A novel order reduction method for linear dynamic systems and its application for designing of PID and lead/lag compensators

Abstract

This paper proposes a new model order reduction technology for the simplification of the complexity of large scale models. The proposed technique is focused on the Mihailov stability approach that guarantees the stability of the reduced model constrained that the complex system is stable. In this scheme, the denominator coefficients of the approximated simplified system are computed by using the Mihailov stability algorithm and the truncation method is used for the determination of coefficients of the numerator polynomial. The effectiveness and efficiency of the proposed approach are illustrated by comparing the step responses of the given system and approximated lower order models. The error indices such as integral square error (ISE), relative integral square error (RISE), integral absolute error (IAE) and integral time weighted absolute error (ITAE) are used as performance indices for comparing the proposed scheme with other existing standard reduced order modeling methods. The obtained reduced model is used for the designing of controllers for the original complex system. A new scheme for the determination of controllers is also proposed for the large scale models with help of reduced order modeling. The proposed technique is validated by applying it to an eighth order flexible-missile control system and a third order fuel control system. The simulation results show the dominance of the proposed methodologies over the latest model diminution techniques available in the literature.

Keywords

Complex system model order reduction reduced order modelling Routh Hurwitz stability transfer function

Introduction

The mathematical modeling of complex dynamical model shows a fundamental role in the various areas of science and engineering such as electrical power systems, mechanical, electromagnetic theory, chemical engineering, ecological, biomedical, and so forth, especially in control system (Fortuna, 1992; Prajapati and Prasad, 2018d). The mathematical model of large scale models may pose problems in its study, synthesis, and identification. Hence, it is preferable to replace it by an equivalent lower order system that preserves the main qualitative features of the large scale model such as time moments, Markov parameters, stability margin, damping ratio, and so forth (Prajapati and Prasad, 2019a, 2019c). The goal of model order reduction is to compute an equivalent model which is simpler than the original complex systems and preserves the essential properties of the original system. The model diminution of higher order dynamical systems is a popular theme within the field of biological models (Javed and Ahmad, 2019; Snowden et al., 2017), control systems (Gautam et al., 2019; Kumar et al., 2016; Vasu et al., 2019), electromagnetic theory (Celo et al., 2005; Ng et al., 2020), mechanical engineering (Ischinger et al., 2019; Lin and Nikravesh, 2006; Veraszto et al., 2020), power systems (Al-Iedani and Gajic, 2020; Luo and Dhople, 2014; Wang and Long, 2020), chemical engineering (Nguyen et al., 2014, 2020; Tong et al., 2014) and so forth.

From the last four decades, several model reduction methodologies in the frequency domain have been discussed for the order diminution of the transfer function of higher order linear time invariant (LTI) dynamic models (Chen et al., 1979; Hutton and Friedland, 1975; Krishnamurthy and Seshadri, 1978; Shamash, 1974; Sinha and Pal, 1990). Among these approaches, Pade approximation (Prajapati and Prasad, 2018a; Shamash, 1974), is a frequently applied system diminution methodology and it is an appropriate scheme for the resembling of the static response of the approximated lower order system and original model. However, this methodology fails to keep the stability of some class of large scale original models into its equivalent reduced model (Ashoor and Singh, 1982; Prajapati et al., 2018). In order to circumvent this drawback several mixed model diminution techniques exist in the literature (Chen et al., 1980; Pal, 1979; Prasad, 2000; Vishwakarma and Prasad, 2008; Wan, 1981). Routh stability (Krishnamurthy and Seshadri, 1978) is another methodology for the diminution of complex linear systems and it is a convenient scheme for the resembling of the transient behavior of the complex system and its equivalent simplified model (Ashoor and Singh, 1982; Prajapati et al., 2018). This approach also has some drawbacks such as non-uniqueness (this method may give a similar reduced order system for different complex models) and unable to keep the dominant roots in the lower order system for the non-minimum higher order plants (Shamash, 1980; Singh, 1979). In order to circumvent these problems, several mixed reduced order modeling methods have been given (Pal, 1979; Prajapati and Prasad, 2018c; Shamash, 1975; Singh et al., 2006).

Routh approximation (Hutton and Friedland, 1975), is also another standard scheme for the diminution of complex linear models in the frequency domain but this technique is restricted for the linear system having strictly proper transfer function (Langholz and Feinmesser, 1978). In Prajapati and Prasad (2018b); Prajapati and Rajendra Prasad (2019); Prasad (2000); and Sarasu and Parthasarathy (1979), mixed methods of model reduction are proposed to circumvent the limitation of Routh approximation method. The authors in Chen et al. (1979) described another approach of model diminution for the simplification of minimum phase higher order models, known as the stability equation method but this method is not appropriate for the non-minimum phase system. Sinha and Pal (1990) presented a model reduction for the simplification of complexity of minimum and non-minimum large scale systems called pole clustering method. This scheme also has some limitations such as it needs tuning factor for the resembling of transient response and gain adjustment factor for acquiring the identical steady state characteristic of the approximated model with the given plant. In Wan (1981), Mihailov stability criterion is applied for obtaining the coefficients of the denominator of the simplified plant and the Pade approximation scheme is applied to compute the coefficients of the numerator. This criterion always provides a stable characteristic polynomial of the approximated lower order system for the stable original plants (Kumar et al., 2011; Prajapati, 2019).

In this contribution, a new model diminution methodology is presented in which denominator polynomial is obtained by Mihailov stability criterion (Kumar et al., 2011; Prajapati, 2019; Wan, 1981) and truncation technique (Shamash, 1981) is applied for the determination of the numerator polynomial. The drawback of the truncation method such as the instability problem for the large scale system of order four or more than four is circumvented by the proposed method because the denominator polynomial is acquired by the Mihailov stability algorithm that guarantees the stability of the reduced model. The proposed algorithm is simple and ensures the stability of the approximated reduced system for the given stable original system. The second objective of this paper is to propose an algorithm for the designing of the controller for the large scale system by using a reduced order model. This algorithm can be used for the designing of controllers of any real time complex systems of any order. This algorithm is validated by using two standard real time control systems available in the literature.

This paper is organized as follows: a new combined method is illustrated in Section 2. In Section 3, a mathematical algorithm is described for the designing of controllers by using a reduced order model. The simplification of some standard higher order systems and comparison of proposed technique with other standard methods are shown in Section 4 and the summary is in Section 5.

