Abstract
Optimal control problems naturally arise in several kinds of applications, including automotive systems. Unfortunately, the solution of such problems—which hinges upon a partial differential differential equation, the so-called Hamilton-Jacobi-Bellman (HJB) pde—might be hard or even impossible to determine in practice. Herein, introducing the notion of Dynamic Value function, we propose a novel technique that consists in the immersion of the given model into an extended state-space in which the solution may be defined in a constructive manner. This leads to a dynamic control law that approximates the optimal policy. The proposed approach is validated by means of a case study arising from the field of combustion engines, namely optimal control of the torque and the speed of a test bench.
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Notes
- 1.
Without loss of generality we suppose that P is a matrix algebraic \(\bar{P}\) solution. In fact, if the Eq. (4.12) holds for some \(\fancyscript{X} \subset \mathbb {R}^{n}\) the statement may be straightforwardly modified accordingly.
- 2.
The notation \((D_{\alpha }^{\varepsilon },\,V^{\varepsilon })\) and \(c^{\varepsilon }(x,\,\xi )\) describes the differential Eq. (4.15) and the function \(V(x,\,\xi )\) in (4.13) with \(\xi \) obtained as the solution of (4.15)–(4.19) and the corresponding approximation error defined as in Lemma 1, respectively.
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© 2014 Springer International Publishing Switzerland
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Sassano, M., Astolfi, A. (2014). Approximate Solution of HJBE and Optimal Control in Internal Combustion Engines. In: Waschl, H., Kolmanovsky, I., Steinbuch, M., del Re, L. (eds) Optimization and Optimal Control in Automotive Systems. Lecture Notes in Control and Information Sciences, vol 455. Springer, Cham. https://doi.org/10.1007/978-3-319-05371-4_4
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DOI: https://doi.org/10.1007/978-3-319-05371-4_4
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