Frequency domain stability analysis of nonlinear active disturbance rejection control system

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Frequency domain stability analysis of nonlinear active disturbance rejection control system Jie Li a, Xiaohui Qi a, Yuanqing Xia b,n, Fan Pu b, Kai Chang a a b

Department of Unmanned Aerial Vehicle Engineering, Mechanical Engineering College, Shijiazhuang 050003, China School of Automation, Beijing Institute of Technology, Beijing 100081, China

art ic l e i nf o

a b s t r a c t

Article history: Received 22 August 2014 Received in revised form 27 September 2014 Accepted 19 November 2014 This paper was recommended for publication by Dong Lili

This paper applies three methods (i.e., root locus analysis, describing function method and extended circle criterion) to approach the frequency domain stability analysis of the fast tool servo system using nonlinear active disturbance rejection control (ADRC) algorithm. Root locus qualitative analysis shows that limit cycle is generated because the gain of the nonlinear function used in ADRC varies with its input. The parameters in the nonlinear function are adjustable to suppress limit cycle. In the process of root locus analysis, the nonlinear function is transformed based on the concept of equivalent gain. Then, frequency domain description of the nonlinear function via describing function is presented and limit cycle quantitative analysis including estimating prediction error is presented, which virtually and theoretically demonstrates that the describing function method cannot guarantee enough precision in this case. Furthermore, absolute stability analysis based on extended circle criterion is investigated as a complement. & 2014 ISA. Published by Elsevier Ltd. All rights reserved.

Keywords: Active disturbance rejection control Nonlinear system Limit cycle Root locus Describing function Circle criterion

1. Introduction The presence of model uncertainties and external disturbances in most industrial control problems is a big challenge for modern control theory with the absence of proper models. Although classical proportional-integral-derivative (PID) control law has its own limitations, it is still dominant even today across various sectors of the entire industry for its error driven rather than model-based. The active disturbance rejection control (ADRC) systematically proposed by Han in [1,2] is a drastic departure from both modern and classical control theory. It is great reflection about ‘control theory: model analysis approach or a direct control approach’ [3] and ‘linear and nonlinear of feedback system’ [4], and so on. On one hand, it inherits from merits of PID that it requires little information about the plant. On the other hand, the ADRC prefers to use nonlinear functions in the design of the observer and the control law, which is potentially much more effective than a linear one and provides surprisingly better results in practice. What is more, it takes from best offering of modern control theory that it uses an extended state observer (ESO) to n

Corresponding author. Tel.: þ 86 10 68914350. E-mail addresses: [email protected] (J. Li), [email protected] (X. Qi), [email protected] (Y. Xia), [email protected] (F. Pu), [email protected] (K. Chang).

estimate and compensate the internal and external disturbances and uncertainties, which is a big breakthrough of ‘internal model theory’ and ‘absolute invariance principle’. Since ADRC was proposed, it has gradually obtained a wide range of practical applications, such as speed control of induction motor drive [5], attitude tracking of rigid spacecraft [6], flexible-joint system [7], robotic uncalibrated hand-eye coordination [8], superconducting RF cavities [9], fractional-order system [10], to name a few. Although theoretical studies of ADRC fall far behind the applications, recently some important theoretical research results have been achieved. For instance, time domain convergence of linear and nonlinear ADRC is proved [11–14], which lays a solid theoretical foundation and provides a strict theoretical support. At the same time, frequency domain analysis of the linear ADRC framework has made much progress [15–18], which is significantly important that the ADRC framework is understood using the almost universal frequency domain analysis languages shared by practicing control engineers, including both bandwidth and stability margins. However, the frequency domain analysis of nonlinear ADRC is much more difficult and has fallen behind linear one. Until recently, frequency domain analysis of nonlinear ADRC via the describing function method has made a breakthrough [19,20]. Nonlinear functions are artificially introduced into the ADRC to make it more effective to inhibit uncertainties and disturbances and improve system dynamics. In return, it may make the system

http://dx.doi.org/10.1016/j.isatra.2014.11.009 0019-0578/& 2014 ISA. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: Li J, et al. Frequency domain stability analysis of nonlinear active disturbance rejection control system. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.11.009i