Basic procedure of proposed method

The proposed method is the combination of Mihailov stability method (Kumar et al., 2011; Prajapati, 2019; Wan, 1981) and truncation method (Shamash, 1981) and it has been illustrated in the following two steps.

Mihailov stability method

The Mihailov stability method is used for the determination of the denominator polynomial of the reduced model, which always ensures the stability of the proposed reduced model. Consider the transfer function of stable higher order model as

G (s) = \frac{N (s)}{D (s)} = \frac{d_{11} + d_{12} s + d_{13} s^{2} + d_{14} s^{3} + \dots + d_{1 n} s^{n - 1}}{e_{11} + e_{12} s + e_{13} s^{2} + e_{14} s^{3} + \dots + e_{1 n} s^{n - 1} + s^{n}}

(1)

where $d_{11}, d_{12} \dots d_{1 n}$ and $e_{11}, e_{12} \dots$ $e_{1 n}$ are the known parameters of the original system (1). Hence, the denominator equation of the original model is

D (s) = e_{11} + e_{12} s + e_{13} s^{2} + \dots + s^{n}

(2)

Replacing $s = jw$ in equation (2) and isolating the real and imaginary portions, it gives

\begin{matrix} D (s) = e_{11} + e_{12} (jw) + e_{13} {(jw)}^{2} + \dots + {(jw)}^{n} \\ = (e_{11} - e_{13} w^{2} + e_{15} w^{4} - \dots e_{1 X} w^{X}) \\ + j (e_{12} w - e_{14} w^{3} + e_{16} w^{5} - \dots e_{1 Y} w^{Y}) \\ = ϕ (w) + j ψ (w) \end{matrix}

(3)

where

ϕ (w) = e_{11} - e_{13} w^{2} + e_{15} w^{4} - \dots e_{1 X} w^{X - 1}

(4)

ψ (w) = e_{12} w - e_{14} w^{3} + e_{16} w^{5} - \dots e_{1 Y} w^{Y - 1}

(5)

{\begin{matrix} X = n - 1 \\ Y = n \end{matrix} when n is even

(6)

{\begin{matrix} X = n \\ Y = n - 1 \end{matrix} when n is odd

(7)

The $ϕ (w)$ and $ψ (w)$ are required in determining the Mihailov frequency behavior of the actual model in which $ψ (w)$ and $ϕ (w)$ show ordinate and abscissa respectively and $w$ changing from 0 to $\infty .$ In case of stable large scale model, the Mihailov frequency curve begin from abscissa at $w = 0$ and crosses the ordinate and abscissa alternatively as $w$ increasing and number of crossings being equal to the number of roots of the characteristic equation, as shown in Figure 1 for a fifth order system.

In order to find stable reduced order system, set $ϕ (w) = 0, ψ (w) = 0$ and determine the crossing frequencies $w_{0} = 0, \pm w_{1}, \pm w_{2}, \dots, \pm w_{n - 1} .$ Assume the transfer function of $rth$ order approximated simplified system is given as follows

R_{r} (s) = \frac{Q_{r} (s)}{P_{r} (s)} = \frac{q_{11} + q_{12} s + q_{13} s^{2} + \dots + q_{1 r} s^{r - 1}}{p_{11} + p_{12} s + p_{13} s^{2} + \dots + p_{1 r} s^{r - 1} + s^{r}}

(8)

Hence, characteristic equation of the approximated simplified system (8) is given as follows

P_{r} (s) = p_{11} + p_{12} s + p_{13} s^{2} + \dots + p_{1 r} s^{r - 1} + s^{r}

(9)

Substituting $s = jw$ in equation (9) and isolating the real and imaginary portions, it gives

\begin{matrix} P_{r} (s) = p_{11} + p_{12} (jw) + p_{13} {(jw)}^{2} + \dots + p_{1 r} {(jw)}^{r - 1} + {(jw)}^{r} \\ = (p_{11} - p_{13} w^{2} + p_{15} w^{4} - \dots p_{1 x} w^{x}) \\ + j (p_{12} w - p_{14} w^{3} + p_{16} w^{5} - \dots p_{1 y} w^{y}) \\ = ξ (w) + j η (w) \end{matrix}

(10)

where

ξ (w) = p_{11} - p_{13} w^{2} + p_{15} w^{4} - \dots p_{1 x} w^{x - 1}

(11)

η (w) = p_{12} w - p_{14} w^{3} + p_{16} w^{5} - \dots p_{1 y} w^{y - 1}

(12)

{\begin{matrix} x = r - 1 \\ y = r \end{matrix} when r is even

(13)

{\begin{matrix} x = r \\ y = r - 1 \end{matrix} when r is odd

(14)

The stable lower order system is acquired by Mihailov stability methodology by the matching of Mihailov frequency characteristic of the actual and reduced models. Therefore, the roots of $ξ (w) = 0 and η (w) = 0$ must be positive real and crossing frequencies are identical as large scale model for the resembling of input/output performance and scattered along the $w$ axis alternatively. Hence, equations (11) and (12) can be written in the terms of roots as shown in equations (15) and (16), respectively

ξ (w) = k_{1} (w^{2} - w_{1}^{2}) (w^{2} - w_{3}^{2}) \dots

(15)

η (w) = k_{2} w (w^{2} - w_{2}^{2}) (w^{2} - w_{4}^{2}) \dots

(16)

where, $k_{1}$ is determined by using relation $ϕ (j w_{0}) = ξ (j w_{0})$ and $k_{2}$ is determined from the relation $ψ (j w_{1}) = η (j w_{1})$ or ${(d ψ / d ϕ)}_{w_{0}} = {(d η / d ξ)}_{w_{0} .}$ After acquiring the denominator equation of the approximated simplified model $P_{r} (jw) = ξ (w) + j η (w)$ , substituting $jw = s$ and it will give final denominator polynomial $P_{r} (s) = p_{11} + p_{12} s + p_{13} s^{2} + \dots + p_{1 r} s^{r - 1} + s^{r}$ . Second step is to find the coefficients of the numerator polynomial of the lower order model by using mathematical algorithm given in Shamash (1981).

