Stability Regions of Fractional-Order PFDß

Controllers with dead-time plant Adil Zulfiqar1, Nisar Ahmed2

Faculty of Electronic Engineering. Ghulam Ishaq Khan Institute of Engineering Sciences and Technology

Topi, Pakistan 2 Faculty of Electronic Engineering.

Ghulam Ishaq Khan Institute of Engineering Sciences and Technology Topi, Pakistan

adilzulfiqar77@yahoo.com (1), nisarahmed@giki.edu.pk (2)

Abstract—The aim of this paper is to show the stability areas of fractional order PID controllers to stabilize the first-order system plus time-delay. This is a new approach generalized from the classical case. First-order open-loop stable and unstable system is consider in this paper. Quasi-polynomial characteristic equation produced by the plant having dead-time. Generalization version of the H. Biehler and Pontryagin theorem are applicable to quasi-polynomials. The problem for the solution of fractional order PID controllers stabilization introduced here is based on determine range of controller's gain for which controller is stable and thus stabilize the plant with dead-time.

Index Terms— First-order plat with dead-time, Hermite-Biehler theorem, Pontryagin theorem, Fractional-order PID controllers, Fractional order systems, Fractional calculus

I. INTRODUCTION PID (Proportionl+Integral+Derivative) controllers operate

the majority of control systems in the world now-a-days. In fact it has been surveyed that in pulp and paper industries 98% of the control loops are controlled by PI controllers [1], and more than 95% PID controllers are used by process control applications [2], Since PID controllers have a great impact on industries and it is authorize that yet a little change in the design of PID controllers could have enormous affect all over the world in the field of control systems. During the last few years remarkable studies have been proposed on the fractional order PID controllers (F0-PID) due to the advances in the field of fractional calculus. F0-PID controllers are the generalized form of the classical PID controllers. The transfer function of such controllers is given as:.

Gf = Gp+^ + GäSß (1) whereGp, G¡ and Gd are the proportional, integral and

derivative gain and aand ß are the integral and derivative order respectively. Using such controllers are advantageous because two more parameters i.e. integral and derivative order are available for design purpose and fractional operator used to design F0-PID provides a powerful tool for the description of storage and ancestry attributes of many materials [3]. The theory and basic concept of fractional calculus (FC) are

presented in [4-5]. During the last few years many scientists took interest in the design and tuning of F0-PID controllers. Menabe in [6] and Li Yuankai in [7] introduced the frequency response and transient response of non-integer integrals and its applications to control systems. Optimal tuning of fractional order controllers are also studied in [8]. The non-integer order techniques for the dynamic systems control and superior performance of the CRONE controllers studied by oustaloup in [9,10]. Generalization of the classical PID controllers namely P I A D P

taking a and ß as integral and derivative order respectively proposed by podlubny in [11], Podlubny showed the better response of FO-PID over classical PID controllers when fractional order systems (F0S) are used. In [12] frequency response of FO-PID controllers is also studied.

Quasi-polynomial characteristic equations produced by the plants whose characteristics possesses dead-time. We will use the generalized interpretation of the H. Biehler theorem applicable to the quasi-polynomial. In [13] for stabilization of the PID controllers extrapolation of H. Biehler theorem was deduced. In this paper FO-PID controllers are used with time-delay plant and extrapolation of H. Biehler theorem will be applied for the stabilization regions of FO-PID controllers. In [14,15] Pontyagin derived H. Biehler theorem and developed the graphical idea to stabilize the time-delay systems.

The rest of the paper is organized as follows. In section II brief history and mathematical description of fractional calculus will be given. In section m basics of fractional order controllers and systems will be considered. In section IV problem will be characterized. In section V theory of dead-time processes will be discussed. In section VI generalized version of Pontryagin and in VII Hermite-Biehler theorem will be considered. In section Vm stability regions of FO-PID controller will be established. Results is discussed in section EX. At the end figures and conclusion will complete the paper.

