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MIMO Repetitive Control of an Active Magnetic Bearing Spindle
7th IFAC Symposium on Mechatronic Systems 7th IFAC Symposium on MechatronicUniversity, Systems UK September 5-8, 2016. Loughborough 7th IFAC IFAC Symposium Symposium on Mechatronic Mechatronic Systems Systems 7th on September 5-8, 2016. Loughborough University,online UK at www.sciencedirect.com Available September 5-8, 2016. Loughborough University, September 5-8, 2016. Loughborough University, UK UK
ScienceDirect IFAC-PapersOnLine 49-21 (2016) 192–199
MIMO Repetitive Control of an Active MIMO MIMO Repetitive Repetitive Control Control of of an an Active Active Magnetic Bearing Spindle Magnetic Bearing Spindle Magnetic Bearing Spindle Sandeep Rai ∗∗ Grant Cavalier †† James Simonelli ‡‡ Sandeep Cavalier Simonelli † § † James Sandeep Rai Rai ∗∗ Grant Grant Cavalier James Simonelli ‡‡ Tsu-Chin Tsao Sandeep Rai Grant Cavalier James Simonelli § Tsu-Chin Tsao § § Tsu-Chin Tsao Tsu-Chin Tsao ∗ ∗ Mechanical and Aerospace Engineering Department, University of and Aerospace Engineering Department, University of ∗ ∗ Mechanical Mechanical Engineering University California,and LosAerospace Angeles, USA (e-mail:Department, [email protected]). Mechanical and Aerospace Engineering Department, University of of Los Angeles, USA (e-mail: [email protected]). † California, California, Los Angeles, USA (e-mail: [email protected]). and Aerospace Engineering Department, University of California, Los Angeles, USA (e-mail: [email protected]). † Mechanical † Mechanical and Aerospace Engineering Department, University of † Mechanical and Angeles, AerospaceUSA Engineering Department, University University of of California, Los (e-mail: [email protected]). Mechanical and Aerospace Engineering Department, Los Angeles, USA (e-mail: [email protected]). ‡ California, California, Los (e-mail: [email protected]). and Angeles, AerospaceUSA Engineering Department, University of California, Los Angeles, USA (e-mail: [email protected]). ‡ Mechanical and Aerospace Engineering Department, University of ‡ ‡ Mechanical Mechanical Engineering University California,and LosAerospace Angeles, USA (e-mail:Department, [email protected]). Mechanical and Aerospace Engineering Department, University of of Los Angeles, USA (e-mail: [email protected]). § California, California, Los Angeles, USA (e-mail: [email protected]). and Aerospace Engineering Department, University of California, Los Angeles, USA (e-mail: [email protected]). § Mechanical § Mechanical and Aerospace Engineering Department, University of § Mechanical and Los Aerospace Engineering Department, University of of California, Angeles, USA (e-mail: [email protected]) Mechanical and Aerospace Engineering Department, University California, Los Angeles, USA (e-mail: [email protected]) California, California, Los Los Angeles, Angeles, USA USA (e-mail: (e-mail: [email protected]) [email protected]) Abstract: We present the identification and control of an open loop unstable MIMO Active Abstract: We present the identification and control of an open loop unstable MIMO Active Abstract: We the identification and of loop unstable MIMO Active Magnetic Bearing Spindle (AMBS) for machining applications. A 20th model is obtained Abstract: We present present the(AMBS) identification and control control of an an open open loop order unstable MIMO Active Magnetic Bearing Spindle for machining applications. A 20th order model is obtained Magnetic Bearing Spindle (AMBS) for machining applications. A 20th order model is obtained from reducing a higher order model identified by the ARX method. For verification purposes, Magnetic Bearing Spindle (AMBS) for machining applications. A 20th order model is obtained from reducing a higher order model identified by the ARX method. For verification purposes, from reducing a identified by ARX purposes, the model is compared to themodel system’s frequency responses obtainedFor by verification the frequency sweep from reducing a higher higher order order model identified by the the ARX method. method. For verification purposes, the model is compared to the system’s frequency responses obtained by frequency sweep the model is model compared to the the system’s frequency responses obtained by the the frequency sweep method. The is used to design a linear quadratic optimal controller, where the weighting the model is compared to system’s frequency responses obtained by the frequency sweep method. The model is used to design a linear quadratic optimal controller, where the weighting method. model is a optimal controller, where the weighting gains areThe tuned by simulation and experiment. A plug-in repetitive controller for asymptotic method. The model is used used to to design design a linear linear quadratic quadratic optimal controller, wherefor theasymptotic weighting gains are tuned by simulation and experiment. A plug-in repetitive controller gains are tuned by simulation and experiment. A plug-in repetitive controller for asymptotic regulation and tracking of signals synchronous to the spindle rotation is designed by formulating gains are tuned by simulation and experiment. A plug-in repetitive controller for asymptotic regulation and tracking of signals synchronous to the rotation is by regulation and tracking of signals synchronous to the spindle spindle rotation is designed designed by formulating formulating the controland design as a of model matching filter to design problem. The model matching’s optimal regulation tracking signals synchronous the spindle rotation is designed by formulating the control design as a model matching filter design problem. The model matching’s the control design as a a model model matching matching filterstabilized design problem. problem. The model respectively matching’s optimal optimal solutions fordesign the non-minimum phase MIMO system are obtained for H∞ the control as filter design The model matching’s optimal solutions for the non-minimum phase MIMO stabilized system are obtained respectively for H∞ solutions for the non-minimum phase MIMO system are obtained respectively for and H2 norm minimizations. Simulation and stabilized experimental results are presented to compare the ∞ solutions for the non-minimum phase MIMO stabilized system are obtained respectively for H H ∞ and H norm minimizations. Simulation and experimental results are presented to compare the 2 and H norm minimizations. Simulation experimental results are presented compare the proposed MIMO repetitive control designand methods and demonstrate the controlto performance. 2 and H norm minimizations. Simulation and experimental results are presented to compare the 2 proposed proposed MIMO MIMO repetitive repetitive control control design design methods methods and and demonstrate demonstrate the the control control performance. performance. proposed MIMO repetitive control design methods and demonstrate the control © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All performance. rights reserved. Keywords: active magnetic bearing, high speed machining, multivariable control, repetitive Keywords: active magnetic bearing, high speed machining, multivariable control, Keywords: activeidentification, magnetic bearing, bearing, high speed speed machining, machining, multivariable multivariable control, control, repetitive repetitive control, system implementation Keywords: active magnetic high repetitive control, system identification, implementation control, system system identification, identification, implementation implementation control, 1. INTRODUCTION the purposes of this paper. Fujita et al. (1993) and Balini 1. INTRODUCTION INTRODUCTION the purposes purposes of of this this paper. paper. Fujita Fujita et et al. al. (1993) (1993) and and Balini Balini 1. the et al.purposes (2011) use a Hpaper. shaping design a 1. INTRODUCTION ∞ loopFujita the of this et approach al. (1993) to and Balini et al. (2011) use a H loop shaping approach to design a ∞ et al. (2011) use a H loop shaping approach to design a robust levitating controller. In most control design consid∞ et al. (2011) use acontroller. H∞ loop In shaping approach to design a levitating most control design considActive magnetic bearings are used in many applications robust robust levitating controller. In most control design considerations, the magnetic bearing can be decoupled, so SISO robust levitating controller. In most control design considActive magnetic bearings are used in many applications erations, the magnetic bearing can be decoupled, so SISO Active are many such as magnetic molecular bearings pumps, compressors, magnetic levitating erations, Active are used used in in magnetic many applications applications the magnetic can be decoupled, so techniques can be usedbearing (Noshadi (2015), Kang and the magnetic bearing canet beal. decoupled, so SISO SISO such as as magnetic molecular bearings pumps, compressors, compressors, levitating erations, can be used (Noshadi et al. (2015), Kang and such molecular pumps, vehicles, and flywheels (Bleuler et al.magnetic (2009)), levitating to name techniques such as molecular pumps, compressors, magnetic levitating techniques can be used (Noshadi et al. (2015), Kang and Tsao (2016)). In Pipeleers et al. (2009), a SISO repetitive techniques can be used (Noshadi et al. (2015), Kang and vehicles, and flywheels (Bleuler et al. (2009)), to name Tsao (2016)). In Pipeleers et al. (2009), a SISO repetitive vehicles, and flywheels (Bleuler et al. (2009)), to name a few. An active magnetic bearing spindle (AMBS) has vehicles, and flywheels (Bleuler et al. (2009)), to name Tsao (2016)). In Pipeleers et al. (2009), a SISO repetitive controller based on the decoupled plant model for an active Tsao (2016)). Inon Pipeleers et al. (2009), a SISO repetitive a few. An active magnetic bearing spindle (AMBS) has controller based the decoupled plant model for an active aaalso few. An active magnetic (AMBS) has been of interest for itsbearing use in spindle high speed machinfew. An of active magnetic (AMBS) has controller controller based onwas the developed decoupled for plant model for for rejection. an active active air bearingbased system disturbance on the decoupled plant model an also been interest for its itsbearing use in in spindle high speed speed machinair bearing system was developed for disturbance rejection. also been of interest use high machining (HSM) because offor itsitspotential benefits of mitigatalso been of interest for use in high speed machinair bearing system developed for disturbance In Nonami and Ito was (1996), a µ synthesis approachrejection. was used air bearing system was developed for disturbance rejection. ing (HSM) because of its potential benefits of mitigatNonami and Ito (1996), aa µ synthesis approach was used ing (HSM) because of wears, thermal-mechanical effects,benefits and creating high In ing (HSM) thermal-mechanical because of of its its potential potential of mitigatmitigatIn Ito synthesis approach was design aand controller for aaµ shaft on a magnetic In Nonami Nonami and Ito (1996), (1996), µflexible synthesis approach was used used ing wears, wears, effects,benefits and creating creating high to to design a controller for a flexible shaft on a magnetic ing thermal-mechanical effects, and high bandwidth and dynamic stiffness for precision machining ing wears, thermal-mechanical effects, and creating high to to design design controller forthe flexible shaft on onina athis magnetic bearing system similar to one considered paper. aa controller for aa flexible shaft magnetic bandwidth and dynamic stiffness for precision machining bearing system similar to the one considered in this paper. bandwidth and precision machining bearing system similar to the one considered in this paper. Chen and Knospe (2007). stiffness However,for AMBS is a challenging bandwidth and dynamic