Truncation method

The numerator polynomial coefficients of the lower order system are computed by the direct truncation technique (Shamash, 1981). In Shamash (1981), the truncation method is applied for computing the numerator and denominator polynomials of the lower order system. But, this method has one drawback in that it does not prove the stability of the approximated simplified model when the order of the reduced model is four or more than four. This drawback can be circumvented by determining the denominator polynomial of the reduced model by any other method that ensures the stability of the reduced model. Due to this, the Mihailov stability technique is used for obtaining the denominator polynomial and the truncation method is used only for computing the numerator polynomial. In the truncation method, the numerator polynomial of the reduced model is determined by retaining the lower order coefficients of the numerator polynomial of the original system and deleting the remaining higher order terms. For the $rth$ order simplified model, the numerator coefficients are obtained by preserving the lower order coefficients of the numerator polynomial of the original from $s^{0} to s^{r - 1}$ , as given in equation (17)

\begin{matrix} Q_{r} (s) = q_{11} + q_{12} s + q_{13} s^{2} + \dots + q_{1 r - 1} s^{r - 1} \\ = d_{11} + d_{12} s + d_{13} s^{2} + \dots + d_{1 r - 1} s^{r - 1} \end{matrix}

(17)

Hence

{\begin{matrix} q_{11} = d_{11} \\ q_{12} = d_{12} \\ q_{13} = d_{13} \\ ⋮ \\ q_{1 r - 1} = d_{1 r - 1} \end{matrix}

(18)

where, $Q_{r} (s)$ is the numerator polynomial and $q_{11}, q_{12}, q_{13}, \dots, q_{1 r - 1}$ are the numerator coefficients of $rth$ order simplified model and $d_{11}, d_{12}, d_{13}, \dots, d_{1 r - 1}$ are the numerator coefficients of the original system.

Controller design algorithm

The design of controller for a large scale system is a lengthy and difficult task. The cost of the design of controller and simulation time is increased proportionally as the complexity of the system increases. To overcome this type of constraints the original higher order system must be substituted by a “good” approximated model and the controller can be designed by using this model. The series controllers are preferably used over the feedback controllers in complex real time model because the large numbers of sensors are needed for sensing the state variables for the synthesis of feedback controllers.

On the basis of required operation of the real time model, a reference system ( $M (s))$ is constructed, so that the closed loop behaviour of the controlled model with unity feedback is comprehensively the same as the response of the reference plant. The controller design algorithm is based on approximate model matching in Padé sense and illustrated by the following few steps.

Step 1: A reference model $M (s)$ is constructed on the basis of the given specifications of the complex real-time system for acquiring the estimated characteristic such that the closed loop characteristic of the controlled model with unity feedback is resembled with that of the reference system. The detailed processes for determining the reference plant from the given specification are illustrated in Peterson and Nassar (1978) and Towill (1970).

Step 2: Obtain an equivalent open loop reference system $\tilde{M} (s)$ . If $M (s)$ is the desired closed loop model (reference system) then the equivalent open-loop transfer function is determined as

\tilde{M} (s) = \frac{M (s)}{1 - M (s)}

(19)

Step 3: Consider the transfer function of a controller, which yields the desired closed loop response as follows (Basilio and Matos, 2002)

G_{c} (s) = \frac{g_{0} + g_{1} s + \dots + g_{k} s^{k}}{h_{0} + h_{1} s + \dots + h_{l} s^{l}}

(20)

Step 4: The unknown parameters of the controller of equation (20) are obtained by equating the characteristic of the open-loop controlled system with the response of the open loop reference model as

G_{c} (s) G (s) = \tilde{M} (s)

(21)

G_{c} (s) = \frac{\tilde{M} (s)}{G (s)} = \frac{\sum_{i = 0}^{\infty} e_{i} s^{i}}{s}

(22)

where $e_{i}$ $(i = 0, 1, 2, \dots, \infty)$ are the Taylor series coefficients about $s = 0,$ and can be acquired by applying moment generating technique (Lal and Mitra, 1974; Shamash, 1974; Zakian, 1973). Also, $G (s)$ is the transfer function of complex system, it can also be substituted by an equivalent lower order model $(R_{r} (s))$ so that the mathematical calculation and simulation time can be minimized. The unknown parameters of the controller are obtained by comparing equations (20) and (22) in Padé sense as follows

\begin{matrix} g_{0} = h_{0} e_{0} \\ g_{1} = h_{0} e_{1} + h_{1} e_{0} \\ g_{2} = h_{0} e_{2} + h_{1} e_{1} + h_{2} e_{0} \\ ⋮ \\ g_{i} = h_{0} e_{i} + h_{1} e_{i - 1} + \dots + h_{i} e_{0} \\ 0 = h_{0} e_{i + 1} + h_{1} e_{i} + \dots + h_{i + 1} e_{0} \\ ⋮ \\ 0 = h_{0} e_{i + j} + h_{1} e_{i + j - 1} + \dots + h_{j} e_{i} \end{matrix}}

(23)

The solution of equation (23) leads to Padé type approximations of the controller having the desired structure.

Step 5: After computing the parameters of full order controller $G_{c} (s)$ from equation (23), the closed loop transfer function are obtained by using equation (24) as

G_{cl} (s) = \frac{G_{c} (s) G (s)}{1 + G_{c} (s) G (s)}

(24)

Step 6: Reduce the large scale system $G (s)$ to $R_{r} (s)$ using the proposed methods and repeat steps 4 and 5. After finding the parameters of lower order controller $G_{cr} (s),$ the closed loop transfer function can be represented as

R_{cl} (s) = \frac{G_{cr} (s) G (s)}{1 + G_{cr} (s) G (s)}

(25)

Illustrative examples

The accuracy of the reduced model obtained using the proposed method is examined by calculating performance error indices given in Narwal and Prasad (2017); Prajapati and Prasad (2019b); and Sikander and Prasad (2015b)

{\begin{matrix} ISE = \int_{0}^{\infty} [y (t) - y_{r} (t))]^{2} dt \\ RISE = \int_{0}^{\infty} [y (t) - y_{r} (t))]^{2} dt / \int_{0}^{\infty} {[\hat{y} (t)]}^{2} dt \end{matrix}

(26)

{\begin{matrix} IAE = \int_{0}^{\infty} | y (t) - y_{r} (t)) | dt \\ ITAE = \int_{0}^{\infty} t | y (t) - y_{r} (t)) | dt \end{matrix}

(27)

where $y (t)$ and $y_{r} (t)$ are step characteristics of the original and simplified plants, respectively. And $\hat{y} (t)$ is the impulse characteristic of the given actual plant.

