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Differentiators play an important role in (continuous) feedback control systems. In particular, the robust and exact second-order differentiator has shown some very interesting properties and it has been used successfully in sliding mode control, in spite of the lack of a Lyapunov based procedure to design its gains. As contribution of this paper, we provide a constructive method to determine a differentiable Lyapunov function for such a differentiator. Moreover, the Lyapunov function is used to provide a procedure to design the differentiator’s parameters. Also, some sets of such parameters are provided. The determination of the positive definiteness of the Lyapunov function and negative definiteness of its derivative is converted to the problem of solving a system of inequalities linear in the parameters of the Lyapunov function candidate and also linear in the gains of the differentiator, but bilinear in both.

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Obtaining precise reducer order battery models is a key component in designing lithium-ion battery management systems. In general, with conventional order reduction techniques, low order is followed by low accuracy. To overcome such limitation, this paper presents a frequency-based constructive method to obtain broad-band low-order approximations of the solid-phase lithium-ion diffusion submodel, the core of the reference pseudo-two-dimensional electrochemical model of lithium-ion cells. The proposed method is a combination of an improvement on the residue grouping technique and of standard balancing and truncation/residualization methods borrowed from control systems theory. As main result, dramatic order reductions from 3000th to 5th-order with minimum accuracy losses are obtained. Taking a well-known case-study of a lithium-ion battery model, it is shown that 5th-order models offer approximations with maximum absolute errors of 0.5%, for characteristic times up to 9.54 × 10 4 s . The proposed method will help to obtain accurate real-time estimations of surface concentration and state of charge of lithium-ion batteries.