## 1 Introduction
This method improves the multi-objective optimization process, as there is no need to select weights for multiple performance criteria (see the early NSGA-II PID gain tuning studies [29; 30; 31; 32] and other multi-objective optimization problems for more details [33]). Compared with a fixed-gain PID controller, performance improvements can be achieved by using scheduled gains for systems with high nonlinearity. Although tuning scheduled gains for the best performance introduces additional challenges over fixed-gain approaches, automatic tuning methods can be developed based on the NSGA-II search algorithm. Two common control optimization processes are shown in Figure 1, where the path with black arrows represents traditional offline processes and the one with blue arrows represents online processes. Based on the identified or physical system model, offline optimization can be conducted with efficient searching in the simulation environment. The optimal controller obtained from simulation study is then used on the test bench or physical system, assuming that the identified (or physical) model is accurate enough to represent the real system. However, as the modeling error can never be avoided, especially for systems with high nonlinear friction, a PID controller obtained from offline optimization may not provide satisfactory performance for the physical system. Although the system modeling accuracy can be improved with increased effort, it is impossible to eliminate the modeling error entirely. The online optimization method applies the NSGA-II algorithm directly during bench tests (see the red arrow in Figure 1). To proceed with an intense search, a large population size with many generations is necessary, resulting in a massive number of test points. However, unlike simulation design, online global search is usually computationally infeasible due to the limitations of real-time microcontroller memory, computational capability, actuator durability, etc. In this paper, a co-optimization strategy is proposed using both offline and online information, as shown by the blue arrow in Figure 1. First, based on the advantages of offline global search, an optimal gain-scheduled PID controller is obtained based on adequate population size and deep generation using the offline model. The offline optimization result is then used as an initial condition for online optimization using an iterative NSGA-II algorithm. The PID controller is automatically tuned online with reduced computational cost based on the proposed iterative NSGA-II algorithm. Thus, the proposed approach combines the comprehensiveness of offline search and the authenticity of online search. In this paper, a highly nonlinear friction EGR valve system is modeled analytically with nonlinear friction, and the scheduled PID gains are tuned based on the proposed NSGA-II optimization algorithm using the offline optimized control gains as the initial setting. The closed-loop system performance is then validated through simulations and experimental studies. The main contribution of this paper is three-fold. First, an offline model-assisted scheduled PID gain tuning method based on the NSGA-II algorithm is proposed. Offline optimization through the simulation study provides a good initial 'guess' for online optimization. Second, the scheduled PID gains are optimized using the NSGA-II algorithm with reduced scheduling parameters based on the test bench input and output responses to make the online optimization feasible in terms of time and actuator durability constraints. Lastly, the coordinated offline-online optimization process is implemented in a microcontroller and the closed-loop system performance is validated by both simulations and experimental studies. The rest of this paper is organized as follows. In Section 2, the detailed system model is depicted, experimentally validated, and used for offline optimization. Section 3 formulates the optimization problem of control gain tuning with defined control gain functions, objective functions, and decision-making criteria.