dynamic precision machining bearing to the one considered in this Chen and and Knospe Knospe (2007). stiffness However,for AMBS is aa challenging challenging testsystem set-upsimilar exhibits significant coupling, so paper. SISO Chen (2007). However, AMBS is mechatronic system, as it presents unstable, nonlinear, and Our Chen and Knospe (2007). However, AMBS is a challenging Our test set-up exhibits significant coupling, so SISO mechatronic system, as it presents unstable, nonlinear, and Our test set-up exhibits significant coupling, so SISO techniques will not work. We propose high-gain feedback mechatronic system, as it presents unstable, nonlinear, and Our test set-up exhibits significant coupling, so SISO coupled-axis system, MIMO open-loop dynamics that require caremechatronic as it presents unstable, nonlinear, and techniques will not work. We propose high-gain feedback coupled-axis MIMO open-loop dynamics that require caretechniques will not work. We propose high-gain feedback control— specifically a MIMO Repetitive Control. Repetcoupled-axis MIMO open-loop dynamics that require caretechniques will not work. We propose high-gain feedback ful modeling, identification, and control to enable the recoupled-axis open-loop dynamics require carespecifically a MIMO Repetitive Control. Repetful modeling, modeling,MIMO identification, and control that to enable enable the re- control— control— specifically MIMO Repetitive Control. Repetcontrol is basedaaon the internal model principle of ful and to the recontrol— specifically MIMO Repetitive Control. Repetalization of theidentification, potential benefits. This paper presents the ful modeling, identification, and control control to enable the the re- itive itive control is based on the internal model principle of alization of the potential benefits. This paper presents itive control is based on the internal model principle of Francis and Wonham (1976) that states that to achieve alization of the potential benefits. This paper presents the control is based on the that internal model principle of methods of and results of real-time control instrumentation, alization the potential benefits.control This paper presents the itive Francis and Wonham (1976) states that to achieve methods and results of real-time instrumentation, Francis and Wonham (1976) that states that to achieve zero tracking error, a model of the reference/disturbance methods and results of real-time control instrumentation, Francis and Wonham (1976) that states that to achieve system identification, and digital control we conducted methods and results of real-time control instrumentation, zero tracking error, a model of the reference/disturbance system identification, identification, and and digital digital control control we we conducted conducted zero error, model of musttracking be contained the controller. Repetitive control system tracking error, aain model of the the reference/disturbance reference/disturbance on a commercial gradeand AMBS for control future applications in zero system identification, digital we conducted must be contained in the controller. Repetitive control on a commercial grade AMBS for future applications in must be contained in the controller. Repetitive control places an internal model at a fundamental frequency on aa commercial grade AMBS for future applications in must be contained in the controller. Repetitive control high-speed machining and machining of irregularly-shaped on commercial grade AMBS for future applications in places an internal model at a fundamental frequency high-speed machining and machining of irregularly-shaped places an internal model at a fundamental frequency and its harmonics. Traditional repetitive control methods high-speed machining and machining of irregularly-shaped an internal model at a fundamental frequency surfaces. machining and machining of irregularly-shaped places high-speed and Traditional repetitive control methods surfaces. and its its harmonics. harmonics. Traditional repetitive controlare methods (Tomizuka et al. (1989), Cuiyan et al. (2004)) SISO surfaces. and its harmonics. Traditional repetitive control methods surfaces. et al. (1989), Cuiyan et al. (2004)) are SISO Control approaches to AMB mainly address establish- (Tomizuka (Tomizuka et al. (1989), Cuiyan et al. (2004)) are SISO formulations, therefore it is not clear how(2004)) to easily extend (Tomizuka et al. (1989), Cuiyan et al. are SISO Control approaches to AMB mainly address establishformulations, therefore it is not clear how to easily extend Control approaches to AMB mainly address establishing stability or compensating unbalanced spindleestablishmotion. these Control approaches to AMB mainly address formulations, therefore it is not clear how to easily extend methods to the multivariable setting. In Kim et al. formulations, therefore it is not clear how to easily extend ing stability or compensating unbalanced spindle motion. these methods to the multivariable setting. In Kim et al. ing stability or compensating spindle motion. Compensation can take the unbalanced form of either minimizing ing stability or compensating unbalanced spindle motion. these methods to the multivariable setting. In Kim et (2004), µ-synthesis is used to design a repetitive controller methods to the multivariable setting. In Kim et al. al. Compensation can can take take the the form form of of either either minimizing minimizing these (2004), µ-synthesis is used to design a repetitive controller Compensation unbalance force transmitted to the housing or Compensation can take thetoform of either minimizing (2004), µ-synthesis is used used to multivariable design aa repetitive repetitive controller and canµ-synthesis be extended to the setting, but it (2004), is to design controller unbalance force transmitted the housing or minimizing and can be extended to the multivariable setting, but it unbalance force transmitted the or minimizing rotor runout Zhou and Shi to (2001). Because precision is and unbalance force transmitted the housing housing or precision minimizing can extended to multivariable setting, it is highly sensitive to the weighting parameters. In but Longcan be be extended to the the multivariable setting, but it rotor runout runout Zhou and Shi Shi to (2001). Because is and is highly sensitive to the weighting parameters. In Longrotor Zhou and (2001). Because precision is valuedrunout above Zhou vibration suppression in the HSM applicarotor and Shi (2001). Because precision is is highly sensitive to the weighting parameters. In Longman (2010), an FIR interpolation method is discussed; is highly sensitive to the weighting parameters. In Longvalued above vibration suppression in the HSM applica(2010), an FIR interpolation method is discussed; valued vibration suppression in HSM tion, weabove will focus exclusively on runout minimization for man valued vibration suppression in the the HSM applicaapplicaman (2010), (2010), an an FIR FIR interpolation interpolation method method is is discussed; discussed; tion, we weabove will focus focus exclusively on runout runout minimization for man tion, will exclusively on minimization for tion, we will focus exclusively on runout minimization for Copyright © 2016, 2016 IFAC 192Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2016 IFAC 192 Copyright 2016 IFAC 192 Peer review© of International Federation of Automatic Copyright ©under 2016 responsibility IFAC 192Control. 10.1016/j.ifacol.2016.10.545
2016 IFAC MECHATRONICS September 5-8, 2016. Loughborough University, Sandeep UK Rai et al. / IFAC-PapersOnLine 49-21 (2016) 192–199
ω∗
Motor Amplifier
AC Induction Motor
Amplifiers
Coils
193
FORWARD SENSORS
REAR SENSORS THRUST AMB
RT Target u
Fmag
y Rotor
Icoils
REAR AMB
y
Forward AMB
AC INDUCTION MOTOR
Fig. 1. Experimental Setup
Fig. 2. Schematic of AMBS System
this method extends naturally to the MIMO setting, but cannot claim optimality and may result in long FIR filters. In this paper, two simple MIMO repetitive controllers are formulated by solving H∞ or H2 model matching problem. For conventional HSM applications, the spindle runout motion must be eliminated to create desirable axissymmetric surfaces. This requires control that rejects the spindle runout motion, which is synchronous to the spindle rotation. Another special machining application that creates irregularly shaped surfaces, termed non-circular or axis-asymmetric machining, requires precise tracking of specified spindle orbital motion. A specific example motivating our investigation here is boring of non-circular holes for wrist pin holes in internal combustion engine pistons. Current production engine piston’s pin bore profiles have elliptical cross sections and tapered diameters to increase the fatigue limit of the pin bore under cyclic combustion loads (Whitacre and Trainer (1986)). To create this noncircular bore shapes, various fast tool servos have been developed to either rapidly move the cutting tip mounted on the rotating boring bar (Cselle (2003), Zhai et al. (2008)), or to move the non-rotating piston (Liang (2013)). AMBS has the potential of performing this task if spindle orbital motion can be controlled precisely to generate tool tip motion for creating the specified non-circular bore. This presents a tracking control problem of a periodic signal whose fundamental frequency is an integer multiple of the spindle speed. This paper addresses the AMBS control problem with consideration of the HSM application. Specifically, we present the following methods and results: (1) A high order ARX system identification of a coupled 4x4 MIMO AMBS, model reduction, and comparison to the empirical frequency response data (2) A stabilizing LQGi controller, tuned for broad bandwidth (3) MIMO Repetitive controller asymptotic regulation and tracking of periodic signals based on model matching of non-minimum phase systems The remainder of this paper is organized as follows: Section 2 describes the AMBS control system instrumentation, system identification, and the stabilizing LQGi tuning; Section 3 presents the MIMO repetitive control design; Section 4 presents the experimental results followed by conclusions in Section 5. 193
2. SYSTEM IDENTIFICATION & STABILIZING CONTROL The experimental system shown in Figure 1 consists of an SKF AMBS instrumented with an induction motor drive, electromagnet power amplifiers, and a digital control system. The system has two radial AMBs and one axial AMB (Figure 2). The radial AMB system considered in this paper is four-input, four-output. National Instruments analog and digital conversion cards were used to interface the five analog gap sensors and ten electromagnetic coil power amplifiers to the digital computer. The system identification, data acquisition, and real-time digital control were realized by LabVIEW Real Time target installed on the computer, sampling at 10 kHz. The axes of the sensors are not co-located with those of the actuators, which introduces strong off-diagonal coupling. The electromagentic force is nonlinear with respect to the coil current and the air gap. Bias currents are applied to the opposing coils to render a linear approximation within ±100 microns from the AMBS centerline. In this paper, linear control is considered for spindle motion within the linear range. Since the system is open loop unstable a stabilizing controller was created and used to perform a closed-loop system identification of the open-loop plant. 2.1 High Order Multivariable ARX Identification A time-domain identification using Pseudo-Random Burst Sequence (PRBS) input data and Autoregressive Exogenous Input (ARX) modeling (Equation 1) was performed to identify the open-loop plant model. The A and B ARX coefficient matrices are both 4x4 while both y(t) and u(t) are 4x1 vectors. Given experimental input and output data, equation 1 can be cast as a ordinary least squares problem.
