Case study on anti-windup compensation - Micro-actuator control in a hard-disk drive

1 Case study on anti-windup compensation - Micro-actuator control in a hard-disk drive Guido Herrmann, Matthew C. Turner, and Ian Postlethwaite Control and Instrumentation Research Group, Department of Engineering, University of Leicester Leicester, LE1 7RH, U.K. gh17@le.ac.uk, mct6@le.ac.uk, ixp@le.ac.uk This chapter demonstrates the use of anti-windup compensation in the control loop of a micro-actuator which is nominally controlled by a linear, discrete robust controller. The micro-actuator is part of a hard disk drive dual-stage servo-control system for positioning of the read/write head. The actuator inputs are constrained to retain the micro-actuator s displacement range of less than.4 µm for mechanical protection. In the first part of the chapter, the anti-windup compensation scheme exemplifies an approach suggested by Weston & Postlethwaite [22]. Here, the scheme is posed as a discrete full-order compensator and the closed loop analysis uses a generalized circle citerion approach. The design of the compensator is posed in LMI-form. In the second part of the chapter, it is shown how the linear micro-actuator control loop with anti-windup compensation is incorporated into the non-linear servo-control scheme for positioning of the read/write head in a hard disk drive. 1.1 Introduction Micro-actuators have been gaining of importance in practical systems during recent years. For instance, it is nowadays the target to integrate actuator, sensor and electronics for powerful computations in one device of micro or nano-scale. These technologies can be useful in optical communication systems, in electromechanical signal processing systems or in healthcare systems, such as BIO-MEMS for microchiplab diagnostics or micro-devices for therapeutic targeting and delivery. Mechanical micro-actuators for nano-positioning have been of interest for the University of Leicester due to its importance in the servo-controller for the positioning of the readwrite head in hard disk data-storage systems [5, 8, 11]. A micro-actuator is being used in a dual-stage control system to achieve high-banwidth positioning control.

2 Guido Herrmann, Matthew C. Turner, and Ian Postlethwaite Fig. 1.1. Schematic of Hard Disk Drive with PZT-actuator Significant research effort on hard-disk drive (HDD)-servo techniques has been invested in the area of dual-stage servo control [17, 18, 14]. The reason for this is the continuous increase in the track density and in storage capacity of HDDs. Recently, a track density of above 42 ktpi (TPI- track per inch) has been demonstrated (see [23, 2]) in a laboratory environment and about 149 ktpi density HDDs are nowadays available in the consumer market. In addition to the significant increase in data density, the increased demand for higher data rates requires the improved performance of the head-positioning servo system in a hard disk drive. A promising way to meet these demands is to augment the conventional voice coil motor (VCM) actuator with a second-stage, high bandwidth micro-actuator. Dual-stage servo systems in HDDs are now a feasible alternative to the single stage VCM-servo system. PZT-based micro-actuators using PZT-elements embedded in the head suspension are popular, e.g. the FUMA -actuator in [19] (Figure 1.1). However, the displacement range of secondary actuators is very limited, typically less than 1-2 µm, and the input signal for the actuator is limited to prevent damage. In dual-stage servo-systems, the two actuators have to deal with the following servo-tasks: seek/settling and track following. Seek/settling control has to ensure a fast movement of the read/write head from one track to another. For track following, high bandwidth controllers are necessary to ensure good error rejection capabilities while counteracting disturbances. Therefore, the primary VCM-actuator is required for larger displacement and the secondary actuator provides large bandwidth. For servo-control of such a dual-stage actuator system, the method of [11, 12] is employed. It is based on a well known decoupled dual-stage controller structure of [13], where design and stability for the primary VCM-control loop and the secondary PZT-control loop are guaranteed independently. It will be shown [11, 12] that the primary and secondary loop remain stable independently regardless of the seek/settling/track-following control method used for the VCM-loop, providing the secondary control loop is stable in the presence of saturation limits. Hence, it seems

