Saturated Output Regulation Approach for Active ... - IEEE Xplore

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Saturated Output Regulation Approach for Active Vibration Control of Thin-Walled Flexible Workpieces With Voice Coil Actuators Hai-Tao Zhang, Senior Member, IEEE, Zhiyong Chen, Senior Member, IEEE, Puwei Chen, Student Member, IEEE, Xiaoming Zhang, Member, IEEE, and Han Ding, Senior Member, IEEE

Abstract—Thin-walled flexible workpieces are known to be the most commonly used flexible elements in mechanical structures and machines in the industries of aerospace, national defense, petrochemistry, and so on. Workpiece machining vibrations induced by machining tools greatly worsen the efficiency and accuracy of milling processes, and hence, vibration attenuation has become a bottleneck for improving the machining quality of thin-walled flexible workpieces. An active vibration control system is hereby established to attenuate vibration by using laser displacement detectors and voice coil motors (VCMs). In this paper, we reveal that the vibration attenuation problem can be theoretically formulated as an output regulation problem subject to saturated actuation of VCMs. Within the formulation, a saturated output regulation algorithm is successfully implemented and its effectiveness is extensively examined by experiments. Index Terms—Manufacturing automation, motion control, output regulation, vibration control.

I. INTRODUCTION HIN-WALLED workpieces are known to be the most commonly used flexible elements in mechanical structures and machines such as aircrafts [1], vehicles [2], [3], and integrated circuit (IC) [4]. Instability of flexible workpieces is usually

T

Manuscript received November 14, 2013; revised April 11, 2015; accepted May 23, 2015. Date of publication June 2, 2015; date of current version February 12, 2016. Recommended by Technical Editor M. Iwasaki. This work was supported by the National Natural Science Foundation of China (NNSFC) under Grants 61322304, 51328501, and 51120155001, the Natural Science Foundation of Hubei Province under Grant 2012FFA009, the Research Fund for the Doctoral Program of Higher Education of China under Grant 20130142110050, and the National Science and Technology Major Projects under Grant 2012ZX04001-012-01-05. (Corresponding author: Zhiyong Chen). H.-T. Zhang is with the School of Automation and the State Key Laboratory of Digital Manufacturing Equipment and Technology, the Key Laboratory of Image Processing and Intelligent Control, Huazhong University of Science and Technology, Wuhan 430074, China (e-mail: [email protected]). Z. Chen is with the School of Electrical Engineering and Computer Science, The University of Newcastle, Callaghan, N.S.W. 2308, Australia (e-mail: [email protected]). P. Chen, X. Zhang, and H. Ding are with the School of Mechanical Science and Engineering and the State Key Laboratory of Digital Manufacturing Equipment and Technology, Huazhong University of Science and Technology, Wuhan 430074, China (e-mail: [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMECH.2015.2440425

caused by external dynamic excitations and random cyclic loads. With the development of manufacturing industry, workpieces become thinner and thus vulnerable to vibrations. Meanwhile, periodic disturbances occur in many engineering control applications, commonly in rotating machinery. So, external machining forces, like cutting, milling, and drilling, easily induce vibrations of workpieces. These unexpected vibrations lead to long-time fatigue and greatly worsen the machining quality of workpieces. Thereby, it is an interesting yet challenging task to develop an effective control method to attenuate vibrations so as to ensure the stability and reliability of thin-walled workpiece machining systems. Some typical examples are reported below. Panels and shells are two main parts of vehicle bodies, which induce most of the vibrations when being excited. Thin-walled pieces, e.g., chassises, hoods, hatchback, and roof panels, of vehicles often experience resonance vibrations when moving on rough roads or being exited by dynamic loads. So, it is desirable if some niched actuators could be embedded in suspension systems or other vehicle modules to attenuate the vibrations. Thereby, a rectangular thin-walled workpiece mounted onto a board was used as a test platform to investigate the vibration dynamics and to design vibration attenuation controllers for panels and shells of vehicle bodies [2]. Similar research on active vibration control (AVC) can be found in [3] for vehicle roof panels as well. As the second example, it was reported in [4] that two voice coil motors (VCMs) are used to excite a thin-walled square metal workpiece mounted on a pallet and to reject the vibrations, respectively. This work in [4] aims at developing a VCM-based vibration controller for IC production machinery and measuring devices sensitivity to vibrations. Similar VCM-based AVC for a thin-walled workpiece supported by four mounts is referred to in [5]. The third example can be found in aeronautical and space structures that often present thin walls containing fuel or other liquids [1]. Five piezoelectric motors (PZMs) were used as actuators to attenuate the fluid-induced vibrations on a thin-walled rectangular aluminum plate. The fourth example can be referred to the vibration control implementation [6] for space vehicles during the ascent phase suffered by intense acoustic fields and mechanically transmitted disturbances. Active damping for the vehicle’s plate structure is achieved by accelerometers and piezoelectric actuators embedded with an H∞ control algorithm.

