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Efficient polyhedral enclosures for the reachable set of nonlinear control systems

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Abstract

This work presents a general theory for the construction of a polyhedral outer approximation of the reachable set (“polyhedral bounds”) of a dynamic system subject to time-varying inputs and uncertain initial conditions. This theory is inspired by the efficient methods for the construction of interval bounds based on comparison theorems. A numerically implementable instance of this theory leads to an auxiliary system of differential equations which can be solved with standard numerical integration methods. Meanwhile, the use of polyhedra provides greater flexibility in defining tight enclosures on the reachable set. These advantages are demonstrated with a few examples, which show that tight bounds can be efficiently computed for general, nonlinear systems. Further, it is demonstrated that the ability to use polyhedra provides a means to meaningfully distinguish between time-varying and constant, but uncertain, inputs.

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Acknowledgments

The authors thank Novartis Pharmaceuticals as part of the Novartis-MIT Center for Continuous Manufacturing for funding this research, and Garrett Dowdy for some suggestions on the technical results.

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Correspondence to Paul I. Barton.

Appendix: Parametric affine relaxations

Appendix: Parametric affine relaxations

This section discusses a method for constructing affine relaxations of the dynamics which satisfy Assumption 4. The method relies on Proposition 5.1 in [17], which is repeated below. It is a general result, and to implement it, one needs specific relaxations of various arithmetic operations. Such operations are listed in Table 1 below.

Proposition 2

(Proposition 5.1 in [17]) Let \(m, n \in \mathbb {N}\). Let \(Z \subset \mathbb {R}^n\) be a nonempty open set, and let \(Y \subset \mathbb {R}^m\). Let \(\mathbf {g} : Z \rightarrow \mathbb {R}^m\) and \(f : Y \rightarrow \mathbb {R}\). Define \({Z}^{\mathbb {I}} = \{ (\mathbf {v},\mathbf {w}) \in \mathbb {R}^n \times \mathbb {R}^n : \mathbf {v}\le \mathbf {w}, [\mathbf {v}, \mathbf {w}] \subset Z \}\), and similarly define \(Y^{\mathbb {I}}\). For \(i \in \{ 1, \dots , m\}\), let \(\mathbf {g}_i^{al}\) and \(\mathbf {g}_i^{au}\) be locally Lipschitz continuous mappings \({Z}^{\mathbb {I}} \rightarrow \mathbb {R}^n\) and \(g_i^{bl}, g_i^{bu}, g_i^L, g_i^U\) be locally Lipschitz continuous mappings \({Z}^{\mathbb {I}} \rightarrow \mathbb {R}\) which satisfy

$$\begin{aligned} \mathbf {g}_i^{al}(\mathbf {v},\mathbf {w}) ^{ T }\mathbf {z}+ g_i^{bl}(\mathbf {v},\mathbf {w})&\le g_i(\mathbf {z}) \le \mathbf {g}_i^{au}(\mathbf {v},\mathbf {w}) ^{ T }\mathbf {z}+ g_i^{bu}(\mathbf {v},\mathbf {w}),\, \forall \mathbf {z}\in [\mathbf {v},\mathbf {w}], \\ g_i^L(\mathbf {v},\mathbf {w})&\le g_i(\mathbf {z}) \le g_i^U(\mathbf {v},\mathbf {w}),\, \forall \mathbf {z}\in [\mathbf {v},\mathbf {w}], \\&\quad [\mathbf {g}^L(\mathbf {v},\mathbf {w}), \mathbf {g}^U(\mathbf {v},\mathbf {w})] \subset Y, \end{aligned}$$

for all \((\mathbf {v},\mathbf {w}) \in {Z}^{\mathbb {I}}\).