Title Suppressed Due to Excessive Length 3 logical to use for the secondary loop an AW compensator to guarantee overall large and small signal stability despite actuator saturation limits. AW compensators are most suitable, as they have been developed to retain the nominal performance of the original control system [9, 7, 2], e.g. high-bandwidth tracking in the servo-control of hard disk drives. This example study on AW-compensation will consider the following issues: 1. controller design for the micro-actuator loop and wind-up problems due actuator limits, 2. discrete anti-windup compensator design, anti-windup compensation for the micro-actuator control loop 3. the micro-actuator control loop as part of the overall seek-settling scheme for the hard disk drive. We will consider for reasons of simplicity the micro-actuator loop at first. 1.2 The micro-actuator control loop and windup-problems The model of a micro-actuator is usually very simple. Ignoring high frequency resonances, the model of the micro-actuator could be considered to be of constant gain only (Figure 1.2). However, in practice these resonances at 6.5 khz and 9.6 khz are significant. Resonances are usually subject to phase and gain uncertainty. The center frequency of a resonance might even shift due to the influence of temperature. This usually forces the control designer to retain the open and closed loop bandwidth lower than 5 % of the smallest significant resonance. Hence, for our micro-actuator, an open loop crossover frequency of not more than 3 khz can be achieved. It must be also noted that hard disk drive technology is usually controlled using digital technology. The sampling frequency is directly correlated to the rotational speed of the hard disk and the servo-information which is placed along the concentric data tracks. The amount of servo-information will limit the available space for user data. Hence, it is necessary to limit the amount of servo-information causing also a limited sampling frequency (Figure 1.3). In our case, we have used a sampling frequency of 27 khz which is a practically relevant sampling frequency in modern hard disk drives. However, this sampling frequency is rather low, since micro-actuators often have also resonance frequencies at 2 khz and above (not depicted here), i.e. above the Nyquist frequency of 13.5 khz. Thus, a hard disk drive servo system is a sampled-data control system, where it is necessary to use discrete or sampled-data control methods to design the servo-controller. In the hard disk drive industry, it is common practice to counteract resonances by simple notch filters. In our case, this may be done by using two discrete notch filters

4 Guido Herrmann, Matthew C. Turner, and Ian Postlethwaite PZT model Log. Magn. [µ m/v] 1 1 1 1 2 1 1 1 2 1 3 1 4 Frequency [Hz] 2 Phase [deg] 2 4 1 1 1 2 1 3 1 4 Frequency [Hz] Fig. 1.2. Continuous model of a PZT-actuator Fig. 1.3. Hard-disk with servo information and data tracks which are digitally implemented (see Figure 1.4). Hence, these notch filters are easily augmented to the input signal of the PZT-actuator via a digital-to-analogue unit (DAU). In general, a saturation is placed between the filter and the DAU to protect the actuator from any damage due to high voltage peaks (see Figure 1.5 for a Matlab respresentation). The saturation limits are also tuned in such a way to protect the micro-actuator from too large displacement. The micro-actuator augmented with the notch filters is easily modelled as a second order system P PZT (z) (see Figure 1.6):

Title Suppressed Due to Excessive Length 5 Notch filters Log. Magn. [µ m/v] 1 1 1 1 2 1 1 1 2 1 3 1 4 Frequency [Hz] Phase [deg] 5 1 15 2 1 1 1 2 1 3 1 4 Frequency [Hz] Fig. 1.4. Frequency response of discrete notch filter sys_notch_d1*sys_notch_d2 PZT NotchFilter Saturation1 Zero Order Hold1 PZT actuator Fig. 1.5. Matlab-Simulink model of PZT-actuator with notch filter and protective saturation element P PZT =.7997z.5363 z 2.785z +.397 (1.1) The first step in the closed loop control design for this micro-actuator is to design a suitable linear controller K(z) based on robust µ-design methods. The design problem is standard. We are minimizing the µ-value for the following control problem with two blocks [ Ws S W u SK ] and [W T T ]: [ ] Ws S [ W u SK WT T ] where S = 1 1+KP PZT and T = KP PZT 1+KP PZT (see Figure 1.7). The weights (Figure 1.8) are given by

