Analyze the theoretical foundations of process control and optimization in chemical engineering systems. Discuss the principles of feedback and feedforward control strategies, including proportional$-$integral$-$derivative $($PID$)$ control, model predictive control $($MPC$),$ and adaptive control techniques. How do factors such as process dynamics, disturbances, and system constraints influence the selection and implementation of control algorithms for improving process stability, performance, and robustness? Evaluate the engineering approaches for optimizing process operations through real$-$time monitoring, data analytics, and advanced control strategies in complex chemical processes.