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ZHANG et al.: ACTIVE VIBRATION CONTROL OF THIN-WALLED FLEXIBLE WORKPIECES

In the literature, the vibration attenuation control problem has been studied using either passive or active control approaches. The passive vibration control approach mainly focuses on mounting passive units on a structure in order to change its dynamic characteristics such as stiffness and/or damping coefficient. However, passive control usually increases the weight of structure, which restricts its further applications [7]. In particular, this approach is efficient at high frequencies but tends to be expensive and bulky at low frequencies [7]–[9], and the effectiveness will be drastically reduced by operation frequency changes of a machining system [10]. Although some recent passive controllers using PZMs connecting to resonant passive electric circuits [11] are light enough, they cannot be used for a sufficiently broad range of frequencies due to the limitation on its internal dynamics. By contrast, the AVC approach is a more prospective approach that uses intelligent actuators (like electrostrictive executors or PZMs) to generate desired forces that absorb the energy caused by unexpected vibrations [9]. By properly tuning the closedloop system dynamics, an AVC design is able to improve the stability region of thin-walled piece machining systems and hence achieve higher machining efficiency. These years have witnessed increasing investigations on AVC for flexible thinwalled pieces. Some representative works are discussed below. Zhang and Sims [12] used an active damping method to attenuate vibrations. Nagaya and Yamazaki [13] presented a method of driving the tool or workpiece move along the opposite direction of the cutting vibrations to directly attenuate the vibrations especially in low frequencies. Jenifene [7] proposed an AVC method for lightly damped dynamic systems, where a delayed position feedback signal is used to design a closed-loop controller. Kar et al. [14] and Zhang et al. [15] applied H∞ robust control approaches for flexible plate structures, which effectively suppress the low-frequency vibrations caused by external disturbances. Tokhi and Hossain [9] and Darus [16] proposed active adaptive control approaches for a flexible beam and a square thin plate, respectively. In both methods, some feedforward control structures are included to enable a precancellation of vibrations at operational points. Tavakolpour et al. [8] proposed a finite-difference model-based AVC approach, where an effective vibration suppression capability is achieved using a piezoelectric actuator embedded with a self-learning feedback controller. Rodriguez-Fortun et al. [17] developed an AVC system using piezoelectric actuator for metrological devices affected by low external loads. The merits of the work lie in admitting a mathematical description for differential flatness. Schultz and Ueda [18] designed a switching control method based on the measurement of resonant frequencies. This approach can maximize the vibration suppression effect by forcing the root of the residual oscillation function to be at a resonance. Jung et al. [19] proposed a receding horizon control method to improve the compression effect of flexible beam structures subject to dynamic loads. However, most of the aforementioned AVC designs have yet to consider expanding the closed-loop stability region for a vibration attenuation system. It motivates an interesting problem of effectively attenuating vibrations with sufficiently large operation range for a thin-walled workpiece machining system

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subject to input saturation and system uncertainties. Actually, some efforts [20]–[25] have devoted to theoretical research on periodic disturbance rejection control with various system constraints. A fuzzy model-based controller was designed in [20] to reject periodical vibrations, with prescribed H∞ performance, in active suspension systems. An improved linear quadratic regulator (LQR) control scheme with input constraints was developed in [21] to increase the closed-loop damping ratio of composite shell structures. An adaptive algorithm [26] is developed based on Fourier series analysis to deal with the so-called regenerative cutting force which causes chatters in machining processes. The model-predictive control (MPC) method [27], [28] was also used to reject vibrations and active noise in, e.g., [22] and [23], with input constraints taken into account. In particular, the MPC design is able to substantially attenuate unexpected vibrations at high sampling rate [29]. Saturated output regulation is another interesting approach for periodic disturbance rejection with input saturations and system uncertainties. It has been systematically studied in the literature (see, e.g. [24], [25]). In this paper, we aim to show that a novel AVC approach based on saturated output regulation theory is able to substantially attenuate vibrations for a thin-walled workpiece machining system using VCMs. More specifically, we first explain how the vibration attenuation problem can be formulated as a linear output regulation problem, which mainly deals with reference tracking and disturbance rejection. Within the framework of output regulation, a kind of control algorithm is discussed. The algorithm, on one hand, can theoretically guarantee exact vibration attenuation; on the other hand, it requires strict technical assumptions and involves complicated theoretical calculation. Therefore, it is critical to verify its practical effectiveness using real experiments. For this purpose, extensive experiments are designed and conducted and the results verify the practical effectiveness in terms of vibration attenuation performance with input saturation. Moreover, the robust performance of the algorithm is also examined, in particular, on model mismatch, parameter drift, frequency variation, etc. Overall, the paper seeks to pave the way from the output regulation control theory to its practical implementation on machining processes of thin-walled workpieces. This paper is organized as follows. In Section II, the complete system architecture is depicted including sensors, actuators, signal processing modules, and communication hardware. The saturated output regulation control algorithm is discussed in Section III. In Section IV, the effectiveness of the control algorithm, embedded in an FPGA, is examined by extensive vibration attenuation experiments on a thin-walled workpiece. Finally, conclusions are drawn in Section V. II. ARCHITECTURE OF AN AVC SYSTEM The architecture of a closed-loop AVC system for a thinwalled aluminum alloy flexible workpiece is illustrated in Fig. 1. The unexpected regenerating vibrations (or disturbances) originated by milling cutters are simulated by VCM 2, which is driven by an input voltage via a National Instruments (NI) PXI-6733 card and Amplifier 2. Vibration displacements are detected by a laser displacement sensor placed underneath the workpiece,