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produce some complex but colorful nonlinear behaviors, such as multiple equilibrium points, limit cycles, bifurcations and chaos. As we all know, stability is the precondition of analyzing and designing a normally working system. Compared with time domain stability criteria, frequency domain stability criteria such as Nyquist stability criterion is often based on open loop system frequency characteristic curve to determine the stability of closed loop system, which is visual and convenient to use. For linear system, the Nyquist stability criterion is a very famous frequency domain method which is widely used to determine the stability of a system. At the same time, there are qualitative and quantitative theories to analyze the frequency domain stability of nonlinear systems. The qualitative theory of dynamical systems provides an overall perspective on the behavior modes of the system, which can be used as a guide for the search of concrete behaviors. In this way, it is complementary to the most conventional quantitative methods such as describing function, circle criterion, Popov, etc. Describing function method is mainly used to predict the existence of limit cycles and their stability. The self-excited oscillation phenomenon (limit cycle) deserves attention since it is apt to occur in any physical nonlinear system. Prolonged oscillations may cause mechanical failure, increase the control error and other undesirable effects. On the other hand, a limit cycle can be desirable, for instance, by providing the vibration that minimizes friction, gap, dead zone and other nonlinear adverse effects in mechanical systems. Therefore, the analysis of limit cycle behavior of nonlinear ADRC is worthy of attention. However, due to inherent approximation of the describing function method, the prediction of limit cycles may cause a large error. As mentioned in [19,20], how to estimate and improve the accuracy of the describing function approximation needs to be further studied and worthy of mention. Since the describing function method is not accurate enough in this case and confined to predict the limit cycle, there is a need to adopt other methods for further studies. Circle criterion is a kind of frequency domain method to analyze the absolute stability of a class of nonlinear system named as Lure systems, which are the feedback systems consist of a linear dynamics with nonlinear feedback constrained by a certain sector bound. As the method is resemblant to the classical Nyquist stability condition, which has good frequency domain geometric meaning and diverse nonlinearities can be treated, it has been widely used in stability and robustness stability analysis of nonlinear systems such as Takagi–Sugeno fuzzy control systems [21], Mamdani fuzzy control systems [22], LTI robust control systems [23] and infinite-dimensional systems [24]. This paper focuses on the frequency domain stability analysis of a second-order fast tool servo system using nonlinear active disturbance rejection control algorithm. We make use of root locus analysis based on the concept of equivalent gain to qualitatively illustrate how the limit cycle is generated and how to avoid it. To get a quantitative result, the describing function method is used and the error caused by the approximation is estimated to guarantee accuracy. As the describing function may not be accurate enough in this case and always confined to predict the limit cycle, the extended circle criterion is adopted to analyze and guarantee absolute stability of the ADRC system as a complement. The rest of this paper is organized as follows: Section 2 introduces the framework and algorithm of the nonlinear ADRC. Section 3 characterizes the nonlinear control system in frequency domain except the nonlinear function, which is transformed based on the concept of equivalent gain. Then, how the limit cycle is generated and how to avoid it are illustrated by root locus analysis. Section 4 characterizes the nonlinear function using describing function and focuses on stability analysis of limit cycle using the describing function method, including error estimate to give a more accurate result. In Section 5, absolute stability analysis based

on extended circle criterion is presented. Finally, some concluding remarks are drawn in Section 6.

2. Nonlinear ADRC framework and algorithm Nonlinear ADRC has been successfully applied to the fast tool servo system [19], which is a typical second-order single-input single-output system and described as y€ ¼  py_  qy þ br

ð1Þ

where y is the regulated output,r is the input force,b is the gain coefficient. For physical meanings of the related parameters please refer to [19]. In the ADRC framework,  py_ qy can be supposed to be the total disturbance that can be estimated and compensated by ESO, which is the core and essence of the ADRC. To this end, the system will be reduced to a simple double-integral plant. Referring to [1], a nonlinear ESO is designed in the form 8 e ¼ z1  y > > > > < z_ 1 ¼ z2  β Ue 01 ð2Þ z_ 2 ¼ z3  β02 Ue þ b0 U u > > > > : z_ 3 ¼  β U falðe; α; δÞ 03 where z1 ,z2 , and z3 are the observer outputs,β 01 ,β 02 , and β 03 are the observer gains,e is the observer error, and b0 is a constant that is approximated to be b. In particular,falðe; α; δÞ is a nonlinear function proposed by Han [1] and defined as ( 1α e=δ jej r δ falðe; α; δÞ ¼ ð3Þ α jej sgnðeÞ jej 4 δ This nonlinear function plays an important role in the newly proposed ADRC framework, due to its characteristics of ‘small error, big gain; big error, small gain’.α and δ are two important parameters to be predetermined. In addition, note that the inputs of the ESO are the system output y and the control signal u, and the output z3 of the ESO provides an estimate about  py_  qy. With a well-tuned observer, the observer output z3 will closely track  py_ qy. Then, the control law is designed as u¼

u0  z 3 b0

ð4Þ

where u0 may employ the simple linear proportional and derivative control law in the form of u0 ¼ kp ðr  z1 Þ þ kd ðr_ z2 Þ

