Active Magnetic Bearings: Robust performance against uncertainty in rotational speed

5th IFAC Symposium on Mechatronic Systems Marriott Boston Cambridge Cambridge, MA, USA, Sept 13-15, 2010

Active Magnetic Bearings: Robust performance against uncertainty in rotational speed ⋆ H. M. N. K. Balini ∗ Jasper Witte ∗ Carsten W. Scherer † Sjoerd Dietz ‡ ∗

Delft Center for Systems and Control, Mekelweg 2, Delft 2628CD, The Netherlands (e-mail: [email protected]). † Department of Mathematics, University of Stuttgart, Pfaffenwaldring 57, D-70569 Stuttgart, Germany (e-mail: [email protected]). ‡ ALTRAN GmbH & Co. KG, Germany (e-mail: [email protected]). Abstract: An unbalanced mass on a rotating spindle causes synchronous vibrations. Active magnetic bearings have been successfully used to mitigate these vibrations for a particular operating speed. This is done by placing a notch in the closed-loop at a frequency corresponding to the rotational speed. In applications such as micro-milling the operational speed of the spindle can very well deviate from the desired set point. These deviations in the rotational speed can be regarded as an uncertainty in the disturbance model affecting the plant. The controller synthesis problem for robust performance against uncertainty in the disturbance model is recently shown to be convex. We design and implement such a robust controller on an experimental AMB setup. Keywords: Active Magnetic Bearings, Mass-imbalances, Robust control, LMIs. 1. INTRODUCTION In recent years, active magnetic bearings (AMBs) are used in a number of industrial applications, namely, hard disk drives, flywheel energy storage systems, and milling machines. In these applications, it is highly desirable to reject sinusoidal disturbances which occur due to an unbalanced mass. It is well known that the frequency of this periodic disturbance is equal to the rotational speed of the spindle, Matsumura et al. (1981), Matsumura and Yoshimoto (1981) and Matsumura et al. (1990). In applications such as micro-milling, the rotational speed of the spindle could deviate by a small percentage from its intended operating speed, owing to tool-workpiece interaction. The variations in the frequency of the sinusoidal disturbance affecting the plant are often unknown. Hence, we regard this variation as an uncertainty in the disturbance model and design a robust controller. In practice, rejecting sinusoidal disturbances of a particular frequency amounts to shaping the closed-loop sensitivity to have a very low or zero gain at the desired frequency. Conventionally, this is achieved by placing a carefully tuned notch-filter in the loop without compromising system stability. Controllers synthesized using µ/H∞ methods use a disturbance model or a weighting function to shape the closed-loop sensitivity. This explicit characterization of the disturbances allows us to consider an uncertainty in the underlying parameters (such as rotational speed). In addition to disturbance rejection, we desire ro⋆ We gratefully acknowledge the support from MicroNed.

978-3-902661-76-0/10/$20.00 © 2010 IFAC

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bustness against uncertainties in the system dynamics. We translate these specifications into an H∞ control problem for robust performance against an uncertain disturbance model. The corresponding synthesis problem is shown to be a convex optimization problem in Dietz et al. (2007). We showed the performance benefits of such a controller on a simulation model of an AMB system in Dietz et al. (2008). In this paper, we design and implement a controller based on Dietz et al. (2007) on an experimental AMB setup, the MBC 500. In general, the control synthesis problem for robust performance is non-convex when the uncertainty affects the plant model. An iterative method known as µ synthesis/DKiteration is often used to obtain a sub-optimal controller. For the problem at hand, the uncertainty block affects the input weighting function alone and not the plant. This allows us to formulate the controller synthesis problem as a convex optimization problem in Linear Matrix Inequalities (LMIs). A detailed derivation of the synthesis algorithm can be found in Dietz et al. (2007). A controller based on H∞ loop shaping design procedure (LSDP) is applied to an experimental AMB system in Fujita et al. (1993a). The designed controller guarantees robustness against uncertainties in the plant dynamics. The LSDP method was extended to reject sinusoidal disturbances of a particular frequency in Fujita et al. (1993b). A Youla parametrization based controller synthesis method is applied to a simulation model in Mohamed and Busch-Vishniac (1996). In both these references the authors choose a controller containing a pole on the imag10.3182/20100913-3-US-2015.00059