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Fig. 2.

Fig. 1. Architecture of the AVC system for a thin-walled flexible workpiece.

which are afterwards fed into an FPGA analog input module via an NI PXI-7852R card which has eight analog inputs with 16-bit resolution ADCs. The developed AVC algorithm is encoded in the FPGA chip. The control signal is afterwards fed into Amplifier 1 to drive VCM 1. Within this architecture, the closedloop AVC system is thus developed, where active vibrations are generated by VCM 1 to counteract the unexpected vibrations of the workpiece. The locations of VCM 1 (the control actuation point), VCM 2 (the disturbance excitation source), and the laser displacement sensor (the measurement output point) are highlighted in Fig. 1. The system runs on NI LabVIEW for its parallel computing capability on large data processing. The layout of the vibration source, the sensor, and the actuator in Fig. 1 is further interpreted as follows. First, the thin-wall workpiece is fixed by four steel mounts at the four corners to maximize vibration magnitudes. In particular, larger vibration magnitudes are detected anywhere closer to the center. So, the displacement sensor is placed close to the center. Then, the vibration source (VCM 2) and the actuator (VCM 1) are placed symmetrically about the sensor. As the two motors are not placed at the exactly same position, the actuator cannot generate forces to exactly counteract those by the vibration source. Two linear VCMs are used in the system described above, due to their small sizes, high bandwidths, high acceleration, and nonbacklash features [4]. This input–output characteristics of linear VCMs also well match the requirement for the AVC design of thin-walled flexible workpieces. However, it is worth mentioning that, as seen in Fig. 1, both of the VCMs cannot pull the workpiece downwards but only push it upwards by hemisphere tips. In order to enable a “pull” action to facilitate complete control actuation, we exert proper initial offsets on the motors. Specifically, the pushing force offset (e.g., corresponding to 0.8V input voltage of the motor) on the workpiece is illustrated in Fig. 2, which causes 0.2-mm initial upward displacement. In this setup, a downward “pulling” force is thus enabled when the driving voltage of motor decreases under 0.8 V. III. SATURATED OUTPUT REGULATION CONTROL ALGORITHM In this section, we will show that the vibration attenuation problem can be theoretically formulated as an output regulation

Setup of a VCM with initial pushing force offset.

problem subject to saturated actuation of VCMs. Within the formulation, a saturated output regulation algorithm is discussed. First, consider a linear system with exosystem [30]: x˙ = Ax + Bu + P ξ ξ˙ = Ωξ y = Cx + Qξ

(1)

with state x ∈ Rn , input u ∈ U ∈ Rm , output y ∈ Rq , and external disturbance ξ ∈ W ∈ Rr . Here, both U and W are compact convex sets describing the allowable inputs and external disturbances. To facilitate the controller development, we assume that (A, B) is stabilizable and all eigenvalues of Ω are simple and have zero real parts. The state-space matrices A, B, C, P , and Q in (1) are to be identified by experiments. In the AVC system of a thin-walled flexible workpiece, the control input u represents the driving voltage for VCM 1, and the vibration ξ ∈ R2 represents the driving voltage and its derivative for VCM 2. In particular, we assume   0 −ω Ω= (2) ω 0 for a class of sinusoidal vibration sources, where ω is the angular frequency. It is noted that the control law adopted in this paper is effective not only for the current single vibration source and single-actuator scenario but also for multiple sources and multiple-actuator scenarios. In terms of the system (1), the control objective is to regulate the actual output y = Cx + Qξ to zero. From the output regulation analysis, the problem is solvable only if there exist compatible matrices Λ and Φ such that [30] ΦΩ = AΦ + BΛ + P CΦ + Q = 0.

(3)

Suppose the matrices A and Ω have no common eigenvalues. Then, there exists a matrix Ψ ∈ Rn ×r such that − AΨ + ΨΩ = −BΛ.

