Replacing $\mathcal{K}_{\infty}$ Function with Leaky ReLU in Barrier Function Design: A Union of Invariant Sets Approach for ReLU-Based Dynamical Systems

A piecewise affine function, denoted by $\mathrm{PWA}(x)$, is represented via an explicit parameterization based on a partition $\mathcal{P}=\{X_{i}\}_{i\in I(\mathcal{P})}$. Each region in the partition corresponds to a set of affine dynamics, described by a collection of matrices $\mathbf{A}_{\mathcal{P}}=\{A_{i}\}_{i\in I(\mathcal{P})}$ and vectors $\mathbf{a}_{\mathcal{P}}=\{a_{i}\}_{i\in I(\mathcal{P})}$. The $\mathrm{PWA}$ function is defined as follows:

where the region (cell) $X_{i}$ is described by

with matrices $E_{i}\in\mathbb{R}^{n_{hi}\times n}$ and vectors $e_{i}\in\mathbb{R}^{n_{hi}}$, defining the boundary hyperplanes of $X_{i}$. Here, $n_{h_{i}}$ denotes the number of hyperplanes in cell $X_{i}$.

To explain the concepts of forward invariant, consider the following nonlinear dynamics:

Consider $f_{cl}$ as locally Lipschitz continuous within the domain $\mathcal{D}\subseteq\mathbb{R}^{n}$. Based on this assumption, for any initial condition $x_{0}\in\mathcal{D}$, there is a time interval $I(x_{0})=[0,\tau_{\text{max}})$ within which a unique solution $x(t)$ to (4) exists, fulfilling the differential equation (4) and the initial condition $x_{0}$ [16].

A key challenge in finding the BF using (6) is choosing the $\mathcal{K}_{\infty}$ function $\alpha(x)$. A method for selecting $\alpha(x)$ will be discussed in the following section.

Definition 3 shows that if there exists an $\alpha(x)$ satisfies constraint (6), then $h(x)$ is a BF, and $S$ remains forward invariant and asymptotically stable in $\mathcal{D}$. Finding a suitable nonlinear $\alpha(x)$ is often challenging and computationally expensive because it often involves searching through a large function space. It is common practice to simplify the analysis using a linear $\alpha(x)$, [14]. However, this methodology may impose limitations on the search for a larger invariant set.

To address these limitations, we propose a Leaky ReLU function as a practical and flexible substitute. Unlike approaches that rely on complex $\mathcal{K}_{\infty}$ functions or trial-and-error selection of linear $\alpha(x)$, the Leaky ReLU function provides an efficient alternative with only a few tunable parameters, enabling more effective barrier function construction for dynamics (4).

Theorem 1 provides insight into how BFs can be constructed by tuning a Leaky ReLU with two parameters rather than looking for a complex $\mathcal{K}_{\infty}$ function. For less conservative behavior, $\alpha_{m}$ could be set to a large value and $\alpha_{1}$ to a small value. The boundedness of $h(x)$ imposes no restrictions if $\mathcal{D}$ is a compact set.

The main goal of this paper is to estimate the invariant set for the dynamical systems identified using ReLU NN or its equivalent $\mathrm{PWA}$ dynamical systems as follows:

where the $\mathrm{PWA}$ function is defined on a pre-selected domain $\mathcal{D}$. In [14], we proposed an approach to estimate the invariant set for dynamical systems (12) with linear $\alpha(x)$. However, using linear $\alpha(x)$ might be challenging. Before discussing how leaky ReLU $\alpha(x)$ can address this challenge, it is necessary to take a look at our work [14].

Our work in [14] consists of parameterizing the BF as a $\mathrm{PWA}$ function on the same partition as the dynamical systems (12). The BF for the cell $X_{i}$ can be parameterized as follows:

where $s_{i}\in\mathcal{R}^{n}$ and $t_{i}\in\mathcal{R}$. $h_{i}(x)$ is a continuous and differentiable function within the interior of the cell.

In [14], we constructed an optimization problem to search for a $\mathrm{PWA}$ invariant set over the domain $\mathcal{D}$. To describe the optimization problem, we must first establish useful index sets. Since our algorithm depends on the partition, we denote all these sets as functions of the partition. We define the index set $I_{\partial\mathcal{D}}$ as follows:

where $I_{\partial\mathcal{D}}(\mathcal{P})$ represents the indices of cells that have a non-empty intersection with the boundary of the $\mathcal{D}$. To determine whether a vertex belongs to the $\partial\mathcal{D}$ or the $\mathrm{Int}\left(\mathcal{D}\right)$, we introduce the following sets:

Here, $I_{b}(\mathcal{P})$ and $I_{\mathrm{Int}}(\mathcal{P})$ are sets of ordered pairs where the first element corresponds to the cell index $i$, and the second element corresponds to the vertex index $k$. Specifically, $I_{b}(\mathcal{P})$ identifies the vertices on the boundary of $\mathcal{D}$ and their associated cells, while $I_{\mathrm{Int}}(\mathcal{P})$ identifies the vertices in the interior of the domain partition $\mathcal{P}$ and their associated cells.

We constructed a linear optimization in [14] as follows to find a certified invariant set with $\alpha(x)=\alpha x$ where $\alpha>0$.

where $\epsilon_{1},\epsilon_{2},\epsilon_{3}>0$. In (17a), $M$ and $N$ denote the number of elements in $I_{b}(\mathcal{P})$ and $I_{\mathrm{Int}}(\mathcal{P})$, respectively. Constraint (17b) ensures that vertices in $I_{b}(\mathcal{P})$ are excluded from the invariant set, while constraint (17c) guides the search algorithm to include all points in $I_{\mathrm{Int}}(\mathcal{P})$.

These constraints, (17b) and (17c), are relaxed, as some vertices in $I_{\mathrm{Int}}(\mathcal{P})$ may not be contained within the invariant set, or the search algorithm may encounter feasibility issues for a given partition. Following the parametrization, constraint (17d) is derived from the general constraint (6). The feasibility of this optimization problem is analyzed in [14]. For further details, please refer to [14].

In [14], we employed a bisection method to obtain a set of parameters $\underline{\alpha}\;=\;\{\alpha_{1},\alpha_{2},\dots,\alpha_{m}\},$ each yielding an associated invariant set. We then retained only the invariant set corresponding to the parameter $\alpha_{k}\in\underline{\alpha}$ that resulted in the largest invariant set, discarding all other sets obtained from the remaining parameters. Although this strategy was effective it is not computationally efficient.

As described earlier, Leaky ReLUs $\alpha(x)$ may address the challenge of using linear $\alpha(x)$ in [14]. However, due to the constraint (17d), the resulting formulation requires mixed-integer programming, which results in high computational costs. Another potential solution is to take advantage of the concept of non-smooth BFs developed in [15], where multiple invariant sets can be merged. The primary challenge in [15] is determining a suitable $\alpha(x)$ that simultaneously satisfies all the necessary barrier-function conditions, which can be difficult in practice. In the next step, we will describe a new algorithm for estimating the invariant set for dynamical systems (12).

This section addresses the challenge of finding suitable $\alpha(x)$ by integrating non-smooth BF with leaky ReLU $\alpha(x)$. This combination aims to produce a possibly larger invariant set compared to [14] without any additional computational cost. To demonstrate how non-smooth BFs can be used with the Leaky ReLU $\alpha(x)$, we must first modify our notation.

As mentioned earlier, the solution to the optimization problem (17) depends on both $\mathcal{P}$ and $\alpha(x)$. Henceforth, we will use the notation $S^{(m)}(\mathcal{P}_{m},\alpha_{m})$ and $h^{(m)}(x,\mathcal{P}_{m},\alpha_{m})$ to reflect this dependency. The rest of the paper assumes $\alpha(x)=\alpha_{m}x$ in $h^{(m)}(x,\mathcal{P}_{m},\alpha_{m})$ unless otherwise stated.

Building on Lemma 1, we introduce the Union of Invariant Sets (UIS) method for constructing invariant sets. Instead of identifying a single Leaky ReLU function $\alpha(x)$ that maximizes the invariant set, UIS forms a unified invariant set by taking the union of multiple invariant sets, each obtained by solving (17) with a distinct linear $\alpha(x)$. This may result in a broader coverage of the state space while maintaining all interior points. The final invariant set is characterized by the barrier function (19) and a Leaky ReLU $\alpha(x)$, integrating information from various valid BFs with linear $\mathcal{K}_{\infty}$ functions systematically. The complete procedure is detailed in Algorithm 1.

As a result of Corollary 1, it is important to note that UIS does not guarantee a larger forward invariant set in all cases; for instance, solutions such as $S(\mathcal{P}_{4},\alpha_{4})$ in Figure 1 do not contribute any new points to the UIS algorithm. However, in comparison to the previous approach proposed in [14], this method offers the advantage of incurring no additional computational expense, while simultaneously allowing for the determination of a potentially larger invariant set.