LPV control of an active vibration isolation system

Transcription

1 29 American Control Conference Hyatt Regency Rierfront, St. Lois, MO, USA Jne 1-12, 29 ThC15.1 LPV control of an actie ibration isolation system W.H.T.M. Aangenent, C.H.A. Criens, M.J.G. an de Molengraft, M.F. Heertjes, and M. Steinbch Department of Mechanical Engineering, Eindhoen Uniersity of Technology Abstract Non-stationary distrbances in motion systems generally limit the closed-loop performance. If these distrbance can be measred, this measrement can be sed to enable a linear parameter arying (LPV) controller to adapt itself to the crrent operating condition, reslting in a closed-loop system with an oerall increased performance. In this paper, this idea is applied to an actie ibration isolation system. Index Terms LPV, ibration isolation, freqency estimator f d payload k b f a y d I. INTRODUCTION In high-precision motion systems, ibrations of nknown freqency and amplitde can seerely limit performance. These ibrations can for instance be indced by floor ibrations that are transferred to the system or by forces that directly act on the system sch as (conditional) flows. Examples of processes where this occrs inclde highresoltion measrement eqipment sch as scanning electron microscopes 1 sed for sb-micron imaging, and photolithographic wafer steppers and scanners 2 sed for fabrication of integrated circits. For these systems, the typical amplitdes of the indced ibrations are of the same magnitde as the dimensions of the measred or manfactred objects, and therefore these ibrations limit performance. To proide sch systems with a ibration free platform, ibration isolation systems are sed, where distrbance isolation is achieed passiely, actiely, or both 3. Figre 1 shows a schematic representation of sch a ibration isolation system. Herein, k, b are the isolator stiffness and damping, respectiely, d represents the floor ibrations and f d denotes the distrbance forces that act directly on the payload. The payload displacement in ertical direction is denoted by y, and f a is the force generated by an actator. The goal of a ibration isolation system is to minimize the effect of the enironmental distrbances d and f d on the ertical elocity = ẏ of the payload. The passie part of the ibration isolation system consists of a heay payload spported by elastic springs and damping nits, reslting in a mechanical low-pass system with typical resonance freqencies between 2 and 5 Hertz. Since the passie system is generally weakly damped, distrbance amplification at the natral freqency is often obsered. Increasing the strctral damping of the system may offer a soltion, howeer, a major disadantage of passie damping is that the distrbance rejection properties deteriorate 4. This is illstrated in Figre 2, where a typical freqency response of the transfer from f d to y is depicted. Moreoer, the passie type is, in principle, nable to attenate distrbances f d that directly act on the payload sch as an exiting force generated in a monted operating machine, an airflow from the air conditioning, Fig. 1. Gain db Schematic representation of a ibration isolation system Fig. 2. Mechanical low-pass characteristic of a passie isolation system with low (black) and high (grey) strctral damping. and acostic excitations. By applying actie damping near the natral freqency of the passie system, i.e., by actie ibration isolation, a significant benefit in ibration isolation can be obtained. In the actie ibration isolation system (AVIS) we consider, actie ibration isolation is achieed by controlled actation of the payload, based on feedback of its payload elocity. The payload elocity is directly measred throgh geophones while the actation is performed by means of Lorentz actators. A pictre of the AVIS is shown in Figre 3(a). In this paper, the payload of the AVIS is sed as an experimental benchmark representing a metrology frame that needs to be isolated from enironmental distrbances. This means that the amplitde of the ertical elocity shold be controlled to be as low as possible. A machine is monted to this metrology frame that performs periodic tasks, which reslt in non-stationary enironmental distrbances f d with freqency content between 4 and 1 Hz, depending on the specific task that is performed. This machine is represented by a rotational imbalance, depicted in Figre 3. The indced distrbances f d are not known beforehand and cannot be /9/$ AACC 373

