proportional stabilization and closed-loop identification of an unstable fractional order process

$: Proportional stabilization and closed-loop identification of an unstable fractional order process$
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Ma

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ARRAA

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ARTICLE IN PRESSG ModelJPC-1741; No. of Pages 8

Journal of Process Control xxx (2014) xxx–xxx

Contents lists available at ScienceDirect

Journal of Process Control

j ourna l ho me pa ge: www.elsev ier .com/ locate / jprocont

roportional stabilization and closed-loop identification of annstable fractional order process

ahsan Tavakoli-Kakhkia,∗, Mohammad Saleh Tavazoeib

Faculty of Electrical Engineering, K.N. Toosi University of Technology, Tehran, IranElectrical Engineering Department, Sharif University of Technology, Tehran, Iran

r t i c l e i n f o

rticle history:eceived 10 September 2012eceived in revised form 6 November 2013ccepted 23 February 2014

a b s t r a c t

This paper deals with proportional stabilization and closed-loop step response identification of the frac-tional order counterparts of the unstable first order plus dead time (FOPDT) processes. At first, thenecessary and sufficient condition for stabilizability of such processes by proportional controllers isfound. Then, by assuming that a process of this kind has been stabilized by a proportional controller

vailable online xxx

eywords:nstable fractional order processlosed-loop identificationtep response

and the step response data of the closed-loop system is available, an algorithm is proposed for estimatingthe order and the parameters of an unstable fractional order model by using the mentioned data.

© 2014 Elsevier Ltd. All rights reserved.

tabilization

. Introduction

The inherent nonlinearity of industrial processes yields in theseystems can have multiple steady states [1,2]. Some of these steadytates, which may be desired operating points for the process,an be unstable [1]. Some typical processes with this property areioreactors, polymerization reactors, exothermic reactors, batcheactors, steam boilers, distillation columns, crystallization pro-esses, and biological systems [3–6].To describe the behavior ofhese processes around their unstable steady states, unstable lin-ar time invariant models are generally used [1]. Using fractionalrder operators, which are originated from the fractional calculus,n constructing the models is a way to enrich these models for bet-er describing the behavior of real-world processes. For example,t has been verified that fractional order models can be effectivelysed in modeling of isotope separation columns [7], bioreactors8], pressurized heavy water reactors [9], liquid/liquid interfaces10], biological systems [11], thermal systems [12,13], and hydro-ogic processes [14]. One significant motivation for applying simpleractional order models in such applications is that these models

Please cite this article in press as: M. Tavakoli-Kakhki, M.S. Tavazoeunstable fractional order process, J. Process Control (2014), http://dx.d

an well approximate the dynamics of many high-order classicalodels [15]. Therefore, simple fractional order models can be good

andidates for describing the dynamics of those systems which

∗ Corresponding author. Tel.: +98 2184062285.E-mail addresses: [email protected], [email protected]

M. Tavakoli-Kakhki), [email protected] (M.S. Tavazoei).

ttp://dx.doi.org/10.1016/j.jprocont.2014.02.019959-1524/© 2014 Elsevier Ltd. All rights reserved.

have been conventionally modeled by high-order classical models.Due to this advantage of fractional order models, finding appropri-ate methods for estimating the order and the parameters of suchmodels is of great importance. Among different methods, thosemethods which are based on using simple-achieved data of thesystem would be valuable methods. For instance, those methodsin which the step response data of the system is used can be appro-priate in practical point of view [16]. Such data can be achievedby doing simple experiments. In [17], some methods have beenproposed for estimating the order and the parameters of stablefractional order models by using the step response data. To com-plete the mentioned work, the aim of this paper is to propose anappropriate method for estimating the order and the parametersof an unstable fractional order model approximating the dynam-ics of an unstable process. To achieve this aim, at first the unstableprocess is stabilized by a proportional controller, and then the stepresponse data of the closed-loop system is used by the proposedmethod to estimate the order and the parameters of an unstablefractional order model. Since for getting the required estimationdata it is necessary to stabilize the process, the stabilizability prob-lem is also investigated in the present work. The considered modelin this paper is

