Limit cycle analysis of active disturbance rejection control system with two nonlinearities.

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Limit cycle analysis of active disturbance rejection control system with two nonlinearities Dan Wu, Ken Chen State Key Laboratory of Tribology, Department of Mechanical Engineering, Tsinghua University, Beijing 100084, China

art ic l e i nf o

a b s t r a c t

Article history: Received 23 April 2013 Received in revised form 25 December 2013 Accepted 3 March 2014 This paper was recommended for publication by Dr. Didier Theilliol

Introduction of nonlinearities to active disturbance rejection control algorithm might have high control efficiency in some situations, but makes the systems with complex nonlinearity. Limit cycle is a typical phenomenon that can be observed in the nonlinear systems, usually causing failure or danger of the systems. This paper approaches the problem of the existence of limit cycles of a second-order fast tool servo system using active disturbance rejection control algorithm with two fal nonlinearities. A frequency domain approach is presented by using describing function technique and transfer function representation to characterize the nonlinear system. The derivations of the describing functions for fal nonlinearities and treatment of two nonlinearities connected in series are given to facilitate the limit cycles analysis. The effects of the parameters of both the nonlinearity and the controller on the limit cycles are presented, indicating that the limit cycles caused by the nonlinearities can be easily suppressed if the parameters are chosen carefully. Simulations in the time domain are performed to assess the prediction accuracy based on the describing function. & 2014 ISA. Published by Elsevier Ltd. All rights reserved.

Keywords: Nonlinear systems Active disturbance rejection control Limit cycle Describing function

1. Introduction Many control problems involve uncertainties in the plant model parameters and external disturbances. The control design aims to deal with such uncertainties and disturbance by different strategies and algorithms in order to tolerate the uncertainties and minimize the negative effect of the disturbances on the system. Active disturbance rejection control (ADRC) is a novel robust control method that was systematically proposed by Han in his pioneer works [1,2]. In contrast to existing model-based designs, the ADRC does not need a precise analytical description of the system, as the unknown parts of dynamics are assumed as the internal disturbance in the plant, which, together with the external disturbance, is denoted as a generalized or total disturbance. Then, an extended state observer (ESO) that extends the system model with an additional and fictitious state variable is proposed to estimate the generalized disturbance and compensate for in the control law. This is a drastic departure because the information about the physical process needed by the controller is derived from the plant input–output data in real time and not from an a priori mathematical model. Therefore, the ADRC is more of a paradigm shift in feedback control system design than a new control design strategy [3]. On the other hand, instead of eliminating the presence of nonlinearity in most of the current control designs for simplicity, the ADRC intentionally introduces the nonlinearities into the design of the observer and the control law. Han has given insight

into the nonlinear feedback mechanism and concluded that a nonlinear ADRC is potentially much more effective than a linear one and provides surprisingly better results in practice. A typical nonlinear ADRC framework is to develop the ESO with two nonlinear gains. In fact, linear control methods rely on the key assumption of small range operation for the linear model to be valid. When the required operation range is large, a linear control controller is likely to perform very poorly or to be unstable, because the nonlinearities in the system cannot be compensated for. Nonlinear controllers may handle the nonlinearities in large range operation directly. Moreover, linear control may require high quality actuators and sensors to produce linear behavior in the specified operation range, while nonlinear control may permit the use of less expensive components with nonlinear characteristics. This will lower operation cost. As for performance optimality, bang– bang type controllers are one of favorite examples, which can produce fast response [4]. Due to active disturbance rejection concept and nonlinear feedback mechanism, the ADRC shows power and attraction to many control problems. This was originally demonstrated through time domain simulations and lately by various engineering applications, such as the trajectory tracking control of a flexible-joint robotic system [5], control for micro-electro-mechanical gyroscopes [6], speed control for permanent magnet synchronous motor servo system [7], and the control design for superconducting radio frequency cavities [8].

http://dx.doi.org/10.1016/j.isatra.2014.03.001 0019-0578/& 2014 ISA. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: Wu D, Chen K. Limit cycle analysis of active disturbance rejection control system with two nonlinearities. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.03.001i

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D. Wu, K. Chen / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

