ProportionalIntegralDerivative (PID) controller is one of the most widely used controllers for its property of simplicity and practicability. In order to design highquality performances PID controllers, an Advanced Fireworks (AFW) algorithm based on selfadaption principle and bimodal Gaussian function is proposed, which is built to optimize the PID controller by parameters tuning. Firstly, a compound index of optimization performance is formulated, and then the extremal optimization method of PID control system is proposed. Secondly, a PID parameters tuning model combined with AFW is built. At last, 5 typical transfer functions are simulated to obtain optimal parameters by AFW and contrast tuning method, such as ZieglerNichols method, Enhanced Fireworks (EFW) algorithm, and Particle Swarm Optimization (PSO). Simulation results show that AFW are effective and are easily implemented methods to solve PID control problems of different transfer functions.
It is not only the simplest but also the most efficient control strategy to solve many realworld control problems by ProportionalIntegralDerivative (PID) control system [
The evolutionary algorithm (EA) principle is a hot research topic in optimization techniques and several of them have been adopted to solve these problems in the past decade, such as PSO (Particle Swarm Optimization) [
A novel evolutionary algorithm named Fireworks Algorithm (FWA) is proposed and enhanced by Tan et al. [
As a consequence, this paper focuses on PID controller parameters tuning with optimization on Advanced Fireworks (AFW) algorithm in the attempt to obtain better performance. Firstly, compound index of PID optimization performance by analyzing PID controller and several traditional performance indices is built. Then, based on the simulation model combined with AFW, 5 typical transfer functions of control system were simulated to obtain optimal parameters by AFW and compared with Enhanced Fireworks (EFW) algorithm, Particle Swarm Optimization (PSO), and Genetic Algorithm (GA). Results show that AFWA is an effective and easily implemented method to solve PID control problems with different transfer functions.
A lot of practice showed that every PID controller parameter (proportional gain, integral gain, and derivative gain) has its own effectiveness to control system clearly, which is very easy to operate and also very effective. The structure of standard control system with PID controller is shown in Figure
The structure of typical control system With PID controller.
For a typical feedback control system, a continuous form of the PID controller is described as follows:
Furthermore, the discrete PID controller is written as follows:
There are several indices measuring the performance of the control system, such as the integral of the absolute value of error (IAE), the integral of time multiplied by the absolute value of error (ITAE), the integral of the square value of error (ISE), and the integral of time multiplied by the square of error (ITSE). These indices did not fully express the control performances comprehensively, so we present another novel index which is more reasonable.
The index we proposed is a hybrid index ITUE which is created by the linear weighted method and combined ITSE output of PID controller and rate of system error into a single index by linear weighted method according to the importance of each index. It is defined as follows:
Furthermore, the discrete performance index of control system is written as follows:
In this section, the methodology for tuning the optimization of PID controllers is described. Firstly, a novel evolutionary algorithm named AFW is proposed, which is constituted of fireworks explosion, mutation explosion, and selection of explosion locations. And then, structure of PID parameters tuning model based on AFW is introduced.
Simulink, developed by MathWorks, is a graphical programming environment for modeling, simulating, and analyzing control systems and offers tight integration with the rest of the MATLAB environment and can either drive MATLAB or be scripted from MATLAB. For Simulink is widely used in automatic control and digital signal processing simulation, testing performance index ITAE and ITUA on PID control system is built as in Figure
Simulink structure of testing performance index on PID control system.
In order to improve computational efficiency of PID parameters tuning, Advanced Fireworks algorithm is proposed.
Advanced Fireworks space is made up by
In the beginning, 1stgeneration location of firework
Fireworks explosion is proposed to generate sparks scattered all around the explosion location of father fireworks, which can be viewed as a searching process around a specific spot. Highquality fireworks generate numerous sparks which centralize the explosion center. That is to say, spot closed to the optimal location needs to be well searched. However, lowquality fireworks generate quite few sparks which are scattered in the space. In order to obtain splendid fireworks explosion, it is wise to set sparks number and explosion amplitude according to gorgeous degree (objective function value) of fireworks.
The number of sparks generated by firework
The amplitude of fireworks explosion by firework
And then,
In order to keep the diversity of sparks, mutation explosion is proposed to generate mutation sparks which are able to escape from original location. Location of mutation spark is generated from and escape from local extremum area as much as possible. So we define
After fireworks explosion and mutation explosion, we already get the location of fireworks, sparks, and mutation sparks. At the same time, the gorgeous degree (objective function value) can be calculated too. Then,
In order to obtain the optimal performance of PID control system on different transfer function, it is necessary to get a set of PID controller parameters (optimal location of element) which promote minimum performance index ITUE/ITAE. Note that filter coefficient may vary under different physical implementation. Without loss of generality, we set filter coefficient
PID parameters tuning model based on Advanced Fireworks algorithm.
Set parameters of AFW, number of fireworks
Initialize sets of PID parameters
Set off fireworks (sets of PID controller parameters) to generate sparks (new sets of PID controller parameters).
Generate mutation sparks (the other new sets of PID controller parameters) according to mutation explosion.
Check out gorgeous degree of all elements (fireworks, sparks, and mutation sparks). That is to say, simulate PID control system (shown as in Figure
Select
If the performance of PID control system meets the requirement or the generation of searching evolution reaches the maximum number of iterations
In order to evaluate the PID controller parameters (
ZN method is performed by setting the integral gain
With the PID parameters tuning model based on AFW, simulation is carried out to test the performance of PID tuning index ITUE and optimizing capacity of AFW on the problem of PID controller parameter tuning. AFW and several compared evolutionary algorithms are applied to optimize the PID controller performance index (ITAE or ITUE) by parameters (the proportional gain
In order to fully compare the performance of different evolutionary algorithms and test performance of different PID controller performance index, it is essential to set similar simulation environment as soon as possible. Population size (partial size in PSO, fireworks size in AFW and EFW) was set as 50 and the maximum evaluation number as 100 for all algorithms on all functions. Simulation time of PID control system was set as 20 seconds, filter coefficient of PID controller was 100, and the weights values of the performance index ITUE were
Main parameters of evolutionary algorithms.
Method  Main parameters 

