Stability definitions

Stability

The equlibrium point is:

  • Stable (in the sense of Lyapunov) if for each $\epsilon\geq0, \exists\delta(\epsilon)>0$ s.t: \begin{equation} \Vert x(0) \Vert < \delta \implies \Vert x(t) \Vert \leq \epsilon,\quad\forall t\geq 0 \end{equation}

  • Unstable if it is not stable

  • Asymptotically stable if it is stablem and $\exists \delta>0$ s.t. \begin{equation} \Vert x(0) \Vert < \delta \implies \lim_{t\to\infty} x(t) = 0 \end{equation}

  • Exponentially stable if $\exists \delta, \alpha,\beta > 0$ s.t. \begin{equation} \Vert x(0) \Vert < \delta \implies \Vert x(t) \Vert < \beta e^{-\alpha t},\quad \forall t \geq 0 \end{equation}

Stability for MPC

The main results on stability of MPC are based on Lyapunov theory.

  • A continuous function $\alpha : [0,a)\to [0,\infty)$ is said to belong to the class $\mathcal{K}$ if it is strictly increasing, and $\alpha(0)= 0$.
  • A continuous function $\alpha : [0,a)\to [0,\infty)$ is said to belong to the class $\mathcal{K}_\infty$ if it belongs to class $\mathcal{K}$, and is unbounded.
  • A continuous function $\beta : [0,a)\times[0,\infty) \to [0,\infty)$ is said to belong to the class $\mathcal{KL}$ if for a fixed $s_* $, the function $\beta(r,s_*) $ belongs to $\mathcal{K}$, and for a fixed $r_* $, the function $\beta(r_{*},s)\to 0$ as $s\to\infty$.

A state $x^\star$ is said to be asymptotically stable for $x(t+1) = f(x)$ on a forward invariant set $\mathcal{Y}$ if \begin{equation} \Vert x(n) - x^\star \Vert \leq \beta(\Vert x(0) - x^\star) \Vert, n) \end{equation} holds for all $x\in\mathcal{Y}$ and $n\in\mathbb{N}.$ To check this condition we use Lyapunov functions. A function $V: Y\to \mathbb{R}$ is called a Lyapunov function if for all $x\in\mathcal{Y}$

  1. There exists functions $\alpha_1, \alpha_2 \in \mathcal{K}_\infty$ such that
  2. There exists a function $\alpha_V \in \mathcal{K}$ such that

Theorem If there for the system $x(t+1) = f(x)$ exists a Lyapunov function $V(x)$ on a forward invariant set $\mathcal{Y}$, then $x_\star$ is an asymptotically stable equlibrium on $\mathcal{Y}$.

Some examples of when stability can be proven:

  • For the infinite horizon case
  • With terminal constraint $x(N) = x^\star$ where $x^\star$ is an equilibrium.
    • Online optimization may become harder
    • If we want a large feasible set $\mathbb{X}_N$ we typically need a large optimization horizon $N$
    • System needs to be controllable to $x^\star$ in finite time
    • Not very often used in industrial practice
  • Add local Lyapunov function $F$ for the equilibrium $x^\star$ as terminal cost, and impose a region $x(N) \in \mathbb{X}_0 $ as a terminal constraint.
    • Yields easier online optimization problems, but additional analytical effort
    • Yields larger feasible sets, but large feasible set still needs a large optimization horizon $N$
    • Does not need exact controllability to $x^\star$
    • Hardly ever used in industrial practice