Adaptive/Robust Backstepping Techniques

• 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

Adaptive Backstepping

Consider a nonlinear system

$$\left\{ \begin{aligned} &\dot{x_1} = x_2 + \theta x_1^2 \\ &\dot {x}_2 = u \end{aligned} \right.$$

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

$$V_1 = \frac{1}{2}z_1^2 + \frac{1}{2}(\widehat{\theta}-\theta)^2$$

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

$$\dot z_1 = z_2 + \alpha_1+\widehat{\theta}z_1^2-(\widehat{\theta}-\theta)z_1^2$$ $$\dot{V}_1 = z_1(z_2 + \alpha_1 + \widehat{\theta}z_1^2)+(\widehat{\theta}-\theta)(\dot{\widehat{\theta}}-z_1z_1^2)$$

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

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

$$\begin{aligned} &\Leftarrow \dot{\widehat{\theta}} = \tau_1 \\ &\Leftarrow \alpha_1 = -z_1 - \widehat{\theta}z_1^2 \end{aligned}$$

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

we obtain:

$$\dot{V}_1 = -z_1^2 + z_1 z_2 + (\widehat{\theta} - \theta)(\dot{\widehat{\theta}}-\tau_1)$$

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

$$\begin{aligned} &\dot{z}_1 = -z_1 + z_2-(\widehat{\theta}-\theta)z_1^2\\ &\dot{z}_2 = u - \frac{\partial a_1}{\partial z_1}\dot{z_1} - \frac{\partial a_1}{\partial \widehat \theta}\dot{\widehat{\theta}} \end{aligned}$$

Choose Lyapunov Function

$$\begin{aligned} &V_2 = V_1 + \frac{1}{2}z_2^2 \\ &\dot{V}_2 = \dot{V}_1 + z_2 \dot{z}_2 \\ &\dot{V}_2 = -z_1^2 + z_2[z_1+u+\frac{\partial \alpha_1}{\partial z_1}(z_1 - z_2)-\frac{\partial \alpha_1}{\partial \widehat{\theta}} \dot{\widehat{\theta}}] +(\widehat{\theta}-\theta)(\dot{\widehat{\theta}}-\tau_1 + z_2\frac{\partial \alpha_1}{\partial z_1}z_1^2) \end{aligned}$$

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

$$\Leftarrow u = -z_1 -z_2 - \frac{\partial \alpha_1}{\partial z_1}(z_1 - z_2) + \frac{\partial \alpha_1}{\partial \widehat{\theta}}\dot{\widehat{\theta}}$$

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

Robust Backstepping

Consider a nonlinear system

$$\left\{ \begin{aligned} &\dot{x_1} = x_2 + \theta x_1^2 \\ &\dot {x}_2 = u \end{aligned} \right.$$

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

$$\dot{x}_1 = \theta x_1^2 + v$$

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

$$v = -c_1x_1 - \frac{\overline{\theta}}{2}(ax_1+\frac{1}{a}x_1^3) \ \ \ \ \ \ \ /\ \ \ \ \ \ \ v = -c_1x_1 - x_1^2\overline{\theta}s(x)$$

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

• $$\dot{V}_1 = -c_1x_1^2 - \frac{1}{2}x_1^2(\frac{\overline \theta}{a}x_1^2-2\theta x_1+a\overline{\theta})$$
• $$\dot{V}_1 = -c_1x_1^2 - x_1^3(\overline{\theta}s(x_1)-\theta)$$

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

$$\begin{aligned} \dot{V}_2 &= x_1[\theta x_1^2+(x_2-v)+v]+(x_2-v)(u-\dot{v})\\ &= x_1(\theta x_1^2+v)+(x_2-v)(u+x_1-\dot{v}) \end{aligned}$$

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

$$\Leftarrow u = -x_1 - c(x_2 - v) + \frac{\partial v}{\partial x_1}(s(x_1,x_2)x_1^2+x_2)$$

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