ð5Þ

where r and r_ are the desired output and derivative, respectively. Both kp and kd are the controller gains. Such a nonlinear ESO in (2) and a linear control law in (4) develop a type of nonlinear ADRC algorithm for a second-order plant, as shown in Fig. 1. Referring to [19], the plant parameters are denoted as p¼ 4.67103/s, q ¼ 3:67  106 =s2 , and b0 ¼ b ¼ 3  107 μm=s2 =V, the observer and controller bandwidths are tuned to ωo ¼ 83:3 krad=s and ωc ¼ 41:7 krad=s, respectively. The parameters are

Fig. 1. Nonlinear ADRC for a second-order plant.

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thus achieved as β01 ¼ 3ω0 ,β 02 ¼ 3ω20 ,β 03 ¼ ω30 =10,kp ¼ ω2c , and kd ¼ 2ωc [25]. Here, the nonlinear observer gain β 03 is reduced to one-tenth of the linear gain.

3

nonlinear dynamic λðe; α; δÞ, as shown in Fig. 5. The expression and coefficients of G(s) are listed in the Appendix A. 3.2. Limit cycle analysis based on root locus

3. Limit cycle analysis based on root locus 3.1. Frequency domain description of ADRC Let's take the nonlinear function falðe; α; δÞ as a link, whose input is e and output is falðe; α; δÞ. Just like linear link, we define equivalent gain of the nonlinear link as follows:

λðe; α; δÞ ¼

falðe; α; δÞ e

ð6Þ

Let's take α ¼ 0:25,δ ¼ 0:1 for an example, and the equivalent gain of falðe; α; δÞ is shown in Fig. 2. Note that the equivalent gain of the λðe; α; δÞ only depends on the input amplitude. There are two features that can be observed from Fig. 2. First,λðe; α; δÞ ¼ δα  1 , if the input amplitude is in the linear range. Second, λðe; α; δÞ decreases as the input amplitude increases, which is intuitively reasonable because the exponent (α o 1) function reduces the ratio of the output to the input. By converting the ADRC algorithm to the frequency domain using the Laplace transform except the nonlinear function falðe; α; δÞ that is transformed by equivalent gain λðe; α; δÞ, the system is described as Fig. 3 (λðe; α; δÞ is replaced by λ). After deduction and simplification, the system in Fig. 3 is changed into a block diagram description, as shown in Fig. 4. All the blocks are derived by the transfer function description listed in the Appendix A. When r ¼0, the system shown in Fig. 4 can be simplified to be a basic feedback structure consisting of a linear dynamic G(s) and a

Based on the simplified structure described in Fig. 5, we try to investigate the mechanism of limit cycle based on the root locus theory. According to the theory of the root locus, the closed-loop gain pole positions change with nonlinear gain λðe; α; δÞ. The root locus of the simplified structure described in Fig. 5 is obtained in Fig. 6. Then we will explain that how the limit cycle is generated and how to avoid it. As shown in Fig. 6, the root locus intersect with the imaginary axis at 1.26105, and the root locus gain λðe; α; δÞ is 67, by which the amplitude of the input e can be determined, that is e ¼ 0:0037. When e 40.0037 (λðe; α; δÞ o 67), thus the poles of the closed-loop system are located in the left half-plane which indicates the system is stable. Then the amplitude of the input e will decrease, and the equivalent gain λðe; α; δÞ will increase, the poles of the closed-loop system will turn up to 1.26105 along with root locus. When eo0.0037 (λðe; α; δÞ 4 67), analogously, the closed-loop system is unstable. Then the amplitude of the input e will increase and the equivalent gain λðe; α; δÞ will decrease, the poles of the closed-loop system will turn down to 1.26105 along with root locus. Because of the system's inertia, it will eventually maintain oscillation with 0:0037 sin ð1:26  105 tÞ. However, if the maximum equivalent gain λðe; α; δÞ is limited to smaller than 67, the system will be stable and will not oscillate. The maximum α1 equivalent gain λðe; α; δÞ ¼ δ , we can adjust both δ and α to meet this requirement. Remark 1. The root locus analysis based on equivalent gain λðe; α; δÞ illustrates how the limit cycle is generated and can provide guidance how to avoid it. Here we just give a qualitative analysis and the results are not accurate. To obtain more accurate results, quantitative analysis will be further investigated. 4. Stability analysis of limit cycle based on the describing function method In this section, we make use of the describing function method to give more accurate limit cycle analysis and error estimate as well.

