Nonlinear processes are very common in process industries, and designing a stabilizing controller is always preferred to maximize the production rate. In this paper, tuning of PID controller for a class of time delayed stable and unstable process models using Particle Swarm Optimization (PSO) algorithm is discussed. The dimension of the search space is only three (
In process industries, many important realtime processing units such as Continuous Stirred Tank Reactor (CSTR), biochemical reactor, and spherical tank system are highly nonlinear in nature. Tuning of controllers to stabilize these nonlinear chemical process loops and impart adequate disturbance rejection is critical because of their complex nature. Based on the operating regions, most of the chemical loops exhibit stable and/or unstable steady states.
Controller tuning is an essential preliminary procedure in almost all industrial process control systems. In control literature, a number of controller structures are available to stabilize stable, unstable, and nonlinear processes [
Despite the significant developments in advanced process control schemes such as predictive control, internal model control, and sliding mode control, PID controllers are still widely used in industrial control application because of their structural simplicity, reputation, robust performance, and easy implementation [
Conventional controller tuning methods proposed by most of the researchers are model dependent, and they require a reduced order models such as firstorder or secondorder process model with time delay. Particularly for unstable systems, the tuning rule proposed for a particular reduced order model will not offer a fitting response for other models (higher order models, model with a positive or negative zero, model with a large delay time to process time constant ratio, etc.). Most of the classical PID tuning methods require numerical computations in order to get the best possible controller parameters. Due to these reasons, in recent years, heuristic algorithmbased controller tuning has greatly attracted the researchers.
From recent literature, it is observed that heuristic algorithmbased optimization procedures have emerged as a powerful tool for finding the solutions for variety of control engineering problems [
In this paper, PID controller parameter tuning is attempted using the PSO algorithm introduced by Kennedy and Eberhart [
The remaining part of the paper is organized as follows. Section
PID controller has a simple structure and is usually available as a packaged form. To perform well with the industrial process problems, the controller should have optimally tuned
Block diagram of closed loop control system.
The closed loop response of the above system can be expressed as
The final steadystate response
Figure
Block diagram of PID controller with prefilter.
Jung et al. reported that, when the filter time constant
Particle Swarm Optimization (PSO) technique, proposed by Kennedy and Eberhart [
In PSO algorithm, the number of parameters to be assigned is very few compared to other natureinspired algorithms. In this, a group of artificial birds is initialized with arbitrary positions
At iteration
In (
Further, for highdimensional problems, dynamical adjusting of inertia weight was introduced by many researchers which can increase the search capabilities of PSO. A review of inertia weight strategies in PSO is given chronologically in subsequent paragraphs.
Eberhart and Shi [
In this paper, we considered constant inertia weightbased velocity updation. The implementation of PSO has the following steps.
For a population size
Objective function values of particles are evaluated using the performance criteria for algorithm convergence.
The objective values obtained above for the initial particles of swarm are set as the initial
The new velocity for each particle is computed using (
The particle position is updated using (
If the stopping criteria are met, positions of particles represented by
The overall performance (speed of convergence, efficiency, and optimization accuracy) of PSO algorithm depends on Objective Function (OF), which monitors the optimization search. The OF is chosen to maximize the domain constrains or to minimize the preference constrains. During the search, without loss of generality, the constrained optimization problem minimizes a scalar function
The minimization problem of preference constrains can be mathematically expressed as
The majority of controller design problems are multiobjective in nature. Multiobjective optimization always provides improved result compared to singleobjective function. Without loss of generality, the multiobjective optimization problem simultaneously minimizes
Weighted sum method is widely adopted by most of the researchers, and it converts the multiobjective problem of minimizing the objectives into a scalar form [
In the control literature, there exist a number of weightedsumbased objective functions [
In many optimization cases, it is very difficult to satisfy all the considered constraints. Hence, there should be some negotiation between the preference constraint parameters without compromising the domain constraint [
Obtainable solutions from search universe
Search boundary for controller parameters is assigned as follows:
The controller design process is to find the optimal values for controller parameters form the search space that minimizes the considered objective function. Figure
PSObased PID controller design.
Figure
PSObased PID controller with prefilter design.
PSObased controller design procedure is developed with number of swarms
The firstorder stable process with the following transfer function model is considered [
PSObased PID tuning is proposed with the method as shown in Figure
Optimized PID parameters for Example

Best values  Iteration number 




0.5  1  58  0.6749  0.6525  0.0228 
2  35  0.6800  0.6415  0.0170  
3  41  0.6904  0.6965  0.0332  
 
0.75  1  45  0.6976  0.6597  0.1136 
2  43  0.8579  0.6827  0.0781  
3  52  0.5779  0.6413  0.0812  
 
