Consider the system

where, are the states, is the control input, and are assumed to be known.

*Objective:* as

# Single-integrator Backstepping
## LaShalle-Yoshizawa Theorem
Let $$x = 0$$ be an equilibrium point of $$\dot{x} = f(x,t)$$ and suppose $$f$$ is locally Lipschitz in $$x$$ uniformly in $$t$$. Let $$V: \mathbb{R}^n \to \mathbb{R}_+$$ be a continuously differentiable, positive define and radially unbounded function $$V(x)$$ such that:
$$
\dot{V} = \frac{\partial V}{\partial x}(x)f(x,t)\leq -W(x) \leq 0,\ \ \ \ \ \ \forall t \geq 0, \ \ \forall x \in \mathbb{R}
$$
Where $$W$$ is a continuous function. Then all solutions of $$\dot{x} = f(x,t)$$ are globally uniformly bounded and satisfy
$$\lim_{t \to \infty} W(x(t)) = 0$$
If $$W(x)$$ is positive define, then equilibrium is **Globally Uniformly Asymptotically Stable**(GUAS)
## Method

*Objective1:*

*Objective2:*

Subsystem , where

Choose Lyapunov Function

add and minus

Regroup

*Objective:* Find Control stabilizing the system at

Lyapunov Candidate:

Let , with

Actual Control Law is

# Conclusions

- The designer can start the design process at the known-stable system and “back out” new controllers that progressively stabilize each outer subsystem.
- Any strict-feedback system can be feedback stabilized using a straightforward procedure.
- Backstepping design doesn’t require a differentiator.