2 (a) (b) Fig. 3. (a) The actie ibration isolator platform inclding the monted rotating imbalance, (b) a close-p of the rotating imbalance. measred directly. As a conseqence, direct distrbance compensation ia feedforward cannot be applied. Althogh other distrbance sorces are present as well, the periodic distrbance generated by the monted machine is expected to dominate the measred otpt. All that is known abot the indced distrbance is that it consists of a single, timearying freqency in the range between 4 and 1 Hz. The objectie is to improe the isolation performance beyond the leel of a commonly sed linear time inariant (LTI) controller. Althogh sch an LTI controller redces the effect of distrbances in the freqency range 4 to 1 Hz, the achieed isolation performance is considered not to be adeqate. To improe the isolation performance, the LTI controller will be adjsted in order to proide additional distrbance redction. A classical soltion to redce distrbances at specific freqencies is the se of an inerted notch filter in the controller to increase the gain at that freqency. Sch a notch filter reslts in a decreased sensitiity for distrbances in a small freqency range arond the center freqency of the notch. Unfortnately, a classical notch filter cannot be applied to the posed problem for two reasons. Firstly, the freqency of the expected periodic distrbance aries in an interal between 4 and 1 Hz, while a fixed notch is only effectie arond a single freqency. Secondly, een if the distrbance signal wold be stationary instead of time-arying (for instance when the monted machine is operating in a stationary condition), the actal distrbance freqency, and hence the target center freqency of the notch, is not known beforehand. In this paper, we propose a soltion to improe the isolation performance that circments these two problems. The problem of the nknown freqency will be tackled by deising an algorithm that identifies the dominating distrbance freqency from aailable measred data from the AVIS. The problem of freqency ariation can be soled by extending controller (2) with a notch of which the center freqency can be aried. This way, a linear parameter arying (LPV) controller is obtained that can adapt the center freqency of the notch to the identified distrbance freqency, reslting in an improed isolation performance at that freqency. The paper is organized as follows. In Section II the spectral analysis method to identify the dominating distrbance freqency is discssed. The design of the LPV controller is done in Section III, while the closed-loop stability is assessed in Section IV. The interconnection of the signal analyzer and the controller is treated in Section V while the proposed approach is simlated in Section VI. Experimental alidation is done in Section VII. Finally, we conclde in Section VIII. II. SPECTRAL ANALYSIS The posed problem reqires a freqency identifier that is able to determine the actal freqency of the distrbance from the measred signals of the AVIS. Since the proposed LPV controller is reqired to adapt itself to the distrbance that acts on the system, the distrbance freqency at the crrent time instant shold be aailable. This information can be obtained throgh time-freqency spectral analysis. Since in practical applications low freqency components often last a long period of time, while high freqency components often appear as short brsts, a so-called mltiresoltion spectrm is desirable. A mltiresoltion spectrm combines a high freqency resoltion (with a corresponding low time resoltion) for low freqencies with a high time resoltion (and hence a low freqency resoltion) for high freqencies. The waelet transform 5 was deeloped to obtain sch a mltiresoltion analysis. Therefore, we se waelet analysis to obtain a spectrm analyzer which proides the dominating distrbance freqency at the crrent time. The waelet that is sed is a trncated Morlet waelet 5 and the real-time implementation of the transformation is inspired by 6. d Σ Fig. 4. III. CONTROLLER DESIGN bs + k Σ f d schematic of the controlled isolation system. The schematic of one of the controlled ibration isolators is depicted in Figre 4, where =f a is the controller otpt, =ẏ is the payload elocity in ertical direction, G denotes the transfer fnction of an isolator modle, and C is the C G y s 3731

3 controller. A forth-order model of an isolator of the AVIS is gien by the transfer fnction 7 G(s) = m 2 s 2 + b 12 s + k 12 m 1 m 2 s 4 + (m 1 + m 2 )(b 12 s 3 + k 12 ), (1) with m 1 = 95 kg, m 2 = 5 kg, b 12 = Nsm 1, and k 12 = Nm 1. The isolator s passie stiffness and damping are k = Nm 1 and b = Nsm 1, respectiely. To redce the effect of distrbances near the resonance freqency, controller C is designed as the complex aled damper 7 ( ) 2 ( ) 2 s ωlp C(s) = k d, (2) s + ω hp s + ω lp which consists of a gain k d, combined with a low-pass filter with ct-off freqency ω lp, and a high-pass filter with ctoff freqency ω hp. The choices for ω hp and ω lp are related to the limitations of the sensors and actators, respectiely. Since the geophones prodce nreliable otpt below.1 Hz, and since actator limitations occr beyond 1 Hz, the ct-off freqencies are chosen as ω hp =.2π rad/s and ω lp =2π rad/s, and the gain is k d =3 1 4 Nsm 1. This controller is designed to presere the desirable properties at high freqencies of the passie isolator, while it significantly redces the effect of distrbances arond the resonance freqency. In Figre 5 the Bode diagram of the ncontrolled plant (i.e., of the passie ibration isolation system) P p (s) = (s) f d (s) = sg(s) 1 + (bs + k)g(s), (3) is compared with that of the the controlled plant (i.e., of the actie ibration isolation system) P a (s) = (s) f d (s) = sg(s) 1 + (bs + k)g(s) + sc(s)g(s). (4) The maximm amplitde occrs arond the resonance freqency, which implies that the plant still is most sensitie to distrbances in this freqency region. Howeer, the sensitiity in this region is redced by abot a factor 17. The price paid is a slightly increased amplitde between 1 and 2 Hz. The freqency of the dominating distrbance that acts on the system can be identified by sing a waelet-based algorithm as discssed in the preios section. This information can be sed to adapt controller (2) in order to improe the isolation performance at that freqency. Sch a controller can be designed ia the LPV synthesis framework 8 sing freqency and parameter dependent weightings expressing the desired performance. For this specific application howeer, it is expected that the desired performance can be achieed by extending controller (2) with an LPV notch filter, the center freqency of which is adapted to the identified distrbance freqency. Therefore, in this section we will design the controller by hand and analyze the closed-loop stability afterwards. An LTI notch filter is described by the transfer fnction H n (s) = s2 + 2β 1 (2πf n )s + (2πf n ) 2 s 2 + 2β 2 (2πf n )s + (2πf n ) 2, (5) Gain db Phase deg Fig. 5. The Bode diagram of the transfer fnction from the enironmental distrbance f d to the measred otpt of the passie isolation system (black) and of the actie isolation system (grey). where f n is the center freqency in Hz, and β 1 and β 2 are parameters that can be sed to alter the redction factor and the width of the notch. Althogh ideally all three parameters f n, β 1, and β 2 shold be adapted depending on the distrbance characteristic, in this paper we only consider the adaptation of the center freqency f n. Seeral factors play a role in the determination of the constant redction factor and width of the notch filter. If only steady-state distrbance rejection arond the center freqency f n is considered, a wide notch with a large redction factor is desirable. Indeed, a wide notch will not only redce distrbances at f n, bt has also good distrbance rejection properties for freqencies near f n, while a larger redction factor reslts in a higher redction. Unfortnately, there are also downsides to sing sch a notch filter. Widening the filter redces the distrbances in a larger area arond the center freqency, bt at the same time reslts in amplification of distrbances in another, larger, freqency region. This is illstrated in Figre 6(a), where the freqency response of the original actiely controlled plant (4) is compared to that of the same controlled plant inclding a notch filter of different widths. The same holds if the redction factor of the notch is increased. Another isse related to the width and redction factor of a notch is its transient response time. A fast transient response of the notch filter is desirable since it allows a fast adaptation of the controller to the crrent distrbance freqency. A wide notch filter has a shorter transient response time than a narrow one, jst as a filter with a smaller redction factor has a shorter transient response time than one with a higher redction factor. The effect on the transient response cased by widening the notch filter is illstrated in Figre 6(b), where the response of the controlled plant to a sinsoidal distrbance of 1 Hz is depicted with the same notch filters of different width. The transient response time of the notch filter depends on the center freqency similarly as the freqency identification time of the real-time freqency analyzer. Therefore, in a 3732

4 Gain db ẏ x (a) Time s (b) Fig. 6. Bode diagram of the original controlled plant (4) (dashed) and two notches of different widths (solid black and grey). (a) Effect of widening the notch on the freqency response, (b) effect of widening the notch on the transient response time. proper oerall controller design, the transient response time of the notch filter for a certain freqency, and the time it takes to identify that freqency shold be matched. We will se a notch that redces the amplitde of the identified sinsoidal distrbance to approximately half of the final redction in 3 periods. Frthermore, we choose a redction factor of 3. The aboe discssion reslts in β 1 =.6 and β 2 =.2. IV. STATE-SPACE REALIZATION AND STABILITY To arrie at a closed-loop state-space system from distrbance inpt f d to measred otpt, we start by representing the ncontrolled AVIS (the passie ibration isolation system) (3) in state-space form as ẋ = Ax + Bf d = Cx, where x R 4 is the state ector, and A,B, C are matrices of appropriate dimensions. The state-space description of the original controller (2) is gien by x c o = A c ox c o + B c o = C c ox c o, where x c o R 4 is the original controller state ector, and A c o,b c o,c c o are matrices of appropriate dimensions. This controller will be agmented with a notch filter that adapts (6) (7) itself to the dominating distrbance. The LTI notch filter (5) can be changed to an LPV notch by sing the center freqency as the schedling ariable, i.e., := 