Example 1: Consider a transfer function of a flexible-missile control system (28) given in Chen et al. (1980a, 1980b)

G (s) = \frac{- s^{6} + 3.06 \times 10^{2} s^{5} - 4.96 \times 10^{4} s^{4} + 3.577 \times 10^{6} s^{3} - 6.303 \times 10^{7} s^{2} - 1.246 \times 10^{10} s + 5.906 \times 10^{11}}{s^{8} + 52.99 s^{7} + 3.05 \times 10^{4} s^{6} + 1.375 \times 10^{6} s^{5} + 1.839 \times 10^{8} s^{4} + 5.232 \times 10^{9} s^{3} + 3.422 \times 10^{11} s^{2} + 2.823 \times 10^{12} s + 1.442 \times 10^{14}}

(28)

The characteristic equation of the original model is given as

\begin{matrix} D (s) = s^{8} + 52.99 s^{7} + 3.05 \times 10^{4} s^{6} \\ + 1.375 \times 10^{6} s^{5} + 1.839 \times 10^{8} s^{4} + 5.232 \times 10^{9} s^{3} \\ + 3.422 \times 10^{11} s^{2} \\ + 2.823 \times 10^{12} s + 1.442 \times 10^{14} \end{matrix}

(29)

Substituting $s = jw$ in equation (29) and isolating the real part and imaginary part and it will give the following equations

\begin{matrix} ϕ (w) = (w^{2} - 591.1193) (w^{2} - 2491.5423) \\ (w^{2} - 4220.8461) (w^{2} - 23196.4923) \end{matrix}

(30)

ψ (w) = w (w^{2} - 646.7532) (w^{2} - 3837.6991) (w^{2} - 21463.8398)

(31)

For the second order simplified system $ξ (w) = k_{1} (w^{2} - 591.1193)$ , $η (w) = k_{2} w$ and $k_{1} = - 2.4394 \times 10^{11}$ and $k_{2} = 1.9977 \times 10^{11}$ . Hence, the characteristic equation coefficients of the second order simplified plant acquired by the Mihailov stability algorithm are

P_{2} (s) = 2.4394 \times 10^{11} s^{2} + 1.9977 \times 10^{11} s + 1.442 \times 10^{14}

(32)

By using the truncation method, the numerator polynomial of the reduced model is evaluated and the second order simplified system computed by using the proposed methodology is

\begin{matrix} R_{2} (s) = \frac{Q_{2} (s)}{P_{2} (s)} \\ = \frac{- 1.246 \times 10^{10} s + 5.906 \times 10^{11}}{2.4394 \times 10^{11} s^{2} + 1.9977 \times 10^{11} s + 1.442 \times 10^{14}} \\ = \frac{- 0.1246 s + 5.906}{2.4394 s^{2} + 1.9977 s + 1442} \end{matrix}

(33)

The step responses of the actual eighth order model and lower order models acquired from the proposed method and other existing methodologies are plotted in Figure 2. This figure shows that the reduced system acquired by the proposed technique precisely retains the dominant characteristics of the original higher order system. From this figure, it can be seen that the response of the proposed reduced model (33) is thoroughly matched with the response of the original system (28). The quantitative analysis of the proposed method and other standard method has been done in Table 1 in terms of various performance indices such as integral square error (ISE), relative integral square error (RISE), integral absolute error (IAE), and integral time weighted absolute error (ITAE). From this table, it can be observed that the proposed technique is giving the lowest values of error indices and which are smaller than the error values acquired by the standard technologies as well as the latest methods. In this table, it is also clear that the lower order system acquired by the proposed method is same as the simplified model obtained by the mixed method of Mihailov stability algorithm and Padé approximation methodology but the proposed technology is simple as compared with this mixed technique because it does not need the computation of time moments. From the above observations, it can be concluded that the proposed technique executes better in obtaining the reduced order model with less mathematical effort. The superiority of the proposed technique to the other popular diminution techniques present in the literature is tabulated in Table 1 by comparing various performance error indices.

Example 2: Consider a ninth-order standard system discuss by several researchers (Desai and Prasad, 2013; Mukherjee and Mittal, 2005; Philip and Pal, 2010; Soloklo and Farsangi, 2015; Tiwari and Kaur, 2017)

G (s) = \frac{s^{4} + 35 s^{3} + 291 s^{2} + 1093 s + 1700}{s^{9} + 9 s^{8} + 66 s^{7} + 294 s^{6} + 1029 s^{5} + 2541 s^{4} + 4684 s^{3} + 5856 s^{2} + 4620 s + 1700}

(34)

Table 1.

Comparison of various model diminution methodologies in terms of ISE, RISE, IAE, and ITAE.

