Convex Co-Design of Control Barrier Function and Safe Feedback Controller Under Input Constraints

TL;DR

The paper addresses safety guarantees for continuous-time linear systems under input constraints by co-designing a Control Barrier Function (CBF) and a safe affine controller within a single convex SOS/SDP framework. It parameterizes the CBF as $b(x)=(x-c)^{\top}\Omega^{-1}(x-c)-1$ and the controller as $u(x)=Y\Omega^{-1}(x-c)+d$, enabling global and local designs that ensure a control-invariant set $\mathcal{B}$ lies inside the safe set $\mathcal{S}$ and remains invariant without requiring explicit backstepping. The contributions include convex programs that handle high/mixed relative degree, and extensions to $L_1$, $L_2$, and $L_\infty$ input constraints, demonstrated on an omni-directional car collision-avoidance example. This approach provides rigorous safety guarantees with computationally tractable SDP-based synthesis suitable for constrained-actuation scenarios in linear systems.

Abstract

We study the problem of co-designing control barrier functions (CBF) and linear state feedback controllers for continuous-time linear systems. We achieve this by means of a single semi-definite optimization program. Our formulation can handle mixed-relative degree problems without requiring an explicit safe controller. Different L-norm based input limitations can be introduced as convex constraints in the proposed program. We demonstrate our results on an omni-directional car numerical example.

Figures (8)

• Figure 1: Simulation results for Case \ref{['cas:1']}.
• Figure 2: Visualization of Case \ref{['cas:2']} in Section \ref{['sec:case']}, where the state $\overline x=[{x}_1,{x}_2]\in\mathbb{R}^2$, $\underline{x}=x_3\in\mathbb{R}$. The safe set $\mathcal{S}:=\{x\in\mathbb{R}^3:s(\overline{x})\ge 0\}$ is a cylinder expanded from a set on $\mathbb{R}^2$ to $\mathbb{R}^3$. The region outside of the blue hyperboloid represents the set $\mathcal{B}:=\{x\in\mathbb{R}^3:b(x)\ge 0\}$, which is control invariant from our construction. The safe set $\mathcal{S}:=\{x\in\mathbb{R}^3:s(\overline{x})\ge 0\}$ is the outside of the inner red cylinder. We can see from the figure that $\mathcal{B}\subseteq \mathcal{S}$.
• Figure 3: Pictorial illustration for Example \ref{['ex:planar']}. The green set $\mathcal{S}^c:=\{x\in\mathbb{R}^2:\overline{x}^2-1{\le} 0\}$ is expanded from a segment on $\mathbb{R}^1$. The designed control invariant set $\mathcal{B}:=\{x\in\mathbb{R}^2:\overline{\Omega}^{-1}\overline{x}^2-\underline{\Omega}\underline{x}^2-1\ge 0$} has been filled in yellow. Intuitively, with a large velocity, i.e. larger $|\underline x|$, the planar car should stay further away from the obstacle, which can be seen by the gap between $\mathcal{B}$ and $\mathcal{S}^c$ being larger for larger $|\overline{x}|$.
• Figure 4: Exact invariant set for the planar car using $u=K_1\overline{x}+K_2\underline{x}$, where $K_1=Y_1\overline{\Omega}^{-1}={2},K_2=Y_2\underline{\Omega}^{-1}={-1}$. The green vertical lines are the boundary of $\mathcal{S}^c$, the set filled in blue is the exact invariant set.
• Figure 5: The union of control invariant set $\mathcal{B}\bigcup\mathcal{B}'$. $\mathcal{B}$ is computed by solving our program \ref{['eq:dqcbflem']} which results in $\overline{\Omega}=1$. Physical considerations for $\mathcal{B}'$ is obtained from the planar car. The union is also control invariant blanchini1999set, and close to the exact invariant set as in Figure \ref{['fig:exact-invariant']}.

Theorems & Definitions (22)

• Definition 1: Forward Invariance
• Definition 2
• Lemma 1
• Definition 3
• Lemma 2: S-procedure
• Theorem 1
• proof
• Example 1
• Proposition 1
• proof