1. Introduction
In almost every industrial application, control of process variables is important for the efficient and safe working of the process. The most recurrent variables which need to be controlled in industries include temperature, level, pressure, and pH [
1]. A typical control system, or rather a good controller, is expected to make the process efficient, suppress the impact of external disturbances on the process, make the process stable, and optimize the entire process. In industries, most of the processes are nonlinear [
2]. To cope with such heavy nonlinearity, the controller is expected to exhibit robustness. Over the last four decades, the proportional integral derivative (PID) controller has held a significant position in process control. This is because it is inexpensive and easy to tune [
3]. Another advantage of the PID is its fixed structure [
4]. The classical methods like those of Ziegler–Nichols and Cohen–Coon focus on manual tuning of the PID controller parameters. Also, there are different PID tuning methods in the frequency domain for open loop stable system without transportation lag or dead time [
5]. Dead time is defined as the time lag in process variable response when the controller output signal is applied. The presence of dead time is undesirable in the control loop. In practical implementation, manual tuning becomes arduous [
6]. In spite of the simple structure, optimal tuning of PID gain parameters is relatively difficult [
7,
8,
9].
To avoid this, various meta-heuristic algorithms are available which make the optimal tuning of PID gains quite easier [
10]. This means we can achieve the desired performance specifications. There are bio-inspired optimization algorithms which can be used to tune the PID controller parameters [
11]. In spite of being tuned using such fine algorithms, the PID controller fails considerably in certain aspects. This is because the PID controller does not give satisfactory results with systems which need setpoint tracking and disturbance rejection simultaneously [
12]. Moreover, it is linear and its performance with nonlinear systems deteriorates. It is mentioned that though the PID controller is widely used because of its simplicity and it cannot meet the requirements needed for better performances unless the controller is tuned properly [
13]. Since PID controllers are fixed gain controllers, they cannot compensate the parameter variation in the plant and cannot adapt to changes in the environment [
14]. Processes with large dead time, inverse response, and nonlinear processes are difficult to control efficiently by the PID [
15]. In PID, there is no direct relationship between the three tuning parameters and setpoint tracking, disturbance rejection ability, and robustness. Therefore, achieving each performance attributes by individual tuning of PID is not straightforward. To overcome this problem, the model predictive controller (MPC) is widely used in industries which employ nonlinear process in particular. It is often implemented in supervisory mode hierarchically above the base PID controller. Therefore, its performance again primarily depends on the PID in the lower level. The PID has to implement the commands received from MPC [
15,
16]. MPC is also implemented as a direct control algorithm, mainly where classical PID structures cannot deliver required control performance, due to difficult dynamics, strong interactions, and active constraints [
17]. The MPC utilises a distinct model of the process to predict the future response of the process. The disturbance rejection offered by MPC is much better when compared to that of PID. However, the MPC suffers from serious drawbacks in that it is complicated and its parameters are tough to calculate [
18].
An alternative control strategy to overcome the shortcomings of the existing controllers was proposed by Kapil Mukati which intended on combining the positive attributes of the PID controller and MPC [
12]. The controller which was proposed was named as RTD-A because a good controller should possess all of the following characteristics, namely: robustness (R), set-point tracking (T), disturbance rejection (D), and overall aggressiveness (A). All the four tuning parameters are normalized to lie within the range of 0 to 1 [
15,
19]. This four factor control technique which was developed aimed at exploiting the modern electronic hardware parts. In addition, focussed at imbibing the best of the qualities of a good controller. Another attempt was made in a paper to analyse and compare the potential of MPC, PID, RTD-A, and cascade controllers using isoflurane with the bispectral index as the controlled variable [
20,
21,
22]. In other work, a fuzzy scheduled RTD-A controller was designed and tested for different nonlinear processes. Some of those were control of pH process and Type I diabetic process [
23]. Later, the RTD-A controller was designed for controlling the industrial process using the second order process with dead time (SOPDT) model. Also, the performance of RTD-A was compared with internal model control (IMC), MPC, and PID. The PID can be tuned using a particle swarm optimization algorithm for DC motor speed control and also it is tuned by fuzzy logic. The overall closed loop performance can be improved by quick optimal tuning of controller [
24].