Table 1 Some arithmetic operations f and their parameterized affine relaxations on \([\mathbf {y}^L, \mathbf {y}^U]\)

Let \(\mathbf {f}^{al}\) and \(\mathbf {f}^{au}\) be locally Lipschitz continuous mappings \({Y}^{\mathbb {I}} \rightarrow \mathbb {R}^m,\) and \(f^{bl}\) and \(f^{bu}\) be locally Lipschitz continuous mappings \({Y}^{\mathbb {I}} \rightarrow \mathbb {R}\) which satisfy

$$\begin{aligned} \mathbf {f}^{al}(\mathbf {v}',\mathbf {w}') ^{ T }\mathbf {y}+ f^{bl}(\mathbf {v}',\mathbf {w}') \le f(\mathbf {y})&\le \mathbf {f}^{au}(\mathbf {v}',\mathbf {w}') ^{ T }\mathbf {y}+ f^{bu}(\mathbf {v}',\mathbf {w}'),\, \forall \mathbf {y}\in [\mathbf {v}',\mathbf {w}'], \end{aligned}$$

for all \((\mathbf {v}',\mathbf {w}') \in {Y}^{\mathbb {I}}\).

Let \(h : Z \rightarrow \mathbb {R}\) be defined by \(h(\mathbf {z}) = f( \mathbf {g}(\mathbf {z}) )\). For \(i \in \{1, \dots , m\}\), let

$$\begin{aligned} \mathbf {h}_i^{al}(\mathbf {v},\mathbf {w})&= {\left\{ \begin{array}{ll} f_i^{al}(\mathbf {g}^L(\mathbf {v},\mathbf {w}), \mathbf {g}^U(\mathbf {v},\mathbf {w})) \mathbf {g}_i^{al}(\mathbf {v},\mathbf {w}) &{}\quad \hbox {if} f_i^{al}(\mathbf {g}^L(\mathbf {v},\mathbf {w}), \mathbf {g}^U(\mathbf {v},\mathbf {w})) \ge 0, \\ f_i^{al}(\mathbf {g}^L(\mathbf {v},\mathbf {w}), \mathbf {g}^U(\mathbf {v},\mathbf {w})) \mathbf {g}_i^{au}(\mathbf {v},\mathbf {w}) &{} \quad \hbox {otherwise}, \end{array}\right. } \\ h_i^{bl}(\mathbf {v},\mathbf {w})&= {\left\{ \begin{array}{ll} f_i^{al}(\mathbf {g}^L(\mathbf {v},\mathbf {w}), \mathbf {g}^U(\mathbf {v},\mathbf {w})) g_i^{bl}(\mathbf {v},\mathbf {w}) &{} \quad \hbox {if} f_i^{al}(\mathbf {g}^L(\mathbf {v},\mathbf {w}), \mathbf {g}^U(\mathbf {v},\mathbf {w})) \ge 0, \\ f_i^{al}(\mathbf {g}^L(\mathbf {v},\mathbf {w}), \mathbf {g}^U(\mathbf {v},\mathbf {w})) g_i^{bu}(\mathbf {v},\mathbf {w}) &{} \quad \hbox {otherwise}, \end{array}\right. }\\ \mathbf {h}_i^{au}(\mathbf {v},\mathbf {w})&= {\left\{ \begin{array}{ll} f_i^{au}(\mathbf {g}^L(\mathbf {v},\mathbf {w}), \mathbf {g}^U(\mathbf {v},\mathbf {w})) \mathbf {g}_i^{au}(\mathbf {v},\mathbf {w}) &{} \quad \hbox {if } f_i^{au}(\mathbf {g}^L(\mathbf {v},\mathbf {w}), \mathbf {g}^U(\mathbf {v},\mathbf {w})) \ge 0, \\ f_i^{au}(\mathbf {g}^L(\mathbf {v},\mathbf {w}), \mathbf {g}^U(\mathbf {v},\mathbf {w})) \mathbf {g}_i^{al}(\mathbf {v},\mathbf {w}) &{} \quad \hbox {otherwise}, \end{array}\right. } \\ h_i^{bu}(\mathbf {v},\mathbf {w})&= {\left\{ \begin{array}{ll} f_i^{au}(\mathbf {g}^L(\mathbf {v},\mathbf {w}), \mathbf {g}^U(\mathbf {v},\mathbf {w})) g_i^{bu}(\mathbf {v},\mathbf {w}) &{} \quad \hbox {if } f_i^{au}(\mathbf {g}^L(\mathbf {v},\mathbf {w}), \mathbf {g}^U(\mathbf {v},\mathbf {w})) \ge 0, \\ f_i^{au}(\mathbf {g}^L(\mathbf {v},\mathbf {w}), \mathbf {g}^U(\mathbf {v},\mathbf {w})) g_i^{bl}(\mathbf {v},\mathbf {w}) &{}\quad \hbox {otherwise}. \end{array}\right. } \end{aligned}$$