1  1  44  0.4885  0.839  0.1013 
2  39  0.6781  0.9102  0.2954  
3  63  1.0119  0.7781  0.1377 
Convergence of particles with optimal solution.
Optimal controller parameters.
Table
Regulatory response for
Controller output for
Similar responses are obtained for the above process model for
Quantitative analysis for Example

Best values 



ISE  IAE 

0.5  1  0  5.5  5.5  9.395  3.065 
2  0  6.5  6.5  9.72  3.118  
3  0.004  3.9  8.5  8.245  2.871  
 
0.75  1  0.0055  4.3  6.6  9.191  3.030 
2  0  8.1  8.1  8.582  2.929  
3  0.0185  4.0  9.8  9.726  3.119  
 
1  1  0.13  2.6  9.5  5.682  2.384 
2  0.112  2.6  10  4.828  2.197  
3  0  6.5  6.5  6.606  2.570 
From Table
Process performance with various
The stable secondorder model discussed by Chiha et al. is considered [
Chiha et al. proposed a Multiobjective Ant Colony Optimization (MACO) algorithm and validated the result with ZieglerNichols (ZN), GA, and ACO tuned PID controller. In this work, we considered the PSO tuned PID controller, and the controller values and its performance measure values are presented in Tables
PID parameters for Example
Sl. Number  Method 




1  ZN  2.808  1.712  1.151 
2  GA  2.391  2.391  1.458 
3  ACO  2.4911  0.8158  1.3540 
4  MACO  2.808  1.021  1.668 
5  PSO (0.75)  1.452  0.4259  0.5084 
Performance measure for Example
Sl. Number  Method  % 


ISE 

1  ZN  31.59%  0.664  4.78  0.854 
2  GA  5.84%  0.676  3.63  0.797 
3  ACO  4.95%  0.701  5.90  0.809 
4  MACO  2.84%  0.700  3.00  0.772 
5  PSO (0.75)  0.00%  2.713  9.836  4.108 
Figure
Regulatory response for
Controller output for
The stable secondorder steadystate model of the bioreactor is presented below [
In this work, PSObased PID is proposed for the above process model with
Controller values for various
Sl. Number  Method 



ISE 

1  ZN 



1.782 
2  GA 



3.148 
3  MOPSO 



1.434 
4  PSO (0.75) 



1.603 
Process response for stable bioreactor model.
Graphical representation of ISE.
The unstable steadystate model of the bioreactor presented below is widely considered by the researchers [
The controller setting by the proposed method for various
Controller settings for Example

Best values  Iteration number 




0.5  1  77 



2  83 


 
3  59 




 
0.75  1  63 



2  76 


 
3  47 


 
 
1  1  62 



2  58 




3  83 



Performance measure values for Example

Best values 



ISE  IAE 

0.5  1  0.434  2.3  29.6  4.497  2.121 
2  0415  2.25  27.5  3.446  1.856  
3  0.454  2.3  25  2.447  1.564  
 
0.75  1  0.409  1.7  26.5  0.75  0.886 
2  0.435  1.5  27  1.09  1.044  
3  0.423  1.5  22.5  0.557  0.746  
 
1  1  0.39  2.56  38.1  1.419  1.19 
2  0.44  2.7  35  1.178  1.05  
3  0.32  2.5  26  1.04  1.020 
Graphical representation of ISE.
In this study, a disturbance of 50% (of setpoint value) is applied at 50 s to analyze the supply disturbance rejection performance. Figure
Output response with PID and PID with prefilter (PPID).
Controller output for PID and PID with prefilter (PPID).
The nonlinear spherical tank system recently discussed by Rajinikanth and Latha is considered to show the realtime implementation of the proposed method [
Figure
Realtime process response for nonlinear spherical tank system.
Initially a reference input of 18 cm is assigned. When the process output reaches a final steadystate value, then the reference input is increased by 2 cm (i.e., 20 cm) at
In the realtime study, the ISE and IAE values are obtained as 455.81 and 196.33, respectively. From this result, it can be noted that the PSO algorithm presents a smooth servo and regulatory response for the spherical tank level control problem and it can be easily implemented in real time using a MATLABsupported realtime process loop.
This paper attempted a controller parameter optimization work with PSO algorithm with various inertia weights. The fixed inertia weight method helps to improve the speed of convergence and also maintains good accuracy in optimized parameters. Proposed weighted sum of multipleobjective function provides the necessary controller parameters and supports enhanced performance for both the reference tracking and disturbance rejection problems. The realtime implementation also confirms the adaptability of the proposed method on the MATLAB supportable realtime hardware. The realtime result with PSO tuned PID offers better result for reference tracking, multiple reference tracking, and disturbance rejection problems.