Lyapunov Optimization

Question

I am new to Lyapunov optimization theory. My problem is that I block on understanding a step : When we want to upper bound the lyapunov drift to minimize it , is there a conventional way to do it ? Sometimes I see people decompose it into subproblems but I don't understand how they proceed.

Answer 1

In Lyapunov optimization theory, one common approach to minimizing the Lyapunov drift is to use a technique called "structured Lyapunov function" or "sum of squares Lyapunov function" approach. This technique involves expressing the Lyapunov function as a sum of squares of polynomials, which allows us to use tools from semidefinite programming (SDP) to optimize the Lyapunov function subject to constraints.

To upper bound the Lyapunov drift, one approach is to decompose it into subproblems and use the structured Lyapunov function approach to solve each subproblem separately. This can be done by decomposing the Lyapunov drift into separate terms that depend on different variables, and then optimizing each term separately subject to constraints.

For example, suppose we have a Lyapunov function $V(x)$ and a dynamics model $dx/dt = f(x)$ where x is the state of the system. The Lyapunov drift is given by:

$dV/dt = ∇V(x)⋅f(x)$

To upper bound this Lyapunov drift, we can decompose it into two terms:

$dV/dt = ∇V(x)⋅f1(x) + ∇V(x)⋅f2(x)$

where f1(x) and f2(x) are two functions that depend on different subsets of the state variables x. We can then use the structured Lyapunov function approach to find upper bounds on each of these two terms subject to constraints.

Specifically, we can define two Lyapunov functions $V1(x)$ and $V2(x)$ such that:

$dV1/dt = ∇V1(x)⋅f1(x) ≤ γ1V1(x)$

$dV2/dt = ∇V2(x)⋅f2(x) ≤ γ2V2(x)$

where $γ1$ and $γ2$ are positive constants that represent the upper bounds on the Lyapunov drifts of the two terms. These Lyapunov functions can be expressed as sums of squares of polynomials, and we can use SDP to optimize them subject to constraints.

By decomposing the Lyapunov drift into separate terms and optimizing each term separately, we can obtain tighter upper bounds on the Lyapunov drift and hence minimize it more efficiently.