Journal of Vibration and Control

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Lowpass filter-based continuous-time approximation of delayed dynamical systems Bo Song and Jian-Qiao Sun Journal of Vibration and Control published online 8 November 2010 DOI: 10.1177/1077546310378432 The online version of this article can be found at: http://jvc.sagepub.com/content/early/2010/10/30/1077546310378432

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Article

Lowpass filter-based continuous-time approximation of delayed dynamical systems

Journal of Vibration and Control 0(0) 1–11 ! The Author(s) 2010 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/1077546310378432 jvc.sagepub.com

Bo Song and Jian-Qiao Sun

Abstract This paper extends our earlier work on continuous-time approximation of time-delayed dynamical systems by introducing a lowpass filter-based approach. The proposed method substantially improves the accuracy of predictions in frequency as well as time domain. It is applicable to linear and nonlinear dynamical systems, and can be readily incorporated with real-time controls. In the paper, we first review the mathematics literature on numerical methods for delayed differential equations including the equivalent abstract Cauchy problem. We show that the mathematics work provides a solid foundation for several well-studied numerical methods for time-delayed dynamical systems in the engineering literature. Examples are presented to show the accuracy of the pole prediction for linear systems, and temporal responses for linear and nonlinear systems. Furthermore, we discuss the bandwidth issue of the method, and demonstrate that many extraneous poles introduced by the discrete approximation of the time-delayed system that do not match any exact poles of the system are still very important and contribute to the accuracy of temporal responses.

Keywords Time-delayed systems, continuous time approximation, abstract Cauchy problem Date received: 22 March 2010; accepted: 19 June 2010

1. Introduction Time delay is a common phenomenon in engineering, economical and biological systems. It is caused by signal transportation and communication lags, feedback delays and retarded hardware responses. Other than a few exceptional cases, time delay is undesirable. Control strategies to eliminate or minimize unwanted effects are often employed. In this paper, we present a new lowpass filter-based continuous-time approximation method for the analysis and control design of linear and nonlinear dynamical systems with time delay. Methods of solution for time-delayed systems have been a subject of many studies. A method that fully discretizes the delayed control system in time domain has been extensively studied. Pinto and Goncalves (2002) have fully discretized a nonlinear single-degreeof-freedom (SDOF) system to study control problems with time delay. Klein and Ramirez (2001) have studied

multiple-degree-of-freedom (MDOF) delayed optimal regulators with a hybrid discretization technique where the state equation is partitioned into discrete and continuous portions. Cai and Huang (2002) have studied optimal vibration controller with a delayed feedback where standard discretization techniques are used. Time-delayed systems have been studied using discretization techniques with an extended state vector. The Smith predictor is a well-known method Smith (1957) that proposes a compensator to stabilize the feedback control designed for the system without time delay. The readers are also referred to two excellent reviews of recent advances in stability and control

School of Engineering, University of California, Merced, CA, USA. Corresponding author: Jian-Qiao Sun, School of Engineering, University of California, Merced, CA, USA Email: [email protected]


2

Journal of Vibration and Control 0(0)