Intentional nonlinearity is artificially introduced into the ADRC to make it more effective in tolerance to uncertainties and disturbance and improvement of system dynamics. In return, it may make the system produce some complex but colorful nonlinear behaviors, such as multiple equilibrium points, limit cycles, bifurcations and chaos. Specially, the limit cycle phenomenon is deserving of attention since it is apt to occur in any physical nonlinear system. A limit cycle can be desirable, for example, by providing the vibration that minimizes frictional effects in mechanical systems. On the other hand, a limit cycle can cause mechanical failure of a control system and other undesirable effects. Previous research paid little attention on the analysis of limit cycle behavior of nonlinear ADRC. This paper attempts to study limit cycle behavior of a nonlinear ADRC with application to a fast tool servo for non-rotationally symmetric turning. In such a machining process, the fast tool servo drives the cutting tool to move back and forth with given amplitude and frequency in order to track the desired motion trajectory [9]. The most typical application of the fast tool servo is ultra-precision machining of micro-structured surfaces, which is much sensitive to undesired limit cycle vibrations between the cutting tool and the workpiece. If limit cycles, or chatters occur during machining, it significantly affects the surface finish quality, causes loss of dimensional accuracy of the workpiece, and accelerates the premature wear, chipping, and failure of the cutting tool [10]. As a result, limit cycles are dangerous and should be avoided in the fast tool servo control design. In order to study the nonlinear ADRC system, the describing function method is used to model the system in the frequency domain and characterize its limit cycles behavior in this paper. Describing function analysis is a widely known technique to study frequency response of nonlinear systems. It is an extension of linear frequency response analysis. In linear systems, transfer functions depend only on the frequency of the input signal. In nonlinear systems, when a specific class of input signal such as a sinusoidal wave is applied to a nonlinear element, one can represent the nonlinear element by a function that depend not only on the frequency, but also on the amplitude of the input signal [4]. The describing function method has been widely used to analyze nonlinear control problems such as predicting limit cycles [11–14], analyzing sliding mode observer dynamics [15], bifurcation analysis of nonlinear systems [16], and investigating the behavior of a fractional order Van der Pol-like oscillator [17]. Previous work [9] has studied the ADRC system with a single separable nonlinearity for simplification, focusing on analysis of stability, tracking and disturbance rejection performances of the system. However, the two-nonlinearity ADRC system might exhibit better control quality [1] but increase complexity of the system. Also, a great deal of the simplicity of the describing function approximate approach is lost when nonlinear effects are presented at more than one station in the system. Therefore, it is necessary to explore the performances of the ADRC system with multiple nonlinearities. Actually, it is quite usual that nonlinearities are presented at more than one station around a control loop. This demands to develop methodology of analyzing nonlinear behavior of the systems with multiple nonlinearities. The rest of the paper is organized as follows. Section 2 gives a simple description of the fast tool servo and the ADRC design with two nonlinearities. Section 3 characterizes the two-nonlinearity ADRC in the frequency domain using both the describing function and the transfer function methods. A time-domain iteration algorithm is also presented to treat two nonlinearities to a single nonlinearity available for analysis of single frequency limit cycles. Section 4 discusses the effect of the controller parameters on the limit cycles. The time-domain simulation is also given to demonstrate the accuracy of limit cycle prediction based on the describing function method. The conclusions in Section 5 close the paper.

2. Design of the fast tool servo controller In this section, a nonlinear active disturbance rejection controller is developed for the fast tool servo applied to ultraprecision machining of micro-structured surfaces. 2.1. Hardware of the fast tool servo The fast tool servo is a typical precision, closed-loop linear motion control system. It consists of a normal-stress electromagnetically driven linear actuator [18], a power amplifier, a capacitance displacement sensor, a commercial Digital Signal Processing (DSP)-based motion control board and a self-developed interface board for implementation of the high speed A/D and D/A conversion as well as the I/O connection with the DSP. The whole system is controlled in real-time using an industrial computer with the host-target architecture [19]. The actuator and the power amplifier dynamics are considered to be the plant dynamics to be controlled. Fig. 1 illustrates the prototype of the developed electromagnetically driven fast tool servo. A theoretical computation and off-line identification reveals that the plant dynamics can be approximated using a secondorder LTI dynamic model in the form of Gp ðsÞ ¼

b ; s2 þ psþ q

ð1Þ

where the plant parameters are denoted as p ¼ 4:67 103 /s, q ¼ 3:67 106 /s2 and b0 ¼ b ¼ 3 107 μm/s2/V. The actuator travel is approximately 50 μm. The measurement resolution of the displacement sensor is 1 nm, and the maximum sampling control rate of the real-time computer is up to 500 kHz. 2.2. Design of nonlinear active disturbance rejection controller Fast tool servo control is a typical precision motion control problem, where the control input must be applied so that the plant output follows desired constant or time varying signals. However, it is usual that there are different kinds of uncertainties in mechanical and electrical components, such as thrust force or torque constant of an actuator, nonlinear friction and stiffness, vibration mode frequencies. This challenges the controller

Fig.1. The picture of the electromagnetic fast tool servo case.



design for achieving both tracking and disturbance rejection performance. The precision motion control problem is closely related to system dynamic inverse. The feed-forward strategy thus easily meets such control requirements. Of course, the problem is more complicated because of the stability and the realization concerns of the inverse systems [20]. Tomizuka [21] proposed a zero phase error tracking control (ZPETC) algorithm to be applied as a feedforward control law, in which the reference input is pre-filtered by the inverse of the closed-loop system model. However, the ZPETC algorithm requires special care when the closed-loop system contains unstable zeros. Also, pre-filtering the reference signal by the inverse of the closed-loop model is susceptible to modeling error. In addition, the combined control framework with feedback and feed-forward compensation is considered as a practical technique for achieving desired performance [22]. A known plant model and sufficient knowledge of the target mechatronic systems should be indispensable from the view of more accurate modelbased feed-forward compensation and more progressive design of feedback controllers. To obtain robust stability, some novel approaches have been proposed, in which disturbance observer design is a kind of powerful tools for disturbance rejection [23,24]. In the ADRC framework, the extended state observer be regarded as a special observer because it estimates not only external disturbance but also system modelling error and parameters variation. By using the ADRC controller structure and parameters tuning methodology, the ADRC concept provides actual simultaneous optimization in both feed-forward and feedback compensators, leading to satisfied control performance of systems. Consequently, this paper proposes to apply ADRC concept to develop the fast tool servo controller. For the sake of simplicity and convenience, the second-order plant is described generally as y€ ¼ f ðy; y_ ; w; tÞ þ bu;

