JPC2013 Robust ID of continuous-time using moments

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Journal of Process Control 23 (2013) 682– 695

Contents lists available at SciVerse ScienceDirect

Journal of Process Control

jou rn al h om epa ge: www.elsev ier .com/ locate / jprocont

obust identification of continuous-time low-order models usingoments of a single rectangular pulse response

oung Chol Kima,∗, Lihua Jinb

School of Electronics Engineering, Chungbuk National University, Cheongju 361-763, Republic of KoreaResearch Institute for Computer and Information Communication, Chungbuk National University, Cheongju 361-763, Republic of Korea

r t i c l e i n f o

rticle history:eceived 15 October 2012eceived in revised form 26 January 2013ccepted 9 March 2013

eywords:arameter identificationethod of moments

irst-order plus time-delay modelecond-order plus time-delay model

a b s t r a c t

This paper presents a simple method for identifying first- and second-order processes with dead-time byusing moments of a single rectangular pulse response in an open-loop system. A closed-form formula isproposed to determine all parameters of four types of process models for a stable linear time-invariantprocess. It is shown that the same approach can be extended to the identification of multi-input,multi-output linear processes. It is demonstrated through a comparative analysis that the proposed iden-tification method results in good accuracy with a noisy output, and is also able to closely approximatevarious high-order processes in those low-order models.

© 2013 Elsevier Ltd. All rights reserved.

. Introduction

It is very common that the dynamics of most industrial processes can be characterized by simple models such as first-order plus timeelay (FOTD), second-order with delay free (SODF), second-order plus time delay (SOTD), and second-order plus a zero with delay freeSOZDF) models. The identification of such low-order models plays an important role in the tuning of PID controllers, which are widelysed in industry because of their simplicity and robustness.

The open-loop identification methods based on the step responses [1–4] are simple and popular. Different methods based on finite-uration pulse responses have also been developed [5–8]. An advantage of pulse testing is that the process input and output return toheir original values after the perturbation. These methods require the determining of extreme values of pulse responses. However, it maye difficult to determine those values if the response data is corrupted by measurement noise. Another disadvantage is that the methodannot be applied to oscillatory systems. A possible way to identify low-order models is to first obtain a high-order model by using a least-quares approach and then reduce it to a low-order model [9]. As an alternative open-loop identification method, Aström and Hägglund10] suggested that FOTD and SOTD models can be obtained by using the method of moments. Gibilaro and Lees [11] reported that for

simple transfer function model defined by a time constant with n multiplicity and a time delay, three parameters of the model can bebtained explicitly by matching the first three moments of the impulse response of the complex and simple models. Moment matching isidely used to reduce the model order [12–14].

This paper will propose a simple technique for identifying four types of low-order models, namely FOTD, SODF, SOTD, and SOZDF, bysing the moments of a single rectangular pulse response in an open-loop system. It is shown that the parameters of the four models cane determined explicitly by calculating only a few moments of output because the ith moments of input are known a priori. In addition,ome examples are provided to verify the effectiveness of the proposed identification method, including its robustness to measurementoise and order mismatch between high-order process and low-order model. To fairly compare performances of the present method with
hose of other ones, several criteria in time and frequency domains are defined. The identification of low-order transfer function modelsas been deeply studied and thus, there are plenty of mature methods [1,9]. Main contributions of the proposed method over this problemre as follows:
∗ Corresponding author. Tel.: +82 43 261 2475; fax: +82 43 272 2475.E-mail addresses: [email protected], [email protected] (Y.C. Kim), [email protected] (L. Jin).

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dx.doi.org/10.1016/j.jprocont.2013.03.002

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Y.C. Kim, L. Jin / Journal of Process Control 23 (2013) 682– 695 683

(i) All parameters of the four types of low-order models are determined by the closed-form formulas as long as a couple of moments of apulse response are calculated using its sampled data.

ii) These formulas are analytically derived.ii) Since the test input is merely a single rectangular pulse, the method can be easily applied to any real processes. Moreover, the pulse

shape is available for not only a rectangular form, but also different ones, for example, half-sine, triangular, displaced cosine forms.iv) Though the proposed method using moment matching is not an optimization technique like a least squared error estimation, it provides

more robust and more accurate identified models than those of some other methods as shown in Section 4.v) The proposed method can be effectively combined with automatic proportional integral derivative (PID) controller tuning because the

pulse type input takes a finite time for a test and makes the system return to the initial state at the terminal time.

The paper is organized as follows: Section 2 provides some definitions and preliminaries. The proposed identification methods areresented in Section 3. Section 4 provides several examples to evaluate the accuracy and robustness of the proposed method. Finally, theonclusion follows in Section 5.

. Definitions and preliminary results

.1. Moments

The ith moment of a real-valued function f (x) is defined as

mi :=∫ ∞

0

xif (x) dx, for i = 0, 1, 2, . . . . (1)

ote that the zero moment of f (x), m0, is equal to its area.The center of gravity (COG) is defined by

� := m1

m0. (2)

onsider a stable linear time-invariant (LTI) process model, G(s), with a single-input u and a single-output y. It is assumed that the systemas no positive zeros.

From (1), the moments of u (t) and y (t) are written by

mui =∫ ∞

0

tiu (t) dt, myi

=∫ ∞

0

tiy (t) dt. (3)

he minimum phase assumption above ensures that the output response does not make undershoot. Therefore, y(t) in (3) can be regardeds a mass distribution.