2πf n. This LPV notch can be represented by arios state-space realizations. It is well-known that the complexity and conseratism of the stability and performance analysis of LPV systems depends on the type of parameter dependence of that system 8. For instance, systems that are affinely parameterized allow for a qadratic stability analysis withot a relaxation gap, and therefore we aim at an affinely parameterized closed-loop system. Since the state-space description of the controlled AVIS is affine in the controller state-space matrices, an affinely parameterized state-space description of the controller reslts in an affinely parameterized state-space description of the controlled AVIS. The LPV notch will be described by the modal canonical state-space representation ẋc A c = () B c() C c() Dc () x c, (8) where x c R2 is the notch state ector and where A c () B c() C c() Dc () = β 2 β2 2+1 β β 2 ( ) ( ) (β 1 β 2) β 2 2 β2 β (β 1 β 2) β 2 2 +β2 β , β2 2 1 β (9) which is indeed affine in. The series connection of (8) and the original controller (7) reslts in the affine parameter dependent controller ẋc where x T c = x ct Ac () B c () C c () D c () Ac () B = c () C c () D c () x ct o T and = xc, (1) A c () Bc () B c oc c () Ac o B c od c () C c o (11) The closed-loop state-space system from distrbance inpt f d to measred otpt is then gien by ẋcl A() B() = C() D() xcl f d., (12) where x T cl = xt x T c T and A BC A() B() c () B = B C() D() c ()C A c (). (13) C Note that (12) is affine in the schedling parameter. Stability of affine parameter dependent systems sch as (12) can be assessed by soling a set of LMIs. The complexity of this set depends on the allowed rate of ariation of the schedling parameter. In or application, the operating condition of the monted machine may be changed arbitrarily 3733

5 and hence the schedling parameter may ary arbitrary fast. Affinely parameterized system (12) with an arbitrary fast arying parameter, is qadratically stable if and only if there exists K =K T sch that 8 A() T K + KA() for all {,}. (14) Or application aims at redcing sinsoidal distrbances within the freqency range 4 to 1 Hz and therefore the schedling parameter will be restricted to lie within the interal 2π4,2π1. The LMI soler SeDMi 9 is sccessflly sed to find a positie definite matrix K R 1 1, which satisfies (14). We now hae that system (12) is stable for arbitrarily fast parameter ariations in the range 2π4,2π1. V. INTERCONNECTION OF THE SPECTRUM ANALYZER AND THE CONTROLLER Since both the real-time spectrm analyzer and the LPV controller (1) hae been designed, they can be interconnected to obtain the complete control setp. Seeral measred signals cold be sed to identify the dominating distrbance freqency. A logical first choice wold be to se the ertical elocity of the AVIS since the goal of the controller is to minimize this elocity. A better alternatie is to se the controller otpt as the inpt to the spectrm analyzer, becase in contrast to the ertical elocity of the AVIS, the amplitde of the distrbance freqency will not decrease de to the sppression of the controller in this signal. As long as the distrbance is acting on the AVIS, its freqency is present in the controller otpt. A schematic oeriew of the complete control system is shown in Figre 7. Spectrm Analyzer Controller AVIS Fig. 7. Schematic representation of the implementation of the controller and the spectrm analyzer. freqency of the dominating distrbance. The distrbance isolation performance is depicted in Figre 8, where the spectrm of the ertical elocity is shown. It is obios that the LPV controller achiees a mch better distrbance rejection than the original controller. A slight increase of the ertical elocity is obsered when the LPV controller is schedled with the estimated freqency instead of the actally applied distrbance freqency, especially at low freqencies. This is cased by the delay that is introdced by the real-time spectrm analyzer. In the following section the Fig. 8. Simlated signal and its spectrm. Dark red indicates no freqency content while ble indicates maximm freqency content. proposed control setp is applied to the experimental setp. VII. EXPERIMENTAL RESULTS In this section, experiments will be performed on the experimental setp as depicted in Figre 3. The rotating imbalance is sed to generate distrbance forces f d that directly act on the payload. The imbalance is represented by a mass m d located at a distance e d from the center of rotation. For constant anglar elocities ω d, the indced distrbance force on the payload eqals f d = m d e d ω 2 d sin(ω d t). (15) VI. SIMULATION To alidate the proposed control setp, a simlation is performed. In this simlation the AVIS is sbjected to a sinsoidal distrbance f d to simlate a distrbance force that acts on the payload. The amplitde of the sinsoid is constant, bt the freqency of the distrbance force is continosly arying. The controller is implemented in three different ways. Dring the first 5 seconds, the original LTI controller (2) is applied. After 5 seconds the LPV controller (1) is