Yi-Qiang Sun·Meng-Shi Jin·Han-Wen Song·Jian Xu

Time-delay identification for vibration systems with multiple feedback

Yi-Qiang Sun1·Meng-Shi Jin1·Han-Wen Song1·Jian Xu1

©The Chinese Society of Theoretical and Applied Mechanics;Institute of Mechanics,Chinese Academy of Sciences and Springer-Verlag Berlin Heidelberg 2016

An approach for time-delay identification is proposed in multiple-degree-of-freedom(MDOF)linear systems with multiple feedback.The applicability of the approach is discussed in detail.Based on the characteristics of frequency domain in feedback controlled system with multiple time-delays,this paper proposes a time-delay identification approach,which is based on the pseudo impedance function of reference point.Treating feedback time-delays as the“frequencies”of the oscillation curve,the time-delays can be obtained from the“frequencies”of the curve.Numerical simulation is conducted to validate the proposed approach. The application scope of the approach is discussed with regard to different forms of feedback.

Multiple time-delays·Parameter identification· Feedback·Frequency response functions

DOI 10.1007/s10409-016-0575-1

1 Introduction

Time-delay generally exists in active control systems,influencing dynamic characteristics of the systems.The influence may lead to vibration suppression,while it can also cause instability.To apply the results of previous papers,the identification of time-delay is essential,since time-delay is a certain value for a specific engineering situation.

Research shows that the increase of time-delay from zero causes a stability switch of the system,which can make a stable system unstable or the opposite[1].Xu et al.[2,3]certified the bifurcation and chaos phenomenon deduced by time-delay,the results showed that the delay can be used as a“switch”to control the motions of a system Sipahi et al.[4]determined the stability robustness map of a general class of higher order linear time-invariant systems with multi-independent delays.Insperger et al.[5–7]proposed the closed-form stability chart determined for the delayed Mathieu equation with and without damping.Wang and Hu[8] derived two criteria of asymptotic stability by means of the frozen time approach and the Kronecker product.

In the application of time-delay,Olgac and Holm-Hansen [9]presented a new concept for active vibration absorption for dynamic structures,called a delayed resonator,which uses time-delay to suppress the vibration of the system. Later,Olgac et al.[10,11]made further effort to study the suppression effects of delayed resonators with different control strategies,which were verified by experiments. Research showed the effectiveness of time-delay in the field of vibration suppression,making the engineering application of time-delay more and more practical.In addition,Liu and Hu[12,13]used time-delay to change a double pendulum into different equilibriums.Satisfactory results were achieved.Sun and Xu[14,15]constructed a 2-degrees-of-freedom(2-DOF)dynamic system,using a time-delayed absorber for vibration suppression.The results indicate that the time-delayed absorber greatly suppressed the resonant response.Wang and Hu[16]presented a modified averaging scheme,which was applied to locate successfully the delay value for a delayed Van der Pol oscillator.In Cai’s work [17],simulation results demonstrated that a control system with time-delayed might become unstable if the control law designed in the case of no time-delayed was used.In their work,Jiang et al.[18]developed a vibration control system for milling,which used a 2-DOF active work piece holdingstage.A state feedback controller was designed,inevitably including the effect of time-delayed.Liu and Hu[19]discussed the instability of a controlled double pendulum and used a discrete delayed linear quadratic controller to suppress the instability which was induced by the group delay of digital filters.

The control design for time-delay systems assumes a precise model(for time-delay parameters in particular).In this sense,parameter identification for time-delay systems is an important and challenging topic[20].Therefore,time-delay identification has long been an active research field in signal processing and system identification.

Most delay identification methods for linear systems are based on analytical principles.In their work,Orlov et al.[21] developed an identifiability analysis for linear time-delay systems with delayed states,control inputs,and measured outputs,all with a finite number of lumped delays.Continuing that work,Orlov et al.[22,23]then developed an adaptive parameter identifier for linear dynamic systems with a finite number of lumped time-delay in the state vector and control input.A study by Drakunov et al.[24]identified the timedelay based on variable structure(SMVS)observer.