Mechatronics'10 Cambridge, MA, USA, Sept 13-15, 2010

inary axis (at the frequency location of the disturbance) from the set of all internally stabilizing H∞ controllers. Such a controller leads to a system zero in the closedloop sensitivity at the desired location, this translates to a perfect rejection of sinusoidal disturbances. In addition to mass imbalance vibrations, AMB spindles experience disturbances due to sensor run-out. An adaptive algorithm to estimate these two disturbance components and cancel their effects is given in Setiawan et al. (2002). A linear regression problem is used to estimate the periodic disturbances. AMB spindles are inherently unstable systems and multi-variable in nature. A generic method to include multi-variable notch-filters in the loop for massimbalance disturbance rejection is given in Herzog et al. (1996). In this paper we design H∞ based controller(s) and use a notch-type weighting function to achieve disturbance rejection at the output. An uncertainty in the rotational or operating speed is characterized by an uncertainty in the frequency location of the notch in the performance weighting function. We use such an uncertain weighting function at the plant output to formulate a robust performance controller synthesis problem. We transform this problem to a controller synthesis problem for robust performance against uncertainty in the input weighting function. We synthesize and implement controllers for 2% and 4% uncertainty in the rotational speed and compare the performance with a nominal design that does not account for any uncertainty. In order to experimentally evaluate the synthesized controllers, we deliberately use a poorly balanced spindle that is suspended by active magnetic bearings. Moreover, the spindle exhibits flexible dynamics and this introduces additional complexity to the controller design problem. The rotors or spindles used in industrial applications are generally well-balanced. They can be considered as rigid for practical purposes, as the flexible modes are absent or occur at very high frequencies. The results presented in this paper serve as a proof of concept for synthesizing robust controllers when one encounters poorly balanced rotors.

Fig. 1. MBC 500 Experimental setup The inputs and outputs to the system are the incremental voltages to the electro-magnets and the displacements of the spindle from the center line respectively. We performed a system identification experiment in closed-loop to obtain input-output data for the dynamics in the horizontal and vertical planes. We used the predictor-based-subspace identification algorithm, PBSIDopt , to obtain state-space models, Balini et al. (2010a). The identified models contain the flexible modes of the spindle which are all neglected to obtain reduced-order models. We design robust controllers using the reduced order models in each plane. For reasons of symmetry and ease of presentation, we explain the design procedure for the horizontal plane alone. The timedomain results presented in a later section are, however, obtained with controllers implemented on both planes. The frequency response for the dynamics including flexible modes in the horizontal plane is shown by the dashed line in Figure 3. The eigenfrequencies of the flexible modes are located around 1090 rad/s and 2385 rad/s. This corresponds to rotational speeds of 10400 and 22800 rpm respectively. The reduced-order model representing only the rigid-body dynamics is shown by a solid line in the same figure.

2. SYSTEM DESCRIPTION The MBC500 experimental setup used in this research is shown in Figure 1. It consists of a horizontally suspended spindle supported by two magnetic bearings at its ends. The system is provided with 4 decentralized SISO controllers which can be switched on/off. These inputoutput ports can be connected to a digital processor via A-D and D-A convertors. The dSPACE DS1103 processor board is used for data-acquisition and robust controller implementation. A schematic view of the MBC500 magnetic bearingspindle-sensor assembly is shown in Figure 2. A pair of opposing electro-magnets, and position sensors are located in the horizontal and vertical planes on either ends of the spindle. The system dynamics in the horizontal and vertical planes are assumed to be decoupled, by neglecting gyroscopic and electro-magnetic coupling. Hence, the entire AMB setup is modeled by two 2×2 MIMO LTI subsystems. 356

Fig. 2. Magnetic bearing-spindle-sensor assembly

3. CONTROLLER DESIGN We first synthesized an H∞ controller using the identified model, that is a model containing the flexible dynamics. The resulting controller is found to be unstable and it could not levitate the spindle from a non-equilibrium position. It is well known that the existing Riccati and