Fig. 2. Equivalent gain curve for the nonlinear function falðe; α; δÞ.

Fig. 4. Block diagram of the nonlinear ADRC system.

Fig. 3. Nonlinear ADRC system described by Laplace transform and equivalent gain.

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output amplitude, including all the harmonics, divide by the input amplitude; secondly, the equivalent gain method can be used for qualitative and quantitative analysis in time and frequency domain, while the describing function method is used for quantitative analysis in frequency domain; thirdly, when the equivalent gain method is used for quantitative analysis, the analysis accuracy depends on the magnitude of the correction, while the describing function method does not require this process. Given the magnitude of the correction need further study, and there is still no error estimation method, therefore, this paper use the describing function method for quantitative analysis.

Fig. 5. Simplified structure of the nonlinear ADRC system when r ¼0.

Let's use describing function φðEÞ instead of equivalent gain λðe; α; δÞ in Fig. 5. Then, the system can be analyzed and synthesized using the traditional frequency response method. The characteristic equation of this system can be written as GðjωÞφðEÞ þ 1 ¼ 0

ð8Þ

We can rewrite it as 1 þ φðEÞ ¼ 0 GðjωÞ

ð9Þ

Based on the famous Nyquist criterion, the closed-loop stability of the system can be analyzed. We can plot both the reverse frequency response function 1=GðjωÞ and the negative describing function  φðEÞ on the complex axis and find out if there are any intersection points of the two curves (Fig. 7). Consider a case that δ ¼ 0:001,α ¼ 0.25, while the rest cases are similar to analyze. Fig. 4 shows the stability analysis result. It is observed that the curve 1=GðjωÞ and the real axis approximately intersect at point B ( 97.19, 0), which indicates that the amplitude of the limit cycle is 2:2 μm and corresponds to a frequency of 149.55 krad/s. In addition, as φðEÞ is real, the plot of  φðEÞ always α1 α1 lies on the real axis.  φðEÞ ranges from  δ to 0. If δ o 97.19, there will not be any intersection with 1=GðjωÞ, which means δ o 0.0022 (α ¼ 0:25). In fact, we can also seek cycle solutions with the harmonic balance equation (8). Rewrite the Eq. (8) as follows:

Fig. 6. Root locus of the system.

fRe½GðjωÞ þjIm½GðjωÞgφðEÞ þ 1 ¼ 0 then we obtain ( Im½GðjωÞ ¼ 0

Fig. 7. Stability analysis based on the Nyquist criterion.

4.1. Limit cycle analysis based on the describing function method Firstly, we obtain the describing function of the nonlinear function, which derived and denoted as φðEÞ with an amplitude of EðsÞ as the input. The derivation is omitted and please refer to [19]. 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3    2 2k4 δ δ 5 2Eα  1 π  5 π 3 7 π 5 φðEÞ ¼ 2 τ  τ þ τ τ 1 þ π E E π 2 12 2 192 2

ð7Þ The features of describing function φðEÞ are similar to equivalent gain λðe; α; δÞ. We should pay attention to the frequency characteristics of describing function φðEÞ, for φðEÞ is a function of the input amplitude alone and there is no phase shift, which is also understandable because the exponent function does not cause a delay in the response to the input. Remark 2. As a matter of fact, both describing function φðEÞ and equivalent gain λðe; α; δÞ are expressions of the input-output gain and restricted by frequency characteristics. However, there are many differences: firstly, the concept is different. The former is the output amplitude of the fundamental harmonic divides by the input amplitude, and the latter is the static characteristics of the

Re½GðjωÞφðEÞ þ1 ¼ 0

ð10Þ

ð11Þ

By the first equation of (11),ω is determined, which is irrelevant to φðEÞ. Then substitute the determined ω into the second equation of (11), the amplitude of the limit cycle E is obtained. However, the accuracy of the results also depends on graphical conditions. If the two curves near the intersection are nearly orthogonal, the analysis results will be accurate. If the two curves near the intersection are almost tangent, it will result in an obvious prediction error from the describing function analysis, and in some cases (depending on the degree of attenuation of the higher harmonics) self-oscillation even will not exist. Unfortunately, in this case, it belongs to the last case. In order to demonstrate the limit cycle prediction in terms of the describing function method, a time domain simulation for this system is adopted. By using the software of MATLAB, the simulations are performed and the results are shown in Fig. 8. From this figure, it is observed that the limit cycle waveform is nearly sinusoidal, and the amplitude and frequency of the fundamental sinusoidal wave are obtained to be 6:53 μm and 144.21 krad/s. Comparing the estimated results given by describing function method with the accurate results obtained by simulation, there is an obvious gap between the above two results. Therefore, it is necessary to estimate the errors and obtain a more accurate result.