ð2Þ

where y is the actuator position output that is measured and to be controlled, y_ and y€ are the velocity and acceleration of the actuator, respectively, u is the input force, b is the coefficient related to the actuator dynamics, and w represents the extraneous unknown input force (known as the external disturbance). In the ADRC framework, f ðy; y_ ; w; tÞ, shortened to f, is the generalized disturbance representing the combined effect of the internal dynamics and external disturbances on the acceleration. The key to the control is to compensate for f, and the system will be reduced to a simple double-integral plant if the value of f can be determined at any given time [2,3]. To this end, the unique ESO is originally proposed to estimate f in real time, which is the core and essence of the ADRC concept and can be found in previous work [1,2] for details. Referring to [1], a nonlinear ESO is designed in the form 8 e ¼ z1 y; f e1 ¼ f alðe; 0:5; δÞ; f e2 ¼ f alðe; 0:25; δÞ > > > > < z_ 1 ¼ z2 β e 1 ; ð3Þ z_ 2 ¼ z3 β 2 f e1 þb0 u > > > > : z_ 3 ¼ β f e2

3

around the origin. δ and α are pre-determined to be constant for α1 various applications. 2δ and k ¼ δ denote the range and slope of the linearity of the fal function. An extreme case is α ¼ 1, in which the fal function changes to be completely linear. The reason that the fal function has been chosen in Eq. (3) is that such a nonlinear function sometimes provides surprisingly better results in practice. For example, the tracking error approaches zero in infinite time when the linear feedback is adopted. However, the tracking error can reach zero much more quickly in finite time if the nonlinear feedback of the form jejα signðeÞ with α o1 is employed. The numerical simulation also shows that the fal nonlinearity can help reduce steady state error remarkably, which avoids an integral control and simplifies the control law [2]. Therefore, the nonlinear function plays an important role in the newly proposed ADRC framework. Additionally, note that the inputs to the ESO are the system output y and the control signal u, and the output z3 of the ESO provides important information about f. It can be shown that with a well-tuned observer, the observer output z3 will closely track f when f is bounded [25]. Then, the control law is designed as u ¼ ðu0 z3 Þ=b0 ;

ð5Þ

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;

ð6Þ

where r and r_ are the desired position and velocity signals. Both kp and kd are the controller gains. Such a nonlinear ESO in Eq. (3) and a linear control law in Eqs. (5) and (6) develop a type of nonlinear ADRC algorithm for a second-order fast linear actuator, as shown in Fig. 2. It is obvious that increasing the gains of the ESO benefits the improvement of disturbance estimation ability and dynamic performances of a closed-loop system. Since digital technology offers many benefits, most of current controllers are implemented in discrete-time domain by a digital computer. The sampling step is limited by the running speed of the digital computer. Moreover, higher gains of the ESO will deteriorate the stability of the system. As a result, it is necessary to tune the gains of the ESO in balance between the stability and disturbance rejection ability. This indicates that the ADRC may deal with uncertainties of the system in a certain range. The quantitative analysis on the estimation ability of the ESO can be found in [26].

3. Frequency domain description of the nonlinear ADRC 3.1. Describing function for the fal nonlinearity Note that the fal nonlinearity is single valued and symmetric about the origin. Consequently, the describing function is a real function of the input amplitude. Because the output of the fal

3

where z1 ; z2 ; z3 are the observer output, β1 ; β2 ; β3 are the observer gains, e is the observer error, and b0 is a constant that is approximated to be b. Specially, fal is a nonlinear function proposed by Han [1] and defined as ( e jej r δ 1 α ¼ ke f alðe; α; δÞ ¼ δ α : ð4Þ jej signðeÞ jej 4 δ This indicates that the variables of fe1 and fe2 reflect and quantitate the nonlinear feedback mechanism of ADRC framework. Obviously, fal function is a continuous power function that is linear

Fig. 2. Configuration of the developed nonlinear ADRC for a two-order fast linear actuator.



