A Novel Mixed Method Using MM-SE with Change in Routhain Array for Model Order Reduction

: The system modelling leads towards the higher-order differential equations. These systems are difficult to analyse. Therefore, for ease and understanding, the conversion of higher to lower order is required. The model order reduction(MOR) is a systematic procedure to tackle these kinds of situations. This paper offers a mixed method for MOR using the modified moment matching (MM) and stability equation (SE) . The modification is applied in the routhain array of MM. The approach has been verified by examining the error between the original, proposed and compared with reduced order available in the literature. The obtained result has been compared on the basis of step response characteristics and the response indices error.


Introduction
The physical machine systems are of higher order .The mathematical complexity increases as the order of the system increases. The design analysis of a complex system is confusing and time-consuming.To remove mathematical complexity and make it simple, model reduction is the solution for higher-order differential equations. Model order reduction (MOR) is a systematic mathematical procedure to reduce the higher-order system(HOS). The reduced-order presents a replica of useful characteristics of the original system. The MOR have extensive application in computational physics, MEMS devices, chemical process, control systems, power systems, circuit designing. Numerous methods are advocated by researchers as per their thoughts and practical situations for solving the HOMs' in the frequency and time domain. Traditional techniques follows systematic mathematical processes to reduced HOS.The Pade approximation is the old technique to reduce the HOS. It gives the reduced order but due to the mathematical uncertainty of the procedural steps, reduced-order may become unstable. The algorithm introduce in [1], equate the time-moment and uses the continued fraction synthesis to make reduced order stable if higher is stable. The computer -oriented algorithm uses the evaluateing the moment of impulse reponse using the MM in [2].A combination of its Markov paprameter and moments given in [3] . The instability occurs because of unpredictable matrix output in the case of the time MM in [2,4] making the reduced-order stable. The unstable reduced model has to regain stability using the pole placement method in [5]. The SE used the pole-zero patterns to obtain the reduced order model(ROM) in [6] with the guarantee of stability in ROM. The other method Routh Approximation [7], the continued fraction method [8] and its inversion is used to obtain the ROM for preserving the stability. The result obtained from reduced-order has an error of approximation. The error of approximation is improved by combining the result of two methods termed the mixed method. The mixed-method consists of two methods, one method to obtain the numerator parameter and the other method is used to obtain the denominator parameter of ROM. The continued fraction third form where the s=0 to inf is proposed in [9]. The problem of instability of ROM is solved using the combination of SE and modified Cauer continued fraction [10,11]. Pade approximation using the Routh-Hurwitz array for stable reduction [12,13]. Dominant Eigen-value concept with continued fraction approach [14,15]. Pade approximation using S.E. proposed by Chen et al in [16] and its improvement in [17]. Factor division with Routh Hurwitz array [18], Eigen spectrum analysis and Cauer Second from [19], Cauer application [20].Eigen spectrum and Pade approximation [21] factor division method with SE in [22]. Stability array with SE method in [23]. Differential equation method used to derive the denominator parameter .Routhain-array method for obtaining the numerator parametd [24] are used to obtain the ROM . In recent times, nature-inspired techniques based on natural phenomena such as genes, swarm, physics and human behaviour based are being used for MOR. These all algorithms are used to minimize the approximation error (response indices especially Integral Square Error) which result out from traditional methods of frequency or time domain. The well-established approach in MOR is a genetic algorithm(G.A.) with Routh-approximation by Li et al. in [25] in which the denominator parameter of reduced-order is calculated by Routh approximation and numerator parameter is calculated by the time delay are identified by G.A.Particle swarm optimization by Z.W. Wang et-al in [26]. A trend is followed by researchers for the implementation of optimization techniques in the MOR field [27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43] to find the unknown ROM. The numerator or denominator parameter in case of mixed-method [44] or to minimize the error between the original system response and reduced system response concluding a new reduced order transfer function.
The MM techniques have been used with Markova parameters for mixed-method in [3,45]. However, its failed analysis of MM with the SE method is in [46]. This manuscript makes an effort to propose a mixed method with using the MMT and SE with changes in the Routhain-array of the MM technique. The proposed mixed method is tested on the selected higher-order systems and the ROM step response characteristics and response indices error are compared to justify the result. The conclusion shows the effectiveness of the novel mixed method.
The paper is well-written and divided into six sections and their subsections. Starting from the Introduction which consists of the literature review, the second section is of followed by problem formulation, the third section is of Methodology chosen for mixed-method with subsection 3.1 for Moment Matching and 3.2 Stability Equation followed by the numerical examples and results with discussion with the last part of conclusion effectively shows the resulting outcome of the proposed methods.

