# stability of nonlinear system internal dynamics example

This question is generated from Example 6.3 in Slotine and Li "Applied Nonlinear Control", Prentice Hall, 1991

A nonlinear system

$$\begin{bmatrix}\dot{x_1}\\\dot{x_2}\end{bmatrix}=\begin{bmatrix}{x_2}^3+u\\u\end{bmatrix}$$ $$y=x_1$$ as the control objective to make $$y$$ track $$y_d$$, and the tracking error is $$e=y-y_d$$.

It is, $$\dot{y}=\dot{x_1}$$ and choosing the control law

$$u=-{x_2}^3-e+\dot{y_d}$$ gives $$\dot{e}+e=0$$ which is stable and converging to zero as the time $$t\rightarrow\infty$$ (or in other words, the error equation has one pole in -1).

The same control input $$u$$ applies also to the second equation of the nonlinear system, representing the internal dynamics. With the choice of $$u$$ as above, $$\dot{x_2}=u$$ yields $$\dot{x_2}+{x_2}^3=\dot{y_d}-e.$$

With the choice of a bounded $$\dot{y_d}$$ and $$e$$ (bounded since $$\dot{e}+e=0$$), then it is $$|\dot{y_d}-e|\leq D,$$ being $$D$$ a positive constant.

From now on my question starts, as I would like to know what are the math steps to derive the following conclusion:

the example concludes that $$|x_2|\leq D^{1/3}$$ (i.e. $$x_2$$ is bounded too!), since $$\dot{x_2}<0$$ when $$x_2>D^{1/3}$$, and $$\dot{x_2}>0$$ when $$x_2. Can someone explain how to derive these inequalities?

The example demonstrates that the chosen control law for $$u$$, i.e. $$u=-{x_2}^3-e+\dot{y_d}$$ -chosen from the first state equation- does not causes the second state $$x_2$$ to become unbounded.

## 1 Answer

Let us rewrite your equation as $$\dot{x}_2 = -x_2^3 + d(t)$$, where $$|d(t)|\le D$$ for all $$t$$. If $$x_2$$ is positive, then $$\dot{x}_2$$ is negative for all $$x_2>D^{1/3}$$. If $$x_2$$ is negative, then $$\dot{x}_2$$ is positive for all $$x_2<-D^{1/3}$$. It implies that the set $$|x_2^3|\le D$$ is invariant and attractive: if the initial condition $$x_2(t_0)$$ belongs to the set, then the trajectory reamins in the set. If the initial condition $$x_2(t_0)$$ is outside the set, then it will converge to the set.

Thus, $$x_2(t)$$ is bounded. However, the claim that $$|x_2|\le D^{1/3}$$ is valid only if you know for sure that the initial condition satisfies this inequality, or if you consider it as $$t \to \infty$$.

• Thank you @Arastas, you made it very clear. So, if I understand well, we are trying to ensure that $x_2$ is always bounded. So we state the condition for $\dot{x_2}>0$ and $\dot{x_2}<0$ (as I assume we don't know $\dot{x_2}$ itself, but we try to estimate what happens in each case)... overall this approach is looking at what happens to $x_2$ when the control law chosen based on controlling $x_1$ is applied – Lello Nov 17 at 22:11
• Yes, you are right. There are also more general ways to study it, i.e., the Lyapunov functions. But for the scalar case is more simple. – Arastas Nov 18 at 9:17