Reduction of Large-Scale Dynamical Systems by Extended Balanced Singular Perturbation Approximation

A simplified approach for model order reduction (MOR) idea is planned for better understanding and explanation of largescale linear dynamical (LSLD) system. Such approaches are designed to well understand the description of the LSLD system based upon the Balanced Singular Perturbation Approximation (BSPA) approach. BSPA is tested for minimum / non-minimal and continuous/discrete-time systems valid for linear time-invariant (LTI) systems. The reduced-order model (ROM) is designed to preserved complete parameters with reasonable accuracy employing MOR. The Proposed approach is based upon retaining the dominant modes (may desirable states) of the system and eliminating comparatively the less significant eigenvalues. As the ROM has been derived from retaining the dominant modes of the largescale linear dynamical stable system, which preserves stability. The strong aspect of the balanced truncation (BT) method is that the steady-state values of the ROM do not match with the original system (OS). The singular perturbation approximation approach (SPA) has been used to remove this drawback. The BSPA has been efficaciously applied on a large-scale system and the outcomes obtained show the efficacy of the approach. The time and frequency response of an approximated system has been also demonstrated by the proposed approach, which proves to be an excellent match as compared to the response obtained by other methods in the literature review with the original system. KeywordsMOR, Large-scale linear dynamical system, Balanced truncation method, Steady state value, Singular perturbation approximation.


Introduction
The major issue in any aspect of a higher dimension systems dynamic behaviour is all over and occurs in different fields, including some engineering applications, e.g. Electrical Power system, control engineering, system design, smart city, transportation device and ecological systems, etc. ( Sikander and Prasad, 2015;Sambariya and Sharma, 2016;Daraghmeh and Qatanani, 2018;Suman and Kumar, 2019). The complex large-scale mathematical system must contain a comprehensive description of the original system (Daraghmeh et al., 2019). This mainly consists of many forms of differential and algebraic equations (Sikander and Prasad, 2015;Kumar et al., 2019). Conventional numerical modelling procedures cannot maintain the various queries as per necessity in applications that are, where the large system must be frequently used to find the solution. It is quite common that after several days, one solution has not yet been obtained. The reduction of the largescale LTI system using MOR has proven to be quite promising to get a stable solution much more quickly. The MOR is a key branch of complex system and control concept, that studies the characteristics of LSLD systems in the need to reduce the trouble while retaining their input-output behaviour intact (Chaturvedi, 2018). The key objective of MOR is to replicate the significant features of the actual system as closely as possible with the ROM comparison, indicating similar behaviours of the actual system. A ROM that precisely represents the prevailing characteristics of the structure under consideration is helpful and sometimes essential. Thus, an increasing essential exists for a systematic procedure to derive a lower order model, which may be called a ROM from the original system such that it preserves almost all essential properties of the system.
Numerous MOR approaches offered have been mostly in two ways that are in the shape of timedomain (TD) and frequency-domain (FD) (Sandberg and Rantzer, 2004). The TD is a form of state space as equation (2) and the FD method associated with transfer function matrix representation (3). They are divided into two basic groups of the system such as the single-input/ output (SISO) and multiple-input/output (MIMO) system (Narwal and Prasad, 2016). Reduction techniques that are used by the researchers have several advantages and disadvantages that are associated with them, as described in the literature search. Also, several common weaknesses along with the methods are that the ROM will become unstable even though the HOS remains stable (Gupta et al., 2019). Moreover, the steady-state matching of the OS with its lower order representation fails most of the time. They are furthermore hampered as the typical high-frequency ranges have poor precision and may have non-minimal phase characteristics (Davison, 1966;Cao et al., 2019) has introduced the first method for MOR in 1966, which re-modified by (Chidambara, 1969;Shamash, 1974) was suggested the Pade approximation method but these reduction methods used by researchers have several pros and cons linked to them. One of the key disadvantages of the methods is that even though the actual system is stable, the ROM is unstable (Hwang and Wang, 1984;Gupta et al., 2019) Also, a variety of approaches have been suggested to approximate large scale system as ROMs, such as Routh approximations (Hutton and Friedland, 1975), stability equation method (Pal, 1983), Mihiailov stability criterion (Wan, 1981), Chebyshev polynomial techniques (Bistritz and Langholz, 1979;Singh et al., 2012), Routh stability array (Pal, 1979), factor division method (Lucas, 1986;Parmar et al., 2007), Hurwitz polynomial approximation (Qin and Cheng, 2003), differentiation method (Bui-Thanh et al., 2008) and truncation method (Prasad et al.,1987) has been stated time to time for the desired response. (Shamash, 1981) have measured the outcome of ROM, with Markov parameter inclusive of the time moment matching to confirm the step response matching. Different methods are based on minimal realization (Lal et al., 1975) which have been suggested and that principle was extended by the Silvermans algorithm . Moreover, the principles of Markov parameters and time moments matching were used to get the ROMs for SISO systems and MIMO systems (Pal, 1983;Shamash, 1975). The main disadvantage of these methods, however, is the fact that a measure of approximate models produced for multi-systems can be equivalent to, or even more than, by which of the OS ( Pal and Ray, 1980). Moreover, several researchers introduced the modified version continued fraction method is reported in (Parthasarathy and Jayasimha, 1982) Routh-Pade approximation (Singh et al., 2008) and the new combined method is introduced by the author (Prajapati and Prasad, 2019a). The BT, reduced system obtained through a balanced realized model subsequently elimination of less controllable and less observable states. It has been found that the model so obtained does not retain the steady-state and DC gain of the LSLD system. (Fernando and Nicholson, 1982a) suggested that weak sub-system elimination can be utilized to preserve the DC gain of the balanced truncation method using the singular perturbation approximation approach (Glover, 1984;Al-Saggaf and Franklin, 1988;Liu and Anderson, 1989;Benner and Schneider, 2010;Tiwari and Kaur, 2018). The BT model has defined the DC gain of a minimal system using a SPA that can be used when the system is stable, minimally, and internally balanced to be reduced. In this paper, the investigator has proposed a hybrid approach with BT and SPA approaches applicable to LSLD system with excellent DC gain matching. The benefit of the approach lies not only to the matching of steady state but its applicability to large-scale dynamical system as well, which has been confirmed over several test systems taken from literature.

