Methodology
This chapter considers the modelling and control issues of a SMIB system. Thus, the work is split into two sections. The first section deals with the reducedorder modelling of a SMIB system considered from the literature. The SMIB presents a higherorder system when modelled. However, to design its controller, a low order model is required due to implementation issues of the controller. The GWO is taken up to reduce the parent SMIB model. Both the numerator and denominator polynomial coefficients are considered as unknowns and are determined with the help of the GWO applying constraints to retain dc gain, minimum phase and stability of the higherorder model. The pseudo random binary sequence (PRBS) was used as the input for both the original and reducedorder models containing unknown numerator terms. The difference between the responses is considered as the error function. Sum of square error (SSE) was minimized to obtain the numerator polynomial. It is worth mentioning here that the GWO was developed to emulate the leadership hierarchy and the hunting activity of grey wolves found in the northern part of America. The detailed algorithm along with its pseudo code can be found in Ref. [9] for interested researchers. Once the reduced system is developed, the second section of this work involves the controller design where the threeterm controller coefficients are determined using the GWO method. The concept of the AMM technique is followed for the controller synthesis. As many as seven heuristic techniques such as Ant lion optimization (ALO), Dragonfly algorithm (DA), Moth flame optimization (MFO), Multi verse optimizer (MVO), Grasshopper optimization algorithm (GOA), Sine cosine algorithm (SCA) and Salp swarm algorithm (SSA) are taken up for comparison to justify the effectiveness of the proposed methodology.
Initially, the reduced order systems are developed with the said methods and the error functions are evaluated. Further, these algorithms are run at least 30 times to provide a statistical measure of the fitness function. Once again, the representations of the minimum, maximum, average and standard deviation are tabulated. Since multiple data sets are compared, the Kruskal Wallis test [44] is carried out to test the validity of the outcome. In addition to this, the /^values are calculated by applying the ranksum test of Wilcoxon [45], which is once again a nonparametric test. Usually, the /^values are considered to be meaningful if they are less than 0.05 for 95% confidence interval. If the /гvalues are found to be greater than 0.05, then the outcomes of the experiments are insignificant. Since the proposed method is compared with manifold metaheuristic techniques, the HolmBonferroni correction [46] is incorporated in the Wilcoxon test to get modified /^values. Moreover, quite a sufficient number as well as relevant timedomain and frequencydomain parameters are assessed for the reduced models in comparison to the original system. As many as five benchmark error performance indices are evaluated to show comparison with some of the recently developed metaheuristic approaches. Thereupon, the controller parameters are estimated with the help of the GWO in the AMM framework. In the controller synthesis problem, the responses of the plantcontroller cascade and the chosen reference model are compared, setting to minimize the integral of square error (ISE) in order to determine the controller gains. In addition, the minimum fitness value is also reported.
Experimental Results and Discussion
A SMIB model, considered from the literature [47], is represented as
Since an eighthorder model is presented, it is quite obvious that the system needs to be reduced to a relatively lower order in order to develop a suitable controller. Applying the GWO technique, the reducedorder model is obtained as follows
In the above model, dc gain, minimumphase feature and stability are preserved. Several methods such as ALO, DA, MFO. MVO, GOA, SCA and SSA are used for comparison whose models are given in Table 10.1. Further, the least fitness function value, SSE in this case, is also reported in this table.
TABLE 10.1
Reduced Model Representations and Their Error Function
Method 
ReducedOrder Models 
SSE 
Proposed 
0.19142 

