Abstract:
In this dissertation, a new heuristic computational intelligence technique has been
developed for the solution for fractional order systems in engineering. These systems
are provided with generic ordinary linear and nonlinear differential equations
involving integer and non-integer order derivatives.
The design scheme consists of two parts, firstly, the strength of feed-forward artificial
neural network (ANN) is exploited for approximate mathematical modeling and
secondly, finding the optimal weights for ANN. The exponential function is used as
an activation function due to availability of its fractional derivative. The linear
combination of these networks defines an unsupervised error for the system. The error
is reduced by selection of appropriate unknown weights, obtained by training the
networks using heuristic techniques. The stochastic techniques applied are based on
nature inspired heuristics like Genetic Algorithm (GA) and Particle Swarm
Optimization (PSO) algorithm. Such global search techniques are hybridized with
efficient local search techniques for rapid convergence. The local optimizers used are
Simulating Annealing (SA) and Pattern Search (PS) techniques.
The methodology is validated by applying to a number of linear and nonlinear
fraction differential equations with known solutions. The well known nonlinear
fractional system in engineering based on Riccati differential equations and Bagley-
Torvik Equations are also solved with the scheme.
The comparative studies are carried out for training of weights for ANN networks
with SA, PS, GA, PSO, GA hybrid with SA (GA-SA), GA hybrid with PS (GA-PS),
PSO hybrid with SA (PSO-SA) and PSO hybrid with PS (PSO-PS) algorithms. It is
found that the GA-SA, GA-PS, PSO-SA and PSO-PS hybrid approaches are the best
stochastic optimizers. The comparison of results is made with available exact
solution, approximate analytic solution and standard numerical solvers. It is found
that in most of the cases the design scheme has produced the results in good
agreement with state of art numerical solvers. The advantage of our approach over
such solvers is that it provides the solution on continuous time inputs with finite
interval instead of predefine discrete grid of inputs. The other perk up of the scheme
in its simplicity of the concept, ease in use, efficiency, and effectiveness.