Let g (t) be the impulse response of G (s). Since

G (s) :=∫ ∞

0

g (t) e−stdt, (4)

he ith moments of g (t) are

mgi

:=∫ ∞

0

tig (t) dt = (−1)i diG (s)

dsi

∣∣∣∣s=0

, for i = 0, 1, 2, . . . . (5)

herefore, the DC gain of G (s) is identical to its zero moment. It is easy to know that the COG of a process model, G (s), has the followingelationship:

�g = �y − �u, (6)

here �u and �y denote the COGs of the input and the output, respectively. From the Taylor series expansion of G (s) about s = 0, it followshat

G (s) =∞∑

k=0

Gk (0)sk

k!=

∞∑k=0

(−1)k mgk

k!sk. (7)

The following theorem states the relationship between the moments of the impulse response of an LTI model, G (s), and those of itsnput and output:

heorem 1. For an LTI system with input u and output y,

myk

=k∑(

k

i

)mg

k−imu

i , for k = 0, 1, 2, . . . , (8)

i=0

here

(ki

)= k!

i! (k − i)!is the binomial coefficient.

6

P

F

C

�

Cr

w

P

s

wn

Tms

84 Y.C. Kim, L. Jin / Journal of Process Control 23 (2013) 682– 695

roof. Use the convolution integral,

y (t) =∫ ∞

0

u (�) g (t − �) d�. (9)

rom (3) and (9) with t − � = x,

myk

=∫ ∞

0

tky (t) dt =∫ ∞

0

u (�)

∫ ∞

0

tkg (t − �) dtd�

=∫ ∞

0

u (�)

∫ ∞

0(x + �)kg (x) dxd� =

∫ ∞

0

u (�)

∫ ∞

0

[k∑

i=0

(k

i

)xk�k−ig (x)

]dxd�.

hanging the integration order yields

myk

=(

k

k

)∫ ∞

0

xkg (x) dx

∫ ∞

0

u (�) d� +(

k

k − 1

)∫ ∞

0

xk−1g (x) dx

∫ ∞

0

�u (�) d�

+(

k

k − 2

)∫ ∞

0

xk−2g (x) dx

∫ ∞

0

�2u (�) d� + · · · +(

k

0

)∫ ∞

0

�ku (�) d�

=k∑

i=0

(k

i

)mg

k−imu

i.

(10)

orollary 1. Moments of the impulse response of an LTI process model can be represented in terms of the moments of its input and output in aecursive manner.

mgk

= 1mu

0

[my

k−

k∑i=0

(k

i

)mg

k−imu

i

], for k = 0, 1, 2, . . . (11)

here mg0 = my

0mu

0.

roof. Rewriting (8) into a vector-matrix form, we have⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

mu0 0 0 · · · 0

mu1 mu

0 0 · · · 0

mu2 2mu

1 mu0 · · · 0

mu3 3mu

2 3mu1 · · · 0

......

.... . .

...

muk

(k

k − 1

)mu

k−1

(k

k − 2

)mu

k−2 · · · mu0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

mg0

mg1

mg2

mg3

...

mgk

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

my0

my1

my2

my3

...

myk

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

.(12)

It is evident that (11) is the solution to (12). �

Now consider the identification problems for the four low-order models by using the moments of input and output of a stable LTIystem. Four types of these models are described by

FOTD : G1 (s) = K

Ts + 1e−Ls, (13)

SODF : G2 (s) = Kω2n

s2 + 2�ωns + ω2n

, (14)

SOTD : G3 (s) = Kω2n

s2 + 2�ωns + ω2n

e−Ls, (15)

SOZDF : G4 (s) = Kω2n (s + z0)

s2 + 2�ωns + ω2n

, (16)

here K /= 0, T > 0, L ≥ 0, � ≥ 0, ωn > 0, K is the process gain, T is the time constant, L is the time-delay, � is the damping ratio, and ωn is theatural frequency of a process.

That is, the problem here is to determine all parameters of the above four low-order models by means of the moment matching method.
Zadeh and Desoer [15] presented the approximation characteristics of the method of moment as follows:
heorem 2. [15] Let S1 and S2 be LTI systems with impulse response g1 and g2, and transfer functions G1 and G2. Assume that the first Noments are equal. Then, for all t, the zero-state responses of S1 and S2 are equal provided that their common input is a polynomial in t of degree

maller than or equal to N.


A

t

t

w

E

T

2

tafsS

w

S

3

tt

3

tr

D0Fig. 1. A single rectangular pulse.

Ismail [16] explained the relationship between moments and real poles. Consider a proper transfer function, G (s), having n real poleshat can be expressed as a partial fraction expansion:

G (s) =n∑

j=1

cj

(s + pj), (17)

here pj and cj are poles and residues of G (s), respectively. From (7),

G (s) = mg0 + (−1) mg

1s + (−1)2

2!mg

2s2 + (−1)3

3!mg

3s3 + · · ·

= G (0) + G′ (0) s + G′′ (0) s2 + G(3) (0) s3 + · · ·.(18)

xpanding (17) into (18) yields

1k!

mgk

= c1

pk+11

+ c2

pk+12

+ · · · + cn

pk+1n

. (19)

his reciprocal relationship between moments and poles emphasizes the dominant poles, which have smaller magnitudes.

.2. Input signal

In the method of moments, the approximated models, Gi (s), for i = 1, 2, 3, 4 are chosen to match G (s) at s = 0. For example, if the firsthree moments are matched, then it follows from (7) that G(0) = Gi(0), G′(0) = G′

i(0), and G′′(0) = G

′′i(0). All the available freedom is used to

chieve an excellent fit about s = 0. The point s = 0 in the s-domain is directly related to the infinite time in the time domain. Consequently,or better approximations, a single rectangular pulse input is introduced in Fig. 1 because the input and its response can go to zero in ahort period of time. However, the pulse shape is not necessarily restricted to the rectangular form. This will be discussed at the end ofection 3.1.

