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Optimal model predictive control solution approximation using Takagi Sugeno for linear and a class of nonlinear systems

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Abstract

The universal approximation property of Takagi-Sugeno fuzzy systems is exploited here to build a fuzzy approximation of the optimal solution of linear and nonlinear model predictive control (MPC). The nonlinear systems considered are affine in the control law. The fuzzy approximator introduced presents some properties not generally shown by the previous approximators. In particular it is constituted by a set of state feedback control laws which are merged to obtain the final nonlinear control law. The constructed control law is very similar to the explicit solution of the linear MPC. The a posteriori stability is analysed based on a novel theoretical test previously published for both the linear and nonlinear MPC. Examples treated in the simulation part have shown the prompt and the good results which prove the effectiveness of the developed control strategy.

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Data availability

The authors confirm that the data supporting the findings of this study are available within the article, for further information contact the first author (hamzaboumaza.umc@gmail.com).

Abbreviations

\(x\left(t\right)\) :

State

\(u\left(t\right)\) :

Input

A :

State matrix

B :

Input matrix

\(F\left(x\left(t\right)\right)\) :

State dependent state matrix

\(G\left(x\left(t\right)\right), \) :

State dependent input matrix

\({J,J}_{N}\) :

Cost function

\(\mathcal{L}\left(.\right)\) :

Step cost function

\(\mathcal{F}\left(.\right)\) :

Final cost

\({\mathbb{U}}\) :

Input constraint set

\({\mathbb{X}}\) :

State constraint set

\(\mathcal{U},\) :

Control sequence

N :

Prediction horizon

Q :

State weight matrix

R :

Input weight matrix

P :

Riccatti solution weight matrix

G :

Grid

L :

Grid size

\({R}_{j}\) :

Fuzzy rule

\({F}_{ij}\) :

Fuzzy set

\({A}_{l}\) :

State matrix in fuzzy model

\({B}_{l}\) :

Control matrix in fuzzy model

\({\mu }_{{F}_{lj}}\left(.\right), \) :

Membership grade associated to\({F}_{ij}\)

\({h}_{j}\left(x(t)\right)\) :

Rule weight

\(\mathbf{A}\) :

The set of extreme matrices

\({\lambda }_{i}\) :

Extreme matrix weight

\({\mathbf{P}}_{q},{\mathbf{P}}_{q}^{^{\prime}}\) :

The set of all products of length q

\(\mathcal{S}\) :

Region

\(\Omega \) :

Approximation region

\({\overline{x} }^{{l}^{*}}, \) :

lth optimal trajectory sequence

\({\overline{u} }^{{l}^{*}}, \) :

Optimal control sequence

\({K}_{l},{K}_{l}^{*}, \) :

lth controller gain and lth optimal controller gain

\(\mathcal{K},{\mathcal{K}}^{*}\) :

Gain and optimal gain matrices

\({u}_{f}^{l}\left(t\right), \) :

Instantaneous controller output

\({\overline{u} }_{f}^{l}\) :

Sequence of controller output

\({\Phi }^{l*}\left(t\right)\) :

Instantaneous data matrix

\(\overline{\Phi }\) :

Overall data base

T :

Discrete time index

t c :

Continuous time index

k :

Prediction index

q :

Stability test

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Acknowledgements

The authors would like to thank the anonymous reviewers for their very useful comments on the original manuscript.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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Authors and Affiliations

  1. Université des Frères Mentouri Constantine 1, BP, 325 Route de Ain El Bey, 25017, Constantine, Algeria

    Hamza Boumaza

  2. Centre de Développement des Technologies Avancées, Cité 20 Août 1956, Baba Hassen, 16081, Algiers, Algeria

    Hamza Boumaza

  3. Ecole Nationale Polytechnique de Constantine Nouvelle ville Universitaire, Ali Mendjeli, Constantine, Algeria

    Khaled Belarbi

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  1. Hamza Boumaza
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1st author: H.Boumaza Phd student; 2nd author: Professor K.Belarbi thesis director.

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Correspondence to Hamza Boumaza.

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For this type of study no human or animals were involved, formal consent is not required.

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Boumaza, H., Belarbi, K. Optimal model predictive control solution approximation using Takagi Sugeno for linear and a class of nonlinear systems. Int. J. Dynam. Control 10, 1265–1278 (2022). https://doi.org/10.1007/s40435-021-00875-4

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