ALO 
3.4751 

DA 
3.3577 

MFO 
2.6270 

MVO 
2.7145 

GOA 
2.3921 

SCA 
1.3534 

SSA 
3.0516 
The bold value represents the minimum error value.
It is seen from Table 10.1 that the reduced system obtained by the proposed approach produces the least SSE. Thus, it can be inferred that the GWO method generated the best reduced model amongst the methods compared. Further, statistical measures of the error function are also studied, and the results of which are shown in Table 10.2. The four most popular indices are considered for the study, namely the lowest, highest, mean and the standard deviation. The first two basically denote the span of the error function while the standard deviation accounts for the stability of the algorithm. The lower the value of the standard deviation, the more stable is the algorithm. The best results i.e. the least value obtained in each column of the table are marked with the aid of bold letters.
From Table 10.2, it is apparent that the GWO outperforms all the reported algorithms in terms of best, worst, average and standard deviation values and hence indicated by bold letters in the table. Only the results of SCA are close to those of the GWO method in terms of the minimum fitness value. The standard deviation of the ALO algorithm is nearly close to that of the proposed technique, indicating that the algorithm is stable enough. Usually, the standard deviation value is less than the average value as found in the table, validating the theoretical concept as well. Since multiple algorithms are used for comparison with our method, the Kruskal Wallis test is carried out as a measure of nonparametric statistical inference for the validity of the results obtained. The test diagram is shown in Figure 10.2.
The Kruskal Wallis test diagram shown in Figure 10.2 clearly indicates that out of the seven metaheuristic algorithms used for comparison, the suggested GWO method proved significant compared to the six algorithms. In only one algorithm, namely Group 7, there seems to be some closeness of the data set. To check further, a Wilcoxon test based on ranksum was conducted on the data samples and the pvalues are reported in Table 10.3. A value less than 0.05 will be considered to be meaningful while a value than 0.05 will be taken up as insignificant. This limit is considered for 95% confidence interval, a popular one amongst the different confidence intervals.
All the pvalues in Table 10.3 are less than 0.05. Thus, the results obtained by the proposed technique are significant with respect to all other algorithms. The same pvalues in the table are merely accidental. A value lower or higher than that may not
TABLE 10.2
Statistical Analysis of the Fitness Function, SSE
Methods 
Lowest 
Highest 
Mean 
Std. Deviation 
Proposed 
0.19142 
0.2635 
0.2133 
0.0083 
ALO 
3.4751 
3.9430 
3.7604 
0.2034 
DA 
3.3577 
4.5830 
3.9883 
0.5096 
MFO 
2.6270 
3.6789 
3.2688 
0.4876 
MVO 
2.7145 
4.2911 
3.4013 
0.5802 
GOA 
2.3921 
4.4221 
3.9015 
0.8536 
SCA 
1.3534 
3.5973 
2.3523 
0.8322 
SSA 
3.0516 
3.9563 
3.6230 
0.3685 
FIGURE 10.2 Kruskal Wallis test for checking the significance of mean ranks.
TABLE 10.3
Calculations Showing pValues Using the Nonparametric Wilcoxon RankSum Test
Algorithm 
ALO 
DA 
MFO 
MVO 
Proposed 
1.0496e12 GOA 
I.0496eI2 SCA 
1.0496e12 SSA 
1 0496e12 
Proposed 
1.0496e12 
I.0496eI2 
1.0496e12 
 
TABLE 10.4
Modified Wilcoxon Test Results with HolmBonferroni Corrections
Algorithm 
pValues after HolmBonferroni Corrections 
hValues 
Proposed 
10" x [0.7347 0.7347 0.6298 0.5248 0.4198 0.3149 0.2099J 
[l 1 1 1 1 1 lj 
be achievable from the given data set. The pvalues are further modified applying HolmBonferroni corrections and quoted in Table 10.4. The Лvalues representing whether the test of hypothesis is true or false are marked in the table, l’s recorded against each hvalue correspond to statistically significant results, whereas 0’s denote insignificant results.
It is observed from the pvalues in the table that they are quite less than 0.05 and hence significant. This is also indicated by the hvalues where the Fs represent meaningful outcomes, and 0’s on the other hand indicate that they are insignificant. Moreover, the important time and frequency domain parameters of the reduced systems are provided in Table 10.5 to make a fair comparison with the original system.
From the results shown in Table 10.5, it is clearly visible that the GWO method produces the closest match to the original higherorder model in terms of both time domain and frequencydomain parameters. The other methods with which the comparison is carried out are only close in terms of overshoot, undershoot, gain and phase margins. The rise time and settling time of the ALO, DA, MFO, MVO, GOA and SCA are really huge, thereby suggesting sluggish response. Only the rise time and the settling time of the SSA method are slightly better. The phase margins of the MFO, GOA and SSA show a wide deviation from that of the original model. The step responses of the original system and the reduced test system are shown in Figure 10.3 to further validate the results shown in Table 10.5 in terms of time domain specifications.
It is quite clear from the time response curves of the original and the reduced systems in Figure 10.3 that the reducedorder model produced by the suggested technique closely matches the parent model. The frequency response is further plotted in Figure 10.4 to check the closeness of the proposed model in terms of the magnitude and phase plot of the Bode diagram.
The Bode diagrams of the parent and reduced system models show a very close resemblance as observed in Figure 10.4. Some of the wellknown errors widely popular in the literature of control are calculated for each of the reduced order test systems and their outcomes are enumerated in Table 10.6. The best rather than the least values reported for each of these errors are indicated with the help of bold letters for the proper understanding of the readers.
TABLE 10.5
Quantitative Time and FrequencyDomain Measures and Their Comparison with the Original System
Test System 
Methods 
Rise Time (s) 
Settling Time (s) 
Overshoot (%) 
Undershoot (%) 
Gain Margin (dB) 
Phase Margin (°) 
Original 
1.0692 
1.5686 
0.7852 
0 
Inf 
90.8510 

