Factor out absolut value

Question

I'm not sure if this is the right question for this platform, but I wanted to give it a try.
I'm facing following issue related to some discontinuous control theory. The basic idea of this example is taken from [1], so it is not my work.
Let's assume there is a second order differential equation given by following equation: $$ \ddot{\theta} = u, $$

where the value $\theta$ is simply integrating the value of $u$ twice.
So further I have "guessed" a function for $u$ $$ u = \begin{cases} 1 & \text{if }s(\theta, \dot{\theta}) < 0 \\ -1 &\text{else}, \end{cases} $$ with the linear function $s$ is given by: $$ s(\theta, \dot{\theta}) = \dot{\theta} + m \; \theta. $$ Next I wanted to see how does this system from the first equation behaves if the function of $u$ is applied. Basically I was checking for the asymptotically stability of this system. Therefore I assumed there is a Lyapunov function candidate and after derivation and some of the known expressions from above equations insert into it I got following equations: $$ \begin{align*} V&= \frac{s^2}{2}\\ \\ \dot{V} &= s \dot{s}\\ &= s \left( m\dot{\theta} + \ddot{\theta} \right)\\ &= s \left( m\dot{\theta} +u \right)\\ &= s \left( m\dot{\theta} -\operatorname{sgn}(s) \right). \end{align*} $$ So up to now it was just some background information, so that you can have an idea what is going on and what the named function and equations are about. Next I have taken, from the above cited book, the relation: $$ s \left( m\dot{\theta} - \operatorname{sgn}(s) \right) < |s|\left(m|\dot{\theta}| -1 \right) <0. $$ So this is my question, what is the idea behind factoring out the term $\operatorname{sgn}(s)$ and leaving the absolute value of $\dot{\theta}$ behind and say this is larger than the previous term?

[1]Edwards, Christopher, and Sarah Spurgeon. Sliding mode control: theory and applications. Crc Press, 1998.

Answer 1

You can see the first part by considering two cases:

1. $s>0$ ($\dot{\theta}>-m\theta$): $$s(m\dot{\theta}-1)\le|s|(m|\dot{\theta}|-1)$$ and
2. $s<0$ ($\dot{\theta}<-m\theta$): $$s(m\dot{\theta}+1)=|s|(m(-\dot{\theta})-1)\le|s|(m|\dot{\theta}|-1)$$

Note that the inequalities are not strict and can turn to equalities if, say, your initial values are such that $s>0$ and $\theta<0$, thus implying $\dot{\theta}>0$.

The second inequality $|s|(m|\dot{\theta}|-1)<0$ is actually the most problematic one. It does not need to hold as we can have $|\dot{\theta}|>1/m$. This means that your sliding mode controller ensures convergence to the sliding mode surface only for the initial values located in a certain neighborhood of $s=0$ given by $|\dot{\theta}|<1/m$.