In control theory, backstepping is a technique developed circa 1990 by Petar V. Kokotovic, and others [1][2] for designing stabilizing controls for a special class of nonlinear dynamical systems.
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.
[3] 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[3] where Also assume that the subsystem is stabilized to the origin (i.e.,
Hence, the process "steps backward" from x out of the strict-feedback form system until the ultimate control u is designed.
This process is known as backstepping because it starts with the requirements on some internal subsystem for stability and progressively steps back out of the system, maintaining stability at each step.
Before describing the backstepping procedure for general strict-feedback form dynamical systems, it is convenient to discuss the approach for a smaller class of strict-feedback form systems.
These systems connect a series of integrators to the input of a system with a known feedback-stabilizing control law, and so the stabilizing approach is known as integrator backstepping.
With a small modification, the integrator backstepping approach can be extended to handle all strict-feedback form systems.
If the system ever reaches the origin, it will remain there forever after.
In this example, backstepping is used to stabilize the single-integrator system in Equation (1) around its equilibrium at the origin.
after the system is started from some arbitrary initial condition.
, it can be used as the upper subsystem in another single-integrator cascade system.
Before discussing the recursive procedure for the general multiple-integrator case, it is instructive to study the recursion present in the two-integrator case.
This recursive procedure can be extended to handle any finite number of integrators.
This claim can be formally proved with mathematical induction.
Hence, any system in this special many-integrator strict-feedback form can be feedback stabilized using a straightforward procedure that can even be automated (e.g., as part of an adaptive control algorithm).
Likewise, they are stabilized by stabilizing the smallest cascaded system and then backstepping to the next cascaded system and repeating the procedure.
Consider the simple strict-feedback system where Rather than designing feedback-stabilizing control
for the upper subsystem is known, the feedback-stabilizing control law from Equation (3) is with gain
So the final feedback-stabilizing control law is with gain
The corresponding Lyapunov function from Equation (2) is Because this strict-feedback system has a feedback-stabilizing control and a corresponding Lyapunov function, it can be cascaded as part of a larger strict-feedback system, and this procedure can be repeated to find the surrounding feedback-stabilizing control.
As in many-integrator backstepping, the single-step procedure can be completed iteratively to stabilize an entire strict-feedback system.
In each step, That is, any strict-feedback system has the recursive structure and can be feedback stabilized by finding the feedback-stabilizing control and Lyapunov function for the single-integrator
and output x) and iterating out from that inner subsystem until the ultimate feedback-stabilizing control u is known.
At iteration i, the equivalent system is By Equation (7), the corresponding feedback-stabilizing control law is with gain
By Equation (8), the corresponding Lyapunov function is By this construction, the ultimate control
(i.e., ultimate control is found at final iteration
Hence, any strict-feedback system can be feedback stabilized using a straightforward procedure that can even be automated (e.g., as part of an adaptive control algorithm).