New algorithm for the design of robust PI controller for

Original Article

New algorithm for the design of robust PI controller for plants with parametric uncertainty

Transactions of the Institute of Measurement and Control 1–9 Ó The Author(s) 2017 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/0142331216685393 journals.sagepub.com/home/tim

Danaboyina Srinivasa Rao1, Mangipudi Siva Kumar1 and Manyala Ramalinga Raju2

Abstract This paper proposes a new algorithm for the design of robust PI controller for plants with parametric uncertainty using new necessary and sufficient stability conditions. Most of the control systems operate under large uncertainty causing degradation of system performance and destabilization. In order to compensate these shortcomings, a robust PI controller is designed based on new necessary and sufficient conditions for stability of a plant with parametric uncertainty, a class of interval polynomial. New necessary and sufficient conditions for the determination of robust stability of interval polynomials have been developed using the results of Routhe’s theorem and Karitonov theorem. A set of inequalities are derived based on these developed new necessary and sufficient conditions to obtain robust controller parameters. The proposed method is simple and involves less computational complexity compared with the available methods in the literature. The efficacy of the proposed methodology is demonstrated with a numerical example for successful implementation.

Keywords Kharitonov’s theorem, Routhe’s theorem, robust stability, interval polynomials, robust controller

Introduction Most of the real plants operate in a wide range of operating conditions. Robustness is then an important feature of the closed loop system. Since control systems operate under large uncertainty, it causes degradation of system performance and destabilization. In this regard, the controller has to be able to stabilize the plant for all operating conditions. This necessitates the design of a robust controller. The problem of designing a robust controller for parametric uncertain plants called interval systems, which have unknown but bounded parameter uncertainties, attracted most of the researchers’ attention. Many methods are available in the literature for the simulation and design of controller for these systems. There has been a great amount of research work on the tuning of P, PI, PID and lag/lead controllers, because these controllers have been widely used in industries for several decades (Astrom and Hagglund, 1993, 1995; Ho et al., 1995; Zhuang and Atherton, 1993; Ziegler and Nichols, 1942). However, many affordable results have recently been reported on computation of all stabilizing P, PI and PID controllers (Ho et al. 1996, 1997a, 1997b, 1997c; Sivakumar et al., 2007). Robust stability analysis for interval systems has been a very important research topic. Expressing the characteristic polynomial by an interval polynomial is an important approach; that is, polynomial by whose coefficient each varies independently in a prescribed interval. The stability analysis of polynomials subjected to parameter uncertainty has received considerable attention after the

celebrated theorem of Kharitonov (Barmish, 1989), which assures robust stability under the condition that four specially constructed ‘extreme polynomials’, called Kharitionov polynomials (Barmish, 1989) are Hurwitz. The problem of robust stability of interval polynomial is also dealt with in Barmish (1989), Deore and Patre (2005, 2007), Chen and Wang (1997), Bartlett et al. (1988), Bhattacharya (1987), Bhattacharya et al. (1995), Dorato (1987), Dorato and Yedavalli (1990) and Patre and Deore (2003). In order to reduce the test of Hurwitz stability of the entire family, several investigations have been presented in the literature. Among these, a few imperative investigations are discussed here. Siva Kumar et al. (2015) proposed an algorithm for the design of a robust PI and PID controller. This method is based on approximating the fuzzy coefficients by the nearest interval system, and then a robust controller is designed using the necessary and sufficient conditions for stability of the interval systems. Siva Kumar et al. (2013) proposed an algorithm based on the Inverse Bilinear Transformation (IBT) to design a robust controller using the necessary and sufficient conditions for a discrete-time interval plants. Ghosh 1

Department of EEE, Gudlavalleru Engineering College, India Department of EEE, UCEK, J.N.T.U. Kakinada, India

2

Corresponding author: Mangipudi Siva Kumar, Department of EEE, Gudlavalleru Engineering College, India. Email: [email protected]