Let \(\mathbf {h}^{al}, \mathbf {h}^{au} : {Z}^{\mathbb {I}} \rightarrow \mathbb {R}^n\) and \(h^{bl}, h^{bu} : {Z}^{\mathbb {I}} \rightarrow \mathbb {R}\) be defined by

$$\begin{aligned}&\mathbf {h}^{al}(\mathbf {v},\mathbf {w}) = \sum _i \mathbf {h}_i^{al}(\mathbf {v},\mathbf {w}),&h^{bl}(\mathbf {v},\mathbf {w}) = f^{bl}(\mathbf {g}^L(\mathbf {v},\mathbf {w}), \mathbf {g}^U(\mathbf {v},\mathbf {w})) + \sum _i h_i^{bl}(\mathbf {v},\mathbf {w}), \\&\mathbf {h}^{au}(\mathbf {v},\mathbf {w}) = \sum _i \mathbf {h}_i^{au}(\mathbf {v},\mathbf {w}),&h^{bu}(\mathbf {v},\mathbf {w}) = f^{bu}(\mathbf {g}^L(\mathbf {v},\mathbf {w}), \mathbf {g}^U(\mathbf {v},\mathbf {w})) + \sum _i h_i^{bu}(\mathbf {v},\mathbf {w}). \end{aligned}$$

Then, \(\mathbf {h}^{al}, \mathbf {h}^{au}, h^{bl}, h^{bu}\) are locally Lipschitz continuous mappings on \({Z}^{\mathbb {I}}\) which satisfy

$$\begin{aligned} \mathbf {h}^{al}(\mathbf {v},\mathbf {w}) ^{ T }\mathbf {z}+ h^{bl}(\mathbf {v},\mathbf {w}) \le h(\mathbf {z}) \le \mathbf {h}^{au}(\mathbf {v},\mathbf {w}) ^{ T }\mathbf {z}+ h^{bu}(\mathbf {v},\mathbf {w}),\, \forall \mathbf {z}\in [\mathbf {v},\mathbf {w}], \end{aligned}$$

for all \((\mathbf {v},\mathbf {w}) \in {Z}^{\mathbb {I}}\).

Constructing relaxations which satisfy Assumption 4 proceed by recursively applying Proposition 2, illustrated by the following example.

Example 1

This example demonstrates how to construct affine relaxations for a simple function (in fact, this function is part of the dynamics of the Lotka–Volterra problem in Sect. 5.1). Let \(f : \mathbb {R} \times \mathbb {R}^2 \ni (p,\mathbf {z}) \mapsto p z_1 (1 - z_2)\). As in the Lotka–Volterra problem, p is an uncertain parameter. The goal is to construct, for any \(p \in [p^L,p^U]\) such that \(p^L > 0\), affine relaxations of \(f(p,\cdot )\) on any interval \([\mathbf {z}^L,\mathbf {z}^U]\) such that \(z_1^L > 0\) and \(z_2^L > 1\) (in the context of Assumption 4, obtaining a nontrivial \(\widetilde{\mathbf {c}}_i^u\) is beneficial only when U is not an interval). Furthermore, one desires that these relaxations are locally Lipschitz continuous with respect to \((\mathbf {z}^L,\mathbf {z}^U)\).