(4)

Next, we define a new state z = x − Φξ and rewrite the system as z˙ = Az + Bu − BΛξ ξ˙ = Ωξ y = Cz.

(5)

ZHANG et al.: ACTIVE VIBRATION CONTROL OF THIN-WALLED FLEXIBLE WORKPIECES

It should be noted that, in the present setup for thin-walled workpiece vibration control experiments, only the measurement output y = Cz is available for feedback control. It motivates us to construct an observer-based controller under the assumption that the pair    A −BΛ ¯ C) ¯ := (A, , [C 0] 0 Ω is observable. More specifically, the observer takes the following form, with two matrices L1 and L2 to be specified: zˆ˙ = Aˆ z + Bu − BΛξˆ − L1 (y − C zˆ) ˙ ξˆ = Ωξˆ − L2 (y − C zˆ)

What is left is to design the controller u for the composite system composed of (5) and (7) such that limt→∞ y(t) = 0. For this purpose, a kind of controller has been proposed in [25], that is,   ˆ Ψ ξ z ˆ − ϕ 1 (8) u = (1 − ϕ1 )Λξˆ + ϕ2 f ϕ2 ⎧ ⎪ ⎪ ⎪ ⎨ 0, ϕ˙ 1 =

⎪ ⎪ ⎪ ⎩ −γϕ1 , ⎧ ⎪ ⎪ ⎨ 0,

ϕ˙ 2 =

⎪ ⎪ ⎩ −γϕ2 ,

zˆ − ϕ1 Ψξˆ ∈ S\SI ϕ2 zˆ − ϕ1 Ψξˆ if ∈ SI ϕ2 if

zˆ − ϕ1 Ψξ ∈ S\SI or ϕ2 ≤ α ϕ2 zˆ − ϕ1 Ψξˆ if ∈ SI and ϕ2 > α ϕ2 if

ϕ1 (t) = ϕ2 (t) = 1, t ∈ [0, T0 ] for some T0 > 0 and α ∈ (0, 1). The parameters used in the controller are explained as follows. 1) The function f and the sets S and SI : In general, f is a specific continuous nonlinear function to be designed, S is the attraction region of the system v˙ = Av + Bf (v),

(9)

and SI is the initial state set of v(t) which converges to the origin. More precisely, let v = z − Ψξ, which is governed by v˙ = Av + Bu

a continuous nonlinear function f : Rn → U such that the equilibrium v = 0 of the system (9) has a region of attraction S ⊂ C. Moreover, the function f is selected such the two assumptions (see [25]) given below are satisfied upon the generalized version of system (9), i.e., v˙ = Av + Bf (v + η1 ) + η2 .

(10)

and let C be the asymptotically null controllable region of the state v. Here, the null controllable region is defined as the set of state v0 that can be driven to origin asymptotically within a finite time by admissible controls (see [24]). Now, it is ready to select

(11)

A1 : For the system (11) with perturbation η1 ≡ 0, there exist d2 > 0 and a set SI containing the origin in its interior, such that SI is an invariant set for all η2 ∞ ≤ d2 . Moreover, for all v0 ∈ SI , limt→∞ v(t) = 0 if η2 ∞ < d2 and limt→∞ η2 (t) = 0. A2 : For the system (11) with perturbation η2 ≡ 0, there exist a set D0 ⊂ Rn and positive numbers k and d1 such that v ∞ ≤ k max(|v0 |∞ , η1 ∞ )

(6)

where the states zˆ and ξˆ are introduced to estimate z and ξ, respectively. Let the state estimation errors be z˜ = z − zˆ, ξ˜ = ˆ The composite system composed of (5) and (6) rewrites ξ − ξ. as (5) and ⎡ ⎤     z˜˙ z˜ A + L1 C −BΛ ⎣ ⎦ = A˜ , A˜ = . (7) ˙ Ω L2 C ξ˜ ξ˜

with

269

v a ≤ k η1 a , ∀v0 ∈ D0 , η1 ∞ ≤ d1 with ∗ a := lim supt→∞ | ∗ (t)|∞ . 2) The matrix L := [LT1 , LT2 ]T . It is proved in [25] that there exist a matrix L := [LT1 , LT2 ]T in (7) and a set DM such that the system composed of (5) and (7) under the controller ˆ u = f (ˆ z − Ψξ)

(12)

satisfies lim [z(t) − Ψξ(t)] = 0, ∀(z0 , ξ0 , z˜0 , ξ˜0 ) ∈ DM .