4

function is symmetric over the four quarters of a period, computation of describing function is done in one quarter. Consider a sinusoidal input eðtÞ ¼ E sin ðωtÞ. If E r δ, then the input remains in the linear range. Therefore, the gain is simply a α1 constant k ¼ δ . If E 4 δ, the output falls into the range of nonlinear exponent. Note that the fal function with α ¼ 0:5 and 0.25 are used in Eq. (3). In above two cases, it is difficult to obtain the analytical solutions to the describing function. Therefore, the four-term Taylor series are employed to approximate the integral values in two cases, respectively. After deduction and simplification, two describing functions of the fal nonlinearity when α ¼ 0:5 and 0.25 are derived and computed according to qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 þ π E20:5 2 π2 τ : N 1 ðEÞ ¼ 2kπ1 τ δE 1 δE 3 1 π 5 7 1π 19 π τ þ τ τ ; 2 2 16 2 6720 2 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 2 2k2 4 δ δ 5 2 π þ 0:75 2 τ τ 1 N 2 ðEÞ ¼ π E E 2 πE 7 3 5 5 π 7 π 31 π τ þ τ τ : ð7Þ 12 2 192 2 32; 256 2 and k2 ¼ 1=δ . Note that this where τ ¼ arcsinðδ=EÞ, k1 ¼ 1=δ result is valid only for E Z δ. For E o δ, N1 ðEÞ ¼ k1 and N2 ðEÞ ¼ k2 , which are the linear gains. It is worthwhile to observe that the describing functions of the fal nonlinearity in Eq. (7) depend only on the input amplitude. The normalized describing functions of N 1 ðEÞ=k1 and N 2 ðEÞ=k2 are shown in Fig. 3 as a function of E=δ, in which δ ¼ 0:1. 0:5

0:75

3.2. Characterization of the nonlinear ADRC in frequency domain By converting the nonlinear ADRC algorithm to the frequency domain using the Laplace transform and describing function, it is possible to obtain from Eqs. (3), (5) and (6) 8 EðsÞ ¼ Z 1 ðsÞ YðsÞ > > > > > Z 1 ðsÞs ¼ Z 2 ðsÞ β1 EðsÞ > < Z 2 ðsÞs ¼ Z 3 ðsÞ β2 N1 ðEÞ þb0 UðsÞ ; ð8Þ > > > Z 3 ðsÞs ¼ β 3 N 2 ðEÞ > > > : b UðsÞ ¼ ðk þ k sÞRðsÞ k Z ðsÞ k Z ðsÞ Z ðsÞ 0

p

d

p 1

d 2

3

where s is the Laplace variable, Z 1 ðsÞ, Z 2 ðsÞ and Z 3 ðsÞ are the outputs of the ESO, EðsÞ is the observer error, YðsÞ is the output of the plant, and UðsÞ is the control input. All of above are in the frequency domain. N 1 ðEÞ and N 2 ðEÞ are the describing functions of the fal nonlinearities with different values of α and an amplitude of EðsÞ as the input. After deduction and simplification, the system in Fig. 1 is changed into a block diagram description, shown in Fig. 4. In Fig. 4, r is the reference input. Gr1 ðsÞ and Gr2 ðsÞ represent regulations of the reference input using the ADRC. Gf ðsÞ describes feedback action of the output Y. G1 ðsÞ and G2 ðsÞ, as well as N 1 ðEÞ

Fig. 4. Block diagram of the nonlinear ADRC with two nonlinearities in the forms of the transfer function and describing function.

Fig. 5. Simplified block diagram for study of limit cycle behavior.

and N 2 ðEÞ denote the effect of observer error, e on the observer outputs. G3 ðsÞ is an integrated description including the plant for simplicity in the next analysis. All of the blocks are derived by the transfer function description listed in the Appendix. The describing function of two nonlinearities, N 1 ðEÞ and N2 ðEÞ are described by Eq. (7). In fact, there is close relation between Figs. 2 and 4 because they describe the developed active disturbance rejection controller in different ways. Fig. 2 gives a simplified description of the controller in the time domain in order to represent the relations among different signals. But Fig. 4 describes the controller in the form of block diagram of frequency domain. Notice from Fig. 4 that the developed ADRC for the secondorder plant is an autonomous, nonlinear system with two nonlinearities. By using the describing function method, it becomes possible to analyze the dynamic behavior of the system in the frequency domain. 3.3. Treatment of two nonlinearities Fig. 4 illustrates a two-nonlinearity closed-loop system. To study single-frequency limit cycle behavior, it is necessary to transform it to a single nonlinearity system and assume no input, resulting in a simplified block diagram, as depicted in Fig. 5. In this figure, Nðx1 ; ωÞ is an equivalent single nonlinearity describing function for the dashed box shown in Fig. 4. It is observed that even if both N1 ðEÞ and N 2 ðEÞ are only amplitude-dependent, Nðx1 ; ωÞ will be both amplitude- and frequency-dependent because of the presumed frequency dependence of G1 ðjωÞ and G2 ðjωÞ. To obtain an accurate Nðx1 ; ωÞ, it is needed to determine the actual first harmonic in x2 when x1 is a pure sinusoid. Assume the input of the equivalent nonlinearity block Nðx1 ; ωÞ to be x1 ðtÞ ¼ A sin ðωtÞ:

ð9Þ

As shown in Fig. 4, the input enters the block N 1 ðEÞ and G1 ðsÞ through the feedback channels and leaves it with changes in both the amplitude and phase at the steady state, denoted as the observer error e, and represented by eðtÞ ¼ E sin ðωt þ θÞ:

Fig. 3. Describing function of the fal nonlinearity in two cases of α ¼ 0:5 and 0.25.

