Novel Parameter Identification by Using a High-Gain Observer With Application to a Gas Turbine...

IEEE TRANSACTIONS ON INDUSTRIAL INFORMATICS, VOL. 4, NO. 4, NOVEMBER 2008 271

Novel Parameter Identification by Using a High-GainObserver With Application to a Gas Turbine Engine

Zhiwei Gao, Senior Member, IEEE, Xuewu Dai, Tim Breikin, and Hong Wang, Senior Member, IEEE

Abstract—In this study, a novel identification technique, that ishigh-gain observer-based identification approach, is proposed forsystems with bounded process and measurement noises. For systemparameters with abnormal changes, an adaptive change detectionand parameter identification algorithm is next presented. The pre-sented technique and algorithm are finally applied to the param-eter identification of the gas turbine engine by using the recordedinput data from the engine test-bed. The identified parameters andthe response curves are desirable. The simulations have proved theeffectiveness of the proposed procedure compared with the pre-vious identification approach.

Index Terms—Bounded noise environment, gas turbine engine,high-gain observer, parameter identification.

I. INTRODUCTION

A S is well-known, system identification is essentiallyimportant for establishing a system model, and it is

a starting point for system analysis and control synthesis.During the past decades, there were a large number of resultsreported in this field (see [1]). Basically, there are two kindsof identification techniques: polynomial model identificationand state–space model identification. The polynomial identi-fication approach (e.g., see [2]–[5]) is particularly suitable fora single-input-single-output (SISO) model. However, for mul-tiple-input-multiple-output (MIMO) systems, the polynomialmodel identification approach may give rise to numericallyill-conditioned mathematical problems. So far, state–spacemodel identification is still a hot research direction for MIMOsystems. Specifically, in the literature by [6]–[8], subspaceidentification technique was adopted, and the resulting iden-tification toolbox n4sid was developed. In the meanwhile,on the basis of different observers/filters, several parameterestimation algorithms were developed by [9]–[11]. However,it is not difficult to find that all the results in [6]–[10] and[11] are based on the noise-free or white noise assumption.Obviously, this kind of noise assumption cannot meet somepractical situations. Recently in [12], based on a high-ordersliding-mode observer, a parameter identification approachwas proposed under the bounded noise assumption rather thanthe white noise assumption. The bounded noise assumption is

Manuscript received February 13, 2008; revised April 29, 2008, June 20,2008, and October 05, 2008. Current version published January 21, 2009. Thiswork was supported in part by the EPSRC grant EP/C015185/1 and in part bythe grants NSFC 60574026 and NSFC 60828007. Paper no. TII-08-02-0014.R3.

Z. Gao is with the School of Electric Engineering and Automation, TianjinUniversity, Tianjin, China (e-mail: [email protected]).

X. Dai, T. Breikin, and H. Wang are with School of Electric and Elec-tronic Engineering, The University of Manchester, Manchester M60 1QD, U.K.(e-mail: [email protected]; [email protected];[email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TII.2008.2007802

obviously popular in many control issues. Unfortunately, onlythe measurement noise was taken into account by [12]. Actu-ally, a process noise always exists in any practical processes.Therefore, this motivates us to develop a novel parameteridentification technique for systems with both bounded processand measurement noises.

High-gain observer technique is a feasible approach for esti-mating or attenuating unknown bounded nonlinearities/uncer-tainties [13]–[15]. Very recently, an interesting high-gain si-multaneous observer was developed by [16] with application inactuator and sensor faults estimation and fault-tolerant controldesign. In this study, by using the high-gain simultaneous ob-server technique with slight modifications, the system state, thebounded output noise and the process uncertainty, composedof the parameter perturbations, are estimated simultaneously.Using the estimated state, estimated process uncertainty and themean method, the parameters are then identified. The dynamicresponse curves of the identified parameters are also given.

For systems with abnormal changes in actuators, sensors andparameters, it is extremely important to detect and isolate faults,and implement fault-tolerant operations. For actuator and sensorfaults, there have been fruitful results in detection, isolation andtolerant-design using different techniques, e.g., see [17]–[25]and the references therein. However, for abnormal parameterchanges, it has not been investigated fully, especially for identi-fying the changed parameters. When the parameters to be iden-tified have obvious changes in different time intervals, it is in-dispensable to detect the abnormal changing points and identifythe parameters in the resulting changing intervals. In this study,a new adaptive identification technique is addressed to simulta-neously detect the abrupt changing points and identify the pa-rameters over the resulting time intervals.

Gas turbine engines are widely used in many fields such asaerospace, marine and power generating etc. (see [26]). It is afundamental and key task to identify the model of the gas turbineengine accurately. In the paper, by using the proposed high-gainobserver-based identification technique and the recorded inputdata from the engine test-bed, simulation study is investigated indetail. The identified parameters are desirable for the gas turbineengine with unexpected bounded process and output noises. Theabnormal change detection and adaptive parameter identifica-tion has also proved effective in this simulated study.

II. HIGH-GAIN OBSERVER DESIGN

Consider the following dynamic system:

(1)

where is a state vector, represents acontrol input vector, is a measurement output vector,

and are input and output noise vectors,

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272 IEEE TRANSACTIONS ON INDUSTRIAL INFORMATICS, VOL. 4, NO. 4, NOVEMBER 2008

respectively; and are known constant matrices, andand are unknown matrices to be determined. Denote

(2)

As a result, an augmented descriptor system can be obtainedfrom (1) and (2) to give

(3)In this study, , , and are all assumed to

bounded. In this context, the following observer can be con-structed:

(4)

where is the state vector of the dynamic systemabove, is the estimate of , ,and are the gain matrices to be designed.

Here we choose

(5)

where is a nonsingular matrix. One thus can calcu-late

(6)

In terms of (2) and (6), it is further derived that

(7)

Using (7), the estimator (4) can be expressed as

(8)

The dynamic equation of the plant (3) can be expressed as

(9)

Letting and subtracting (8) from (9), onehas

(10)

From (10), one can choose a high-gain to reduce the effectfrom on the estimation error dynamics.

According to the work by Gao et al. [16], the high-gain matrixcan be computed as

(11)

where is solved from the following Lyapunov equation:

(12)

with satisfying.

Choose the Lyapunov function as , and it isderived that similar to [16]

(13)

It is also concluded from (12) that will decrease whenincreases in terms of [16]. Therefore from (13), the influencesof , , and can be reduced significantly by increasing .

Further from (13), one can derive the bound of the steadyerror dynamics

(14)

Therefore, by increasing the value of , the bound of can bemade as small as desired.

Remark 1: By letting , the aug-mented system (3) is constructed. Therefore, the observer (4),stemming form the design technique proposed by [16], can beused to estimate the state , the input uncertainty and theoutput noise simultaneously

(15)

(16)

(17)

It is noted that, the observer (4) does exist without any con-straints since . It is worthy to point out that the estimatesof and pave the way for the parameter identificationin the succeeding section.

III. NOVEL PARAMETER IDENTIFICATION TECHNIQUES

A. Observer-Based Parameter Identification

The estimates of and will be used to identify theunknown parameters and . The and its estimate

can be expressed as

(18)

(19)

where ; andare the estimates of and , respectively.

Subtracting (19) from (18), one can derive that

(20)where . Since thesteady errors of and both tend to desired small by

GAO et al.: NOVEL PARAMETER IDENTIFICATION BY USING A HIGH-GAIN OBSERVER 273

selecting sufficiently large , one can conclude thatand as time tends to infinity.

From (19), one has

(21)

where

......

. . ....

...

where represents the th row of thematrix , is the parameter to be identified.

An important formula (21) has been given. However, thereare identified parameters in the equations describedby (21). Therefore, (21) cannot be solved uniquely. Motivatedby the work [12] for continuous-system parameter identificationand with slight modifications, the following equations can beconstructed in terms of (21):

...

(22)

where , , is some constant timeinterval, and .

Remark 2: is used to form a number of independent (22)such that a unique solution can be obtained. Because theadopted observer is a high-gain observer, the estimation errorcan reach and keep within a desired small bound very quickly.When we choose to construct the sampling points at( ) in the steady stable region, the estima-tion error can be kept to be desired small. Moreover, it is notedthat the following facts: 1) and are given dimensions; 2)waiting for a bit longer time before taking the sampling pointsto construct independent equations; 3) choosing a smaller .Then all the sample points at ( ) canbe within the steady stable region. In other words, alland ( ) can well trackand ( ), respectively.

Equations (21) and (22) can be grouped into the one-order linear algebraic system as follows:

......

(23)

The solution to (23) is

(24)

By using the well-known Cramer’s rule (e.g., see [27]), one has

(25)

where is the th component of the vector , and isthe matrix obtained by replacing the entries in the th columnof by the entries in the matrix .

In order to avoid possible jumps at some points due to thenumerical computation, a way is to take the integral to give thefollowing formula:

(26)

where , is the end time of the current integralwindow, and is the selected integral length. Therefore, theresponse curve of can be given as increases along thetime axis. In (26), the least-square idea proposed by [28] is usedactually.

In order to further smooth the response curve, we can take themean calculation over the time interval as follows:

(27)

where is the identified parameter obtained from (26), andis the length to take the mean calculation.The identified parameter is assumed to stay constantly

(not to change abnormally) over the whole simulation interval,the identified parameter can be thus calculated as

(28)

where is the starting time, and is the endtime of the simulation.

Now it is ready to give the following identification algorithm.

Algorithm 1. High-gain observer-based identification (HOI)algorithm.

1) Construct the augmented matrices as in (2).2) Select the observer gains as in (5) and calculate and

in terms of (6).3) Solve the Lyapunov (12) and compute the high-gain

matrix according to (11).4) Implement the real-time estimation using (4), and obtain

the estimates of , and in the forms of (15)–(17).5) Construct (21).6) Select a time interval and construct (22).7) Calculate the identified parameters ,

, according to the integral (26),and plot the parameter curves.

8) From (27), further calculate the refined identifiedparameter , , and plot theresponse curves of the parameters.


9) Finally, a set of constant identified parameters is givenby (28).

B. Adaptive Parameter Identification

In the last subsection, the case is concerned where the iden-tified system parameters do not have obvious changes or thechanges are tolerable under the whole identification interval.However, if some parameter changes in and are ob-vious in different time intervals, the obtained identified param-eters in terms of (28) will be inaccurate. Moreover, in practicalsystems, the system parameters may change abnormally due tothe age or the accidents of the components. Therefore, it is veryimportant to detect the changes and identify the changed param-eters simultaneously for improving identification accuracy andsystem reliability. In this study, the simultaneous change detec-tion and parameter identification is called adaptive parameteridentification.

For a system parameter with a single abnormal changingpoint only, the constant threshold method can work well whenthe system runs at some known steady state. If a series of abruptchanges happen, it is more likely that such an approach canonly detect the first change, but fail to detect the succeedingchanges, due to the constant threshold.

In this study, we assume that the considered system parame-ters may have multiple intolerable abrupt change points. In orderto solve this issue, adaptive threshold is adopted for detectinga sequence of changes. The basic concept is to recalculate thethreshold after detecting each change. The adaptive threshold isexpressed as the tolerable maximal relative varying ratio withrespect to the nominal value of the identified parameter. Actu-ally, the nominal value is defined as the mean value of the iden-tified parameter under some selected steady period. Firstly, weshould calculate the nominal value over a selected initial period,and determine the first threshold. If any a real-time parametercalculated by (27) is beyond the threshold, and keeps the over-flow over a confirmed time interval, a change alarm is given.By calculating the new nominal value after the change, and de-termine a new threshold, the change detection process can keepgoing. The adaptive change detection and parameter identifica-tion technique can be described by the following algorithm.

Algorithm 2. Adaptive parameter identification.

1) Choose such that

where is the selected time length for the nominalparameter identification; , , , and are defined asbefore.

2) Calculate the nominal value of the identified parameter as

(29)

where is given by (26).3) Set the adaptive thresholds for the changes detection as

follows:

ifif

(30)

and

ifif

(31)

where ; is the tolerable maximal relativevarying ratio with resect to the nominal value ; and

are the up and low boundaries, respectively.4) Select as the so-called confirm window for the change.

If the current time satisfies

(32)where the real-time parameter is given by (27), thena change is detected at the time .

5) If any change is detected, calculate the following meanvalue:

(33)

Set , and go back to step 2.6) Otherwise, continue step 3 until the simulation ends. Then

go to next step.7) Calculate the following mean value:

(34)

where is the end time of the simulation.

IV. PARAMETER IDENTIFICATION FOR A GAS TURBINE SYSTEM

A gas turbine model is characterized by

(35)

where the input is the mass flow rate to the combustionchamber, the components of the output are the low-pres-sure and high-pressure shaft speeds, respectively. The coeffi-cient matrices

(36)

are known. In the meanwhile, the matrices and , due tothe modeling accuracy, the aging of physical apparatus and thenoise effect, are the parameters to be estimated. In this simula-tion, we assume

(37)

A. Parameter Identification: Standard n4sid Algorithm

In order to make comparison, we firstly identify the parame-ters using the standard n4sid toolbox.

1) Case Study: Noises Free: For the gas turbine system (35),the responses of the input and output data are shown in Figs. 1


Fig. 1. Input data of the gas turbine engine.

Fig. 2. Output data of the gas turbine engine: noise free.

TABLE IIDENTIFIED PARAMETERS VIA STANDARD N4SID TECHNIQUE

and 2, respectively. The input data are gathered from the en-gine test-bed at the normal engine operating point [26], and theoutput data are the model responses with noise free (that is,

and ). Using the standard system identificationtoolbox n4sid with the input and output data, one can obtain theperturbed parameters and as shown in third row (seeIdentification 1) of Table I. Comparing the identified parame-ters in the third row with the true parameters in the second row,one can see that the standard subspace identification approachis obviously effective for systems without noises.

2) Case Study: Subject to Noises: Let

(38)

Fig. 3. Output data of the gas turbine engine: subject to noises.

which is corrupted by a random signal with the variance 0.0001;and

where corrupted by a random signalwith the variance 0.00005.

The output response of the system subjected to the abovenoises is exhibited by Fig. 3. Using the standard system identi-fication toolbox n4sid, one can obtain the identified parametersas shown in the fourth row (see Identification 2) of Table I. Onecan see that the identified parameters are not acceptable com-pared with the real parameters as shown in the second row. Itis not strange, because the standard subspace identification ap-proach is only valid for systems under white noise environment.

B. Novel Parameter Identification Approach

Now we use the high-gain observer-based identification(HOI) approach.

1) Design of the High-Gain Observer: The augmented plantin the form of (3) is constructed with the equation at the bottomof the next page.

Choose

Select , and the high gain is computed as

From Figs. 4 and 5, one can see that the responses of , ,and their estimated curves almost, respectively, coincide, whichmeans that the tracking properties are desirable via the high-gainobserver.


Fig. 4. States and their estimates: subject to noises.

Fig. 5. Input uncertainties and their estimates: subject to noises.

2) High-Gain Observer-Based Identification (HOI): Byusing the proposed algorithm 1, the responses of the identified

parameters for systems subjected to noises are exhibited inFig. 6. For comparison, the parameter responses for systemswithout noises are given in Fig. 7. One can see that, in bothcases, the dynamic responses of the identified parameterscan track the true perturbed parameter values. It is noticedthat the curves of Figs. 6(b) and 7(b) are more smooth thanFigs. 6(a) and 7(a), respectively. It owes to the mean calcula-tion obviously. Moreover, the constant values of the identifiedparameters are given as shown in Table II. Compared withthe true values (see the second row of Table II), the identifiedparameters for systems without noises (see HOI 1, the thirdrow of Table II) and subjected to noises (see HOI 2, the fourthrow of Table II) are both desirable.

3) Comparison of Identification Accuracies for Different :In the above high-gain observer-based design technique, wehave chosen so that the identified parameters are de-sirable. Here for comparison, we keep as the same as before,but choose a smaller , that is, . In this case, the observergain can be computed as

The responses of the states , the input uncertainties , and theirestimates are exhibited by Figs. 8 and 9, respectively. The re-sulting identified parameters are given in the fifth row (see HOI3) of Table II. One can see that the identified accuracies of theparameters by taking are much lower than the case when

. It has been indicated by the simulated curves thatthe estimated errors of in Fig. 9 ( ) are clearly worsethan those in Fig. 5 ( ), although the estimated errorsof in Figs. 4 and 8 do not make obvious differences. As a re-sult, the identified accuracies can be improved significantly byincreasing , which completely agrees with the proposed theo-retic results [see (14) and (20)].


Fig. 6. Responses of the identified parameters: subject to noises.

Fig. 7. Responses of the identified parameters: noise free.

Fig. 8. States and their estimates: � � �. Fig. 9. Input uncertainties and their estimates: � � �.


Fig. 10. Responses of the identified parameters with changing intervals: � � ��.

TABLE IIIDENTIFIED PARAMETERS VIA THE PROPOSED TECHNIQUE

TABLE IIIIDENTIFIED PARAMETER CHANGES: SUBJECTED TO NOISES

4) Adaptive Parameter Identification: In this case, we as-sume the perturbed parameters with obvious changes at differenttime intervals as follows:

(39)

where and are defined in (37).Choose and the same as before. From (39), there

are three abnormal changing points in and , which occurat 25 s, 45 s, and 50 s, respectively. Using the proposed algo-rithm 2, the three changing points are detected respectively at27.247 s, 49.97 s, and 54.708 s for systems subjected to noises.The adaptive identified parameters are shown in Table III. Onecan see that the parameters under four time intervals are iden-tified adaptively and satisfactorily. For systems without noises,the three changing points are detected at 27.27 s, 50.024 s, and

TABLE IVIDENTIFIED PARAMETER CHANGES: NOISE FREE

54.709 s, respectively. The desired identified parameters aregiven by Table IV. Moreover, the dynamic responses of theidentified parameters are given by Fig. 10. In consequence, theperformance of the adaptive changes detection and parametersidentification is desirable.

V. CONCLUSIONS

In this paper, on the basis of the high-gain observer technique,the integral mean method and the adaptive threshold approach,two novel parameters identification algorithms have been pro-posed. The presented algorithms are effective for systems underbounded process and measurement noises environment, whichis more realistic in practical cases. The proposed procedureshave been applied to the simulated study of the gas turbine en-gine. The simulated curves and the identified parameters areboth desirable. It is worthy to point out that the proposed iden-tification algorithms are valid for any initial matrices and ,even when and are both null matrices.

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Zhiwei Gao (SM’08) received the B.E. degree in in-dustrial automation and the M.E. and Ph.D. degreesin systems engineering, all from Tianjin University,Tianjin, China, in 1987, 1993, and 1996, respectively.

In 2001, he was awarded the professorship at theSchool of Electric Engineering and Automation,Tianjin University. In 2004, he was granted theAlexander von Humboldt Research Fellowship,Germany. His current research interests includemodeling, identification, fault diagnosis, fault-tol-erant design, estimation, filtering, and systems

biology.Dr. Gao was the guest editor for the International Journal on Automatic Con-

trol and System Engineering and is presently the Editor-in-Chief for the Interna-tional Journal of Systems Signal Control and Engineering Applications. He hasbeen an associate editor for the IEEE TRANSACTIONS ON CONTROL SYSTEMS

TECHNOLOGY.

Xuewu Dai received the B.E. degree in electronicengineering and the M.S. degree in computer engi-neering from the Southwest University, Chongqing,China. He is currently pursuing the Ph.D. degreeat the Control Systems Center, University of Man-chester, Manchester, U.K.

His research interests include system identifica-tion, condition monitoring, fuzzy neural networks,and distributed control systems.

Tim Breikin received the electrical engineering andPh.D. degrees from Ufa State Aviation TechnicalUniversity, Ufa, Russia.

He is presently a Lecturer at the School of Elec-trical and Electronic Engineering, University of Man-chester, Manchester, U.K. His research interests areindustrial application of control theory and systemidentification.

Hong Wang (SM’05) received the B.Sc. degreefrom the Huainan University of Mining Engineering,China, and the M.E. and Ph.D. degrees from theHuazhong University of Science and Technology,China, in 1982, 1984, and 1987, respectively.

He is currently a Professor of process control atthe School of Electrical and Electronic Engineering,University of Manchester, Manchester, U.K. Hisresearch interests include stochastic distributioncontrol, fault diagnosis, nonlinear control, anddata-based modeling.

Prof. Wang was an Associate Editor for the IEEE TRANSACTIONS ON

AUTOMATIC CONTROL and is an editorial board member for the Transactions ofthe Institute of Measurement and Control and for Optimal Control Applicationsand Methods.

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