Mechatronics'10 Cambridge, MA, USA, Sept 13-15, 2010

et al. (1981) and Matsumura and Yoshimoto (1981). In practice, disturbance attenuation is achieved by shaping the closed-loop sensitivity. We choose a first order weighting function Wy (see below) to specify the desired sensitivity profile. For rejecting sinusoidal disturbances at a particular rotational speed (say θ × 60/2π rpm), we desire a notch in the closed-loop sensitivity at the corresponding frequency of θ rad/s. To specify this, we combine Wy with the weighting function Wn , where s2 + ζ1 s + θ2 s/M + ωB I. (2) I and Wn = 2 Wy = s + ωB A s + ζ2 s + θ2 {z } |

Bode Diagram From: In(1)

From: In(2)

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Fig. 3. Flexible model, nominal model and uncertainty weight LMI based solvers for H∞ control often return an unstable controller. The flexible dynamics occur as pole-zero pairs around the imaginary axis. The synthesized H∞ controllers have a tendency to cancel such pole-zero pairs and this is particularly visible in standard S/SK designs, Sefton and Glover (1990). We observe that the zeros of the flexible AMB spindle are canceled by the unstable poles of the synthesized H∞ controller. It is however possible to implement to unstable controller for overall improvement in performance. The AMB-spindle can be first levitated using a stable controller and then the unstable controller can be gradually switched on. The procedure relies on the Youla parametrization of the constituent controllers and is explained in detail in Balini et al. (2010b). In this paper we intend to restrict ourselves to stable controllers. Hence we use a reduced order model that is obtained by neglecting the flexible dynamics. The neglected dynamics is accounted for by means of an additive uncertainty. We desire the controller to be robustly stabilizing against this uncertainty. We are not duly concerned about robust performance against this uncertainty. This is due to the fact that our performance requirements are strict in the low frequency region and the additive uncertainty affects only the high frequency dynamics. The uncertain plant model is then Gunc = G + W∆ ∆, k∆k∞ < 1. (1) For the horizontal plane dynamics, Figure 3 shows the identified flexible model by the dash-dotted line, the nominal plant model G by the solid line and the additive uncertainty weight W∆ by the dashed line. For simplicity, the weighting function W∆ only accounts for uncertainties in the diagonal blocks of the system. Note that the rigidbody behavior is prominent in the low-frequency region. Neglecting the flexible dynamics only influences the offdiagonal blocks of the system. 3.1 Nominal Performance specification An unbalanced mass can be regarded as a sinusoidal disturbance acting on the output of the plant, Matsumura 357

An upper bound for the desired closed-loop sensitivity is obtained by frequency response of the inverse of the weighting function Wn Wy , as shown in Figure 4. The parameters of the weighting functions used for our controller synthesis are given in Table 1. Bode Diagram 5 0 −5 −10 Magnitude (dB)

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Fig. 4. Inverse of the performance weight Table 1. θ = θ0 250

ζ1 50

ζ2 5

A 0.01

M 1.5

ωB 1

Nominal Design We first synthesize a nominal controller K0 without considering any uncertainty in the rotational speed. The design objectives are robust stability against uncertain system dynamics and nominal performance for disturbance rejection. In other words, the bounds on the H∞ norm of the closed-loop transfer matrices are specified as • Robust stability kW∆ K0 (I + GK0 )−1 k∞ ≤ 1, • Nominal performance k(Wn Wy )(I + GK0 )−1 k∞ ≤ 1. The interconnection shown in Figure 5 is used to synthesize a controller K0 . The output weight W∆ accounts for the additive uncertainty affecting the plant. Moreover, it aids in shaping the frequency response of the controller to have desired roll-off. We desire a notch in the closed-loop sensitivity at a frequency of θ = θ0 (250 rad/s or 2387 rpm). Note that the performance weight Wn Wy is placed on the output. If the weight is placed on the disturbance

Mechatronics'10 Cambridge, MA, USA, Sept 13-15, 2010

input channel w1 , it puts an unnecessary bound on the controller gains. The additional input channel w2 aids in providing robust stability against uncertain plant dynamics.

