This document describes a study applying a recursive orthogonal least squares (ROLS) algorithm to develop a neural network model for a continuous fermenter system using a nonlinear model predictive control (NMPC) approach. The ROLS algorithm is used to update the weighting matrix to model the nonlinear, multivariable system dynamics. The NMPC controller using this model is tested on the fermenter by changing setpoints. It is shown to reach new setpoints faster than alternative controllers like IMC, PI, and FLC controllers.

1. CHEMCON-2014
IICHE
Date: 27-30 December 2014
*Arpit Jain
1
Arpit Jain1
*, Himanshu Tyagi1
, Akshay Jha2
, Imran Rahman3
1
IIT Roorkee, Roorkee-247667, India,
2
IIT Guwahati, Guwahati-781039, India
3
National Chemical Laboratory, Pune -411008, India
Introduction
This paper delves into the application of Recursive
Orthogonal Least Squares (ROLS) algorithm for multi-
input, single-output systems to update the weighting
matrix of a radial basis function network. Specifically, the
network model predicts the one step ahead value of the
control variable and this is used in the framework of
Newton-Raphson method to determine the control action.
The nonlinear model-predictive control (NMPC) using
above algorithm is applied on a continuous fermenter.
The training data was generated by exciting the system
using a Pseudo Random Binary Signal (PRBS). Its
results demonstrate its effectiveness in capturing the
dynamics of data associated with non-linear systems.
NMPC (ROLS) Algorithm
In neural modelling a NARX model is usually adopted to
represent non-linear dynamic systems. In the MIMO case
the following form of the NARX model is considered,
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
The RBF network model for this proposed model is
developed:
( ) ̂ ( ) ( ) ∑ ( ) ( )
Now, through ROLS algorithm we can update
weight matrix at each step [3] for the training data.
Newton-Rapshon method is used to update the value
of the manipulated variable as shown below.
( ) ( )
( )
( )
Case Study
A continuous fermenter consists of a stirred tank that is
equipped with an inlet for feed and outlet for output
stream. It is assumed that the volume of the fermenter is
constant and the contents are well mixed. In the
fermentation process, due to the uncertainty of
parameters, a general theory for design of nonlinear
feedback controller [2] is used. It is represented by the
following nonlinear ordinary differential equations.
̇
̇ ( ) (
⁄
)
̇ ( )
( ( ))
( ⁄ )
Nominal values of the model parameters and variables in
the above equations and operating conditions are shown
in the nomenclature.
Identification of system
The recursive orthogonal least square method has been
utilized for identifying the system representing the
dynamics of fermenter. Fro convenience the process
input-output data of the system was collected using the
phenomenological model. Specifically, the manipulated
variable is varied randomly using PRBS and its effect on
the controlled variable is monitored. A set of 1400 input-
output data is generated by an open loop simulation of
this continuous fermenter using a sample time of TSAMPLE
=0.03h and a switching probability of PS =0.05 with the
dilution rate varying randomly at each transition. Lags in
input and output are taken to be 2 and
Setpoint Tracking
The control objective is to maintain X (cell concentration)
at the specified set point. The NMPC (ROLS) controller
response for the changes in setpoint, from 6.0 g/l to 7.0
g/l and from 6.0 g/l to 5.0 g/l is shown in Figure 1 and 2
respectively. For the former case, the NMPC’s closed
loop cell concentration is attained within 4h (Fig 1)
whereas for the latter case, it is attained within 1.7 h (Fig
2)). A comparison is done in Table 1 of the performance,
in terms of time taken to reach the desired setpoint in the
NON-LINEAR MODEL PREDICTIVE CONTROLLER BASED ON RECURSIVE
ORTHOGONAL LEAST SQUARES

2. CHEMCON-2014
IICHE
Date: 27-30 December 2014
2
event of a setpoint change, of non-linear model-predictive
control scheme based on IMC along with PI control[4] as
well as FLC [3].
Figure 1. Response to NMPC(ROLS) to change in
setpoint from 6 g/l to 7 g/l
Figure 2. Response to NMPC (ROLS) to change
in setpoint from 6 g/l to 5 g/l
Table 1. Time to reach the setpoint (SP) for
different controllers
Controller SP- 7 g/l SP- 5 g/l SP- 6 g/l
1 NMPC(ROLS) 4 hr 1.7 hr 0 hr
2 IMC 6.5 hr 6.6 hr 15 hr
3 PI 27 hr 17.5 hr 30 hr
4 FLC 4 hr 5 hr -
Conclusion
The paper presents a nonlinear model based control
algorithm and results of its application to fermenter. Here
the process model and control is developed using
recursive orthogonal least square method. In terms of
closed loop performance, an improvement over the earlier
reported result was demonstrated .
References
[1]D.L.Yu, J.B. Gomm and D. Williams(1997), Neural
Processing Letters 5:167-176.
[2]E. P. Nahas, M. A. Henson and D. E.Seborg(1992),
Computers and Chemical Engineering, 16,12, 1039-1057.
[3]Imran Rahman, S.S. Tambe and B. D. Kulkarni,
(2003)ISPEC’03:322-327, IIT, Mumbai
[4]Qiuping Hu, Gade Pandu Rangaiah(1998),Chemical
Engineering Science,53, 3041-3049.
Nomenclature
1 Dilution rate (D) 0.202
2 Substrate inhibition constant (Ki) 22 g
3 Substrate saturation constant(Km) 1.2 g
4 Process or product concentration (P) 19.14 g
5 Substrate concentration in the outlet
stream (S)
5.0 g
6 Substrate concentration in the feed
(Sf)
20.0 g
7 Cell concentration (X) 6.0 g
8 Cell mass yield (Yx/s) 0.4 g/g
9 Kinetic parameter (α) 2.2 g/g
10 Kinetic parameter (β) 0.2
11 Maximum growth rate (µm) 0.48