• Uncertainties

Structured(or parametric) uncertainties
Corresponds to inaccuracies on terms actually included in the model
Unstructured uncertainties(or unmodeled dynamics)
Corresponds to inaccuracies on(i.e., underestimation of) the system order

Consider a nonlinear system

where the parameter $\theta$ is unknown, we assume that $\theta$ is constant.

Objective: $x_1 \to 0$ as $t \to \infty$

For this system the adaptive controller design is in two steps.

Step 1

We introduce $z_1 = x_1$ and $z_2 = x_2 - \alpha_1$ and consider $\alpha$ as a control to stabilize the $z_1$-system with respect to the Lyapunov function

The $z_1$-system and the corresponding $\dot{V}_1$ are:

let $\tau_1 = z_1 z_1^2$, and if $z_2 = 0$

Objective: $\dot{V}_1 = -z_1^2$

Substitute $\tau_1(z_1)$ and $\alpha_1(z_1, \widehat{\theta})$

we obtain:

where $z_1 z_2 + (\widehat{\theta} - \theta)(\dot{\widehat{\theta}}-\tau_1)$ should be taken off.

Step 2

With $z_1 = x_1, z_2 = x_2 - \alpha_1$, the original system has been transformed into

Choose Lyapunov Function

Objective: $\dot{V}_2 = -z_1^2-z_2^2$

where $\dot{\widehat{\theta}} = \tau_1-z_1\frac{\partial \alpha_1}{\partial z_1}z_1^2$

Robust Backstepping

Consider a nonlinear system

we assume that the unknown parameter $\theta$ belongs to a known interval $|\theta| < \overline \theta$

Objective: $x_1 \to 0$ as $t \to \infty$

For this system the robust controller design is in two steps.

Step 1

To apply backstepping, we design a $v$-controller for

Choose Lyapunov function $V_1 = \frac{1}{2}x_1^2$
Several different designs are possible:

where $s(x)$ is a smooth switching function.

Step 2

We now employ backstepping to design $u$-controllers for the second order system.
Choose Lyapunov function: $V_2 = \frac{1}{2}x_1^2+\frac{1}{2}(x_2-v)^2$
Choose$v = -c_1x_1 - \frac{\overline{\theta}}{2}(ax_1+\frac{1}{a}x_1^3)$
and we have

Objective: $\dot{V_2} \leq 0$

We should choose $s(x_1,x_2)$ to satisfy $(x_2-v)\frac{\partial v}{\partial x_1} x_1^2(s(x_1,x_2)-\theta) \leq 0$:

• When $(x_2 - v)\frac{\partial v}{\partial x_1}\geq 0$, $s(x_1,x_2) = -\overline{\theta}$
• When $(x_2 - v)\frac{\partial v}{\partial x_1} < 0$, $s(x_1,x_2) = \overline{\theta}$