Problem Formulation
Consider the LTI higher-order system in transfer function form given in Equation (1) Here, (s) a N and (s) b D are the numerator and denominator polynomial of a higher-order system. The reducedorder system represented in Equation (2) 1 0 0 (s) (s) (s) In the above Equation (

Methodologies
The moment matching and stability equation are used to develop the novel method. The moment matching method is used to determine the denominator parameter and the stability equation used to determine the numerator parameter of the ROM.

3.1.Moment Matching
The moment matching method is given in [2], according to the method consider a system having transfer function in the ratio of two polynomial     The reduced-order numerator parameter is represented by 0

3.2.Stability Equation
The Stability Equation (SE) method is selected from the [6]. The traditional method gives the guarantee of stability of the reduced-order which is calculated as mentioned. Let the transfer function of a HOS given in Equation (3) be written as in Equation (9)       Then the reduced system represented as in Equation (17

Routhain Approach
The moment matching has the Routhain array approach for obtaining a reduced order. This can be clarified from [2] and mentioning it as case-1 in our experiment analysis.

Case-1
The higher-order physical system may be represented in transfer function form as in Equation. (3). Here, are numerator parameter and is the denominator parameter of the higher-order system. Divide the higher-order transfer function 0 a to make it unity.
The new coefficient of " s " is formed as for the numerator and for denominator parameters. Equation (3) changes as Equation The denominator 1, 2 1, , A A L and numerator 0 1 2, , , B B B L parameter are used to create the routhain array as mentioned in Table 1. The routhain parameter used to find the lower order using [2]the following matrix Equation. (20).
The 11 c , 12 c , 21 c and 22 c represents the upper left hand, upper right hand lower left hand and lower right-hand submatrices of the coefficient matrix. The reduced second order from the above Equation (21) is given by  (21) gives the numerator and Equation (22)

Case-2 Proposed approach
In the experimental case, we reduced-order obtained the following steps as given in Equation (24) The new coefficient of " s " is formed as The coefficient's of "s" denominator and numerator put in routhain array as mentioned in Table 2

Proposed Mixed-Mixed Method
The numerator parameter of Equation (24) and denominator parameter in Equation (17) are mixed in order to create a mixed method given in Equation (28)   21 22 The mixed-method in Equation (28) is not resulted in satisfactory using the MM and SE. Therefore, experimental case 2 is implemented. The reduced-order numerator parameter used from the SE from Equation (17) and denominator parameter from the reduced moment matching in transfer function Equation (27), resulted in mixedmethod mentioned in Equation (29 The proposed mixed-method approach is justified using the implementation of a higher-order system as mentioned in section 5.

