1.0 System Analysis

This report details the design and evolution of a complete Adaptive Cruise Control (ACC) system, from initial dynamic modeling to a final implementation using Model Predictive Control (MPC).

---

1.1 Vehicle Dynamics Model

The foundation of the project is a longitudinal vehicle dynamics model. The net force on the vehicle dictates its acceleration according to Newton's second law: $$m \cdot a_x = F_{prop} - F_{brake} - F_{resist}$$ Where the total resistive force ($F_{resist}$) is the sum of rolling resistance and aerodynamic drag:

• Rolling Resistance ($F_{roll}$): $$F_{roll} = C_{rr} \cdot m \cdot g$$
• Aerodynamic Drag ($F_{aero}$): $$F_{aero} = 0.5 \cdot \rho \cdot C_d \cdot A \cdot V_{ego}^2$$

---

2.0 Control System Evolution

The control strategy evolved through several stages to meet performance requirements.

2.1 Initial Approach: PID Speed Controller

The initial controller was a standard PID designed to track a driver-set speed ($V_{set}$). To systematically tune this controller, the non-linear vehicle model was linearized around a nominal operating speed, resulting in a first-order transfer function that described the vehicle's acceleration dynamics. While this allowed for systematic tuning, the resulting fixed-gain controller performed poorly outside of its specific design-point velocity.

2.2 Hierarchical State-Based Controller

To handle multiple driving scenarios, a hierarchical architecture was developed using Stateflow. This high-level controller determines the operational mode (NoTarget or Following) and outputs the primary control objective: the desired acceleration (a_des_final).

• In NoTarget mode, a_des_final is calculated to achieve the driver's set speed.
• In Following mode, a_des_final is calculated to maintain a safe following distance based on the time-gap policy: $$D_{des} = D_{standstill} + T_{gap} \cdot V_{ego}$$

2.3 Final Controller: MPC Mathematical Formulation

To robustly track the a_des_final command while respecting physical limits, the low-level PID was replaced with a Model Predictive Controller (MPC). The MPC's objective is to calculate the optimal propulsion force ($F_{prop}$) at each time step. This is achieved by solving a constrained optimization problem based on a predictive model.

• Prediction Model: The controller uses a discrete-time state-space model to predict future acceleration: $$a_x(k+1) = A \cdot a_x(k) + B \cdot F_{prop}(k)$$ Where $k$ is the current time step, $a_x$ is the vehicle's actual acceleration, and $A, B$ are matrices derived from the linearized physical model.

• Optimization Problem: At each time step k, the MPC calculates the optimal sequence of future force commands by minimizing a cost function J. This function balances tracking performance against passenger comfort (minimizing jerk). $$\min_{F_{prop}} J(k) = \sum_{i=1}^{N_p} w_{accel} (a_{x,pred}(k+i) - a_{des_final}(k+i))^2 + \sum_{i=0}^{N_c-1} w_{jerk} (\Delta F_{prop}(k+i))^2$$ Where:

  • $(a_{x,pred} - a_{des_final})^2$ is the squared tracking error for acceleration.
  • $(\Delta F_{prop})^2$ is the squared change in force, which penalizes jerk.
  • $w_{accel}$ and $w_{jerk}$ are the tuning weights that define the trade-off between performance and comfort. This minimization is performed subject to the vehicle's physical constraints:
  • $0 \le F_{prop} \le F_{max}$
  • $a_{min} \le a_x \le a_{max}$
  • $\Delta F_{min} \le \Delta F_{prop} \le \Delta F_{max}$

• Receding Horizon Principle: The controller calculates the optimal plan over the future horizon but only implements the very first force command, $F_{prop}(k)$. At the next time step, it measures the new vehicle state and repeats the entire optimization to generate a new, updated plan. This makes the controller robust to real-world disturbances.

---

3.0 Conclusion

The final system architecture, a high-level Stateflow controller providing a reference to a low-level MPC, is robust, high-performing, and correctly models the advanced control strategies used in modern automotive applications. It successfully balances the logical complexity of mode-switching with the mathematical rigor of constrained, optimal control.