This thesis presents a methodology for the development of control barrier functions (CBFs) through a backstepping inspired approach. Given a set defined as the superlevel set of a function, h, the main result is a constructive means for generating control barrier functions that guarantee forward invariance of this set. In particular, if the function defining the set has relative degree n, an iterative methodology utilizing higher order derivatives of h provably results in a control barrier function that can be explicitly derived. To demonstrate these formal results, they are applied in the context of bipedal robotic walking. Physical constraints, e.g., joint limits, are represented by control barrier functions and unified with control objectives expressed through control Lyapunov functions (CLFs) via quadratic program (QP) based controllers. The end result is the generation of stable walking satisfying physical realizability constraints for a model of the bipedal robot AMBER2.

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We present a new theoretical framework for designing linear parameter varying (LPV) controllers in the polynomial chaos (PC) framework. We assume the scheduling variable to be random and apply the PC approach to synthesize the controller for the resulting linear stochastic dynamical system. Two algorithms are presented that minimize the performance objective with respect to the stochastic system. The first algorithm is based on the Galerkin projection, and the second algorithm is based on the stochastic collocation. LPV controllers from both algorithms are shown to compare with Monte Carlo method and ROLMIP for a nonlinear missile system.

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