The input can be expressed as

u (t) = A [H(t) − H(t − D)] , (20)

here A /= 0, D > 0, and H(t) ={

1, t ≥ 00, t < 0

is the Heaviside unit function.

Then moments and the COG of u (t) are obtained by

muk = 1

(k + 1)AD(k+1), �u = 1

2D. (21)

ince mg0 = G (0) from (5) and my

0 = mg0mu

0 from (8), the DC gain of the LTI process is

G (0) = my0

mu0

. (22)

. Main results

In this section, the identification formulas of low-order models of a single input, single output (SISO) LTI process are represented usinghe moments of a single rectangular pulse response. First, the four types of models in (13)–(16) will be considered. Then it will be shownhat the proposed method can be directly extended to the identification of multi-input, multi-output (MIMO) LTI systems.

.1. Identifications of four low-order models

For the four models in (13)–(16), namely, FOTD, SODF, SOTD, and SOZDF, their parameterizations can be determined by only the firsthree or four moments of a simple pulse response in which they are calculated numerically. Here, let the kth moment of a rectangular pulseesponse be my

k.

6

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w

P

S

F

S

F

T

F

S

I(

ab

P


roposition 1. For the FOTD model in (13), the parameters can be determined by

(i) K = my0

AD, (23)

(ii) T =

√my

2

my0

−(

my1

my0

)2

− 112

D2, (24)

(iii) L = my1

my0

− 12

D − T , (25)

here the hat “ˆ” indicates the estimated value.

roof. For the FOTD process in (13), it follows from (8) that

my0 = mg

0mu0,

my1 = mg

1mu0 + mg

0mu1,

my2 = mg

2mu0 + 2mg

1mu1 + mg

0mu2.

(26)

ubstituting (13) into (5) yields

mg0 = G1 (0) = K,

mg1 = (−1)

dG1 (s)ds

∣∣∣s=0

= K (L + T) ,

mg2 = (−1)2 d2G1 (s)

ds2

∣∣∣∣s=0

= K[(L + T)2 + T2

].

(27)

rom (21), the moments of the input pulse are already known as

mu0 = AD, mu

1 = 12

AD2, and mu2 = 1

3AD3. (28)

ubstituting (27) and (28) into (26), we have

my0 = ADK,

my1 = ADK

(12

D + L + T)

,

my2 = ADK

[13

D2 + (D + L + T) (L + T) + T2]

.

(29)

rom (26) and (27), the DC gain of G1 (s) is

K = G1 (0) = mg0 = my

0mu

0= my

0AD

. (30)

he COG of input and output are determined as

�u = mu1

mu0

= 12

D, �y = my1

my0

= 12

D + L + T. (31)

rom (31), it follows that

�y − �u = �g = L + T. (32)

ubstituting (32) into (29) yields

T2 = my2

my0

−(

my1

my0

)2

− 112

D2. (33)

t is evident from (31) that the right-hand side of (33) is positive. Therefore, (33) results in (24). Since T is now known, (25) is derived from32). �

To identify an SODF process, one may use the method in [7]. However, this method can be applied only to an underdamped SODF systemnd the DC gain must be known previously. An advantage of the proposed method is that the model parameters can be determined foroth the underdamped and overdamped cases.

roposition 2. For the SODF model in (14), the parameters can be determined by

(i) K = my0

AD, (34)

P

I

F

T

�

P

w

a

Nu


(ii) ωn =√√√√ 2

2(

my1

my0

)2

− my2

my0

− Dmy

1my

0

+ 13 D2

, (35)

(iii) � = 12

(my

1

my0

− 12

D

)ωn. (36)

roof. Applying (8) to (14) as in the proof of Proposition 1, and the first three moments of the output are obtained as follows:

my0 = ADK, (37)

my1 = ADK

(12

D + 2�

ωn

), (38)

my2 = ADK

[13

D2 + D2�

ωn+ 2(

2�

ωn

)2− 2

(1

ω2n

)]

= my0

[13

D2 + 2(

12

D + 2�

ωn

)2− D

(12

D + 2�

ωn

)− 2

(1

ω2n

)].

(39)

t is obvious that (37) is identical to (34). The COG of the output is obtained by

�y = my1

my0

= 12

D + 2�

ωn. (40)

rom (39) and (40), it follows that

2

ω2n

= 13

D2 + 2(

�y)2 − D�y − my

2

my0

. (41)

he right-hand side of (39) is positive. Therefore, (36) is derived by (41). Since ωn is now known, (38) yields

� = 12

(�y − 1

2D)

ωn. (42)

roposition 3. For the SOTD model in (15), the parameters can be determined by

(i) K = my0

AD, (43)

(ii) L = c1 − �∗1, (44)

(iii) ωn =√

2(�∗

1

)2 + c21 − c2

, (45)

(iv) � = 12

ωn�∗1, (46)

here �∗1 is a real positive solution to the cubic equation

�31 + 3

(c2

1 − c2)

�1 +(

2c31 − 3c1c2 + c3

)= 0, (47)

nd

c1 := mg1

mg0

= �y − �u,

c2 := mg2

mg0

= my2

my0

− 2�u(

�y − �u)

− mu2

mu0

,

c := mg3 = my

3 − 3�u my2 −

(3�y − 6�u

) mu2 − 6

(�u)2 (

�y − �u)

− mu3 .

(48)

3mg

0 my0 my

0mu

0 mu0

ote that the parameters c1, c2, and c3 are known constants because mgi

can be calculated by means of the moments of input and outputsing (11).

