Backstepping

In control theory, backstepping is a technique developed circa 1990 by Petar V. Kokotovic and others for designing stabilizing controls for a special class of nonlinear dynamical systems. These systems are built from subsystems that radiate out from an irreducible subsystem that can be stabilized using some other method. Because of this recursive structure, the designer can start the design process at the known-stable system and "back out" new controllers that progressively stabilize each outer subsystem. The process terminates when the final external control is reached. Hence, this process is known as backstepping. The backstepping approach provides a recursive method for stabilizing the origin of a system in strict-feedback form. That is, consider a system of the form where with , are scalars, u is a scalar input to the system, vanish at the origin (i.e., ), are nonzero over the domain of interest (i.e., for ). Also assume that the subsystem is stabilized to the origin (i.e., ) by some known control such that . It is also assumed that a Lyapunov function for this stable subsystem is known. That is, this x subsystem is stabilized by some other method and backstepping extends its stability to the shell around it. In systems of this strict-feedback form around a stable x subsystem, The backstepping-designed control input u has its most immediate stabilizing impact on state . The state then acts like a stabilizing control on the state before it. This process continues so that each state is stabilized by the fictitious "control" . The backstepping approach determines how to stabilize the x subsystem using , and then proceeds with determining how to make the next state drive to the control required to stabilize x. Hence, the process "steps backward" from x out of the strict-feedback form system until the ultimate control u is designed. It is given that the smaller (i.e., lower-order) subsystem is already stabilized to the origin by some control where . That is, choice of to stabilize this system must occur using some other method.