  This paper proposes a constructive approach to safety control of nonlinear
cascade systems subject to multiple state constraints. New design ingredients
include a unified characterization of safety and stability for systematic
designs of safety controllers, and a novel technique of reshaping the feasible
sets of quadratically constrained quadratic programming induced from safety
control. The proposed method guarantees Lipschitz continuity of virtual control
laws, enabling a stepwise constructive design. A refined nonlinear small-gain
synthesis is employed to address the nonlinear uncertain interconnections
between the resulting subsystems corresponding to different virtual control
laws, and to guarantee the achievement of the safety control objective. When
the safety constraints are removed, the proposed approach coincides with the
standard constructive nonlinear control. The proposed safety-control algorithm
is experimentally validated in a testbed involving a vertical takeoff and
landing (VTOL) vehicle taking off in narrow spaces.