With regard to the identification for time-delay,little theoretical work other than Hu[25]was published,he identified displacement and velocity time-delays of a single-degree-of-freedom(SDOF)system in paper.The condition was first derived for the case when two frequency response functions with different time-delays were equal at certain frequencies. The work also pointed out that if the system was subject to a harmonic input with frequency ω,the identified time-delay might be deviated from the real values by 2 pπ/ω,where p is a positive integer.Afterwards,the identifiability of short time-delay was analyzed and some identifiability conditions were provided.

The work by Gawthrop and Nihtila[26]derived an algorithm for the exact least-squares identification of an approximate continuous-time time-delay system.Then Elnaggar et al.[27]presented an algorithm for the recursive estimation of both process parameters and delay.The work by Sipahi et al.[4]used an efficient and exhaustive frequency-sweeping technique to determine the time-delays.And the work by Drakunov et al.[24]proposed several algorithms for delay identification based on variable structure observers.

Currently,using optimization algorithms to estimate the time-delay is a major time-delay identification method.However,in this paper,modal analysis method is applied to identify the time-delay.To be more specific,after time-delay feedback is applied to the system containing no time-delay, the original system will be changed into another system with time-delay.Whether time-delay exists or not,both systems can be investigated by adopting modal analysis.The difference between the two systems is the time-delay.

Based on Hu’s work[25],we developed an approach to identify single time-delay[28].Paper[25]mentioned the problem that only time-delays with small amount could be identified.However,in this paper,the approach in Ref.[28] is improved to identify multiple feedback time-delays,effectively solving the problem mentioned in Ref.[25].Furthermore,this paper also discusses the applicability of the approach.

Based on frequency response functions(FRFs)of a dynamic system,this paper gradually derives FRFs of a system with multiple feedback time-delays.According to the characteristics of transfer functions,an approach for time delay identification is proposed whose application fields are also given.The paper is organized as follows.Section 2,starting from open-loop system,describes characteristics of FRFs in an open-loop system and controlled system with multiple feedback time-delays.In Sect.3,the pseudo impedance function is defined,based on which an identification approach is proposed.Numerical examples are presented in Sect.4 to show the effectiveness of the proposed approach.Feedback systems are classified.The applicability is discussed in Sect.5.Conclusions are provided in Sect.6.

2 Characteristics of FRFs of a time-delayed system

FRFs of linear systems can be defined as the ratio between the Fourier transform of output variable and input variable whose initial values are assumed as zero.

Dynamic equations in open-loop multiple-degree-of-freedom(MDOF)linear systems can be expressed as

where M,C,K,and F(t)represents mass matrix,damping matrix,stiffness matrix,and input vector,respectively.and X(t)represents acceleration,velocity,and displacement response,respectively.Applying the Fourier transform to Eq.(1)under zero initial condition,the following equation can be obtained.

According to the definition of FRFs in linear systems, impedance matrix of the system can be obtained as

For a MDOF discrete dynamical system,the impedance matrix of an open-loop system can be expressed as follows,

The FRF matrix can be written as

In Eq.(5),det(Z(jω))is the determinant of the impedance matrix.Under the condition where physical parameters are constants,det(Z(jω))is a polynomial function containing the term of jω.adj(Z(jω))is the adjoint of impedance matrix.Apq(jω)is algebraic complement of impedance matrix Z(jω),is a polynomial function containing jω.

Taking displacement feedback control as an example and applying feedback time-delays to the system,Eq.(1)can be rewritten as

Applying the Fourier transform to Eq.(6)and merging like terms,the dynamic equations of feedback system with time-delays are in the frequency domain.

From Eq.(7),impedance matrix of the time-delay system can be expressed as follows.

Equation(8)is the impedance matrix of a system with feedback time-delays.According to matrix theory,the FRFs matrix of systems with time-delay feedback can be expressed as

Applying the response signal of point r after different time-delays to all the measured points,feedback gain matrix G(jω,τ)in Eq.(8)will be like the following.

Impedance matrix of systems with time-delay feedback will be expressed as

The elements of column r in the impedance matrix(11)are different from those in impedance matrix(4).Since time-delays only exist in column r,the algebraic complements of columns in systems with time-delays are the same as those in open-loop systems,while algebraic complements of other elements are all changed by time-delays.