6 Guido Herrmann, Matthew C. Turner, and Ian Postlethwaite Discrete PZT model with notch Log. Magn. [µ m/v] 1 1 1 2 1 1 1 2 1 3 1 4 Frequency [Hz] 2 Phase [deg] 2 4 1 1 1 2 1 3 1 4 Frequency [Hz] Fig. 1.6. Zero-order hold discretization of PZT-model with notch (line); Second order model of PZT-actuator (dash) Fig. 1.7. µ-control problem W T = 1.3637(z +.4294), W s = z +.6843.25664(z 2.578), W u =.1. z.965 The design of the controller is easily achieved using the µ-toolbox of Matlab. However, a small trick has to be used. Matlab does not provide µ-tools for discrete control problems. Hence, the discrete control problem has to be converted into a continuoustime problem using a bilinear transformation [3, 5]. Note that the bilinear transfor-

Title Suppressed Due to Excessive Length 7 µ design weights Log. Magn. [µ m/v] 1 1 1 1 1 1 2 1 3 1 4 Frequency [Hz] Phase [deg] 5 5 1 15 2 1 1 1 2 1 3 1 4 Frequency [Hz] Fig. 1.8. µ-control problem weights (W T (line), W s (dashed)) mation creates an equivalent continuous-time µ-problem. The discrete controller can be obtained from the solution of the continuous-time control problem by using again the bilinear transformation approach. The discrete controller will be always stabilizing and µ-(sub)-optimal with the same µ-value as obtained from the continuous-time design. Thus, for our design problem, a controller is obtained using DK-iteration for a µ-value of 2. Balanced truncation methods allow to obtain a third order controller which perfectly matches the originally designed controller. The discrete controller of third order is: K(z) = 6.584z3 1.43z 2 + 3.15z 2.79 z 3 +.459z 2.9169z.4211 The control-loop is easily simulated considering a sine sweep in the frequency range from 1 Hz to 5 Hz as demand for the position. For small demand amplitude of.2 µm, the controller is well able to follow this demand without significant phase lag but with a small amplitude loss (Figure 1.9). However, for large demand amplitude of 1 µm, the actuator saturation limit of 1.54 V is activated, limiting also the actuator displacement range. Thus, in this case, the controller is not able to track the controller demand as the actuator displacement is limited to a magnitude of less than.5 µm. Hence, the actuator starts to oscillate and the controller features a lag (Figure 1.1). These two issues of the resonance and the lag can be easily resolved by introducing an anti-windup compensator to recover quickly from the performance loss due to saturation. The anti-windup compensator of interest for us will be discussed next.

8 Guido Herrmann, Matthew C. Turner, and Ian Postlethwaite.2.15.1.5 time [s].5.1.15.2.1.2.3.4.5.6.7.8.9.1 postion [µm] Fig. 1.9. Control result for.2 µm demand amplitude (demand (dashed), actuator-position (line)) 1.8.6.4.2 time [s].2.4.6.8 1.1.2.3.4.5.6.7.8.9.1 postion [µm] Fig. 1.1. Control result for 1 µm demand amplitude (demand (dashed), actuator-position (line)) 1.3 Anti-windup compensation for discrete linear control systems We consider the system in Figure 1.11 as introduced in [21, 22] for continuoustime anti-windup (AW) compensation and in [7] for discrete-time control systems.

