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. ''

Backstepping approach

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 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 V_x for this stable subsystem is known. That is, this x subsystem is stabilized by some other method and backstepping extends its stability to the \textbf{z} shell around it. In systems of this strict-feedback form around a stable x subsystem, The backstepping approach determines how to stabilize the x subsystem using z_1, and then proceeds with determining how to make the next state z_2 drive z_1 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.

Recursive Control Design Overview

x to the origin. 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. Because then the resulting system has an equilibrium at the origin (i.e., where, z_1=0, z_2=0, ..., z_{k-1}=0, and z_k=0) that is globally asymptotically stable.

Integrator Backstepping

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.

Single-integrator Equilibrium

Consider the dynamical system where and z_1 is a scalar. This system is a cascade connection of an integrator with the x subsystem (i.e., the input u enters an integrator, and the integral z_1 enters the x subsystem). We assume that, and so if u_1=0, and z_1 = 0, then So the origin is an equilibrium (i.e., a stationary point) of the system. If the system ever reaches the origin, it will remain there forever after.

Single-integrator Backstepping

In this example, backstepping is used to stabilize the single-integrator system in Equation around its equilibrium at the origin. To be less precise, we wish to design a control law that ensures that the states return to after the system is started from some arbitrary initial condition. x subsystem is stable (in the sense of Lyapunov). Roughly speaking, this notion of stability means that: x subsystem. As the x states of the system move away from the origin, the energy also grows. x states must decay toward. That is, the origin will be a stable equilibrium of the system – the x states will continuously approach the origin as time increases. x subsystem. x and z_1 and functions f_x and g_x come from the system. The function u_x comes from our known-stable subsystem. The gain parameter k_1 > 0 affects the convergence rate or our system. Under this control law, our system is stable at the origin. So because this system is feedback stabilized by and has Lyapunov function with, it can be used as the upper subsystem in another single-integrator cascade system.

Motivating Example: Two-integrator Backstepping

Before discussing the recursive procedure for the general multiple-integrator case, it is instructive to study the recursion present in the two-integrator case. That is, consider the dynamical system where and z_1 and z_2 are scalars. This system is a cascade connection of the single-integrator system in Equation with another integrator (i.e., the input u_2 enters through an integrator, and the output of that integrator enters the system in Equation by its u_1 input). By letting then the two-integrator system in Equation becomes the single-integrator system By the single-integrator procedure, the control law stabilizes the upper z_2-to- y subsystem using the Lyapunov function, and so Equation is a new single-integrator system that is structurally equivalent to the single-integrator system in Equation. So a stabilizing control u_2 can be found using the same single-integrator procedure that was used to find u_1.

Many-integrator backstepping

In the two-integrator case, the upper single-integrator subsystem was stabilized yielding a new single-integrator system that can be similarly stabilized. This recursive procedure can be extended to handle any finite number of integrators. This claim can be formally proved with mathematical induction. Here, a stabilized multiple-integrator system is built up from subsystems of already-stabilized multiple-integrator subsystems. x are fed back to the input u_x by the control law, then the output states (and the Lyapunov function) return to the origin after a single perturbation (e.g., after a nonzero initial condition or a sharp disturbance). This subsystem is stabilized by feedback control law u_x. x . The resulting augmented dynamical system is x to input u_1 according to the control law x will return to z_1 = 0 and after a single perturbation. This subsystem is stabilized by feedback control law u_1, and the corresponding Lyapunov function from Equation is x . The resulting augmented dynamical system is x to input u_2 according to the control law x will return to z_1 = 0, z_2 = 0, and after a single perturbation. This subsystem is stabilized by feedback control law u_2, and the corresponding Lyapunov function is x . The resulting augmented dynamical system is x to input u_3 according to the control law x will return to z_1 = 0, z_2 = 0, z_3 = 0, and after a single perturbation. This subsystem is stabilized by feedback control law u_3, and the corresponding Lyapunov function is x ) and iterating out from that inner subsystem until the ultimate feedback-stabilizing control u is known. At iteration i, the equivalent system is 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).

Generic Backstepping

Systems in the special strict-feedback form have a recursive structure similar to the many-integrator system structure. Likewise, they are stabilized by stabilizing the smallest cascaded system and then backstepping to the next cascaded system and repeating the procedure. So it is critical to develop a single-step procedure; that procedure can be recursively applied to cover the many-step case. Fortunately, due to the requirements on the functions in the strict-feedback form, each single-step system can be rendered by feedback to a single-integrator system, and that single-integrator system can be stabilized using methods discussed above.

Single-step Procedure

Consider the simple strict-feedback system where x and z_1,. Rather than designing feedback-stabilizing control u_1 directly, introduce a new control u_{a1} (to be designed later) and use control law which is possible because g_1 \neq 0. So the system in Equation is which simplifies to This new u_{a1}-to- x system matches the single-integrator cascade system in Equation. Assuming that a feedback-stabilizing control law and Lyapunov function for the upper subsystem is known, the feedback-stabilizing control law from Equation is with gain k_1 > 0. So the final feedback-stabilizing control law is with gain k_1 > 0. The corresponding Lyapunov function from Equation 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.

Many-step Procedure

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 subsystem (i.e., with input z_2 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, the corresponding feedback-stabilizing control law is with gain k_i > 0. By Equation , the corresponding Lyapunov function is By this construction, the ultimate control (i.e., ultimate control is found at final iteration i=k). 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).