2 (1985) presented that a pure gain compensator c(s) = K stabilizes the entire interval plant family if, and only if, it stabilizes a distinguished set of eight of the extreme plants. Hollot and Fang (1990) worked on the experimental set up developed by Ghosh (1985), but allowed the controller to be first order. They prove that to robustly stabilize the extreme family, it is necessary and sufficient to stabilize the set of extreme plants that are obtained by taking all possible combinations of extreme values of the plant numerator coefficient with extreme values of the plant denominator coefficients. If the plant numerator has degree m and the plant denominator is monotonic with degree n, the number of extreme plants can be high as Next = 2m-n + 1 in Barmish et al. (1992). They proved that, it is necessary and sufficient to stabilize only 16 of the extreme plants. A complete survey of these extreme points is given in Barmish and Kang (1993). In Patre and Deore (2003), a necessary and sufficient condition for interval polynomials is proposed using the results of Nie (1976) for fixed polynomials. Patre and Deore (2011) have considered a two degree-of-freedom for interval process plant with interval time delay to guarantee both robust stability and performance. They designed a robust controller using the necessary and sufficient conditions for a chemical process plant with delay subjected to unknown, but bounded parameter uncertainties referred to an interval process plant with interval time delay. Yogesh (2011) presented simple necessary conditions for interval polynomials based on the concept of Routhe’s theorem and Kharitonov’s theorem. However, these conditions are not sufficient to design a robust stabilizing controller for interval plants. To avoid this drawback, in this paper new necessary and sufficient conditions for the stability of interval polynomial are developed, and then a PI controller is designed for an interval plant based on the proposed new necessary and sufficient stability conditions. These conditions are used to derive a set of inequalities in terms of controller parameters. These inequality constraints are solved by MATLAB Optimization tool box (2010) to obtain control parameters. This method is simple and involves less computational complexity compared with the method presented in Deore and Patre (2005). This efficacy of this method is illustrated through a typical numerical example available in the literature. The paper is organized as follows: Section 2 states a review of necessary and sufficient conditions for stability of Interval polynomial. Section 3 describes the proposed new necessary and sufficient conditions for robust stability of Interval polynomial. In Section 4, a robust controller design methodology based on developed new necessary and sufficient conditions is proposed to ensure robust stability of interval process systems. In Section 5, the proposed method is applied to design a robust PI controller for a jet-engine system. The conclusion is given in Section 6.

Review of necessary and sufficient conditions for stability of interval polynomial TheoremP1 (Barmish, 1989): An interval polynomial family AðsÞ = ni= 0 ½xi ; yi si with invariant degree is robustly stable if, and only if, its four Kharitonov’s polynomials are stable.

Transactions of the Institute of Measurement and Control Consider a real coefficient polynomial of degree n of the form AðsÞ =

Xn i=0

ai si ; ai 2 ½xi ; yi ; for yi  xi . 0

ð1Þ

Where the real coefficients ai take arbitrary value in the closed interval ½xi ; yi  are strictly Hurwitz if, and only if, the following four polynomials called Kharitonov polynomials are strictly Hurwitz: A1 ðsÞ = x0 + x1 s + y2 s2 + y3 s3 + x4 s4 +       A2 ðsÞ = x0 + y1 s + y2 s2 + x3 s3 + x4 s4 +       A3 ðsÞ = y0 + x1 s + x2 s2 + y3 s3 + y4 s4 +      

ð2Þ

A4 ðsÞ = y0 + y1 s + x2 s2 + x3 s3 + y4 s4 +      

New necessary and sufficient conditions for robust stability of interval polynomial The stability of linear systems whose parameters are constant can be determined by Routh Hurwitz criterion. But, when uncertainties are present in the system parameters, Routh criterion cannot be directly applied. When there are uncertainties in system parameters, the necessary conditions for stability can be determined by using Kharitonov’s theorem. According to Anderson et al. (1989), the necessary and sufficient condition for robust stability of interval polynomials for order n = 1 and n = 2 is positive lower bounds on the coefficients of an interval polynomial. Therefore, consider an interval polynomial of order n = 1 AðsÞ =

X1 i=0

ai si ; where ai 2 ½xi ; yi :

AðsÞ = a1 s + a0 = ½x1 ; y1 s + ½x0 ; y0 : Therefore, as per Anderson et al. (1989), the robust stability condition is x1 . 0 and x0 . 0 i:e: xi . 0

for i = 0; 1:

Similarly, for order n = 2 X2

a si = a2 s2 + a1 s + a0 i=0 i = ½x2 ; y2 s2 + ½x1 ; y1 s + ½x0 ; y0 :

AðsÞ =

Therefore, the robust stability condition is x2 . 0; x1 . 0 and x0 . 0 i:e: xi . 0 for i = 0; 1; 2: The sufficient conditions for robust stability of interval polynomials for degree greater than 3, is to use R-H criterion for all four Kharitonov’s polynomials. This will lead to an increase in computational complexity for designing a robust stabilizing controller of interval plants for an order greater than 3. This necessitates developing new necessary and sufficient conditions for robust stability of interval plants with less computational complexity. Therefore, to avoid these drawbacks, in this paper, new necessary and sufficient conditions for robust stability of interval polynomial are proposed for

3 n . 3. These stability conditions are developed using Karitonov theorem (Barmish, 1989) and Routhe’s theorem (Guillemin, 1962). The new necessary and sufficient conditions for robust stability of interval polynomials for n . 3 are presented here. Lemma 1 Consider a real Hurwitz polynomial P(s) of the form PðsÞ = pn sn + pn1 sn1 + pn2 sn2 +          + p1 s + p0 ; i = 0; 1; 2;          ; n ð3Þ

A1 =

ð4Þ

1 ½PðsÞ  PðsÞjs2 = x 2s

ð5Þ

P1 =

Theorem 2 (Guillemin, 1962): For stability of PðsÞ, the two polynomials P0 and P1 formed by the alternate coefficients of a Hurwitz polynomial in accordance with Equations (4) and (5) must have negative real zeros.

yi  xi . 0 for i = 0; 1; 2; 3;          ; n:

For fourth-order interval polynomial (n = 4). Consider the fourth-order interval polynomial as AðsÞ = a4 s4 + a3 s3 + a2 s2 + a1 s + a0 Where a0 2 ½x0 ; y0 ; a1 2 ½x1 ; y1 ; a2 2 ½x2 ; y2 ; a3 2 ½x3 ; y3 ; and a4 2 ½x4 ; y4 

ð11Þ

A3 ðsÞ = y4 s4 + y3 s3 + x2 s2 + x1 s + y0

ð12Þ

A4 ðsÞ = y4 s4 + x3 s3 + x2 s2 + y1 s + y0

ð13Þ

1 1 ½A ðsÞ  A1 ðsÞjs2 = x = y3 x + x1 2s

ð14Þ ð15Þ

According to Theorem 2, for stability of A1 ðsÞ; the above two polynomials A10 and A11 must have negative real zeros. Therefore, from Equations (14) and (15), the stability conditions for polynomial A1 ðsÞ are y22 . 4x0 x4

ð16Þ

x1 \0 y3

ð17Þ

Similarly, applying Lemma 1 to Equation (11), the A2 ðsÞ can be represented into two polynomials A20 and A21 as given below. 1 A20 = ½A2 ðsÞ + A2 ðsÞjs2 = x = x4 x2 + y2 x + x0 2

ð18Þ

1 2 ½A ðsÞ  A2 ðsÞjs2 = x = x3 x + y1 2s

ð19Þ

According to Theorem 2, for stability of A2 ðsÞ; the above two polynomials A20 and A21 must have negative real zeros. Therefore, from Equations (18) and (19) the stability conditions for polynomial A2 ðsÞ are

ð6Þ

y22 . 4x0 x4

ð20Þ

ð7Þ

y1 \0 x3

ð21Þ

Using Lemma 1, The AðsÞ can be represented into two polynomials A0 and A1 as given below. 1 A0 = ½AðsÞ + Aðsjs2 = x = a4 x2 + a2 x + a0 2