y(t) +
N
k=1
Ak y(t − k) =
N
k=0
Bk u(t − k)
(1)
In this application, a high order (≈ 120) ARX model was used to solve the least squares problem, which was then put into observer form (Equation 2). This translates the inputoutput form into a high order state space system, which is suitable for a balanced model reduction. Figure 3, shows the Hankel Singular Values of the high order ARX model from which a 20th order model was chosen. Interestingly, a high order model followed by a model reduction was seen to give a much better fit than simply solving for a low order
2016 IFAC MECHATRONICS 194 Sandeep September 5-8, 2016. Loughborough University, UK Rai et al. / IFAC-PapersOnLine 49-21 (2016) 192–199
2.2 Linear Quadratic Control Design
Hankel Singular Values (State Contributions) 0.8
Stable modes
In order to stabilize the open loop plant and attenuate low frequency disturbances, a Linear-Quadratic-Gaussian with Integral action (LQGi) controller was designed. The main design considerations for the LQGi controller were bandwidth, saturation, and stiffness. Obviously, higher inner loop bandwidth is desired, but comes at a cost of saturation and robustness. Hardware considerations dictated the use of a more conservative controller for the initial lift of the rotor; once the rotor is safely levitated, a more aggressive controller can be applied.
0.7
0.6
State Energy
0.5
0.4
0.3
0.2
0.1
0 10
20
30 State
40
50
60
Fig. 3. First 60 Hankel Singular Values of High Order State Space Model
ARX model. Furthermore, the least squares solution can be computed efficiently with the large amounts of data. In order to assess the effectiveness of the identification model, four separate sine sweep experiments were performed. Each experiment excited only one input of the closed loop system, to obtain frequency response data for frequencies ranging from 10 to 3500 Hz. The reference, plant input, and plant output were recorded for each frequency and processed to obtain a matrix of frequency responses, from r to u and from r to y (figure 7), as shown in Equation 3. ˆ ur (jωk ) = Um (jωk ) , h l,m Rl (jωk )
l = 1...4, m = 1...4
−1 ˆ yr (jωk ) H ˆ ur (jωk ) ˆ yu (jωk ) = H H
(3)
LQGi control is well known, but the design process is briefly summarized here for completeness. Given a plant as in Equation 5, the overall LQGi controller is given by Equation 6. The inputs to the controller are the plant output and the error, r − y. A p Bp P (z) = (5) Cp 0
Ap − LCp − Bp Kx −Bp Ki L 0 0 I 0 I C(z) = (6) −Kx −Ki 0 0 The controller contains K and L which are the state feedback and observer gains, respectively. By the separation principle, the state feedback and observer gains can be designed separately by solving an Algebraic Riccati Equation (ARE). However, to incorporate integral action the plant needs to be augmented before solving the Riccati Equation as in Equation 7. The notation used here is that dlqr(A, B, Q, R) denotes the solution of the discrete-time ARE with state and output weighting matrices Q and R. The observer gain is independently solved as in Equation 8, then the overall controller is formed. The weighting matrices are used as tuning parameters.
(4) [Kx Ki ] = dlqr(
The open loop frequency response was then obtained through point-by-point matrix inversion, as shown in Equation 4. Figure 4 shows the raw Frequency Response Data (FRD) and the reduced state-space model’s Bode Magnitude plots. From this data, it was determined that the reduced model was accurate from DC to approximately 1kHz. The open loop frequency responses show resonant modes of the rotor at 1.1 kHz and 2 kHz, which appear strongly in the cross terms and make a decoupling transformation difficult to obtain. It is because of this difficulty in decoupling the input-output pairs that a truly MIMO control design approach is necessary.
−A1 I
0
0 .. . .. . .. . 0
I .. .
−A2 . . . A= .. . . .
. −AN
.. ..
.
. ···
0 ··· ··· . .. · · · .. . .. .. . . .. .. . . .. .. . . ··· ···
0 .. . .. .
, 0
I 0
(7) (8)
By combining the plant (Eq. 5) and the LQGi (Eq. 6), the closed loop plant G(z) can be obtained. Ap − Bp Kx −Bp Ki 0 −Cp I I G(z) = Cp 0 0
(9)
The Q and R weighting are design parameters. The order of the Q matrix is the order of the plant plus the four integrator states. Therefore, Q is partitioned as
B1 − A 1 B 0 B2 − A 2 B 0 .. . , B= . .. .. . BN − A N B 0
194
Ap 0 B , p , Qk , Rk ) −Cp I 0
LT = dlqr(ATp , CpT , Ql , Rl )
C = I 0 0 ··· ··· 0 ,
D = B0
(2)
2016 IFAC MECHATRONICS September 5-8, 2016. Loughborough University, Sandeep UK Rai et al. / IFAC-PapersOnLine 49-21 (2016) 192–199
195
Bode Diagram
To: Out(4)
To: Out(3)
Magnitude (dB)
To: Out(2)
To: Out(1)
From: In(1)
From: In(2)
From: In(3)
From: In(4)
Raw FRD Open-Loop Model
50 0 -50 50 0 -50 50 0 -50 50 0 -50 100
1k
100
1k
100
1k
100
1k
Frequency (Hz)
Fig. 4. Bode Magnitude Plot of Open Loop System
-10 -20
Gain [dB]
-30
-50
Position (um)
-70 -80 -90 10 -1
10 0
30
-60
Agressive LQGi Start up LQGi
10 0
10 1
10 2
10 3
1.72 1.74 Time (secs) V24
20 10 0 1.7
10 4
Frequency [Hz]
30
20
1.7
-40
V13 Position (um)
Position (um)
30
1.72 1.74 Time (secs)
W13
20 10 0 1.7 30
Position (um)
0
1.72 1.74 Time (secs) W24
20 10
Experiment Simulation
0 1.7
1.72 1.74 Time (secs)
Fig. 5. Maximum Singular Value of the Closed Loop Frequency Response
Fig. 6. Step Responses of LQGi Controller, Experiment and Simulation
in Equation 10, where α is the weight associated with the plant states and β corresponds to the integrator states. This reduces the parameters in the Q matrix to two, and when coupled with the regularization of the R matrix to the identity, we have only those two variables parameterizing the LQGi design. We found this to be an adequate amount of control to design a useable inner-loop controller.
design, Figure 6 shows good agreement between simulation and experimental results.
Q=
αI 0 0 βI
R=I
(10) (11)
Figure 5 summarizes the results of the inner-loop control design. As seen, the aggressive LQGi controller has a bandwidth of approximately 100 Hz and attenuates the resonant modes. Given actuator constraints, this was the highest comfortable bandwidth for a broadband controller, and is well within the first resonant mode. As a final verification of the system identification and controller 195
3. MIMO REPETITIVE CONTROL When the switch in Figure 7 is closed, the repetitive controller is added on to the existing LQGi control. Similar to single variable case, the main objective is to produce infinite gain at a desired frequency to track references or reject disturbances at its harmonics. This is accomplished by the internal model in Equation 12, where N is the number of samples constituting one period of the desired frequency. D(z) =
1 I zN − 1
(12)
In order to design a repetitive controller, filters F (z) and Q(z) need to be determined. The design of these filters is much more difficult when the plant is multivariable; methods such as ZPETC are single-variable formulations, and cannot be used.
2016 IFAC MECHATRONICS 196 Sandeep September 5-8, 2016. Loughborough University, UK Rai et al. / IFAC-PapersOnLine 49-21 (2016) 192–199
From Tsao (1994), it is known that the optimal F (z) is such that the optimal cost in Equation 14 is an allpass filter, and that increasing the preview length, N2 , decreases the error of the model matching problem.
z −N2
z −N1
Σ r
Q(z)
F (z) C(z)
Σ
u
P (z)
y
Σ
3.2 H2 Factorization Based Fig. 7. Block Diagram of the Plug-in Repetitive Control It is known that for the plug-in repetitive controller to be stable, the inequality in Equation 13 is a sufficient condition. (z −N2 I − F (z)G(z))Q(z) ∞ < 1
(13)
G(z) is the nominal closed-loop transfer function, from reference input to output with the LQGi controller given in equation 9. From the stability equation, it is easy to see that the inequality is satisfied when F (z) = G−1 (z). However, when the system is non-minimum phase, there will be an unstable pole-zero cancellation, so a na¨ıve inversion will not work. In this paper, two methods are proposed to design a MIMO inversion filter F (z): an H∞ approach, and an H2 inner-outer factorization approach. 3.1 H∞ minimization
Instead of solving the H∞ model matching problem for the stabilizing inversion filter, the H2 model matching problem can be solved: min (z −N2 M (z) − F (z)G(z)) 2 (16) F (z)∈RH∞
The solution to the H2 optimal problem is given in Equation 17: −N2 −1 Fopt = G−1 Gi (z)M (z)]+ (17) o (z)[z where the plant can be factored into its inner and outer systems H(z) = Gi (z)Go (z); the outer (minimum phase) part can be easily inverted, and the inversion of the inner part approximated by a FIR filter. A state-space realization of the inner and outer factors can be easily obtained by solving a generalized Riccati equation Ionescu and Oara (1996). FIR approximation of the unstable inverse of the inner factor is done by conjugating the conjugate system (Equation 18).
