Particle Swarm Optimization of a Recurrent Neural Network Control for an Underactuated Rotary Crane with Particle Filter Based State Estimation - Developing Concepts in Applied Intelligence - page 52

for nonlinear function approximation, such as less computational cost and faster con-
vergence [9]. In this paper, attention is paid to the use of PSO as a training algorithm to
construct an RNN-based controller for a nonlinear system.
On the other hand, due to the fact that in dynamic system control the system state is
not always observable completely, state estimation is thus an important prerequisite to
make the operation possible, safe and economical. For example, sensing the velocities
in mechanical systems is hard or may require more sensors with high cost. We thus
assume that the velocity states of the system are not observed and hence use a particle
filter (PF) to estimate the velocities. Known as a stochastic estimation method, PF has
emerged as a very potential technique during the last decade as it is applicable for
nonlinear and non-Gaussian systems [10], [11].
In order to validate the method, this study considers a rotary crane to be the control
object, which is a reduced two degree-of-freedom model with underactuated behavior.
Underactuated systems, which are characterized by the fact that they process fewer ac-
tuators than degrees of freedom, have attracted much research during the last decades
due to it pervasive applications. Such systems generate interesting but difficult control
problems as they usually exhibit nonholonomic behavior and complex dynamics. As
widely used in industry, crane systems have been the subject of several studies, present-
ing various methods for load swing control including classical and fuzzy approaches
[12], [13]. The crane system being considered herein is hard to stabilize quickly and
also known to be a nonholonomic system [13]. As is well-known, the class of nonholo-
nomic systems cannot be asymptotically stabilized by continuous time-invariant static
feedback [14]. Several approaches have been introduced for such systems, presenting
time varying, discontinuous, dynamic state feedback control methods [14]. Generally,
these methods aim to deal with asymptotic stabilization, considering the ideal system
model while assuming that all system states are observed. This paper addresses the
control of such a system with supposing that there exist unknown states (velocities).

2 Nonlinear Control Using a Recurrent Neural Network and

Particle Swarm Optimization with Particle Filter Estimation

Considering a discrete, nonlinear, non-Gaussian system represented by:

x k = f k ( x k − 1 , η k ) (1)

y k = g k ( x k , ξ k ) (2)

where y k is the vector of observations, x k is the state vector at time k ( k = 0 , 1 , 2 ... );

f k ( . ) is the (known) state transition function, g k ( . ) is the (known) observation function;

η k and ξ k are noise vectors. The control task is to regulate the system from an arbitrary

initial point x init to a desired point x ref with an assumption that there exist unobserved
states. To do so, an RNN is used as a state-feedback controller which is optimized by
PSO and the latent states are estimated by a PF. The schematic of the control system is
shown in Fig. 1, which also includes a limiter for dealing with the constraints involving
to the control input ( | u |≤ u max or | u |≤ u max ) .