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Note that we have defined ρðωÞ only for ω at which 1=GðjωÞ lies outside the critical circle for all k¼ 3,5,…; that is, for ω in the set

 βα ð20Þ Ω ¼ ωjρðωÞ 4 2 0

On any connected subset Ω of Ω, define

σ ð ωÞ ¼

ððβ  αÞ=2Þ2 ρðωÞ ððβ  αÞ=2Þ

ð21Þ

The positive quantity σ ðωÞ is an upper bound on the error term

ξφ.

Lemma 1. Under the stated assumptions above, Z 2σ ðωÞa 2 ω 2π =ω 2 ; 8 ω A Ω0 yh ðtÞdt r π 0 βα δφ r σ ðωÞ; 8 ω A Ω0

Fig. 8. Limit cycle behavior obtained by the time domain simulation.

4.2. Error estimate method The following error estimate method is based on the Lure system with nonlinearity belonging to the sector ½k1 ; k2 . Assuming that the output only contains odd harmonics and chooses the time origin such that the phase of the first harmonic is zero, that is yðtÞ ¼ a sin ωt þ yh ðtÞ

ð12Þ

yh ðtÞ is higher harmonics. Introducing the function 2ω πa

φða; yh Þ ¼ j

Z π =ω 0

φða sin ωt þ yh ðtÞÞexpð  jωtÞdt

ð13Þ

Then, we have the exact harmonic balance equation 1 þ φðE; yh Þ ¼ 0 GðjωÞ

ð14Þ

We can rewrite it as 1 þ φðEÞ ¼ ξφ GðjωÞ

ð15Þ

where

δφ ¼ φðEÞ  φðE; yh Þ 2ω πa

φða; yh Þ ¼ j

Z π =ω 0

ð16Þ

φða sin ωt þ yh ðtÞÞexpð  jωtÞdt

ð17Þ

When yh ¼0, φnðE; 0Þ ¼ φðEÞ, thus ξφ ¼ 0, and (15) reduces to (8). Therefore, the harmonic balance Eq. (8) is an approximate version of the exact Eq. (15). The error term ξφ cannot be found exactly, but its size can often be estimated. Next step is to find an upper bound on ξφ. To that end, let us define two functions ρðωÞ and σ ðωÞ. Start by drawing the locus of 1=GðjωÞ in the complex plane. On the same graph, draw a critical circle with the interval [  k2 ,  k1 ] on the real axis as a diameter. Now consider an ω such that the points on the locus 1=GðjωÞ corresponding to kω(k 41 and odd) lie outside the critical circle. The distance from any one of these points to the center of the critical circle is k þ 1 ð18Þ 2 GðjkωÞ Define

ρðωÞ ¼

k þ 1 k 4 1;k odd 2 GðjkωÞ inf

ð19Þ

ð22Þ ð23Þ

The proof of Lemma 1 is available in [26]; therefore, it is omitted. These two inequalities allow us to calculate the upper bound of ξφ and higher harmonics yh ðtÞ. 0 For each ω A Ω  Ω, we can draw such an error circle. The 0 envelope of all error circles over the connected set Ω forms an uncertainty band. The reason for choosing a subset of Ω is that, as ω approaches the boundary of Ω, the error circles become arbitrarily large and cease to give any useful information. The 0 subset Ω should be chosen with the objective of drawing a narrow band. We are going to look at the intersections of the uncertainty band with the locus of φðEÞ. Actually, we can find error bounds by examining the intersection. Let E1 and E2 be the amplitudes corresponding to the intersections of the boundary of the uncertainty band with the  φðEÞ locus. Let ω1 and ω2 be the frequencies corresponding to the error circles of radii σ ðω1 Þ and σ ðω2 Þ, which are tangent to the  φðEÞ locus on either side of it. Define a rectangle Γ in the ðω; EÞ plane by

