Abstract
We present a correct-by-design method of state-dependent control synthesis for linear discrete-time switching systems. Given an objective region R of the state space, the method builds a capture set S and a control which steers any element of S into R. The method works by iterated backward reachability from R. More precisely, S is given as a parametric extension of R, and the maximum value of the parameter is solved by linear programming. The method can also be used to synthesize a stability control which maintains indefinitely within R all the states starting at R. We explain how the synthesis method can be performed in a distributed manner. The method has been implemented and successfully applied to the synthesis of a distributed control of a concrete floor heating system with 11 rooms and \(2^{11}=2048\) switching modes.
Partly supported by EU project Cassting (FP7-601148).
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Notes
- 1.
This separability technique is made possible by the fact that the difference equation \(x_1(t+1)=f_1(x_1(t),x_2(t),u_1)\) (see Sect. 2.1) does not involve the control mode \(u_2\).
- 2.
Actually, we will consider in the examples that \((R_1+a)\) is a product of intervals of the form \([\ell -a,m]\) where the interval is extended only at its lower end, but the method is strictly identical.
- 3.
If \(x(t)\in R\), then \(x(t)\in r_{i,j}\) for some \((i,j)\in I_1\times I_2\), hence \(x(t+k)=f(x,\pi _{i,j})\in R\) for some \(k\le K\).
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Le Coënt, A., Fribourg, L., Markey, N., De Vuyst, F., Chamoin, L. (2016). Distributed Synthesis of State-Dependent Switching Control. In: Larsen, K., Potapov, I., Srba, J. (eds) Reachability Problems. RP 2016. Lecture Notes in Computer Science(), vol 9899. Springer, Cham. https://doi.org/10.1007/978-3-319-45994-3_9
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