CBF_ACC

Discription

This collection of MATLAB scripts intends to study the performance of state-constrained controllers utilizing control barrier functions in the context of adaptive cruise control.

Problem Formulation

We consider a case in which two vehicles, modeled as point masses, are moving in a straight line. The following vehicle is equipped with an ACC and the lead vehicle drives with constant speed $v_0$.

• Control Objective: Cruising at a given speed $v_d$ for the following vehicle.
• Safety Objective: Ensure that the distance $D$ is not violating the following safety constraint:

$$\begin{equation} D \geq T_h v \end{equation}$$

with $T_h$ being the look-ahead time.

The dynamics of the system can be defined as follows:

$$\begin{equation} \begin{bmatrix} \dot{x} \\ \dot{v}\\ \dot{D} \\ \end{bmatrix} = \begin{bmatrix} v \\ - \frac{1}{m} F_r(v)\\ v_0 - v \\ \end{bmatrix} + \begin{bmatrix} 0\\ \frac{1}{m}\\ 0 \\ \end{bmatrix} F_w \end{equation}$$

with $x$ being the position, $m$ the mass and $v$ the speed of the controlled vehicle. The control input $u$ of the ACC is defined as the wheel force $F_w$, while the aerodyanmic drag is given as $F_r$, which is defined as:

$$\begin{equation} F_r(v) = f_0 + f_1 v + f_2 v^2 \end{equation}$$

Concept of Control Lyapunov Functions

We consider a control affine plant of the form:

$$\begin{equation} \dot{x}(t) = f(x(t)) + g(x) (u(t)) \end{equation}$$

where $x(t) \in \mathbf{R}^{n}$ is a measurable state vector and $u(t) \in \mathbf{R}^{m}$ is a control input vector. The control input is assumed to be magnitude limited by $\vert u_0 \vert$, which leads to the following closed set for the control input space

$$\begin{equation} \mathcal{U} = \begin{Bmatrix} u \in \mathbf{R}^{m} : - u_0 \geq u(t) \geq u_0 \end{Bmatrix} \end{equation}$$

For which we define the control objective of globally stabilizing the considered system to a point $x^∗ = 0$, and hence imposing $x(t) \rightarrow 0$. This can be achieved by finding a feedback control law $k(x)$ that drives a positive-definite continuous function $V : D \in \mathbf{R}^{m} \rightarrow R \geq 0$.

The system is globally stabilizable if there exists the class $\mathcal{K}_{\infty}$ functions $\alpha_1$, $\alpha_2$, $\alpha_3$ such that

$$\begin{equation} \alpha_1 (\lVert x \rVert) \leq V(x) \leq \alpha_2 (\lVert x \rVert) \end{equation}$$

$$\begin{equation} \inf_{u \in U} \dot{V}(x, u) \leq - \alpha_3 (\lVert x \rVert) \end{equation}$$

where $\alpha_3(h(x)$ is often chosen to be

$$\begin{equation} \alpha_3(\lVert x \rVert) = \lambda V(x) \end{equation}$$

with $\lambda$ being $\lambda > 0$. Since any controller that respects the above requirements can ensure stability, there is no need to explicitly construct the feedback controller

$$\begin{equation} k(x) = \begin{Bmatrix} u \in \mathbf{R}^{m} : \dot{V}(x, u) = [L_f V (x) + L_g V (x)u] ≤ −\lambda V (x) \end{Bmatrix} \end{equation}$$

One can define the following positive definite control Lyapunov function (CLF) $V(x,u)$:

$$\begin{equation} \inf_{u \in U} L_f V (x) + L_g V (x)u \leq - \alpha_3 (\lVert x \rVert) \end{equation}$$

Concept of Control Barrier Functions

We consider again a control affine plant of the form:

$$\begin{equation} \dot{x}(t) = f(x(t)) + g(x) (u(t)) \end{equation}$$

where $x(t) \in \mathbf{R}^{n}$ is a measurable state vector and $u(t) \in \mathbf{R}^{m}$ is a control input vector. The control input is assumed to be magnitude limited by $\vert u_0 \vert$, which leads to the following closed set for the control input space

$$\begin{equation} \mathcal{U} = \begin{Bmatrix} u \in \mathbf{R}^{m} : - u_0 \geq u(t) \geq u_0 \end{Bmatrix} \end{equation}$$

A closed set $\mathcal{C} \in \mathbf{R}^n$, which we consider as a safe set, is defined in the following form:

$$\begin{equation} \mathcal{C} = \begin{Bmatrix} x \in \mathbf{R}^n : h(x) \geq 0 \end{Bmatrix} \end{equation}$$

with $h: \mathbf{R}^n \times \mathbf{R}^p \to \mathbf{R}$ being a continuously differentiable function, called control barrier function (CBF).

The CBF can ensure for the presented control affine system that for any initial condition $x_0 := x(t_0) \in \mathcal{C}$, that $x(t)$ stays within $\mathcal{C}$ for any $t$, if there exists an extended class $\mathcal{K}$ functions $\alpha$ such that for all $x \in Int(\mathcal{C})$

$$\begin{equation} \sup_{u \in U} [L_f B(x) + L_g B(x) u + \alpha(h(x))] \geq 0 \end{equation}$$

where $\alpha(h(x)$ often chosen to be

$$\begin{equation} \alpha(h(x)) = \gamma h(x) \end{equation}$$

with $\gamma$ being $\gamma > 0$.

Pointwise CLF-CBF-QP Controller

By using a QP-based approach, it is possible to unify both CLF-based "performance objectives" and CBF-based "safety considerations". Using a quadratic programming formulation, the solution for $u$ can be found by solving:

$$\begin{align} &\min_{u \in \mathcal{U}} \begin{aligned}[t] &\frac{1}{2} u^T H u + F u \end{aligned} \\ &\text{s.t.} \notag \\ & L_f V(x) + L_g V(x)u + c_3 V(x) - \delta \leq 0, \notag\\ & L_f h(x) + L_g h(x)u - \alpha (h(x)) \leq 0, \notag \end{align}$$

with $\delta$ being a slack or relaxation variable that ensures a feasible QP by relaxing the condition on stability to guarantee safety.

Simulation Study

After each simulation run, a plot with results is given out. An example of such a plot is given here:

Dependencies

The scripts use external libraries, which need to be installed.

• Export_fig

Further, the following MATLAB toolboxes are needed:

• Optimization Toolbox

The software was tested with MATLAB 2020b under Windows 11 Home.

License

The contents of this repository are covered under the MIT License.

References

We kindly acknowledge the following papers, which have been the foundation of the here presented scripts:

• [1] J. Zeng, B. Zhang, and K. Sreenath, “Safety-Critical Model Predictive Control with Discrete-Time Control Barrier Function,” in 2021 American Control Conference (ACC), May 2021, pp. 3882–3889.
• [2] A. D. Ames, X. Xu, J. W. Grizzle and P. Tabuada, "Control Barrier Function Based Quadratic Programs for Safety Critical Systems," in IEEE Transactions on Automatic Control, vol. 62, no. 8, pp. 3861-3876.