Abstract
This chapter focuses on analytical solutions for the powersplit control problem defined in the previous chapter. Two well known solution concepts will be considered: the method of Lagrange multipliers and Pontryagin’s Minimum Principle. Both concepts have in common that they obtain the necessary conditions for an optimal solution by evaluating where the first differential of the augmented objective function becomes equal to zero. Illustrative examples are used to introduce each solution concept. Next, analytical expressions will be derived that solve the optimal powersplit problem under consideration. This chapter ends with a summary for both solution concepts and demonstrates the coherence between their respective optimality conditions.
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- 1.
In this example the optimal solution will only be found if a 2+b 2≤d 2. If this condition is not satisfied, there will be no constraint active (i.e., the minimum of the function emerges outside the circle with radius d) and the trivial solution holds: miny=c.
- 2.
In the automotive industry λ is inextricable related to the ICE air/fuel ratio. It is desirable to use an alternative symbol and in this application area the symbol p is used instead of λ.
- 3.
Only in situations of firm braking we have P r ≪0 such that φ 1 becomes non-negative.
- 4.
Note that mathematical conditions, e.g., F(t) should be continuous and has continuous partial derivatives, are neglected here for simplicity.
- 5.
Some mathematical background is neglected here, but in general this requires that J ∗ has continuous second partial derivatives and also the dynamical system equation f has continuous first derivatives.
- 6.
Also in this derivation some assumptions are left implicit.
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de Jager, B., van Keulen, T., Kessels, J. (2013). Analytical Solution Methods. In: Optimal Control of Hybrid Vehicles. Advances in Industrial Control. Springer, London. https://doi.org/10.1007/978-1-4471-5076-3_4
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