H. MATHEMATICAL BACKGROUND OF FRACTIONAL CALCULUS

At the end of 17th century the history of fractional calculus started and origin of fractional calculus was mainly due to exchange of letters. From one of those letters Leibniz wrote a

978 978-1-4799-3457-7/13/$31.00 ©2013 IEEE

letter to L'Hôpital and put a following question [16]: "Can we generalize integer order derivatives to non-integer order derivatives?" The question was strange and aroused L'Hôpital and he replied with another question that: "What will happen if order of derivative becomes 1/2"? Leibniz replied to L'Hôpital with historical words dated 30th

September 1695: "It will lead to paradox from which one day useful consequences can be drawn". Now-a-days on the basis of these letters many scientists believed that the precise birthday of fractional calculus is 30th

September 1695 and considered Gotfried Leibniz as father of fractional calculus [17].

/ dr

a n = dtr

1

Uw ,Re(r > 0) ,Re(r = 0) ] ,Re(r < 0),

(2)

where a and t in the above formula are terminals of operator and r represents the order of derivative. Following are some definitions commonly used in FC.

A. Grunwald-Letnimov's Definition

j v m = iimh h~r x ^ f ( - i y Q / ( t - j h ) (3)

in this formula [(t — a) /h] is integral part and (y) is binomial coefficient.

B. Riemann-Liouville's Definition 1 dn ft f(T)

*Ftrm=-r - J — 1 Ja ft-T - dr r(n-r) dtn Ja (t-z)r~n+1

for (n-l<r<n) and where T(.) is the Gamma function.

C. Cauto 's Definition

for (n-l<r<n). r ( n - r ) Ja ( t - T ) r " n + 1 d.T

(4)

(5)

m. FO-CONTROLLERS AND SYSTEMS Fractional order systems are those systems which can be

well described by fractional integrals and derivatives and fractional order controllers are those that can control fractional order systems. Better explanation of storage and ancestry properties of many materials can be done by fractional integrals and derivatives. Following equations show the continuous LTI model of fractional system.

B(Fao^-ai)y(t) = C(f"OA /?m)u(t) (6) Where B and C are fractional derivative laws and y and u are output and input respectively. Or explicitly

dnF^yCt) + an_1FCCn-1y(t)+.. +a0Fa°y(t) = bmF^mu(t) + bm_iFß^~iu{t)+.. +b0Fß°u(t) (7) Equation (7) is in time domain. Transfer function of fractional order system is given below:

tffcO) = aicsak+aic-1sak-i+..+a1sai+a0sao (8)

IV. PROBLEM CHARACTERIZATION Systems having step response shown in figure (1) are

commonly known first order system with time delay as shown below mathematically.

mm

Figure 1 Open loop step response.

P ( s ) = - A - e l+Ts -is (9)

where k represents the gain of system, L and T represents the time delay and time constant of the system.

Figure 2 Feedback Control System. Feedback control system is shown in figure (2) consists of a controller C(s) and a plant G(s) y,r,u are output input and controlled output respectively. The controller used in this figure is fractional order as shown in equation (1) and our purpose is to determine all the controller's gain for which given closed loop system is stable.

V. THEORY FOR DEAD-TIME PLANTS Many control applications involve dead-time in its

characteristic, such as heated tank with long pipe, and transport mass or energy etc. additional lag is introduced in the system phase by dead-time, which causes to decrease phase and gain margin of the transfer function results in making difficult control of the system. These dead-time (time delay) produces dynamic models with characteristic equation of the following form.

r (s ) = b(s) + e~sTlm1 (s) + e_ s T 2m2(s) + - + e'^m^Çs) (10)

Whereas b(s) and m¡(s), I = 1,2,... are the polynomials with real coefficients.

Quasi-polynomial characteristic equation consists of variable in the form s and es which is used for the stability of closed loop system when system characterized by time delay. Following are two assumptions used to characterize this class of systems. Bi) deg[b(s)]=p and deg[m¡(s)] < p for all I = 1,2,... B2) 0 < Ti < T2 < - < Tk

Assumptions given above are used to define quasi-polynomial with a non-zero principal term, non-zero means that coefficient of the term containing highset power of s and es is non-zero. Quasi-polynomial without principal term characterized by infinite roots with positive real parts that

make closed loop system unstable. Considering the previous equation (10) esTn term has infinite zeros, thus we assume another equation equivalent to (10) of the following form.

r*(s) = e ^ r ( s ) = e ^ b { s ) + e ^ - ^ m ^ s ) + es(Tn-T^m2(s)+..+mn(s) (11)

The condition for the stability of the system having the characteristic equation like (10) is that all the zeros of equation (11) lie in the open left half plane [18].