Title Suppressed Due to Excessive Length 9 Fig. 1.11. Conditioning with M(z) Fig. 1.12. Equivalent representation of conditioning with M(z) P(z) is the discrete plant and K(z) is the nominal linear discrete controller. The AWcompensator is given by the blocks, M(z) I and PM(z). The anti-windup compensation is designed using the free parameter M(z). The control signal is constrained by the limits of a saturation function. The saturation function is defined as sat(u) := [sat 1 (u 1 ),...,sat m (u m ))] T (1.2) where sat i (u i ) := sign(u i ) min{ u i,ū i }, where ū i > is the i th saturation limit. The following identity holds Dz(u) = u sat(u) (1.3) where Dz(u) is the deadzone function. For the deadzone nonlinearity Dz(u), it follows that there exists a diagonal matrix W such that

1 Guido Herrmann, Matthew C. Turner, and Ian Postlethwaite Dz(u) W(u Dz(u)), u R m (1.4) This inequality will be used later for stability analysis employing the S-procedure [1]. This is the discrete-time version of the continuous time configuration introduced in [21, 22]. As in [21, 22], it is straightforward to show that, using the identity (1.3), Figure 1.11 can be re-drawn as Figure 1.12. The representation in both of these diagrams essentially casts the anti-windup problem as the problem of choosing an appropriate M(z). However, notice that Figure 1.12 reveals an attractive decoupling into nominal linear system, nonlinear loop and disturbance filter. As noted in [22], most linear conditioning schemes can be interpreted in the framework of Figure 1.11. In [7], the mapping T : u lin y d was picked as a measure of the anti-windup compensator s performance (Figure 1.12). We shall choose to minimise the l 2 gain, T i,l2, in our anti-windup synthesis. The detectable plant, P(z), has the following state-space description { xp (k + 1) = A P(z) p x p (k) + B p u m (k) y(k) = C p x p (k) + D p u m (k) (1.5) where x p R n p is the plant state, u m R m is the actual control input to the plant, y R q is the output which is fed back to the controller. We are making the assumption that the plant P(z) is asymptotically stable is λ max (A p ) < 1 for global stability of the non-linear control system with anti-windup compensation. As in [22] for continuous-time, M(z) can be chosen as a coprime factor of P(z). So if P(z) = N(z)M 1 (z), we can search for a coprime factor of P(z) such that the anti-windup closed-loop has the best performance in terms of the gain of T. This approach is also related to that of [16] and, to a lesser extent, that of [15]. To achieve full-order stabilisation we would like to choose coprime factors, which share the same state space and are of order equal to that of P(z). Employing Figure 1.12, such coprime factorisations can be characterised by [ ] M(z) I N(z) A p + B p F B p F C p + D p F D p (1.6) where ũ(k) = Dz(u lin (k) u d (k)). Note that these equations are parameterised by the free parameter F and therefore we attempt to choose F such that T i,l2 is minimised. Theorem 1. There exists a dynamic compensator Θ(z) = [M(z) I (P(z)M(z)) ] which solves strongly the anti-windup problem if there exist matrices Q >,U =

Title Suppressed Due to Excessive Length 11 diag(µ 1,... µ m ) >,L R (m+q) m and a scalar γ > such that the following linear matrix inequality is satisfied Q L QC p + L D p QA p + L B p 2U I UD p UB p γi γi < (1.7) Q Furthermore, if this inequality is satisfied, a suitable F for the coprime factorisation (1.6) achieving T i,l2 < γ, is given by F = LQ 1. Proof: Let us choose a Lyapunov function candidate as V (k) = x(k) Px(k) >. We define the Lyapunov difference as V (k) := V (k + 1) V (k). Next we consider the function Ṽ (k) which is defined as Ṽ (k) := V (k) + 2ũ(k) W[u lin (k) u d (k) ũ(k)] + 1 γ y d(k) 2 γ u lin (k) 2 (1.8) This function is a combination of the Lyapunov difference (first term), the sector bounds from (1.4) associated with the deadzone nonlinearity (second term), and two final terms which ensure we have a certain level of l 2 performance. If we can ensure that equation (1.8) is negative definite, we have 1. Asymptotic stability When u lin (k) = and Ṽ (k) = V (k) + 2ũ(k) W[u lin (k) u d (k) ũ(k)] + 1 }{{} γ y d(k) 2 < (1.9) }{{} for [x(k) ũ(k) u lin (k)] asymptotic stability is implied from V (k) <. 2. l 2 gain < γ. Summing Ṽ (k) from to gives: k= V (k) + 2 k= As k= V (k) = V ( ) V () we get ũ(k) W[u lin (k) u d (k) ũ(k)] + 1 γ y d 2 l 2 γ u lin 2 l 2 < (1.1)