Proposed Methodology for System Reduction 2.1 Problem Formulation
Supposing an LTI SISO system having order 'n' is given as where m is less than n and numerator polynomial (ni) are constants coefficient and denominator polynomial ( i d ) of the original system, correspondingly it is supposed that state of the system is such as all the roots lie in the left side of the s-map.
The difficulty lies in finding an equation (6), which in some way has approximated the original system and preserves essential parameters of the original system. Such a solution corresponds as closely as possible to the solution of the system for a similar form of input (Antoulas et al., 2018).
The MOR problem consists of finding an approximate system of the reduced system described via where, rn , () of order r, r is very less than n, which is approximates OS and , ii nd are constants coefficient of numerator polynomial and denominator polynomial of the ROM. These parameters are to be obtained while MOR of an OS to ROM, such that the important properties of OS to preserved with less amount of error (Prajapati and Prasad, 2019b).

Proposed Hybrid Technique for Approximation
In this section, we address the order reducing procedure for higher-dimensional systems resulting in a hybrid approach using BT and balanced SPA. In the BT method, all balanced systems are separated into two parts as a slow and fast mode by defining the lower Hankel singular values (HSV) as fast mode, with the others defined as a slow mode. First, the derivative of all states equal to zero in fast mode can be obtained by defining a reduced system. The main aim of structure preservation in the ROM is to preserve dominant frequencies of the original system. Hence, to preserve dominant dynamic modes in the reduced system.