A2 ðsÞ = x4 s4 + x3 s3 + y2 s2 + y1 s + x0

1 A1 =

2 A1 =

Sufficient stability conditions for an interval polynomials

ð10Þ

1 1 1 1 2 A0 = ½A ðsÞ + A ðsÞjs2 = x = x4 x + y2 x + x0 2

Consider an interval polynomial of order n . 3 of the form

where ai 2 ½xi ; yi  for i = 0; 1; 2; 3;          ; n: The necessary conditions for an interval polynomial to be stable is given as

A1 ðsÞ = x4 s4 + y3 s3 + y2 s2 + x1 s + x0

Applying Lemma 1 to Equation (10), the A1 ðsÞ can be represented into two polynomials A10 and A11 as given below:

Necessary stability conditions for an interval polynomials AðsÞ = an sn + an1 sn1 +          + a1 s + a0 ;

ð9Þ

According to Theorem 2, for robust stability of interval polynomial AðsÞ, the above polynomials A0 and A1 must have negative real zeros i.e. a22 . 4a0 a4 and a1 a3 \0 Applying Kharitonov’s theorem to Equation (6), the interval polynomial AðsÞ, can be represented into the following four polynomials:

Where pi is real and positive, p0 . 0 If any complex number Z such that Re . 0; jf ðzÞj . jf ðzÞj; moreover, jf ðzÞjz on C . jf ðzÞjz on C ; where C is a Closed contour, then according to Routhe’s theorem (Guillemin, 1962) the following two polynomials can be formulated. 1 P0 = ½PðsÞ + PðsÞjs2 = x 2

1 ½AðsÞ  Aðsjs2 = x = a3 x + a1 2s

ð8Þ

Similarly, applying Lemma 1 to Equation (12), the A3 ðsÞ can be represented into two polynomials A30 and A31 as given below. 1 A30 = ½A3 ðsÞ + A3 ðsÞjs2 = x = y4 x2 + x2 x + y0 2

ð22Þ

4

Transactions of the Institute of Measurement and Control A31 =

1 3 ½A ðsÞ  A3 ðsÞjs2 = x = y3 x + x1 2s

ð23Þ

According to Theorem 2, for stability of A3 ðsÞ; the above two polynomials A30 and A31 must have negative real zeros. Therefore, from Equations (22) and (23) the stability conditions for polynomial A3 ðsÞ are x22 . 4y0 y4

ð24Þ

x1 \0 y3

ð25Þ

A41 =

1 4 ½A ðsÞ  A4 ðsÞjs2 = x = x3 x + y1 2s

. 4y0 y4

y1 \0 x3

ð26Þ ð27Þ

A2 ðsÞ = y5 s5 + x4 s4 + x3 s3 + y2 s2 + y1 s + x0 A3 ðsÞ = x5 s5 + y4 s4 + y3 s3 + x2 s2 + x1 s + y0 and

ð29Þ

x22 . 4y0 y4

ð30Þ

y1 \0 x3

ð31Þ

For fifth-order interval polynomial (n = 5). Consider the fifthorder interval polynomial as ð32Þ

Where a0 2 ½x0 ; y0 ; a1 2 ½x1 ; y1 ; ð33Þ a2 2 ½x2 ; y2 ; a3 2 ½x3 ; y3 ; a4 2 ½x4 ; y4  and a5 2 ½x5 ; y5  Using Lemma 1, The AðsÞ can be represented into two polynomials A0 and A1 as given below 1 A0 = ½AðsÞ + Aðsjs2 = x = a4 x2 + a2 x + a0 2

ð36Þ

A4 ðsÞ = y5 s5 + y4 s4 + x3 s3 + x2 s2 + y1 s + y0 Applying Lemma 1 and Theorem 2 to all Kharitonov’s polynomials, the stability conditions for the fifth-order interval polynomial are obtained and they are: A1 ðsÞ : y22 . 4x0 x4 and y23 . 4x1 x5 A2 ðsÞ : y22 . 4x0 x4 and x23 . 4y1 y5 A3 ðsÞ : x22 . 4y0 y4 and y23 . 4x1 x5