studies of time-delayed dynamical systems (Gu and Niculescu, 2003; Richard, 2003). A method using Chebyshev polynomials to approximate general nonlinear functions of time has been developed to handle linear and nonlinear time-delayed dynamical systems with periodic coefficients (Ma et al., 2003, 2005; Deshmukh et al., 2006, 2008). The method has also been applied to study optimal control problems. A temporal finite-element method has been proposed by Garg et al. (2007) to study the stability of time-delayed systems with parametric excitations. The work reported by Kalmar-Nagy (2005) makes use of the piecewise exact solution of linear differential equations with a single time delay to create a map in order to study the stability of the system. The semi-discretization (SD) is a well-established method in the literature and used widely in structural and fluid mechanics governed by partial differential equations (PDEs) (Pfeiffer and Marquardt, 1996; Leugering, 2000). The method has been applied to delayed deterministic dynamical systems by Insperger and Stepan (2001, 2002). The method has been extended to control systems with delayed feedback (Sheng et al., 2004; Sheng and Sun, 2005). The effect of various higher-order approximations in SD on the computational efficiency and accuracy has been examined by Elbeyli and Sun (2004). The continuous-time approximation (CTA) method is an extension of the method of SD and provides an alternative to handle systems with multiple independent time delays (Sun, 2009). The CTA method has been applied to study control problems of the time-delayed linear dynamical systems, and stochastic dynamical systems with time delay. It turns out that the CTA and SD methods can be derived, in a uniform manner, from the abstract Cauchy problem of delayed differential equations (DDEs). Numerical methods based on the abstract Cauchy problem for computing the right-most characteristic roots of DDEs are presented by Engelborghs and Roose (2002) and Breda et al. (2004, 2005). The convergence and stability of the method with the Chebyshev polynomial expansion of the delayed response are discussed. The abstract Cauchy problem can be stated in terms of a PDE, which is open to various numerical methods for solutions. A finite-difference method to solve the differential-difference equation of the time-delayed system and the stability of the method are presented by Bellen and Maset (2000), and a method of lines for solving the PDE of the timedelayed system is investigated by Maset (2003) and Koto (2004). These methods are the same as the CTA method (Sun, 2009). The higher-order Runge–Kutta methods and their convergence are studied by Maset (2003). The implicit–explicit (IMEX) linear multistep

Runge–Kutta method for DDEs is studied by Koto (2009). Bellen and Zennaro (2003) present a comprehensive discussion of numerical methods for DDEs up to 2003. Sun (2009) proposed the CTA method based on the physical intuition and guided by the goal to create additional states to approximate the infinitedimensional system. The abstract Cauchy problem of DDEs in the mathematical literature provides a rigorous foundation for the CTA method. In this paper, we present a new numerical method for the analysis and control design of time-delayed dynamical systems. The method is an extension of the CTA approach, and is motivated by real-time applications. We assume that the delayed response is measured, and contains noise. A lowpass filter is applied to the response, followed by differentiation. This is a standard approach to take derivatives of the measured signals in real-time applications. The method, which we call lowpass filter-based CTA (LPCTA), leads to an IMEX numerical integration algorithm for DDEs. In this paper we investigate the effectiveness of the LPCTA method in frequency and time domains. The rest of the paper is organized as follows. In Section 2, we review the abstract Cauchy problem, and point out that many numerical methods can be derived in this framework. In Section 3, we present the LPCTA method. Section 4 shows numerical examples of the LPCTA method applied to linear and nonlinear time-delayed systems. We also discuss the effect of the poles obtained by the LPCTA method on the temporal response of the time-delayed system. Section 5 concludes the paper.

2. Abstract Cauchy problem for DDEs The discussion in this section follows closely the method of Bellen and Zennaro (2003). Consider an n-dimensional system with k discrete time delays _ ¼ A0 xðtÞ þ xðtÞ

k X

Al xðt l Þ fðxðtÞÞ,

l¼1

xðÞ ¼ uðÞ,

2 ½, 0,

ð1Þ

where x(t) 2 Rn, A0, P A1, . . ., Ak 2 Rnn, 0 ¼ 0 < 1 < k ¼ and fðxðtÞÞ ¼ kl¼0 Al xðt l Þ. The solution operator T(t) (t 0) of the system (1) is defined by TðtÞuðÞ ¼ xðt þ Þ,

uðÞ 2 X,

ð2Þ

where the Banach space X ¼ C([, 0], Rn) is endowed with the maximum norm jjuðÞjj ¼ max juðÞj, 2½, 0


u 2 X,

ð3Þ

Song and Sun

3

and x(t þ ) denotes the solution of Equation (1) with the initial condition u() 2 X. The family {T(t)}t>0 is a C0-semigroup with an infinitesimal generator A : D(A) X ! X given by A’ ¼

d’ðÞ , d

u 2 DðAÞ,

where the domain D(A) is defined as d’ðÞ DðAÞ ¼ u 2 X : 2 X and d ) k X ¼ Al uðl Þ :

