On unifying control barrier and Lyapunov functions using QP and Sontag's formula with an application to tumor dynamics

TL;DR

A hybrid continuous control law is introduced that recovers Sontag's formula locally and is illustrated in stabilization of a 3D tumor model, subject to physiological constraints, which yields useful insights into optimal cancer treatment design.

Abstract

A common tool in system theory for formulating control laws that achieve local asymptotic stability are Control Lyapunov functions (CLFs), while Control Barrier functions (CBFs) are typically employed to enforce safety constraints. Combining these two types of functions is of interest, because it leads to stabilizing controllers with safety guarantees. A common approach to merge CLFs and CBFs is to solve an optimization problem where both CLF and CBF inequalities are imposed as constraints. In this paper, we show via an example from the literature that this approach can lead to undesirable behavior (i.e., slow convergence and oscillating inputs). Then, we propose a novel cost function that penalizes the deviation from Sontag's formula by using a state-dependent weighting matrix. We show that by minimizing the developed cost function subject to a CBF constraint, local asymptotic stability is obtained with an explicit domain of attraction, without using a CLF constraint. To deal with vanishing properties of the weight matrix as the state approaches the equilibrium, we introduce a hybrid continuous control law that recovers Sontag's formula locally. The effectiveness of the developed hybrid stabilizing control law based on CLFs and CBFs is illustrated in stabilization of a 3D tumor model, subject to physiological constraints (i.e., all states must be positive), which yields useful insights into optimal cancer treatment design.

Figures (6)

• Figure 1: Example with the standard CLF-CBF-QP problem, for different $p$ with the values of $p$ defined in the color bar.
• Figure 2: Graphical illustration of the sets $\mathcal{W}^\ast$, $\mathcal{C}$ and $\mathcal{A}_{\mathcal{W}\mathcal{C}}$.
• Figure 3: Example \ref{['eq:ex system']} with the introduced hybrid S-CBF-QP control law for different $\gamma$ with the values of $\gamma$ defined in the color bar.
• Figure 4: Stabilization of equilibrium representing tumor dormancy with the introduced hybrid S-CBF-QP control law for different $\gamma$ with the values of $\gamma$ defined in the color bar.
• Figure :

Theorems & Definitions (15)

• Definition II.1: Khalil:1173048
• Definition II.2: Khalil:1173048
• Definition II.3: ames2019control
• Proposition II.4: ames2014control
• Definition II.5: Khalil:1173048
• Definition II.6: krstic1998stabilization
• Lemma II.7
• proof
• Lemma III.1
• proof