Motivated by the stability condition, the H∞ problem (Equation 14) can be solved for the inversion filter F (z). min
F (z)∈RH∞
(z −N2 M (z) − F (z)G(z)) ∞
(14)
Clearly, this is a model matching problem from r to e, as shown in Figure 8. If G(z) has a state space realization (Ap , Bp , Cp , Dp ) given by Equation 9, and z −N2 M (z) is given by (Am , Bm , Cm , 0), then Equation 15 is the generalized plant and the model matching problem can be solved by standard H∞ solvers. M (z)z −N2
r
F (z)
G(z)
G−1 i
=
G−T i
∗ 1 z
(18)
The conjugate system can be expressed as in Equation 19. Most importantly, it is stable, since the unstable poles of the inner factor are mirrored across the unit circle. Therefore, it can be put into infinite impulse response form. G−T i
1 −A−T Ci−T A−T i i = T B T A−T DiT − BiT A−T z i Ci i i ˜ A˜ B = ˜ ˜ C D k ∞ ˜ ˜ 1 +D C˜ A˜k−1 B = z
(19) (20) (21)
k=1
e
Fig. 8. Model Matching Interpretation of Stability Condition 0 Bp Ap 0 xp (k) xp (k + 1) xm (k + 1) 0 Am Bm 0 xm (k) (15) = C −C e 0 Dp r p m r u 0 0 I 0 N2 is a tuning parameter that depends on how close the non-minimum phase zeros of the system are to the unit circle. For the magnetic bearing system, there are nonminimum phase zeroes so this delay length must be nonzero. M (z) is a low pass filter that limits the bandwidth of the inversion filter, so that high frequency noise will not be amplified, which would cause noisy actuation and possibly instability.
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Then the conjugate system can be truncated and conjugated, as in Equation 22. ∗ N 1 −T ˜ T = Kf ir (z) ˜ T (A˜T )k−1 C˜ T (z)k + D ≈ Gi B z k=1 (22) This results in a non-casual filter Kf ir (z) of order N , but the delays in Equation 16 make the first N2 coefficients casual. The final filter that solves the H2 problem (Equation 23) is the stable inversion of the inner factor, cascaded with the inversion of outer factor. Once the filter is designed, it must be substituted into the stability condition to verify that it satisfies the inequality. F (z) = G−1 o (z)Kf ir (z)
(23)
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q(z) = (.25z + .5 + .25z −1 )Nq
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Ideally, a properly designed inverse filter guarantees the stability of the repetitive control system and internal models at the necessary frequencies. However, at high frequencies, the model is usually not accurate, which can cause instability or undesirable behavior. The Q filter, which severely attenuates high frequencies, can provide robustness to this mismatch. For the magnetic bearing system under consideration, the Q filter is comprised of four SISO filters (Equation 24) in block diagonal form. Q(z) is noncausal, but the preview steps can be emulated.
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Fig. 9. Maximum Singular Value of Stability Condition Figure 9 plots the maximum singular value of the stability condition as a function of frequency. Equation 14 requires that this value be less than unity for all frequencies, which is shown to be satisfied for both the H∞ and Factorizationbased controllers. 4. EXPERIMENTAL RESULTS To verify the disturbance rejection of the controllers on the physical system, the spindle was spun at 400 Hz. The control used a real-time target, sampling at 10 kHz. Both repetitive controllers used a 5th order Q filter with a cutoff at 800 Hz, which was designed with the 1 kHz bandwidth where the model is valid in mind. The H∞ design required 10 preview steps, while the factorization approach required 20—both are well within the period of the fundamental frequency of the repetitive controller, so the difference is not relevant. The disturbance caused by the spindle spinning at 400 Hz amounts to 24,000 RPM—well in the high speed machining region. Figure 10 shows both time-domain and spectral results of this experiment. The repetitive controller was switched on a 0.5 seconds. The red spectrum is the LQGi controller levitating the spindle with no disturbance injected, which is the baseline against which the others are compared. The only disturbance injected was at 400Hz; the other spikes are harmonics of the 60Hz mains hum, which neither controller was designed to reject. The harmonic at 300Hz is rejected by coincidence— the fundamental frequency of the Repetitive controllers internal model was set at 100Hz. The spectral plot show that both repetitive controller have control action at integers of 100 Hz as is expected, but tapers off after 600 Hz due to the Q filter. 197
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Fig. 10. 400Hz Disturbance Rejection Results Controller H∞ Factorization
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Table 1. Comparison of H∞ and Factorization Methods Results are similar in the tracking experiment: the main spectral error component is at the target frequency of 400Hz, but the second-largest component is the harmonics of the mains hum. Tracking involves the spindle moving outside of the identified region. The controllers are stable and able to track the reference with less than one micron of error, tracking a 15µm (peak-to-peak) reference. Tracking performance is good, without amplifying sideband frequencies. Results of both experiments are summarized in Table 1. The H∞ controller slightly outperforms the factorization based controller in terms of RMS error. Both control design methods are acceptable for repetitive internal model type controllers. The experimental data shows that significant harmonic errors at 60 Hz and its higher harmonics, caused by the AC electric power, were not accounted for. These components could have been eliminated by adding another repetitive control loop with the internal model at these frequencies. 5. CONCLUSIONS Based on the experience in the control design and experimental investigation for the AMBS reported in this paper, the following concluding remarks are made:
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LQGi Only
further investigation on system identification and control design and this is of future research interest.
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The authors are grateful to Dr. Larry Liang of FederalMogul Corporation for providing valuable information on AMBS for high-speed machining applications.
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Fig. 11. 400Hz Tracking Results (1) The ARX method conducted within stabilized feedback loops followed by balanced truncation model reduction was efficient and effective in obtaining the open loop unstable MIMO AMBS dynamic model, which agrees well with the system’s frequency response data. (2) Based on the identified model, the optimal linear quadratic control with integral action was effectively tuned by only two scalar weighting gains—one on the plant’s states and one on the integrators—with the control weighting normalized to identity. (3) The plug-in MIMO repetitive controller was designed by optimal model matching, and either the H∞ or H2 norm minimization was effective, with similar performance in experiment. (4) Control experiments on the AMBS with 10kHz sampling implemented by the LabVIEW RealTime Target have demonstrated the LQGi and repetitive control performance in rejecting errors synchronous to the 400 Hz spindle speed rotation. (5) Repetitive control for multiple periods can be employed to reject AC electric power disturbances. The existing SISO approaches (P´erez-Arancibia et al. (2010),Kalyanam and Tsao (2012)) should be extended for the present MIMO system. These results also suggest that the AMBS, together with the control methods investigated herein, can be applied to high-speed machining applications where the synchronous spindle error motion is prevalent. The machining process, depending on the cutting conditions, may introduce additional disturbances and unmodeled dynamics that require 198
Balini, H., Scherer, C.W., and Witte, J. (2011). Performance enhancement for amb systems using unstable controllers. IEEE Transactions on Control Systems Technology, 19(6), 1479–1492. Bleuler, H., Cole, M., Keogh, P., Larsonneur, R., Maslen, E., Okada, Y., Schweitzer, G., Traxler, A., Schweitzer, G., Maslen, E.H., et al. (2009). Magnetic bearings: theory, design, and application to rotating machinery. Springer Science & Business Media. Chen, M. and Knospe, C.R. (2007). Control approaches to the suppression of machining chatter using active magnetic bearings. Control Systems Technology, IEEE Transactions on, 15(2), 220–232. Cselle, T. (2003). Piezoelectric adjusting mechanism. US Patent 6,586,862. Cuiyan, L., Dongchun, Z., and Xianyi, Z. (2004). A survey of repetitive control. In IEEE/RSJ international conference on intelligent robots and systems. Francis, B.A. and Wonham, W.M. (1976). The internal model principle of control theory. Automatica, 12(5), 457–465. Fujita, M., Hatake, K., Matsumura, F., and Uchida, K. (1993). Experiments on the loop shaping based h, control of a magnetic bearing. In American Control Conference, 1993, 8–13. IEEE. Ionescu, V. and Oara, C. (1996). Spectral and innerouter factorizations for discrete-time systems. IEEE transactions on Automatic Control, 41(12), 1840–1845. Kalyanam, K. and Tsao, T.C. (2012). Two-period repetitive and adaptive control for repeatable and nonrepeatable runout compensation in disk drive track following. IEEE/ASME Transactions on Mechatronics, 17(4), 756–766. Kang, C. and Tsao, T.C. (2016). Control of magnetic bearings for rotor unbalance with plug-in time-varying resonators. Journal of Dynamic Systems, Measurement, and Control, 138(1), 011001. Kim, B.S., Li, J., and Tsao, T.C. (2004). Twoparameter robust repetitive control with application to a novel dual-stage actuator for noncircular machining. IEEE/ASME Transactions on mechatronics, 9(4), 644– 652. Liang, J. (2013). Method of forming piston pin holes and boring system therefor. US Patent 8,506,215. Longman, R.W. (2010). On the theory and design of linear repetitive control systemsg. European Journal of Control, 16(5), 447–496. Nonami, K. and Ito, T. (1996). µ synthesis of flexible rotor-magnetic bearing systems. IEEE Transactions on Control Systems Technology, 4(5), 503–512. Noshadi, A., Shi, J., Lee, W.S., Shi, P., and Kalam, A. (2015). System identification and robust control of multi-input multi-output active magnetic bearing systems.
