H2 Guaranteed Cost Control in Track-Following Servos

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1 Chapter 14 H2 Guarantee Cost Control in Track-Following Servos Richar Conway, Jianbin Nie, an Roberto Horowitz Abstract. This chapter presents two new control synthesis approaches for ual-stage track-following servo systems. Both approaches are base on H 2 guarantee cost analysis, in which an upper boun on the worst-case H 2 performance of a iscretetime system with gain-boune unstructure causal LTI uncertainty is etermine by solving either a semi-efinite program (SDP) or several Riccati equations. We review the results of a paper on H 2 guarantee cost analysis an a paper on optimal full information H 2 guarantee cost control an then use these results to evelop two output feeback control synthesis approaches. The first approach is base entirely on the solution of SDPs whereas the secon approach exploits Riccati equation structure to reuce the number an complexity of the SDPs that nee to be solve. Throughout the paper, we apply the analysis an control techniques to a har isk rive moel with a PZT-actuate suspension an emonstrate that the approaches that exploit Riccati equation structure are faster an at least as accurate as their SDP counterparts Introuction For several ecaes now, the areal storage ensity of har isk rives(hdds) has been oubling roughly every 18 months, in accorance with Kryer s law. As the storage ensity is pushe higher, the concentric tracks on the isk which contain ata must be pushe closer together, which requires much more accurate control of the rea/write hea. Currently available har rives can store 2 TB of ata on a 3.5 rive with three platters. This correspons to an areal ata ensity of 600 gigabit/in 2. The current goal of the magnetic recoring inustry is to achieve an areal storage ensity of 4 terabit/in 2.Itisexpectethatthetrackwith require to achieve this Richar Conway Jianbin Nie RobertoHorowitz Department of Mechanical Engineering, University of California, Berkeley rconway345@berkeley.eu,njbin@berkeley.eu horowitz@me.berkeley.eu E. Eleftheriou & S.O.R. Moheimani (Es.): Cntrl. Tech. for Emerging Micro/Nanoscale Sys. LNCIS 413, pp springerlink.com c Springer-Verlag Berlin Heielberg 2011

2 236 R. Conway, J. Nie, an R. Horowitz Hea Suspension Pivot Voice Coil Motor (VCM) Disk Spinle Motor E-block Data Track Fig Schematic of a typical HDD ata ensity is 25 nm. To achieve this specification for track-following control, in which the rea/write hea is maintaine as close to the center of a given ata track as possible, the 3σ value of the close-loop position error signal (PES) shoul be less than 2.5 nm. To help achieve this goal, the use of a seconary actuator has been propose to give increase precision in rea/write hea positioning. There are three classes of seconary actuators: actuate suspensions (6), actuate sliers (7), an actuate heas (17). Each of these propose seconary actuator classes correspons to a ifferent actuator location in Fig In the actuate hea configuration, a microactuator (MA) actuates the rea/write hea with respect to the slier mounte at the tip of the suspension. In the actuate slier configuration, an MA irectly actuates the hea/slier assembly with respect to the suspension. For both of these configurations, it is ifficult to esign an MA which can be easily incorporate into the manufacture an assembly of a HDD on a large scale. In the actuate suspension configuration, the MA actuates the suspension with respect to the E-block. This seconary actuator scheme is the least ifficult to esign an has been incorporate into some consumer proucts. We will use this seconary actuation scheme in this paper. 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 gives robust performance over a set of HDDs. One framework for solving this problem is guarantee cost control. This methoology is a control esign methoology whose objectives involve worst-case quaratic time omain costs over a moele set of parametric uncertainty. Both the state feeback synthesis problem an the output feeback synthesis problem can be solve for iscrete-time systems by using semiefinite programs (SDPs) convex optimization involving linear matrix inequalities (LMIs) as is one in (14) an (18), respectively. 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

