H 2 GUARANTEED COST CONTROL DESIGN AND IMPLEMENTATION FOR DUAL-STAGE HARD DISK DRIVE TRACK-FOLLOWING SERVOS. Richard Conway

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1 Proceeings of the ASME 211 Dynamic Systems an Control Conference DSCC211 October 31 - November 2, 211, Arlington, VA, USA DSCC H 2 GUARANTEED COST CONTROL DESIGN AND IMPLEMENTATION FOR DUAL-STAGE HARD DISK DRIVE TRACK-FOLLOWING SERVOS Jianbin Nie Richar Conway Computer Mechanics Laboratory Computer Mechanics Laboratory Department of Mechanical Engineering Department of Mechanical Engineering University of California University of California Berkeley, California 9472 Berkeley, California njbin.ustc@gmail.com rconway345@gmail.com Roberto Horowitz Professor of Mechanical Engineering University of California Berkeley, California horowitz@berkeley.eu ABSTRACT This paper iscusses the esign an implementation of H 2 guarantee cost control for ual-stage har isk rive trackfollowing servo systems. The propose approach is base on H 2 guarantee cost analysis, in which an upper boun on the worst-case H 2 performance of a iscrete-time system with gainboune unstructure uncertainty is etermine via several Riccati equations. Subsequently, the output feeback H 2 guarantee cost control synthesis algorithm is presente by exploiting Riccati equation structure to reuce the number an complexity of the semi-efinite programs (SDPs) that nee to be solve. The presente control synthesis methoology is then applie to a har isk rive with a PZT-actuate suspension. Experimental results on the actual isk rive valiate our control esign. 1 INTRODUCTION Since the first magnetic rive was invente in the 195s by IBM, the areal ensity of har isk rives (HDDs) has been oubling roughly every 18 months [1], in accorance with Moore s law. As the storage ensity is pushe higher, the concentric tracks on the isk must be pushe closer together, which requires much more accurate positioning control of magnetic rea/write heas. The current goal of the magnetic recoring inustry is to achieve an areal ensity of 4 terabits/in 2, which implies that the track with is require to be 25 nm. As a result, the 3σ value of the close-loop position error signal (PES) uring trackfollowing control must be less than 2.5 nm. In orer to achieve the continuous increase of the ata storage ensity, ual-stage actuation (DSA) which combines the Aress all corresponence to this author. traitional voice coil motor (VCM) an an aitional microactuator (MA) has been propose as a means of enhancing servo tracking performance by increasing the servo banwith [2]. The configurations of ual-stage actuators can be categorize into three groups accoring to the location of the seconary actuator: actuate suspension, actuate slier an actuate hea. In this paper, we will focus on the control esign of DSA servo systems with actuate suspensions. As mentione earlier, the relevant performance metric in a HDD is the stanar eviation of the PES. Since the square H 2 norm of a system can be interprete as the sum of variances of the system outputs uner the assumption that the system is riven by inepenent white zero mean Gaussian signals with unit covariance, the H 2 norm is a useful performance metric for HDDs. In aition, since there ten to be large variations in HDD ynamics ue to variations in manufacture an assembly, it is not enough to achieve the esire level of performance for a single plant; the controller must guarantee the esire level of performance for a large set of