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P´erez-Arancibia, N.O., Tsao, T.C., and Gibson, J.S. (2010). A new method for synthesizing multiple-period adaptive–repetitive controllers and its application to the control of hard disk drives. Automatica, 46(7), 1186– 1195. Pipeleers, G., Demeulenaere, B., Al-Bender, F., De Schutter, J., and Swevers, J. (2009). Optimal performance tradeoffs in repetitive control: Experimental validation on an active air bearing setup. IEEE Transactions on Control Systems Technology, 17(4), 970–979. Tomizuka, M., Tsao, T.C., and Chew, K.K. (1989). Analysis and synthesis of discrete-time repetitive controllers. Journal of Dynamic Systems, Measurement, and Control, 111(3), 353–358. Tsao, T.C. (1994). Optimal feed-forward digital tracking controller design. Journal of dynamic systems, measurement, and control, 116(4), 583–592. Whitacre, J.P. and Trainer, H. (1986). Piston pin bore treatments for high performance diesel engines. Technical report, SAE Technical Paper. Zhai, P., Zhang, C.r., Wang, X.l., Qin, L., and Qin, Y.z. (2008). Novel mechanism for boring non-cylinder piston pinhole based on giant magnetostrictive materials. Journal of Shanghai University (English Edition), 12, 363–367. Zhou, S. and Shi, J. (2001). Active balancing and vibration control of rotating machinery: a survey. Shock and Vibration Digest, 33(5), 361–371.
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MIMO Repetitive Control of an Active MIMO MIMO Repetitive Repetitive Control Control of of an an Active Active Magnetic Bearing Spindle Magnetic Bearing Spindle Magnetic Bearing Spindle Sandeep Rai ∗∗ Grant Cavalier †† James Simonelli ‡‡ Sandeep Cavalier Simonelli † § † James Sandeep Rai Rai ∗∗ Grant Grant Cavalier James Simonelli ‡‡ Tsu-Chin Tsao Sandeep Rai Grant Cavalier James Simonelli § Tsu-Chin Tsao § § Tsu-Chin Tsao Tsu-Chin Tsao ∗ ∗ Mechanical and Aerospace Engineering Department, University of and Aerospace Engineering Department, University of ∗ ∗ Mechanical Mechanical Engineering University California,and LosAerospace Angeles, USA (e-mail:Department, [email protected]). Mechanical and Aerospace Engineering Department, University of of Los Angeles, USA (e-mail: [email protected]). † California, California, Los Angeles, USA (e-mail: [email protected]). and Aerospace Engineering Department, University of California, Los Angeles, USA (e-mail: [email protected]). † Mechanical † Mechanical and Aerospace Engineering Department, University of † Mechanical and Angeles, AerospaceUSA Engineering Department, University University of of California, Los (e-mail: [email protected]). Mechanical and Aerospace Engineering Department, Los Angeles, USA (e-mail: [email protected]). ‡ California, California, Los (e-mail: [email protected]). and Angeles, AerospaceUSA Engineering Department, University of California, Los Angeles, USA (e-mail: [email protected]). ‡ Mechanical and Aerospace Engineering Department, University of ‡ ‡ Mechanical Mechanical Engineering University California,and LosAerospace Angeles, USA (e-mail:Department, [email protected]). Mechanical and Aerospace Engineering Department, University of of Los Angeles, USA (e-mail: [email protected]). § California, California, Los Angeles, USA (e-mail: [email protected]). and Aerospace Engineering Department, University of California, Los Angeles, USA (e-mail: [email protected]). § Mechanical § Mechanical and Aerospace Engineering Department, University of § Mechanical and Los Aerospace Engineering Department, University of of California, Angeles, USA (e-mail: [email protected]) Mechanical and Aerospace Engineering Department, University California, Los Angeles, USA (e-mail: [email protected]) California, California, Los Los Angeles, Angeles, USA USA (e-mail: (e-mail: [email protected]) [email protected]) Abstract: We present the identification and control of an open loop unstable MIMO Active Abstract: We present the identification and control of an open loop unstable MIMO Active Abstract: We the identification and of loop unstable MIMO Active Magnetic Bearing Spindle (AMBS) for machining applications. A 20th model is obtained Abstract: We present present the(AMBS) identification and control control of an an open open loop order unstable MIMO Active Magnetic Bearing Spindle for machining applications. A 20th order model is obtained Magnetic Bearing Spindle (AMBS) for machining applications. A 20th order model is obtained from reducing a higher order model identified by the ARX method. For verification purposes, Magnetic Bearing Spindle (AMBS) for machining applications. A 20th order model is obtained from reducing a higher order model identified by the ARX method. For verification purposes, from reducing a identified by ARX purposes, the model is compared to themodel system’s frequency responses obtainedFor by verification the frequency sweep from reducing a higher higher order order model identified by the the ARX method. method. For verification purposes, the model is compared to the system’s frequency responses obtained by frequency sweep the model is model compared to the the system’s frequency responses obtained by the the frequency sweep method. The is used to design a linear quadratic optimal controller, where the weighting the model is compared to system’s frequency responses obtained by the frequency sweep method. The model is used to design a linear quadratic optimal controller, where the weighting method. model is a optimal controller, where the weighting gains areThe tuned by simulation and experiment. A plug-in repetitive controller for asymptotic method. The model is used used to to design design a linear linear quadratic quadratic optimal controller, wherefor theasymptotic weighting gains are tuned by simulation and experiment. A plug-in repetitive controller gains are tuned by simulation and experiment. A plug-in repetitive controller for asymptotic regulation and tracking of signals synchronous to the spindle rotation is designed by formulating gains are tuned by simulation and experiment. A plug-in repetitive controller for asymptotic regulation and tracking of signals synchronous to the rotation is by regulation and tracking of signals synchronous to the spindle spindle rotation is designed designed by formulating formulating the controland design as a of model matching filter to design problem. The model matching’s optimal regulation tracking signals synchronous the spindle rotation is designed by formulating the control design as a model matching filter design problem. The model matching’s the control design as a a model model matching matching filterstabilized design problem. problem. The model respectively matching’s optimal optimal solutions fordesign the non-minimum phase MIMO system are obtained for H∞ the control as filter design The model matching’s optimal solutions for the non-minimum phase MIMO stabilized system are obtained respectively for H∞ solutions for the non-minimum phase MIMO system are obtained respectively for and H2 norm minimizations. Simulation and stabilized experimental results are presented to compare the ∞ solutions for the non-minimum phase MIMO stabilized system are obtained respectively for H H ∞ and H norm minimizations. Simulation and experimental results are presented to compare the 2 and H norm minimizations. Simulation experimental results are presented compare the proposed MIMO repetitive control designand methods and demonstrate the controlto performance. 2 and H norm minimizations. Simulation and experimental results are presented to compare the 2 proposed proposed MIMO MIMO repetitive repetitive control control design design methods methods and and demonstrate demonstrate the the control control performance. performance. proposed MIMO repetitive control design methods and demonstrate the control © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All performance. rights reserved. Keywords: active magnetic bearing, high speed machining, multivariable control, repetitive Keywords: active magnetic bearing, high speed machining, multivariable control, Keywords: activeidentification, magnetic bearing, bearing, high speed speed machining, machining, multivariable multivariable control, control, repetitive repetitive control, system implementation Keywords: active magnetic high repetitive control, system identification, implementation control, system system identification, identification, implementation implementation control, 1. INTRODUCTION the purposes of this paper. Fujita et al. (1993) and Balini 1. INTRODUCTION INTRODUCTION the purposes purposes of of this this paper. paper. Fujita Fujita et et al. al. (1993) (1993) and and Balini Balini 1. the et al.purposes (2011) use a Hpaper. shaping design a 1. INTRODUCTION ∞ loopFujita the of this et approach al. (1993) to and Balini et al. (2011) use a H loop shaping approach to design a ∞ et al. (2011) use a H loop shaping approach to design a robust levitating controller. In most control design consid∞ et al. (2011) use acontroller. H∞ loop In shaping approach to design a levitating most control design considActive magnetic bearings are used in many applications robust robust levitating controller. In most control design considerations, the magnetic bearing can be decoupled, so SISO robust levitating controller. In most control design considActive magnetic bearings are used in many applications erations, the magnetic bearing can be decoupled, so SISO Active are many such as magnetic molecular bearings pumps, compressors, magnetic levitating erations, Active are used used in in magnetic many applications applications the magnetic can be decoupled, so techniques can be usedbearing (Noshadi (2015), Kang and the magnetic bearing canet beal. decoupled, so SISO SISO such as as magnetic molecular bearings pumps, compressors, compressors, levitating erations, can be used (Noshadi et al. (2015), Kang and such molecular pumps, vehicles, and flywheels (Bleuler et al.magnetic (2009)), levitating to name techniques such as molecular pumps, compressors, magnetic levitating techniques can be used (Noshadi et al. (2015), Kang and Tsao (2016)). In Pipeleers et al. (2009), a SISO repetitive techniques can be used (Noshadi et al. (2015), Kang and vehicles, and flywheels (Bleuler et al. (2009)), to name Tsao (2016)). In Pipeleers et al. (2009), a SISO repetitive vehicles, and flywheels (Bleuler et al. (2009)), to name a few. An active magnetic bearing spindle (AMBS) has vehicles, and flywheels (Bleuler et al. (2009)), to name Tsao (2016)). In Pipeleers et al. (2009), a SISO repetitive controller based on the decoupled plant model for an active Tsao (2016)). Inon Pipeleers et al. (2009), a SISO repetitive a few. An active magnetic bearing spindle (AMBS) has controller based the decoupled plant model for an active aaalso few. An active magnetic (AMBS) has been of interest for itsbearing use in spindle high speed machinfew. An of active magnetic (AMBS) has controller controller based onwas the developed decoupled for plant model for for rejection. an active active air bearingbased system disturbance on the decoupled plant model an also been interest for its itsbearing use in in spindle high speed speed machinair bearing system was developed for disturbance rejection. also been of interest use high machining (HSM) because offor itsitspotential benefits of mitigatalso been of interest for use in high speed machinair bearing system developed for disturbance In Nonami and Ito was (1996), a µ synthesis approachrejection. was used air bearing system was developed for disturbance rejection. ing (HSM) because of its potential benefits of mitigatNonami and Ito (1996), aa µ synthesis approach was used ing (HSM) because of wears, thermal-mechanical effects,benefits and creating high In ing (HSM) thermal-mechanical because of of its its potential potential of mitigatmitigatIn Ito synthesis approach was design aand controller for aaµ shaft on a magnetic In Nonami Nonami and Ito (1996), (1996), µflexible synthesis approach was used used ing wears, wears, effects,benefits and creating creating high to to design a controller for a flexible shaft on a magnetic ing thermal-mechanical effects, and high bandwidth and dynamic stiffness for precision machining ing wears, thermal-mechanical