According to the position of the algebraic complement in the companion matrix,the numerator of row r in the FRFs does not include the information on time-delays.Take Hrn(jω,τ) as an example,

Other numerators in the FRFs contain part of the information on the time-delays.Take Hnn(jω,τ)as an example,

On the right side of Eq.(14),the numerator contains all the information on time-delays except τn.

According to the characteristics of FRFs described by Eq. (13),an identification method can be constructed by directly making use of FRFs to identify time-delays.

3 Time-delay identification approach

Based on the characteristics of the FRF(13),the pseudo impedance function can be defined as the inverse of the FRF between two arbitrary points.

Use“～”to show the difference between the pseudo impedance function and impedance function.Use the right superscript“D”to show the difference between the open-loop system and the time-delay controlled system.

Based on the definition of Eq.(15),the denominator, which does not contain time-delays,can be obtained by taking the inverse of an arbitrary FRF in row r,while the numerator contains all the time-delays.Take Hrn(jω,τ)as an example,

Fig.1 5-DOF model

It is known from Eq.(17)that after separating the real part from the imaginary part in the difference curve between pseudo impedance functions before and after feedback time-delays,two curves can be obtained that contain the information of all the time-delays.

For the oscillation signal in the time domain,frequency is defined as the oscillations per unit time.Hertz is the unit. However,in this paper,oscillation data in the frequency domain is used to identify time-delays.According to the principles of Fourier transform,after transforming signals in the time domain,data in the frequency domain can be obtained.This paper sets that data in the time domain are obtained by transforming oscillation data in the frequency domain.Specifically speaking,Eq.(17)shows that the difference between the impedance function and pseudo impedance function is a trigonometric function whose independence variable isω,while the time-delayτpacts as the“frequency”of the trigonometric function.The quotation marks are used because the time-delay τpplays the role of the“frequency”in the frequency domain,but actually it has the dimension of time.

The difference function is in complex field,which contains real part and imaginary part,both of which are trigonometric functions with τpbeing the“frequency”.Either real part or imaginary part can be separated from the difference function.

Fig.2 FRF curves of the open-loop system

It is commonly known that the Fourier transform is used to extract the frequencies of data in the time domain.However, datas of the difference curve are in the frequency domain. This means that if the Fourier transform is applied to the data of either the real part or the imaginary part of the difference curve,the“frequency”in the frequency domain will be obtained.As mentioned above,the“frequency”in the frequency domain is the time-delay τp.

The proposed time-delay identification approach,based on full understanding of characteristics of the frequency domain in systems with time-delay feedback,directly uses FRFs to identify time-delays.Compared with current optimization algorithms,the proposed approach is very robust.

4 Numerical simulation

This section will take a 5-DOF system as an example to conduct a numerical simulation,using the proposed identification approach.The mechanical model in Fig.1 is as follows,with Eq.(1)being the dynamic equation.

Take point M3as the reference point and apply its displacement signal after different time-delays to the three degrees of freedom.Assume that τ1=0.075 s,τ2=0.12 s,τ3=τ4= 0 s,τ5=0.05 s.FRF curves of the reference point can be obtained,as is shown in Fig.3.

Fig.3 FRF curves of the reference point before and after time-delay feedback

The blue solid line is the FRF curve of point M3before time-delay feedback control,while the black dotted line is the one after control.Obviously,time-delays generate more extreme points in FRF curve of the reference point.As is shown in Fig.4,the difference curves of the real part and imaginary part of the pseudo impedance function can be separately obtained by the inverse of the two curves in Fig.3.