ð24Þ Γ ¼ ðω; aÞjω1 o ω o ω2 ; a1 o E o a2 The rectangle Γ contains the point ðω; EÞ for which the loci of 1=GðjωÞ and  φðEÞ intersect, that is, the solution of the exact harmonic balance Eq. (14). It turns out that if certain regularity conditions hold, then it is possible to show that (14) has a solution in the closure of Γ . These regularity conditions are d d φðEÞ a0; Im½GðjωÞ a0 ð25Þ da d ω E ¼ as ω ¼ ωs Finally, note that at high frequencies for which all harmonics (including the first) have the corresponding 1=GðjωÞ points outside the critical circle, we do not need to draw the uncertainty band. Therefore, we define a set

 ~ ¼ ωj k1 þ k2 þ 1 4 k1  k2 ; k ¼ 1; 3; 5; ⋯ ð26Þ Ω 2 GðjkωÞ 2 and take the smallest frequency in Ω  as the largest frequency in Ω  , then decrease ω until the error circles become uncomfortably large. The next theorem is on the justification of the describing function method. Theorem 1. Consider the Lure system with nonlinearity belonging to the sector ½k1 ; k2 . Draw the loci of 1=GðjωÞ and  φðEÞ in the complex plane and construct the critical circle and the band of uncertainty as described earlier. Then,

 the system has no half-wave symmetric periodic solutions with ~, fundamental frequency ω A Ω

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 the system has no half-wave symmetric periodic solutions with ~ if the corresponding error circle fundamental frequency ω A Ω does not intersect the  φðEÞ locus,  for each complete intersection defining a set Γ in the ðω; EÞ plane, there is at least one half-wave symmetric periodic solution. yðtÞ ¼ a sin ωt þ yh ðtÞ with ðω; EÞ in Γ and yh ðtÞ satisfies the bound of (23). Proof. The proof of Theorem 1 is available in [26], therefore, it is omitted. Remark 3. Note that Theorem 1 gives a sufficient condition for oscillation and a sufficient condition for nonoscillation. Between the two conditions, there is an area of ambiguity where we cannot reach conclusions of oscillation or nonoscillation. 4.3. Application of the error estimate method

Fig. 10. Uncertainty band.

To apply the error estimate method, the following two theorems are presented. Theorem 2. The system in Fig. 5 is equivalent to the one in Fig. 9, which is called Lure system. Proof. As α; δ are predetermined, then falðe; α; δÞ is abbreviated to falðeÞ,λðe; α; δÞ is abbreviated to λðeÞ. We get ( 1α 1=δ jej r δ falðeÞ ¼ λðeÞ ¼ ð27Þ α  1 e jej sgnðeÞ jej 4 δ

λð  eÞ ¼ λðeÞ

ð28Þ

eðtÞ ¼ rðtÞ yðtÞ ¼  yðtÞ

ð29Þ

y ¼ falð yÞ U GðsÞ ¼  yU λð  yÞ U GðsÞ ¼  y U λðyÞGðsÞ ¼ falðyÞ U GðsÞ ð30Þ Thus, these two nonlinear systems are actually equivalent by (30). Theorem 3. The following inequality holds, which means that falðyÞ belongs to the sector ½k1 ; k2 , k1 y2 r y Uf alðyÞ r k2 y2 ; 8 y A R where k1 ¼ 0,k2 ¼ δ Proof. We have lim yα  1 r

y-1

0r

α1

.

falðyÞ falð0Þ α1 rδ y 0

falðyÞ α1 rδ y

0 r falðyÞy r δ

ð31Þ

ð32Þ

α1 2 y

ð33Þ

that is k1 y r y U falðyÞ r k2 y , which means the falðyÞ belongs to the α1 sector ½0; δ . Based on Theorems 1–3, the limit circle analysis error caused by the approximation is estimated to guarantee the accuracy of the describing function. 2

2

Fig. 9. Equivalent feedback structure of the system.

Draw the loci of 1=GðjωÞ and  φðEÞ in the complex plane and construct the critical circle and the band of uncertainty as described earlier, as shown in Fig. 10. In this case, we obtain a1 ¼ 0:002 μm; a2 ¼ 0:007 μm; ω1 ¼ 98500 rad=s; ω2 ¼ 155200 rad=s ð34Þ

Γ ¼ ðω; aÞj98500 o ω o 155200; 0:002 oa o 0:007 d d φðEÞ a 0; Im½GðjωÞ a0 da dω E ¼ 0:0022 ω ¼ 149550