The Steps of Order Reduction Algorithm using the Balanced Truncation Method to be Followed.
Steps: In this step, both Grammians such as controllability and observability of the OS are determined as follows: Instead of calculating the two Grammian via integral equations, solve the Lyapunov equations (9) and (10) Assumption. The n th -order dimensional system is an asymptotically stable system and also minimal. Moreover, the state-space system equation (1) or the pair ( , ) oo AB states controllable if and only if the n × nm state controllability matrix and pair ( , ) oo AC are observable if the np×n observability matrix (Moore, 1981).  (Gugercin and Antoulas, 2004;Imran et al., 2014).  In control philosophy, eigenvalues express system stability, although HSV describes the "energy" of each state in the system.  Numerically we express as a stable state-space system equation (1) (Sikander and Prasad, 2017).

By assumption, both Grammians
 This is also a significant action of the minimality of realization of the original system is the diminishing positive number such that  The diagonal matrix (  ) if such a matrix realization exists (Lall et al., 2002;Gugercin and Antoulas, 2004;Segalman, 2007).
 Compute (Cholesky) factors (CF) of the Grammians are often obtained by this factorization according to (Pernebo and Silverman, 1982;Datta, 2004  Compute SVD, the T oc QQ is singular value decomposition of Grammians, also known as SVD of the system, found as follows (Datta, 2004).
where, U and V are a vector, define as left and right singular. Also, unitary matrices (orthogonal).
This system matric can be transformed into the balanced model by a similarity transformation matrices W, which can be achieved as follows (Datta, 2004;Gugercin, 2008;Imran et al., 2014).
The original system has been completely balanced, which is partitioned as: where, 11 A is part of a strong subsystem and 1  are () r r r n  matrixes.
We call this reduced order model a balanced system approximation of direct-truncation (DT). There are some well-known results on approximation ( Liu and Anderson, 1989).
Lemma A. (Pernebo and Silverman, 1982) The subsystem matrix ( , , ) i  is both completely controllable and observable (Fernando and Nicholson,1982a). Now, let us focus on applying of the SPA to the reduced model reduction of an LTI system (Fernando and Nicholson, 1982b).
Equation (19) has been attained as a minimal realised model containing strong and weakly subsystems. Thus, the SPA may be effortlessly applied to subsystems of Equation (19). In the BT model, reduced (r) balanced states are retained which are completely controllable and observable so balanced states are preserved, and remaining weakly controllable and/or observable states are truncated. SPA is used to maintain the DC gain of the original system in the model (Safonov and Chiang, 1989;Kumar et al., 2012). The concerned researcher may referee to (Kokotovic et al., 1976) for more indications of the method.
As the balanced realised system determine, can be re-write in the form of given as Again, re-write is equation form where,  is a positive small perturbational parameter of singular perturbation approximation approach (Fernando and Nicholson, 1983;Gajic and Lelic, 2001).
By comparing the derivative of the weakly subsystem to zero below, the BSPA model can be achieved (Kumar et al., 2012;Guiver, 2019 The various numerical tests to demonstrate the technique will be discussed in the preceding section.

Error Investigation of Reduced-Order Models
If the reduced system has been obtained from the original system, then the modeling error transfer function is well-defined via The actual infinity norm ( H  ) error bound in th r a ROM may be calculated by taking H  of () respectively, Furthermore, the actual amount of error bound is continuously less than or equal to the theoretical amount of error bound. Bound of the system. (Datta, 2004;Antoulas, 2005 Also, the comparison of the response has been carried out on the basis of the unit step response. The performance of ROM obtained is also compared on the basis of measured by calculating the performance indices, the accurateness of the proposed approach is known as the integral square error (ISE) between the transient sections of the actual system and the reduced-order model. The proposed scheme is used to achieve the best values of the coefficients in Equation 6. , for reduce model, by diminishing the performance indices ISE, also, to evaluate the performance of the reduced model in comparison with the original system and the reduced systems in terms of integral absolute error (IAE) (Singh et al., 2004;Sikander and Prasad, 2015;Tiwari and Kaur, 2018) described by the following equation