ð4Þ

du ð0Þ d ð5Þ

l¼0

The equation k X du ð0Þ ¼ Al uðl Þ, d l¼0

ð6Þ

is known as the splicing condition. When u() satisfies the splicing condition, the response x(t) of the DDE is C1-continuous. When the splicing condition is not satisfied, x(t) is C0-continuous (Bellen and Zennaro, 2003). The discontinuity in u() propagates to the solution x(t) making it nonsmooth. The system (1) can be restated as an abstract Cauchy problem in terms of the infinitesimal generator dxðt þ Þ ¼ AðÞxðt þ Þ, t 4 0, dt xðÞ ¼ uðÞ, 2 ½, 0:

ð7Þ

It should be pointed out that, in general, the infinitesimal operator A is a function of time delay index . Hence, this simple looking linear system implicitly lives in an infinite dimensional state space. Introduce a function vðt, Þ ¼ xðt þ Þ,

t 0, 0:

ð8Þ

Equations (4) and (7) lead to a hyperbolic PDE for v(t, ) @v @v ðt, Þ ¼ ðt, Þ, @t @

ð9Þ

with a boundary condition @v ðt, 0Þ ¼ fðvðt, 0ÞÞ, @

t 0,

ð10Þ

and an initial condition vð0, Þ ¼ uðhÞ,

2 ½, 0:

ð11Þ

The abstract Cauchy problem (7) and the PDE (9) do not contain time delay explicitly and are amenable to various numerical methods of integration (Bellen and Maset, 2000; Bellen and Zennaro, 2003; Maset, 2003; Koto, 2009). The methods of SD and CTA can also be derived from the abstract Cauchy problem (7) and the PDE (9) (Elbeyli and Sun, 2004; Sun, 2009). Next, we construct a discrete approximation of A. Consider a mesh N ¼ { i, i ¼ 0, 1, . . ., N} of N þ 1 points in [0, ] such that 0 ¼ 0 < 1 < < N ¼ . The continuous space X is replaced by the space XN of the discrete functions defined on the mesh N. That is, x(t þ ) is discretized into a block-vector yðtÞ ¼ ½xðtÞ, xðt 1 Þ, . . . , xðt N ÞT T y0 ðtÞ, y1 ðtÞ, y2 ðtÞ, . . . , yN ðtÞ :

ð12Þ

Let (LNy)() be the unique Rn(Nþ1)-valued interpolating polynomial of degree N with (LNy)( N,i) ¼ yi(t). In particular, (LNy)( N,0) ¼ y0(t) ¼ x(t). The infinitesimal generator A is approximated by a spectral differentiation matrix AN determined by the following equations _ ¼ fðLN yðN,0 ÞÞ ¼ ðAN yðtÞÞ0 , y_ 0 ¼ xðtÞ d ðLN yÞ ðN,i Þ ¼ ðAN yðtÞÞi , i ¼ 1, . . . , N: y_ i ¼ d

ð13Þ

It should be noted that the term (ANy(t))0 in the first equation should be interpreted in the sense of the operator as defined by f(LNy( N,0)). Other terms (ANy(t))i (i ¼ 1, . . ., N) are matrix multiplications. The system (7) now reads y_ ðtÞ¼ AN yðtÞ:

ð14Þ

Note that y(t) is n(N þ 1) 1, and AN is n(N þ 1) n(N þ 1). The initial condition reads y(0) ¼ [u(0), u( N,1), . . ., u( N,N)]T. In a nutshell, Equation (14) is a CTA of the system (1), which is the same as that presented in Sun (2009). A detailed matrix representation of AN with the Chebyshev external nodes on [, 0] is presented in Breda et al. (2004, 2005).

2.1. Convergence with Chebyshev nodes Let B(, ) be a closed ball in C centered at with radius where is the eigenvalue of the original system. It is shown by Breda et al. (2004, 2005) that when the Chebyshev external nodes on [, 0] are used for (LNy)(), the maximum error emax of the collocation polynomial is bounded above by C0 C1 N emax pffiffiffiffi jjujj, N N


ð15Þ

4


where C0 and C1 are constants determined by and , but independent of N. Furthermore, max j i j N ,

1 i

i ¼ 1, . . . , ,

Define a parameter r ¼ 1/(pT). Then H(z) can be rewritten as

ð16Þ HðzÞ ¼

where i denotes the eigenvalue of the matrix AN of multiplicity n that matches the exact eigenvalue of the original system, and 1= !1= C2 1 C1 N pffiffiffiffi N ¼ , C3 N N

