Design of FOPID controller for higher order continuous interval system using improved approximation ensuring stability

This contribution deals with the design of a fractional-order proportional-integral-derivative (FOPID) controller through reduce-order modeling for continuous interval systems. First, a higher order interval plant (HOIP) is considered. The reduced-order interval plant (ROIP) for considered HOIP is derived by multipoint Padé approximation integrated with Routh table. Then, FOPID controller is designed for ROIP to satisfy the phase margin and gain cross over frequency. Thus obtained FOPID controller is implemented on HOIP also to validate the performance of designed FOPID on HOIP. A single-input-single-output (SISO) test system is taken up to elaborate the entire process of controller design. The outcomes affirm the validity of the designed FOPID controller. The designed FOPID controller produced stable results retaining the phase margin and gain cross-over frequency when implemented on HOIP. The results further proved that FOPID controller is working efficiently for ROIP and HOIP.


Introduction
The burgeoning demands for accurate depiction and simulation of system dynamics require complex and higher order mathematical models [25]. In many industrial processes such as chemical plants, steam generators, nuclear reactors, air flight systems etc., the order of dynamic system may be quite large. As the order of plant increases, the controller design becomes complex and high order controllers need to be designed [4]. The realization of these controllers demands huge expenditure and poses the probable troubles of reliability, commissioning and maintenance. Therefore, control engineers prefer to work with low order controllers [9].
While modelling any engineering system, a nominal mathematical description (also known as nominal model) is generally obtained. The coefficients of numerator and denominator parameters do have fixed values in case of nominal model. This is due to obtaining the nominal model at certain operating point. In case, the operating point of system shifts due to any of the reasons like, aging, parameter variations, sensor noises, environmental changes; it becomes to difficult to describe the dynamics of system using earlier obtained nominal model. To tackle these problems, a model in which system parameters varies within certain range may be obtained. The model in which parameters varies in certain range is known as interval system. Recently, various engineering systems are modelled as interval systems [3]. Also, various techniques for model reduction of interval systems are presented in the literature [17].
The design of controllers [19] through interval arithmetic is extended to the interval systems. The design of controller using minimization of ISE via evolutionary approaches is appeared in literature [13]. In this technique, ISE of closed loop system is minimized using two interactive genetic algorithms for interval plants while the stability is ensured using Kharitonov polynomials. In [10], Hsu and Lin presented a technique based on hybrid particle swarm optimizer for designing of digital controller. Hsu et al. [11] presented a digital controller using resemblance of extremal gain/phase margin criteria for uncertain interval systems.
The controller design methodology based on twodegrees-of-freedom criterion for interval processes is also appeared in [19]. The pre-filter guarantees the robust performance and the stability is ensured using Hurwitz theorem [19] through making use of Kharitonov theorem [7,24,26]. Yeroglu and Tan [28] designed FOPID controller by combining the Bode envelopes with design criteria for the interval plants. The articles [19,28] designed the controller for original systems. In case, the order of system is high, it is desirable to reduce the order of system for controller design. So, the controller design using reducedorder modelling should also be investigated in detail for interval systems.
In this contribution, a fractional-order proportionalintegral-derivative (FOPID) controller [1,22] is designed for reduced-order interval plant (ROIP) of higher-order continuous interval plant (HOIP) to satisfy the criteria of phase margin and gain cross over frequency. Then, this FOPID controller is implemented on HOIP. Multipoint Padé approximation integrated with Routh table derives the ROIP. The Routh table is utilized to derive the denominator of the interval model whereas multipoint Padé approximation is exploited for derivation of numerator [5]. Addition of Routh table assures for stability of ROIP. A SISO system is adopted to illustrate the entire process of controller designing. The results confirm the validity of the suggested method for designing of FOPID controller.
To present the proposed technique extensively, the paper comprises seven sections. Beginning with introductory part of the problem, the preliminaries are introduced in Sect. 2. The fractional-order calculus and FOPID controller are discussed in this section. The problem description is included in Sect. 3. The design procedure of FOPID controllers is given in Sect. 4. Section 5 covers order reduction procedure of high-order plant. Section 6 provides an illustrative test system, and lastly, Sect. 7 summarizes the work.

Preliminaries
The basic description of fractional calculus and fractional order systems is elaborated in this section. This section also describes FOPID controller.

Fractional order calculus
In fractional calculus, the continuous integro-differential operator [18] of order r is defined as with the limits of the operation bounded by u and v. The integration and differentiation of any order can be performed which can assume any value; rational, irrational or complex. The frequently used integro-differential definitions are given as follows.
The Riemann-Liouville (RL) definition [1] is given as where r takes any values between n − 1 and n, and (.) is a Gamma function.
The Grunwald-Letnikov definition [9] of differentiation is given as where (−1) q r q are the binomial coefficients C (r) q , (q = 0, 1, …) and [.] denotes the integer part. The expressions given in [9] are used to obtain the binomial coefficients.

The Laplace transform of fractional order systems
The Laplace transformations of the integro-differential expressions [9] can be given as where r is an integer number and (n − 1) < r < n . If all the derivatives of f (t) are zero, the expression given by (5) can be written as Consider a SISO system given as where output and input of the system are designated as y(t) and u(t) , respectively. By taking Laplace transform [9] of (7), following transfer function is obtained

Fractional-order proportional-integral-derivative (FOPID) controller
The FOPID controller is given as which is commonly known as PI w D z controller, where the real numbers w and z, respectively, are designated as orders of integration and differentiation. Clearly, FOPID controllers have two additional variables when compared to classical PID controller [6,8,15,21,23]. Due to this, FOPID controllers can be designed such that these perform better control and are less sensitive to parameter variations [20].

Problem description
Consider an nth-order continuous interval system Ḡ (s) for closed-loop system of Fig. 1 given as a n d̂ n y(t) dt̂ n +â n−1 d̂ n−1 y(t) The interval polynomial Ā (s) of the numerator has interval coefficients a p , a p for (p = 0, 1, … , n − 1) , with a p and a p as the lower and upper limits, respectively. Similarly, the interval polynomial B (s) of denominator has interval coefficients b p , b p for (p = 0, 1, … , n) , with combined with two design criteria b p and b p as the lower and upper limits, respectively. Figure 2 depicts the feedback system without controller. HOIP is reduced to ROIP and is represented in Fig. 3. In Fig. 3, the kth-order model Ĝ (s) of G(s) is given as The interval polynomial Ĉ (s) of the numerator has interval coefficients c p , c p for (p = 0, 1, … , k − 1) with c p and c p as the lower and upper limits, respectively. Similarly, the interval polynomial D (s) of the denominator has interval coefficients d p , d p for (p = 0, 1, ⋯ , k) with d p and d p as the lower and upper limits, respectively.
The system in Fig. 2 is approximated by that of Fig. 3, where H(s) is assumed to be same in both the cases. Fig. 4 represents ROIP with controller C (s) that is designed for ROIP. Thus designed controller is implemented on HOIP as shown in Fig 5. (10) [a 0 , a 0 ] + [a 1 , a 1 ]s + [a 2 , a 2 ]s 2 + ⋯ + [a n−1 , a n−1 ]s n−1 Fig. 1 The feedback control having controller

Design of fractional order controller
In this section, design of PI w D z controller for ROIP is presented. Five unknown parameters of PI w D z controller are calculated to satisfy gain margin and phase margin of interval system. All the design parameters of PI w D z controller are computed by solving two non-linear equations. Bode envelopes of (12) are combined with two design criteria to obtain the PI w D z controller parameters [28]. The significant frequency domain characteristics for design of FOPID controller are namely; phase margin ( pm ) and gain cross-over frequency ( gc ). pm and gc constitute key measures of robustness and stability. Therefore, the FOPID controller is designed to obtain the desired pm and gc . The expressions [16,27] given in (13) and (14) should be satisfied for a specified gc and pm .

Design of fractional order controller by use of Bode envelope
The transfer functions given in (15) and (16), respectively, offer the minimum and maximum plots of the gain.
Similarly, the transfer functions given by (17) and (18), respectively, provide the minimum and maximum plots of the phase.

Derivation of approximant for high order interval system
The high-order interval system given in (10) is approximated to (11) by multipoint Padé approximation integrated with Routh table [5,14]. The inclusion of Routh table for reduction of (10) results in stable approximation of (11).

Derivation of the denominator
D(s) , the denominator of the interval model, is derived from the Routh array given in Table 1. The Routh array is formed from the denominator of (10). The denominator of (10) is given as The entries in top two rows of Table 1 are taken from (23), while the entries in the successive rows are completed using (24).
where The entries of row numbers (n + 1 − k) and (n + 2 − k) of Table 1 are utilized to obtain the kth-order denominator, D (s) , of the interval model as given below

SISO test system
The proposed methodology is illustrated with a thirdorder test system. A third-order SISO test system [12] is considered as The desired model for (37) is written as

Determination of denominator
Using (24), Table 3 for denominator of (37) becomes Direct truncation of Table 3 gives the denominator of (38) Routh like tables, Table 4 for (41) and (42), and Table 5      The similar entries of the third rows of Table 4 and Table 5 are equated to obtain (44).

Design of FOPID controller
The FOPID controller is designed for (43). For this problem, (16) and (18) are identified as Gain cross-over frequency gc = 3.1 rad/sec and phase margin pm = 80 are the given design specifications of the system. For these criteria, (23) and (24)  The FOPID controller for the specified design criteria is obtained as The step responses of feedback system of Fig. 5 having HOIP (i.e. (37)) and controller (i.e. (50)) are depicted in Fig. 6. In Fig. 6, LL denotes lower limits of HOIP and UL represents upper limits of HOIP. The Bode plots of � C(s)Ḡ(s) , given in Fig. 7, offers the stable response. In this case also, LL and UL denote, respectively, lower limits and upper limits. From Figs. 6 and 7, it is clearly visible that the application of FOPID controller in cascade with HOIP assures the stable responses. Therefore, conclusion can be made that proposed technique has utility in designing FOPID controller for interval systems using reduced-order modeling.

Conclusion
This article proposed the design of fractional-order proportional-integral-derivative (FOPID) controller for higher order continuous interval system on the basis of reduced-order modeling. For controller design, higher order interval plant (HOIP) is first converted into reduced-order interval plant (ROIP). Multipoint Padé approximation interlaced with Routh table is used for deriving ROIP. After obtaining the ROIP, it is utilized for FOPID controller design. Phase margin and gain crossover frequency are the basis of design of FOPID controller. Thus designed FOPID controller is also implemented on higher order interval plant. The results showed that FOPID controller working efficiently when implemented on HOIP. The key feature of this article is that it is addressing the design of control using reduced-order modelling. However, as far as limitations of work proposed are considered, ROIP has to be obtained first for HOIP before proceeding to controller design. The future work of this contribution lies in design of controller for MIMO continuous and discrete interval systems.     6 Step responses of feedback control having controller and HOIP

Fig. 7 Bode plot of C (s)Ḡ(s)
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