Application of Heavy and Underestimated Dynamic Models in Adaptive Receding Horizon Control Without Constraints




Newton-Raphson Algorithm, Gradient Descent Algorithm, Reduced Gradient Algorithm, Receding Horizon Control, Fixed Point Iteration-based Adaptive Control


In the heuristic “Adaptive Receding Horizon Controller” (ARHC) the available dynamic model of the controlled system usually is placed in the role of a constraint under which various cost functions can be minimized over a horizon. A possible secure design can be making calculations for a “heavy dynamic model” that may produce high dynamical burden that is efficiently penalized by the cost functions and instead of the original nominal trajectory results a “deformed” one that can be realized by the controlled system of “less heavy dynamics”. In the lack of accurate system model a fixed point iterationbased adaptive approach is suggested for the precise realization of this deformed trajectory. To reduce the computational burden of the control the usual approach in which the dynamic model is considered as constraint and Lagrange-multipliers are introduced as co-state variables is evaded. The heavy dynamic model is directly built in the cost and the computationally greedy Reduced Gradient Algorithm is replaced by a transition between the simple and fast Newton-Raphson and the slower Gradient Descent algorithms (GDA). In the paper simulation examples are presented for two dynamically coupled van der Pol oscillators as a strongly nonlinear system. The comparative use of simple nondifferentiable and differentiable cost functions is considered, too.


A. Atinga, A. Wirtu, and J. K. Tar, “Adaptive receding horizon control for nonlinear systems exemplified by two coupled van der Pol oscillators,” in Proceedings of the IEEE 16th International Symposium on Applied Computational Intelligence and Informatics SACI 2022, Timisoara, Romania. IEEE, 2022, pp. 317–322

B. van der Pol, “Forced oscillations in a circuit with non-linear resistance (reception with reactive triode),” The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, vol. 7, no. 3, pp. 65–80, January, 1927.

R. Bellman, “Dynamic programming and a new formalism in the calculus of variations,” Proc. Natl. Acad. Sci., vol. 40, no. 4, pp. 231–235, April, 1954.

——, Dynamic Programming. Princeton Univ. Press, Princeton, N. J., 1957.

B. Armstrong, O. Khatib, and J. Burdick, “The explicit dynamic model and internal parameters of the PUMA 560 arm,” Proc. IEEE Conf. On Robotics and Automation 1986, pp. 510–518, 1986.

J. Richalet, A. Rault, J. Testud, and J. Papon, “Model predictive heuristic control: Applications to industrial processes,” Automatica, vol. 14, no. 5, pp. 413–428, September, 1978.

J. Lagrange, J. Binet, and J. Garnier, M´ecanique analytique (Analytical Mechanics) (Eds. J.P.M. Binet and J.G. Garnier). Ve Courcier, Paris, 1811.

J. Riccati, “Animadversiones in aequationes differentiales secundi gradus (observations regarding differential equations of the second order),”Actorum Eruditorum, quae Lipsiae publicantur, Supplementa,, vol. 8, pp. 66–73, 1724.

A. Laub, A Schur Method for Solving Algebraic Riccati Equations (LIDS-P 859 Research Report). MIT Libraries, Document Services, After 1979.

R. Kalman, “Contribution to the theory of optimal control,” Boletin Sociedad Matematica Mexicana, vol. 5, no. 1, pp. 102–119, April, 1960.

S. Boyd, L. Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in Systems and Control Theory. SIAM books, Philadelphia, 1994.

W. Hamilton, “On a general method in dynamics,” Philosophical Transactions of the Royal Society, part II for 1834, pp. 247–308, 1834.

——, “Second essay on a general method in dynamics,” Philosophical Transactions of the Royal Society, part I for 1835, pp. 95–144, 1835.

V. Arnold, Mathematical Methods of Classical Mechanics. Springer - Verlag, 1989.

L. S. Lasdon, A. D. Waren, A. Jain, and M. Ratner, “Design and testing of a generalized reduced gradient code for Nonlinear Programming,” ACM Transactions on Mathematical Software (TOMS), vol. 4, no. 1, pp. 34–50, March, 1978.

D. Fylstra, L. Lasdon, J. Watson, and A. Waren, “Design and use of the Microsoft EXCEL Solver,” Interfaces, vol. 28, no. 5, pp. 29–55, September-October, 1998.

J. Gram, “U¨ ber die Entwickelung reeler Funktionen in Reihen mittelst der Methode der kleinsten Quadrate,” Journal f¨ur die reine und angewandte Mathematik, vol. 94, pp. 71–73, 1883.

E. Schmidt, “Zur Theorie der linearen und nichtlinearen Integralgleichungen. I. Teil: Entwicklung willk¨urlicher Funktionen nach Systemen vorgeschriebener,” Matematische Annalen, vol. 63, p. 442, December, 1907.

P.-S. Laplace, Th´eorie Analytique des Probabilit´es, in: Oeuvres completes de Laplace (Analytical Theory of Probabilities, in: Laplace’s Colleceted Works)/ publi´es sous les auspices de l’Acad´emie des sciences, par MM. les secr´etaires perp´etuel. Gauthier-Villars, Paris, 1886.

H. Issa and J. Tar, “Speeding up the Reduced Gradient Method for constrained optimization,” Proc. of the IEEE 19th World Symposium on Applied Machine Intelligence and Informatics (SAMI 2021), January 21-23, Herl’any, Slovakia, pp. 485–490, 2021.

H. Issa, H. Khan, and J. K. Tar, “Suboptimal adaptive Receding Horizon Control using simplified nonlinear programming,” in 2021 IEEE 25th International Conference on Intelligent Engineering Systems (INES), July 2021, pp. 000 221–000 228.

J. Raphson, Analysis aequationum universalis (Analysis of Universal Equations). typis TB prostant venales apud (printed for sale by) A. & I. Churchill, 1702.

T. J. Ypma, “Historical development of the Newton-Raphson method”, SIAM Review, vol. 37, no. 4, pp. 531–551, December, 1995.

B. Lantos and Z. Bod´o, “High level kinematic and low level nonlinear dynamic control of unmanned ground vehicles,” Acta Polytechnica Hungarica, vol. 16, no. 1, pp. 97–117, April, 2019

B. Csan´adi, P. Galambos, J. Tar, G. Gy¨or¨ok, and A. Serester, “A novel, abstract rotation-based fixed point transformation in adaptive control,” In the Proc. of the 2018 IEEE International Conference on Systems, Man, and Cybernetics (SMC2018), October 7-10, 2018, Miyazaki, Japan, pp. 2577–2582, 2018.

S. Banach, “Sur les op´erations dans les ensembles abstraits et leur application aux ´equations int´egrales (About the Operations in the Abstract Sets and Their Application to Integral Equations),” Fund. Math., vol. 3, pp. 133–181, 1922.

O. Rodrigues, “Des lois g´eometriques qui regissent les d´eplacements d’un syst´eme solide dans l’ espace, et de la variation des coordonn´ees provenant de ces d´eplacement consid´er´ees ind´ependent des causes qui peuvent les produire (Geometric laws which govern the displacements of a solid system in space: and the variation of the coordinates coming from these displacements considered independently of the causes which can produce them),” J. Math. Pures Appl., vol. 5, pp. 380–440, 1840




How to Cite

“Application of Heavy and Underestimated Dynamic Models in Adaptive Receding Horizon Control Without Constraints”, Syst. Theor. Control Comput. J., vol. 2, no. 2, pp. 1–8, Dec. 2022, doi: 10.52846/stccj.2022.2.2.36.