12 Guido Herrmann, Matthew C. Turner, and Ian Postlethwaite V ( ) V () + 2 ũ(k) W[u lin (k) u d (k) ũ(k)] + 1 }{{} k= γ y d 2 l 2 γ u lin 2 l 2 < > }{{} (1.11) for [x(k) ũ(k) u lin (k)], which implies y d 2 l 2 < γ 2 u lin 2 l 2 +V () and therefore T i,l2 < γ. This procedure of extending the Lyapunov difference by some extra non-negative terms is well established as S-procedure [1]. Strictly speaking m + 1 terms are included. For the sector non-linearity (1.4), m terms are included, while the m + 1-th term is contributed by the l 2 -gain constraint. It is well known that the inclusion of one term only, creates an equivalent inequality for both conditions to be satisfied. However, once more than one term is considered, sufficiency is only achieved and conservatism is introduced. Furthermore, the sector bound as given in (1.4) may also introduce conservatism as discussed earlier. These facts are well-known and accepted trade-offs [1, 4] as they allow practically feasible solutions (e.g. [1]). Substituting for x(k),u d (k) and y d (k) in (1.8), we have that Ṽ (k) < if and only if where x(k) ũ(k) u lin (k) V F11 V F12 V F22 W γi } {{ } V F x(k) ũ(k) u lin (k) <, [x(k) ũ(k) u lin (k)] (1.12) V F11 = (A p + B p F) P(A p + B p F) P + 1 γ (C p + D p F) (C p + D p F) (1.13) V F12 = (A p + B p F) PB p F W + 1 γ (C p + D p F) D p (1.14) V F22 = 2W + 1 γ D pd p + B ppb p. (1.15) The remainder of the proof has to show that V F < is equivalent to (1.7). This follows by standard Schur complement and congruence transformation arguments. The Schur complement implies that (1.12) holds if and only if P F W C p + F D p A p + F B p 2W W D p B p γi γi < (1.16) P 1

Title Suppressed Due to Excessive Length 13 The next step is to multiply the left and right of the matrix inequality with so that P 1 W 1 I I < (1.17) I P 1 P 1 F P 1 C p + P 1 F D p P 1 A p + P 1 F B p 2W 1 I W 1 D p W 1 B p γi γi < (1.18) P 1 If we define L = FP 1, U = W 1 and Q = P 1, then (1.7) follows. Hence, considering equation (1.9), internal stability, follows. The non-linear operator T i,l2 has a finite l 2 gain with an upper bound of γ > which is easily implied from equation (1.11). Furthermore, note that as there is no direct feedthrough term in the nonlinear loop, well-posedness is ensured. 1.4 Anti-windup compensation for the micro-actuator It is now straightforward to design an anti-windup compensator for the linear control scheme of Section 1.2. The model used for controller design (1.1) delivers an antiwindup compensator of (1.6) as given by the state space representation of the second order system [ ].785.6195 A p =, B.5 p = [.5 and the suitable state feedback gain F as known from (1.6): ], C p = [.1599.2145 ], D p = F = [ 1.243 1.7618 ]. The l 2 gain has been minimized to a value of γ =.2616.