Numerical Experiments
Test System 1: Consider a FOM model for successfully applied the proposed reduction method. It is a large-scale dynamical system of a benchmarks problem of order 1006, This test illustration is taken from (Chahlaoui and Van Dooren, 2002;2005 Figure 2. This bar graph displays the optimum reduction order. The number of singular values that dominate non-zero is assumed to be that as an order of the original system. The first-eight singular values here are important, and then the ninth singular values quickly decay. Consequently, the order of reduction has been preferred as an eighth order.
Also, the infinity norm of the modelling error has been computed to show the effectiveness of the method. It is depicted in Table 1. It is understood that the actual error values are lower than the theoretical error values which may due to the underdamped nature of the system or due to poor transient response in an attempt to remove the steady-state error. Also, to achieve exact approximation and confirms the efficacy of the method. In addition, the response is also compared on the basis of performance indices as shown in Table 2 using the proposed approach, the ISE and IAE value is found the lowest value than another method. performance indices have been calculated to show the efficacy of the BSPA approach.
Test System 2: We now consider a Continuous-LTI and stable system of a simple mechanical system of 6 th order .for comparison purposes, this test system is taken from (Gawronski and Juang, 1990).
Bar chart HSV for the original system has been calculated and depicted in Figure 8. Which is show the optimal order reduction. Relatively, the number of non-zero dominant singular values non-zero dominant is regarded as the model of the reduction scheme. The first-second singular values are most significant here and afterward, the third singular values have decayed suddenly, becoming insignificant. So, the order of reduction is obvious as the second order.   Figure 7 and Figure 10 have shown both responses, time, and frequency of the reduced-order model with the original system and also compared. Respectively and it is understood that the response of ROM approximates the results of the OS very closely with no steady-state error in time response and exactly matches in frequency response. Furthermore, another time response of the OS and ROM for 200s second has been also shown in Figure 9. It is the magnification (zoom) of both systems response to views the transient behaviour comparison. It is again evidently Demonstrating that the reduced system to be a close characteristic with all essential parameters of the original system. Also, the infinity norm ( H  ) of the modelling error has been computed to show the effectiveness of the method. It is tabulated in Table 1. It is observed that the actual value of error bound is lower than the theoretical value of error bound which may due to the underdamped nature of the system or due to poor transient response in an attempt to eliminate the steady-state error. Also, to achieve exact approximation and confirms the efficacy of the method. In addition, the response is also compared on the basis of performance indices as shown in Table 2 using the proposed approach, the ISE and IAE value is found the lowest value than another method. performance indices have been calculated to show the efficacy and accuracy of the proposed approach.

Result and Discussion
In this discussion, the time and frequency responses of the ROM with OS are shown in the above figures. All mathematical experiments have been performed using Matrix Laboratory R2019a (Academic use). The time and frequency responses of the OS and ROM in Figures of the for-Test systems. This proposed method is excellent in comparison to the use of the BT method and other methods. The infinity norm of the modelling error has been also computed and results are depicted in Table 1. it is seen that to an excellent precise approximation with a minor error of the OS with the ROM. again, it is observed that the actual value of error bound is obtained lower than the theoretical value of error bound which may due to the underdamped nature of the system or due to poor transient response in an attempt to remove the steady-state error. In addition, the closeness and accuracy of the response have been measured through Performance Indices, presented in Table  2. The ISE and IAE value by the proposed approach is much lower than those compared with BT methods. This shows the superiority of the proposed approach.

Conclusion
In this paper, a new proposed approach of ROM of large-scale dynamic LTI systems, based upon BSPA methodology has been presented. It is superior to all of the traditional techniques or other approaches. This way weakness has been eliminated by the SPA from BT. The proposed approaches are the result of the hybridization of BT and the SPA method. This method has been used to effectively present a few test systems of the large-scale dynamic LTI systems. Furthermore, the time-frequency response of the proposed approach based on BSPA to gives ROM more precise and exact approximation than other approaches. In additional, comparing the error standard with existing works demonstrates the precision, reliability and enhanced efficiency of the presented system. This approach is more effective when successfully applied to a large-scale system.