## 1 Introduction  
For many valve actuation systems, designing a closed-loop valve position control system is challenging due to its nonlinear frictions, especially for valves that must operate in high temperature environments, such as exhaust gas recirculation (EGR) valves. One approach is to apply advanced model-based control strategies (for example, sliding mode control [1], linear parameter-varying control [2; 3; 4], etc.) to deal with valve system nonlinearity. However, it is difficult to obtain an accurate nonlinear friction model. Thus, the advanced model-based control approach usually requires long development cycles and tremendous modeling efforts by experienced engineers [5; 6]. Moreover, such advanced model-based controllers are often difficult to tune, validate, and implement. On the other hand, classical proportional-integral-derivative (PID) control continues to dominate in valve system control due to its advantages of simplicity and easy tuning/calibration. System nonlinearity with varying operating conditions is one of main challenges in PID controller tuning. For example, test results show that a well-calibrated PID EGR valve controller can have a small steady-state error under multiple operational conditions; however, under small valve opening conditions, the relative steady-state error of the valve can increase significantly [5]. The main reason for this is that PID controllers are driven by the position error. Under small opening conditions, the control signal generated by a fixed-gain PID might not be large enough to overcome the high static friction force. To improve PID controller performance, gain-scheduled PID control has been developed with different scheduling strategies [7; 8; 9], which allows the PID control law to be adapted for systems that experience nonlinear dynamics. However, optimizing the scheduled PID gains to obtain the best performance is challenging. The classical tuning method combines the Ziegler-Nichols [10] method and manual tuning in a heuristic method, making it difficult to achieve optimality, especially for nonlinear dynamic systems with gain-scheduled PID controllers. More advanced optimal design methods for PID control have been developed, including nonlinear PID [11; 12; 13; 14; 15; 16], fuzzy PID [17; 18; 19; 20; 21], etc. This paper investigates how to automatically search and tune scheduled PID gains by utilizing system model knowledge and online data to optimize the closed-loop system performance of a nonlinear valve. Genetic algorithms (GA) [22; 23] can be used to solve the optimization problem of control gain tuning, especially for highly nonlinear systems, based on biological principles of natural selection and regeneration. Although GA search (optimization) does not guarantee achieving the global optimal solution in theory, it conducts a model-free search over a large parameter space. For the traditional single-objective genetic algorithm, there is only one predefined objective function to be optimized. However, to evaluate closed-loop system performance, performance indices such as the overshoot, settling time, and steady-state error need to be considered simultaneously with their trade-off relationships. One method is to combine these into a weighted sum to form a single objective optimization problem [24; 25; 26]. In this way, the obtained optimal solution is dependent on the selected weightings. With advancements in multi-objective optimization methods [27], this limitation can be overcome. The Non-Dominated Sorting in Genetic Algorithms-II (NSGA-II) method [28] is one of the most well-known computationally efficient multi-objective optimization algorithms. The NSGA-II algorithm provides a Pareto-front set of solutions, allowing the decision-maker to observe the trade-off relations among different objectives and select a solution based on the desired performance requirements.