Implementation and Result analysis
The method is applied on the LTI SISO system given as Example 1 and Example 2 5.1. Example 1: . Consider a sixth-order system from [39] as mentioned in transfer function format given in Equation ( The higher-order mentioned in Equation (30) is reduced according to the steps from Equation (19)(20)(21)(22)(23)(24)(25). The routhain array for obtaining the reduced-order given in Table 3 The reduced order using the MM case-1 is given in Equation ( The reduced mixed method using the MM and SE as mentioned in Equation (30) given in Equation (35)   The step response characteristics of the Mixed method not follows the original system and reduced Therefore the experimental case-2 is applied to the original system in Equation (31) to obtain a well-mixed method.
Equation (39) Equation (38) gives the numerator parameter and denominator parameter in Equation (37) The proposed reduced-order follows using the SE and MM is given in Equation (43). The reduced-order follows the step response of the original system approximately   The obtained reduced order from case-1 and using case-2 all are compared in the table and compared with the proposed mixed method. The obtained reduced order of HOS as in Equation (32)   The reponse characteritics of the system is given in Table 5 along with the reduced order. The ST min, ST max and peak is approximately same as that of the original system   Figure 2 shows that the proposed order retains stability and its step response characteristics shows that response is similar to HOS. Table 5 concludes that the proposed reduced-order approximately replicates the original higher-order system. The justification of the reduced order is verified by comparing it with the reduced order available in the literature. Figure 3 shows the step reponse comparison of the proposed and reduced order available in the literature.  In Table 6 it is clear that the proposed reduced-order system has the approximately same response as that of the original system and competitively competes with the response from literature. To justify the response of the proposed reduced order of the randomly chosen higher-order system from the literature have a minimum error between them so, the calculation of Integral time absolute error (ITAE), Integral absolute error (IAE) and Integral square error (ISE) is calculated and compared in Table 7. The response indices, i.e Integral Square Error (ISE) penalizes the larger errors more than the smaller error. This gives a more conservative response and the reduced system return faster to the set point. Integral Absolute Error (IAE) essentially takes the absolute value of the error. The negative area is accounted for when the IAE is used and diminishing the problem encountered with Integral error and Integral Time Absolute Error (ITAE) is the absolute value of error multiplied by the time, this results in error existing overtime being penalized even they may be small and result in more heavy damped response and these errors are given as Integral Square Error (ISE) is given in Eq. (42) Integral Absolute Error (IAE) is given in Eq. (43) Integral Time Absolute Error (ITAE) is given in Eq. (44) In response indices, Eq.   G t is the step response of the higher-order and   r G t is the response of the lower order.
The obtained reduced order is given in Equation (49 The obtained reduced-order using the SE is as follows in Equation (50

Considering the case-2
Considering the fourth order from Equation (46) To reduce the higher-order using the steps from Equation (26)(27), offered routhain Table 2. Table 9 is the routhain array for case-2.
The reduced-order obtained using the case-1 is given in Equation (54 The reduced-order using the proposed mixed method is obtained and compared in Table 10 Figure 4 gives the step response characteristics of the proposed reduced-order and reduced-order using the given methodologies in the manuscript. The proposed reduced-order attains the rise time in the shortest times period, which is 0.8365 and settles early as compared with others, which are 1.3940 after that, the response line follows the original system path.

Figure 4:
Step response Characteristics of Higher Order with proposed reduced-order The proposed reduced-order pole-zero plotting with the original higher-order system, second-order reduced using the moment matching, 2 nd order reduced using the SE, 2 nd order using the Improved MM, 2 nd order using the improved SE and proposed reduced-order using the SE and Moment matching is given in Figure 5. Step response characteristics of HOS with proposed reduced-order and reducedorder from literature The step response comparative analysis of the HOS 2 is in table 9. The result obtained is competitive with the response from literature and it can be observed from the step response characteristics Figure 6. The response indices of system 2 are available in Table 11 which is also competitive, however, the ITAE, IAE and ISE are calculated and comparison is given in Table 12.  The response indices from the proposed ITAE is 1.681, IAE is 0.7082 and ISE is 0.185 which is better than the 3 rd Cauer form, Prasad and Pal, S Mukherjee and competitive with other reduced orders from literature.

Conclusion
The rigorous study of a new method for MOR has been carried out in this article. The technique unifies of two strategies i.e time moment matching with the concept of modification in Routhain array and stability equation. The modifications in the routhain array from descending to ascending way of coefficient of "s" make systematic changes in the value of 0