Training for Region of Attraction Identification

Satisfying the minimization and decrease conditions within the training region (or any region around the fixed point, however small) is sufficient for proving local stability. In many cases, however, we desire an estimate of the region of attraction, rather than simply a guarantee of local stability.

Any compact sublevel set wherein the minimization and decrease conditions are satisfied is an inner estimate of the region of attraction. Therefore, we can restrict training for those conditions to only within a predetermined sublevel set $\{ x : V(x) \le \rho \}$. To do so, define a LyapunovDecreaseCondition as usual and then pass it through the make_RoA_aware function, which returns an analogous RoAAwareDecreaseCondition.

NeuralLyapunov.make_RoA_aware — Function

make_RoA_aware(cond; ρ, out_of_RoA_penalty, sigmoid)

Add awareness of the region of attraction (RoA) estimation task to the supplied AbstractLyapunovDecreaseCondition.

When estimating the region of attraction using a Lyapunov function, the decrease condition only needs to be met within a bounded sublevel set $\{ x : V(x) ≤ ρ \}$. The returned RoAAwareDecreaseCondition enforces the decrease condition represented by cond only in that sublevel set.

Arguments

cond::AbstractLyapunovDecreaseCondition: specifies the loss to be applied when

$V(x) ≤ ρ$.

Keyword Arguments

ρ: the target level such that the RoA will be $\{ x : V(x) ≤ ρ \}$, defaults to 1.
out_of_RoA_penalty::Function: specifies the loss to be applied when $V(x) > ρ$, defaults to no loss.
sigmoid::Function: approximately one when the input is positive and approximately zero when the input is negative, defaults to unit step function.

The loss applied to samples $x$ such that $V(x) > ρ$ is $\lvert \texttt{out\_of\_RoA\_penalty}(V(x), V̇(x), x, x_0, ρ) \rvert^2$.

The sigmoid function allows for a smooth transition between the $V(x) ≤ ρ$ case and the $V(x) > ρ$ case, by combining the above equations into one:

$\texttt{sigmoid}(ρ - V(x)) (\text{in-RoA expression}) + \texttt{sigmoid}(V(x) - ρ) (\text{out-of-RoA expression}) = 0$.

Note that a hard transition, which only enforces the in-RoA equation when $V(x) ≤ ρ$ and the out-of-RoA equation when $V(x) > ρ$ can be provided by a sigmoid which is exactly one when the input is nonnegative and exactly zero when the input is negative. As such, the default value is sigmoid(t) = t ≥ zero(t).

See also: RoAAwareDecreaseCondition

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NeuralLyapunov.RoAAwareDecreaseCondition — Type

RoAAwareDecreaseCondition(cond, sigmoid, ρ, out_of_RoA_penalty)

Specifies the form of the Lyapunov decrease condition to be used, training for a region of attraction estimate of $\{ x : V(x) ≤ ρ \}$.

Fields

cond::AbstractLyapunovDecreaseCondition: specifies the loss to be applied when $V(x) ≤ ρ$.
sigmoid::Function: approximately one when the input is positive and approximately zero when the input is negative.
ρ: the level of the sublevel set forming the estimate of the region of attraction.
out_of_RoA_penalty::Function: a loss function to be applied penalizing points outside the sublevel set forming the region of attraction estimate.

Training conditions

Inside the region of attraction (i.e., when $V(x) ≤ ρ$), the loss function specified by cond will be applied. Outside the region of attraction (i.e., when $V(x) > ρ$), the loss function specified by out_of_RoA_penalty will be applied.

The sigmoid function allows for a smooth transition between the $V(x) ≤ ρ$ case and the $V(x) > ρ$ case, by combining the different losses into one:

$\texttt{sigmoid}(ρ - V(x)) (\text{in-RoA expression}) + \texttt{sigmoid}(V(x) - ρ) (\text{out-of-RoA expression}) = 0$.

If the dynamics truly have a fixed point at $x_0$ and $V̇(x)$ is truly the rate of decrease of $V(x)$ along the dynamics, then $V̇(x_0)$ will be $0$ and there is no need to train for $V̇(x_0) = 0$.

Examples:

Asymptotic decrease can be enforced by requiring $V̇(x) ≤ -C \lVert x - x_0 \rVert^2$, for some positive $C$, which can be represented by a LyapunovDecreaseCondition with

rate_metric = (V, dVdt) -> dVdt
strength = (x, x0) -> C * (x - x0) ⋅ (x - x0)

Exponential decrease of rate $k$ is proven by $V̇(x) ≤ - k V(x)$, which can be represented by a LyapunovDecreaseCondition with

rate_metric = (V, dVdt) -> dVdt + k * V
strength = (x, x0) -> 0.0

Enforcing either condition only in the region of attraction and not penalizing any points outside that region would correspond to

out_of_RoA_penalty = (V, dVdt, state, fixed_point, ρ) -> 0.0

whereas an example of a penalty that decays farther in state space from the fixed point is

out_of_RoA_penalty = (V, dVdt, x, x0, ρ) -> 1.0 / ((x - x0) ⋅ (x - x0))

Note that this penalty could also depend on values of $V$ and $V̇$ at various points, as well as $ρ$.

In any of these cases, the rectified linear unit rectifier = (t) -> max(zero(t), t) exactly represents the inequality, but differentiable approximations of this function may be employed.

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