AFW/EFW 

mutation spark number 



PSO  Inertia weight 
individual speed limitation is [−0.5, 0.5] 
In order to evaluate the quality of the controller, step response is applied in this section. Step response is the time behavior of the outputs of a general system when its inputs change from zero to one in a very short time. Then several quantities related to time behavior may be obtained, such as overshoot
Simulate the PID parameters tuning model based on AFWA with 5 typical transfer functions, respectively, in which filter coefficient is set as 100 according to testing benchmarks. Results of best ITAE, best ITUE, and their corresponding PID controller parameters are given as in Table
Statistical results.
Function  Pa.  ZN  AFW  EFW  PSO  

ITAE  ITUE  ITAE  ITUE  ITAE  ITUE  ITAE  ITUE  
( 

5.225  0.4536 


0.0814  0.3649  0.0814  0.3672 

13.3021  0.6771  0.2594  30  0.2873  30  0.2791  

1.2164  7.5184  2.9116  30  3.1724  30  3.31  

12.0285  7.5108  2.9064  22.2751  3.1694  22.275  3.2908  

—  0.489  1.31  0.735  1.2  0.735  1.16  

4.36%  0%  0%  4.74%  0%  4.74%  0.0649%  


( 

0.05974 


0.1046  0.0016  0.1043  0.0016  0.1223 

1.1504  23.1311  0.5909  8.6633  0.7817  8.7425  2.7446  

0.7772  0  0  30  0  30  9.2965  

0.0705  2.0809  0.052  0.36304  0.09986  0.3646  0.1514  

0.587  0.0643  0.288  0.155  0.342  0.154  0.375  

10.4%  49.7%  0%  25.2%  0%  25.3%  8.12%  


( 

23.29  7.8  0.0042  0.1211 


0.0061  0.1226 

0.0271  21.6408  1.3742  30  1.1759  30  1.4559  

0.0043  2.1830  11.7829  24.2796  9.2061  30  13.6554  

0.0866  0.74008  0.12561  15.2678  0.105  3.1136  0.14221  

23.9  0.0938  0.67  0.0718  0.808  0.0697  0.59  

7.41%  43.5%  3.77%  75.7%  4.26%  57.3%  3.9%  


( 

13.19  2.938 


0.4358 

0.4358 


0.7484  5.2644  2.8763  30  2.8741  30  2.8886  

0.0020  0  0  0  0  0  0  

2.6791  29.9807  14.6236  30  14.623  30  14.6463  

21.1  2.4  1.07  2.85  2.41  2.85  2.39  

5.04%  0.784%  0.0349%  18.9%  0.778%  18.9%  0.804%  


( 

5.67  0.9095 


1.1880  0.6974  1.1880  0.6985 

21.5801  1.7623  0.94238  30  0.95494  30  1.0341  

0.92192  2.508  1.2476  11.3251  1.2849  11.3253  1.3078  

30  29.5197  14.6483  30  15.0704  30  15.1394  

0.706  2.73  1.18  6.07  2.66  6.07  2.54  

10%  0%  0%  26.1%  0.0874%  26.1%  0.449% 
Based on PID control parameters tuned by ZN method and optimal parameters (
PID parameters tuning model based on Advanced Fireworks algorithm.
Function 1
Function 2
Function 3
Function 4
Function 5
PID parameters tuning model based on Advanced Fireworks algorithm.
Function 1
Function 2
Function 3
Function 4
Function 5
It is shown that AFW almost has the best performance index value of 2 different performance indexes ITAE and ITUE in PID parameters tuning with transfer functions as (1) (2) (4) (5) or has little difference with the best performance index value but much better than the other evolutionary algorithms with transfer functions as (2) (3) according to Table
Besides, curves with hybrid performance index ITUE get small overshoot and small rate of curve changing than cures with ITAE in each transfer function or each evolutionary algorithm according to Table
In this paper, PID control system parameters tuning problem is transformed into a class of 3dimensional parameter optimization problem based on performance indexes ITUE and ITAE. In order to design highquality performances PID controller, an Advanced Fireworks (AFW) algorithm is built to optimize the PID controller by parameters tuning. Extremal optimization method of PID control system was got by analyzing PID controller and building compound index of optimization performance. Then build a PID parameters tuning model combined with AFW. At last, 5 typical transfer functions of control system were simulated to obtain optimal parameters by AFW and were compared with ZN tuning method, Enhanced Fireworks (EFW) algorithm, and Particle Swarm Optimization (PSO). The simulation results show that AFW algorithms are effective and easily implemented methods to solve PID control problems of different transfer functions.
The authors declare that they have no competing interests.
This work is supported by National Natural Science Foundation of China under Grants nos. 71171199 and 61472443.