G(s) = ke−Ls

˛where 0 < < 1, T > 0, and k /= 0, (1)

i, Proportional stabilization and closed-loop identification of anoi.org/10.1016/j.jprocont.2014.02.019

Ts − 1

which can be considered as the fractional counterpart of the unsta-ble first-order plus dead time (FOPDT) models [18]. More precisely,the following questions will be answered in this paper:

dx.doi.org/10.1016/j.jprocont.2014.02.019


http://www.sciencedirect.com/science/journal/09591524

http://www.elsevier.com/locate/jprocont

mailto:[email protected]




ARTICLE IN PRESSG ModelJJPC-1741; No. of Pages 8

2 M. Tavakoli-Kakhki, M.S. Tavazoei / Journal of Process Control xxx (2014) xxx–xxx

( )G s( ) cC s k=+ r y

_

(

(

dp

2

austst

T

G

wo

L

Picps

1

w{lTF

G

f

ϕ

s(˛�/2) < 1

s(˛�/2) = 1

s(˛�/2) > 1

, (6)

Fig. 2. Polar plot of G(jω) where L < L*.

(2) cannot be stabilized by a proportional controller. Now, we wantto find the critical time delay by which the polar plot of G(jω) variesfrom the first case to the second case. For this critical time delay,

Fig. 1. A simple feedback control system structure.

1) What is the necessary and sufficient condition for stabilizabilityof a system modeled in the form (1) by a proportional con-troller?

2) If a system in the form (1) is stabilized by a proportional con-troller, how the order and the parameters of this system can beestimated from the noisy step response data of the closed-loopsystem?

The first question will be answered in Section 2. Section 3evotes to finding the answer of the second question. Finally, theaper is concluded in Section 4.

. Stabilizability by a proportional controller

Consider the control system structure shown in Fig. 1, andssume that the process in this structure can be modeled by thenstable fractional order transfer function G(s) in form (1). In thisection, it is investigated that is there a proportional controller inhe form C(s) = kc which guarantees the stability of the closed-loopystem shown in Fig. 1? To this end, at first consider the followingheorem.

heorem 1. System

(s) = e−Ls

s˛ − 1, (2)

here 0 < < 1 is stabilizable by a proportional controller if andnly if

< �(1 − ˛)(

2 cos(

˛�

2

))−1/˛

. (3)

roof. Consider the closed-loop system shown in Fig. 1 where G(s)s given by (2). According to [19: Theorem 3.1], we know that thislosed-loop system is BIBO stable if and only if it does not have anyole in region

{s ∈ C|Re(s)≥0

}. In the other words, this closed-loop

ystem is BIBO stable if equation

+ kcG(s) = 0, (4)

here G(s) is given by (2), does not have any solution in regions ∈ C|Re(s)≥0}. Therefore, the stability of the mentioned closed-oop system can be checked by the Nyquist stability criterion [20].o this end, let us investigate the shape of the polar plot of G(jω).rom (2) it is concluded that

(jω) = cos(Lω) − j sin(Lω)ω˛ cos(˛�/2) + jω˛ sin(˛�/2) − 1

, (5)

or ω ≥ 0. (5) results in

(ω) � �G(jω) =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

−Lω + tan−1 ω˛ sin(˛�/2)1 − ω˛ cos(˛�/2)

− � if ω˛ co

−Lω − �

2if ω˛ co

−Lω + tan−1 ω˛ sin(˛�/2)1 − ω˛ cos(˛�/2)

if ω˛ co


According to (6),

dϕ(ω)dω

= −L + ˛ω˛−1 sin(˛�/2)1 − 2ω˛ cos(˛�/2) + ω2˛

. (7)

Fig. 3. Polar plot of G(jω) where L > L*.

