Abstract
This paper reviews the Higher Order Nonlinear Units (HONUs) and their fundamental supervised sample-by-sample and batch learning algorithms for data-driven controller learning when only measured data are known about the plant. We recall recently introduced conjugate gradient batch learning for weakly nonlinear plant identification with HONUs and we compare its performance to classical Levenberg-Marquard (LM). Further, we recall recursive least square (RLS) adaptation and compare its performance to L-M learning both for plant approximation and controller tuning. Further, a model reference adaptive control (MRAC) strategy with efficient controller learning for linear and weakly nonlinear plants is proposed with static HONUs that avoids recurrent computations, and its potentials and limitations with respect to plant nonlinearity are discussed. Recently developed stability approach for recurrent HONUs and for closed control loops with linear plant and nonlinear (HONU) controller is recalled and discussed in connotation stability of the adaptive closed control loop.
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Abbreviations
- CG:
-
Conjugate Gradient algorithm
- CNU:
-
Cubic Neural Unit (HONU r = 3)
- \( {\mathbf{colx}} \) :
-
Long column vector of polynomial terms
- d :
-
Desired value (setpoint)
- LNU:
-
Linear Neural Unit (HONU r = 1)
- QNU:
-
Quadratic Neural Unit (HONU r = 2)
- k :
-
Discrete index of time
- L-M:
-
Levenberg-Marquardt batch learning algorithm
- n,m :
-
Length of vector \( {\mathbf{x}},\xi \)
- \( n_{y} ,n_{u} \) :
-
Length of recent history of y or u [samples]
- \( r,\gamma \) :
-
Order of polynomial nonlinearity (plant, controller)
- \( r_{o} \) :
-
Control input gain at plant input
- T :
-
Vector transposition
- u :
-
Control input
- \( {\mathbf{w}},{\mathbf{v}} \) :
-
Long row vectors of all neural weights (plant, controller)
- \( {\mathbf{x}},\xi \) :
-
Augmented input vector to HONU (plant, controller)
- \( \tilde{y} \) :
-
Neural output from HONU
- y :
-
Controlled output variable (measured)
- \( y_{ref} \) :
-
Reference model output
- e :
-
Error between real output and HONU
- \( e_{ref} \) :
-
Error between reference model and control loop
- \( \mu \) :
-
Learning rate
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Acknowledgements
Authors acknowledge support from the EU Operational Programme Research, Development and Education, and from the Center of Advanced Aerospace Technology (CZ.02.1.01/0.0/0.0/16_019/0000826), and the Japanese JSPS KAKENHI Grant Number 15J05402.
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Benes, P.M., Bukovsky, I., Vesely, M., Voracek, J., Ichiji, K., Homma, N. (2019). Framework for Discrete-Time Model Reference Adaptive Control of Weakly Nonlinear Systems with HONUs. In: Sabourin, C., Merelo, J.J., Madani, K., Warwick, K. (eds) Computational Intelligence. IJCCI 2017. Studies in Computational Intelligence, vol 829. Springer, Cham. https://doi.org/10.1007/978-3-030-16469-0_13
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