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replaced by a NN model with a known structure but a number of unknown parameters and a modeling
error component. With regard to the network nonlinearities, the unknown parameters may appear both
linearly and nonlinearly, changing the original issue into a nonlinear robust adaptive control problem.

3.1. Neural network and the control of dynamic nonlinear systems
The characteristic of neural networks is that they are quite parallel. They can speed up computations and
assist in the solving of issues that need much processing. Since NNs have nonlinear representations and can
respond to changes in the environment, they easily reflect physical conditions like industrial processes and
their control, whereas precise mathematical models are harder to construct.

One of the few theoretical frameworks for employing NNs for the controllability and stability of dynamical
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systems has been established by Levin and Narendra . Their research is limited to feedforward MLNs with
dynamic BP and nonlinear systems with full state information access. Figure 3 presents the proposed
architecture of the NNs. Equation (24) considers a system at a discrete-time index k,

where x(k) ∈ χ ⊂ R , u(k) ∈ U ⊂ R and f(0,0) = 0 so that x = 0 is an equilibrium. Conditions are given, in
r
n
Equation (25), under which the two following NNs can be trained to feedback linearize and stabilize the
system.

The results are extended to non-feedback linearizable systems. If the controllability matrix around the
origin has a full rank, a methodology and conditions for training a single NN to directly stabilize the system
around the origin have been devised. Narendra and Parthasarathy use NNs to create various
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identification and controller structures. Although the MLNs represent static nonlinear maps and the RNNs
represent nonlinear dynamic feedback systems, they suggest that the feedforward MLNs and RNNs are
comparable. They describe four network models of varying complexity for identifying and controlling
nonlinear dynamical systems using basic examples.

Sontag proposed an article where he tried to explore the capabilities and the ultimate limitations of
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alternative NN architectures . He suggests that NNs with two hidden layers may be used to stabilize
nonlinear systems in general. Intuitively, the conclusion contradicts NNs approximation theories, which
claim that single hidden layer NNs are universal approximators. Sontag’s solutions are based on the
description of the control issue as an inverse kinematics problem rather than an approximation problem.

In 1990, Barto drew an interesting parallel between connectionist learning approaches and those
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investigated in the well-established field of classical adaptive control. When utilizing NNs to address a
parameter estimate problem, the representations are frequently chosen based on how nervous systems
represent information. In contrast, in a traditional method, issue representation options are made based on
the physics of the problem. As opposed to conventional methods, a connectionist approach is dependent on
the structure of the network and the correlation between the connectionist weights. A traditional controller
may readily include a priori information; however, in NNs, it is often an input-output connection. In both