6

Pt

w

S

Tc

T

S

S

w

T

w

T


roof. Recall that K /= 0, T > 0, L ≥ 0, � ≥ 0, and ωn > 0 are assumed for the model (15). According to Corollary 1, the first four moments ofhe impulse response of the SOTD model can be represented in terms of the moments of the input and output as follows:

mg0 = my

0mu

0,

mg1 = 1

mu0

(my1 − mg

0mu1),

mg2 = 1

mu0

(my2 − 2mg

1mu1 − mg

0mu2),

mg3 = 1

mu0

(my3 − 3mg

2mu1 − 3mg

1mu2 − mg

0mu3),

(49)

here muk

for k = 0, 1, 2, 3 are given by (21). Applying (5) to (49), it follows that

mg0 = K,

mg1 = K

(L + 2�

ωn

),

mg2 = K

{(L + 2�

ωn

)2+[(

2�

ωn

)2− 2

(1

ω2n

)]},

mg3 = K

{(L + 2�

ωn

)3+ 3

[(2�

ωn

)2− 2

(1

ω2n

)][L + 2

(2�

ωn

)]−(

2�

ωn

)3}

.

(50)

ince mg0 = G3 (0) = K = my

0mu

0and mu

0 = AD, (43) holds. Substituting (50) into (48) yields

c1 = mg1

mg0

= L + 2�

ωn,

c2 = mg2

mg0

=(

L + 2�

ωn

)2+[(

2�

ωn

)2− 2

(1

ω2n

)],

c3 = mg3

mg0

=(

L + 2�

ωn

)3−(

2�

ωn

)3+ 3

[(2�

ωn

)2− 2

(1

ω2n

)][L + 2

(2�

ωn

)].

(51)

he identification problem is boiled down to the problem of solving (51) relative to three parameters{

�, ωn, L}

. Here the parameters c1,2, and c3 are obtained using (49). The following new parameters are defined:

�1 := 2�

ωn, �2 := L + 2�

ωn, �3 := 1

ω2n

. (52)

hen (51) has the following constraint:

c2 − c21 = �2

1 − 2�3. (53)

ubstituting (53) into (51), the following cubic equation with respect to �1 is derived.

�31 + 3(c2

1 − c2)�1 + (c3 − 3c1c2 + 2c31) = 0. (54)

implifying (54) yields

�31 + d1�1 + d0 = 0, (55)

here

d1 = 3(c21 − c2), d0 = (c3 − 3c1c2 + 2c3

1). (56)

he solution to the cubic equation (55) is as follows:⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

�11 = Q + R,

�12 = −12

(Q + R) − j

√3

2(Q − R) ,

�13 = −12

(Q + R) + j

√3

2(Q − R) ,

(57)

here √ √2 3

√ √2 3

Q =3

−d0

2+ d0

4+ d1

27, R =

3−d0

2− d0

4+ d1

27. (58)

here are three types of solutions depending on the discriminant � = d20

4 + d31

27 :

(

L

(

P

w

E

Pta

H

T


(i) one root is real and two complex conjugate if � > 0,(ii) all roots are real and at least two are equal if � = 0,iii) all roots are real and distinct if � < 0.

et the real solution to (55) be �∗1. Then (44) is derived by (51) and (52). From (53),

1

ω2n

= 12

(�∗1 + c2

1 − c2). (59)

59) is identical to (45). Subsequently, (46) is determined by (52). �

roposition 4. For the SOZDF model in (16), the parameters can be determined by

(i) K = 6mg0mg

1mg2 − 6(mg

1)3 − (mg0)2mg

3

6(mg1)2 − 3mg

0mg2

, (60)

(ii) z0 = my0

ADK= mg

0

K, (61)

(iii) ωn = 1√ϑ∗

2

, (62)

(iv) � = 12

ϑ∗1ωn, (63)

here

ϑ∗1 = 3mg

1mg2 − mg

0mg3

6(mg1)2 − 3mg

0mg2

=my

3my0 − 3my

2my1 − 3D(my

1)2 + 32

D2my1my

0 + 14

D3(my0)2

3my2my

0 − 6(my1)2 + 3Dmy

1my0 − D2(my

0)2,

(64)

ϑ∗2 = 3(mg

2)2 − 2mg1mg

3

12(mg1)2 − 6mg

0mg2

=2my

3my1 − Dmy

3my0 − 3(my

2)2 + 3Dmy2my

1 + 12

D2my2my

0

6my2my

0 − 12(my1)2 + 6Dmy

1my0 − 2D2(my

0)2

+−2D2(my

1)2 + 12

D3my1my

0 − 112

D4(my0)2

6my2my

0 − 12(my1)2 + 6Dmy

1my0 − 2D2(my

0)2.

(65)

quivalently, K can be determined by another formula:

(v) K = 1AD

[(ϑ∗

1 + 12

D)

my0 − my

1

]. (66)

roof. Using (11), the first four moments of the impulse response of the SOZDF model (16) can be determined by means of the moments ofhe pulse response as in (49). It means that mg

k, k = 0, 1, 2, . . . are known values. Applying (5) to (16), these moments can be also described

s follows:

mg0 = Kz0, (67)

mg1 = K

[z0

(2�

ωn

)− 1]

, (68)

mg2 = 2K

{z0

[(2�

ωn

)2−(

1

ω2n

)]−(

2�

ωn

)}, (69)

mg3 = 6K

{z0

[(2�

ωn

)3− 2(

2�

ωn

)(1

ω2n

)]−[(

2�

ωn

)2−(

1

ω2n

)]}. (70)

ere, new unknown variables are defined as follows:

ϑ1 := 2�, ϑ2 := 1

2. (71)