Bode Diagram 5

0

Robust Performance specification

−1

(W W )

In many practical applications minor variations in the operating speed of the spindle are common. In milling application the tool-work piece interaction provides an uncertain environment to maintain a fixed rotational speed. Moreover an accurate measurement of the rotational speed is not always possible. From experimental data, it is shown in a later section that controllers with a notch at a predetermined location (rotational speed) substantially underperform at small deviations from the intended operating speeds. Hence, it is highly desirable to design controllers that give robust performance against uncertainty in the operating speed. This can be achieved by introducing a uncertainty in the frequency location of the notch in the weighting function Wn . We now view θ as an uncertain real parameter θ = θ0 (1 + rδ), |δ| < 1, (3) where r is a percentage bound on the deviation from its nominal value θ0 and δ is a unit-norm bounded uncertainty. For a few samples of the inverse of the uncertain weighting function, the frequency response around θ0 is shown in Figure 6 (with θ0 = 250 and r = 0.02). Clearly, a closed-loop sensitivity which lies below all the frequency response curves in Figure 6 achieves robust performance against uncertainty in the rotational speed. 3.2 Design for robust performance: against δ The sinusoidal disturbance acting on the plant can be modeled as the output of a disturbance filter. The disturbance filter can be chosen as an autonomous system with non-zero initial condition or it can be a second-order system that is excited by an external input. The frequency of the sinusoidal disturbance acts a parameter for this disturbance filter. A second-order system generating a pure sinusoid is marginally stable with poles on the imaginary axis. For practical purposes we can perturb the poles of such a system to be in the left half plane. We regard such a disturbance generating system as the output weighting function for controller synthesis. The weighting function contains the frequency of the sinusoidal disturbance as a parameter. Moreover, it is an uncertain parameter due to small variations in the rotational speed about a nominal value. It is therefore useful to pull out the uncertainty δ, in the rotational speed or disturbance frequency, from the disturbance filter Wn . Thus Wn is represented as an LFT W¦

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Fig. 6. Inverse of the uncertain performance weight interconnection of ∆r = δI with Wnu = wnu I. A nonunique state-space description of the transfer matrix wnu is ! p     ζ − ζ2 ˙ξ = −ζ2 −θ0 ξ + − θ0 r p0 p+ 1 v,(4) θ0 0 0 0 θ0 r p ! θ0 r 0 ξ, (5) q= p θ0 r 0 z = (1 0) ξ + v.

(6)

We use the interconnection shown in Figure 7 to synthesize a controller for robust performance against the uncertainty δ in the operating speed. The interconnection encompasses the robust stability requirement against uncertain high frequency dynamics in the plant model (by virtue of W∆ ). The formulas used for controller synthesis are based on Dietz et al. (2007) and discussed in the next section. W¦

z1 p

w1 w2 K

u -

G

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¦r v

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Fig. 7. Interconnection for robust controller synthesis: K 4. CONTROLLER SYNTHESIS The interconnection in Figure 7 can be transformed to a generic plant-controller interconnection within the generalized plant framework as shown in Figure 8a. The uncer˜ r affects only the output weighting function tainty block ∆ ˜ ˜ that guarantees W . We desire to synthesize a controller K a robust performance level of γ for the closed-loop system, that is ˜r ⋆ W ˜ )(P˜ ⋆ K)k ˜ ∞ ≤ γ, ∀k∆ ˜ r k < 1. k(∆ (7) The controller synthesis algorithm of Dietz et al. (2007) considers a similar interconnection wherein the uncertain

w1 w2 K0 u -

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n

r = 0.02 r = 0.04

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Fig. 5. Interconnection for controller synthesis: K0 358

Mechatronics'10 Cambridge, MA, USA, Sept 13-15, 2010

qf

~ ¦r

pf

~ W

vf

zf yf (a)

~ P

wf

~ K

uf

z

q

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v

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The matrix Π ∈ Π is assumed to be a static multiplier. The set Π allows one to capture various types of non-linearities or time-varying operators, Megretski and Rantzer (1997), Fan et al. (1991). In particular, for a single, possibly timevarying parameter δ ∈ [−1, 1], repeated such that ∆ = δI, condition (11) holds for all elements Π of    −D G ′ ΠDG : : D ≻ 0, G = −G (12) G′ D

p

w

u

K (b)