t→∞

With the parameters selected in the aforementioned procedure, it is proved in [25] that the system composed of (5) and (7), under the controller (8) with a properly selected γ, satisfies limt→∞ z(t) = 0, and hence, limt→∞ y(t) = 0, for any (z0 , ξ0 , z˜0 , ξ˜0 ) ∈ DM . Moreover, the control input u(t) is always bounded. As a result, the saturated output regulation problem is solved. Next, we will have some discussion about the practical issues in implementing the controller (8). The aforementioned controller (8) can theoretically guarantee exact vibration attenuation under the initial condition (z0 , ξ0 , z˜0 , ξ˜0 ) ∈ DM . Essentially, the initial condition requires the magnitude of disturbance signals and the initial tracking errors be sufficiently small, depending on the size of control saturation. However, this condition is somewhat strict and not easy to analytically calculated and it is usually unavoidably conservative. Therefore, it is critical to verify the practical effectiveness of the controller using real experiments. Rather than analytically calculating the initial condition based on a specified control saturation threshold, we use a more feasible setup in real experiments. More precisely, we first specify the initial values, including the magnitude of disturbance signals and the initial tracking errors, and then test controllers with various saturation thresholds. To test controllers with various saturation thresholds, we make some slight modification on the controller (8) such that the saturation threshold can be explicitly tuned. One method is ˆ i.e., to add a clipping saturation function h on the term Λξ,   zˆ − ϕ1 Ψξˆ ˆ . (13) u = (1 − ϕ1 )h(Λξ) + ϕ2 f ϕ2

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The other method is to add a clipping saturation function h on the final control input, i.e.,    ˆ Ψ ξ z ˆ − ϕ 1 u = h (1 − ϕ1 )Λξˆ + ϕ2 f . (14) ϕ2 Both methods will be examined in experiments. In the aforementioned output regulation framework, it is assumed that the frequency of vibration is exactly known. The next practical issue is the robustness of the controller with respect to frequency variation of the vibration source ξ, which is often encountered in engineering applications. Specifically, the second equation of (1) rewrites ξ˙ = (Ω + ΔΩ)ξ. For instance, if the frequency f in (2) is changed to (1 + β)f with β ∈ R, then ΔΩ = βΩ. Therefore, (4) becomes −A(Ψ+ΔΨ)+(Ψ + ΔΨ)(Ω + ΔΩ) = −B(Λ+ ΔΛ)

(15)

with ΔΛ = B + ΦΔΩ according to (3). Here, the symbol “+” denotes matrix pseudoinverse. In this case, (6) becomes zˆ˙ = Aˆ z + Bu − B(Λ + ΔΛ)ξˆ − L1 (y − C zˆ) ˙ ξˆ = (Ω + ΔΩ)ξˆ − L2 (y − C zˆ).

(16)

Experiments will be designed to verify the controller (8) [or (13)] for the system (16). For vibrations containing multiple frequencies, we first take a dominating frequency (e.g., the resonance frequency of the workpiece obtained by an impact experiment) to design the controller parameters and then apply the aforementioned robustness analysis by treating other frequencies as variations. IV. EXPERIMENTS In this section, we design experiments to verify the effectiveness of the proposed controller by addressing the practical issues raised in the previous section. To collect data for identification, we conduct AVC experiments on the thin-walled aluminum alloy workpiece vibration system as shown in Fig. 1. Throughout the paper, the sampling period is 1 ms. We take f = 10 Hz as a case study. First, as shown in Fig. 3(a), a sinusoidal input voltage ξ0 (t) = 0.15 sin(2πf t)V is fed to VCM 2, which excites vibrations y0 (t) = 0.062 sin(2πf t + φ0 )mm. At the 5.2th second, a sinusoidal input voltage u(t) = [0.2 sin(2πf t + φ1 )]V is fed to VCM 1, which attenuates the vibration magnitude from 0.062 to 0.048 mm. We next use the sup-space method to achieve the model (1) with the following parameters:     −0.0224 0.0479 5.9012 A= , B= −0.0174 −0.0937 −1.7342   0.8374 −3.6405 P = −0.2496 1.1226 C = [0.1241,

0.4240],

Q = [0.5684,

0.1934].

(17)

As shown in the medium panel of Fig. 3(b), the average of the modeling error |y − ym |/ max(y) along the entire process is 0.072%, which is much smaller than the average error 0.23%

Fig. 3. (a) Identification experiments at f = 10 Hz. (b) Identification model output y m (upper graph) and modeling error em := |y − y m |/ max(y) with Q (middle graph) and without Q (bottom graph).

without the direct ξ–y gain matrix Q as shown in the lower panel of Fig. 3(b). To validate the generality of the model (1), we also examined the modeling error at the range of 11–20 Hz. All results show that the model with Q has remarkably smaller modeling error than the one without Q. Next, we carry out experiments to examine the proposed AVC algorithm. The detailed calculation procedure for controller parameters is elaborated below, with f = 10 Hz as a case study. Recall that our objective is to drive the measurement output y = Cz + Qξ to zero while maximizing the stability region for the state v = z − Ψξ. Based on the identified matrices (17), we calculate Λ = [−0.1237, 0.6098], ⎡

Φ=⎣

−0.3614

−0.1230

−1.2348

−0.4201

⎤

⎡

⎦, Ψ = ⎣

0.0573 −0.0168

0.0116 −0.0034

⎤ ⎦

by (3) and (4). For  > 0 ( is generally a small positive number, here we pick  = 0.02), the Riccati Equation AT Π + ΠA − ΠBB T Π + I = 0 has the solution ⎡

Π=⎣

0.0075

0.0028

−0.0028

0.0120

⎤

⎦.