3 14 H2 Guarantee Cost Control in Track-Following Servos 237 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. This chapter reviews the basic results of H 2 guarantee cost analysis (2), which use the techniques of guarantee cost control to yiel an upper boun on the worstcase H 2 performance of a system with ynamic unstructure uncertainty. Using this characterization of performance, we then review the solution of the corresponing full information control problem (3) a generalization of the state feeback control problem an use the results to generate two heuristics for solving the output feeback control problem. For all of the problems in this chapter, we consier two approaches: an approach base on solving SDPs an an approach base on solving Riccati equations. Parallel to reviewing an eveloping the relevant theory, we apply these techniques to the esign an analysis of HDD track-following controllers. For all numerical experiments in this paper, we use a 2.2 GHz Intel Core 2 Duo processor with 2 GB RAM running MATLAB (with multithreae computation isable) uner 32-bit Winows Vista. We solve SDPs two ways in this paper: using SeDuMi (16) with YALMIP (12) an using the mincx comman in the Robust Control Toolbox for MATLAB without YALMIP. Through these HDD track-following control examples, we emonstrate the computational avantages of the approach base on Riccati equation solutions Preliminaries In this chapter, we will enote the spectral norm (i.e. the maximum singular value) an the Frobenius norm of a matrix M respectively as M an M F.Wewillsay that M is Schur if all of its eigenvalues lie strictly insie the unit isk in the complex plane. The operator iag takes several matrices an stacks them iagonally: M 1 0 iag[m 1,...,M n ]=... 0 M n. (14.1) Positive efiniteness (resp. semi-efiniteness) of a symmetric matrix X will be enote by X 0(resp.X 0), an a in a symmetric matrix will represent a block which follows from symmetry. For given (A,B,Q,R,S),whereQ = Q T an R = R T,weefinethefunctions R (A,B,Q,R,S) (P) := A T PA + Q (A T PB + S)(B T PB + R) 1 (B T PA + S T ) (14.2a) K (A,B,Q,R,S) (P) := (B T PB + R) 1 (B T PA + S T ). (14.2b) We will make the notation more compact in the remainer of the paper by respectively enoting these functions as R φ (P) an K φ (P) where φ is an appropriately efine 5-tuple. Note that the equation R φ (P) =P is a iscrete algebraic Riccati

4 238 R. Conway, J. Nie, an R. Horowitz equation (DARE). If R φ (P)=P = P T an A + BK φ (P) is Schur, then P is calle a stabilizing solution of the DARE. Throughout the paper, we will implicitly use the property that if a DARE has a stabilizing solution, it is unique (11). Amatrixpair(A,B) will be calle -stabilizable if K such that A+BK is Schur. Amatrixpair(A,C) will be calle -etectable if L such that A + LC is Schur. For agivenstableancausalltisystemg, itsh 2 an H norms will respectively be enote as G 2 an G.FortwocausalLTIsystemsG 1 an G 2,weenotethe lower linear fractional transformation (LFT) of G 1 by G 2 (shown in Fig. 14.2(a)) as F l (G 1,G 2 ).WewillenotetheupperLFTofG 1 by G 2 (shown in Fig. 14.2(b)) as F u (G 1,G 2 ). Fig Linear fractional transformations (LFTs) G 1 G 2 (a) Lower LFT G 2 G 1 (b) Upper LFT 14.2 Har Disk Drive Moel In this section, we present the HDD moel we will be using throughout this chapter. The HDD we are consiering has the PZT-actuate suspension shown in Fig. 14.3, which is a Vector moel suspension provie to us by Hutchinson Technology Inc. In our setup, we use a laser Doppler vibrometer (LDV) to measure the absolute raial isplacement of the slier. The control circuits inclue a Texas Instrument TMS320C6713 DSP boar an an in-house mae analog boar with a 12-bit ADC, a12-bitdac,avoltageamplifiertorivethema,anacurrentamplifiertorive the voice coil motor. The DSP sampling perio is santhecontroller elay, which inclues the ADC an DAC conversion elay an the DSP computation elay, is 6 µs. A hole was cut through the case of the rive to allow the LDV laser to shine on the slier. It shoul be note that these moifications affect the response of the rive an may have etrimentally affecte the attainable performance of the servo system. The block iagram of our HDD setup is shown in Fig an the relevant signals an their units are liste in Table In this block iagram, we treat the ynamics from the two control inputs to the hea isplacement as a single block Fig Picture of the Vector moel PZT-actuate suspension use in our experimental setup

