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A Novel Identification-Based Convex Control Scheme via Recurrent High-Order Neural Networks: An Application to the Internal Combustion Engine

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Abstract

This paper proposes an identification-based nonlinear control scheme which casts the plant model as a recurrent high-order neural network. The model thus obtained consists on polynomials of a fixed number of nonlinearities, a fact that is exploited by transforming it into an exact tensor-product representation whose nested convex sums may increase with the network order while preserving the number of different interpolating functions. Convexity is then used along the direct Lyapunov method to find conditions for controller design in the form of linear matrix inequalities or sum-of-squares; thanks to the fixed number of nonlinearities, they can be made progressively more relaxed while preventing the computational burden usually associated with Pólya-like relaxations. The control law thus obtained is a generalization of the well-known parallel distributed compensation; its effectiveness is illustrated in academic examples and an internal combustion engine setup.

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Notes

  1. This is called a tensor-product model and can be embedded into a quasi-LPV one if the interpolating convex functions \(\omega _i(\cdot )\) are state-dependent; these functions are usually referred as weights, but this denomination is reserved in this paper to the RHONN synaptic weights [44, 45].

  2. Usually referred as the premise vector in Takagi–Sugeno (TS) fuzzy contexts [9].

  3. A polynomial p(x) is SOS if there exist polynomials \(f_i(x)\), \(i\in \{1,2,\ldots ,M\}\) such that \(p(x)=\sum _{i=1}^Mf^2_i(x)\).

  4. A RHONN of order r is complete if it contains every possible monomial in \(y_j\) of orders \(1,2,\ldots ,r\) in vector (2).

  5. If the inputs of a RHONN do not pass trough sigmoid functions \(u=\nu \); this is a common practice for affine-in-control systems [43].

  6. Indeed, it can be formulated as an output regulation problem via an exosystem \(\dot{\bar{w}}=0\), \(\bar{w}(0)=\eta _{ss}\), and mappings \(\gamma (\bar{w})=\nu _{ss}\), \(\pi (\bar{w})=\bar{w}\) [8]; convex control (Sect. 2.2) can still be applied [48, 49].

  7. Should the origin \(\eta _s=0\) not be in the compact set \({\mathcal {C}}\) due to x being mapped to \(\eta \) and further to \(\eta _s\), it can always be enlarged as to contain it. Recall that Lyapunov theory will otherwise not hold unless a neighbourhood (i.e., \({\mathcal {C}}\)) of the origin is provided.

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Acknowledgements

This work has been supported by the CONACYT scholarship 456977, the postdoctoral fellowship for CVU 366627, the CONACYT sabbatical fellowship for CVU 202355, the ITSON PFCE 2017/18, and the ECOS Nord SEP-CONACYT-ANUIES Project (Mexico 291309 / France M17M08).

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Authors and Affiliations

  1. Department of Electrical and Electronics Enginnering, Sonora Institute of Technology, Ciudad Obregón, Mexico

    Carlos Armenta, Víctor Estrada-Manzo & Miguel Bernal

  2. CNRS, UMR 8201 - LAMIH, Univ. Valenciennes, 59313, Valenciennes, France

    Thomas Laurain

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Correspondence to Miguel Bernal.

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Armenta, C., Laurain, T., Estrada-Manzo, V. et al. A Novel Identification-Based Convex Control Scheme via Recurrent High-Order Neural Networks: An Application to the Internal Combustion Engine. Neural Process Lett 51, 303–324 (2020). https://doi.org/10.1007/s11063-019-10095-9

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