## 3 LB-MPC in Path Tracking of Mobile Platforms
### Learning and Optimizing for Controller
#### 3.2.1 Learning and Optimizing for Control Precision
\begin{table}
\begin{tabular}{c c} \hline \hline
**Evaluation Contents** & **Expression Formula** \\ \hline The system dynamics in discrete time & \(x(k+1)=f_{t}(x(k),u(k),k,w(k),\theta_{t})\) \\ The system state & \(x(k)\) The system state \\ The system dynamics in discrete time & \(u(k)\) The applied input at time \(k\) \\  & \(w(k)\) The noise in the system \\  & \(\theta_{t}\) The parametric uncertainty of the system \\ \hline The cost or objective function & \(J_{t}=E\left(\sum\limits_{k=0}^{\overline{N}}l_{t}(x(k),u(k),k)\right)\) \\  & \(\overline{N}\) The possibly infinite horizon \\ \hline Both the system dynamics and the prediction model in LB-MPC under uncertain disturbances & \(f(x,u,k,\theta,w)=f_{n}(x,u,k)+f_{1}(x,u,k,\theta,w)\) \\ \hline The parameterization of the cost function and constraints in optimal control problem & \(f_{n}\) The nominal system model \\  & \(I(x_{i},u_{i},\theta_{t})\) The cost function \\  & \(\mathcal{X}(\theta_{X})\) The constraints of system state \\  & \(\mathcal{U}(\theta_{t})\) The constraints of applied input \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The parameterized version of an MPC problem [31]. compensate the deviation between real dynamics and linear predictive dynamics, resulting in an improved controller with fewer evaluations [101]. The controller is designed using the current linear dynamic model and the parameterized cost function on the basis of the Bayesian optimization. This method evaluates the controller performance gap with the actual physical device in the closed-loop operation, and iteratively updates the dynamic model on the basis of this information to improve the controller performance [102]. The inverse optimization control algorithm is used to learn the appropriate cost function parameters of MPC from the human demonstrated data [103]. The motion generated by the path tracking controller matches the specific characteristics of human-generated motion and avoids massive parameter adjustments. In the consideration of the online estimation and adjustment of control parameters, a framework composed of real-time parameter estimators and feedback control strategies can improve the path tracking performance in mobile platforms [104]. The terminal set is an important design parameter in MPC. The MPC reduces the bad effects caused by the limitation of the prediction range by using terminal cost functions and constraints. The data are collected through the ML methods to improve the terminal components. A large terminal set leads to a large area, and it is quick and feasible in solving the MPC problem in this area. The state convex hull terminated on the trajectory of the origin is proven to be control invariant [105]. On this basis, a method for constructing a terminal set from a given trajectory solves the optimization problem required for parameterized offline calculation terminal controllers [106]. The final terminal controller is a solution for state-dependent optimization problem. For constrained uncertain systems, a robust learning model predictive controller is used for collecting the data from iterative tasks and estimating the current value function, which meets the system constraints accurately [107]. The terminal security set and the terminal cost function for iterative learning are designed to improve the closed-loop tracking performance of the controller [108]. This method estimates the unknown system parameters and generates high-performance state trajectory at the same time. The iterative learning has been further studied, and a task decomposition method for iterative learning model predictive control has been proposed [109]. This method quickly converges to the local optimal minimum time trajectory as compared to simple methods. It is necessary for the controller to ensure that the mobile platforms accurately track the predefined path under uncertain disturbances.