ð10Þ

After deduction, the amplitude and phase of the signal, e(t) are derived as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E ¼ A= 1 þ 2a1 N 1 ðEÞ cos φ1 þ a1 2 N1 ðEÞ2 ; ð11Þ



θ ¼ arcsin

N 1 ðEÞa1 E sin φ1 ; A

ð12Þ

where a1 and φ1 are the amplitude and phase angle of G1 ðsÞ, respectively. Next work is to seek solutions of Eqs. (11) and (12). Note that when the input signal x1 represented by Eq. (9) is given, both A and ω are determined, resulting in both a1 and φ1 known. Since Eq. (11) is an implicit equation, the iteration algorithm is proposed to calculate the value of E. First, a desired value of A is set and an initial value of E is given. Then the initial value of E is used to compute A according to Eq. (11). It is required to compute repeatedly by changing the value of E until the corresponding value of A approaches the desired value. To obtain good approximation, relative computation accuracy is set as one ten thousandth. After E is determined, it is easily to get the value of θ by using Eq. (12). Next, the signal e(t) continues along N 2 ðEÞ and G2 ðsÞ in serial, as well as one unit feed-forward in parallel. The resulting steady state output signal x2 is expressed as x2 ðtÞ ¼ B sin ðωt þ ΦÞ:

ð13Þ

If define the amplitude and phase angle of G2 ðsÞ as a2 and φ2 , one can obtain from Fig. 4

N ðEÞa2 sin ðθ þ φ2 Þ sin θ ; ΦðA; ωÞ ¼ arctan 2 N 2 ðEÞa2 cos ðθ þ φ2 Þ cos θ BðA; ωÞ ¼

EN 2 ðEÞa2 sin ðθ þ φ2 Þ E sin θ : sin ðΦðA; ωÞÞ

ð14Þ

This indicates that both B and Φ are frequency- and amplitudedependent on the input signal. When Eqs. (7), (11) and (12) are substituted into Eq. (14), the describing function of the equivalent nonlinearity Nðx1 ; ωÞ, namely NðA; ωÞ can be derived as NðA; ωÞ ¼ KðA; ωÞejΦðA;ωÞ ¼

BðA; ωÞ jΦðA;ωÞ e ; A

ð15Þ

where KðA; ωÞ and ΦðA; ωÞ are the amplitude and phase of the describing function NðA; ωÞ. In addition, both N1 ðEÞ and N 2 ðEÞ are related with the fal nonlinearity parameter δ, which makes NðA; ωÞ dependent on δ.

4. Limit cycles analysis 4.1. Graphical limit cycle determination Consider the nonlinear system as represented by Fig. 4, which is a quasi-linearized system. To validate a limit cycle study, the nonlinear element Nðx1 ; ωÞ is characterized by its describing function in Eq. (15), and the linear element by its resulting frequency response function LðjωÞ that is represented by LðjωÞ ¼ G3 ðjωÞGf ðjωÞ:

ð16Þ

Clearly, requirements for the describing function method to be applicable are that LðjωÞ is low-pass filter and no resonant peaks occur in its frequency response, so that the periodic input to the nonlinearity block is nearly sinusoidal. Obviously, both G3 ðjωÞ and Gf ðjωÞ depicted in the appendix are strictly proper transfer functions whose numerators and denominators are polynomials without common factors and the relative order of LðjωÞ is firstorder. Linear theory is now applied to the above quasi-linearized system, whose characteristic equation is 1 þ NðA; ωÞLðjωÞ ¼ 0;

