On unifying control barrier and Lyapunov functions using QP and Sontag's formula with an application to tumor dynamics
arxiv(2024)
摘要
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.
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