5 14 H2 Guarantee Cost Control in Track-Following Servos 239 w r w a q y h w n G r ¾ n u v u p ¾ a G p W K r - + y Fig HDD block iagram Table 14.1 HDD signals Signal Description Units r Disturbances on the hea position (8) nm u p PZT actuator control signal V u v Voice coil motor control signal V w a Airflow isturbances (normalize) w n PES sensor noise (normalize) w r Disturbances on the hea position (normalize) y PES nm y h Hea isplacement relative to the track center nm Table 14.2 Moel parameters for G p moe, i [ ] ai,1 B a i i,2 [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]

6 240 R. Conway, J. Nie, an R. Horowitz to take into account the knowlege that both actuators can excite the same vibration moes in the suspension. Exploiting this knowlege allows us to form a moel which oes not have reunant states resulting from incluing two copies of the suspension vibration moes. The block is an unknown stable causal LTI system which satisfies 1. This block, along with W,characterizestheoutputynamic multiplicative uncertainty on G p. To construct a iscrete-time moel of our system, we use the methoology of (13). To fin the moel of G p,wefirstobtainefrequencyresponsesofoursystem from u v to y an u p to y. Usingweighteleastsquares,weseparatelyfita continuous-time moel to each of these frequency responses, which we then combine into a single moel an use common moe ientification (4) to eliminate reunant copies of the suspension vibration moes. We then iscretize this moel with the 6 µs elayoneachofitstwoinputstoyielthemoelforg p.thismoel is given by G p (z)= [ ] z i=1 [ ] ( [ ]) 1 ai, zi B a i,2 0 i (14.3) where the moel parameters are as liste in Table Because there are two poles at z = 0 one for each input channel the statespace moel of G p has 14 states. The poles at z = 0wereintroucebythe iscretization of the continuous-time input elay. The six vibration moes in (14.3) are orere from lowest to highest resonance frequency. The Boe plot of this moel is shown in Fig The weighting for the ynamic multiplicative uncertainty of G p,givenby W = z z 0.465, (14.4) was chosen so that the uncertain moel envelope the experimental frequency response of G p.theboemagnitueplotofw is shown in Fig. 14.6(a). Since is asisosystem,upperanlowerbounsonthemagnitueofeachinput/outputpair in G p can be easily compute one frequency at a time. Doing so yiels the upper an lower bouns on the Boe magnitue plots of G p shown in Fig The values σ a = an σ n = 1.3 wereeterminebymatchingthepowerspectrum ensity of the open loop slier motion respectively at low an high frequency. The isturbances on the hea position are characterize by G r (z) = [ 10 ]( [ ]) 1 [ ] zi [ 11 ]( [ ]) 1 [ ] zi. (14.5) Figure 14.6(b) shows the Boe magnitue plot of G r.inaitiontocapturingthe effect of isturbances on the hea position, this moel of G r also captures the

7 14 H2 Guarantee Cost Control in Track-Following Servos Magnitue (B) u p y h u v y h Phase (eg) u p y h u v y h Frequency (Hz) 10 4 Fig Boe plot of G p Magnitue (B) Magnitue (B) Frequency (Hz) Frequency (Hz) (a) W (b) G r Fig Boe magnitue plots of W an G r Magnitue (B) Frequency (Hz) Magnitue (B) Frequency (Hz) (a) u v y h (b) u p y h Fig Nominal Boe magnitue plots for G p along with pointwise upper an lower bouns over moele uncertainty

8 242 R. Conway, J. Nie, an R. Horowitz low-frequency rift in the LDV position measurements resulting from integration of velocity measurements. The secon-orer moe near 1 khz in this moel captures the effect of isk moes between 1 khz an 3 khz. These isturbances, although realistic for our experimental setup, are larger than the isturbances typically foun in a HDD. First of all, the measurement noise of the LDV is somewhat larger than the measurement noise of the PES. Moreover, as we previously mentione, the LDV has a significant low-frequency rift. These two factors along with the mechanical moifications of the rive significantly eteriorate the achievable level of close-loop performance. With some manipulation, the blocks in Fig can be groupe to form the LFT representation in Fig In this form G H has 19 states. For the remainer of this chapter, we will use the balance realization of G H for analysis an control esign. Fig LFT representation of HDD moel yh v u p q y G H K wa h i r uv w n u p 14.3 H2 Guarantee Cost Analysis In this section, we review the results of (2) on H 2 guarantee cost analysis. In particular, we first present an SDP for etermining the H 2 guarantee cost performance of a given system an then show that this convex optimization can be efficiently solve using nonlinear convex optimization involving Riccati equation solutions. For the sake of brevity an clarity of presentation, we o not present the proofs here; intereste reaers shoul rea the paper cite above Semi-efinite Programming Approach Before consiering systems with uncertainty, we first consier a given iscrete-time LTI system G with known state-space realization [ A G G B ] G C G D. (14.6) G Awell-knowncharacterizationoftheH 2 norm is that G 2 2 < γ if an only if there exist P 0anW such that tr{w} < γ W B T G PB G + DT G D G P A T G PA G +CT G C G. (14.7a) (14.7b) (14.7c)