effects, and creating high to to design design controller forthe flexible shaft on onina athis magnetic bearing system similar to one considered paper. aa controller for aa flexible shaft magnetic bandwidth and dynamic stiffness for precision machining bearing system similar to the one considered in this paper. bandwidth and precision machining bearing system similar to the one considered in this paper. Chen and Knospe (2007). stiffness However,for AMBS is a challenging bandwidth and dynamic dynamic precision machining bearing to the one considered in this Chen and and Knospe Knospe (2007). stiffness However,for AMBS is aa challenging challenging testsystem set-upsimilar exhibits significant coupling, so paper. SISO Chen (2007). However, AMBS is mechatronic system, as it presents unstable, nonlinear, and Our Chen and Knospe (2007). However, AMBS is a challenging Our test set-up exhibits significant coupling, so SISO mechatronic system, as it presents unstable, nonlinear, and Our test set-up exhibits significant coupling, so SISO techniques will not work. We propose high-gain feedback mechatronic system, as it presents unstable, nonlinear, and Our test set-up exhibits significant coupling, so SISO coupled-axis system, MIMO open-loop dynamics that require caremechatronic as it presents unstable, nonlinear, and techniques will not work. We propose high-gain feedback coupled-axis MIMO open-loop dynamics that require caretechniques will not work. We propose high-gain feedback control— specifically a MIMO Repetitive Control. Repetcoupled-axis MIMO open-loop dynamics that require caretechniques will not work. We propose high-gain feedback ful modeling, identification, and control to enable the recoupled-axis open-loop dynamics require carespecifically a MIMO Repetitive Control. Repetful modeling, modeling,MIMO identification, and control that to enable enable the re- control— control— specifically MIMO Repetitive Control. Repetcontrol is basedaaon the internal model principle of ful and to the recontrol— specifically MIMO Repetitive Control. Repetalization of theidentification, potential benefits. This paper presents the ful modeling, identification, and control control to enable the the re- itive itive control is based on the internal model principle of alization of the potential benefits. This paper presents itive control is based on the internal model principle of Francis and Wonham (1976) that states that to achieve alization of the potential benefits. This paper presents the control is based on the that internal model principle of methods of and results of real-time control instrumentation, alization the potential benefits.control This paper presents the itive Francis and Wonham (1976) states that to achieve methods and results of real-time instrumentation, Francis and Wonham (1976) that states that to achieve zero tracking error, a model of the reference/disturbance methods and results of real-time control instrumentation, Francis and Wonham (1976) that states that to achieve system identification, and digital control we conducted methods and results of real-time control instrumentation, zero tracking error, a model of the reference/disturbance system identification, identification, and and digital digital control control we we conducted conducted zero error, model of musttracking be contained the controller. Repetitive control system tracking error, aain model of the the reference/disturbance reference/disturbance on a commercial gradeand AMBS for control future applications in zero system identification, digital we conducted must be contained in the controller. Repetitive control on a commercial grade AMBS for future applications in must be contained in the controller. Repetitive control places an internal model at a fundamental frequency on aa commercial grade AMBS for future applications in must be contained in the controller. Repetitive control high-speed machining and machining of irregularly-shaped on commercial grade AMBS for future applications in places an internal model at a fundamental frequency high-speed machining and machining of irregularly-shaped places an internal model at a fundamental frequency and its harmonics. Traditional repetitive control methods high-speed machining and machining of irregularly-shaped an internal model at a fundamental frequency surfaces. machining and machining of irregularly-shaped places high-speed and Traditional repetitive control methods surfaces. and its its harmonics. harmonics. Traditional repetitive controlare methods (Tomizuka et al. (1989), Cuiyan et al. (2004)) SISO surfaces. and its harmonics. Traditional repetitive control methods surfaces. et al. (1989), Cuiyan et al. (2004)) are SISO Control approaches to AMB mainly address establish- (Tomizuka (Tomizuka et al. (1989), Cuiyan et al. (2004)) are SISO formulations, therefore it is not clear how(2004)) to easily extend (Tomizuka et al. (1989), Cuiyan et al. are SISO Control approaches to AMB mainly address establishformulations, therefore it is not clear how to easily extend Control approaches to AMB mainly address establishing stability or compensating unbalanced spindleestablishmotion. these Control approaches to AMB mainly address formulations, therefore it is not clear how to easily extend methods to the multivariable setting. In Kim et al. formulations, therefore it is not clear how to easily extend ing stability or compensating unbalanced spindle motion. these methods to the multivariable setting. In Kim et al. ing stability or compensating spindle motion. Compensation can take the unbalanced form of either minimizing ing stability or compensating unbalanced spindle motion. these methods to the multivariable setting. In Kim et (2004), µ-synthesis is used to design a repetitive controller methods to the multivariable setting. In Kim et al. al. Compensation can can take take the the form form of of either either minimizing minimizing these (2004), µ-synthesis is used to design a repetitive controller Compensation unbalance force transmitted to the housing or Compensation can take thetoform of either minimizing (2004), µ-synthesis is used used to multivariable design aa repetitive repetitive controller and canµ-synthesis be extended to the setting, but it (2004), is to design controller unbalance force transmitted the housing or minimizing and can be extended to the multivariable setting, but it unbalance force transmitted the or minimizing rotor runout Zhou and Shi to (2001). Because precision is and unbalance force transmitted the housing housing or precision minimizing can extended to multivariable setting, it is highly sensitive to the weighting parameters. In but Longcan be be extended to the the multivariable setting, but it rotor runout runout Zhou and Shi Shi to (2001). Because is and is highly sensitive to the weighting parameters. In Longrotor Zhou and (2001). Because precision is valuedrunout above Zhou vibration suppression in the HSM applicarotor and Shi (2001). Because precision is is highly sensitive to the weighting parameters. In Longman (2010), an FIR interpolation method is discussed; is highly sensitive to the weighting parameters. In Longvalued above vibration suppression in the HSM applica(2010), an FIR interpolation method is discussed; valued vibration suppression in HSM tion, weabove will focus exclusively on runout minimization for man valued vibration suppression in the the HSM applicaapplicaman (2010), (2010), an an FIR FIR interpolation interpolation method method is is discussed; discussed; tion, we weabove will focus focus exclusively on runout runout minimization for man tion, will exclusively on minimization for tion, we will focus exclusively on runout minimization for Copyright © 2016, 2016 IFAC 192Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2016 IFAC 192 Copyright 2016 IFAC 192 Peer review© of International Federation of Automatic Copyright ©under 2016 responsibility IFAC 192Control. 10.1016/j.ifacol.2016.10.545
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ω∗
Motor Amplifier
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y Rotor
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Fig. 1. Experimental Setup
Fig. 2. Schematic of AMBS System
this method extends naturally to the MIMO setting, but cannot claim optimality and may result in long FIR filters. In this paper, two simple MIMO repetitive controllers are formulated by solving H∞ or H2 model matching problem. For conventional HSM applications, the spindle runout motion must be eliminated to create desirable axissymmetric surfaces. This requires control that rejects the spindle runout motion, which is synchronous to the spindle rotation. Another special machining application that creates irregularly shaped surfaces, termed non-circular or axis-asymmetric machining, requires precise tracking of specified spindle orbital motion. A specific example motivating our investigation here is boring of non-circular holes for wrist pin holes in internal combustion engine pistons. Current production engine piston’s pin bore profiles have elliptical cross sections and tapered diameters to increase the fatigue limit of the pin bore under cyclic combustion loads (Whitacre and Trainer (1986)). To create this noncircular bore shapes, various fast tool servos have been developed to either rapidly move the cutting tip mounted on the rotating boring bar (Cselle (2003), Zhai et al. (2008)), or to move the non-rotating piston (Liang (2013)). AMBS has the potential of performing this task if spindle orbital motion can be controlled precisely to generate tool tip motion for creating the specified non-circular bore. This presents a tracking control problem of a periodic signal whose fundamental frequency is an integer multiple of the spindle speed. This paper addresses the AMBS control problem with consideration of the HSM application. Specifically, we present the following methods and results: (1) A high order ARX system identification of a coupled 4x4 MIMO AMBS, model reduction, and comparison to the empirical frequency response data (2) A stabilizing LQGi controller, tuned for broad bandwidth (3) MIMO Repetitive controller asymptotic regulation and tracking of periodic signals based on model matching of non-minimum phase systems The remainder of this paper is organized as follows: Section 2 describes the AMBS control system instrumentation, system identification, and the stabilizing LQGi tuning; Section 3 presents the MIMO repetitive control design; Section 4 presents the experimental results followed by conclusions in Section 5. 193
2. SYSTEM IDENTIFICATION & STABILIZING CONTROL The experimental system shown in Figure 1 consists of an SKF AMBS instrumented with an induction motor drive, electromagnet power amplifiers, and a digital control system. The system has two radial AMBs and one axial AMB (Figure 2). The radial AMB system considered in this paper is four-input, four-output. National Instruments analog and digital conversion cards were used to interface the five analog gap sensors and ten electromagnetic coil power amplifiers to the digital computer. The system identification, data acquisition, and real-time digital control were realized by LabVIEW Real Time target installed on the computer, sampling at 10 kHz. The axes of the sensors are not co-located with those of the actuators, which introduces strong off-diagonal coupling. The electromagentic force is nonlinear with respect to the coil current and the air gap. Bias currents are applied to the opposing coils to render a linear approximation within ±100 microns from the AMBS centerline. In this paper, linear control is considered for spindle motion within the linear range. Since the system is open loop unstable a stabilizing controller was created and used to perform a closed-loop system identification of the open-loop plant. 2.1 High Order Multivariable ARX Identification A time-domain identification using Pseudo-Random Burst Sequence (PRBS) input data and Autoregressive Exogenous Input (ARX) modeling (Equation 1) was performed to identify the open-loop plant model. The A and B ARX coefficient matrices are both 4x4 while both y(t) and u(t) are 4x1 vectors. Given experimental input and output data, equation 1 can be cast as a ordinary least squares problem.