ð17Þ

with C2 and C3 ¼ C3() are constants. It should be noted that this convergence analysis does not apply to the extraneous eigenvalues introduced by the discretization, which do not match any true eigenvalues of the original system. It also does not explicitly address the accuracy of temporal response prediction, although better prediction of more dominant poles usually leads to more accurate temporal responses. We numerically examine this issue in the example section, and further show that these extraneous eigenvalues are not negligible, and contribute to temporal responses. For the convergence and stability analysis when the finite difference and Runge–Kutta methods are used to derive Equation (14), the readers are referred to Maset (2003), Bellen and Maset (2000), Koto (2009), and Bellen and Zennaro (2003).

3. Lowpass filter-based CTA We now present a new method to discretize the system (1). Let T be the sample time of a digital system, and p > 0 be a bandwidth parameter of an anti-aliasing lowpass filter. The derivative of a measured signal can be computed by passing the measurement through the following transfer function, HðsÞ ¼

p s: sþp

ð18Þ

2ðz 1Þ : Tðz þ 1Þ

ð19Þ

ð21Þ

_ ¼ HðzÞxðzÞ, or in the digital time In the z-domain, xðzÞ domain, 1 1 1 _ þ _ 1Þ ¼ ðxðnÞ xðn 1ÞÞ: þ r xðnÞ r xðn 2 2 T ð22Þ This relationship can be adopted for the CTA method (Sun, 2009). Equation (22) is applied to the delayed response vector and reads 1 1 _ i Þ þ _ iþ1 Þ þ r xðt r xðt 2 2 1 ¼ ½xðt i Þ xðt iþ1 Þ,

ð23Þ

where ¼ /N. Here i ¼ i (i ¼ 0, 1, 2, . . ., N). We have selected the sample time T ¼ . Equation (18) represents a common practice in signal processing and is used in many electronic instruments in which the derivatives of measured signals are needed. In the current content, however, we turn around to make use of it to create an algorithm for discretizing Equation (7) over the delayed time domain in order to arrive at Equation (14). It turns out that Equation (18) presents a very accurate IMEX formula for derivatives. Reconsider the system (1) with a control input, x_ ¼ f ðxðtÞÞ þ BuðtÞ,

ð24Þ

where u 2 Rm, and B 2 Rnm is the control influence matrix. The equation for the block-vector y(t) defined previously reads

_ HðsÞxðsÞ. Consider Tustin’s approximation Hence, xðsÞ s¼

1 z1 : T 12 þ r z þ 12 r

HN y_ ðtÞ ¼ FN ðyðtÞÞ þ GN uðtÞ:

ð25Þ

For the linear system

The digital version of the transfer function with Tustin’s approximation reads

x_ ¼ A0 xðtÞ þ A xðt Þ þ BuðtÞ,

ð26Þ

HN y_ ðtÞ ¼ FL yðtÞ þ GN uðtÞ,

ð27Þ

we have 2pðz 1Þ HðzÞ ¼ : ð2 þ pTÞz þ ð pT 2Þ

ð20Þ


Song and Sun

5

where 2

I

0

6 1 þ r I 1 r I 6 2 2 6 6 6 HN ¼ 6 0 6 . 6 .. 4 2

0 3

0

0

0 .. .

..

0

. 1 2

.. .

þr I

1 2

0

r I

B 607 6 7 7 GN ¼ 6 6 .. 7, 4.5 0

3 7 7 7 7 7, 7 7 7 5

ð28Þ

2

3 6 1 y1 ðtÞ y2 ðtÞ 7 6 7 7, FN ¼ 6 .. 6 7 4 5 . 1 yN ðtÞ yNþ1 ðtÞ 3 2 0 0 A A0 6 1 I 1 I 0 0 7 7 6 7 6 6 .. .. 7 FL ¼ 6 . . 7 7: 6 0 7 6 . . 6 .. .. 0 7 5 4 1 1 0 0 I I