Fig. 8. Generalized plant-weight-controller interconnections weighting function sits on the input channel as shown in ˜ r is known to be timeFigure 8b. When the uncertainty ∆ invariant the interconnections in Figures 8a and 8b can be shown to be equivalent. The synthesis algorithm presented in Dietz et al. (2007) uses integral quadratic constraints (IQCs) to describe the uncertainty set and hence can even accommodate time-varying uncertainties. For implementation on the MBC 500 experimental setup we design controllers under the assumption of time-invariant uncertainties. Using Theorem 1 we first synthesize a controller K for the interconnection shown in Figure 8b. The controller ˜ for the problem at hand is simply obtained as K ˜ = KT . K The state-space formulas for the controllers are related as   T   T AK BK ˜ := AK CK . and K (8) K := T T CK DK BK DK Theorem 1. For the interconnections shown in Figures 8a ˜ = WT, ∆ ˜ r = ∆T are timeand 8b, assume P˜ = P T , W invariant norm bounded uncertainties k∆k∞ < 1, then a controller K stabilizes the plant P and k(P ⋆ K)(∆ ⋆ W )k∞ ≤ γ (9) T ˜ if and only if a controller K = K stabilizes the plant P˜ and ˜r ⋆ W ˜ )(P˜ ⋆ K)k ˜ ∞ ≤ γ. k(∆ (10) 4.1 State-space formulae for robust controller For the interconnection shown in Figure 8b assume the state- space matrices of the LTI plant P and the weighting function W to be   A Bv Bu P =  Cz Dzv Dzu  and Cy Dyv 0     AW Bp Bw Wqp Wqw =  Cq Dqp Dqw  , W= Wvp Wvw Cv Dvp Dvw

where A ∈ Rn×n and AW ∈ RnW ×nW . All eigenvalues of AW are assumed to lie in the left half plane. The uncertainty block ∆ is characterized by the so called IQC (Integral Quadratic Constraints). It belongs to a set ∆, which in-turn is captured by a structured matrix Π, called the multiplier. The set ∆ contains 0 as an element. The input-output channels of the uncertainty block satisfy the following quadratic constraint.    

∆q ∆q  ,Π  0, ∀q ∈ L2+ , ∀∆ ∈ ∆. L2+ q q (11) 359

Note that a time-invariant uncertainty is described by a subset of ΠDG and hence we use the same for our controller synthesis. The controller K is obtained from a solution to the LMIs given in Theorem 2. Theorem 2. Suppose we are given the interconnection in Figure 8b, the uncertainty set ∆ and scalings Π ∈ Π ¯ L, M ¯ be partitioned satisfying (11). Let matrices T, X, K, as     X11 X12 T11 T12 (13) , X= T = ′ ′ X12 X22 T12 T22 and       ¯1 M ¯2 ¯ L1 M M L= ¯2 , ¯1 K ¯ = K L2 K ′ ¯ ¯ ¯ 2, M ¯ 2 , L′ in which T11 , X11 , K1 , M1 , L1 and T22 , X22 , K 2 have n and nW columns respectively. Then, there exists a controller such that the robust H∞ -norm from w → z ¯ L, M ¯ , N } and Π ∈ Π is at most γ if there exists {T, X, K, for which   I 0 0   0 I 0 0  A B p Bw    I 0  ′ I 0 0 0  0 ≺ 0, (14) ( .. )    0 0 Π 0  Ce Dqp Dqw     0 0 0 J 0 0 I Cz Dzp Dzw   I T12 T11 0 0 T22   0 T22 (15)  I 0 X X  ≻ 0, 11 12 ′ T12 T22 X21 X22 where A, Cz are given in (*) at the top of the page, ! −γI 0 1 J= and 0 I γ   Bv Dvp + T12 Bp + Bu N Dyp T22 Bp   Bp =  , X11 Bv Dvp + X12 Bp + L1 Dyp   X21 Bv Dvp + X22 Bp + L2 Dyp  Bv Dvw + T12 Bw + Bu N Dyw T22 Bw   (16) Bw =  , X11 Bv Dvw + X12 Bw + L1 Dyw  X21 Bv Dvw + X22 Bw + L2 Dyw Dzp = Dzv Dvp + Dzu N Dyv Dvp , Dzw = Dzv Dvw + Dzu N Dyv Dvw , Ce = ( 0 Cq 0 Cq ) . Note that all boldface symbols depend on the decision variables in an affine fashion. Moreover, an application of the Schur complement formula renders condition (14) affine in {γ, Cz , Dzp , Dzw } which allows to infimize γ and compute sub-optimal controllers. The controller reconstruction formulae are taken from Scherer et al. (1997), where further details can be found. In order to reconstruct the controller matrices let