HDDs. Thus, we are intereste in fining a controller which can prouce as small H 2 norm of close-loop systems as possible over a set of HDDs [3,4]. To characterize the worst-case performance of a set of HDDs, we use its H 2 guarantee cost an upper boun on the worst-case H 2 performance of a system with unstructure uncertainty. Using this characterization, we focus on the H 2 guarantee cost control synthesis. In [5], the authors evelope two control synthesis algorithms respectively through the solution of iscrete Riccati equations (REs) an entirely base on the solution of a sequence of semi-efinite programs (SDPs). Since the control synthesis algorithm using iscrete REs are often significantly more computationally efficient an numerically robust than the one entirely using SDPs, 1 Copyright 211 by ASME Downloae From: on 1/11/214 Terms of Use:

2 we utilize the algorithm via the solution of REs presente in [5] to synthesize H 2 guarantee cost control. For such a control synthesis algorithm, the output feeback H 2 guarantee cost control problem is reuce to a full control problem by exploiting Riccati equation structure to reuce the number an complexity of SDPs that nee to be solve. In orer to evaluate the effectiveness of the presente control technique, we apply it to a har isk rive with a PZT-actuate suspension. Moreover, the esigne controller was implemente on this actual har isk rive with a laser Doppler vibrometer (LDV) to measure the feeback signal [6]. The paper is organize as follows. Section 2 escribes our ual-stage HDD servo system. In Section 3, the H 2 guarantee cost analysis is provie. The control synthesis algorithm is presente in Section 4. The control implementation is iscusse in Section 5. The conclusion is given in Section 6. 2 MODELING OF DUAL-STAGE HDD SERVOS Figure 1 shows the picture of a PZT-actuate suspension provie by Hutchinson Technology, Inc. (HTI). Two yellow PZT actuators are place near the root of the suspension. They generate a push-pull action when riven by ifferential voltages. Meanwhile, a leverage mechanism is utilize to convert an amplify this small actuation isplacement into large hea motion. u v u p w a ¾ a q G p W w r y h w n r - ¾ n G r + K Figure 2: MODELING OF DUAL-STAGE HDD SERVO CON- TROL SYSTEM Magnitue (B) Phase (eg) Experiment ata Ientifie moel Frequency (Hz) y Figure 3: VCM FREQUENCY RESPONSE Figure 1: A PICTURE OF A PZT-ACTUATED SUSPENSION (PROVIDED BY HUTCHINSON TECHNOLOGY, INC..) 2.1 System Ientification In this paper, the DSA servo system is moele as the block iagram shown in Fig. 2 in the same way as [6]. In Fig. 2, the winage is moele as a white noise with scale w a ; track runout r ue to isk vibrations is moele as a color noise; the measurement noise is assume to be a white noise with scale w n. Notice that w a, w r an w n have unit variance. The etails of the moeling of σ a, G r an σ n for the ual-stage servo system are emonstrate in [6]. The two actuators are represente by a nominal ual-input-single-output (DISO) system G p with the output multiplicative uncertainty. W is the uncertainty weighting function an the unstructure uncertainty is boune with 1. As illustrate in [6], we separately fitte a continuous-time moel to each of these frequency responses of the VCM an MA, Magnitue (B) Phase (eg) 2 4 Experiment ata Ientifie moel Figure 4: SPONSE Frequency (Hz) PZT MICRO-ACTUATOR FREQUENCY RE- which are shown in Fig 3 an Fig. 4 respectively. Then, we combine the fitte moels into a single moel an use common moe ientification [3] to eliminate reunant copies of the suspension vibration moes. As a result, the continuous-time moel 2 Copyright 211 by ASME Downloae From: on 1/11/214 Terms of Use:

3 Table 1: Moel PARAMETERS FOR G p moe, i ω i ( 1 5 ) ζ i B i e e e e e e e e e e6 y Wu * u v v = z Wu * u p p 2 y q G K u w a w= w r w n Figure 5: DUAL-STAGE HDD SERVOS IN LFT REPRESEN- TATION. for the nominal plant G p is given by G p (s) = [ ] [ ] ( [ ]) 1 2ζi ω + 1 si i 1 ω 2 B i=1 i i (1) where the moel parameters are as liste in Table Dual-stage Servo Control Systems We rewrite the ual-stage servo system as the linear fractional transformation (LFT) shown in Fig. 5. In Fig. 5, z 2 is performance monitoring output with the control input weighting values of W uv an W up for VCM an MA respectively. Suppose, the open-loop system G an the controller K have the following q z 2 Figure 6: UPPER LFT REPRESENTATION WITH UNCER- TAINTY. state-space realization: A B 1 B 2 B 3 [ ] G C 1 D 11 D 12 D 13 C 2 D 21 D 22 D 23, K Ac B c. (2) C c D c C 3 D 31 D 32 3 H 2 GUARANTEED COST ANALYSIS As mentione in Section 1, it is esirable to esign a controller to achieve robust performance, i.e. the esigne controller is able to guarantee an aequate level of performance for a set of har isk rives. In this section, we utilize the analysis of the robust H 2 performance of an uncertain iscrete-time linear time-invariant (LTI) system in [5] to establish an upper boun on the worst-case H 2 performance of an uncertain system. Such an upper boun will be employe to formulate H 2 guarantee cost control problem in next section. 3.1 H 2 Guarantee Cost We consier the uncertain system Ḡ an its uncertainty as illustrate in Fig. 6. Throughout this paper, we enote the upper LFT of Ḡ by as F u (Ḡ, ), as shown in Fig. 6. Suppose, is a norm-boune unstructure uncertainty with 1 an Ḡ has the state-space realization w Ā B 1 B 2 Ḡ C 1 D 11 D 12. (3) C 2 D 21 D 22 From [7], we know that F u (Ḡ, ) 2 2 < γ, 1 if there exist τ >,P,W,V such that tr{w} < γ an MḠ(τ,P,W,V ) := P V T Ā B 1 B 2 τi C 1 D 11 D 12 V W C 2 D 21 D 22 P Ā B 1 B 2 τi C 1 D 11 D 12. I C 2 D 21 D 22 T This sufficient conition for F u (Ḡ, ) 2 2 < γ, 1 motivates us to consier the following optimization J τ (Ḡ) := inf tr{w} s.t. P, MḠ(τ,P,W,V ). (4) P,W,V 3 Copyright 211 by ASME Downloae From: on 1/11/214 Terms of Use:

4 When (4) is infeasible for a particular value of τ >, we use the convention that J τ (Ḡ) =. Then, J τ (Ḡ) turns out to be an upper boun of the worst-case H 2 norm with the uncertainty, i.e., J τ (Ḡ) sup 1 F u (Ḡ, ) 2 2. In orer to reuce the ifference between the upper boun an the worst-case H 2 norm, we consier the following optimization to minimize the upper boun: inf J τ(ḡ). (5) τ> Notice that the square root of the value of (5) is calle H 2 guarantee cost for the system Ḡ shown in Fig. 6. In aition, the H 2 guarantee cost given by (5) is finite if an only if the system is robustly stable an this can be checke using a single Riccati equation [5]. 3.2 H 2 Guarantee Cost Computation For a given system Ḡ in (3), we efine Q := τ C T 1 C 1 + C T 2 C 2, R := τ( D T 11 D 11 I) + D T 21 D 21 S := τ C T 1 D 11 + C T 2 D 21, Q W := τ D T 12 D 12 + D T 22 D 22 S W := τ D T 12 D 11 + D T 22 D 21, ψ := ( B 2, B 1, Q W, R, S W ) ϕ := (Ā, B, Q, R, S) R ϕ (P) := ĀT PĀ + Q (Ā T P B + S)( B T P B + R) 1 ( B T PĀ + S T ) K ϕ (P) := ( B T P B + R) 1 ( B T PĀ + S T ) A ϕ (P) := Ā + BK ϕ (P) In [5], we have evelope an algorithm to compute the upper boun J τ (Ḡ) as follows. Algorithm 1. The following algorithm computes J τ (Ḡ) uner the assumption that A is Schur. 