Abstract
In a previous work, we explained how Euler’s method for computing approximate solutions of systems of ordinary differential equations can be used to synthesize safety controllers for sampled switched systems. We continue here this line of research by showing how Euler’s method can also be used for synthesizing safety controllers in a distributed manner. The global system is seen as an interconnection of two (or more) sub-systems where, for each component, the sub-state corresponding to the other component is seen as an “input”; the method exploits (a variant of) the notions of incremental input-to-state stability ( \(\delta \) -ISS) and ISS Lyapunov function. We illustrate this distributed control synthesis method on a building ventilation example.
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Notes
- 1.
- 2.
Given an initial point \(x\in R\), the induced control \(\sigma \) corresponds to a sequence of patterns \(\pi _{i_1},\pi _{i_2},\dots \) defined as follows: Since \(x\in R\), there exists a a point \(\tilde{x}_{i_1}\) with \(1\le i_1\le m\) such that \(x\in B(\tilde{x}_{i_1},\delta ^0)\); then using pattern \(\pi _{i_1}\), one has: \(\phi _{\pi _{i_1}}(k_{i_1}\tau ;x)\in R\). Let \(x'=\phi _{\pi _{i_1}}(k_{i_1}\tau ;x)\); there exists a point \(\tilde{x}_{i_2}\) with \(1\le i_2\le m\) such that \(x'\in B(\tilde{x}_{i_2},\delta ^0)\), etc.
- 3.
So \(T=T_1\times T_2\) with: \(T_1=B(c_1,\varDelta )\), \(T_2=B(c_2,\varDelta )\).
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Le Coënt, A., Alexandre dit Sandretto, J., Chapoutot, A., Fribourg, L., De Vuyst, F., Chamoin, L. (2017). Distributed Control Synthesis Using Euler’s Method. In: Hague, M., Potapov, I. (eds) Reachability Problems. RP 2017. Lecture Notes in Computer Science(), vol 10506. Springer, Cham. https://doi.org/10.1007/978-3-319-67089-8_9
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