## 3 LB-MPC in Path Tracking of Mobile Platforms
### Learning and Optimizing for Controller Output under Uncertain Disturbances
#### 3.3.2 Learning and Optimizing Controller Output Based on Security Framework  
The complex and high-dimensional control performance is achieved without the prior knowledge of the system by general learning control technique, especially the reinforcement learning control technique [121]. However, most of the learning control techniques are unable to ensure the security constraints with the physical limitations, especially during the learning iterations. In order for the aforementioned problem to be solved, the security framework was proposed in the control theory [122]. The centralized linear system security framework ensures the security by matching the learning-based input with the initial input of the MPC law online. The MPC law drives the system to a known secure terminal set. The model predictive security filter is applied to control the state and input space [123], as shown in Figure 11. The filter transforms the constrained dynamic system into an unconstrained security system, and any reinforcement learning algorithm can be applied in the security system without any constraint. The security system is established by constantly updating the security strategy. The security strategy is based on MPC formulation using a data-driven prediction model and considering state- and input-dependent uncertainties. The filter ensures the vehicle security during aggressive maneuvers, which was demonstrated by the applications for assisted manual driving and deep imitation learning using a miniature remote-controlled vehicle [124]. Figure 10: The control framework of RLB-MPC. As compared to the centralized linear system security framework, the distributed model predictive security certification scheme ensures the state and input constraint satisfaction when applying any learning-based control algorithm to an uncertain distributed linear system with dynamic couplings [125]. In addition, two different sets of distributed system security have been proposed on the basis of the latest research findings regarding structural invariant sets [126]. Different sets are different in their dynamic allocation of local sets and provide different trade-offs between the required communication and the realized set size. The synthesis of a security set and control law offer improved scalability by relying on the approximations based on convex optimization problems [127]. For non-linear and potentially larger-scale systems with security certification, a security framework is proposed to improve the learning-based and insecure control strategies. Furthermore, a probabilistic model predictive security authentication that can be combined with any reinforcement learning algorithm is proposed, relying on the Bayesian scheme. This model provides security assurance in terms of state and input [128]. The control of complex systems faces a trade-off between high performance and security assurance, which in particular limits the application of learning-based methods in security-critical systems. Ensuring the security constraints under physical limitations  
Figure 11: The model predictive security filter. On the basis of the current state \(x(k)\), a learning-based algorithm provides a control input \(u_{\zeta}(k)=\pi_{\zeta}(k,x(k))\), which is processed by the security filter, i.e., \(u(k)=\pi_{\zeta}(k,x(k),u_{\zeta}(k))\), and is applied to the real system. The detailed working of a model predictive security filter is presented at the bottom of the illustrations, which shows the state of the system at time \(k\) with a secure backup plan for a shorter horizon obtained from the solution at time \(k\)-1 (depicted in brown), as well as areas with poor model quality (depicted in green). An arbitrary learning input \(u_{\zeta}(k)\) can pass through the model predictive security filter if a feasible solution towards the terminal secure set \(S^{t}\) is obtained (depicted in green).