  Complex control systems are often described in a layered fashion, represented
as higher-order systems where the inputs appear after a chain of integrators.
While Control Barrier Functions (CBFs) have proven to be powerful tools for
safety-critical controller design of nonlinear systems, their application to
higher-order systems adds complexity to the controller synthesis process -- it
necessitates dynamically extending the CBF to include higher order terms, which
consequently modifies the safe set in complex ways. We propose an alternative
approach for addressing safety of higher-order systems through Control Barrier
Function Backstepping. Drawing inspiration from the method of Lyapunov
backstepping, we provide a constructive framework for synthesizing
safety-critical controllers and CBFs for higher-order systems from a top-level
dynamics safety specification and controller design. Furthermore, we integrate
the proposed method with Lyapunov backstepping, allowing the tasks of stability
and safety to be expressed individually but achieved jointly. We demonstrate
the efficacy of this approach in simulation.
