To illustrate the concept of backstepping, or integrator backstepping, we start with a simple second-order system:

where (x₁, x₂) ∈ ²is the state, g(x₁) ≠0 for x₁ in some domain D that defines the operating envelope, and u ∈ is the control input. The objective is to design a feedback control algorithm to cause x₁(t) to converge to y_d (t). In this section, we assume that both ƒ(x₁) and g(x₁) are known functions.

The key idea behind the backstepping procedure is that the tracking problem would be solved if the control input u could force x₂(t) to satisfy

with k₁ > 0. In this case satisfies , which implies that x₁(t) converses to y_d(t). This is equivalent to treating x₂ as a virtual control input for the x₁ subsystem. Therefore, we introduce the virtual control variable , which is defined as

By adding and subtracting in (5.48) we obtain

If we let z₁ = x₁– y_dthen z₁ satisfies

Now, consider a coordinate transformation

whose derivative is given by

With this change of variables, we have rewritten the original system (5.48)-(5.49) as the tracking error dynamics:

The main, and key difference, between the original system (5.48)-(5.49) and the modified system (5.51)-(5.52) is that the modified system has an equilibrium at the origin and the z₁ dynamics of that equilibrium are asymptotically stable when z₂ = 0 and v = 0.

Now consider the Lyapunov function

whose time derivative along the solutions of (5.51)-(5.52) is given by

If we select the modified control input as

which shows that the equilibrium point (z₁, z₂) = (0,0) of the closed-loop tracking error dynamics is globally asymptotically stable.

From the definition of v we conclude (by combining (5.50) and (5.53)) that the feedback control law u given by

results in a globally asymptotically stable origin for the (z₁, z₂) system that ensures perfect tracking of y_d by x₁, assuming of course, that g(x₁) is bounded away from zero for all x₁ ∈ .

Some remarks:

• Even with a simple second-order system, the feedback control algorithm (5.54) becomes quite complex. Once the backstepping procedure gets extended to the n-th order case, it becomes considerably more complex. In fact, as we will see, for the n-th order case, the feedback control law is usually not written in a closed form, as in (5.54), but recursively based on a so-called backstepping procedure, which has as many steps as the number of state variables.
  
  
• A key assumption in the above backstepping procedure is that both ƒ(x₁) and g(x₁) are known exactly. In the case where they are partially or completely unknown then it may be appropriate that these functions be estimated online, which is the topic of discussion in the next two chapters.