The process resembles the construction of an interval enclosure of the range of f via interval arithmetic, and indeed part of the method involves interval arithmetic. Evaluation of f is broken down into a sequence of auxiliary variables called “factors,” which can be expressed as simple arithmetic operations on previously computed factors. An interval enclosure and affine relaxation of each factor can also be computed, and following the rules in Proposition 2 and Table 1, the affine relaxations will also be locally Lipschitz continuous in the manner desired. See Table 2 for the factored expression. Note that factor \(v_3\), corresponding to the parameter p, is initialized with the trivial affine relaxations \(\mathbf {0} ^{ T }\mathbf {z}+ p^L \le p \le \mathbf {0} ^{ T }\mathbf {z}+ p^U\). This ensures that the final relaxations obtained are valid for all \(p \in [p^L,p^U]\). Also, note that the restrictions \(z_1^L > 0, z_2^L > 1\), and \(p^L > 0\), simplify the evaluation and preclude the need to consider the different “branches” when constructing the affine relaxations for factors \(v_5\) and \(v_6\), as indicated in Proposition 2 (for example, this implies that \({1/2}(v_4^L + v_4^U) < 0\)). Although in general, the different cases must be taken into account.

The final factor, \(v_6\), gives the value of f, and thus one also has

$$\begin{aligned} (\mathbf {v}_6^{al})^{ T }\mathbf {z}+ v_6^{bl} \le f(p,\mathbf {z}) \le (\mathbf {v}_6^{au})^{ T }\mathbf {z}+ v_6^{bu} \end{aligned}$$

for all \((p,\mathbf {z}) \in [p^L,p^U] \times [\mathbf {z}^L,\mathbf {z}^U]\). However, by virtue of Proposition 2, \(\mathbf {v}_6^{al}, \mathbf {v}_6^{au}, v_6^{bl}, v_6^{bu}\) can be considered locally Lipschitz continuous functions with respect to \((\mathbf {z}^L,\mathbf {z}^U)\).

Table 2 Factored expression, corresponding interval enclosures, and corresponding affine relaxations for Example 1

More generally, Assumption 4 requires parametric relaxations which are continuous on \(T \times D_x^{\mathbb {I}}\). To include dependence on t, one could construct relaxations with respect to t as well, over the degenerate interval [tt]. Then, it is clear that the final relaxations would be locally Lipschitz continuous on \((T \times D_x)^{\mathbb {I}}\). The following lemmata shows that this yields the desired properties.

Lemma 6

Assume \(m, n, p \in \mathbb {N}\). Let \(C \subset \mathbb {R}^m\) be nonempty and compact and \(D \subset \mathbb {R}^n\) be nonempty. Let \(\mathbf {g} : C \times D \rightarrow \mathbb {R}^p\) be locally Lipschitz continuous. Then for all \(\mathbf {z}\in D\), there exists a neighborhood \(N(\mathbf {z})\) and \(L > 0\) such that for all \((\mathbf {y},\mathbf {z}_1), (\mathbf {y},\mathbf {z}_2) \in C \times N(\mathbf {z}) \cap D\)

$$\begin{aligned} \left\| \mathbf {g}(\mathbf {y},\mathbf {z}_1) - \mathbf {g}(\mathbf {y},\mathbf {z}_2) \right\| \le L \left\| \mathbf {z}_1 - \mathbf {z}_2 \right\| . \end{aligned}$$

Proof

Choose \(\mathbf {z}\in D\). For each \(\mathbf {y}\in C\), let \(N(\mathbf {y},\mathbf {z})\) be a neighborhood of \((\mathbf {y},\mathbf {z})\) such that \(\mathbf {g}\) is Lipschitz continuous on \(N(\mathbf {y},\mathbf {z}) \cap (C \times D)\), with corresponding Lipschitz constant \(L(\mathbf {y})\). However, this collection of open sets forms an open cover of \(C \times \{ \mathbf {z}\}\), which is compact, and thus we can choose a finite number of these neighborhoods \(\{ N(\mathbf {y}_i, \mathbf {z}) : 1 \le i \le k\}\), such that their union, \(\widetilde{N}\), contains \(C \times \{ \mathbf {z}\}\). Let L be the (finite) maximum of the corresponding Lipschitz constants (i.e. \(L = \max \{ L(\mathbf {y}_i) : 1 \le i \le k \}\)). Note that \(\widetilde{N}\) is an open set, and \(\mathbf {g}\) is Lipschitz continuous on \(\widetilde{N} \cap (C \times D)\) with Lipschitz constant L.