ωn ωn

hen (68)–(70) can be equivalently reformed as follows:

mg1 = mg

0ϑ1 − K, mg2 = 2mg

1ϑ1 − 2mg0ϑ2, mg

3 = 3mg2ϑ1 − 6mg

1ϑ2, (72)

6

o

T

So

S

Rtnio

3

w

ot

T


r in the matrix form,⎡⎢⎣

−1 mg0 0

0 2mg1 −2mg

0

0 3mg2 −6mg

1

⎤⎥⎦⎡⎣ K

ϑ1

ϑ2

⎤⎦ =

⎡⎢⎣

mg1

mg2

mg3

⎤⎥⎦ . (73)

he solution to (73) is given by

K∗ = 6mg0mg

1mg2 − 6(mg

1)3 − (mg0)2mg

3

6(mg1)2 − 3mg

0mg2

, (74)

ϑ∗1 = mg

3mg0 − 3mg

2mg1

3mg2mg

0 − 6(

mg1

)2, (75)

ϑ∗2 =

2mg3mg

1 − 3(

mg2

)2

6mg2mg

0 − 12(

mg1

)2. (76)

ince K is now known, (67) yields (61). Substituting (75) and (76) into (71) leads to (62) and (63). From (7) and (8), the first moment of theutput is given as

my1 =[(

2�

ωn

)+ 1

2D]

my0 − ADK. (77)

ubstituting ϑ∗1 into (77) yields (66). �

emark 1. In this identification method, the pulse shape is not restricted to rectangular but allowed different forms such as half-sine,riangular, displaced cosine, and so on. For such a pulse input, its moments can be easily determined by either mathematical calculation orumerical integration of the sampled data. Corollary 1 states that the moments of impulse response, mg

kfor k = 0, 1, . . ., are always obtained

n terms of the moments of input and output. Therefore, from (27), (50), and (67)–(70), it is easy to derive similar formulas for identificationf the model parameters to those propositions proposed in Section 3.1.

.2. Identification of MIMO process

The transfer function of a stable LTI process with n inputs and m outputs is given by

G (s) =

⎡⎢⎢⎢⎢⎢⎢⎢⎣

G11 (s) G12 (s) · · · G1n (s)

G21 (s) G22 (s) · · · G2n (s)

......

...

Gm1 (s) Gn2 (s) · · · Gmn (s)

⎤⎥⎥⎥⎥⎥⎥⎥⎦

. (78)

The ith output is written by

Yi (s) =n∑

j=1

Gij (s) Uj (s) , for i = 1, 2, . . . , m. (79)

Applying Theorem 1 to (79), the kth moment of yi (t) is determined by

myik =

n∑j=1

k∑l=0

(k

l

)mgij

k−lmujl , for i = 1, 2, . . . , m and k = 0, 1, 2, . . . , (80)

here gij is the impulse response of Gij (s). For example, if G (s) is a 2 × 2 model, then the first moments of outputs are as follows.

my11 = mg11

1 mu10 + mg11

0 mu11 + mg12

1 mu20 + mg12

0 mu21 , (81)

my21 = mg21

1 mu10 + mg21

0 mu11 + mg22

1 mu20 + mg22

0 mu21 . (82)

A simple open-loop approach to multi-variable identification is to deal with the MIMO system as a set of SISO problems. To do this, onlyne input is applied to the process at a time while the other inputs are zero. If all the remaining inputs except for uj are set to zeros, thenhe ith output is

Yi (s) = Gij (s) Uj (s) . (83)

herefore, the parameter of Gij (s) can be identified by using the same method as in the SISO case in Section 3.1.

3

o

3

TIbcmo

Wfso

3

wTu

wtof

3

mhlma

3

wpmv

T

4

ae


.3. Pulse selection and numerical integration

A guide will be discussed for selecting input pulse and step size in the numerical integration of moments. In addition, the effectivenessf the proposed method to measurement noise and order mismatch between high-order process and low-order model will be addressed.

.3.1. Magnitude and width of input pulseRecall that the actual process is generally nonlinear, and that a low-order model that is linearized around an operating level is obtained.

his means that the magnitude of the pulse should not be so high to keep the process response within a linear range. According to theSA-S26 standard [17], it is recommended that the amplitude of the input pulse test signal be 10% of the actual input span of the systemeing tested. When a pulse-type testing is employed, the degree of success depends on the width of the pulse. The pulse width must behosen such that the system is properly excited. If the pulse width is long compared with the response of the system, the system is onlyoderately excited and thus may lose high frequency information; whereas, if it is too narrow, then the system may be forced into a region

f nonlinear operation.The Fourier transform of the input pulse shown in Fig. 1 is

U (ω) = A

ω

{sin (ωD) − j [1 − cos (ωD)]

}. (84)

hen a frequency is ω = 2D , U (ω) goes to zero. Therefore, the smaller the D selected, the higher the frequency to which G (jω) can be

ound. The proper pulse width should possess the characteristic of producing an output peak rising to 50–70% of the input peak with aystem gain of unity [17]. In [18], a good rule of thumb is to keep the width of the pulse to less than about half the smallest time constantf interest.

.3.2. Step size �t in numerical integrationMoments of u (t) and y (t) must be computed by the numerical sum through the use of acquired experimental data. That is,

mui =

NT∑k=1

tiu (k�t) , myi

=NT∑

k=1

tiy (k�t) , (85)

here �t and NT denote the step size and the number of data points, respectively. Let the terminal time of the integral in (3) be Tt. Then,t = (NT − 1) �t. It is evident that the smaller the step size is, the better the accuracy becomes. A conventional criterion for selecting thepper limit of data points is that the step size �t should be such that

ωmax�t ≤

2, (86)

here ωmax denotes the maximum frequency at which the frequency band of input pulse contains [18]. It is a merit of the pulse test thathe terminal time of the integral can be easily determined because the input is a single rectangular pulse of finite width D. A disadvantagef this method is that it is very difficult to evaluate higher-order moments because tk should be multiplied by a signal in the integralorm.