Since 0 < < 1, it is deduced that limω→0+

(dϕ(ω)/dω) = +∞. This

means that ϕ(ω) will be an increasing function for small values ofω ≥ 0. On the other hand, according to (7) we have lim

ω→+∞(dϕ/dω) =

−∞. Hence, there is ω = ω0 such that ϕ(ω) is a decreasing functionfor ω > ω0. Paying attention to the mentioned points, two differentcases schematically shown in Figs. 2 and 3 may occur for the polarplot of G(jω). In the first case (Fig. 2), the polar plot begins from point(−1,0), and after that intersects the real axis at a point denoted by(R0,0) which places in the right side of the beginning point (−1,0).According to the Nyquist stability criterion, in such a case system (2)is stabilized by proportional controller C(s) = kc if 1 < kc < 1/R0. Inthe second case (Fig. 3), the polar plot begins from point (−1,0) andafter that intersects the real axis at a point denoted by (R0,0) whichsettles in the left side of the beginning point (−1,0). According tothe Nyquist stability criterion, it is resulted that in this case system




M. Tavakoli-Kakhki, M.S. Tavazoei / Journal of Process Control xxx (2014) xxx–xxx 3

wc

R

R

f

s

a

−

au

−

s

−

w

c

c

s

2

Fig. 4. Polar plot of G(jω) where L = L*.

hich is denoted by L*, we have (R0,0) = (−1,0) (see Fig. 4), andonsequently ω = ω∗ > 0 is found such that

e(G(jω∗)) = −1 and Im(G(jω∗)) = 0. (8)

From (5),

e(G(jω)) = − cos(Lω) + ω˛ cos((˛�/2) + Lω)1 − 2ω˛ cos(˛�/2) + ω2˛

and

Im(G(jω)) = sin(Lω) − ω˛ sin((˛�/2) + Lω)1 − 2ω˛ cos(˛�/2) + ω2˛

, (9)

or ω ≥ 0. Therefore from (8) and (9), two equalities

in(L∗ω∗) − (ω∗)˛ sin(

˛�

2+ Lω∗

)= 0 (10)

nd

cos(L∗ω∗) + (ω∗)˛ cos(

˛�

2+ L∗ω∗

)

= −1 + 2(ω∗)˛ cos(

˛�

2

)− (ω∗)2˛ (11)

re obtained. By multiplying (11) in sin((˛�/2) + L∗ω∗) and makingse of (10) it is concluded that

sin(

˛�

2

)= sin

(−˛�

2+ L∗ω∗

)− (ω∗)˛ sin(L∗ω∗). (12)

By multiplying the both sides of (12) in sin((˛�/2) + L∗ω∗) andubstituting (ω∗)˛ sin((˛�/2) + L∗ω∗) from (10), it is deduced that

sin(

˛�

2

)sin

(˛�

2+ L∗ω∗

)

= sin(

−˛�

2+ L∗ω∗

)sin

(˛�

2+ L∗ω∗

)− sin2(L∗ω∗), (13)

hich results in

os(L∗ω∗) + cos(˛�) − cos(˛� + L∗ω∗) = 1. (14)

Relation (14) can be rewritten as

os(L∗ω∗)(1 − cos(˛�)) + sin(˛�) sin(L∗ω∗) = 1 − cos(˛�). (15)

By considering equalities 1 − cos(˛�) = 2 sin2(˛�/2) andin(˛�) = 2 sin(˛�/2) cos(˛�/2), from (15) it is concluded that( ) [ ( ) ( )]


sin˛�

2cos(L∗ω∗) sin

˛�

2+ sin(L∗ω∗) cos

˛�

2

= 2 sin2(

˛�

2

), (16)

Fig. 5. Stabilizability region for system (2) (system (2) is stabilizable by a propor-tional controller if and only if the pair (˛,L) locates on the gray region).

which results in

sin(

˛�

2+ L∗ω∗

)= sin

(˛�

2

). (17)

Relations (10) and (17) brings about ω∗ =(sin(L∗ω∗)/ sin(˛�/2))1/˛. Therefore, according to (17)

L∗ = x∗(

sin(˛�/2)sin(x∗)

)1/˛

, (18)

where x = x* is the first positive solution of equation

sin(x + (˛�/2)) = sin(˛�/2). (19)

From (19), it is concluded that

x∗ = (1 − ˛)�. (20)

By substituting x* from (20) in (18), it is deduced that

L∗ = �(1 − ˛)(

2 cos(

˛�

2

))−1/˛

. (21)

Note that we know that where L = 0 proportional stabilizabilityof system (2) is possible. Therefore, it is resulted that system (2) isstabilizable by a proportional controller if 0 ≤ L < L∗, and it is notstabilizable by such a controller if L ≥ L*.