1. Fin the stabilizing solution of the DARE P = R ϕ (P ) 2. Compute the Cholesky factorization LL T = ( B T 1 P B 1 + R) 3. Ǩ = L\( B T 1 P B 2 + S T W ) 4. J τ (Ḡ) = sum(p ( B 2 B T 2 )) + τ D 12 2 F + D 22 2 F + Ǩ 2 F If either of the first two steps fail, then J τ (Ḡ) =. Notice that represents the Haamar prouct (i.e. elementwise multiplication) for two matrices, the operator sum will take the sum of all elements of a matrix, an F enotes the Frobenius norm of a matrix *. In aition, we say a matrix is Schur if all of its eigenvalues lie strictly insie the unit isk in the complex plane. In [5], we also have evelope an algorithm to compute τ (J τ(ḡ)) as follows. Algorithm 2. The following algorithm computes τ (J τ(ḡ)). 1. Use Algorithm 1 to compute P,L,Ǩ,J τ (Ḡ) 2. K ϕ (P ) = L T \(L\( B T 1 P Ā + S T )) 3. A ϕ (P ) = Ā + B 1 K ϕ (P ) 4. Q Lyap = ( C 1 + D 11 K ϕ (P )) T ( C 1 + D 11 K ϕ (P )) K ϕ (P ) T K ϕ (P ) 5. Solve the iscrete Lyapunov equation P τ = A ϕ (P ) T P τ A ϕ (P ) + Q Lyap 6. K ψ (P ) = L T \Ǩ 7. A ψ (P ) = B 2 + B 1 K ψ (P ) ( )) 8. τ (J τ (Ḡ)) = sum( P τ A ψ (P )A ψ (P ) T + D 12 + D 11 K ψ (P ) 2 F K ψ(p ) 2 F For the control synthesis algorithm presente in next section, the following algorithm has been evelope in [5] to compute the H 2 guarantee cost in (5). Algorithm 3. The following algorithm computes the H 2 guarantee cost of Ḡ. 1. Check Finiteness of the H 2 Guarantee Cost: Verify that the system is robustly stable using a Riccati equation to check its H norm. 2. Fin Initial Interval: Choose α > 1. Starting from k =, iterate over k until a value of τ = α k is foun such that J τ (Ḡ) an (/τ)(j τ (Ḡ)) >. Denote this value of value of τ by τ u ; this correspons to an upper boun on the optimal value of τ. If τ u = 1, then is a lower boun on the optimal value of τ, otherwise τ u /α is a lower boun. 3. Bisection: Solve the equation (/τ)(j τ (Ḡ)) = over τ using bisection. Whenever J τ (Ḡ) =, this correspons to a lower boun on the optimal value of τ. In this algorithm, each evaluation of J τ (Ḡ) is one using Algorithm 1 an each evaluation of (/τ)(j τ (Ḡ)) when J τ (Ḡ) is one using Algorithm 2. 4 H 2 GUARANTEED COST CONTROL SYNTHESIS 4.1 Control Problem Formulation With the H 2 guarantee cost analysis results in Section 3, we are intereste in fining a controller to minimize the upper boun in (4) so as to achieve the optimal H 2 guarantee cost for the robust performance. Consequently, the H 2 guarantee cost control esign for the servo system shown in Fig. 5 can be transforme to the following optimization inf K inf τ> J τ(f l (G,K)). (6) 4 Copyright 211 by ASME Downloae From: on 1/11/214 Terms of Use:

5 z 2 q y G fi K Figure 7: FULL INFORMATION CONTROL STRUCTURE. 4.2 Full Information Control Algorithm Before we present the output feeback control synthesis algorithm, we consier the optimal control (in terms of J τ ) of the interconnection shown in Fig. 7. In this iagram, we let G fi have the state-space realization A B 1 B 2 B 3 C 1 D 11 D 12 D 13 G f i C 2 D 21 D 22 D 23 I. (7) I I u From the state-space realization in (7), we learn that the feeback controller has the full access to the plant state an isturbances. Thus, the control esign of G f i is calle full information control problem. As iscusse in [5], we consier the optimal full information control problem corresponing to solving the optimization problem J f i,ε := inf K J (ε 1 ) (F l(g f i,k)). (8) A few quantities which will be important in this section are the combinations of state-space matrices B [1,3] := [ [ ] [ ] ] C1 D11 B 1 B 3, C :=, D C 1 := 2 D 21 [ ] [ ] [ ] D12 D13 D11 D D 2 :=, D D 3 :=, D 22 D [1,3] := D 21 D 23 an the parameters Q := C1 T C 1 + εc2 T C 2, S := C1 T [ ] [ ] D11 D 13 + εc T 2 D21 D 23 Q := D T 12D 12 + εd T 22D 22, S := D T [ ] [ ] 12 D11 D 13 + εd T 22 D21 D 23 [ D T R := 11 D 11 I D T 11 D ] [ 13 D T D T 13 D + ε 21 D 21 D T 21 D ] D T 23 D 23 ϕ := (A, B [1,3], Q, R, S), w ψ := (B 2, B [1,3], Q, R, S) Throughout this paper, the symbol enotes the transpose of the corresponing element at its transpose position. In this section, we will make the following assumptions: (A1) D T 3 D 3 is invertible (A2) (A,B( 3 ) is stabilizable [ ]) A λi B3 (A3) im Ker =, λ C satisfying λ 1. C D 3 Notice that the operator Ker( ) represents the kernel (i.e. null space) of the matrix *. These regularity conitions are analogous to those require for the esign of a linear quaratic regulator or full information H controller using iscrete Riccati equations [8]. With these formulas in place, we first present two algorithms which respectively compute J f i,ε an its erivative with respect to ε. Algorithm 4. The following algorithm computes J f i,ε, the corresponing optimal controller, an the corresponing optimal close-loop system uner assumptions (A1) (A3). 1. Fin the stabilizing solution of the DARE R ϕ (P) = P 2. Compute the Cholesky factorization T22 T T 22 = B T 3 P B 3 + D T 13 D 13 + εd T 23 D T 21 = T22 T \(BT 3 P B 1 + D T 13 D 11 + εd T 23 D 21) 4. Compute the Cholesky factorization T11 T T 11 = T21 T T 21 + I B T 1 P B 1 D T 11 D 11 εd T 21 D Form the optimal controller gains: K o x = T 22 \(T T 22\(B T 3 P A + D T 13C 1 + εd T 23C 2 )) K o = T 22\T 21 K o w = T 22 \(T T 22\(B T 3 P B 2 + D T 13D 12 + εd T 23D 22 )) 6. Form the close-loop state-space matrices A cl = A + B 3 Kx o, B cl 1 = B 1 + B 3 K o, Bcl 2 = B 2 + B 3 Kw o C1 cl = C 1 + D 13 Kx o, D cl 11 = D 11 + D 13 K o, Dcl 12 = D 12 + D 13 Kw o C2 cl = C 2 + D 23 Kx o, D cl 21 = D 21 + D 23 K o, Dcl 22 = D 22 + D 23 Kw o 7. Verify that ( A cl is Schur ) 8. Ǩ = T11 T \ (B cl 1 )T P B cl 2 + (Dcl 11 )T D cl 12 + ε(dcl 21 )T D cl J f i,ε = ε (sum(p 1 [B cl ) Ǩ 2 F 2 (Bcl 2 )T ]) + D cl 12 2 F + ε Dcl 22 2 F + If steps 1, 2, or 4 fail or if A cl is foun to be not Schur in step 7, then J f i,ε = an there is no optimizing controller. Algorithm 5. The following algorithm computes J f i,ε /ε. 1. Use Algorithm 4 to compute the following quantities: P, T 11, T 21, T 22, Kx o, K o, Ko w, A cl, B cl 1, Bcl 2, Ccl 1, Ccl 2, Dcl 11, Dcl 12, D cl 21, Dcl 22, Ǩ, an J f i,ε 5 Copyright 211 by ASME Downloae From: on 1/11/214 Terms of Use:

6 2. Compute the quantities ( ) K x = T 11 \ T11\[(B T cl 1 ) T P A + (D cl 11) T C 1 + ε(d cl 21) T C 2 ] ( ) K w = T 11 \ T11\[(B T cl 1 ) T P B 2 + (D cl 11) T D 12 + ε(d cl 21) T D 22 ] 3. Compute the quantities 4. Compute the quantities K x = K o x + K o K x K w = K o w + K o K w A ϕ (P ) = A + B 1 K x + B 3 K x, A ψ (P ) = B 2 + B 1 K w + B 3 K w Č = C 2 + D 21 K x + D 23 K x, Ď = D 22 + D 21 K w + D 23 K w 5. Using the MATLAB function lyapchol, solve for the Cholesky factor U in the iscrete Lyapunov equation U T U = A ϕ (P ) T (U T U)A ϕ (P ) +Č T Č 6. J f i,ε /ε = ε 1 ( UA ψ (P ) 2 F + Ď 2 F J fi,ε) With these results in place, we can easily solve (8) using the following algorithm: Algorithm 6. The following algorithm computes the optimal H 2 guarantee cost of the close-loop system along with an optimal controller. 