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271

Fig. 4. (a) State invariant sets SI (inner solid ellipse) and S (middle dashed ellipse) and asymptotically null controllable region C (outer dashdotted ellipse); (b) Invariant set Se of the state estimation error e.

Now, the state feedback gain is F = −σB T Π = [−0.0059, 0.0007] with σ selected as 0.15. Afterwards, by using the Kalman filter on the system (6), we design the observer matrices L1 = [0.6186, −0.1714]T × 10−3 and L2 = [−136.49, 39.95]T . In the output feedback controller (13), we set both h and f as a saturation function 0.2sat(·) V with the threshold value of 0.2 V. Moreover, we select controller parameters α = 0.1, γ = 0.0043, d1 = 0.2, and d2 = 0.1 in the following procedure. First, the parameters α and γ should be small positive numbers which determine the aggressiveness of the controller (13). If the control signal oscillates too quickly between the upper and lower bounds when facing large overshoots, α and γ should increase to make the control signal less aggressive. On the other hand, d1 and d2 give upper bounds of state estimation errors of z and ξ, which should be small positive values too. The larger value of d1 and d2 , the larger state estimation errors are allowed. To show the stability region of the closed-loop system, the initial invariant set SI (v T Xv ≤ 1), the closed-loop invariant set S, and the asymptotically null controllable region C are depicted in Fig 4(a). Meanwhile, the invariant set Se of the state estimate ˆ is shown in Fig. 4(b). error e = v − vˆ (with vˆ = zˆ − Ψξ) The output regulation performance of the AVC (13) to the thin-walled workpiece with the parameters designed above is shown in Fig. 5(a) and (b). It was observed that the vibration magnitudes were attenuated by 67% and 78% at the excitation frequencies of 10 Hz [see Fig. 5(a)] and 14 Hz [see Fig. 5(b)], respectively. We also examined the controller on 15 and 17 Hz, and the attenuation rate is 83% and 74%, respectively. In any case, the developed output regulation control method substantially attenuated vibrations as expected. The controller performed best near the resonance frequency of the workpiece around 15 Hz. Note that the vibrations could not be completely eliminated due to their inherent multiple frequency components and modeling errors. Now, we are ready to check the robustness of the present AVC (13) with respect to model uncertainty of the plant. We take the 10-Hz case as an example and deliberately add model mismatch to the identified state matrices A and B, denoted as A and B , respectively. Specifically,  A =

−0.015

0.0479

−0.0174

−0.0937



 , B =

5.5 −1.7

Fig. 5. Closed-loop control performances at 10 Hz (a), 14 Hz (b) , and 10 Hz with model uncertainty (c), with vibration magnitudes reduced by 67%, 78%, and 59%, respectively. The input voltage magnitude of the vibration source VCM 2 keeps at 0.2 V, and the allowable input range [−0.2, 0.2] V for the actuator VCM 1 are marked by dashed lines in the bottom graphs.

which leads to that 

 Π =

0.0084

0.0032

0.0032

−0.01221



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Fig. 6. Vibration attenuation performance with different maximal voltages of VCM 1. In particular, the vibration source VCM 2 is more powerful than the actuator VCM 1.