In this paper, the RTD-A controller parameters
θR,
θT,
θD, and
θA are found out by different meta-heuristic algorithms. In order to find out the best optimization technique available, initially, a comparison study was made among eight different optimization techniques namely the genetic algorithm (GA), simulated annealing (SA), the gravitational search algorithm (GSA) [
25], ant colony optimization (ACO), bacterial foraging (BF), cuckoo search (CS), harmonic search (HS), and grey wolf optimization (GWO) [
26]. These algorithms were used to tune the gains of the PID controller and its performance was analysed for a linear process. Here, grey wolf optimization (GWO) and the genetic algorithm (GA) were used individually to find the RTD-A controller tuning parameters. These two algorithms were chosen after a comparative analysis was made between eight different heuristic optimization techniques. The complete study is presented in
Section 4.1.1 and experimental results given in
Section 5.1. This tuned controller was later used to control the level of the conical tank and first-order model with dead time (FOPDT) process. The gain scheduling was done because it is needed in industrial processes whose dynamics keep changing with variables [
27].
The contribution of this paper, in particular, is to demonstrate that RTD-A controller parameters tuned using grey wolf optimization (GWO) exhibit peerless performance in controlling nonlinear processes. This means that the tuning of the parameters will become easier and a robust controller is developed which can handle any disturbances and setpoint changes [
15].
The paper is organized in the following way:
Section 2 describes the RTD-A controller basics,
Section 3 briefs about the optimization techniques used, the process considered and its description is given in
Section 4 and the results and conclusion are mentioned in
Section 5 and
Section 6, respectively.
2. RTD-A Controller Basics
The RTD-A control strategy comprises of the following three major components [
15].
Prediction of process output
Updating of model predictions
Calculation of control action.
According to the actual RTD-A scheme, any process be it linear or nonlinear, firstly has to be represented by a simple first-order with time delay process (FOPDT). The non-linear process can be approximated into linear model using Taylor’s series expansion [
28].
This FOPDT model can be generalized as:
where
K is process gain,
is dead time or transportation lag,
is time constant,
u(
s) is process input, and
y(
s) is process output. The Discrete model for Equation (1) is given in Equation (2).
where,
a =
e−Δt/τ, Δ
t is the sampling time,
b =
K(1 −
e−Δt/τ) and
m = round(
α/Δ
t)
From Equation (2), the output prediction for next
m steps can be written as in Equation (3).
where,
µ is the weighted sum of past control action taken during
m period and
u(
k) is the current control action. There will be only one control
u(
k) allowed from the current time instant and is given in Equation (4).
Equation (5) is obtained by considering the predicted process output over next
N-steps.
where
The idyllic model prediction requires model error
that explicitly decomposed into two parts such as effect of model mismatch
em(
k) and unmeasured disturbance
eD(
k) in Equation (8). Therefore, the total error is shown in Equation (7),
where
where
θR lies between 0 and 1. The controller has the robustness ability to handle the plant uncertainties. The future disturbance effect is predicted according to Equation (9).
for
, Where
Here
θD is the disturbance rejection controller tuning parameter. Finally, by solving the Equation (5) and Equation (9) the updated N-step model output prediction is given by Equation (10).
A plant with the setpoint trajectory
has the control action to minimize the error. At a discrete time interval, the control action will be updated. If the desired final output is considered as
yd(
k), then the desired trajectory to follow
y*(
k) is given by Equation (11).
The setpoint tracking tuning parameter
θT is introduced which varies from 0 to 1. The value of
N will be calculated by the controller aggressiveness tuning parameter
θA given in Equation (12).
The RTD-A controller output u(k) is given in Equation (13). The calculated controller output is applied to the plant.
Where
.
The RTD-A parameters are tuned using optimization techniques like genetic algorithm (GA), simulated annealing (SA) and grey wolf optimization (GWO).
The parameters are tuned in such a way that the performance index integral absolute error (IAE) is minimized. The error is the difference between the actual process output and the desired setpoint to be reached. This is obtained by considering the first-order linear model.
The block diagram representation of the basic RTD-A controller is shown in
Figure 1.
For the sake of simplicity, the main steps involved in the RTD-A control scheme is listed. They are:
- (1)
The depiction of the process model in terms of first-order dynamics.
- (2)
Determination of control system specifications. For example, firstly the sampling time, then the discretized form of model parameters, later, the computation of the tuning parameter for each controller performance attributes.
- (3)
Lastly, the calculation of control vector.
The entire block diagram can be explained in simple words in the following way. It consists of a controller and a plant. The setpoint
yd that has to be tracked or rather achieved. As the RTD-A controller is composed of a predictor, a model of the plant is present in the controller. The difference between the model’s output and the plant’s output is used to determine the robustness parameter. The plant output is measured using necessary sensors for monitoring which also provides feedback to the controller. The four basic blocks that are used to calculate the controller action
u(
k) are reference trajectory, control action calculator, current disturbance effect and future disturbance effect predictor. The plant may have low disturbance in the input side or output side and it may contain noise. The reference trajectory and the control action calculator block are in the forward loop. The current disturbance and future disturbance effect predictor form the feedback loop [
29].