1. Check Regularity Conitions: Verify that assumptions (A1) (A3) hol. 2. Fin Initial Interval: Choose α > 1. Check if J fi,ε an J f i,ε /ε < when ε = 1. If so, start from k = 1 an increment k until either or these conitions fail to be met when ε = α k. Denoting the corresponing value of ε as ε u, there exists an optimal value of ε in the interval (α 1 ε u,ε u ). If instea either J f i,ε = or J f i,ε /ε > when ε = 1, start from k = 1 an increment k until J f i,ε an J f i,ε /ε < when ε = α k. Denoting the corresponing value of ε as ε l, there exists an optimal value of ε in the interval (ε l,αε l ). 3. Bisection: Solve J f i,ε /ε = over ε using bisection. Whenever J f i,ε = for a particular value of ε, this value of ε is an upper boun on the optimal value of ε. In this algorithm, each evaluation of J f i,ε is one using Algorithm 4 an each evaluation of J f i,ε /ε when J f i,ε is one using Algorithm Full Control Algorithm In [5], the output feeback control problem in (6) is reuce to the following optimal full control problem infj 1 (F l (G 4, K)) (9) K where G 4 has the following the state-space realization A + B 1 K x B 1 T 1 11 B 2 + B 1 K w I G 4 T 22 K x T 22 K o T 11 1 T 22 K w T 22 (1) C 3 + D 31 K x D 31 T11 1 D 32 + D 31 K w where the parameters K x, K w, K x, an K w are efine in Algorithm 5. It is not currently known whether or not the optimal full control problem can be solve using Riccati equations. Therefore, to solve this problem, we will resort to the SDP approach. In particular, we have prove that the optimization problem (9) is equivalent to the optimization problem inf tr{w} s.t. (11a) P,W,V,ˆL x,l v P T11 T T 11 V W PǍ + ˆL x Č PB 1 + ˆL x D 31 P ˇB + ˆL x Ď P T 22 (L v Č K x ) T 22 (L v D 31 K o) T 22(L v Ď K w ) I (11b) where [ ] [ ] Ǎ ˇB A + B1 K := x B 2 + B 1 K w. (12) Č Ď C 3 + D 31 K x D 32 + D 31 K w Consequentially, for any P,W,V, ˆL x,l v that satisfy (11b), an output feeback controller which achieves J τ (F l (G,K)) J f i,ε + ε 1 tr{w} can be reconstructe from A c = A + B 1 K x + B 3 K x + (P 1 ˆL x B 3 L v )(C 3 + D 31 K x ) [ Ac P K 1 ] ˆL x B 3 L v. (13) L v (C 3 + D 31 K x ) K x L v 4.4 Output Feeback Control Algorithm This section gives a heuristic for solving (6) using the results presente so far in this paper. Algorithm 7. The following algorithm is a heuristic for solving the optimal output feeback control problem (6). 1. Fin Initial Value of τ (a) Full Information Controller Design: Using Algorithm 6, esign an optimal full information controller. (b) Fin Feasible Value of τ: Choose α >. For the final values etermine uring the last full information controller esign, solve (11) using an SDP solver. If 6 Copyright 211 by ASME Downloae From: on 1/11/214 Terms of Use:

7 the optimization was feasible, reconstruct the corresponing output feeback controller K using (13). If the optimization was not feasible, set τ ατ, esign a full information controller for ε = τ 1 using Algorithm 4, an reo this step. (c) Close-Loop System Analysis (Fixe K): Form the close-loop system F l (G,K) an analyze its H 2 guarantee cost performance using Algorithm Controller Design (a) Output Feeback Controller Design (Fixe τ): For the value of τ > foun in the previous close-loop system analysis step, solve (11) using an SDP solver an reconstruct the corresponing controller K using (13). (b) Close-Loop System Analysis (Fixe K): Form the close-loop