According to Theorem 1, in Fig. 5 those pairs of (˛,L) have beenspecified for which system (2) can be stabilizable by a propor-tional controller. Note that → 1 results in L* → 1. This means thatthe first-order system G(s) = e−Ls/(s − 1) is stabilizable by a pro-portional controller if and only if L < L* = 1. This result has beenpreviously obtained in [21: Corollary 1].

Example 1. Consider the fractional order system

G(s) = e−Ls

s0.9 − 1. (22)

From Theorem 1, it is deduced that this system is stabilizableby a proportional controller if L < L* = 1.1425. For instance, this sys-tem can be stabilized by a proportional controller if L = 1.1. Fig. 6which shows the polar plot of G(jω) for L = 1.1 confirms this point.From this figure, it is concluded that G(s) can be stabilized by theproportional controller C(s) = kc if 0.9808 < 1/kc < 1. Fig. 7 showsthe step response of the closed-loop system where kc = 1/0.99for different cases L = 1.1 (stable closed-loop system) and L = 1.15(unstable closed-loop system). It is worth noting that the upperbound L* = 1.1425 was found on the time delay L in the nomi-nal case. Clearly in the presence of uncertainty on the model of


the process, which may induced by disturbances or errors in esti-mation of parameters [22: Ch. 2], such an upper bound shouldbe decreased for guaranteeing the proportional stabilizability. Tomake the point more clear, assume that the upper bound of the



4 M. Tavakoli-Kakhki, M.S. Tavazoei / Journal of Process Control xxx (2014) xxx–xxx

Table 1Stabilizability conditions where the upper bound of the relative model error in Example 1 is given by �ω0.9 for different values of �.

0

L

rwnfcossibvfwrbncot

ttttT

rt0

Fn

� 0.1 0.2 0.3

Condition L ∈ [0, 1.00) L ∈ [0, 0.87) L ∈ [0, 0.74)

elative model error is given by �ω0.9, i.e. |�G(jω)/G(jω)| ≤ �ω0.9

here G(s) defined by (22) denotes the transfer function of theominal process, �G(s) indicates the difference between transfer

unctions of the actual and nominal processes, and � is a knownonstant [23]. By an approach similar to that presented in the prooff Theorem 1 and considering the perturbed polar plots (for twoamples, see Fig. 8), the upper bounds on the time delay to ensuretabilization by proportional controllers can be found (Obviouslyf � = 0, the upper bound is L* = 1.1425). These upper bounds haveeen numerically calculated and brought in Table 1 for differentalues of �. This table confirms that if the value of � increases,or a smaller range of time delays the proportional stabilizationill be guaranteed. Increasing the value of � not only reduces the

ange of admissible time delays in the proportional stabilization,ut also reduces the range of stabilizing gain kc. For example, in theominal case (L,�) = (0.7,0), (22) is stabilized by the proportionalontroller C(s) = kc if 0.6025 < 1/kc < 1 (see the nominal polar plotf Fig. 8(a)). But in the case (L,�) = (0.7,0.1) the stability is guaran-eed if 0.7298 < 1/kc < 1 (see the perturbed polar plot in Fig. 8(a)).