y(t) +
N
k=1
Ak y(t − k) =
N
k=0
Bk u(t − k)
(1)
In this application, a high order (≈ 120) ARX model was used to solve the least squares problem, which was then put into observer form (Equation 2). This translates the inputoutput form into a high order state space system, which is suitable for a balanced model reduction. Figure 3, shows the Hankel Singular Values of the high order ARX model from which a 20th order model was chosen. Interestingly, a high order model followed by a model reduction was seen to give a much better fit than simply solving for a low order
2016 IFAC MECHATRONICS 194 Sandeep September 5-8, 2016. Loughborough University, UK Rai et al. / IFAC-PapersOnLine 49-21 (2016) 192–199
2.2 Linear Quadratic Control Design
Hankel Singular Values (State Contributions) 0.8
Stable modes
In order to stabilize the open loop plant and attenuate low frequency disturbances, a Linear-Quadratic-Gaussian with Integral action (LQGi) controller was designed. The main design considerations for the LQGi controller were bandwidth, saturation, and stiffness. Obviously, higher inner loop bandwidth is desired, but comes at a cost of saturation and robustness. Hardware considerations dictated the use of a more conservative controller for the initial lift of the rotor; once the rotor is safely levitated, a more aggressive controller can be applied.
0.7
0.6
State Energy
0.5
0.4
0.3
0.2
0.1
0 10
20
30 State
40
50
60
Fig. 3. First 60 Hankel Singular Values of High Order State Space Model
ARX model. Furthermore, the least squares solution can be computed efficiently with the large amounts of data. In order to assess the effectiveness of the identification model, four separate sine sweep experiments were performed. Each experiment excited only one input of the closed loop system, to obtain frequency response data for frequencies ranging from 10 to 3500 Hz. The reference, plant input, and plant output were recorded for each frequency and processed to obtain a matrix of frequency responses, from r to u and from r to y (figure 7), as shown in Equation 3. ˆ ur (jωk ) = Um (jωk ) , h l,m Rl (jωk )
l = 1...4, m = 1...4
−1 ˆ yr (jωk ) H ˆ ur (jωk ) ˆ yu (jωk ) = H H
(3)
LQGi control is well known, but the design process is briefly summarized here for completeness. Given a plant as in Equation 5, the overall LQGi controller is given by Equation 6. The inputs to the controller are the plant output and the error, r − y. A p Bp P (z) = (5) Cp 0
Ap − LCp − Bp Kx −Bp Ki L 0 0 I 0 I C(z) = (6) −Kx −Ki 0 0 The controller contains K and L which are the state feedback and observer gains, respectively. By the separation principle, the state feedback and observer gains can be designed separately by solving an Algebraic Riccati Equation (ARE). However, to incorporate integral action the plant needs to be augmented before solving the Riccati Equation as in Equation 7. The notation used here is that dlqr(A, B, Q, R) denotes the solution of the discrete-time ARE with state and output weighting matrices Q and R. The observer gain is independently solved as in Equation 8, then the overall controller is formed. The weighting matrices are used as tuning parameters.
(4) [Kx Ki ] = dlqr(
The open loop frequency response was then obtained through point-by-point matrix inversion, as shown in Equation 4. Figure 4 shows the raw Frequency Response Data (FRD) and the reduced state-space model’s Bode Magnitude plots. From this data, it was determined that the reduced model was accurate from DC to approximately 1kHz. The open loop frequency responses show resonant modes of the rotor at 1.1 kHz and 2 kHz, which appear strongly in the cross terms and make a decoupling transformation difficult to obtain. It is because of this difficulty in decoupling the input-output pairs that a truly MIMO control design approach is necessary.
−A1 I
0
0 .. . .. . .. . 0
I .. .
−A2 . . . A= .. . . .
. −AN
.. ..
.
. ···
0 ··· ··· . .. · · · .. . .. .. . . .. .. . . .. .. . . ··· ···
0 .. . .. .
, 0
I 0
(7) (8)
By combining the plant (Eq. 5) and the LQGi (Eq. 6), the closed loop plant G(z) can be obtained. Ap − Bp Kx −Bp Ki 0 −Cp I I G(z) = Cp 0 0
(9)
The Q and R weighting are design parameters. The order of the Q matrix is the order of the plant plus the four integrator states. Therefore, Q is partitioned as
B1 − A 1 B 0 B2 − A 2 B 0 .. . , B= . .. .. . BN − A N B 0
194
Ap 0 B , p , Qk , Rk ) −Cp I 0
LT = dlqr(ATp , CpT , Ql , Rl )
C = I 0 0 ··· ··· 0 ,
D = B0
(2)
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195
Bode Diagram
To: Out(4)
To: Out(3)
Magnitude (dB)
To: Out(2)
To: Out(1)
From: In(1)
From: In(2)
From: In(3)
From: In(4)
Raw FRD Open-Loop Model
50 0 -50 50 0 -50 50 0 -50 50 0 -50 100
1k
100
1k
100
1k
100
1k
Frequency (Hz)
Fig. 4. Bode Magnitude Plot of Open Loop System
-10 -20
Gain [dB]
-30
-50
Position (um)
-70 -80 -90 10 -1
10 0
30
-60
Agressive LQGi Start up LQGi
10 0
10 1
10 2
10 3
1.72 1.74 Time (secs) V24
20 10 0 1.7
10 4
Frequency [Hz]
30
20
1.7
-40
V13 Position (um)
Position (um)
30
1.72 1.74 Time (secs)
W13
20 10 0 1.7 30
Position (um)
0
1.72 1.74 Time (secs) W24
20 10
Experiment Simulation
0 1.7
1.72 1.74 Time (secs)
Fig. 5. Maximum Singular Value of the Closed Loop Frequency Response
Fig. 6. Step Responses of LQGi Controller, Experiment and Simulation
in Equation 10, where α is the weight associated with the plant states and β corresponds to the integrator states. This reduces the parameters in the Q matrix to two, and when coupled with the regularization of the R matrix to the identity, we have only those two variables parameterizing the LQGi design. We found this to be an adequate amount of control to design a useable inner-loop controller.
design, Figure 6 shows good agreement between simulation and experimental results.
Q=
αI 0 0 βI
R=I
(10) (11)
Figure 5 summarizes the results of the inner-loop control design. As seen, the aggressive LQGi controller has a bandwidth of approximately 100 Hz and attenuates the resonant modes. Given actuator constraints, this was the highest comfortable bandwidth for a broadband controller, and is well within the first resonant mode. As a final verification of the system identification and controller 195
3. MIMO REPETITIVE CONTROL When the switch in Figure 7 is closed, the repetitive controller is added on to the existing LQGi control. Similar to single variable case, the main objective is to produce infinite gain at a desired frequency to track references or reject disturbances at its harmonics. This is accomplished by the internal model in Equation 12, where N is the number of samples constituting one period of the desired frequency. D(z) =
1 I zN − 1
(12)
In order to design a repetitive controller, filters F (z) and Q(z) need to be determined. The design of these filters is much more difficult when the plant is multivariable; methods such as ZPETC are single-variable formulations, and cannot be used.
2016 IFAC MECHATRONICS 196 Sandeep September 5-8, 2016. Loughborough University, UK Rai et al. / IFAC-PapersOnLine 49-21 (2016) 192–199
From Tsao (1994), it is known that the optimal F (z) is such that the optimal cost in Equation 14 is an allpass filter, and that increasing the preview length, N2 , decreases the error of the model matching problem.
z −N2
z −N1
Σ r
Q(z)
F (z) C(z)
Σ
u
P (z)
y
Σ
3.2 H2 Factorization Based Fig. 7. Block Diagram of the Plug-in Repetitive Control It is known that for the plug-in repetitive controller to be stable, the inequality in Equation 13 is a sufficient condition. (z −N2 I − F (z)G(z))Q(z) ∞ < 1
(13)
G(z) is the nominal closed-loop transfer function, from reference input to output with the LQGi controller given in equation 9. From the stability equation, it is easy to see that the inequality is satisfied when F (z) = G−1 (z). However, when the system is non-minimum phase, there will be an unstable pole-zero cancellation, so a na¨ıve inversion will not work. In this paper, two methods are proposed to design a MIMO inversion filter F (z): an H∞ approach, and an H2 inner-outer factorization approach. 3.1 H∞ minimization
Instead of solving the H∞ model matching problem for the stabilizing inversion filter, the H2 model matching problem can be solved: min (z −N2 M (z) − F (z)G(z)) 2 (16) F (z)∈RH∞
The solution to the H2 optimal problem is given in Equation 17: −N2 −1 Fopt = G−1 Gi (z)M (z)]+ (17) o (z)[z where the plant can be factored into its inner and outer systems H(z) = Gi (z)Go (z); the outer (minimum phase) part can be easily inverted, and the inversion of the inner part approximated by a FIR filter. A state-space realization of the inner and outer factors can be easily obtained by solving a generalized Riccati equation Ionescu and Oara (1996). FIR approximation of the unstable inverse of the inner factor is done by conjugating the conjugate system (Equation 18).