ð35Þ ð36Þ

According to Theorem 1, the ADRC system with the given plant and parameters has a periodic solution. The frequency of oscillation ω belongs to the interval [98500,155200] rad/s, and the amplitude of the first harmonic at the input of the nonlinearity belongs to the interval [0.002, 0.007]μm. Further, according to the inequalities (22), we know that the higher harmonic yh ðtÞ satisfies 2σ ðωÞa 2 R βα ω=π 02π=ω y2h ðtÞdt r 0:46 ð37Þ r R ω=π 02π=ω ð0:0022 U sin ð149550tÞÞ2 dt 4:84e  6 According to (37), the maximum is 0.46, which indicates that the higher harmonic may not be neglected. The previous simulation has also demonstrated this speculation. What is more, both possible interval of the oscillation's frequency and amplitude are a little big. Therefore, describing function method should be justified to guarantee accuracy for such ADRC systems. The result of error estimate indicates that describing function method cannot guarantee enough precision in this case and always confined to predict the limit cycle. What is more, since Theorem 1 presents sufficient conditions, there exist cases that the system stability cannot be determined under certain parameters or parameter perturbation of the plant. To that end, circle criterion will be adopted to ensure the absolute stability of the ADRC system as a complement.

5. Absolute stability analysis based on extended circle criterion According to Theorems 2 and 3, stability analysis of the fast tool servo system using nonlinear ADRC can be transformed into a Lure's problem, which indicates the circle criterion may be applied. However, as GðsÞ is not Hurwitz with one eigenvalue at the origin, the circle criterion cannot be applied directly. To solve this problem, an extended circle criterion is needed.

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We have previously obtained by describing function method α1 1α that if δ o 97.19, i.e.,δ 40.01, the system will be stable. 1α While if δ 40.1, the system is absolute stable via the extended circle criterion. Compared with the results of the describing function method, we find that the result got by the describing function method is not accurate, while the one got by the extended circle criterion is conservative. Remark 4. Circle criterion provides sufficient conditions, which are conservative. Further analysis is needed to reduce conservation, combining with the existing achievements. On the other hand, the above stability analysis is based on a normal ideal system. To be in line with the actual situation, robust stability analysis is needed. Still, the characteristics of frequency domain and time domain stability analysis are well worthy of comparison. In a word, more deep research works need to be carried out. Fig. 11. The Nyquist plot of G(s).

6. Conclusions 5.1. Extended circle criterion Here, we present an extended circle criterion, for which the linear dynamics GðsÞ possess n eigenvalues at the origin and no eigenvalues with positive real parts. Suppose the plant is described by the following state equations: x_ ¼ Ax þ Bu y ¼ Cx

ð38Þ

where A A Rnn ,B; x A Rn1 , and C A R1n . We assume that the pair (A, B) is controllable, and the pair (A, C) is observable. Lemma 2. Suppose that the system matrix A is a Hurwitz matrix, i.e., all the eigenvalues of A have negative real parts, and that nonlinear dynamics belongs to the sector ½0; k. Then, a sufficient condition for the global asymptotical stability of the Lure system is given by: Re½1 þ kgðjωÞ 4 0;

8ωAR

ð39Þ

1

where gðsÞ ¼ CðsI  AÞ B. Proof. The proof of Lemma 2 is available in [27], therefore, it is omitted. Theorem 4 (Extended circle criterion). Suppose that the system matrix A of the linear dynamics of the Lure system has no eigenvalues on the imaginary axis except for n eigenvalues at the origin and no eigenvalues with positive real parts, and that nonlinear dynamics belongs to the sector ½0; k. Two sufficient conditions for the global asymptotical stability of this Lure system are: I. as ω moves from 0  1 to 0 þ 1 , the plot of gðjωÞ traces n clockwise semicircles with infinite radius about the origin; II. the Nyquist plot of gðjωÞ lies to the right of the vertical line defined by ReðsÞ ¼  1=k. Proof. If the first condition is true, then the system matrix A is Hurwitz. Furthermore, if the second condition is true, the system is global asymptotical stability according to Lemma 2.

