Abstract
In this paper, a Chattering Free Sliding Mode Control (CFSMC) with observer based adaptive Radial Basis Function Neural Network (RBFNN) has been designed for first-order transfer function model of temperature trajectory in a fixed bed reactor. The steady-state behavior and effect of different operating parameters such as feed velocity and temperature on the operation of the fixed bed reactor have been discussed. Due to RBFNN’s capability to map the nonlinear dynamics online through self-learning ability, it is combined with CFSMC to reduce the chattering behavior. The adaptive RBFNN has been used to approximate the nonlinear dynamic behavior of the fixed bed reactor. To predict the states of the system, high gain observer based on adaptive RBFNN has been used. Design parameter of the observer has been estimated using Hurwitz polynomial. The effect of neuron number on the mapping error and the effect of space discretization step on modeling error have also been discussed. To decrease the chattering generated by the Sliding Mode Controller (SMC) in the temperature trajectory tracking, an equivalent control term is neglected from the final controller. It has two main advantages: one is the reduction in chattering behavior which is the main drawback of SMC and the second is the reduction of the high gain requirement. The SMC is used to overcome against external disturbance, load variation, variation in key parameters and model mismatch. To make the simulation realistic, constraints have been applied to control input and input rate. For guaranteeing the system stability, Lyapunov theorem has been applied. To show the suitability of the hybrid controller, a comparison has been carried out between the hybrid and PID controller. To quantify the performance, Integral Time Weighted Absolute Error (ITAE) has been estimated. Under the condition of existing model errors and external disturbances, simulation study of the control of the fixed bed reactor shows that the hybrid control algorithm consisting of sliding mode control and observer-based adaptive RBFNN performs well both for tracking the temperature trajectory and reducing the chattering.
Nomenclature
- As
heat transfer surface area,m2
- B
width vector
concentration of plant ethyl acetate/observer, mol/m3
- cij
center vector
- Cp
specific heat of reaction, J/(mol. K)
- d
disturbance
diffusivity of ethyl acetate, m2/s
error of plant model and observer
- E
activation energy, J/(mol. K)
- E1
error function for neural network
model uncertainty in f
- fx
nonlinear term due to Arrhenius law
- G(s)
transfer function of PID
model uncertainty in g
- hj
Gaussian function
heat of reaction,J/mol
- k
reaction rate constant, s−1
- kmix
thermal conductivity of solution, W/(m. K)
- l
Hurwitz polynomial matrix
- L
length of reactor, m
- R
ideal gas constant, J/(mol. K)
sliding surface of plant model and observer
- T
time, s
temperature of plant/observer, K
setpoint temperature trajectory, K
equivalent sliding mode control of plant model and observer
feedback and robust control term
overall heat transfer coefficient, W/(m2.K)
- v
velocity of reactant, m/s
- vo
void fraction
volume of reactor,m3
- w
weight vector
- W
semi representation of temperature equation
- x
state vector
- y/ym
plant/RBF model output
- z
distance along the length of reactor, m
- Greek Symbol
momentum rate
robustness parameter
high gain parameter
chattering parameter
learning rate
Hurwitz polynomial parameter
tuning parameter
density of solution
- τ
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