A hybrid bacterial foraging and modified particle swarm optimization for model order reduction

This paper study the model reduction procedures used for the reduction of large-scale dynamic models into a smaller one through some sort of differential and algebraic equations. A confirmed relevance between these two models exists, and it shows same characteristics under study. These reduction procedures are generally utilized for mitigating computational complexity, facilitating system analysis, and thence reducing time and costs. This paper comes out with a study showing the impact of the consolidation between the Bacterial-Foraging (BF) and Modified particle swarm optimization (MPSO) for the reduced order model (ROM). The proposed hybrid algorithm (BF-MPSO) is comprehensively compared with the BF and MPSO algorithms; a comparison is also made with selected existing techniques.


INTRODUCTION
Scientists and engineers are often challenged with the analysis, design, and synthesis of real-life problems due to the regularly increasing size of system models showing up by the present technology and societal and environmental processes. In such studies, the initial step is the refinement of a mathematical model which can be an alternative to the real problem. Modelling and controlling of complex-dynamicsystems (CDS) is the farthest essential areas of study in many engineering fields and sciences [1]- [3]. In various cases and engineering applications, the dynamic system model under the study can be complicated to some extent and pose challenges when used. Where there is high and complex mathematical model show exactly the problem at hand, but it is not suitable for the numerical simulation. To overcoming this problem, model order reduction (MOR) approach is used, which aims to convert a system model from higher order to a lower order to facilitate the computational complexity of such problem and has lately been intensively sophisticated for use with piecemeal more CDS inclusive both optimization and control [4]- [5] .
Various popular MOR methods for linear and nonlinear large-scale dynamical systems, are available in the researches for MOR [6]- [8]. The need for new innovative and advanced approaches is justified. Theories of evolutional computation are proposed [9] and mathematically formulated as a new way to model and control of CDS [10]- [12].
All the above-mentioned methods didn't take into account the merge between the BF and PSO for the reduced order model. This paper comes out with a study showing the impact of the consolidation between the BF and PSO for the reduced order model. As well as, the results are counterweight with the original BF and with the proposed MPSO.

BF algorithm
Recently, BF has become increasingly suitable as a global optimization technique in science and engineering subjects [13], [14]. The basic structural details of the BF algorithm are depicted in Figure 1.

MPSO algorithm
PSO has seen many changes since it's introduced by Eberhart and Kennedy [15]. Whenever researchers learn about this technology, new versions are discovered, incorporated into new applications. PSO is a populace grounded streamlining tool in which the framework is set through various arbitrarypossibility elements famous as particles. Every particle takes a position ( ) i X t and speed (V ) i t , which are refreshed by the accompanying equations: Gbest the best achieved by particle Different approaches are beneficial in texts for adjusting [16]- [19]. The proposed MPSO described as follows: Step 1: Identify the factors of PSO.
Step 2: Create an initial populace with M particles.
Step 3: estimate the fitness of each particle.
Step 4: Update according to one of the strategies proposed by us in [19].
Step 5: Update i t X and i t V for each particle by using Equations (8) and (9).
Step 6: Check the termination conditions.

BF-MPSO algorithm
To improve the performance; recent methods combine the PSO and BF algorithms together [20]- [22]. Here we propose a hybrid algorithm (combining the features of BF and the proposed MPSO (BF-MPSO) to acquire a ROM as shown in Figure 2.

RESULTS AND ANALYSIS
In this segment, the proposed advancement strategies are attempting to limit the ISE as indicated in (3). The proposed techniques were actualized on a Pentium IV 3-GHz PC in the MATLAB 2010 condition. The exhibitions of the BF, MPSO, and BF-MPSO calculations were assessed utilizing consistent estimations of the underlying elements proclaimed in Table 1. The step-responses of the full and ROMs are displayed in Figure 3(a). Likewise, to assess the feature of the model in the frequency space Figure 3(b) shows the frequency-amplitude attributes of the full and ROMs. Keeping in mind the ISE and mean square error were computed, to compare the proposed method with different ROMs, as appeared in Table 2. A comparison for the conjunction of the fitness function with the number of generations for the two MPSO schemes is presented in Figure 4. Likewise, Figure 5 displays the variant of the minimum fitness with the number of chemotactic steps.
The step responses of the full and ROMs are presented in Figure 6(a). Also, Figure 6(b) displays the frequency-amplitude attributes of the full and ROMs. Also, the ISE and mean square error were computed, to compare the proposed method with different ROMs, as appeared in Table 3. A comparison for the conjunction of the fitness function with the number of generations for the two MPSO schemes is presented in Figure 7.   Example 3: A system model specified in [9], which is 6th order 2-input 2-output: By utilizing the second procedure of the MPSO calculation, the ROM system 2 () Gswas:  The step responses of the full and ROMs are displayed in Figure 9(a). Likewise, Figure 9(b) displays the frequency-amplitude features of the full and ROMs. To compare the proposed method with different ROMs, the ISE and mean square error were computed, as shown in Table 4.

CONCLUSION
In this paper, we presented a comparative study of three algorithms for ROM optimization problems, namely: MPSO, BFO, and MPSO_BF. From Figures 3, 6, and 9, unmistakably observed that the suggested techniques keep up steady state value and stability in the ROMs, while Figures 4 and 7 delineate that the convergence speed of the second MPSO strategy is the fastest among the two strategies. Figures 5  and 8 illustrate that the speed of convergence and additionally the precision of the proposed BF-MPSO is better than that of BF. In addition, these algorithms can use a smaller number of chemotactic steps, which makes them faster. At long last, the information showed in Tables 2, 3, and 4 exhibits that the proposed calculation performs well in contrast with other accessible procedures.