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404 H. Li and Z. Zhang

Table 31.1 Main parameters of the DI135 diesel

Cylinder diameter D (mm) 135

Piston stoke S (mm) 140

Piston swept volume V s(L) 2.0

Compression ratio " 17.5

Declare power Pe (KW) 14.7

Declare speed n (r/min) 1,500

Bosh Pump Type A ®10

Fuel nozzle 5  0:32  150

Injection starting pressure P

(MPa) 22

Fuel supply advance angle 

(CA) 15

The vibration signal transducer was located on the cylinder head to monitor the

combustion process. At the same time, there is not any vibration signal inﬂuence

from other cylinder effect just for single cylinder diesel engine. Thus, the vibration

signal measured from a cylinder head can be considered a classiﬁcation signal for

the working condition, including combustion and fuel injection process, for a single

cylinder. It is very useful to estimate the combustion process. It is usually used to

classify different working conditions.

31.4.2 Vibration Signal Analysis

The monitored vibration signal was obtained from four working conditions, normal

condition (NC), nozzle fault condition (NF), inlet fault (IF), and reduced injec-

tion fuel angle (RI). The Vibration signals were monitored with speed 1,500 rpm

with 25% load. The sampling frequency and analysis frequency are 25,600 and

10,000 Hz, respectively. The time domain signal corresponding to the combustion

process for different working conditions is shown in Fig. 31.6 according to diesel

engine dynamics analysis. TDC is used to determine the time domain information

for vibration signal. For the combustion process, the total monitored signal length

is 512. It is obvious that the vibration signals are very similar in time domain. It is

unable to separate different working conditions just according to vibration signal.

31.4.3 Hilbert Spectrum Recognition

Based on empirical mode decomposition and Hilbert–Huang transform, Hilbert

spectrum can be obtained for different working conditions shown in Fig. 31.7.The

HS is a broad band frequency signal as shown in Fig. 31.7. The energy is also con-

centrated around TDC. At the same time, there is not much difference for the HS

in different working conditions although there is difference in the time-frequency

image. But it is not clear to distinguish different conditions.

31 Mechanical Dynamics and Time-Frequency Image Processing 405

Fig. 31.6 Time domain signal in different conditions; (a)NC,(b)NF,(c)IF,(d)RI

Fig. 31.7 Time-frequency image for different working condition: (a)NC,(b)NF,(c)IF,(d)RI

406 H. Li and Z. Zhang

Fig. 31.8 Recalculation time-frequency image for different working condition: (a)NC,(b)NF,

(c)IF,(d)RI

As stated above, Hilbert spectrum can be looked as a two-dimension time-

frequency image. Thus, image processing technology can be used for feature ex-

traction and fault diagnosis. At the same time, it is better to give more concentration

for the energy distribution. That small energy can be removed, which can improve

the expression for Hilbert spectrum. HS is recalculated and a new time-frequency

image is determined in this research. After energy distribution recalculation, new

time-frequency image can be obtained shown in Fig. 31.8.

31.4.4 ED Discriminant

It is much clearer for different diesel engine working condition classiﬁcation af-

ter the time-frequency image recalculation. The energy is also as not dispersed as

shown in Fig. 31.7. It is more suitable for condition recognition. For quantitative de-

scription, ED is used in this research for different condition classiﬁcation. There is

much difference for different working cycles although they are in the same working

31 Mechanical Dynamics and Time-Frequency Image Processing 407

Fig. 31.9 Time domain vibration signal for diesel engine cylinder head

Table 31.2 Classiﬁcation results for different working conditions

ED1 ED2 ED3 ED4 Classiﬁcation result

Test 1 0.3146 5.6163 5.6614 5.6650 NC

Test 2 5.5007 0.9064 5.3414 5.3844 NF

Test 3 5.6607 5.5260 0.8699 5.3159 IF

Test 4 5.6844 5.5224 5.3372 0.7945 RI

condition shown in Fig. 31.9. Cyclostationary is very common for diesel engine [8].