fðyðtÞÞ

ð29Þ

4. Examples

Note that HN is a lower triangular matrix and is non-singular as long as r 6¼ 1/2. When r ¼ 1/2, the LPCTA is reduced to be the backward finite difference method, which is unstable for the time-delayed system (Sun, 2009). When r ! 0 or the bandwidth of the lowpass filter p ! 1, the left-hand side of Equation (23) reads 1 _ iþ1 Þ xðt _ iþ1=2 Þ: _ i Þ þ xðt ½xðt 2

time-delayed dynamical system is captured by the numerical solution. Note that the response of timedelayed dynamical systems contains infinitely highfrequency components. Since the lowpass filter should preserve the fidelity of high-frequency components in the numerical solution, the bandwidth p must be much larger than 1/. Note that FL for linear systems can be illconditioned when N is large. However, the dynamics of the approximate system is determined by H1 N FL . When N is large, we have to adjust the bandwidth ratio r accordingly. Once a desired bandwidth of the solution is decided, two parameters N and p can be chosen properly to achieve the solution that is accurate within the specified frequency range. Equations (25) and (27) are based on a first-order lowpass filter. Higher-order lowpass filters of various type can also be applied. For general and rigorous discussions of convergence and stability of various numerical integration algorithms for time-delayed systems, the readers are referred to Maset (2003), Bellen and Maset (2000), Koto (2009), and Bellen and Zennaro (2003). The convergence analysis of the LPCTA method is being pursued and will be reported in the future. This paper presents numerical examples to demonstrate its effectiveness.

In this section, we examine the effectiveness and accuracy of the LPCTA method. To measure the accuracy of the LPCTA solution, we define a normalized root mean square (RMS) error over a duration of time T1

eRMS

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z Z 1 T1 1 T1 2 ¼ ðxðtÞ xAP ðtÞÞ dt xðtÞ2 dt, T1 0 T1 0 ð31Þ

ð30Þ

This implies that Equation (23) represents the central difference approximation of the delayed response, which is consistent with the Tustin transformation in Equation (19). It has been shown by Sun (2009) that the central difference approximation gives accurate prediction of temporal responses, and performs poorly in the frequency domain. Numerical examples presented later show that for 0 < r < 1/2, the proposed LPCTA method improves the performance of the CTA in frequency as well as in time domain. Recall that T ¼ and r ¼ 1/(p). Note that 1/ defines the bandwidth of the discretization and p denotes the bandwidth of the lowpass filter. The parameter r is therefore a bandwidth ratio. The bandwidth 1/ determines how high the frequency of the

where x(t) is obtained from the direct numerical integration and xAP(t) denotes the solution obtained with an approximation method.

4.1. First-order systems (I) Consider a linear system with one time delay _ ¼ 5xðtÞ þ 0:5xðt =2Þ: xðtÞ

ð32Þ

We take r ¼ 0.001 for all the cases. Figure 1 shows all of the poles of Equation (32) obtained by the LPCTA method. In the middle of Figure 1, a zoom view shows the excellent agreement between the exact and predicted poles of low frequencies.


6


4

2

x104

3

1.5

2

Im (λ)

x (t)

100

1

50

1

0

0

0.5

−50 −100 −5

−1

−4

−3

−2

0

−1

0

−2 −3

−0.5 0

−4 −7

−6

−5

−4

−3 Re (λ)

−2

−1

5

0 x104

Figure 1. The exact and approximate poles predicted by the lowpass filter-based continuous-time approximation (LPCTA) method with N ¼ 29 for Equation (32). ‘x’: the exact poles; ‘þ’: the predicted poles.

10 Time (s)

15

20

Figure 3. The temporal response of Equation (33). ‘- - -’: the direct integration; ‘—’: the lowpass filter-based continuous-time approximation (LPCTA) method with N ¼ 26. The agreement between the results is excellent.

the bandwidth of the method 1/ is sufficiently high, the RMS errors of the LPCTA method are smaller than that of the method with Chebyshev nodes. This example illustrates that the LPCTA method and the method with Chebyshev nodes are comparable. (II) Now, we consider a nonlinear system with one time delay

101

eRMS (%)

100

_ ¼ 5xðtÞ þ 0:5xðt =2Þ xðtÞ3 þ sinð!tÞ: ð33Þ xðtÞ 10−1

10−2 100

101

102

103

N

Figure 2. Variation of the normalized root mean square (RMS) errors for the system in Equation (32) with N for the lowpass filter-based continuous-time approximation (LPCTA) method and the method with Chebyshev nodes. ‘’: the LPCTA method; ‘x’: the method with Chebyshev nodes.