Reduction method	Lower order system	ISE	RISE	IAE	ITAE
Hankel norm approx. technique (Gu, 2005)	$\frac{- 0.0947 s + 0.795}{s^{2} + 0.5854 s + 608.7}$	78.00 $\times 10^{- 4}$	0.0492	2.7805	136.615
Balanced truncation method (Moore, 1981), Schur decomposition method (Kumar et al., 2012)	$\frac{- 0.1384 s + 1.875}{s^{2} + 0.9322 s + 605.4}$	9.8645 $\times 10^{- 4}$	0.0063	0.9910	49.9728
Routh Hurwitz and Padé approx. (Pal, 1979; Shamash, 1975)	$\frac{- 0.23535 s + 5.906}{1.8794 s^{2} + 1.1903 s + 1442}$	5.4693 $\times 10^{- 4}$	0.0035	0.1441	0.4195
Routh Hurwitz and factor division algorithm (Singh et al., 2006), Improved Routh Hurwitz (Prajapati and Prasad, 2018c)	$\frac{- 0.23534 s + 5.906}{1.8794 s^{2} + 1.1903 s + 1442}$	5.4692 $\times 10^{- 4}$	0.0035	0.1441	0.4195
Routh Hurwitz method (Krishnamurthy and Seshadri, 1978)	$\frac{4.3787 \times 10^{- 5} s + 5.906}{1.8794 s^{2} + 1.1903 s + 1442}$	4.6355 $\times 10^{- 4}$	0.0029	0.1232	0.3064
Modified pole clustering method and modified Padé approx. (Singh et al., 2016)	$\frac{2.8024}{s^{2} + 1.0788 s + 6.8422 \times 10^{2}}$	4.0735 $\times 10^{- 4}$	0.0026	0.1064	0.1991
Modified pole clustering method and factor division algorithm (Sikander and Prasad, 2017)	$\frac{- 1.585 \times 10^{- 4} s + 2.8024}{s^{2} + 1.0788 s + 6.8422 \times 10^{2}}$	4.0715 $\times 10^{- 4}$	0.0026	0.1064	0.1991
Modified pole clustering method and Padé approx. (Vishwakarma and Prasad, 2011)	$\frac{- 0.1096 s + 2.8024}{s^{2} + 1.0788 s + 6.8422 \times 10^{2}}$	3.7474 $\times 10^{- 4}$	0.0024	0.0960	0.1822
Pole clustering method and Padé approx. (Vishwakarma and Prasad, 2008)	$\frac{- 0.3603 s + 9.1013}{s^{2} + 2.4046 s + 2.2221 \times 10^{3}}$	3.4588 $\times 10^{- 4}$	0.0022	0.1041	0.2167
Improved pole clustering technique (Komarasamy et al., 2011)	$\frac{- 0.1096 s + 2.8024}{s^{2} + 1.0788 s + 6.8422 \times 10^{2}}$	3.4588 $\times 10^{- 4}$	0.0022	0.1041	0.2167
Routh approximation technique (Hutton and Friedland, 1975)	$\frac{- 0.1663 s + 7.88}{s^{2} + 37.67 s + 1924}$	2.4165 $\times 10^{- 4}$	0.0015	0.0884	0.1803
Routh and Padé approx. (Lepschy and Viaro, 1982), Improved Routh approx. (Prajapati and Prasad, 2018b), Routh approx. and factor division method (Prajapati and Prasad, 2019c)	$\frac{- 0.1662 s + 7.8801}{s^{2} + 37.67 s + 1924}$	2.4164 $\times 10^{- 4}$	0.0015	0.0884	0.1778
Differentiation method and Padé approx. (Kranthi et al., 2013)	$\frac{- 4.26 s + 119.06}{2.464 s^{2} + 142.3 s + 29070}$	2.3979 $\times 10^{- 4}$	0.0015	0.0882	0.1804
Differentiation technique (Gutman et al., 1982)	$\frac{- 0.4187 s + 119.1}{2.464 s^{2} + 142.3 s + 29070}$	2.3424 $\times 10^{- 4}$	0.0015	0.0885	0.2403
Stability equation criterion and continued-fraction method (Chen et al., 1980a)	$\frac{- 2.67 \times 10^{- 4} s + 2.4209}{s^{2} + 11.5726 s + 591.1932}$	2.2688 $\times 10^{- 4}$	0.0014	0.0875	0.2174
Truncation methodology (Shamash, 1981)	$\frac{- 0.1246 s + 5.906}{3.422 s^{2} + 28.23 s + 1442}$	2.1574 $\times 10^{- 4}$	1.4 $\times 10^{- 3}$	0.0848	0.1768
Factor division and stability equation technique (Sikander and Prasad, 2015a)	$\frac{- 0.1066 s + 5.906}{2.4399 s^{2} + 28.23 s + 1442}$	2.1459 $\times 10^{- 4}$	0.0014	0.0852	0.1763
Stability equation (Chen et al., 1979), Padé approx. and stability equation (TC Chen et al., 1980b)	$\frac{- 0.1246 s + 5.906}{2.4399 \times 10^{11} s^{2} + 28.23 s + 1442}$	2.1272 $\times 10^{- 4}$	0.0013	0.0849	0.1762
Padé approximation (Shamash, 1974)	$\frac{- 0.0626 s + 1.99}{s^{2} + 4.479 s + 485.8}$	1.9682 $\times 10^{- 4}$	1.211 $\times 10^{- 3}$	0.0831	0.2077
Time moment matching (Zakian, 1973)	$\frac{- 1.2886 \times 10^{- 4} s + 4.09 \times 10^{- 3}}{2.0585 \times 10^{- 3} s^{2} + 9.2198 \times 10^{- 3} s + 1}$	1.9682 $\times 10^{- 4}$	$1.21 \times 10^{- 3}$	0.0830	0.206
Mihailov stability and improved Pade approx. method (Prajapati et al., 2020)	$\frac{5.906}{2.4394 s^{2} + 1.9977 s + 1442}$	1.4225 $\times 10^{- 4}$	9.022 $\times 10^{- 4}$	0.0557	0.1108
Proposed technique, Pade approx. and Mihailov stability method (Wan, 1981)	$\frac{- 0.1246 s + 5.906}{2.4394 s^{2} + 1.9977 s + 1442}$	6.3371 $\times 10^{- 5}$	4.0194 $\times 10^{- 4}$	0.0431	0.1294

The denominator equation of the system given in equation (33) is

\begin{matrix} D (s) = s^{9} + 9 s^{8} + 66 s^{7} + 294 s^{6} + 1029 s^{5} + 2541 s^{4} \\ + 4684 s^{3} + 5856 s^{2} + 4620 s + 1700 \end{matrix}

(35)

Substituting $s = jw$ and isolating the real part and imaginary part and it gives

\begin{matrix} ϕ (w) = (w^{2} - 0.3379) (w^{2} - 3.0847) \\ (w^{2} - 8.912) (w^{2} - 20.332) \end{matrix}

(36)

\begin{matrix} ψ (w) = w (w^{2} - 1.3557) (w^{2} - 5.561) \\ (w^{2} - 13.4195) (w^{2} - 45.6634) \end{matrix}

(37)

For the third-order approximated lower order system $ξ (w) = k_{1} (w^{2} - 0.3379)$ , $η (w) = k_{2} w (w^{2} - 1.3557)$ and $k_{1} = - 5031$ and $k_{2} = 3099$ . Hence, the characteristic equation of the third-order approximated simplified model acquired by the Mihailov stability algorithm is given in equation (38)

P_{3} (s) = 3099 s^{3} + 5031 s^{2} + 4201 s + 1700

(38)

By using truncation method, the numerator polynomial of the reduced model is evaluated and the third order simplified system obtained by using proposed methodology is given in equation (39)

R_{3} (s) = \frac{Q_{3} (s)}{P_{3} (s)} = \frac{291 s^{2} + 1093 s + 1700}{3099 s^{3} + 5031 s^{2} + 4201 s + 1700}

(39)

Comparison of the step responses for the higher order and reduced order systems determined by the proposed technique as well as other technologies are shown in Figure 3. This figure demonstrates that the lower order model (39) acquired by the proposed technique precisely retains the characteristics of the original system (34). In Table 2, it is shown that the performance error indices values of the proposed method are the lowest values compared with some standard methods (Chen et al., 1979; Gu, 2005; Krishnamurthy and Seshadri, 1978; Moore, 1981; Wan, 1981), recently proposed methods (Kranthi et al., 2013; Kumar et al., 2012; Prajapati et al., 2020; Prajapati and Prasad, 2018c) and optimization methods (Desai and Prasad, 2013; Vishwakarma and Prasad, 2009). The accuracy and effectiveness of the proposed scheme to the other popular diminution schemes present in the literature are tabulated in Table 2 by comparing various performance error indices.

Example 3: Consider a fuel control system taken from Aguirre (1991) having transfer function and reference system as

G (s) = \frac{0.4299 s^{2} + 0.601 s + 0.1069}{s^{3} + 0.7026 s^{2} + 0.8746 s + 0.1107}

(40)

M (s) = \frac{0.16}{s^{2} + s + 0.16}

(41)

Table 2.

Comparison of various model diminution methodologies in terms of ISE, RISE, IAE, and ITAE.

Reduction method	Lower order system	ISE	RISE	IAE	ITAE
Hankel norm approx. technique (Gu, 2005)	$\frac{0.5263 s^{2} - 1.286 s + 2.825}{s^{3} + 2.039 s^{2} + 4.402 s + 2.521}$	13	2.9465	116.81	6027.1
Pade approximation and differentiation technique (Kranthi et al., 2013)	$\frac{2.933 s^{2} + 33.05 s + 102.8}{3.372 s^{3} + 29.51 s^{2} + 93.14 s + 102.8}$	9.2321	1.9621	14.894	20.9372
Differentiation technique (Gutman et al., 1982)	$\frac{2.933 s^{2} + 33.05 s + 102.8}{3.372 s^{3} + 29.51 s^{2} + 93.14 s + 102.8}$	9.2321	1.9621	14.894	20.9372
Balanced truncation method (Moore, 1981), Schur decomposition method (Kumar et al., 2012)	$\frac{0.1405 s^{2} - 0.8492 s + 1.881}{s^{3} + 1.575 s^{2} + 3.523 s + 1.717}$	8.5453	1.8161	90.295	4777.3
Modified balanced truncation method (Huang et al., 2013)	$\frac{0.1405 s^{2} - 0.8492 s + 1.717}{s^{3} + 1.575 s^{2} + 3.523 s + 1.717}$	4.3331	0.9209	14.073	36.9938
Routh Hurwitz method (Krishnamurthy and Seshadri, 1978)	$\frac{259.8 s^{2} + 864 s + 1700}{913.9 s^{3} + 1500 s^{2} + 3023 s + 1700}$	3.9217	0.8335	10.864	21.6615
Response matching technique (Mukherjee and Mittal, 2005)	$\frac{0.2945 s^{2} - 2.202 s + 2.32}{s^{3} + 2.501 s^{2} + 4.77 s + 2.32}$	1.6057	0.3413	9.3606	28.6362
Routh Hurwitz and Padé approx. (Pal, 1979; Shamash, 1975), Routh Hurwitz and factor division algorithm (Singh et al., 2006), Improved Routh Hurwitz (Prajapati and Prasad, 2018c)	$\frac{- 751.695 s^{2} - 504 s + 1700}{913.9 s^{3} + 1500 s^{2} + 3023 s + 1700}$	1.4852	0.3156	8.8486	26.2855
Truncation methodology (Shamash, 1981)	$\frac{291 s^{2} + 1093 s + 1700}{4684 s^{3} + 5856 s^{2} + 4620 s + 1700}$	0.7825	0.1663	7.5642	35.289
Mihailov stability and Pade approximation method (Wan, 1981)	$\frac{335.3018 s^{2} + 674 s + 1700}{3099 s^{2} s^{3} + 5031 s^{2} + 4201 s + 1700}$	0.7078	0.1504	6.4704	24.228
Modified pole clustering method and genetic algorithm (Vishwakarma and Prasad, 2009)	$\frac{- 0.264 s^{2} + 0.483 s + 0.751}{s^{3} + 2.195 s^{2} + 2.346 s + 0.751}$	0.6591	0.1401	6.292	24.138
Stability equation and big bang big crunch method (Desai and Prasad, 2013)	$\frac{0.0789 s^{2} + 0.3142 s + 0.493}{s^{3} + 1.3 s^{2} + 1.44 s + 0.493}$	0.6205	0.1319	6.2929	23.265
Mihailov stability and improved Pade approximation method (Prajapati et al., 2020)	$\frac{674 s + 1700}{3099 s^{2} s^{3} + 5031 s^{2} + 4201 s + 1700}$	0.6032	0.1282	6.0996	25.411
Stability equation (Chen et al., 1979)	$\frac{285 s^{2} + 1093 s + 1700}{3048 s^{3} + 5031 s^{2} + 4620 s + 1700}$	0.581	0.1235	4.9845	13.641
Proposed technique	$\frac{291 s^{2} + 1093 s + 1700}{3099 s^{2} s^{3} + 5031 s^{2} + 4201 s + 1700}$	0.567	0.1205	5.9323	21.842

The open loop transfer function of the reference system (41) is obtained as

\tilde{M} (s) = \frac{M (s)}{1 - M (s)} = \frac{0.16}{s (s + 1)}

(42)

By using the original plant (40), the compensator is acquired as follows

\begin{matrix} G_{c} (s) = \frac{\tilde{M} (s)}{G (s)} = \frac{e_{0} + e_{1} s + e_{2} s^{2}}{s} = \frac{K (1 + K_{1} s)}{s (1 + K_{2} s)} \\ = \frac{0.1657 + 0.2118 s - 1.949 s^{2}}{s} \end{matrix}

(43)

Hence, unknown parameters of the compensators (43) are obtained as $K = 0.1657, K_{1} = 10.4791, K_{2} = 9.2005$ . The corresponding closed loop transfer function is obtained as

\begin{matrix} R_{cl} (s) = \frac{G (s) G_{c} (s)}{1 + G (s) G_{c} (s)} \\ = \frac{0.7464 s^{3} + 1.115 s^{2} + 0.2852 s + 0.0177}{9.201 s^{5} + 7.464 s^{4} + 9.496 s^{3} + 3.008 s^{2} + 0.3959 s + 0.0177} \end{matrix}

(44)

The lower order system evaluated by the proposed methodology is given in equation (45)

R_{2} (s) = \frac{0.601 s + 0.1069}{0.7024 s^{2} + 0.8498 s + 0.1107}

(45)

By using the proposed reduced order model, the controller parameters are obtained as follows

\begin{matrix} G_{cr} (s) = \frac{\tilde{M} (s)}{R_{r} (s)} = \frac{e_{0} + e_{1} s + e_{2} s^{2}}{s} \\ = \frac{K (1 + K_{1} s)}{s (1 + K_{2} s)} = \frac{0.1657 + 0.1747 s - 1.0372 s^{2}}{s} \end{matrix}

(46)

Hence, the compensator parameters are $K = 0.1657, K_{1} = 6.9910, K_{2} = 5.9365$ . The close loop transfer of the original system with the compensator is given as follows

\begin{matrix} R_{cl} (s) = \frac{G (s) G_{cr} (s)}{1 + G (s) G_{cr} (s)} \\ = \frac{0.498 s^{3} + 0.7674 s^{2} + 0.2234 s + 0.0177}{5.936 s^{5} + 5.171 s^{4} + 6.393 s^{3} + 2.299 s^{2} + 0.3341 s + 0.0177} \end{matrix}

(47)

Figure 4 shows the closed loop original plant with compensators obtained by using various model diminution techniques. These compensators are computed by using the large scale original system and lower order systems. From this comparison, it is clear that the characteristic of the closed loop system (47) strongly resembles that of the reference model (41) in all time ranges. In Figure 5, comparison of frequency domain characteristics of closed loop systems are shown. In this figure, it is clear that the characteristic of closed loop system with compensator obtained by the original system is fully overlapped with the response of closed loop system with compensator obtained by the proposed reduced model. Table 3 displays the time domain specifications of the closed-loop plants with compensators. From comparative analysis, it can be observed that the time domain properties of the closed loop plant with the compensator acquired by using proposed reduced model are approximately similar to that of the closed-loop system in which compensator is acquired by using the actual system and these specifications are approximately similar to that of the reference model. Hence, the proposed method generates reduced order models and controllers which are very helpful for quick understanding of the large scale systems, simplifying the controller design, decreasing calculation complexity, making the faster simulation and reducing the requirements of storage space.

Example 4: Consider the flexible-missile control model and the rigid body loop given by an eighth order transfer function once again for the design of PID controller so that it gives natural frequency 10 rad/sec and damping ratio 20.

G (s) = \frac{- 1 (s + 53) (s - 53) (s^{2} - 152.2 s + 14500) (s^{2} + 153.8 s + 14500)}{(s^{2} + s + 605) (s^{2} + 45.5 s + 2660) (s^{2} + 2.51 s + 3900) (s^{2} + 3.99 s + 22980)}

(48)

Table 3.

Comparison of time domain specifications of closed loop plant with compensator.

Reduced techniques	Reduced model	Compensator $(K, K_{1}, K_{2})$	Rise time	Settling time	Peak overshoot	Peak time
—	Reference model	—	11.5604	20.9988	0	35.5749
—	Original system	0.1657, 10.4791, 9.2005	12.211	19.9322	0.2069	32.0869
Proposed method	$\frac{0.601 s + 0.1069}{0.7024 s^{2} + 0.8498 s + 0.1107}$	0.1657, 6.9910, 5.9365	12.6907	20.6907	0.3805	33.1961
Balanced trunc. (Moore, 1981), Schur decomposition method (Safonov and Chiang, 1989)	$\frac{0.4141 s + 0.6029}{s^{2} + 0.6281 s + 0.8005}$	0.2124, 1.4080, 2.3102	9.0962	28.9431	4.1222	20.2088
Stability equation method (Chen et al., 1979), stability equation and factor division method (Sikander and Prasad, 2015a)	$\frac{0.601 s + 0.1069}{0.7024 s^{2} + 0.8746 s + 0.1107}$	0.1657, 7.3352, 6.0566	12.7358	21.3274	0.2015	34.3573
Truncation method (Shamash, 1981)	$\frac{0.601 s + 0.1069}{0.7026 s^{2} + 0.8746 s + 0.1107}$	0.1657, 7.3337, 6.0552	12.7362	21.3281	0.2016	34.3576
Routh approx. (Hutton and Friedland, 1975), Routh approx. and factor division method (Prajapati and Rajendra Prasad, 2019)	$\frac{1.043 s + 0.1856}{s^{2} + 1.518 s + 0.1922}$	0.1657, 8.2240, 6.9456	12.5341	20.8615	0.1796	33.8298
Differentiation method (Gutman et al., 1982)	$\frac{0.9015 s + 0.3207}{0.7026 s^{2} + 1.749 s + 0.3321}$	0.1657, 5.7443, 4.2888	13.4505	22.6630	0.2147	38.8081
Padé approx. (Shamash, 1974)	$\frac{2.358 s + 0.4204}{s^{2} + 3.434 s + 0.4353}$	0.1657, 10.4763, 9.1965	12.2231	19.9376	0.2055	32.0875
Routh stability method (Krishnamurthy and Seshadri, 1978)	$\frac{0.601 s + 0.1069}{0.7026 s^{2} + 0.717 s + 0.1107}$	0.1657, 11.4718, 11.6169	12.4321	18.7246	1.7164	28.3222
Routh stability and factor division (Singh et al., 2006), improved Routh stability method (Prajapati and Prasad, 2018c)	$\frac{0.4488 s + 0.1069}{0.7026 s^{2} + 0.717 s + 0.1107}$	0.1657, 11.4718, 11.6169	12.4321	18.7246	1.7164	28.3222
Modified balanced truncation (Huang et al., 2013)	$\frac{0.4141 s + 0.7733}{s^{2} + 0.6281 s + 0.8005}$	0.1656, 1.7352, 2.4860	11.579	31.9822	2.7060	26.3217
Hankel norm method (Glover, 1984)	$\frac{0.1274 s + 0.8212}{s^{2} + 0.7206 s + 1.09}$	0.2124, 2.2021, 2.6965	9.3944	28.3768	3.1200	20.6982
Pade approx. and Differentiation method (Kranthi et al., 2013)	$\frac{0.9582 s + 0.3207}{0.7026 s^{2} + 1.749 s + 0.3321}$	0.1657, 5.9486, 4.67	13.1694	21.9838	0.2756	35.0147

By using natural frequency 10 rad/sec and damping ratio 20.0, the reference model is constructed as follows

M (s) = \frac{100}{s^{2} + 400 s + 100}

(49)

Hence, the open loop reference system is

\tilde{M} (s) = \frac{M (s)}{1 - M (s)} = \frac{100}{s (s + 400)}

(50)

By using the original plant, the PID controller is acquired as follows

\begin{array}{l} G_{c} (s) = \frac{\tilde{M} (s)}{G (s)} = \frac{e_{0} + e_{1} s + e_{2} s^{2}}{s} = \frac{K_{i} + K_{p} s + K_{d} s^{2}}{s} \\ = \frac{61.0396 + 2.3301 s + 0.1979 s^{2}}{s} \end{array}

(51)

Hence, $K_{p} = 2.3301, K_{i} = 61.0396, K_{d} = 0.1979$

The approximant obtained by the proposed method is

R_{2} (s) = \frac{- 2.353 \times 10^{10} s + 5.906 \times 10^{11}}{1.879 \times 10^{11} s^{2} + 1.19 \times 10^{11} s + 1.442 \times 10^{14}}

(52)

By using the proposed lower order system, the PID controller is determined as follows

\begin{matrix} G_{cr} (s) = \frac{\tilde{M} (s)}{R_{r} (s)} = \frac{e_{0} + e_{1} s + e_{2} s^{2}}{s} = \frac{K_{i} + K_{p} s + K_{d} s^{2}}{s} \\ = \frac{61.0396 + 2.3301 s + 0.1727 s^{2}}{s} \end{matrix}

(53)

Hence, the unknown compensator parameters are $K_{p} = 2.3301, K_{i} = 61.0396, K_{d} = 0.1727$ . The closed loop transfer of the actual plant with the controller computed by using the simplified model is given as follows

R_{cl} (s) = \frac{G (s) G_{cr} (s)}{1 + G (s) G_{cr} (s)}

(54)

The time response comparison of the closed loop transfer function of the actual plant with PID controllers computed by approximated systems with the reference system is given in Figure 6. In this figure, it is clear that all the time response characteristics are nearly matching in both static region and transient region with reference model except the response of the model in which the controller is computed by using approximant obtained by the balanced truncation algorithm. In Table 4, the time domain specification of the closed loop model with PID controllers is tabulated. This table shows that the properties of the closed loop model with the controller computed by the actual plant are almost matched with the properties of the closed loop model with controller determined by using simplified plants. However, the determination of controller by using approximated system is comparatively easier than the determination of controller by using high dimensional plant. It is also clear that the properties of the closed loop model with the controller are approximately similar to the reference system (49).

Table 4.

Comparison of time domain specification of the closed loop plant with PID controller.

System reduction techniques	Approximated model	PID controller $(K_{p}, K_{i}, K_{d})$	Rise time	Settling time	Peak time
—	Reference model	—	8.7826	15.641	42.1570
—	Original system	2.3301, 61.0396, 0.1979	8.8144	15.6444	42.1569
Proposed method	$\frac{- 0.1246 s + 5.906}{2.4394 s^{2} + 1.9977 s + 1442}$	1.2197, 61.0396, 0.1292	8.7471	15.5893	41.9661
Padé approximation (Shamash, 1974)	$\frac{- 0.0626 s + 1.99}{s^{2} + 4.479 s + 485.8}$	2.3299, 61.0301, 0.1979	8.8153	15.6470	42.1635
Factor division and generalized pole clustering method (Prajapati and Prasad, 2019a)	$\frac{- 0.1282 + 3.268}{s^{2} + 1.15 s + 797.9}$	2.33, 61.0404, 0.1681	8.8008	15.6403	42.1576
Routh approximation (Hutton and Friedland, 1975)	$\frac{- 0.1663 s + 7.88}{s^{2} + 37.67 s + 1924}$	2.3307, 61.0406, 0.0783	8.7823	15.6518	42.1615
Stability equation (Chen et al., 1979), factor division and stability equation (Sikander and Prasad, 2015a)	$\frac{- 0.1246 s + 5.906}{2.44 s^{2} + 28.23 s + 1442}$	2.3301, 61.0396, 0.1498	8.7986	15.6364	42.1590
Routh stability (Krishnamurthy and Seshadri, 1978)	$\frac{4.379 \times 10^{- 5} s + 5.906}{1.879 s^{2} + 1.19 s + 1442}$	−0.1027, 61.0396, 0.0798	8.6949	15.5256	41.7375
Modified pole clustering and improved Padé approx. (Singh et al., 2016)	$\frac{2.802}{s^{2} + 1.079 s + 684.2}$	−0.0564, 61.0388, 0.0894	8.6976	15.5278	41.746
Improved pole clustering (Komarasamy et al., 2011)	$\frac{- 0.1096 s + 2.802}{s^{2} + 1.079 s + 684.2}$	2.3308, 61.0388, 0.1805	8.8051	15.643	42.158
Balanced truncation (Moore, 1981), Schur decomposition method (Safonov and Chiang, 1989)	$\frac{- 0.1384 s + 1.875}{s^{2} + 0.9322 s + 605.4}$	5.8807, 80.72, 0.5676	6.8452	11.8563	19.841
Improved balanced truncation (Prajapati and Prasad, 2019b)	$\frac{- 0.097 s + 2.479}{s^{2} + 0.9322 s + 605.4}$	2.3293, 61.0405, 0.1921	8.8107	15.6437	42.1564

Conclusion

In this paper, an effective combined model order reduction method for simplifying the complexity of a linear continuous time higher order single input single output system is proposed. In this method, a stable and accurate reduced model is obtained by using the Mihailov stability criterion and truncation method. The Mihailov stability technique is applied to compute the denominator polynomial coefficients of the approximated lower order system, and the coefficients of the numerator polynomial are determined by using the truncation method. The proposed method has circumvented the limitation of the truncation method and has the inbuilt feature of retaining stability, static and dynamic characteristics of the actual system. The performance error indices of the proposed reduced models are comparatively lower than those of lower order models obtained by the notable model order reduction techniques available in the literature.

A new algorithm for the design of the controller is also proposed. In this algorithm, the proposed reduced model is used for the design of the controller and finally, this controller is implemented with the original large scale system. The effectiveness and accuracy of the proposed algorithm have been illustrated by applying it to the flexible missile control system and fuel control model. A comparison of the simulation results indicates the benchmark of the illustrated algorithm as a new potential approach to reduced order modeling and controller design. Hence, the lower order system and controller determined by the proposed techniques validate the study, synthesis and development of a controller for the large scale systems with less computational effort. However, the proposed works can be further extended as: (1) the simplification of MIMO continuous time plants, discrete time models and controller design; (2) the simplification of linear interval plants and controller design; and (3) the approximation of continuous time delay models and controller design.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Arvind Kumar Prajapati

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