F = −σ(B )T Π = [−0.0061, 0.0006], L 1 = [0.87, −0.35]T ×10−2 , and L 2 = [−136.4930, 39.9522]T . The corresponding control performance is shown in Fig. 5(c), where the other controller parameters are the same as those in Fig. 5(a). It was observed that the proposed AVC (13) still works with vibration attenuation rate of 59%. The robustness with respect to model uncertainty is thus verified. Next, we examine the performance of the AVC (13) to attenuate more powerful external vibrations generated by VCM 2 at 14 Hz as shown in Fig. 6. We set the maximal voltage of VCM 2 w ¯ = 0.30 V, and the maximal voltage of VCM 1 u ¯ decreases to 0.25 V [see Fig. 6(a)], 0.2 V [see Fig. 6(b)] and ¯ ξ] ¯ and U = [−¯ 0.15 V [see Fig. 6(c)]. Here, ξ ∈ W = [−ξ, u, u ¯]. The associated vibration attenuation rates of Fig. 6(a)–(c) are 62.0%, 42.2%, and 35.8%, respectively. The more detailed relationship between the attenuation rate and the maximal input voltage of VCM 1 is shown in Fig. 6(d). It was observed that the present AVC has substantial vibration attenuation effect if W is moderately larger than U . But if W grows too larger than U , then the present AVC (13) does not work satisfactorily. For instance, if the smallest acceptable attenuate rate is 50%, then the maximal tolerant vibration source power of VCM 2 can be (ξ¯ − u ¯)/¯ u = 37.3% larger than the power of VCM 1. It is of interest to check how initial tracking errors affect the control performance. In realistic industrial applications, it is not easy to exactly set the initial states. Nevertheless, we can use an indirect setup for this purpose. In particular, we fix the maximal voltage of VCM 1 at 1.5 V and change the maximal voltage of VCM 2 w ¯ from 3.0 V down to 2.5 V [see Fig. 7(a)], 2.0 V [see Fig. 7(b)], 1.5 V [see Fig. 7(c)], and 1.0 V. The associated vibration attenuation rates are 24.7%, 44.9%, 60.6%, 76.5%, and 79.6%, respectively [see Fig. 7(d)]. In this setup, VCM 2 is still the vibration source to the system, but it gives different initial tracking errors (0.01, −0.04, and 0 mm in Fig. 7(a)–(c), respectively) at the moment (regarded as the initial time) when the controller is switched ON. Although these initial errors cannot be directly specified, we repeated the experiments 20 times (corresponding to different initial errors) and achieved similar

Fig. 7. Vibration attenuation performance with different maximal voltages of the vibration source VCM 2. Here, the maximal voltage of VCM 1 is fixed at 0.15 V.

performance. It is thus illustrated that the disturbances have more significant influence on control performance than initial tracking errors. To compare the performance of the AVCs (13) and (14) suffered by external impulse perturbations, we exerted two 2.5-N clicks on the upper surface of the workpiece at the 9.5th second and 19.5th second in Fig. 8(a) and (b), respectively. It was observed that: 1) the settling times of the AVC (13) (147 and 98 ms) are 29% shorter than those of the AVC (14) (206 and 139 ms); and 2) the overshoots of the AVC (13) are 12% lower than those of the AVC (14) . That is because the AVC (13) makes better use of the input saturations, and the input boundary activation period is 31% lower than that of the AVC (14). Afterwards, to examine the robustness to constant disturbances, as seen in Fig. 8(c), we moved the laser displacement sensor 0.02 mm downwards twice at the 14th second and the 28th second, respectively. In this way, two constant output disturbances of 0.02 mm were thus introduced, but again, under the proposed controller, the influence settled down in no more than 100 ms. The robustness to external disturbances is thus demonstrated. To show the robustness of the proposed AVC (13) with respect to frequency variance of the vibration source, we compared its performance with a conventional controller based on LQR design shown in Fig. 9(a) and (b). In this experiment, the controllers are designed with the nominal vibration frequency f = 15 Hz, but the real vibration frequencies varied in the sequence of {18 → 17 → 16 → 15 → 14} Hz, which are marked by vertical dashed lines in Fig. 9. In the LQR design, the state feedback gain K = [0.7012, −0.0846] and the observer gain L = [0.0534, −0.9925]T are calculated using the Kalman filter. One of the essential differences between LQR and the proposed AVC method lies in that the former is based on the linear model (1) with Q = 0, while the latter adopts a more accurate Q. It was observed that the vibrations attenuation rate of the proposed AVC (13) is remarkably greater than that of LQR with respect to frequency variations. More specifically, the vibration attenuation rates of AVC and LQR are 65–83% and 36–66%, respectively. In both cases shown in Fig. 9(a) and

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Fig. 8. Click disturbance response comparison of (a) the AVC (13) and (b) the AVC (14). (c) Constant press disturbance response of the AVC (13) at 15 Hz. In this experiment, the allowable input range [−0.3, 0.3]V were applied for the actuator VCM 1, and range [−0.35, 3.5] V were applied for the vibration source VCM 2.

Fig. 9. Robust performance with respect to frequency variations from 18 Hz down to 14 Hz. (a) LQR method. (b) AVC. (c) AVC with frequency feedback.