Reduced 
Proposed 
1.1967 
5.1427 
2.2227 
0 
Inf 
92.7294 
ALO 
273.2268 
486.4874 
0 
0 
Inf 
93.1089 

DA 
409.1763 
728.0001 
0 
0 
Inf 
96.3845 

MFO 
104.1833 
182.9161 
0 
0 
Inf 
156.4374 

MVO 
70.0453 
140.6177 
0 
0 
Inf 
95.5164 

GOA 
1.3223e+03 
2.3453e+03 
0 
0 
Inf 
109.8870 

SCA 
40.9063 
336.3791 
0 
0 
Inf 
93.3197 

SSA 
17.3216 
30.5515 
0 
0 
Inf 
135.1152 
FIGURE 10.3 Step response matching of parent and reduced models using the GWO technique.
FIGURE 10.4 Bode responses of the original and their reduced systems obtained by applying the proposed method.
TABLE 10.6
Representation of Popular Error Indices Using Different Methods
Methods 
IAE 
ITAE 
ISE 
USE 
H_{in}( Norm 
Proposed 
0.0677 
0.0503 
0.0046 
0.0037 
0.1136 
ALO 
2.2142 
1.2810 
4.5083 
2.4893 
3.3427 
DA 
2.2095 
1.2789 
4.4813 
2.4750 
3.3155 
MFO 
2.0110 
1.1807 
3.7716 
2.1086 
2.9623 
MVO 
1.0732 
0.8342 
1.1443 
0.9601 
1.6727 
GOA 
2.1770 
1.2630 
4.3310 
2.3989 
3.2233 
SCA 
0.3876 
0.3214 
0.1766 
0.1672 
0.8452 
SSA 
1.8383 
1.0611 
3.1013 
1.7002 
2.7011 
From Table 10.6, it is observed that the proposed GWO technique surpasses all other algorithms in terms of the error indices considered. The least value in each column is thus indicated with the help of bold letters. A reference model is then selected as per [48] to constitute the controller design of the reduced SMIB model. Generally, the response of the plant and controller with unknown parameters, connected in cascade, is compared to the response of the reference model. The objective is set in such a way that the plant and controller combination follows the response of the reference model approximately. The ISE is minimized to determine the controller parameters whose inputoutput relationship is defined by
The controller gains are thus determined and the results are shown in Table 10.7 along with the fitness function values. Similar to the model order reduction problem, in the controller design problem, the proposed methodology is compared with
TABLE 10.7
Different Controller Gains and Their Fitness Values
Methods 
K, 
K, 
Ко 
fmin 
Proposed 
0.12589 
0.048128 
3.0368e06 
6.6008e—09 
ALO 
2.8592 
1.0965 
1.1993e—06 
0.00015018 
DA 
5.0000 
2.4788 
3.3968e06 
0.00020168 
MFO 
0.095166 
3.2931 
5.0366e06 
1.741e—07 
MVO 
0.05926 
0.014016 
0.00063065 
0.00017156 
GOA 
2.9233 
5.0000 
0.21605 
0.38052 
SCA 
0.10152 
0.075265 
4.6343e05 
4.5961e07 
SSA 
4.2953 
0.78715 
0.0010865 
0.016981 
some of the widely cited heuristic techniques. The least fitness values reported in this table give an indication of the best controller parameters obtained, marked in bold.
In Table 10.7, the GWO method reports the least ISE value and hence the best choice for the controller amongst all other compared techniques. MFO and SCA methods only provide close match. The methods such as ALO, DA and MVO also yield satisfactory results. The results of GOA and SSA are far away from the best reported results. The one producing the best fitness value is also expected to closely match the response of the reference model. Thus, the GWO technique successfully addresses the modelling and control of SMIB test systems. Multiple test runs could also be performed for the controller synthesis part. In that case, parametric and nonparametric tests could be conducted to get better assessments. Moreover, new optimisation techniques such as the whale optimisation algorithm (WOA), Harris hawks optimisation (HHO), equilibrium optimizer (EO), marine predator algorithm (MPA) etc. may be applied to get still better performance of modelling and control aspects of SMIB systems. Obvious variants such as chaotic form, oppositionbased methods and new hybrid combinations of these algorithms could be employed for the reducedorder modelling and controller synthesis problem.