Now, suppose that the proportional controller C(s) = kc is usedo control of system (1). For this case, the characteristic equation ofhe closed-loop system will be Ts˛ − 1 + kcke−Ls = 0. By defininghe new variable s � T1/˛s, the mentioned characteristic equa-ion is transformed to s˛ − 1 + kcke−LT−1/˛s = 0. Clearly, equations˛ − 1 + kcke−Ls = 0 does not have any solution with non-negative


eal part if and only if equation s˛ − 1 + kcke−LT−1/˛s = 0 has no solu-ion in such a region. On the other hand, s˛ − 1 + kcke−LT−1/˛s =

can be considered as the characteristic equation of a

ig. 6. (a) Polar plot of G(jω) in Example 1 where L = 1.1, (b) the zoomed plot forear zero frequencies.

.4 0.5 0.6 0.7

∈ [0, 0.62) L ∈ [0, 0.51) L ∈ [0, 0.39) L ∈ [0, 0.28)

closed-loop system with the open loop transfer function kckG(s),where G(s) = e−LT−1/˛s/(s˛ − 1). Consequently, the following corol-lary is concluded from Theorem 1.

Corollary 1. The unstable fractional order system (1) is stabiliz-able by a proportional controller if and only if L < T1/˛L∗ where L*is defined by (21).

According to Corollary 1, the ratio of L and T1/˛ specifies whetherthe unstable fractional order system (1) can be stabilized by a pro-portional controller. If such a ratio is less than L*, stabilization bya proportional controller will be possible. Otherwise, system (1)cannot be stabilized by a proportional controller. It is worth notingthat the value of L* is less than 1.16 (see Fig. 5). This means that forprocesses with dominant time delays stabilization by proportionalcontrollers will not be possible.

As mentioned before, lim˛→1

L∗ = 1. By considering this point, from

Corollary 1 it is deduced that the FOPDT process

G(s) = ke−Ls

Ts − 1(T > 0) (23)

is stabilizable by a proportional controller if and only if T/L > 1 (Thisresult is compatible with that previously presented in [24: Theorem7.3.2]). In such a case, the stabilizing range of controller parameter


kc can be obtained from the arguments presented in the proof ofTheorem 1. In this proof, it is deduced the frequency correspond-ing to the intersection point (R0,0), denoted by ω = ωR0 , is the first

Fig. 7. Unit step response of the closed-loop system in Example 1 where kc = 1/0.99and G(s) is given by (22) for different cases (a) L = 1.1 and (b) L = 1.15.


ARTICLE ING ModelJJPC-1741; No. of Pages 8

M. Tavakoli-Kakhki, M.S. Tavazoei / Journal

Fig. 8. Nominal and perturbed polar plots in Example 1 for two cases (a)(tp

pe

t

t

scs

ai

w

t

7

3f

sr

L,�) = (0.7,0.1) and (b) (L,�) = (1.1,0.1). In the first case, stabilization by propor-ional controllers is possible, whereas in the second case there is no guarantee forroportional stabilization.

ositive solution of equation Im(G(jω)) = 0. According to (23), thisquation is equivalent to

an(Lω) = ωT. (24)

Also, R0 =∣∣G(jω)

∣∣ω=ωR0

. From this equality and (24), it is found

hat R0 = (k/(√

T2ω2R0

+ 1)). On the other hand, from the Nyquist

tability criterion we know that (23) is stabilized by proportionalontroller C(s) = kc if 1 < kkc < 1/R0. Consequently if T/L > 1, (23) istabilized by proportional controller C(s) = kc where

1k

< kc <

√T2ω2

R0+ 1

k, (25)

nd ω = ωR0 is the first positive solution of equation (24). By defin-ng z � Lω, the above condition can be rewritten in the form

1k

< kc <T

kL

√z2

1 + L2

T2, (26)

here z = z1 is the first positive solution of equation

an(z) = T

Lz. (27)

Condition (26) is the same as that presented in [24: Theorem.3.2].

. Estimation of the order and parameters of an unstableractional order model


In the previous section, it was shown that the fractional orderystem (1) can be stabilized by a proportional controller if the crite-ia presented in Corollary 1 holds. In this section, the aim is to

PRESSof Process Control xxx (2014) xxx–xxx 5

present an algorithm for estimating the order and the parametersof system (1) stabilized by a proportional controller based on usingthe step response data of the closed-loop system. To this end, somebasic results about the estimation of the order and the parame-ters are obtained in Sections 3.1 and 3.2, respectively. Finally, theproposed algorithm is given in Section 3.3. Also, in this subsectionan example is presented to show the effectiveness of the proposedalgorithm.

