  Deep learning using neural networks has revolutionized machine learning and
put artificial intelligence into everyday life. In order to introduce
self-learning to dynamic systems other than neural networks, we extend the
Brandt-Lin learning algorithm of neural networks to a large class of dynamic
systems. This extension is possible because the Brandt-Lin algorithm does not
require a dedicated step to back-propagate the errors in neural networks. To
this end, we first generalize signal-flow graphs so that they can be used to
model nonlinear systems as well as linear systems. We then derive the extended
Brandt-Lin algorithm that can be used to adapt the weights of branches in
generalized signal-flow graphs. We show the applications of the new algorithm
by applying it to adaptive PID control. In particular, we derive a new
adaptation law for PID controllers. We verify the effectiveness of the method
using simulations for linear and nonlinear plants, stable as well as unstable
plants.