ð17Þ

5

which can be written as a particularly useful form of LðjωÞ ¼

1 : NðA; ωÞ

ð18Þ

Solutions of this equation yield the amplitudes and frequencies of the loop limit cycles. Certainly, if the above equation has no solutions, then the nonlinear system has no limit cycles. The usual solution is to find the intersection points between two curves of LðjωÞ and 1=NðA; ωÞ in the complex plane. However, the describing function NðA; ωÞ depends on both input amplitude and frequency. Consequently, there are generally an infinite number of intersection points between the LðjωÞ curve and the 1=NðA; ωÞ curves. Only the intersection points with matched frequency ω indicate limit cycles. To avoid the complexity of matching frequencies at intersection points, it may be simple to consider the graphical solution of Eq. (17) directly, based on the plots of NðA; ωÞLðjωÞ. With A fixed and ω varying from 0 to 1, a curve representing NðA; ωÞLðjωÞ can be obtained. Different values of A correspond to a family of curves. If there is a curve passing through the point ( 1,0) in the complex plane, a limit cycle exists. The amplitude of the limit cycle is the value of A for the curve, and the frequency of the limit cycle is the value of ω at the point ( 1,0). Obviously, this method is more straightforward and only requires repetitive computation of the NðA; ωÞLðjωÞ to generate the family of curves, which may be handled easily by computers. 4.2. Limit cycle determination Consider the fast tool servo control example in Section 2. First, it is required to tune the controller parameters. In the ADRC algorithm, only two parameters that are the observer and controller bandwidths are to be tuned. Generally, higher bandwidths result in better performance of tracking and disturbance rejection, which inversely increase the ripple threshold for the control signal due to the presence of sensor noise. Therefore, the tuning parameters should reach a tradeoff between the bandwidths and the control signal ripple that does not significantly impair actuator activity. In the control example, the sampling control step h is 4 μs and the required controller bandwidth is more than 10 kHz. Referring to previous studies [2,3], the observer and controller bandwidths are tuned to ω0 ¼ 1=ð3hÞ ¼83.3 krad/s and ωc ¼ 0:5ω0 ¼ 41.7 krad/s. The parameters are thus achieved as β1 ¼ 3ω0 , β2 ¼ 3ω0 2 , β3 ¼ ω0 3 , kp ¼ ωc 2 and kd ¼ 2ωc . Fig. 6 shows two families of curves NðA; ωÞLðjωÞ, in which the fal nonlinearity parameter δ is 0.01 and 0.5, respectively. It is observed that the family of curves approach the point ( 1,0) gradually with the increase of the values of A in two cases. In spite of this, the two families of curves don't go through the point ( 1,0), indicating no limit cycles occurs. Another observation in Fig. 6a is that the two curves NðA; ωÞLðjωÞ coincide in low frequency range at A¼ 0.01 and 0.1. This is due to the values of E are smaller than δ within the low frequency range in two cases, leading to the nonlinear function falling into the linear range completely. The similar phenomenon may be also observed in two cases of A¼0.1 and 1 in Fig. 6b. Especially, when A ¼0.01 and 0.1, corresponding two curves NðA; ωÞLðjωÞ completely coincide throughout frequency range. This can be explained by the fact that all values of E in whole frequency range make the nonlinear function in linear range. In such a condition, the describing function NðA; ωÞ may be considered as an accurate transfer function representation independent on the input amplitude. The ESO parameters play remarkable roles on the controller performance. To study the effects of these parameters on the limit cycles of the nonlinear fast tool servo system, the NðA; ωÞLðjωÞ curve family with various values of β 2 and β3 are plotted in Fig. 7



6

2

Imag Axis

N ( A, ω ) L( jω ) 0

A=0.01 A=0.1 A=1 A=10 A=100 A=1000

-2

-4

ω

-1

-0.8 -0.6 Real Axis

-0.4

-0.2

0

2

Imag Axis

N ( A, ω ) L( jω ) 0

ω

-2

-4

A=0.01 A=0.1 A=1 A=10 A=100 A=1000

-1

-0.8

-0.6 -0.4 Real Axis

-0.2

0

Fig. 6. Graphic limit cycles determination in terms of polar plot representations: (a) δ ¼ 0:01 and (b) δ ¼ 0:5.

Fig. 7. Effects of ESO parameters on the limit cycles: (a) varying β2 and (b) varying β3 .

where A¼1, δ ¼ 0:01, β2 ¼ 3ω0 2 and β 3 ¼ ω0 3 . Additionally, suppose that the plant is not subject to parameter variations, so that the p, q and b are fixed at their respective nominal values. It is noted from Fig. 7a that the NðA; ωÞLðjωÞ curve moves left gradually and even encircles the point ( 1,0) when β2 is decreased. This means that there must be a value β2 make the NðA; ωÞLðjωÞ curve pass through the point ( 1,0) and thus limit cycles may occur. The same observation is easily obtained in Fig. 7b. Increasing β3 will increase the possibility of limit cycles. This also implies that the limit cycles may be suppressed efficiently if β2 and β3 are tuned carefully. To demonstrate the results shown in Fig. 7, it is necessary to explore the time-domain tracking performance in order to determine whether the system will converge to the desired position under the condition where there is no limit cycle. As a result, the parameters with A¼1, δ ¼ 0:01, β 1 ¼ 3ω0 , β2 ¼ 3ω0 2 and β 3 ¼ ω0 3 are adopted to ensure the system without limit cycle. Fig. 8 show that both the actual position and velocity signals track the desired ones very well.

