Consider the system

where, $x,y \in \mathbb{R}$ are the states, $u \in \mathbb{R}$ is the control input, and $f,g$ are assumed to be known.

Objective: $x \to 0$ as $t \to \infty$

# 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: $x \to 0$
Objective2: $\eta \to \eta_d$

Subsystem $\dot{x} = F(x)$, where $F(x) \triangleq f(x)+g(x)\eta_d$
Choose Lyapunov Function $V_x$

add and minus $g(x)\eta_d$

Regroup

Objective: Find Control $v_1$ stabilizing the system at $e_1 = 0$

Lyapunov Candidate:

Let $v_1 = -\frac{\partial V_x}{\partial x}g(x) - k_1 e_1$, with $k_1 > 0$

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.