Mechatronics'10 Cambridge, MA, USA, Sept 13-15, 2010

A Cz



Y =

  

=



¯1 AT11 + Bu M 0

¯2 −AT12 + Bv Cv + T12 AW + Bu M T22 AW

¯1 K

¯2 K

¯1 Cz T11 + Dzu M

¯2 −Cz T12 + Dzv Cv + Dzu M

−1 ′ −1 T11 + T12 T22 T12 −T12 T22 −1 ′ −1 −T22 T12 T22 T



(17)

and find matrices U, V such that U V = I − XY . Then with     −1 ˆ ¯ I 0 K K ′ ¯ ˆ = M T12 T22 M

the controller matrices can be obtained as DK := N, −T ˆ − DK CX)U ˜ CK := (M , ˜ K ), BK := V −1 (L − Y BD ˆ − V BK CX ˜ AK := V −1 K
−T T ˜ ˜ K C)X ˜ −Y BCK U − Y (A˜ + BD U .

A + Bu N Cy Bv Cv + T12 AW + Bu N Dyv Cv 0 T22 AW X11 A + L1 Cy X11 Bv Cv + X12 AW + L1 Dyv Cv X21 A + L2 Cy X21 Bv Cv + X22 AW + L2 Dyv Cv Cz + Dzu N Cy Dzv Cv + Dzu N Dyv Cv

   

(*)

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5. SYNTHESIS AND EXPERIMENTAL RESULTS

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γ - Horizontal 1.95 2.1 2.2

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Table 2. r 0 0.02 0.04

Fig. 9. Maximum singular values of the closed-loop sensitivity.

Singular Values

For controller synthesis we use the performance weights given in (2) and the parameters in Table 1. We first synthesize a controller K0 without considering any uncertainty in the disturbance model. We then synthesize controllers for robust performance against a maximum uncertainty of ±2% and ±4% in the rotational speed of the spindle. This is done by taking the value of r as 0.02 and 0.04 in (3). The synthesis inequalities given in (14) and (15) are linear in the decision variables for fixed values of the performance index γ. We synthesize the controllers for fixed values of γ that are given in Table 2. It is however possible to iteratively reduce the achievable γ by running a bisection algorithm. γ - Vertical 1.91 2.2 2.3

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The maximum singular values of the closed-loop sensitivity with each of the controllers are shown in Figure 9. For frequencies close to θ0 (250 rad/s or 2387 rpm), the same singular values are shown in Figure 10. The vertical lines indicate the bounds (2% and 4% ) on the deviation from the nominal rotational speed. From this figure observe that the notches for the robust designs are wider and deeper in comparison to the nominal design. Clearly the robust designs give higher attenuation than the nominal design for the entire region of uncertainty (given by r). This indicates that the disturbance attenuation for any rotational speed in the range [θ0 (1 − r), θ0 (1 + r)] is better for the robust controllers than the attenuation with the nominal controller at an operating speed of θ0 . 5.2 Experimental results To evaluate the designed controllers we implement them on the dSPACE DS1103 processor board which closes the loop 360

2.4

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Fig. 10. Maximum singular values of the closed-loop sensitivity close to θ0 . with the MBC 500 setup. The continuous time controllers are discretized using a Tustin transformation with a sampling time of 2.5×10−5 s. We measure the displacements of the spindle while operating at rotational speeds of 2387, 2430 and 2480 rpm with each controller. The rotational speeds of 2430 and 2480 rpm correspond to +2% and +4% deviation respectively, from the nominal rotational speed of 2387 rpm. The displacements on one end of the spindle while operating at the nominal rotational speed are shown in Figure 11. The displacements for rotational speeds 2430 and 2480 rpm are shown in Figures 12 and 13 respectively. From the figures, the disturbance attenuation at the nominal rotational speed is marginally better for the robust designs than the nominal design. For higher rotational speeds the nominal design shows a deteriora-

Mechatronics'10 Cambridge, MA, USA, Sept 13-15, 2010

Displacements @2387 rpm: Left End 60

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Fig. 13. Displacements at left end for 2480 rpm.