We claim that there exists a \(\delta > 0\) such that \(C \times N_{\delta }(\mathbf {z}) \subset \widetilde{N}\) (where \(N_{\delta }(\mathbf {z})\) is viewed as a subset of \(\mathbb {R}^n\)). This follows from, for instance, Lemma 1 in Section 5 of [10]. The argument is that the complement of \(\widetilde{N}, \widetilde{N}^C\), is closed and disjoint from \(C \times \{\mathbf {z}\}\), and so there exists a \(\delta > 0\) such that the distance between any point in \(C \times \{ \mathbf {z}\}\) and any point in \(\widetilde{N}^C\) is greater than \(\delta \). This implies that \(C \times N_{\delta }(\mathbf {z})\) is disjoint from \(\widetilde{N}^C\), which in turn implies \(C \times N_{\delta }(\mathbf {z}) \subset \widetilde{N}\). The result follows from Lipschitz continuity on \((C \times N_{\delta }(\mathbf {z}) ) \cap ( C \times D) = C \times N_{\delta }(\mathbf {z}) \cap D\). \(\square \)

Lemma 7

Assume \(n, p \in \mathbb {N}\). Let \(T \subset \mathbb {R}\) be nonempty and compact and \(D \subset \mathbb {R}^n\) be nonempty. Define \((T \times D)^{\mathbb {I}} = \{ (s,\mathbf {v},t,\mathbf {w}) : s\le t, \mathbf {v}\le \mathbf {w}, [s,t] \times [\mathbf {v},\mathbf {w}] \subset T \times D \}\) and define \(D^{\mathbb {I}}\) similarly. Let \(\widetilde{\mathbf {g}} : (T \times D)^{\mathbb {I}} \rightarrow \mathbb {R}^p\) be locally Lipschitz continuous. Define \(\mathbf {g} : T \times D^{\mathbb {I}} \rightarrow \mathbb {R}^p\) by \(\mathbf {g}(t,\mathbf {v},\mathbf {w}) = \widetilde{\mathbf {g}}(t,\mathbf {v},t,\mathbf {w})\). Then \(\mathbf {g}\) is continuous, and for all \((\mathbf {v},\mathbf {w}) \in D^{\mathbb {I}}\) there exists a neighborhood \(N(\mathbf {v},\mathbf {w})\) and \(L > 0\) such that for all \((t,\mathbf {v}_1,\mathbf {w}_1), (t,\mathbf {v}_2,\mathbf {w}_2) \in T \times N(\mathbf {v},\mathbf {w}) \cap D^{\mathbb {I}}\)

$$\begin{aligned} \left\| \mathbf {g}(t, \mathbf {v}_1, \mathbf {w}_1) - \mathbf {g}(t,\mathbf {v}_2, \mathbf {w}_2) \right\| \le L \left\| (\mathbf {v}_1, \mathbf {w}_1) - (\mathbf {v}_2, \mathbf {w}_2) \right\| . \end{aligned}$$

Proof

The mapping \(\mathbf {h} : (t,\mathbf {v},\mathbf {w}) \mapsto (t, \mathbf {v}, t, \mathbf {w})\) is Lipschitz continuous, and so \(\mathbf {g}\), as the composition of locally Lipschitz \(\widetilde{\mathbf {g}}\) and \(\mathbf {h}\) is locally Lipschitz continuous on \(T \times D^{\mathbb {I}}\). Applying Lemma 6 we obtain the desired result. \(\square \)

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Harwood, S.M., Barton, P.I. Efficient polyhedral enclosures for the reachable set of nonlinear control systems. Math. Control Signals Syst. 28, 8 (2016). https://doi.org/10.1007/s00498-015-0153-2

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