOPC. To further highlight superiorities of the proposed approach, comparison with another typical CTOPC approach [32; 33], which considers both current and high-derivative tracking errors proposed by Professor [PERSON], is also conducted. The main contributions of this paper are twofold. 1.