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In order to evaluate the controllers at all the rotational speeds inside the uncertain region, we slowly accelerate the spindle from a speed of 2300 rpm to 2500 rpm at a rate of 4 rpm/s. The displacements at one end of the spindle are shown in Figure 14. The robust designs clearly outperform the nominal controller for the entire operating speed regime. This verifies the singular values curves in Figure 10 for the closed-loop sensitivity.

Displacements @2480 rpm: Left End 60

y (µm)

tion in performance as the amplitude of the displacement increases remarkably. The robust designs however show only a slightly increased amplitude. The attenuation with the robust designs is clearly far superior to the nominal design at 2430 and 2480 rpm. At a rotational speed of 2430 rpm the robust designs show similar attenuation levels. For a rotational speed of 2480 rpm, the robust design for r = 0.04 clearly gives far better disturbance attenuation.

0.02

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Fig. 11. Displacements at left end for 2387 rpm.

Fig. 14. Displacements at left end during acceleration from 2300 to 2500 rpm.

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disturbance attenuation for our operating speeds of interest. The price for this increased performance is paid by losing robustness of the system as indicated by the higher values of performance index γ. We synthesized and implemented a controller for 5% uncertainty in θ. This controller however failed to stabilize the spindle in its equilibrium position. The loss in robustness is indicated by a visibly high peak in the closed-loop sensitivity. Such a peak can also be seen for our robust designs in Figure 9.

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Fig. 12. Displacements at left end for 2430 rpm. 5.3 Discussion In this paper, we used a new controller synthesis algorithm to ensure robustness against uncertainty in the rotational speed of the spindle. The robust controller synthesized for 4% uncertainty in the operating speed gave the best 361

It is observed that the synthesized robust controller essentially leads to a wider and deeper notch in the closed-loop sensitivity. On the other hand, the robust performance index γ increases for higher values of uncertainty, given in Table 2. This indicates that the price for having a deeper and wider notch in the sensitivity is paid by compromising the performance in other channels. A standard H∞ controller with similar characteristics can be obtained by using a carefully chosen weighting function. This often involves a trial and error type of tuning for weighting functions. For using SISO controllers on each individual channels, one needs to hand-tune the notches in a number of loops. In contrast, the synthesis algorithm used in this paper is a systematic approach to design MIMO controllers

Mechatronics'10 Cambridge, MA, USA, Sept 13-15, 2010

and provides essential insights into robust stability and performance. In order to replicate the performance of the robust controller with r = 4%, we tuned the parameters of the weighting function Wn and synthesized a nominal H∞ controller. A nominal design using the interconnection in Figure 5 with ζ1 = 210 and ζ2 = 2.5 gives the same performance as the robust design. The maximum singular values for the closed-loop sensitivity are shown in Figure 15. Our implementation of both the controllers on the MBC 500 set up show strikingly similar disturbance attenuation profiles for all the rotational speeds of interest. The tuning of Wn required little effort on our part as we had apriori insight into the desired shape of the weighting function (via the robust design). 1

10

r=0.04 tuned

0

Singular Values

10

−1

10

−2

10

2.3

10

2.4

10 Frequency (rad/s)

2.5

10

Fig. 15. Maximum singular values of the closed-loop sensitivity close to θ0 with the tuned controller. 6. CONCLUSIONS In this paper we synthesized and experimentally evaluated controllers for robust performance against uncertainty in the operating speed of an AMB spindle. We used the generalized plant framework to provide additional robustness for stability against uncertain system dynamics. REFERENCES Balini, H.M.N.K., Houtzager, I., Witte, J., and Scherer, C.W. (2010a). Subspace Identification and Robust Control of an AMB. In American Control Conference. USA. Balini, H.M.N.K., Scherer, C.W., and Witte, J. (2010b). Performance Enhancement for AMB Systems Using Unstable H∞ Controllers. IEEE Trans. Control Systems Technology - accepted. Dietz, S.G., Scherer, C.W., and K¨ oro˘ glu, H. (2007). Robust control against disturbance model uncertainty: a convex sollution. 46th IEEE Conference on Decision and Control, 842–847. New Orleans, USA. Dietz, S., Balini, H.M.N.K., and Scherer, C.W. (2008). Robust control against disturbance model uncertainty in active magnetic bearings. In Proc. of 11th International Symposium on Magnetic Bearings. Nara, Japan. 362

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