system F l (G,K) an analyze its H 2 guarantee cost performance using Algorithm 3. Return to step 2a. In our implementation, we use α = 1. We use two stopping criteria in this algorithm. If the number of output feeback controller optimizations (i.e. the number of times steps 1b an 2a have been execute) excees 3 or if J [i 1] o f /J [i] o f 1 < 1 4 where J [i] o f is the cost reporte the ith time step 2b executes, we terminate the algorithm. We also terminate the algorithm if the SDP solver claims infeasibility in step 2a. Magnitue (abs) Boe Diagram Uncertainty weighting function Frequency (Hz) Figure 8: PLANT UNCERTAINTY WEIGHTING FUNCTION W was utilize to measure the absolute raial isplacement of the slier. The resolution of the LDV is 2 nm for the measurement gain of.5 µ/v. The control circuits inclue a Texas Instrument TMS32C6713 DSP boar an an in-house mae analog boar with a 12-bit ADC, a 12-bit DAC, a voltage amplifier to rive the MA, an a current amplifier to rive the VCM. The DSP sampling frequency is 71.4 KHz in this paper. An the input elay incluing ADC an DAC conversion elay an DSP computation elay is 6 µs. A hole was cut through the case of the rive to make laser go into the rive. 5 CONTROL IMPLEMENTATION 5.1 H 2 Guarantee Cost Control Design For the ual-stage track-following servo esign shown in Fig. 2, we nee to esign a proper uncertainty weighting function W so that the moeling of the uncertain plant, (1+W )G p, covers all plant variations but is not too conservative. Base on the servo system ientification in [6], the uncertainty weighting function was selecte as shown in Fig. 8. The selecte uncertainty weighting function emonstrates that the real plant coul have an unstructure uncertainty of a ±2% gain variation at low frequency an a ±54% gain variation at high frequency respectively from the nominal plant. With the control input weighting values of W uv = W up =.4, the H 2 guarantee cost control is synthesize by using Algorithm 7. As emonstrate in [7, 9], the propose control algorithm, by exploiting the Riccati equation structure to ecrease the number an complexity of SDPs, is able to increase the computation spee an accuracy compare to the approaches [1] solving LMIs. 5.2 Experimental Stuy Figure 9 shows a picture of the experimental setup. A PZTactuate suspension shown in Fig. 1 was assemble to an arm of the E-block of a commercial RPM isk rive. An LDV PZT-actuate suspension Amplifier LDV DSP Figure 9: DSA EXPERIMENT SETUP ) was calculate using the controller transfer function an measure open-loop plant transfer functions. From the experiment results, we see the preicte an observe close-loop sensitivity functions agree very well Then, the synthesize controller was implemente on our experiment setup illustrate in Fig. 9. In orer to valiate our control esign, the close-loop sensitivity function by using the sinusoial sweeping signal was measure. The observe sensitivity function is inicate by the black line in Fig. 1, while the preicte one is inicate by the re line. Note that the preicte sensitivity function (T s = 1 1+G p K 7 Copyright 211 by ASME Downloae From: on 1/11/214 Terms of Use:

8 especially at frequencies below 1 KHz. The preicte sensitivity function at frequencies above 1 KHz is very noisy, because the measure MA frequency response is quite noise, as shown in Fig. 4. From the results in Fig. 1, we also see that the observe sensitivity transfer function matches the base line (in the sense of average) of the preicte one from 1 KHz to 3 KHz. In aition, the experiment results emonstrate that the esigne controller using our presente H 2 guarantee cost control synthesis algorithm prouce a relatively-high gain-crossover frequency of 3 KHz for the error rejection function. Magnitue (B) Observe Boe Diagram Preicte Frequency (Hz) Figure 1: EXPERIMENT RESULTS FOR ERROR REJEC- TION FUNCTIONS 6 CONCLUSION In this paper, we first presente an upper boun, H 2 guarantee cost, for the worst-case H 2 performance of a iscrete-time system with a norm-boune unstructure uncertainty. Base on the H 2 guarantee cost analysis an the corresponing optimal full information control, we reviewe the output feeback H 2 guarantee cost control algorithm by exploiting Riccati equation structure to reuce the number an complexity of the SDPs that nee to be solve. Consequently, the presente control algorithm is able to increase the computation spee an accuracy compare to the approaches entirely base on semi-efinite programs. Then, we applie the propose control algorithm to ualstage HDD track-following servo systems. The experiment results by implementing the synthesize controller on an actual isk rive with a PZT-actuate suspension emonstrate the effectiveness of the presente H 2 guarantee cost control algorithm an valiate our control esign. Specifically, the esigne ualstage servo by our propose control synthesis algorithm attaine a relatively-high gain-crossover frequency of 3 KHz for the error rejection function. ACKNOWLEDGMENT The authors thank Western Digital Inc. an Hutchinson Technology Inc. with parts an valuable information to perform this stuy. This work was performe with support from the Information Storage Inustry Consortium (INSIC) an UC Berkeley Computer Mechanics Laboratory (CML). REFERENCES [1] Abramovitch, D., an Franklin, G., 22. A brief history of isk rive control. IEEE Control Systems Magazine, 22(3), pp [2] Li, Y., an Horowitz, R., 21. Mechatronics of electrostatic microactuators for computer isk rive ual-stage servo systems. IEEE/ASME Trans. Mechatronics, 6(2), pp [3] Conway, R., Felix, S., an Horowitz, R., 27. Moel reuction an parametric uncertainty ientification for robust h2 control synthesis for ual-stage har isk rives. IEEE Transactions on Magnetics, 43(9), pp [4] Peaucelle, D., Ebihara, Y., an Arzelier, D., 28. Robust H 2 perfomance of iscrete-time perioic systems: Lmis with reuce imensions. In Proc. of the 17th Worl Congress, Vol. 17, International Feration of Automatic Control, IFAC. [5] Conway, R., 211. Discrete-Time H 2 Guarantee Cost Control. PhD Thesis, Univeristy of California Berkeley, Berkeley, CA. [6] Nie, J., an Horowitz, R., 29. Design an implementation of ual-stage track-following control for har isk rives. In Proc. of the 29 Dynamic Systems an Control Conference. [7] Conway, R., an Horowitz, R., 29. Analysis of h 2 guarantee cost performance. In Proc. of the 29 Dynamic Systems an Control Conference. [8] Peters, M. A., an Iglesias, P. A., Minimum Entropy Control for Time-Varying Systems. Systems & Control: Founations & Applications. Birkhäuser, Boston, MA. [9] Conway, R., an Horowitz, R., 21. Optimal full information H 2 guarantee cost control of iscrete-time systems. In Proc. of the 21 Dynamic Systems an Control Conference. [1] Conway, R., an Horowitz, R., 21. Guarantee cost control for linear perioically time-varying systems with structure uncertainty an a generalize H 2 objective. Mechatronics, 2(1), pp Copyright 211 by ASME Downloae From: on 1/11/214 Terms of Use: