Approximate Hamilton-Jacobi Reachability Analysis for a Class of Two-Timescale Systems, with Application to Biological Models

TL;DR

This work uses prior work on singularly perturbed differential games to identify a class of systems which can be readily reduced in HJR, and relates these results to the quantities of interest in HJR.

Abstract

Hamilton-Jacobi reachability (HJR) is an exciting framework used for control of safety-critical systems with nonlinear and possibly uncertain dynamics. However, HJR suffers from the curse of dimensionality, with computation times growing exponentially in the dimension of the system state. Many autonomous and controlled systems involve dynamics that evolve on multiple timescales, and for these systems, singular perturbation methods can be used for model reduction. However, such methods are more challenging to apply in HJR due to the presence of an underlying differential game. In this work, we leverage prior work on singularly perturbed differential games to identify a class of systems which can be readily reduced, and we relate these results to the quantities of interest in HJR. We demonstrate the utility of our results on two examples involving biological systems, where dynamics fitting the identified class are frequently encountered.

Figures (3)

• Figure 1: Depiction of a bacterium, engineered with the circuit of interest. Solid arrow represents chemical conversion of T to T* via K; dashed arrow represents down-regulation of T production by T* via binding to the promoter of G; dotted arrows represent activity modulation by control and disturbance signals. In particular, $\mathbf{u}$ (representing the inducer I) modulates the ability of T* to bind to the promoter of G, $\mathbf{d}_1$ modulates kinase activity, $\mathbf{d}_2$ modulates the cell growth rate, and $\mathbf{d}_3$ modulates the activity of G.
• Figure 2: Contour plots at time $t = -0.5$ of the value function $V_\varepsilon(t,z,y)$ of the SP system \ref{['eqn:small-example-1']}-\ref{['eqn:small-example-2']}. The target set is $\mathcal{S} = (0.25,0.75)$, and the terminal payoff function is taken to be $\ell(z) = \min\{10 (|z - 0.5| - 0.25),3\}$. The intersection of this system's BRS with the compact set $\mathcal{Z} \times \mathcal{Y} = [0,1]^2$ is the region inside the black curve. The inner and outer bounds from \ref{['eqn:main-theorem-bound']}, with $\eta = 0.1$, are the regions inside the yellow and magenta curves, respectively. Parameters were $\mathcal{U} = [0.1,1]$, $\mathcal{D} = [0.5,2]^3$, $\alpha = 1$. (Top) Results for $\varepsilon = 1$. Note that the bounds do not hold because $\varepsilon$ is not sufficiently small. (Bottom) Results for $\varepsilon = 0.01$. Note that the bounds now do hold as $\varepsilon$ is sufficiently small, so the system is approximately one-dimensional.
• Figure 3: (Top) Example MRN, here with $N = 20$. Nodes in the network represent metabolites, and edges represent reactions. The upstream molecule is $\text{m}_1$ (coral), which is rapidly converted in metabolites that are themselves ultimately metabolized into the downstream molecule p (cyan). Edge thickness is proportional to the reaction's rate coefficient. If $T$ is the adjacency matrix of the graph, with $T_{ij}$ being the weight of the edge from node $j$ to $i$, one obtains the corresponding matrix $A_{\text{MRN}}$ in \ref{['eqn:big-system-fast']} by taking the submatrix of $T$ corresponding to all nodes other than p and subtracting from each diagonal element the corresponding column sum. One obtains the corresponding matrix $C_{\text{MRN}}$ in \ref{['eqn:big-system-slow-1']} by taking the row of $T$ corresponding to node p and subsequently eliminating the element of this row corresponding to node p. (Bottom) Contour plot of the value function $\bar{V}$ for the reduced model \ref{['eqn:big-system-rm-1']}-\ref{['eqn:big-system-rm-3']}, in which the metabolic network is as shown above. Here $t = -3$, $\mathcal{U} = [0,1]^2$, $\mathcal{D} = [0.9,1.1]$, and the weights of the edges in the metabolic network were sampled from a unit uniform distribution. Only the $z_1 = 0$ slice of the value function is shown. The target set $\mathcal{S}$ is the interior of the solid grey line, while the sub-level sets of the reduced value function for levels $\pm\eta = \pm0.5$ lie below the dashed magenta and dotted yellow lines, respectively. Trajectories of the SP dynamics \ref{['eqn:big-system-slow-1']}-\ref{['eqn:big-system-slow-3']} with $\varepsilon = 0.01$ are shown as a white lines, with the initial state marked with an $\bigcirc$, and the final with an $\times$. The initial states are $\mathbf{z}_1(t) = 0$, $\mathbf{z}_2(t) = 0.025$, $\mathbf{z}_3(t) = 0.1$, $\mathbf{y}(t) = (0,\dots,0)$ and $\mathbf{z}_1(t) = 0$, $\mathbf{z}_2(t) = 0.15$, $\mathbf{z}_3(t) = 0.1$, $\mathbf{y}(t) = (0,\dots,0)$. The optimal control law was chosen based on the value function computed for the reduced model, and the disturbance was chosen as a random signal of points uniformly sampled from $\mathcal{D}$. The target is $\mathcal{S} = \mathbb{R} \times (.4,.6) \times (.4,.6)$ and the payoff function is $\ell(z) = \min\{\max\{10(z_2 - 0.5)-1, 10(z_3 - 0.5) - 1\}, 4 \}$.

Theorems & Definitions (16)

• Definition 1
• Lemma 1
• Theorem 1
• proof
• Corollary 1
• Remark 1
• Lemma 2
• proof
• Lemma 3
• proof