From a famous control paper titled 'Input-to-state stability for discrete-time nonlinear systems', the following holds true: given a discrete-time system
$$x(k+1)=f(x(k),u(k))$$
let $V(x)=|x|^2$ where $|\cdot|$ indicates L2 norm, given that
$$V(x(k+1))-V(x(k))\leq-aV(x(k))+b|u|^2,$$
where $a,b>0$, then $V(x)$ is a ISS-Lyapunov function for the system and the following inequality can be established (the definition of Input-to-State stability):
$$|x(k)|\leq\beta(|x(0)|,k)+\gamma(\lVert u\rVert_\infty),$$
where $\lVert u\rVert_\infty=\text{sup}(|u(k)|:k\in\mathbb{Z}_+)<\infty$, $\gamma:\mathbb{R}_{\geq0}\rightarrow\mathbb{R}_{\geq0}$ is a class $\textit{K}$ function (i.e. $\gamma$ is continuous, strictly increasing and $\gamma(0)=0$) and $\beta:\mathbb{R}_{\geq0}\times\mathbb{R}_{\geq0}\rightarrow\mathbb{R}_{\geq0}$ is a class $\textit{KL}$ function (i.e. $\beta(s,t)$ is a stricly increasing function with respect to $s$ and a decreasing function with respect to $t$, and $\beta(s,t)\rightarrow0$ as $t\rightarrow0$).
My question is how to obtain the expression of $\beta$ and $\gamma$ in the above context? They should be in the form of $k$, $a$, $b$, I have tried to read the paper but could not figure out the exact expression.
Thanks!
Best Answer
OK, after some further reading it seems that there is a general form of the solution:
$$ |x(k)|\leq\left(1-\left(1-\rho\right)a\right)^{\frac{k}{2}}|x(0)|+\left(\frac{b}{a\rho}\right)^\frac{1}{2}|u|_\infty, $$ given that $0<a<1$ and for an arbitrary $\rho$ such that $0<\rho<1$.
Let me know if this is not correct.