Figure 3 compares the temporal responses by the direct integration and by the LPCTA method with N ¼ 26 and ! ¼ 1. The initial condition is x(0) ¼ 2. The agreement between the responses is excellent. Figure 4 shows the normalized RMS errors as a function of excitation frequency !. From the figure, we see that the method with Chebyshev nodes consistently has smaller RMS errors than that of the LPCTA method. (III) Consider a nonlinear system with two time delays _ ¼ xðtÞ þ xðt 1Þ 0:5xðt =2Þ x3 ðtÞ xðtÞ þ sinð!tÞ:

Next, we compare the LPCTA method and the method represented by Equation (14) with Chebyshev nodes (Breda et al., 2004, 2005). Figure 2 shows the normalized RMS errors of the solution for Equation (32) as a function of N. When N is small, or the bandwidth of the method 1/ is small, the RMS errors of the method with Chebyshev nodes are consistently better than that of the LPCTA method. However, when

ð34Þ

Figure 5 shows all the poles of Equation (34) obtained by the LPCTA method. Figure 6 shows the responses by the direct integration and by the LPCTA method with N ¼ 26 and ! ¼ 1. The initial condition is x(0) ¼ 0. The agreement between the responses is excellent. Figure 7 shows the normalized RMS errors for the LPCTA method and the method with Chebyshev nodes


Song and Sun

7

0.8

100

0.6

0.2 x (t)

eRMS (%)

0.4

10−1

0 −0.2 −0.4 −0.6

10−2 10−4

10−2

100

102

104

−0.8

106

15

10

5

0

ω

20

Time (s)

Figure 4. Variation of the normalized root mean square (RMS) errors with ! for Equation (33). ‘’: the lowpass filter-based continuous-time approximation (LPCTA) method with N ¼ 26; ‘x’: the method with Chebyshev nodes.

Figure 6. The temporal response of Equation (34). ‘- - -’: the direct integration; ‘—’ the lowpass filter-based continuous-time approximation (LPCTA) method with N ¼ 26. The agreement between the results is excellent.

100

x105 5

10−1 eRMS (%)

200

Im (λ)

100 0

0

10−2

–100 –200 –5

–4

–3

–2

–1

10−3

0

10−4

–5 −10

−8

−6

−4 Re (λ)

−2

10−4

10−2

100

0 x105

Figure 5. The exact and approximate poles predicted by the lowpass filter-based continuous-time approximation (LPCTA) method with N ¼ 29 for the linear part of Equation (34). ‘x’: the exact poles; ‘þ’: the predicted poles.

as a function of the driving frequency !. From the figure, we can see that the RMS errors of both of the methods have the same trend, and peak around ! ¼ 1. At ! ¼ 105, the RMS errors have a sharp jump. We note that ! ¼ 105 is larger than the highest frequency of the poles obtained by the approximate methods. In other words, this frequency is out of the bandwidth of the discrete solution. In order to obtain accurate solutions for even higher frequencies, we must increase N or the bandwidth of the method.

102

104

106

ω

Figure 7. Variation of the normalized root mean square (RMS) errors with ! of Equation (34). ‘’: the lowpass filter-based continuous-time approximation (LPCTA) method; ‘x’: the method with Chebyshev nodes. N ¼ 26 is for both the methods.

4.2. Second-order systems (I) Consider a linear system with one time delay

_ ¼ xðtÞ

0 1 0 xðtÞ þ 4 0:2 0:5

0 xðt =2Þ, 0:5 ð35Þ

where x ¼ [x1(t), x2(t)]T. We select r ¼ 0.0001 and N ¼ 28. Figure 8 shows all of the poles of Equation (35) obtained by the LPCTA


8


0

x 104 x1 (t)

1

1 0

0.5

60 40

−1

Im (λ)

20

0

5

0

5

10

15

20

10

15

20

0

0

0

−20 −60 −4

−3

−2

1

x2 (t)

−40 0

−0.5

0.5 0

−0.5 −1

−1 −2

−1.5

−1 Re (λ)

−0.5

0 x 104

Figure 8. The exact and approximate poles of Equation (35) predicted by the lowpass filter-based continuous-time approximation (LPCTA) method with N ¼ 28. ‘x’: the exact poles; ‘þ’: the predicted poles.