14 Guido Herrmann, Matthew C. Turner, and Ian Postlethwaite 1.8.6.4.2 time [s].2.4.6.8 1.1.2.3.4.5.6.7.8.9.1 postion [µm] Fig. 1.13. Control result for 1 µm demand amplitude (demand (dashed), actuator-position (line)) The application of this computed AW-compensator to the control system of Section 1.2 with saturation limit improves the closed loop (Figure 1.13). The oscillations observed without AW-compensation are minimized and the lag seems to be eradicated (Figure 1.13 and 1.1) for the sine sweep with 1 µm amplitude. Thus, AW-compensation can achieve a significantly improved tracking performance considering the fact that the micro-actuator has a displacement limit of about ±.5µm. 1.5 The micro-actuator control loop as part of a hard-disk drive servo-system The micro-controller is to be incorporated into the control scheme for the servocontrol of a hard disk drive actuator. The micro-actuator is directly embedded in the suspension of the voice-coil motor (VCM) actuator (Figure 1.1). A linear model of the actuator is given by the superposition of the two models of the PZT-actuator, P PZT, and the VCM-actuator, P VCM : [P PZT P VCM ] The VCM actuator can be modeled as a double integrator which is affected by high frequency resonances and the dynamics of friction in low frequency. The VCMactuator is usually controlled by a (proximate)-time-optimal control scheme [3, 6,

Title Suppressed Due to Excessive Length 15 Fig. 1.14. Simplified seek/settle/trackfollowing scheme Fig. 1.15. Equivalent representation of the servo scheme 11] which achieves fast seeking for fast positioning of the read/write head, while a smooth transfer to a linear track-following scheme needs to be achieved for trackfollowing (see [3, 6, 11] for a more detailed explanation). To achieve a higher trackfollowing controller bandwidth, the secondary PZT-actuator is used. The constraints on both actuators (in particular for the PZT-actuator) need to be incorporated into the servo-control scheme. The approach is to use a decoupled control scheme as from [13], which incorporates the traditional time-optimal seek-settling scheme for the VCM-actuator [6, 11] and it allows an independent controller design for the PZTmicro-actuator so that the controller design of Sections 1.2-1.3 can be reused. The

16 Guido Herrmann, Matthew C. Turner, and Ian Postlethwaite Fig. 1.16. Simplified seek/settle/trackfollowing scheme with observer 2.5 2 postion [µm] 1.5 1.5.5.1.15.2.25.3.35.4 time [s] Fig. 1.17. Control result for 2 µm seek step without AW-compensation (demand (dashed), actuator-position (line)) control scheme works on the assumption that the absolute position of the PZT-microactuator is measurable (see Figure 1.14). Employing the superposition principle, it is easily seen that the control scheme of Figure 1.14 is equivalent to the controller configuration of Figure 1.15, where the control loops for the PZT and the VCM-actuator are decoupled. Hence, both controllers can be designed independently in terms of stability. For the implementation of the scheme, it is necessary to use an observer which obtains the absolute position

Title Suppressed Due to Excessive Length 17 2.5 2 postion [µm] 1.5 1.5.5.1.15.2.25.3.35.4 time [s] Fig. 1.18. Control result for 2 µm seek step with AW-compensation(demand (dashed), actuator-position (line)) of the PZT-actuator (see Figure 1.16). It is easily verified that observer and controller separate due to the linearity of the plant. Using now a non-linear seek-settle control scheme for the VCM-actuator as presented in [6, 11] and the linear control loop from Sections 1.2-1.3 with and without anti-windup compensation for the controller of the PZT-actuator, it is easily verified that the AW-compensation scheme significantly improves on seeking performance (see Figures 1.17 and 1.18). The controller with AW-compensation prevents for a 2 µm-seek step a very large overshoot and allows fast settling to assume track following of the read/write head as quickly as possible. 1.6 Summary The control system of a PZT-micro-actuator has been investigated for its feasible range, i.e. the available displacement range of the micro-actuator. It has been shown that anti-windup compensation can improve tracking of demands which are not feasible. It limits controller lags and suppresses high frequency resonances for a practically valid hard disk drive micro-actuator. In connection with the recently established dual-stage track-seek/following scheme [6, 11], it has been shown that the AW-compensator suppresses large overshoots and enhances settling speed.

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