ð37Þ

A4 ðsÞ : x22 . 4y0 y4 and x23 . 4y1 y5

ð28Þ

Remark 1: If the stability condition x22 . 4y0 y4 for the Kharitonov polynomials A3 ðsÞ and A4 ðsÞ is satisfied, then the stability condition y22 . 4x0 x4 for the Kharitonov polynomials A1 ðsÞ and A2 ðsÞ is also satisfied. 1 Similarly, if the stability condition y x3 \0 for the Kharitonov polynomials A2 ðsÞ and A4 ðsÞ is satisfied, then the 1 stability condition x y3 \0 for the Kharitonov polynomials 1 3 A ðsÞ and A ðsÞ is also satisfied. Thus, the sufficient conditions for robust stability of fourth order interval polynomial are

AðsÞ = a5 s5 + a4 s4 + a3 s3 + a2 s2 + a1 s + a0

ð35Þ

According to Theorem 2, for robust stability of interval polynomial AðsÞ, the above polynomials A0 and A1 must have negative real zeros; that is, a22 . 4a0 a4 and a23 . 4a1 a5 : Applying Kharitonov’s theorem to Equation (32), the interval polynomial A(s), can be represented into the following four polynomials

According to Theorem 2, for stability of A4 ðsÞ, the above two polynomials A40 and A41 must have negative real zeros. Therefore, from Equations (26) and (27) the stability conditions for polynomial A4 ðsÞ are x22

1 ½AðsÞ  Aðsjs2 = x = a5 x2 + a3 x + a1 2s

A1 ðsÞ = x5 s5 + x4 s4 + y3 s3 + y2 s2 + x1 s + x0

Similarly, applying Lemma 1 to Equation (13), the A4 ðsÞ can be represented into two polynomials A40 and A41 as given below. 1 4 4 4 2 A0 = ½A ðsÞ + A ðsÞjs2 = x = y4 x + x2 x + y0 2

A1 =

ð34Þ

Remark 2: From Equation (37), the sufficient conditions for robust stability of fifth order interval polynomial are: x22 . 4y0 y4

ð38Þ

x23 . 4y1 y5

ð39Þ

For sixth-order interval polynomial (n = 6). Consider the sixthorder interval polynomial as AðsÞ = a6 s6 + a5 s5 + a4 s4 + a3 s3 + a2 s2 + a1 s + a0

ð40Þ

Where a0 2 ½x0 ; y0 ; a1 2 ½x1 ; y1 ; a2 2 ½x2 ; y2 ; a3 2 ½x3 ; y3 ; a4 2 ½x4 ; y4 ; a5 2 ½x5 ; y5  and a6 2 ½x6 ; y6 

ð41Þ

Using Lemma 1, The AðsÞ can be represented into two polynomials A0 and A1 as given below: 1 A0 = ½AðsÞ + Aðsjs2 = x = a6 x3 + a4 x2 + a2 x + a0 2 A1 =

1 ½AðsÞ  Aðsjs2 = x = a5 x2 + a3 x + a1 2s

ð42Þ ð43Þ

According to the Theorem 2, for robust stability of interval polynomial AðsÞ, the above polynomials A0 and A1 must have negative real zeros i.e. a22 . 3a0 a4 and 2 a3 . 4a1 a5 : Applying Kharitonov’s theorem to Equation (40), the interval polynomial AðsÞ can be represented into the following four polynomials:

5 Table 1. Robust stability conditions for various higher order interval polynomials. Order of the polynomial

Robust stability conditions

n¼4

x22 .4y0 y4 and

y1 \0: x3 2 2 x2 .4y0 y4 and x3 .4y1 y5 . x22 .3y0 y4 and x32 .4y1 y5 : x22 .3y0 y4 and x32 .3y1 y5 . y2 \0. x32 .3y1 y5 , x32 .y1 y5 and x6 y2 y3 2 2 y4 .4x0 x8 , y5 .4x1 x9 , \0 and \0 x6 x7 y 3 y42 .4x0 x8 , y52 .4x1 x9 , y62 .4x2 x10 and \0 x7

n¼5 n¼6 n¼7 n¼8 n¼9 n ¼ 10

3

6

5

4

3

2

4

6

5

4

3

2

A ðsÞ = x6 s + x5 s + y4 s + y3 s + x2 s + x1 s + y0 and

ð44Þ

A ðsÞ = x6 s + y5 s + y4 s + x3 s + x2 s + y1 s + y0

A ðsÞ : 2

A ðsÞ : A3 ðsÞ : 4

A ðsÞ :

y22 y22 x22 x22

. 3x0 x4 and . 3x0 x4 and . 3y0 y4 and . 3y0 y4 and

y23 x23 y23 x23

Consider a plant with parametric uncertainty represented by its transfer function as Gðs; a; bÞ =