VI. HERMITE-BIEHLER THEOREM Consider equation (11) in the following form.

r*(s) = rr*(w) + jr*(w) fors = jw (12) Where r* (w) and r¿"(w) represents the real and imaginary parts of quasi-polynomial respectively. Using the assumptions given above equation (12) is stable if and only if following two conditions fulfill.

1) r* (w) and rt* (w) have only real roots and these roots are interlaced.

2) r>(w')r r*(w) - rf (w')rr'*(w) > 0 for w = w' from (-00,+00)

Where bar (') shows the derivative w.r.t w. From the 2nd

condition it is analytical assumption that vector r*(/w) rotates continuously in the positive direction with positive velocity as w varies from —oo to +oo.

Vn. PONTYAGIN THEOREM Let A and B denote the highest power of s and es

respectively in r*(/w). Let S be the constant such that coefficients of the highest degree terms in rr*(w) and rf(w) do not disappear at w = S. Thus folloeing interval is necessary and sufficient for the equations rf (w) = 0 and rr*(w) = 0 have only real roots i.e.

—2mn + S < w < 2mn + 5 , m = 1,2,3.. (13) Thus rf (w) and rr*(w) have exactly 4mA + B roots, beginning with adequate large m. If characteristic equation has fractional order A and B then rf(w) and r*(w) must have exactly 4m([A] + 1) + [m] + 1 roots.

VIH. STABILITY REGIONS OF F0-PIADB

Characteristic equation of the closed-loop system shown in figure (2) with F0-PIaD^ controller is given below.

r(s) = (kGt + kGpsa + kGdsa+P)e~Ls + (1 + Ts)sa

(14) H. Biehler and Pontyagin theorem will be applied on equation (14) to solve stability problem and find all stabilizing regions of PIaDp controller. Let's begin with quasi-polynomial r(s) as follows:

r*(s) = kGi + kGpsa + fcGdsa+^ + (1 + Ts)saeLs = m(s) + ¿>(s)eis (15) Where deg[b(s)] = a + 1 and deg[m(s)] = a + ß conditions like a + ß < a + l=>ß < 1 and L > 0 must be verified to fulfil Bj and B2. Equation (15) can be written as follows if we choose a = x/y and ß = zjy.

"(s) = sx/y [fcGdsy + kGp + (1 + r s )e i s ] + kGt

(16)

+

(17)

Putting P = Ls in above equation we get:

r'(P) = QX,y[kGdQKkGp + (l + T ^

kGt Thus for P = jw, r*(Jw) becomes:

r-(jw) = ^ f k G d + ( f f k G p + ( f f

* (cosw — j wsinw + jsinw +jj wcosw) + kGt (18) Real and imaginary parts of above equation are given below:

X+z

y I f i X~hZ —

+ (kGp + cosw - jwsinwj |fle \w\* -(sinw +

-wcosw | / m | o > j | \w\y ffisign(w) (19)

I f í C + Z \ i x+z x+z

/mj0)~J| y sign(w)

+ (kGp + cosw — —wsinw * sign(w)

+(sinw + ~ wcosw j Im [0')y]| My Q)y (20) It is possible to realize from (19) & (20) that fixing integral and derivative order x, y, z real part (19) depends only on three parameters (Gp, Gt, Gd), but imaginary part (20) depends only on two i.e. (Gp, Gd) the following consecutive step are given to verify that roots of real and imaginary parts of r*(/w) are interlaced. For this purpose values of real part are calculated in the zeros of the imaginary part of r*(/w). For w * 0 the real part can be written as follows:

rr*(w) = k j Re {(/)ï}| \w\y * [Gp + a(w)Gd +

(21) b(w)Gt + c(w)] Where

Í X+Z-

0) y

b(w) = l\Re[(j)y] O S and

c(w) = ^ [(cosw — ţ wsin\vj —

Csinw + jwc0sw)sign(w)]

Since imaginary part of r*(/w) is an odd function it always possesses root at w = 0, thus for w = w0 = 0

r / K , ) = kGi (22) For w = Wj 0

X

rr'(jw) = k\üe {(/)'} | M* Q" * [Gp + a(w)Gd + b(yr)Gi + c(w)] (23) Thus following conditions should hold in order to verify interlace property between real and imaginary roots of r * (jw).

r r * ( w 0 ) > 0 ^ / f > 0 (23)

And

( - i y r r ' ( w j ) > 0 ^ ( - i y a ( w j ) G d +

( - 1 Vb(wj)Gt + (.-iyc(wj)>(-iyGp (24) Real part of r*(Jw) related to these conditions and define a volume in the space (Gv, Gt, Gd). 2nd condition of H. Biehler theorem ri*(w )rr*(u' ) - ri*(w )rr.'*(w ) > 0 for some w = w defined another volume in the space.