sed, where the schedling parameter is taken to be the actally applied sinsoidal distrbance f d. Finally, after 1 seconds, the controller implementation from Figre 7 is sed and the schedling parameter is replaced with the estimated These distrbance forces are sinsoids with freqency ω d/2π Hz and an amplitde that is proportional to ωd 2. The rotational elocity of the mass can ary in time, bt is restricted between 4 and 1 reoltions per second. As mentioned before, enironmental distrbances are present as well, bt distrbances from the rotating mass dominate the distrbance spectrm. Dring this experiment, the rotational elocity ω d of the imbalance is controlled to ary continosly between 4 and 1 reoltions per second. The ariation is obtained by sing a sinsoidal reference elocity profile for the imbalance with a freqency of.2 Hz. The first half of the experiment, p to 5 seconds, the original LTI controller (2) is applied to be 3734

6 able to qantify the performance improement of the LPV controller. After 5 seconds the LPV controller (1) is sed together with the real-time spectrm analyzer. Figre 9 shows the estimated freqency of the dominating distrbance by the real-time spectrm analyzer. The measred ertical elocity of the AVIS and the off-line compted time-freqency spectrm are depicted in Figre 1. Seeral obserations can be made from this figre. The oerall error is decreased by a factor 3 to 5, depending on the distrbance freqency. When the LPV controller is schedled to increase the performance arond 1 Hz an increase in the error leel in the freqency range 1 to 15 Hz can be obsered. This is in agreement with Figre 6, where it was shown that the application of a notch arond 1 Hz reslts in amplification of the enironmental distrbances in that range. As a final obseration we note that there is a freqency component arond 13 Hz in the ertical elocity of the payload. Althogh this cannot be explained from the first principles model of the isolator (1), freqency response measrements show that indeed a resonance freqency is present in this range. It is clear from this experiment that the LPV controller can offer a major increase in performance Fig. 9. Time s Estimated freqency. VIII. CONCLUSIONS In this paper we proposed a nonlinear controller setp for the actie control of a ibration isolation system. This setp consists of two parts: (i) a real-time mltiresoltion spectrm analyzer that is able to identify the crrently dominating distrbance, and (ii) an LPV controller that adapts itself to the aailable distrbance information. This reslted in a control system that is able to adapt itself to the crrent operating condition reslting in a closed-loop system with an oerall increased performance when compared to an LTI controller. The part of the controller that was schedled according to the identified distrbance freqency was a notch filter. Sch a notch filter has a certain width and redction factor. These design parameters shold be chosen with the application and expected distrbance ariation in mind. A wide notch with a large redction factor reslts in good distrbance redction properties arond the center freqency of the notch. Howeer, at the same time enironmental distrbances in other freqency regions will be amplified by sch a notch. A too narrow notch with a small redction factor on the other hand may not be adeqate to redce the dominating distrbance at the center freqency. The proposed control system was alidated by experiments that were performed on an actie ibration isolation system. Compared to the originally designed LTI controller, the LPV control system was able to decrease the error by a factor 3 to 5 in case the distrbance spectrm is dominated by a single sinsoid with arying freqency. REFERENCES 1 C. Jlian Chen, Introdction to Scanning Tnneling Microscopy. Oxford Uniersity Press, H. Yoshioka, Y. Takahashi, K. Katayama, T. Imazawa, and N. Mrai, An actie microibration isolation system for hi-tech manfactring facilities, Jornal of Vibration and Acostics, ol. 123, no. 2, pp , L. Zo and J.-J. Slotine, Actie control ibration isolation sing dynamic manifold, The Jornal of the Acostical Society of America, ol. 122, no. 6, p. 3148, E. Riin, Passie ibration isolation. New York: ASME, P. Flandrin, Time-Freqency/Time-scale analysis. London, UK: Academic Press, M. Vrhel, C. Lee, and M. Unser, Rapid comptation of the continos waelet transform by obliqe projections, IEEE Trans. Signal Processing, ol. 45, no. 4, pp , M. Heertjes, N. an de Wow, and W. Heemels, Switching control in actie ibration isolation, in Proceedings of the 6th EUROMECH Nonlinear Oscillations Conference (ENOC), Saint Petersbrg, Rssia, C. Scherer, Robst mixed control and LPV control with fll block scalings, in Adances in linear matrix ineqality methods in control, L. Ghaoi and S. Niclesc, Eds. Philadelphia: Siam, J. Strm, Using SeDMi 1.2, a Matlab toolbox for optimization oer symmetric cones, Optimization methods and software, ol , pp , Fig. 1. Measred signal and its spectrm. Dark red indicates no freqency content while ble indicates maximm freqency content. 3735