From the view of limit cycle stability analysis, it is noted from Fig. 10 that the points near the intersection P and along the increasing A side of the curve 1=NðA; ωÞ are not encircled by the curve LðjωÞ. It deduces that the limit cycle is stable according to the limit cycle criterion [4]. In order to demonstrate the limit cycle prediction in terms of describing function method, a time-domain simulation of the nonlinear ADRC fast tool servo system is necessary. By using the software of MATLAB, the simulations are performed and the results are given in Fig. 11. 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 1.606 mm and 144,511 rad/s. The third-harmonic amplitude is only 1 percent of the fundamental at the system output. Note that there is difference in both amplitude and frequency between the results of nonlinear simulation and the proposed describing function analysis. The inaccuracy of the limit cycle prediction is obvious and perhaps to be expected, due to the approximate nature of the describing function method itself. In fact, the inaccuracy of the predicted amplitude and frequency of a limit cycle depends on how well the nonlinear system satisfies the assumptions of the describing function method. On the one hand, satisfaction of the low-pass filter hypothesis requirement is the keystone of describing function success. An interesting result is that as the order of the linear portion of the system increases, the error in describing function application is reduced. In this case, the linear element LðjωÞ is a fourth-order system and the relative order is first-order, which has good lowpass filter performance. On the other hand, graphical conditions are necessary. From Fig. 10, it is observed that the linear element LðjωÞ and the nonlinear element 1=NðA; ωÞ loci are almost tangent, resulting in an obvious prediction error from the describing function analysis. This is because the effect of neglected higher harmonics may cause the change of the intersection situations, particularly when filtering in the linear element is weak. Therefore, how to estimate and improve the accuracy of the describing function approximation may be further work and worthy of mention. It must consider the residual harmonics of the nonlinear output ignored in the actual describing function development. In

4.3. Stability analysis of limit cycles Consider a specified situation in which δ ¼ 0:01, β3 ¼ 10ω0 3 , and other parameters are as same as those in Section 4.2. Such parameters are so chosen that a nonlinear system with limit cycles may be yielded. Fig. 9 illustrates a family of NðA; ωÞLðjωÞcurves on the complex plane with ω as running parameter and A fixed for each curve. The results show that when A is 1.548 mm, the NðA; ωÞLðjωÞ curve just passes through the point ( 1,0), indicating the existence of a limit cycle. The value of ω at the point ( 1,0) is 144,460 rad/s that is the frequency of the limit cycle. To determine the stability of the limit cycle, the plots of frequency response LðjωÞ and inverse describing function 1=NðA; ωÞ when ω ¼ 144; 460 rad=s are given in Fig. 10. There is one intersection point P in this figure, predicting that the system has one limit cycle. The frequency of the indicated limit cycle is read to be 144,460 rad/s from the calibrated LðjωÞ locus and its amplitude is read to be 1.548 mm from the calibrated 1=NðA; ωÞ locus.



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Fig.10. Graphic limit cycle stability analysis.

Fig. 11. The limit cycle behavior of the nonlinear ADRC fast tool servo system obtained by the time-domain simulation.

addition, the filtering influence of the linear elements must be assessed. These harmonics, along with the linear elements determine whether, and to what degree, describing function solution of a problem will be successful.

5. Conclusions

Fig. 8. Difference between the desired and actual signals: (a) the desired and actual positions, (b) the output z1of the ESO and the actual position, (c) the desired and actual velocity.

Fig. 9. A family of NðA; ωÞLðjωÞ curves for determination of limit cycles.

The control technique to achieve higher tracking motion accuracy has always been an active research topic. The controller design should take into account the effects of uncertainties coming from the inertial load, friction, cogging, force ripple, and electrical parameters as well as nonlinear or unknown external disturbances that arise in general motion control systems. This leads to various approaches to deal with the uncertainties. As a novel and practical approach, ADRC behaves excellently in many applications, which is thus regarded as a new feedback control design paradigm. This paper developed an active disturbance rejection controller with fal nonlinearities for the second-order fast tool servo to achieve better tracking and disturbance rejection performances. To analyze the limit cycles behavior of the system with two fal nonlinearities, the classical transfer function and describing function techniques are employed to characterize the nonlinear system in the frequency domain. A time domain iteration algorithm is presented to transfer the two-nonlinearity system to a single nonlinearity system. This makes it possible obtain a more accurate solution to studying single frequency limit cycle behavior. The analysis results show that the nonlinear parameter and the extended state observer gains have important influences on the limit cycle behavior. If these parameters are tuned carefully, the limit cycle can be suppressed efficiently. The time domain simulation supports the conclusion, which demonstrates the applicability of the approach for studying the limit cycles. However, due to inherent approximation of the describing function technique, the inaccuracy of prediction of limit cycles is considerable and not surprising. Therefore, further work is to explore the accuracy problems and quantitative application



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conditions while using the describing function technique. Moreover, other methods, such as the Lyapunov method will be considered to study the stability of the nonlinear ADRC systems. AS a new design paradigm, ADRC is unavoidably faced with obvious challenges. Future work on both theoretical study and practical applications is in progress to make ADRC mature and perfect. Acknowledgements The authors gratefully acknowledge financial support from the National Nature Science Foundation of China under Grant no. 51175278 and the State Key Laboratory of Tribology of China under Grant no. SKLT12C02. Appendix The transfer function description of the blocks in Fig. 3. Gr1 ðsÞ ¼