IX. RESULTS

To calculate the set of controller's gains to ensure the close-loop system stability following procedures are developed. The first method based on Pontryagin theorem which ensures real roots of rf according to m by defining the range of values of Gp and Gd. The second method defines the set of controller's parameters to verify interlace property of real or imaginary roots of r*(/w). Fractional order PD is considered in order to elaborate developed procedures. Thus for this we take G¡ = 0 and a = 0 and derivative order selected is 1/3. Parameters of plant also selected as fc = 1, T = 2, L = 1.2. Our purpose here is to find range of values of Gp and Gd so that closed loop system is stable. Putting G i = 0 and a = 0 in equation (18), thus

( ]w\z/y T

— ) kGd +kGp + (cosw — - wsinw + T

jsinw + j - wcosw) (25) Separating real and imaginary roots of above equation we get:

rr*(w) = kGd\Re{(j)z/y}\\w\z/y g ) " * + (kGp + T cosw — - wsinw) (26)

r¿*(w) = kGd |/m j(/>j | \w\y sign(w) + (sinw + jwcosw) (27) We can see from above real and imaginary parts that factor QY'y has y complex solutions, but solution for characteristic equation would be common one with LCM. Thus derivative action 1/3 can be selected for z = 1, y = 3 or z = 2, y = 6 and

V3 so on. Complex solution of 0')1/3 is + The rñ

common solution among these is — + .5. Following table I shows the solution of PD for different values.

TABLE 1 COMPLEX SOLUTION OF ( J ) Z / Y

Putting the common solution in real and imaginary parts of above equations, we get:

rr*(w) = kGd — Iwl1/3 Í-J + (kGp + cosw — - wsinw) (27)

i r¡(w) = kGd j\w\3 Q)3 sign(w) + (sinw +

T jwcosw) (28)

It is clear from above equations that real part depends on both Gd and Gp but imaginary part depends only on Gd. Applying this procedure following figure shows the range of Gd according to the value of S.

0 1 2 3 4 5 6 S Figure 3 Gävs S first procedure applied to PD controller for z=l andy=3.

The next procedure is to determine controller's gain Gd, Gp to verify the interlace property between real and imaginary roots. For this procedure calculate the real roots of real part r* in the zero of imaginary part r¡. By putting G¡ = 0, a = 0 in equation (21) and successive equations till (22) for w * 0 following equation of real part is:

rr*(w) = k[Gp + a(w)Gd + c (w) ] (29) Where

/ l \ z / y

a(w) = \Re{(jy/y}\\w\^[l) ,

b(w) = ^ [(cosw — ^ wsinw)]

Since imaginary part rf(w) is an odd function and possesses root at w = 0 thus for w = w0 = 0 real part at w0 becomes:

rr*(w) = kGp + 1 (30) Thus to verify interlace property following condition should hold.

PD 1/3 pD2/6 p D3/9 p D4/12

- / +/ - / ±/ V 3

±T+J. 5 V 3

±T+J. 5 V 3

±T+J. 5 ± 1

± — 7-5

±.9848

- j . 1736

V 3 „

±T±J.s

±.342

— j. 9 3 9 7

V 3 ± 5 ± J ~

±.6428

- j. 7 6 6

rr* (wo) > 0 =>kGv + 1 > 0 =>GV > -1/k Following figure shows the range of values of Gv and Gd

verify the 2nd condition of Hermite-Biehler theorem.

(31) that

Figure 4 PDz¡y controller stability regions forz = l,y = 3.

Figure 5 shows the superiority of fractional order controllers over integer order controllers. We can see that figure 5a shows the points of all type of controllers i.e. P, PI, PD, PD, where as figure 5b shows the plane form of controllers and we can vary the order of derivative and integral so that maximum performance can be met and we get more region for stability of first order plus dead time systems as it is mentioned in problem.