.3.3. Robustness against the order-mismatch between process and modelAs mentioned earlier, in most cases of PID controller designs, the processes are modeled as a FOTD or a SOTD transfer function. Otherwise,

odel reduction must necessarily be performed. In this context, there are two approaches: One is to design a high-order controller for aigh-order process model and then reduce the controller to a low-order form; the other is to first reduce the process model to an appropriate

ow-order form so that the resulting model based-controller will be in low-order form. The method of moments is one of the most popularodel reduction techniques. In this regards, identifying a low-order model by using moment matching is equivalent to directly obtaining

reduced-order form of a high-order process. This argument will be demonstrated in the next section.

.3.4. Effect of measurement noiseOutput data are usually corrupted by measurement noise, which can be always described by

y(t) = G(p)u(t) + (t), (87)

here p is the time-domain differential operator and (t) represents measurement noise and disturbances of all natures. There are manyossibilities that depend on the method used to model the disturbance (t). More general approaches for identification of continuous-timeodels are discussed in [19]. In this paper, we consider simple processes where the effect of disturbance is so small that it is negligible and

(t) is a measurement noise modeled by white Gaussian with zero mean. Then y(t) and (t) are stochastic processes. Taking the expectedalue of its moments, it follows that

E{myi} = E{

∫ ∞

0

tiy(t)dt} =∫ ∞

0

tiE{y(t)}dt = myi. (88)

herefore, the method of moments is robust to measurement noise, which can be modeled by a white Gaussian.

. Comparative analysis: accuracy and robustness

For the purpose of comparing the performances of the proposed identification method with those of other methods, several examplesre considered. First, the effect of the moment method to noise is discussed, and the robustness to the mismatch in the model order isxamined. The performances are assessed by using several criteria in both time and frequency domains, which are defined as follows.


Table 1Identified FOTD models in example 1.

Methods G (s)

Without noise With noise (SNR = 20)

Aström’s model 12s+1 e−3s 1.0718

2.0094s+1 e−2.9510s

r

wre

w

ps

4

E

tasNimviar

Ea

León de la Barra’s model 0.99981.9995s+1 e−2.9996s 1.0832

2.1834s+1 e−3.0126s

The proposed FOTD model 11.9991s+1 e−3.0008s 0.9994

1.9775s+1 e−3.0169s

A. Time domain accuracyLet the step and impulse responses of the identified model be ys (t) and yi (t), respectively, and let the measured data be ys (t) and yi (t),

espectively. The accuracy of the identification model in time domain is evaluated by the integral of the absolute errors (IAE):

Es :=NT∑

k=1

∣∣ys (k�t) − ys (k�t)∣∣ , (89)

Ei :=NT∑

k=1

∣∣yi (k�t) − yi (k�t)∣∣ . (90)

B. Frequency domain accuracyLet the identified model of the actual process G (s) be G (s). Then the frequency responses of G (s) and G (s) are

G (jω) = GK (ω) ej�(ω), (91)

G (jω) = GK (ω) ej�(ω). (92)

The IAE for the magnitude and the phase between the actual process and identified model are calculated as follows:

EK :=Nf∑

k=1

∣∣10log10GK (2k�f ) − 10log10GK (2k�f )∣∣ , (93)

E� :=Nf∑

k=1

∣∣� (2k�f ) − � (2k�f )∣∣ , (94)

herein Nf and �f are the number of frequency response data and the step size of frequency, respectively. Let the upper bound of frequencyange to be tested be ωu, then, ωu =

(Nf − 1

)�f . The frequency domain estimation error can be also measured by the following worst case

rror (WCE) [4]:

EW := maxω∈[0,ωc ]

{∣∣∣∣G (jω) − G (jω)G (jω)

∣∣∣∣}

× 100%, (95)

here ωc is the phase crossover frequency, that is, ∠G (jωc) = −.In all the examples in this section, a value of 1 is chosen for the amplitude of input pulse so that it equals the system gain of unity. The

ulse width (D) is selected such that tr1 ≤ D ≤ tr2, where tr1 and tr2 are the instances of time when the output attains 50 and 70 % of its finalteady state, respectively.

.1. Effects of the noise

xample 1. Consider a FOTD process

G (s) = 12s + 1

e−3s.

The proposed method is compared with other methods using extremal data points reported in [20] and [8]. For a fair comparison,he finite-duration pulse inputs with the same total energy are used. It is seen that the two response times, tr1 and tr2, of the processre 4.4 and 5.5, respectively. An input pulse with A = 1, D = 5, � = 1.2, and � = 0.7D is chosen, where � and � denote the multiplier andeparation time between two rectangular pulses, specifying a double rectangular pulse in [8]. The data length and the step size interval are

= 30,000 and �t = 0.001[s]. In addition, a white Gaussian noise with SNR = 20 was added by the process outputs. These outputs are shownn Fig. 2. The moments of outputs computed using (85) are my

0 = 5.0, my1 = 37.499, and my

2 = 311.635 when noise is absent, and my0 = 4.997,

y1 = 37.451, and my

2 = 310.651 with noise, respectively. The models identified with and without noise are given in Table 1. The resultingalues of error criteria are given in Table 2. For the noise-free models, all three methods provide good identification. However, when noises included in the output, the other two methods show poor accuracy, whereas the proposed method gets the best accuracy in both time
nd frequency domains. Moreover, it shows good consistency for the cases where the output data is either free of noise or noisy. For theseeasons, the proposed method is more robust to noise than Aström and León de la Barra’s methods.
xample 2. It is shown that the proposed method can accurately identify a second-order process, regardless of whether it includes delay,nd regardless of whether it is overdamped or underdamped.