N ðs; aÞ Dðs; bÞ

ð48Þ

N ðs; aÞ = a0 + a1 s + a2 s2 + a3 s3 + a4 s4 +       + am1 sm1 + am sm Dðs; bÞ = b0 + b1 s + b2 s2 + b3 s3

Applying Lemma 1 and Theorem 2 to all Kharitonov’s polynomials, the stability conditions for the sixth-order interval polynomial are obtained and they are: 1

Design procedure for robust PI controller

Where the numerator and denominator polynomials are of the form

A1 ðsÞ = y6 s6 + x5 s5 + x4 s4 + y3 s3 + y2 s2 + x1 s + x0 A2 ðsÞ = y6 s6 + y5 s5 + x4 s4 + x3 s3 + y2 s2 + y1 s + x0

Figure 1. Block diagram of interval Plant with a robust controller.

ð49Þ

+ b4 s4 +       + bn1 sn1 + bn sn + a 2 A = ½a i ; ai  for i = 0; 1; 2;       ; m + b 2 B = ½b i ; bi  for i = 0; 1; 2;       ; n

. 4x1 x5 . 4y1 y5 . 4x1 x5

ð45Þ

. 4y1 y5

Remark 3: From Equation (45), the sufficient conditions for robust stability of sixth order interval polynomial are: x22 . 3y0 y4

ð46Þ

x23 . 4y1 y5

ð47Þ

+ 2 + Where bn = ½1; 1 and the bounds a2 i ; ai ; bi and bi are specified a priori and m  n. Let the stabilizing PI controller transfer function be given by

CPI ðsÞ = K1 +

K2 Nc ðsÞ = Dc ðsÞ s

ð50Þ

Now the system with robust controller for Parametric Uncertainty is as shown in Figure 1. Then, the closed loop transfer function with PI controller can be defined as Nc ðsÞN ðs; aÞ Nc ðsÞN ðs; aÞ + Dc ðsÞDðs; bÞ

Similarly, the robust stability conditions are derived for various higher order polynomials. The robust stability conditions for higher-order interval polynomials are represented in Table 1. New necessary and sufficient conditions for the stability of an interval polynomial of order greater than three are developed in this section. The developed stability conditions have the following advantages:

The Characteristic equation of this system with PI controller is given as

1. The stability of interval polynomials can be determined easily using these developed stability conditions without formulating the four Kharitonov polynomials, unlike Kharitonov’s theorem. 2. Using these stability conditions, a robust PI/PID controller can be designed easily that can stabilize the given plant under large uncertainty. 3. As it requires a smaller number of stability conditions, the number of computations required for the development of stabilizing controller are fewer when compared with the other methods.

Where, N ðs; aÞ and Dðs; bÞ are the numerator and denominator polynomials of the plant considered respectively and Nc ðsÞ and Dc ðsÞ are the numerator and denominator polynomials of PI controller transfer function respectively. This PI controller robustly stabilizes the interval plants family, if for all a 2 A and b e B, then the characteristic polynomial of closed loop transfer function given in Equation (52) has all zeros have negative real values . Now, by applying the necessary and sufficient conditions, given in Table 1, to the closed-loop polynomial Dðs; a; bÞ, it leads to a set of inequalities in terms of controller parameters. Then these inequalities can be solved by using MATLAB-Optimization tool box (2010) so

T ðsÞ =

Dðs; a; bÞ = Nc ðsÞN ðs; aÞ + Dc ðsÞDðs; bÞ

ð51Þ

ð52Þ

6

Transactions of the Institute of Measurement and Control

Table 2. Variation of k1 and k2 for different values of e for proposed method and existing method. Controller set

1 2 3 4

e

Proposed method

0.5 1 1.5 2

Method given Deore and Patre (2005)

K1

K2

K1

K2

0.1097 0.2194 0.3291 0.4388

0.0007 0.0014 0.0018 0.0023

0.0028 0.0057 0.0085 0.00113

0.0005 0.0011 0.0016 0.0021

 P as to minimize the objective function J = nj= 1 Kj j: to obtain controller parameters. Then, after obtaining the Controller parameters, using Equation (2), form four sets of Kharitonov’s polynomials (Barmish, 1989) to check the stability and the closed-loop step response and then to verify the results.

Inequality constraints:  4:558K1 + e\0 3920K2  9409 + e\0  980K1 + e\0  940K2 + e\0

Design of robust controller for jet engine In this section, a design procedure for a robust PI controller of a plant with parametric uncertainty is illustrated. Consider a jet engine (Deore and Patre, 2005) whose transfer function with parametric uncertainty is given by Gðs; a; bÞ =

a0 b3 s3 + b2 s2 + b3 s

Where a0 2 ½940 ; 980 ; b3 2 ½1 ; 1; b2 2 ½215 ; 230 and b1 2 ½97 ; 107: The given Jet-Engine is stable interval system. In order to improve its time domain specifications, it is desired to design a PI controller. Then the transfer function of the PI controller is given by CPI ðsÞ = K1 +

K2 Nc ðsÞ = Dc ðsÞ s

The CPI(s) will stabilize the given plant, if the closed loop interval polynomial is stable. Then the closed loop transfer function with PI controller becomes T ðsÞ =

The controller parameters K1 and K2 are restricted to small values by choosing the objective function J properly. The linear programming problem consists of two decision variables and four constraints. In the linear programming problem, the purpose of using a small positive number e is to formulate feasible set closed. The feasible parameter region will be influenced by the value of p. In this work, ‘‘FEMINCON’’ is used to minimize the objective function J in MATLAB. It finds a constrained minimum of a function of several variables. It attempts to solve problems of minimization subject to linear as well as nonlinear constraints (Mathworks Inc, 2010). The various values of K1 and K2 for different values of e are shown in Table 2. The values of the controller parameter K1 and K2 are increased by increasing e, which shows the sensitivity of controller parameters with respect to feasible set parameter e. The closed loop step response of system with PI controller for both proposed method (k1 = 0.3291 and k2 = 0.0018) and the method given in Deore and Patre (2005) (k1 = 0.0085 and k2 = 0.0016) are shown in Figure 2 and Figure 3 with controller set 3(e = 1.5) respectively. The step response

½940K1 ; 980K1 s + ½940K2 ; 980K2  ½1 ; 1s4 + ½215 ; 230s3 + ½97 ; 107s2 + ½940K1 ; 980K1 s + ½940K2 ; 980K2 

From the above equation, the characteristic equation of closed loop interval system with PI controller can be taken as Dðs; a; bÞ = ½1 ; 1s4 + ½215 ; 230s3 + ½97 ; 107s2 + ½940K1 ; 980K1 s + ½940K2 ; 980K2 

ð54Þ

By applying the new necessary and sufficient conditions from Equations (1), (30) and (31) to the closed loop polynomial (54), a set of inequality constraints are obtained. In order to make these set of constraints into a feasible closed set, a small positive number ‘e’ is introduced in to the constraints. Hence, the optimization problem can be stated P as to find K1 and K2 such that objective function J = nj= 1 Kj j: is minimized, subjected to the following constraints.

ð53Þ

comparison of four extreme plants with PI controller obtained by the proposed method and the method given in Deore and Patre (2005) is shown in Figure 4. The time domain specifications of Figure 4 are shown in Table 3, which describes the efficacy of the proposed method for the design of robust PI controller. It has been observed from the Figure 4 that the designed PI controller robustly stabilizes the plant very quickly when compared with the method given in Deore and Patre (2005). It is observed from Table 3 that the time domain specifications, such as peak over shoot, settling time, rise time and peak time have been improved when compared with the existing method of Deore and Patre (2005). The decrease in

7 Table 3. Time domain specifications for proposed and existing method. Name of the polynomial

First Second Third Fourth

Proposed method

Existing method (Deore and Patre, 2005)

% peak overshoot MP

Peak time tp(sec)

Rise time tr(sec)

Settling time ts(sec)

% peak overshoot MP

Peak time tp(sec)

Rise time tr(sec)

Settling time ts(sec)

53.6 53.0 57.0 56.6

2.77 2.59 2.81 2.63

1.05 0.99 1.04 0.981

16.9 15.8 19.5 16.4

65.5 62.3 69.0 65.4

23.0 22.4 21.4 21.0

7.93 7.05 7.39 7.31

159 155 170 146

Figure 2. Closed loop step response with PI controller for all extreme plants (set-3) using proposed method.

Figure 3. Closed loop step response with PI controller for all extreme plants (set-3) using the method given in Deore and Patre (2005).

percentage peak over shoot indicates, that the designed PI controller robustly stabilizes the dynamic response of the system. It has been observed, that the relative settling time of the

system with proposed PI controller is less than the existing method (Deore and Patre, 2005). As the relative settling time is the measure of relative stability, thereby lower settling time

8

Transactions of the Institute of Measurement and Control

Figure 4. Closed loop step response comparison of PI controller.

will improve the relative stability of the system. It is also observed, that the relative rise time and peak time of the system with proposed PI controller is lower than the method given in Deore and Patre (2005). As the relative rise time of the system indicates the measure of speed of the response, the speed of the response of the system with the proposed controller is improved. This shows the efficacy of the proposed method in terms of time domain specifications. The proposed method is simple and requires fewer computations as it uses a smaller number of inequality constraints than the method given by Deore and Patre (2005). Thus, the developed PI controller using new necessary and sufficient conditions of interval polynomial robustly stabilizes the jet engine. The proposed new stability conditions can be implemented easily for determining the stability of higher order interval plants.

Conclusions In this paper, a new algorithm for the design of robust PI controller for interval plants with parametric uncertainty is proposed. The proposed method uses the developed new necessary and sufficient conditions for stability of interval polynomial to derive a set of inequalities in terms of controller parameters. These inequalities can be solved to obtain robust controller parameters. The proposed PI controller procedure is illustrated through a typical numerical example. It has been observed from the simulation results that the designed PI controller stabilizes the plant with fewer time domain parameters than the existing method. The stability of interval polynomials can be judged easily using these developed new necessary and sufficient conditions without formulating the four Kharitonov polynomials like Kharitonov’s theorem. Hence, we conclude that the designed PI controller obtained from the developed new necessary and sufficient conditions stabilizes the plant superior than the methods available in the literature. The proposed method is simple and requires fewer computations as the number of inequality constraints is smaller than the existing method. This shows the efficacy of

the proposed PI controller using the developed new necessary and sufficient stability conditions for an interval polynomial.

Declaration of conflicting interests The authors declare that there is no conflict of interest.

Funding This research received no specific grant from any funding agency in the public, commercial or not-for-profit sectors.

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Appendix Nomenclature K1 K2 A(s) P(s) A1(s) A2(s) A3(s) A4(s) P0 and P1 A0 and A1 A10 and A11 A20 and A21 A30 and A31 A40 and A41 G(s, a, b) N(s, a) D(s, b) CPI(s) T(s) D(s, a, b)

Proportional gain Integral gain Interval polynomial Real Hurwitz polynomial First Kharitonov’s polynomial of A(s) Second Kharitonov’s polynomial of A(s) Third Kharitonov’s polynomial of A(s) Fourth Kharitonov’s polynomial of A(s) Even real Hurwitz polynomials of P(s) Even real Hurwitz polynomials of A(s) Even real Hurwitz polynomials of A1(s) Even real Hurwitz polynomials of A2 (s) Even real Hurwitz polynomials of A3 (s) Even real Hurwitz polynomials of A4 (s) Transfer function of interval plant Numerator polynomial of G(s, a, b) Denominator polynomial of G(s, a, b) Transfer function of PI controller Closed loop transfer function of interval plant. Denominator polynomial of T(s)