kd s þ kp s2 þ ðkd þ β1 Þs þ kd β 1 þ kp

Gr2 ðsÞ ¼

kd s þ kp kd s þðkd β 1 þkp Þ

G1 ðsÞ ¼ G2 ðsÞ ¼

β2

s2 þ ðkd þ β 1 Þs þ kd β1 þ kp

β3

kd s2 þ ðkd β1 þ kp Þs

G3 ðsÞ ¼

bkd s þ bðkd β1 þ kp Þ b0 s2 þ ðb0 p þ bkd Þs þ ðb0 q þbkp Þ

Gf ðsÞ ¼

s2 þkd s þ kp s2 þðkd þ β 1 Þs þ kd β1 þ kp

References [1] Han J. Active disturbance rejection control technique—the technique for estimating and compensating the uncertainties. National Defense Industry Press; 2008 (in Chinese). [2] Han J. From PID to active disturbance rejection control. IEEE Trans Ind Electron 2009;56(3):900–6. [3] Gao Z. Active disturbance rejection control: a paradigm shift in feedback control system design. In: Proceedings of American Control Conference; 2006. p. 2399–405.

[4] Slotine J-J E, Li W. Applied nonlinear control. Prentice-Hall; 1991. [5] Talole SE, Kolhe JP, Phadke SB. Extended-state-observer-based control of flexible-joint system with experimental validation. IEEE Trans Ind Electron 2010;57(4):1411–9. [6] Zheng Q, Dong L, Lee DH, Gao Z. Active disturbance rejection control for MEMS gyroscopes. IEEE Trans Control Syst Technol 2009;17(6):1432–8. [7] Liu H, Li S. Speed control for PMSM servo system using predictive functional control and extended state observer. IEEE Trans Ind Electron 2012;59(2): 1171–83. [8] Vincent J, Morris D, Usher N, Gao Z, Zhao S, Nicoletti A, Zheng Q. On active disturbance rejection based control design for superconducting RF cavities. Nucl Instrum Methods Phys Res, Sect A 2011;643(1):11–6. [9] Wu D, Chen K. Frequency domain analysis of nonlinear active disturbance rejection control via the describing function method. IEEE Trans Ind Electron 2013. [10] Altintas Y, Weck M. Chatter stability of metal cutting and grinding. CIRP Ann. Manuf. Technol. 2004;53:619–42. [11] Perng JW. Describing function analysis of uncertain fuzzy vehicle control systems. Neural Comput Appl 2012;21(3):555–63. [12] Kim MS, Chung SC. Friction identification of ball-screw driven servomechanisms through the limit cycle analysis. Mechatronics 2006;16:131–40. [13] Boudy F, Durox D, Schuller T, Jomaas G, Candel S. Describing function analysis of limit cycles in a multiple flame combustor. J Eng Gas Turb Power 2011;133 (6):061502. [14] Wang YJ. Characterization and quenching of friction-induced limit cycles of electro-hydraulic servovalve control systems with transport delay. ISA Trans 2010;49(4):489–500. [15] Boikoy I, Castellanosz MI, Fridman L. Describing function analysis of secondorder sliding mode observers. Int J Syst Sci 2007;38(10):817–24. [16] Valentina L, Michele B, Marco G. On the application of the describing function technique to the bifurcation analysis of nonlinear systems. IEEE Trans Circuits Syst Express Briefs 2007;54(4):343–7. [17] Attari M, Haeri M, Tavazoei MS. Analysis of a fractional order Van der Pol-like oscillator via describing function method. Nonlinear Dyn 2010;61(1-2): 265–74. [18] Wu D, Xie X, Zhou S. Design of a normal stress electromagnetic fast linear actuator. IEEE Trans Magn 2010;46(4):1007–14. [19] Wu D, Zhou S, Xie X. Design and control of an electromagnetic fast tool servo with high bandwidth. IET Electron Power Appl 2011;5(2):217–23. [20] Tsao TC. Optimal feed-forward digital tracking controller design. ASME J Dyn Syst, Meas Control 1994;116(4):583–92. [21] Tomizuka M, Chen MS, Renn S, Tsao TC. Tool positioning for noncircular cutting with lathe. ASME J Dyn Syst, Meas Control 1987;109(2):176–9. [22] Makoto I, Kenta S, Yoshihiro M. High-precision motion control techniques: a promising approach to improving motion performance. IEEE Ind Electron Mag 2012;6(1):32–40. [23] Kobayashi H, Katsura S, Ohnishi K. An analysis of parameter variations of disturbance observer for motion control. IEEE Trans Ind Electron 2007;54 (6):3413–21. [24] Bang JS, Shim H, Park SK, Seo JH. Robust tracking and vibration suppression for a two-inertia system by combining backstepping approach with disturbance observer. IEEE Trans Ind Electron 2010;57(9):3197–206. [25] Yang X, Huang Y. Capabilities of extended state observer for estimating uncertainties. In: Proceeding of American Control Conference Jun.; 2009. pp. 3700–3705. [26] Yi Huang, Wenchao Xue. Active disturbance rejection control: methodology, applications and theoretical analysis. J Syst Sci Math Sci 2012;32(10): 1287–307.


Active disturbance rejection control: methodology and theoretical analysis.

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