In order to see the periodicity of difference curves clearly, the curves in Fig.3 will be divided into two parts.Figure 4a is the difference curve in interval 0–80 Hz.Figure 4b is in interval 80–160 Hz.Since the difference in neighborhood of two anti-resonances in the FRF curve of H35is so great, complete wave shapes of the interval is not shown in the figures,considering overall visibility of the graphs.As is mentioned in Sect.3,the difference curve contains a real part and an imaginary part,both of which are trigonometric functions,where ω is the independent variable and τpis the“frequency”.Take the real part as an example.The real part, which contains the“frequency”τp,can be separated from the difference curve.Applying the Fourier transform to the data of the real part,the“frequency”of the real part can be obtained.This means that the time-delayτpcan be identified. Since the imaginary part also contains the“frequency”τp,the same procedure can be done to identify the time-delayτp.The identified time-delays are shown in Fig.5,using the real part and imaginary part,respectively.The result is shown in Fig.5.

The two curves shown above are the fast Fourier transform (FFT)results obtained from the real part and imaginary part of the difference curve,respectively.The abscissa values of the three peaks are 0.0475,0.0735,and 0.12 s.The maximum absolute error is 0.0015 s.The maximum relative error is 3%.Since there are no surplus time-delays,the result is dependable.

5 Discussion on application scope of the approach

The form of feedback described in matrix(11)is only one type of feedback in MDOF systems.It is obvious that with this type of feedback,the feedback time-delays are identified by the approach proposed in this paper as long as an arbitrary FRF of row r is obtained.In other words,if the system satisfies two conditions,time-delays can be identified by the proposed approach.One condition is that there are some FRFs whose numerators do not contain time-delays. The other is that the determinant of the impedance matrix only includes linear superposition of exponential term with time-delays.In the following,a classified discussion will be made with regard to specific forms of feedback.

For feedback control systems,a reference point can be defined as the measuring point which picks up the feedback control signal;define the point upon which feedback signal is applied as the feedback point.According to the number of feedback points and reference points,systems with multiple time-delays can be classified into four categories:multiple reference points and multiple feedback points(MRMF),multiple reference points and a single feedback point(MRSF),a single reference point and multiple feedback points(SRMF), and a single reference point and a single feedback point (SRSF).In the following,discussion on applicability of the proposed approach will be made for different conditions.

5.1 MRMF

Fig.4 a Curves of the pseudo impedance function of the reference point before and after control(0–80 Hz).b Curves of the pseudo impedance function of reference points before and after control(80–160 Hz);

Consider a general working condition with feedback.Assume every measuring point is a reference point.Applying a signal after n different time-delays to n measuring points of the system,the impedance matrix will become Every element in impedance matrix(18)contains time-delays so that we know that the determinant of the matrix contains n2time-delays.Every algebraic complement contains(n−1)2terms with time-delays.Under this working condition,numerators not containing terms for time-delays will not appear in the FRFs.Also,numerators and denominators in the FRFs will appear not only in forms of linear superposition with exponential terms containing time-delays, but also in forms of product of exponential terms containing time-delays,which leads to superposition of time-delays.In this type of feedback system,the identification approach proposed in this paper is invalid.

Considering a simplified system of the above system, assume time-delays only exist in row i and column n in the impedance matrix.Under this feedback condition,the impedance matrix is expressed as

Fig.5 FFT result of the difference curve

According to the form of impedance matrix,among all the FRFs,only the numerator ofdoes not contain time-delays.The expression is as follows

We know from Eq.(20),under this working condition,the form of product of exponential terms with time-delays will appear in the determinant of feedback time-delay matrix,thus leading to the appearance of extraneous time-delays in identification results.The extraneous time-delays are induced by the addition of multiple feedback time-delays.This means although FRFs not containing time-delays do exist,the time-delays cannot be accurately identified.

Simplify the system further and assume there are only two reference points and two feedback points.Assume that point i and point n are the original feedback,which means the signal is applied to the point itself after different time-delays.The impedance matrix is as follows

The determinant of the impedance matrix is

where

5.2 MRSF

Simplifying the feedback points into one point,the system can be called a time-delay system with multiple reference points and single feedback point.As for this type of system, assume point 1 is the feedback point.All the measuring points are assumed to be reference points.Applying the signal of every reference point after time-delays to point1,gain matrix G will only have one column of data.

The corresponding impedance matrix with time-delays can be expressed as follows

The determinant of the impedance matrix above can be written as

Seen from matrix(25),algebraic complement of the elements in the first row do not contain time-delays,which means the elements in the first row of the FRF matrix has the characteristic of not including time-delays in numerators.The determinant of the impedance matrix only contains linear superposition of exponential terms with time-delays,which satisfies the two identification conditions proposed above. When point 1 is the input point,which means the input point coincides with the feedback point,the pseudo impedance function not containing time-delays in the denominators is constructed.Take point n as an example.

In short,as for time-delay systems with MRSF,there is a column of FRFs from which the time-delays can be accurately identified using the proposed approach.

5.3 SRMF

Simplify the reference points described in Eq.(18)into one single reference point.The examples described in Sects.2 and 3 belong to this type of working condition.We know from above that there is a row of FRFs from which time-delays can be identified by using the proposed approach.

Considering experimental conditions,a column of FRFs will be obtained if an excitation point is assumed.In controlled systems with multiple reference points and a single feedback point,time-delays can be accurately identified only when the excitation point coincides with the feedback point, while for controlled systems with a single reference point and multiple feedback points,the FRFs used to identify the time-delays are those of the reference point.As long as the excitation point exists,there is no constraint on its position.

5.4 SRSF

Reduce the number of reference points and feedback points to only one so that the system will be with SRSF.Assume measuring point r is the reference point.Signal is applied to measuring point1 after different time-delays.The impedance matrix of the controlled system will be

Now the determinant of the impedance matrix will be expressed as

We see from matrix(29),the algebraic complements of elements in row 1 and column r will not contain time-delays, whose corresponding FRFs satisfy the two conditions proposed above.Takeandas examples.

It can be seen from Eq.(33),for this type of feedback,with feedback gain being a constant and FRFobtained, the difference curves of real and imaginary parts will appear as the superposition of a set of simple harmonic waves with the same amplitude.

Conduct FFT to the above formula and the feedback time-delays and gains can be identified.

If FRFs are to be used to identify time-delays,two conditions should be satisfied.One is to obtain FRFs whose numerators do not include time-delays.The other is to guarantee there is no product of exponential terms containing time-delays in the determinant of the impedance matrix.The pseudo impedance matrix can be constructed if condition one is satisfied;it can be guaranteed that there will not be fake time-delays in the identified time-delays if condition two is satisfied.The analytical expression of the time-delays can be obtained if both conditions are satisfied.

6 Conclusions and discussion

This paper studied the identification of parameters in systems with multiple feedback time-delays.The characteristics of FRFs for a time-delay system are discussed firstly.Starting from the definition of FRFs,this paper describes the impedance matrix and FRF matrix of an open-loop MDOF linear system.Imposing time-delays to the dynamic system,the impedance matrix and FRF matrix of the controlled MDOF linear system are obtained.The impact of time-delays for the two matrices are clarified.The dynamic characteristics of the time-delay controlled system are presented.

Secondly,the pseudo impedance functions of both controlled and uncontrolled systems are defined.As time-delay does not change the denominator of pseudo impedance functions on reference points,identification becomes possible. The difference between pseudo impedance functions of controlled and uncontrolled systems on reference points is the response derived by time-delays.The difference is a curve in frequency domain.Actually,the“frequencies”of this curve are the values of time-delays.A simulation of a 5-DOF system is provided to verify the above conclusions. The comparison between the results shows the faithfulness of the approach.

At last,this paper discussed the application scope of the approach.According to the number of feedback points and reference points,time-delay controlled systems are classified into four categories.In view of the four different systems,the applicability of the approach are discussed.Two identifiability conditions are proposed.One is to obtain FRFs whose numerators do not include time-delays.The other is to guarantee there is no product of exponential terms containing time-delays in the determinant of impedance matrix,providing a new basis for identifiability of time-delay systems.

Acknowledgments This work was supported by the National Natural Science Foundation of China(Grant 11272235).

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✉ Han-Wen Song hwsong@tongji.edu.cn

1School of Aerospace Engineering and Applied Mechanics, Tongji University,Shanghai 200092,China

4 December 2015/Revised:28 February 2016/Accepted:20 April 2016/Published online:20 September 2016