Backstepping

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In control theory backstepping is a technique for designing controls for nonlinear systems developed around 1990 by Petar V. Kokotovic and others.^[1] It is a recursive technique in which one designs feedback controls and finds Lyapounov functions for a set of n increasingly complex systems, the last system being the one we are interested in. An integrator is added at each step, and one may "backstep" through the cascaded chain to arrive at the true control law.

The backstepping approach involves the control of a particular structure:

$\dot{x} = f(x) + g(x)\xi$

$\dot{\xi} = u$

where $x(t) \in \mathbb R^n$ is the state vector and $u(t)\in \mathbb R^p$ is the vector of inputs. Here, the actual control input cascades down through a series of integrators $\dot{\xi}$ , and the control is first designed for a subsystem, then one may "backstep" through the cascaded chain to arrive at the true control law.

[edit] Control Design

Consider the example of stabilizing $(x,ξ)$ to $(0,0)$ . Choosing $ξ = φ(x)$ , with $φ(0) = 0$ , we can rewrite the system as

$\dot{x} = f(x) + g(x)\phi(x)$

We also assume that there is a Lyapunov function $V (x) > 0$ such that

$\dot{V}=\frac{\partial V}{\partial x}(f(x)+g(x)\phi(x)) <= - W(x)$

where $W (x)$ is a positive definite function. Rewriting the original system, we get

$\dot{x} = (f(x) + g(x)\phi(x))+g(x)(\xi-\phi(x))$

$\dot{\xi} = u$

A change of variable from $(x,ξ)$ to $(x, z)$ with $z = ξ - φ(x)$ gives

$\dot{x} = (f(x) + g(x)\phi(x))+g(x)z$

$\dot{z} = u-\dot{\phi}$

Choosing $u = v + \dot{\phi}$ gives

$\dot{x} = (f(x) + g(x)\phi(x))+g(x)z$

$\dot{z} = v$

Defining the augmented Lyapunov function candidate

$V_a(x,z)=V(x)+\frac{1}{2}z^2$

and checking that

$\dot{V}_a = \frac{\partial V}{\partial x}(f(x) + g(x)\phi(x))+ \frac{\partial V}{\partial x}g(x)z+zv <= -W(x)+ \frac{\partial V}{\partial x}g(x)z+zv$