As a novel and practical approach, ADRC behaves excellently in many applications such as the fast tool servo system presented in this paper. Stability is the precondition of analyzing and designing a normally working system, and the frequency domain method is an important tool for practicing control engineers. This paper applies three methods (i.e., root locus analysis, describing function method and extended circle criterion) to approach the frequency domain stability analysis of the fast tool servo system. Firstly, root locus qualitative analysis shows that limit cycle is generated because the gain of the nonlinear function varies with its input. δ and α are two important adjustable parameters to suppress limit cycle when the other bandwidth based parameters are determined. Then, quantitative limit cycle analysis is presented by describing function method, but the graphical conditions indicate that it may not be accurate enough, which is then demonstrated by time domain simulation. To guarantee the accuracy of the describing function method, the error is estimated subsequently, which theoretically further demonstrates that describing function method cannot guarantee enough precision in this case. Further, absolute stability analysis based on the extended circle criterion is investigated as a complement. The above methods used in this paper can be extended to stability analysis of other plants using ADRC with one nonlinearity. As digital control is the mainstream and trend, and nonlinear ADRC is more suitable for digital control, we should investigate the influence of discretization on performance in nonlinear ADRC control systems. Although stability analysis in time domain has made some progress [28], to the best of authors' knowledge, there is not any analysis in frequency domain for nonlinear ADRC discrete system. Still, the describing function method cannot be used for discrete system, thus other methods such as circle criterion should be adopted. Future work will focus on the analysis of the ADRC containing multiple nonlinearities and in discrete framework. This analysis is much more difficult due to the coupling of the linearity and nonlinearity. In brief, frequency domain analysis and synthesis for nonlinear ADRC still has a long way to go.

5.2. Absolute stability analysis based on extended circle criterion Acknowledgment Draw the Nyquist plot of GðjωÞ in the complex plane, as shown in Fig. 11. We conclude that the Nyquist plot of GðjωÞ lies to the right of the vertical line defined by ReðsÞ ¼  0:1. At the same time, as ω moves from 0  1 to 0 þ 1 , the plot of GðjωÞ traces one clockwise semicircles with infinite radius about the origin. If  1=k2 ¼ 1α δ o  0:1, according to Theorem 4, the system is absolutely stable.

The authors would like to thank the referees for their valuable and helpful comments which have improved the presentation. The work was supported by the National Basic Research Program of China (973 Program) (2012CB720000), the National Natural Science Foundation of China (61225015), Foundation for Innovative Research Groups of the National Natural Science Foundation of China (Grant no. 61321002).

Please cite this article as: Li J, et al. Frequency domain stability analysis of nonlinear active disturbance rejection control system. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.11.009i

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Appendix A The transfer function description of the blocks in Fig. 2. kd s þkp s2 þ ðkd þ β01 Þs þ ðkd β01 þ kp þ β 02 Þ k s3 þ ðkp þ kd β01 Þs2 þðkp β 01 þ kd β02 Þs þ kp β 02 Gr2 ðsÞ ¼ d s2 þ ðkd þ β 01 Þs þ ðkd β01 þ kp þ β 02 Þ

Gr1 ðsÞ ¼

s2 þ kd s þ kp d þ β 01 Þs þ ðkd β 01 þ kp þ β 02 Þ ðkp β01 þ kd β02 Þs þ kp β02 Gf 1 ðsÞ ¼ 2 s þ ðkd þ β 01 Þs þ ðkd β01 þ kp þ β 02 Þ

Gf ðsÞ ¼

Gc ðsÞ ¼

s2 þðk

β03 s

b s2 þ ps þ q Gp ðsÞ Gb ðsÞ ¼ b0 s þ Gp ðsÞGf 1 ðsÞ Gp ðsÞ ¼

Expression of G(s) 1 A0 s2 þ A1 s þ A2 GðsÞ ¼ U s B0 s4 þ B1 s3 þB2 s2 þ B3 s þ B4 Coefficients in G(s) A0 ¼ bβ03 A1 ¼ bβ03 kd A2 ¼ bβ03 kp B0 ¼ b0 B1 ¼ b0 ðp þ kd þ β01 Þ B2 ¼ b0 ðq þ β 01 kd þ kp þ β02 þ β01 p þ kd pÞ B3 ¼ b0 ðβ 01 q þ kd q þ kp p þ β01 kd p þ β02 pÞ þ bðβ 01 kp þ β02 kd Þ B4 ¼ b0 qðβ01 kd þ kp β02 Þ þbβ02 kp References [1] Han J. Active disturbance rejection control technique – the technique for estimating and compensating the uncertainties. Beijing: National Defense Industry Press; 2008. [2] Han J. From pid to active disturbance rejection control. IEEE Trans Ind Electron 2009;56(3):900–6. [3] Han J. Control theory, is it a model analysis approach or a direct control approach? J Syst Sci Math Sci 1989;9(4):328–35. [4] Han J. Linear and nonlinear of feedback system. ControlDecis 1988;3(2):27–32. [5] Feng G, Liu Y, Huang L. A new robust algorithm to improve the dynamic performance on the speed control of induction motor drive. IEEE Trans Power Electron 2004;19(6):1614–27.

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Please cite this article as: Li J, et al. Frequency domain stability analysis of nonlinear active disturbance rejection control system. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.11.009i