3.1. Order estimation

Let us begin this subsection by presenting the fractional integraldefinition which is useful in the procedure of the order estimation.The integral of order � , where 0 < � ∈ R, is defined as follows [25].

0It� x(t) = 1

� (�)

∫ t

0

(t − )�−1x()d. (28)

The Laplace transform of 0It� x(t) is given by s−� X(s) where X(s)

denotes the Laplace transform of x(t) [25]. Now, consider the fol-lowing theorem.

Theorem 2. Assume that y(t) denotes the unit step response of astable closed-loop system in the from shown in Fig. 1 where G(s) isgiven by (1). In this case, 0I�

t (y(∞) − y(t)) tends to a nonzero finitevalue where t → ∞ if and only if � = ˛.

Proof. The transfer function of the closed-loop system is as fol-lows.

T(s) = kcke−Ls

Ts˛ − 1 + kcke−Ls. (29)

Hence, if the closed-loop system is stable we have

y(∞) = kck

kck − 1(30)

Note that stability of the closed-loop system guaranteeskck /= 1, since otherwise s = 0 will be a root for the closed-loopsystem, and consequently this system cannot be BIBO stable insuch a case. By considering this point and doing a straightforwardcalculation the Laplace transform of y(∞) − y(t) is determined asfollows.

0I�t (y(∞) − y(t)) = kck(−1 + e−Ls + Ts˛)

s� (kck − 1)(Ts˛ − 1 + kcke−Ls). (31)

According to the generalized final value theorem [26], (31)results in

limt→∞

(0I�t (y(∞) − y(t))) = kck

(kck − 1)2lims→0

−1 + e−Ls + Ts˛

s� . (32)

Now applying the L’Hopital rule in the right hand sideof (32) results in lim

t→∞(0I�

t (y(∞) − y(t))) = 0 if 0 < � < <

1, limt→∞

(0I�t (y(∞) − y(t))) = kckT/(kck − 1)2 if 0 < � = < 1, and

limt→∞

(0I�t (y(∞) − y(t))) = ∞ if 0 < < � < 1.

In the proof of Theorem 2 it has been revealed that when t → ∞the value of 0I�

t (y(∞) − y(t)) tends to zero if � < and tends to infin-ity if < � . Also, for t → ∞ this fractional integral tends to a nonzerofinite value if � = ˛. According to these results, if the step responseof the closed-loop system shown in Fig. 1 is available, the fractionalorder can be estimated by using the following algorithm.

Algorithm 1. Data: The unit step response y(t) of the closed-loopsystem shown in Fig. 1


Aim: Estimation of order in model (1)Algorithm steps:Step I: In the structure shown in Fig. 9, find � l and �h such that

limt→∞

f�l(t) = 0 and lim

t→∞f�h

(t) = ∞.


ARTICLE ING ModelJJPC-1741; No. of Pages 8

6 M. Tavakoli-Kakhki, M.S. Tavazoei / Journal

∞

r

3

trs

•

•

•

[trft

3d

so

T

T

s

T

s

T

3d

T

Fig. 9. Estimation of by fractional order integration.

Step II: Let �m = 0.5(�l + �h). If limt→∞

f�m (t) = c where 0 < |c| <

, then = �m. Otherwise, go to Step III.Step III: If lim

t→∞f�m (t) = 0 replace � l with �m and if lim

t→∞f�m (t) = ∞

eplace �h with �m. Go to Step 2.