(b), the best performance happened at the nominal frequency of 15 Hz, which is also the resonance frequency of the workpiece. These frequency variations correspond to ΔΩ = βΩ with Ω given in (2) and −0.06 ≤ β ≤ 0.2. To further investigate the allowable degree of uncertainty, we conducted the experiments with f ∈ [8, 13] ∪ [19, 22] Hz. It was observed that, if the mini-

mal acceptable vibration attenuation rate is 50%, then the largest allowable ΔΩ satisfies ΔΩ ∞ / Ω ∞ = 33.3% with the closest frequencies at f = 10 Hz (attenuate rate 43%) and 20 Hz (attenuate rate 47%). It is also worth mentioning that, as shown in Fig. 9(c), the control performance was further improved when an adaptive law was added to tune the controller parameters

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Fig. 10. Spectra of the displacement signals of AVC and LQR at (a) 10 Hz and (b) 14 Hz.

online based on the real vibration frequency which was assumed measurable for feedback. Finally, the spectra of the displacement signals of Fig. 5(a) and (b), using fast Fourier transform, are shown in Fig. 10. It was observed that the proposed AVC works effectively not only at the designed frequencies, i.e., f = 10 and 14 Hz, but also at other frequencies. The spectra of the displacement signals using LQR are also plotted in Fig. 10 as a comparison. The vibration attenuation superiority of AVC over LQR is thus verified in frequency domain. V. CONCLUSION To attenuate the vibrations of thin-walled workpieces during machining processes, we have established a closed-loop AVC system for thin-walled aluminum alloy workpieces. In this system, laser displacement detectors and VCMs are adopted as the sensors and actuators, respectively. To improve the vibration attenuation performance, we have implemented a saturated output regulation control algorithm with robustness to excitation frequency variations. Extensive closed-loop control experiments have verified the feasibility and advantages of the present AVC design. REFERENCES [1] S. Carra, M. Amabili, R. Ohayon, and P. M. Hutin, “Active vibration control of a thin rectangular plate in air or in contact with water in presence of tonal primary disturbance,” Aerosp. Sci. Technol., vol. 12, pp. 54–61, 2008.

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Puwei Chen (S’12) received the B.E. degree from Fuzhou University, Fuzhou, China, in 2011, and the M.E. degree in mechatronic engineering from the Huazhong University of Science and Technology, Wuhan, China, in 2014. His research interests include active chatter mitigation control in multiaxis machining processes.

Hai-Tao Zhang (M’07–SM’13) received the B.E. and Ph.D. degrees from the University of Science and Technology of China, Hefei, China, in 2000 and 2005, respectively. During January–December 2007, he was a Postdoctoral Researcher with the University of Cambridge, Cambridge, U.K. Since 2005, he has been with the Huazhong University of Science and Technology, Wuhan, China, where he was an Associate Professor from 2005 to 2010 and has been a Full Professor since 2010. His research interests include model predictive control, multiaxis machining, nanomanufacturing control, and multiagent systems control. Dr. Zhang has been an Associate Editor of Nature Scientific Reports and Asian Journal of Control, since 2013, a Conference Editorial Board Member of the IEEE Control Systems Society (CSS) and the Chair of IEEE CSS Wuhan Chapter.

Xiaoming Zhang (M’11) received the B.E. degree from Jilin University, Changchun, China, and the Ph.D. degree from Shanghai Jiaotong University, Shanghai, China, in 2001 and 2009, respectively. During 2010–2011, he was an Alexander von Humboldt Research Fellow with Technische Universitat ¨ Darmstadt, Darmstadt, Germany. Since 2011, he has been with the Huazhong University of Science and Technology, Wuhan, China, where he is currently an Associate Professor of mechanical engineering. His research interests include engineering mechanics and machining dynamics.

Zhiyong Chen (S’03–SM’13) received the B.S. degree from the University of Science and Technology of China, Hefei, China, in 2000, and the M.S. and Ph.D. degrees from the Chinese University of Hong Kong, Shatin, Hong Kong, in 2002 and 2005, respectively. He was as a Research Associate with the University of Virginia, Charlottesville, VA, USA, from 2005 to 2006. He joined the University of Newcastle, Callaghan, N.S.W., Australia, in 2006, where he is currently an Associate Professor. His current research interests include systems and control, networked systems, and mechanical systems. Dr. Chen is an Associate Editor of the IEEE TRANSACTIONS ON AUTOMATIC CONTROL, IEEE TRANSACTIONS ON CYBERNETICS, and International Journal of Robust and Nonlinear Control.

Han Ding (M’97–SM’00) received the Ph.D. degree from the Huazhong University of Science and Technology (HUST), Wuhan, China, in 1989. Supported by the Alexander von Humboldt Foundation, he was with the University of Stuttgart, Stuttgart, Germany from 1993 to 1994. He has been a Professor at HUST since 1997, where he is currently the Director of the State Key Lab of Digital Manufacturing Equipment and Technology. He was a “Cheung Kong” Chair Professor at Shanghai Jiao Tong University from 2001 to 2006. He became a Member of the Chinese Academy of Sciences in 2013. His research interests include robotics, multiaxis machining, and control engineering. Dr. Ding served as an Associate Editor of the IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING (TASE) from 2004 to 2007. He is currently an Editor of IEEE TASE and a Senior Editor of IEEE ROBOTICS AND AUTOMATION LETTERS.