2 x1 (t)

Time (s)

Figure 10. The temporal responses of Equation (36). ‘- - -’: the direct integration; ‘—’: the lowpass filter-based continuous-time approximation (LPCTA) method with N ¼ 26. The initial conditions are x1(0) ¼ 0 and x2(0) ¼ 0. The agreement between the responses is excellent.

(II) Consider now a nonlinear forced system

1

0 −1 −2

0

5

10

15

20

0 0 _ ¼ xðtÞ xðtÞ þ xðt 2Þ 1 0:2 1 1

0 : þ 3 x1 ðtÞ þ sinð!tÞ 0

1

ð36Þ

x2 (t)

4 2 0 −2 −4

0

5

10

15

20

Time (s)

Figure 9. The temporal responses of Equation (35). ‘- - -’: the direct integration; ‘—’: the lowpass filter-based continuous-time approximation (LPCTA) method with N ¼ 26. The initial conditions are x1(0) ¼ 2 and x2(0) ¼ 0. The agreement between the responses is excellent.

method. The zoom view in the figure shows that the agreement between the exact and predicted poles is excellent at lower frequencies. Temporal responses are presented in Figure 9. The figure shows that the responses obtained by the direct integration and by the LPCTA method with N ¼ 26 are in excellent agreement. The initial conditions are taken to be x1(0) ¼ 2 and x2(0) ¼ 0.

Figure 10 compares the responses by the direct integration and by the LPCTA method with N ¼ 26. The initial conditions are x1(0) ¼ 0 and x2(0) ¼ 0. The agreement between the responses is excellent. Figure 11 shows the RMS errors of the LPCTA method and the method with Chebyshev nodes as a function of the driving frequency both with N ¼ 26. We observe very similar trends of the RMS error to that in Figure 7.

4.3. Effect of the predicted poles As shown in the previous examples, the discrete methods can accurately predict the poles of the timedelayed system at low frequencies, and introduce extraneous poles that do not match any exact poles. The question is what the effects of all of the poles on the system response are. To find the answer to this question, we consider a linear system. The discretized system with the LPCTA method takes the form of Equation (27). Let zi be the right eigenvectors of H1 N FL AN associated with the eigenvalues or poles i, and vi be the left


Song and Sun

9

800

100

600

10−2

400

40

200

20

Im (λ)

eRMS (%)

10−1

10−3

0

0

−20

−200

−40 −5

−4

−3

−2

−1

0

−400 10−4

−600 10−5 10−4

10−2

100

102

104

−800 −1400 −1200 −1000 −800 −600 −400 −200 Re (λ)

106

ω

Figure 11. Variation of the normalized root mean square (RMS) errors with ! of Equation (36). ‘’: the lowpass filterbased continuous-time approximation (LPCTA) method with N ¼ 26; ‘x’: the method with Chebyshev nodes.

Figure 12. The exact and approximate poles of Equation (32) predicted by the lowpass filter-based continuous-time approximation (LPCTA) method with N ¼ 26. The sub-figure shows the predicted poles retained to compute the results at the bottom of Figure 13. ‘x’: the exact poles; ‘þ’: the predicted poles.

eigenvectors of AN. The solution of Equation (27) can be expressed in terms of the characteristic function yðtÞ ¼ (ðtÞyð0Þ þ 0

2

t

(ðtÞ(T ðsÞH1 N GN ðsÞuðsÞ ds, ð37Þ

x (t)

Z

0

1 0

where −1

(ðtÞ ¼

nðNþ1Þ X

zi vTi ei t :