Table 2Estimation errors in example 1.

Methods Without noise With noise (SNR = 20)

Es Ei EK E� EW Es Ei EK E� EW

Aström’s 0 0 0 0 0 6.168 0.164 2.351 0.251 0.072León de la Barra’s 0.016 0.001 0 0 0.0002 6.287 0.798 1.753 1.259 0.083The proposed 0.002 0.502 0 0 0.0003 0.064 0.536 0.005 0.006 0.007

Table 3Exact process and identified SODF models in example 2.

Over-damped Under-damped

4

rDawi

4

r

4

omG

E

F

Exact process 1s2+4s+1

1s2+0.4s+1

Identified model 1.00021.0462s2+3.9998s+1

0.99931.3002s2+0.3754s+1

.1.1. SODF processesFirst, the two SODF systems in (14) are considered, the exact parameters of which are K = 1, ωn = 1, and damping ratios of � = 2 and � = 0.2,

espectively. It is seen that two response times, tr1 and tr2, of the overdamped process are 2.95 and 4.91, respectively. For the model, A = 1, = 4, N = 20, 000, and �t = 0.002[s] are chosen. Similarly, the values for the underdamped model are selected as A = 1, D = 1.2, N = 40, 000,nd �t = 0.001[s] since tr1 = 1.13 and tr2 = 1.41. For the purpose of comparison of noise effect, two cases, noise-free and white Gaussian noiseith SNR = 20 are considered. By applying Proposition 2 to these processes, we have obtained the identified models in Table 3. The results

ndicate the accuracy of the proposed identification method.

.1.2. SOTD processNow, consider the above overdamped process with the delay L = 2. That is,

G (s) = 1s2 + 4s + 1

e−2s.

In this example, the parameters are selected as A = 1, D = 5, N = 40, 000, and �t = 0.001[s] since tr1 = 4.9 and tr2 = 6.7. After collecting pulseesponse data including noise and applying Proposition 3, the following model is obtained:

G (s) = 11.899s2 + 4.206s + 1

e−1.79s.

.2. Robustness to the mismatch in the model order

In general, practical processes may be of high-order. For a fair comparison of robustness of identification methods with respect torder mismatch between the actual process and the model, a noise-free process is considered here. The models identified by the proposedethod are compared with those in [9], which were obtained by model reduction methods using Skogestad’s half rule [21], Isaksson andraebe’s method [22], and Huang’s step response data [23], respectively.

xample 3. Consider the high-order process of minimum-phase dynamics used in [9] as follows:

G (s) = (15s + 1)2 (4s + 1) (2s + 1)

(20s + 1)3(10s + 1)3(5s + 1)3(0.5s + 1)3.

0 5 10 15 20 25 30−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1Output Responses

Time (sec)

Am

plit

ud

e

M1M2M3

M3

M2

M1

ig. 2. Process outputs of three different inputs for the FOTD system in example 1 (M1: Aström’s method, M2: León de la Barra’s method, M3: the proposed FOTD method).


Table 4Identified models and estimation errors in example 3.

Methods G (s) Es Ei EK E� EW

Skogestad’s 1(20s+1)(15s+1) e−35.5s 1.70 0.10 5.63 4.21 0.59

Isaksson and Graebe’s 25.5s+12642s2+73.25s+1

5.75 0.14 5.14 28.0 0.35

Huang’s 144.46s+1 e−27.38s 2.01 0.10 1.81 10.4 0.16

The proposed FOTD 133.25s+1 e−37.3s 1.18 0.10 3.07 2.44 0.46

The proposed SODF 11932s2+70.5s+1

2.50 0.09 6.91 8.87 0.49

The proposed SOTD 1695.3s2+49.96s+1

e−20.5s 0.29 0.02 0.20 1.48 0.05

The proposed SOZDF −13.59s+1974.2s2+56.91s+1

0.98 0.08 0.48 5.66 0.05

0 100 200 300 400−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

0.025Impulse Responses

Time (sec)

Am

plit

ud

e

TrueM1M2M3

FM

Imt

E

Ft

ig. 3. Impulse responses of the actual process and the identified models for the high-order system in example 3 (M1: Huang’s method, M2: the proposed SOTD method,3: the proposed SOZDF method).

The parameters specifying input pulse are chosen as A = 1 and D = 80. Here the simulation parameters are N = 4, 000 and �t = 0.1[s].dentified models and estimation errors are given in Table 4. Both time and frequency responses of the true process and the three identified

odels are shown in Figs. 3 and 4, respectively. The SOTD and SOZDF models identified using the proposed method are much more accuratehan the others in both time and frequency domains.

xample 4. Consider the minimum phase high-order process with coincident poles used in [9]:

G (s) = 1

(s + 1)8.

−25

−20

−15

−10

−5

0

Mag

nit

ud

e (d

B)

10−3

10 −2

10 −1

−360

−270

−180

−90

0

Ph

ase

(deg

)

Bode Diagrams

Frequency (rad/sec)

TrueM1M2M3

ig. 4. Bode diagrams of the actual process and the identified models for the high-order system in example 3 (M1: Huang’s method, M2: the proposed SOTD method, M3:he proposed SOZDF method).


Table 5Identified models and estimation errors in example 4.

Methods G (s) Es Ei EK E� EW

Skogestad’s 1(1.5s+1)(s+1) e−5.5s 2.23 1.51 0.023 0.0012 0.0240

Isaksson and Graebe’s 114.52s2+5.02s+1

7.62 1.67 0.048 0.2663 0.0497

Huang’s 14.24s+1 e−3.88s 2.59 1.25 0.040 0.0083 0.0420

Proposed FOTD 12.828s+1 e−5.17s 1.45 1.32 0.0013 0.0045 0.0013

Proposed SODF 128s2+8s+1

4.61 1.53 0.0319 0.0406 0.0332

tm

5

SftmofWulotuo

A

b

R

[[

[[[[[[[[[[[[

Proposed SOTD 16.232s2+4.524s+1

e−3.48s 0.54 0.40 0.0005 0.0002 0.0005

Proposed SOZDF −2s+112s2+6s+1

2.22 1.39 0.0035 0.0015 0.0036

The parameters of input pulse are chosen as A = 1 and D = 8. Here, the simulation parameters are N = 4, 000 and �t = 0.01[s]. Table 5 showshe identified models and estimation errors. The results indicate that the proposed methods increase accuracy, and that the proposed SOTD

odel is an excellent approximate of the given high-order process.

. Concluding remarks

In classical control design, the empirical identification of an actual process makes extensive use of simple models, such as FOTD, SODF,OTD, and SOZDF models. This paper proposed a simple open-loop identification method that could determine the parameters of theseour models in the closed-form formulas by computing only a few moments of output. The test input is solely a single rectangular pulse andhus can be easily applied to any real processes. All the propositions for identification formulas have been proven analytically. The moment

ethod developed for a SISO LTI process can be directly extended to the identification of a MIMO LTI process. The accuracy and robustnessf the proposed method were demonstrated through a comparative analysis. For this, several criteria were set and defined in time andrequency domains, namely, the IAE of the step and impulse responses, the IAE of the frequency magnitude and phase responses, and the

CE of the frequency response, respectively. The proposed identification method using moments is more accurate than those methodssing extreme points of pulse response when the output data is corrupted by noise. Furthermore, the proposed method provides excellent

ow-order approximations for the various types of high-order processes in [9]. The definition of moments requires positive input andutput, and hence this identification method cannot be applied to the nonminimum-phase process. In particular, it should be emphasizedhat this method is a unified identification method for continuous-time first- and second-order models because the same formula can besed regardless of whether the processes are overdamped or underdamped, regardless of whether they have delay or not, and regardlessf their orders.

cknowledgements

This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) fundedy the Ministry of Education, Science and Technology (grant No. 2012-0007503).

eferences

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17 (1978) 237–241.[4] Q.G. Wang, Y. Zhang, Robust identification of continuous systems with dead-time from step responses, Automatica 37 (2001) 377–390.[5] K.J. Aström, T. Hägglund, Automatic tuning of simple regulators with specification on phase angle and amplitude margins, Automatica 20 (1984) 645–651.[6] S.H. Hwang, S.T. Lai, Use of two-stage least-squares algorithms for identification of continuous systems with time-delay based on pulse responses, Automatica 40 (2004)

1561–1568.[7] B.A. León de la Barra, M. Mossberg, Identification of under-damped second-order systems using finite-duration rectangular pulse inputs, in: Proceedings of the 2007

American Control Conference, NY, USA.[8] B.A. León de la Barra, Lihua Jin, Y.C. Kim, M. Mossberg, Identification of first-order time-delay systems using two different pulse inputs, in: Proceedings of the 17th IFAC

World Congress, Seoul, Korea.[9] A. Visioli, Practical PID Control, Springer, London, 2006.10] K.J. Aström, T. Hägglund, Automatic Tuning of PID Controllers, ISA, Research Triangle Park, 1988.11] L.G. Gibilaro, F.P. Lees, The reduction of complex transfer function models to simple models using the methhod of moments, Chemical Engineering Science 24 (1969)

85–93.12] A.C. Antoulas, Approximation of Large-scale Dynamical Systems, SIAM, Philadelphia, 2005.13] R. Eid, Time Domain Model Reduction by Moment Matching, Ph. D. Thesis, Technical University Munich, Munich, 2009.14] A. Astolfi, Model reduction by moment matching for linear and nonlinear systems, IEEE Transactions on Automatic Control 55 (2010) 2321–2336.15] I.A. Zadeh, C.A. Desoer, Linear System Theory, McGraw-Hill, New York, 1963.16] Y.I. Ismail, Efficient model order reduction via multi-node moment matching, in: Proceedings of the International Conference on Computer Aided Design, CA, USA, 2002.17] ISA-S26-1968 standard: dynamic response testing of process control instrumentation, Instrument society of America, 1966.18] P.B. Deshpande, R.H. Ash, Elements of Computer Process Control with Advanced Control Applications, ISA, Research Triangle Park, 1981.
19] H. Garnier, L. Wang (Eds.), Identification of Continuous-time Models from Sampled Data, Springer-Verlag, London, 2008.20] K.J. Aström, T. Hägglund, Advanced PID Control, ISA, Research Triangle Park, 2006.21] S. Skogestad, Simple analytic rules for model reduction and PID controller tuning, Journal of Process Control 13 (2003) 291–309.22] A.J. Isaksson, S.F. Graebe, Analytical PID parameter expressions for higher order systems, Automatica 35 (1999) 1121–1130.23] B. Huang, A pragmatic approach towards assessment of control loop performance, International Journal of Adaptive Control and Signal Processing 17 (2003) 589–608.

JPC2013 Robust ID of continuous-time using moments

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