.2. Estimation of parameters

In [27], a procedure is presented for the closed-loop identifica-ion of an unstable FOPDT process by using the closed-loop stepesponse data. The mentioned procedure has the following threeteps:

Finding the values of T(s)∣∣s=ε

, T ′(s)∣∣s=ε

, and T ′′(s)∣∣s=ε

by using theclosed-loop step response data where ε is a positive constantFinding the values of G(s)

∣∣s=ε

, G′(s)∣∣s=ε

, and G′′(s)∣∣s=ε

from the

values of T(s)∣∣s=ε

, T ′(s)∣∣s=ε

, and T ′′(s)∣∣s=ε

Finding the system parameters from the values of G(s)∣∣s=ε

,

G′(s)∣∣s=ε


.

In this subsection, the third step of the procedure presented in27] is modified such that it can be used for the closed-loop iden-ification of the processes described by (1). In the rest, at first theequired relations for the first and the second steps are rewrittenrom [27]. Then, the required relations for estimating the parame-ers from the values of G(s)

∣∣s=ε

, G′(s)∣∣s=ε


are derived.

.2.1. Finding the values of T(s) and its first and seconderivatives for s = ε > 0

Assume that y(t) is the unit step response of the closed-loopystem. If Y(s) denotes the Laplace transform of unit step responsef the closed-loop system, we have T(s) = sY(s) which results in

(s) = s

∫ ∞

0

e−sty(t)dt. (33)

Therefore, the value of T(s) at s = ε > 0 is obtained as follows.

(ε) = ε

∫ ∞

0

e−εty(t)dt (34)

By taking the first derivative from the both sides of (33) andubstituting s = ε > 0, it is concluded that

′(ε) =∫ ∞

0

(1 − εt)e−εty(t)dt. (35)

Also, taking the second derivative from both sides of (33) andubstituting s = ε > 0 result in

′′(ε) =∫ ∞

0

(εt − 2)te−εty(t)dt. (36)

.2.2. Finding the values of G(s) and its first and seconderivatives for s = ε > 0


Note that

(s) = kcG(s)1 + kcG(s)

. (37)

PRESSof Process Control xxx (2014) xxx–xxx

According to (37),

G(s) = T(s)kc (1 − T(s))

. (38)

From (38), the following relations are deduced:

G(ε) = T(ε)kc(1 − T(ε))

. (39)

G′(ε) = T ′(ε)

kc(1 − T(ε))2. (40)

G′′(ε) = T ′′(ε)

kc(1 − T(ε))2+ 2

(T ′(ε))2

kc(1 − T(ε))3. (41)

3.2.3. Finding the parameters from the values of G(s)∣∣s=ε

,

G′(s)∣∣s=ε


By taking the first derivative from the both sides of (1), we have

G′(s) = −Lk

Ts˛ − 1e−Ls − k˛Ts˛−1

(Ts˛ − 1)2e−Ls. (42)

From (1) and (42), it is deduced that

G′(s)G(s)

= −L − ˛Ts˛−1

Ts˛ − 1. (43)

Taking the first derivative from the both sides of (43) results in

G′′(s)G(s) − (G′(s))2

G2(s)= T2˛s2˛−2 + Ts˛−2(˛2 − ˛)

(Ts˛ − 1)2. (44)

By substituting s = ε > 0 in (44), it is concluded that the systemparameter T can be obtained by solving the following quadraticequation

cT2 + bT + a = 0, (45)

where

a = (G′(ε))2 − G′′(ε)G(ε)G2(ε)

, (46)

b = ε˛−2(˛2 − ˛) − 2aε˛, (47)

and

c = ˛ε2˛−2 + ε2˛a. (48)

Also, by substituting s = ε > 0 in (43) it can be concluded that

L = −G′(ε)G(ε)

− ˛Tε˛−1

Tε˛ − 1. (49)

Moreover, by substituting s = ε > 0 in (1) it can be deduced that

k = (Tε˛ − 1)G(ε)e−Lε. (50)

3.3. The proposed algorithm

Benefiting from the results of Sections 3.1 and 3.2, the follow-ing algorithm can be proposed for estimation the order and theparameters of the unstable fractional order model (1) by using theclosed-loop step response data.

Algorithm 2. Data: The unit step response y(t) of the closed-loopsystem shown in Fig. 1, the controller parameter kc, and constantε > 0

Aim: Estimation of order � and parameters k, T, and L in model(1)


Algorithm steps:Step I: Estimate the order � by applying Algorithm 1.Step II: Compute T(�), T ′(ε), and T ′′(ε) from (34), (35), and (36),

respectively.



M. Tavakoli-Kakhki, M.S. Tavazoei / Journal of Process Control xxx (2014) xxx–xxx 7

Table 2Approximated models for different values of ε in Example 2.

ε 0.1 0.15 0.2

Approximated models G(s) = 0.9987e−0.8351s

2.0309s0.7−1G(s) = 0.9993e−0.8523s

2.0345s0.7−1G(s) = 1.0006e−0.8704s

2.0370s0.7−1

FS

r

(

opfi(ime(

oto

Et(

G

aocrFttbtev

E

ig. 10. Noisy unit step response of the closed-loop system in Example 2 withNR = 45.

Step III: Compute G(�), G′(ε), and G′′(ε) from (39), (40), and (41),espectively.

Step IV: Estimate the parameter T by solving the quadratic Eq.45).

Step V: Estimate the parameter L from (49).Step VI: Estimate the parameter k from (50).

In Algorithm 2, ε is an arbitrary positive constant. Also, insteadf using Relation (50) in Step VI of the above algorithm to obtainarameter k, this parameter can be estimated by noticing to thenal value of the unit step response of the closed-loop systemRelation (30)). Moreover, since parameter T is estimated by solv-ng a quadratic equation, two positive values for this parameter

ay be obtained. In such a case, the acceptable value for param-ter T is the value which is more consistent with the value ofkck − 1)2 lim

t→∞(0I˛

t (y(∞) − y(t)))/kck which according to the proof

f Theorem 2 should be equal to T. The following example showshe effectiveness of Algorithm 2 when the noisy step response dataf the closed-loop system is available.

xample 2. Assume that the noisy unit step response data ofhe closed-loop system shown in Fig. 1 with signal-to-noise ratioSNR)1 45 is available where

(s) = 12s0.7 − 1

e−0.8s, (51)

nd kc = 1.25 (See the response shown in Fig. 10). For estimating therder ˛, the preliminary value of pair (� l,�h) in Algorithm 1 has beenhosen as (0.2,1). The first and the second iterations of this algo-ithm respectively result in (� l,�h) = (0.6,1) and (� l,�h) = (0.6,0.8).inally, in the third iteration the order is estimated as 0.7 (seehe related numerical results in Fig. 11). It is worth mentioninghat the fractional integrals to obtain these results are calculatedased on the numerical method proposed in [29]. Now, by applyinghe remained steps of Algorithm 2 the system parameters are alsostimated. Table 2 presents the approximated models for different


alues of ε.

1 The signal-to-noise ratio (SNR) is calculated based on the definition used in [28:q. (22)].

Fig. 11. f� (t) = 0I�t (y(∞) − y(t)) for different values of integral order � in Example

2.

4. Conclusions

In this paper, at first it was analytically shown that under whichcriteria the unstable fractional order system (1) can be stabilized bya proportional controller. Then, it was assumed that the mentionedunstable system has been stabilized by a proportional controller,and the noisy step response data of the closed-loop system is avail-able. Considering these assumptions, an algorithm was proposedfor estimating the order and the parameters of (1) by using thementioned data of the closed-loop system. Due to the significantrole of the integral based operations in the proposed estimationalgorithm, this algorithm would be robust against the measure-ment noise. This point was confirmed by the numerical simulationresults. As a limitation, the proposed algorithm can estimate theorder and the parameters only when the process model is in theform (1). Extending this algorithm in order to be used for estimat-ing the orders and parameters of more-complex unstable fractionalorder models can be considered as an interesting topic for futureresearch works.

Acknowledgment

This work was supported by the Iran National Science Founda-tion (INSF).


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proportional stabilization and closed-loop identification of an unstable fractional order process

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