M t1

,1 = 11 PD

\

PID • ^ = 1 ,

PD i

PID •

1 P PI A P 1 PI f o A=1 O A=1

(a) (b) Figure 5 Comparison between integer order (a) andfractional order (b)

controllers

CONCLUSION

Every practical application of control system faces time delay in its characteristics. Time delay systems produce quasi-polynomial type characteristic equation having s and es terms. Generalized version of H. Biehler and Pontryagin theorem is used for the solution of such equations. Using non-integer order PIaDp controllers for stability of time delay plants, we have more parameters to tune in equivalence with integer order PID controllers. With the help of non-integer order PIaDp controllers we can fulfil five different specifications of closed loop systems taking advantage of fractional order i.e. integral order a and derivative order ß. Range of stability regions increases using fractional order PI aDp in contrast with classical PID controllers.

REFERENCES

[1]. W. L. Bialkowski, "Control of the pulp and paper making process," in The Control Handbook, W. S. Levine, Ed. New York: IEEE Press, 1996, pp. 1219-1242.

[2]. K. Astrom and T. Hägglund, PID Controllers: Theory, Design, and Tuning. Research Triangle Park, NC: Instrument Society of America, 1995.

[3]. R. Caponetto, G. Dongola, L. Fortuna, I. Petras, Fractional Order Systems: Modeling and Control Applications, World Scientific, Series A, vol. 72

[4], K.B. Oldham and J. Spanier, The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order (Dover Publication, Mineóla, 2006)

[5]. R. E. Guitierrez, J. M. Rosario, J. T. Machado, Fractional Order Calculus: Basic Concepts and Engineering Applications, Hindawi Publishing Corporation, Mathematic Problems in Engineering volume 2010 article ID 375858, 19 pages

[6]. S. Manabe. The Non-integer Integral and its Application to Control Systems. ETJ of Japan, 6(3/4):83.87, 1961.

[7]. Wang Jifeng and Li Yuankai, Frequency Domain Analysis and applications for fractional order control systems. 7th

International Symposium on Measurement Technology and Intelligent Instruments, 13 (2005) 268-273,2005

[8], C. A. Monje, B. M. Vinagre, Y. Chen, V. Feliu, P. Lanusse and J. Sabatier, "Optimal Tuning for Fractional P f l f Controllers" Control Engineering PracticeVolume 16, Issue 7, July 2008, Pages 798-812.

[9]. Oustaloup, J. Sabatier, and P. Lanusse. From Fractal Robustness to the CRONE Control. Fractional Calculus and Applied Analysis: An International Journal for Theory and Applications, 2( 1) : 1.3 0,

[10]. Oustaloup and B. Mathieu. La Commande CRONE: du Scalaire au Multivariable. Hermès, Paris, 1999.

[11]. Podlubny. Fractional-Order Systems and PID-Controllers. IEEE Transaction on Automatic Control, 44(1):208.214, January 1999.

[12], B. M. Vinagre, I. Podlubny, L. Dorcak, and V. Feliu. On Fractional PID Controllers: A Frequency Domain Approach. In IFAC Workshop on Digital Control. Past, Present and Future of PID Control, pages 53.58, Terrasa, Spain, April 2000

[13]. Datta, M. T. Ho, and S. P. Bhattacharyya, Structure and Synthesis of PID Controllers. London, U.K.: Springer-Verlag, 2000

[14], L. S. Pontryagin, "On the zeros of some elementary transcendental function" (in English), Amer. Math. Society Translation, vol. 2, pp. 95-110, 1955.

[15]. J. S. Karmarkar and D. D. Siljak, "Stability analysis of systems with time delay," Proc. IEE, vol. 117, no. 7, pp. 1421-1424, July 1970

[16]. K.B. Oldham and J. Spanier, The Fractional Calculus. [17]. New York: Academic Press, 1974. B. Ross, "Fractional

calculus," Mathemat. Mag., vol. 50, no. 3, pp. 115-122, 1977.

[18], Kharitonov, V. L., and Zhabko, A. P. (1994). Robust stability of time-delay systems, IEEE Transaction Automatic Control, 39,12, pp. 2388-2397.