Page 827 - Mechanical Engineers' Handbook (Volume 2)

818 Neural Networks in Feedback Control Systems

applications in closed-loop control are fundamentally different from open-loop applications
such as classification and image processing. The basic multilayer NN tuning strategy is
backpropagation. 17 Basic problems that had to be addressed for closed-loop NN control 33,44
included weight initialization for feedback stability, determining the gradients needed for
backpropagation tuning, determining what to backpropagate, obviating the need for prelim-
inary off-line tuning, modifying backprop so that it tunes the weights forward through time,
and providing efficient computer code for implementation. These issues have since been
addressed by many approaches.
Initial work in NN was for system identification and identification-based indirect control.
In closed-loop control applications, it is necessary to show the stability of the tracking error
as well as boundedness of the NN weight estimation errors. Proofs for internal stability,
bounded NN weights (e.g., bounded control signals), guaranteed tracking performance, and
robustness were absent in early works. Uncertainty as to how to initialize the NN weights
led to the necessity for ‘‘preliminary off-line tuning.’’ Work on off-line learning was for-
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malized by Kawato. Off-line learning can yield important structural information.
Subsequent work in NNs for control addressed closed-loop system structure and stability
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issues. Work by Sussmann and Albertini and Sontag was important in determining system
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properties of NNs (e.g., minimality and uniqueness of the ideal NN weights, observability
of dynamic NNs). The seminal work of Narendra and Parthasarathy 9,10 had an emphasis on
finding the gradients needed for backprop tuning in feedback systems, which, when the plant
dynamics are included, become recurrent nets. In recurrent nets, these gradients themselves
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satisfy difference or differential equations, so they are difficult to find. Sadegh showed that
knowing an approximate plant Jacobian is often good enough to guarantee suitable closed-
loop performance.
The approximation properties of NN 2,3 are basic to their feedback controls applications.
Based on this and analysis of the error dynamics, various modifications to backprop were
presented that guaranteed closed-loop stability as well as weight error boundedness. These
are akin to terms added in adaptive control to make algorithms robust to high-frequency
unmodeled dynamics. Sanner and Slotine used radial basis functions in control and showed
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how to select the NN basis functions, Polycarpou and Ioannou 56,57 used a projection method
for weight updates, and Lewis and Syrmos 37 used backprop with an e-modification term. 18
All this work used NNs that are linear in the unknown parameter. In linear NNs, the problem
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is relegated to determining activation functions that form a basis set (e.g., RBF and func-
5
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tional link programmable network (FLPN) ). It was shown by Sanner and Slotine how to
systematically derive stable NN controllers using approximation theory and basis functions.
Barron 58 has shown that using NNs that are linear in the tunable parameters gives a funda-
mental limitation of the approximation accuracy to the order of 1/L 2/ n , where L is the number
of hidden layer neurons and n is the number of inputs. Nonlinear-in-the-parameters NNs
overcome this difficulty and were first used by Chen and Khalil, who used backprop with
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deadzone weight tuning, and Lewis et al., who used Narendra’s e-modification term in the
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backprop. In nonlinear-in-the-parameters NNs, the basis is automatically selected online by
tuning the first-layer weights and thresholds. Multilayer NNs were rigorously used for
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discrete-time control by Jagannathan and Lewis. Polycarpou derived NN controllers that
do not assume known bounds on the ideal weights. Dynamic/recurrent NNs were used for
control by Rovithakis and Christodoulou, 63 Poznyak, 64 Rovithakis, 65 who considered multi-
plicative disturbances; Zhang and Wang, 66 and others.
Most stability results on NN control have been local in nature, and global stability has
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been treated by Kwan et al., and others. Recently, NN control has been used in conjunction
with other control approaches to extend the class of systems that yields to nonparametric
control. Calise and coworkers 30,68,69 used NNs in conjunction with dynamic inversion to