Motivated by the stability condition, the H∞ problem (Equation 14) can be solved for the inversion filter F (z). min
F (z)∈RH∞
(z −N2 M (z) − F (z)G(z)) ∞
(14)
Clearly, this is a model matching problem from r to e, as shown in Figure 8. If G(z) has a state space realization (Ap , Bp , Cp , Dp ) given by Equation 9, and z −N2 M (z) is given by (Am , Bm , Cm , 0), then Equation 15 is the generalized plant and the model matching problem can be solved by standard H∞ solvers. M (z)z −N2
r
F (z)
G(z)
G−1 i
=
G−T i
∗ 1 z
(18)
The conjugate system can be expressed as in Equation 19. Most importantly, it is stable, since the unstable poles of the inner factor are mirrored across the unit circle. Therefore, it can be put into infinite impulse response form. G−T i
1 −A−T Ci−T A−T i i = T B T A−T DiT − BiT A−T z i Ci i i ˜ A˜ B = ˜ ˜ C D k ∞ ˜ ˜ 1 +D C˜ A˜k−1 B = z
(19) (20) (21)
k=1
e
Fig. 8. Model Matching Interpretation of Stability Condition 0 Bp Ap 0 xp (k) xp (k + 1) xm (k + 1) 0 Am Bm 0 xm (k) (15) = C −C e 0 Dp r p m r u 0 0 I 0 N2 is a tuning parameter that depends on how close the non-minimum phase zeros of the system are to the unit circle. For the magnetic bearing system, there are nonminimum phase zeroes so this delay length must be nonzero. M (z) is a low pass filter that limits the bandwidth of the inversion filter, so that high frequency noise will not be amplified, which would cause noisy actuation and possibly instability.
196
Then the conjugate system can be truncated and conjugated, as in Equation 22. ∗ N 1 −T ˜ T = Kf ir (z) ˜ T (A˜T )k−1 C˜ T (z)k + D ≈ Gi B z k=1 (22) This results in a non-casual filter Kf ir (z) of order N , but the delays in Equation 16 make the first N2 coefficients casual. The final filter that solves the H2 problem (Equation 23) is the stable inversion of the inner factor, cascaded with the inversion of outer factor. Once the filter is designed, it must be substituted into the stability condition to verify that it satisfies the inequality. F (z) = G−1 o (z)Kf ir (z)
(23)
2016 IFAC MECHATRONICS September 5-8, 2016. Loughborough University, Sandeep UK Rai et al. / IFAC-PapersOnLine 49-21 (2016) 192–199
q(z) = (.25z + .5 + .25z −1 )Nq
Position (um)
5 H
0.48
0.49
0.5
-40
(24)
∞
0
-5 0.47
Power (dB)
Ideally, a properly designed inverse filter guarantees the stability of the repetitive control system and internal models at the necessary frequencies. However, at high frequencies, the model is usually not accurate, which can cause instability or undesirable behavior. The Q filter, which severely attenuates high frequencies, can provide robustness to this mismatch. For the magnetic bearing system under consideration, the Q filter is comprised of four SISO filters (Equation 24) in block diagonal form. Q(z) is noncausal, but the preview steps can be emulated.
197
0.51
0.52 0.53 Time (sec) Spectrum of Error
0.54
0.55
0.56
0.57
H2 Factorization LQGi No Disturbance
-60 -80 -100 -120
0
200
400
600 Frequency (Hz)
800
1000
1200
0 5
Factorization H∞
Position (um)
−5
Gain (dB)
−10
−15
H
∞
0
-5 0.47
0.48
0.49
0.5
−20
0.51
0.52 0.53 Time (sec)
0.54
0.55
0.56
0.57
-40 Power (dB)
−25
−30
−35 0 10
1
10
2
10 Frequency (rad/s)
3
10
4
Fig. 9. Maximum Singular Value of Stability Condition Figure 9 plots the maximum singular value of the stability condition as a function of frequency. Equation 14 requires that this value be less than unity for all frequencies, which is shown to be satisfied for both the H∞ and Factorizationbased controllers. 4. EXPERIMENTAL RESULTS To verify the disturbance rejection of the controllers on the physical system, the spindle was spun at 400 Hz. The control used a real-time target, sampling at 10 kHz. Both repetitive controllers used a 5th order Q filter with a cutoff at 800 Hz, which was designed with the 1 kHz bandwidth where the model is valid in mind. The H∞ design required 10 preview steps, while the factorization approach required 20—both are well within the period of the fundamental frequency of the repetitive controller, so the difference is not relevant. The disturbance caused by the spindle spinning at 400 Hz amounts to 24,000 RPM—well in the high speed machining region. Figure 10 shows both time-domain and spectral results of this experiment. The repetitive controller was switched on a 0.5 seconds. The red spectrum is the LQGi controller levitating the spindle with no disturbance injected, which is the baseline against which the others are compared. The only disturbance injected was at 400Hz; the other spikes are harmonics of the 60Hz mains hum, which neither controller was designed to reject. The harmonic at 300Hz is rejected by coincidence— the fundamental frequency of the Repetitive controllers internal model was set at 100Hz. The spectral plot show that both repetitive controller have control action at integers of 100 Hz as is expected, but tapers off after 600 Hz due to the Q filter. 197
∞
LQGi No Disturbance
-80 -100 -120
10
H
-60
0
200
400
600 Frequency (Hz)
800
1000
1200
Fig. 10. 400Hz Disturbance Rejection Results Controller H∞ Factorization
RMSe [µm] 0.68 0.82
Tracking
Controller H∞ Factorization
RMSe [µm] 0.46 0.52
Rejection
Table 1. Comparison of H∞ and Factorization Methods Results are similar in the tracking experiment: the main spectral error component is at the target frequency of 400Hz, but the second-largest component is the harmonics of the mains hum. Tracking involves the spindle moving outside of the identified region. The controllers are stable and able to track the reference with less than one micron of error, tracking a 15µm (peak-to-peak) reference. Tracking performance is good, without amplifying sideband frequencies. Results of both experiments are summarized in Table 1. The H∞ controller slightly outperforms the factorization based controller in terms of RMS error. Both control design methods are acceptable for repetitive internal model type controllers. The experimental data shows that significant harmonic errors at 60 Hz and its higher harmonics, caused by the AC electric power, were not accounted for. These components could have been eliminated by adding another repetitive control loop with the internal model at these frequencies. 5. CONCLUSIONS Based on the experience in the control design and experimental investigation for the AMBS reported in this paper, the following concluding remarks are made:
2016 IFAC MECHATRONICS 198 Sandeep September 5-8, 2016. Loughborough University, UK Rai et al. / IFAC-PapersOnLine 49-21 (2016) 192–199
LQGi Only
further investigation on system identification and control design and this is of future research interest.
Repetitive
Position
5
Position (um)
Position (um)
Reference
0 -5 0.05
0.055 0.06 0.065 Time (sec)
5
6. ACKNOWLEDGMENT
0
The authors are grateful to Dr. Larry Liang of FederalMogul Corporation for providing valuable information on AMBS for high-speed machining applications.
-5
0.07
3
3.005 3.01 3.015 Time (sec)
3.02
Power (dB)
-40 H
-60
REFERENCES
2
LQGi Levitating
-80 -100 -120
0
200
400
600 Frequency (Hz)
800
Position
LQGi Only
1000
1200
Repetitive
Position (um)
Position (um)
Reference
5 0 -5 0.1
0.105 0.11 0.115 Time (sec)
0.12
5 0 -5 3
3.005 3.01 3.015 Time (sec)
3.02
Power (dB)
-40 H
-60
∞
LQGi Levitating
-80 -100 -120
0
200
400
600 Frequency (Hz)
800
1000
1200
Fig. 11. 400Hz Tracking Results (1) The ARX method conducted within stabilized feedback loops followed by balanced truncation model reduction was efficient and effective in obtaining the open loop unstable MIMO AMBS dynamic model, which agrees well with the system’s frequency response data. (2) Based on the identified model, the optimal linear quadratic control with integral action was effectively tuned by only two scalar weighting gains—one on the plant’s states and one on the integrators—with the control weighting normalized to identity. (3) The plug-in MIMO repetitive controller was designed by optimal model matching, and either the H∞ or H2 norm minimization was effective, with similar performance in experiment. (4) Control experiments on the AMBS with 10kHz sampling implemented by the LabVIEW RealTime Target have demonstrated the LQGi and repetitive control performance in rejecting errors synchronous to the 400 Hz spindle speed rotation. (5) Repetitive control for multiple periods can be employed to reject AC electric power disturbances. The existing SISO approaches (P´erez-Arancibia et al. (2010),Kalyanam and Tsao (2012)) should be extended for the present MIMO system. These results also suggest that the AMBS, together with the control methods investigated herein, can be applied to high-speed machining applications where the synchronous spindle error motion is prevalent. The machining process, depending on the cutting conditions, may introduce additional disturbances and unmodeled dynamics that require 198
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