0

1

2

0

1

2

3

4

5

3

4

5

3

ð38Þ

i¼1

In this solution, we can selectively discard the terms associated with certain poles in order to study their effect on the solution. When some terms in the function ((t) are neglected, y(t) no longer satisfies the initial condition. In the numerical examples reported in the following, we scale the remaining terms to meet the initial condition. Consider now the homogeneous response of Equation (32). Figure 12 show the exact and predicted poles obtained by the LPCTA method with N ¼ 26 for Equation (32). The zoom view in the figure shows the approximate dominant poles that are retained for the results in Figure 13. The top part of Figure 13 shows the response by the LPCTA method with all of the poles. The temporal response agrees with the exact solution very well. The bottom part of Figure 13 shows the response by using the retained dominant poles. In this case, the response does not match the exact solution initially, but catches up later. This is because the response is determined mainly by the dominant poles as t ! 1. We observe that many poles

x (t)

2 1 0 −1

Time (s)

Figure 13. The temporal responses of Equation (32). Top: the solution obtained with all of the approximate poles of the lowpass filter-based continuous-time approximation (LPCTA) method shown in Figure 12. Bottom: the solution obtained with the retained approximate poles shown in the middle of Figure 12. ‘—’: the direct integration; ‘- - -’: the LPCTA method with N ¼ 26.

obtained by the LPCTA method do not even have a match with the exact poles, and yet they still contribute to the temporal response of the system. Figure 14 shows the exact and predicted poles obtained by the LPCTA method with N ¼ 26 for Equation (33) with e ¼ 0. The zoom view in the figure shows the approximate dominant poles that are retained for the results in Figure 15. The top part of


10

Journal of Vibration and Control 0(0) discussed before, when the forcing frequency is larger than the highest frequency of the predicted poles, the accuracy of the prediction will deteriorate further.

800 600 400

100

Im (λ)

200

5. Conclusion

50 0

0

−50

−200

−100 −5

−4

−3

−2

−1

0

−400 −600 −800 −1400 −1200 −1000 −800 −600 −400 −200

0

Re (λ)

Figure 14. The exact and approximate poles of Equation (33) predicted by the lowpass filter-based continuous-time approximation (LPCTA) method with N ¼ 26. The middle figure shows the predicted poles retained to compute the results in Figure 15. ‘x’: the exact poles; ‘þ’: the predicted poles.

0.2 x (t)

0.1 0 −0.1 −0.2

0

1

2

3

4

5

We have presented the method of lowpass filter based continuous time approximation for time-delayed systems. The method can handle linear and nonlinear systems effectively. The effect of the predicted poles obtained by the LPCTA method on the temporal response is discussed in detail. It is interesting to note that although many poles predicted by discrete methods are extraneous, and do not have a match of the exact poles, they contribute significantly to the temporal response of the system. The LPCTA method has two factors determining its bandwidth. These are the discretization time interval of the delayed response, and the lowpass filter bandwidth parameter p. For linear systems, the bandwidth of the method can be ultimately described by the highest frequency of the predicted poles. When the forcing frequency is smaller than the highest frequency of the predicted poles systems, the predicted forced response matches the results of the direct numerical integration well. When the forcing frequency is larger than the highest frequency of the predicted poles of the system, the accuracy of the prediction deteriorates. Finally, we point out that the current method can also predict unstable poles of the time-delayed systems. Numerical examples of this kind will be presented elsewhere.

0.02 x (t)

0.01

References

0 −0.01 −0.02

0

0.5

1

1.5

2

Time (s)

Figure 15. The temporal responses of Equation (33). ‘—’: the direct integration; ‘- - -’: the lowpass filter-based continuous-time approximation (LPCTA) method with the retained poles shown in the middle of Figure 14. Top: e ¼ 0 and ! ¼ 10. Bottom: e ¼ 0 and ! ¼ 100.

Figure 15 shows the response by the LPCTA method with all of the poles. The response agrees with the exact solution very well (e ¼ 0, ! ¼ 10). The bottom part of Figure 15 shows the response with the retained dominant poles (e ¼ 0, ! ¼ 100). Note that the forcing frequency ! ¼ 100 is less than the highest frequency of the predicted poles. In this case, the predicted response does not match the exact solution well because the high-frequency predicted poles are neglected. As we

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Journal of Vibration and Control

Journal of Vibration and Control

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