It is not accurate just according to one cycle vibration signal. To reduce cyclosta-

tionary effect for pattern classiﬁcation, multicycle time-frequency image average is

used. Based on ﬁve time-frequency image average, four standard images can be ob-

tained. These images can be looked on as the typical image for different working

conditions. Then, ED is used to classify different working conditions for test sam-

ples based on above calculation equation (31.8). It can give a quantitative description

for pattern recognition. The distance with different standard image is shown in Ta-

ble 31.2 for different working condition classiﬁcation. Its corresponding condition

is also given in Table 31.2 according to ED.

According to Table 31.2, the smaller is ED, the more similar the two images.

That means the test sample is similar with standard image. The time image reﬂects

machine working condition. It also means the similar of two monitored conditions.

It can be recognized that two monitored signals are in the same condition if ED

is very small. Therefore, ED is used to estimate the similar of two images. In this

research, a threshold ED

is deﬁned equal to 1. When ED between the standard im-

age and test image is smaller or equal to ED

, it means that the test sample is same

as the standard image. They are in the same working condition. Otherwise, they are

in different working conditions. ED can be used to estimate the similarity between

test sample and standard sample. According to ED, sample “Test 1” can be accu-

rately classiﬁed as NC where ED is smaller than 1. It is also the same as practical

condition. For other working condition classiﬁcation, it can have the identical re-

sult. It is obvious that different conditions can be recognized according to ED. This

method is convenient for practical vibration signal analysis and pattern recognition

with determined working conditions.

408 H. Li and Z. Zhang

31.5 Conclusions

Mechanical dynamics analysis is very important for machine pattern recognition

and condition monitoring. It is the basis for engine vibration signal analysis. Diesel

engine structure dynamics and impulse response characteristics are investigated in

detail in this research. It is very important for time domain vibration signal determi-

nation for analysis and pattern classiﬁcation. It can effectively reduce the calculation

and improve the accuracy of pattern recognition based on mechanical dynamics

analysis.

Hilbert spectrum is used to construct time-frequency distribution, which is very

effective for nonstationary and nonlinear signal analysis. Time-frequency distribu-

tion can be looked as a two-dimension image. Thus, image recognition method is

investigated in this research. Euclidean distance is used to classify engines accord-

ing to different working conditions. Different working condition classiﬁcation of

diesel engine is as an example to testify the effectiveness of this method. According

to the classiﬁcation result analysis, it can be concluded that this method is suitable

for diesel engine pattern recognition and fault diagnosis.

This developed method is much more suitable for separated structure analysis

and fault diagnosis of diesel engine. For multicylinder with single body diesel en-

gine, its structure dynamics should be further investigated.

Acknowledgments The support from Chinese National Science Foundation (Grant No.

50805014) for this research is gratefully acknowledged. The ﬁrst author also wishes to acknowl-

edge to ﬁnancial aid from State key laboratory of mechanical system and vibration, Shanghai Jiao

Tong University (Grant No. VSN-2008-04) and State Key Laboratory of Structural Analysis for

Industrial Equipment, Dalian University of Technology (Grant No. GZ0817).

References

1. Logan K, Inozu B, Roy P, Hetet JF, Chesse P, Tauzia X (2002) Real-time marine diesel engine

simulation for fault diagnosis. Mar Technol 39(1):21–28

2. Zhou PL, Li HK (2005) Pattern recognition on diesel engine working conditions by wavelet

Kullback–Leibler distance method. Proc Inst Mech Eng Part C J Mech Eng Sci 219(9):879–888

3. Huang NE, Shen Z, Long SR, Wu MLC, Shih HH, Zheng QN, Yen NC, Tung CC, Liu HH (1998)

The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary

time series analysis. Proc R Soc Lon Ser A Math Phys Eng Sci 454:903–995

4. Li HK, Zhou PL, Ma XJ (2006) Marine diesel engine fault diagnosis by using improved Hilbert

spectrum. J Ship Res 4:378–387

5. Ma XJ, Yu B, Zhang ZX (2000) A new approach to time-frequency analysis – local wave

method. J Vib Eng 25(s):219–224

6. Wang WJ, McFadden PD (1993) Early detection of gear failure by vibration analysis-II: inter-

pretation of the timefrequency distribution using image processing techniques. Mech Syst Signal

Process 7(3):205–215

7. Lv C, Wang GZ (2005) Noise fault diagnosis based on time-frequency domain model. J Vib

Shock (in Chinese) 24(2):54–57

8. Antoni J, Daniere J, et al (2002) Effective vibration analysis of IC engines using

cyclostationarity. Part I – a methodology for condition monitoring. J Sound Vib 257(5): 815–837

Chapter 32

Input Design for Systems Under Identiﬁcation

as Applied to Ultrasonic Transducers

Nishant Unnikrishnan, Yicheng Pan, Marco Schoen, Ajay Mahajan,

Jarlen Don, and Tsuchin Chu

Abstract An input design system identiﬁcation (IDSI) method is outlined in this

chapter based on input/output data gathered from random excitation of a system

so as to excite all modes. The data set is then used to compose a new set of more

focused input data, from which the system is excited again and identiﬁed. In this

chapter, the input design method is used for the system identiﬁcation of an ultra-

sonic sensor pair (transmitter–receiver) so as to obtain an accurate model for it.

This model is essential for the analysis of a 3D position estimation system that uses

ultrasonic transducers. A single transmitter is attached to the point of interest, and its

position is triangulated based on signals received at multiple receivers ﬁxed in a 3D

environment. An accurate model for the response of the ultrasonic receivers is es-

sential in the eventual optimization of the entire system. This chapter only presents

results for a single pair (called the system in this chapter), but the results will be

applicable for the entire 3D system. Results are given that show the comparison of

the actual output signal of the system and the output of the model obtained from

the IDSI method as well as the identiﬁability indicator and an identiﬁed state-space

model. It is shown that the IDSI improves system identiﬁcation even in the presence

of excessive noise.

32.1 Introduction

The ﬁeld of system identiﬁcation has been developed for a multitude of purposes.

As each discipline gives its motivation, there is a common goal: one wants to in-

fer accurately the characteristics of a particular system from its input/output data.

There exists a great multitude of proposed and developed identiﬁcation methods

for linear, stochastic, and autonomous systems [1–4]. The main focus of this work

T. Chu (



)

Department of Mechanical Engineering and Energy Processes, Southern Illinois

University Carbondale, 1230 Lincoln Drive, Carbondale, IL 62901-6603, USA

e-mail: tchu@siu.edu

A.C.J. Luo (ed.), Dynamical Systems: Discontinuity, Stochasticity and Time-Delay,

DOI 10.1007/978-1-4419-5754-2

32,

 Springer Science+Business Media, LLC 2010

409

410 N. Unnikrishnan et al.

is the parametric system identiﬁcation methods, in particular the Observer/Kalman

ﬁlter Identiﬁcation method (OKID) [2] and the indirect closed-loop system identi-

ﬁcation method [5, 6]. In general, one seeks to obtain a system representation that

is as accurate and close to the original system as possible. The system identiﬁca-

tion methods used here employ autoregressive with exogenous input (ARX) model

representations and least-squares estimation for the determination of its parameters.

The objective of this work is to present a new input design which improves the ac-

curacy of the identiﬁcation results of ultrasonic sensors. Experimental results show

that the new method is effective in identifying ultrasonic transducers with high pro-

cess and/or measurement noises.

32.2 Input Design Formulation for ARX Models

Consider a stochastic, discrete time, autonomous system given in state space form:

kC1

D Ax

C Bu

C w

D Cx

C Du

C v

(32.1)

where x 2<

n1

, u 2<

s1

,andy 2<

m1

are state, input and output vectors,

respectively; w

is the process noise, v

is the measurement noise; ŒA;B;C;Dare

the state-space parameters. Sequences w

and v

are assumed to be Gaussian, white,

zero-mean, and stationary with covariance matrices Q and R respectively. Deﬁning

the steady state Kalman ﬁlter gain K D APC

.CPC

C R/

1

with P being the

unique positive deﬁnite symmetric solution of the algebraic Riccati equation, (32.1)

can be rewritten using the innovation "

kC1

D A Ox

C Bu

C K"

D C Ox

C Du

C "

(32.2)

Equation (32.2) is known as the steady state ﬁlter innovation model, where

D y

 C Ox

contains the new information. The steady state Kalman ﬁlter

gain K is guaranteed if the system is detectable and .A; Q

1=2

/ is stabilizable.

Rewriting (32.2) in input/output description and using

A D A.I

 KC/, one

can derive an ARX model representation

iD1

i1

AKy

ki

iD1

i1

ki

C "

(32.3)

provided that A.I

 KC / is asymptotically stable, and the model order p is large

enough such that

Š 0. Deﬁning the matrix coefﬁcient a

and b

D C

i1

AK and b

D C

i1

B (32.4)

32 Input Design for Systems Applied to Ultrasonic Transducers 411

Then the ARX model for such a system can be given as

iD1

ki

iD1

ki

C "

(32.5)

Since the system model is unknown, the parameter matrices for the ARX

model (a

and b

) have to be estimated from the input/output data. Note that

the ﬁnite ARX models given in (32.3)and(32.5) represent a truncated inﬁnite

transversal.

If one considers each summation of the ﬁnite ARX model series as a contributor

for the current estimate of the output, the ARX model of an open-loop system can

be represented in matrix format as follows:

00 0

0  0

00a

 0

000 a

k1

k2

k3

kp

00 0

0  0

00b

 0

000 b

k1

k2

k3

kp

(

32.6)

with

iD1

(32.7)

If one wants to design a new input sequence to yield a new ARX model with p

model order, one can use (32.6) to write

 D ƒy

C Qu (32.8)

where

 D

;ƒD

00 0

0  0

00a

 0

000 a

kp

1

kp

2

kp

3

kp

;D

00 0

0  0

00b

 0

000 b

412 N. Unnikrishnan et al.

and

Qu D

kp

1

kp

2

kp

3

kp

The coefﬁcients a

and b

of the ARX model are matrices for multiple input and

multiple output systems. Since each element of (32.8) represents a contribution to

the current output, the inﬂuence of this summation can be minimized by setting

 D 0 (32.9)

For a linear time-invariant system, the matrices ƒ and  are constant. If ƒ and 

are known and  is invertible, one can solve for the input which satisﬁes (32.9)

Qu D

1

ƒy

kp

j

Db

1

kp

j

;jD 1;2;:::;p (32.10)

where p D p  p

Using (32.4) and rewriting ƒ and  in (32.8) yields

ƒ D

AK 00 0

AK 0  0

00C

AK  0

00 0 C

p1

(32.11)

 D

B0 0 0

B0 0

00C

B  0

00 0 C

p1

(32.12)

If  is invertible, and using (32.10), the revised input Qu can be written as

kp

j

D.C

1

AKy

kp

j

for i D 1;2;:::;p (32.13)

32 Input Design for Systems Applied to Ultrasonic Transducers 413

If C 2<

non

, B 2<

nni

and no D ni D n,wehave

kp

j

DB

1

AKy

kp

j

(32.14)

32.3 System Identiﬁcation Algorithm

A number of different parametric system identiﬁcation algorithm can be employed

for the proposed input design. The one used in this work is the Observer Kalman

Filter System Identiﬁcation Algorithm (OKID).

Consider an output vector My D



pC1

::: y

L1



and the information

matrix







where L is the data length. Forming the following overall information matrix from

experimental input/output data:

V D

p1

::: u

L1



p1



p2

::: 

L2



::: 

Lp1

(32.15)

and the Markov Parameter sequence

Y D



B ::: C

p1



(32.16)

where

B D



B  KD K



. Then one can write

My D

V C " C C

Ox (32.17a)

where Ox D



::: Ox

Lp2



and " D



pC1

pC2

::: "

L1



If p is large enough, the term C

becomes zero and one can write

My D

V C " (32.17b)

Employing a least-square method, one can compute the sequence

Y , which is the

observer Markov parameters:

Y DMy





1

(32.18)