9 14 H2 Guarantee Cost Control in Track-Following Servos 243 In this context, however, it is beneficial for us to consier an alternate characterization which says that G 2 2 < γ if an only if there exist P 0,W,V such that tr{w} < γ (14.8a) [ ] [ PV T A G B ] T [ ][ G P 0 A G B ] G VW C G D G 0 I C G D. (14.8b) G It shoul be note that eliminating V from (14.8b) using the matrix elimination technique (see, e.g., (1)) yiels (14.7b) (14.7c). This alternate characterization is more suitable for two reasons. First, as will be iscusse later in this section, it will allow us to consier a richer set of system uncertainty moels. Secon, it will allow us to use the matrix variable elimination technique to erive an optimal control scheme in Sect. 14.4, which will be important in our approaches to the output feeback problem evelope in Sect Fig LFT representation of uncertain system q p ¹G w We now turn our attention to analyzing the H 2 performance of the system interconnection shown in Fig where Ḡ has the state-space realization Ā B 1 B 2 Ḡ C 1 D 11 D 12 (14.9) C 2 D 21 D 22 an is a real matrix satisfying 1. Closing the loop yiels [ Ā + B 1 (I D 11 ) 1 C 1 B 2 + B 1 (I D 11 ) 1 ] D 12 F u (Ḡ, ) C 2 + D 21 (I D 11 ) 1 C 1 D 22 + D 21 (I D 11 ) 1 D 12 [ ] A B =:. (14.10) C D We are thus intereste in etermining if F u (Ḡ, ) 2 2 < γ, 1. Using the characterization of the H 2 norm given by (14.8), we woul like to know if there exists P 0,W,V such that tr{w} < γ an [ PV T VW ] [ ] T [ ][ ] A B P 0 A B, 1. (14.11) C D 0 I C D It shoul be note that, although the state-space matrices are a function of, the analysis variables (P, W,anV)arenot.ApplyingtheS-proceure(see,e.g.,(1))to

10 244 R. Conway, J. Nie, an R. Horowitz (14.11) yiels the equivalent conition that there exists τ > 0,P 0,W,V such that tr{w} < γ an M (τ,p,w,v ) := P 0 V T 0 τi 0 V 0 W Ā B 1 B 2 C 1 D 11 D 12 C 2 D 21 D 22 P 0 0 Ā B 1 B 2 0 τi 0 C 1 D 11 D I C 2 D 21 D 22 (14.12) By Schur complements, this is equivalent to the existence of τ,p,w,v such that P 0 τi M (τ,p,w,v) := V 0 W PĀ P B 1 P B 2 P 0. (14.13) τ C 1 τ D 11 τ D 12 0 τi C 2 D 21 D I These two new conitions, which are equivalent to each other, remove the epenence of the matrices on at the expense of introucing an extra scalar parameter, τ. Sincethematricesinthetwonewconitionsonotepenon, theygiveus computationally tractable means to verify that γ is an upper boun on the worst-case H 2 performance of F u (Ḡ, ) when is a real matrix satisfying 1. Rewriting these feasibility problems as an optimization problems to fin the smallest upper boun of this type yiels T inf tr{w} τ>0,p 0,W,V s.t. M (τ,p,w,v) 0 (14.14) inf τ,p,w,v s.t. M (τ,p,w,v ) 0. (14.15) We will refer to the square root of the value of these optimization problems as the H 2 guarantee cost of Ḡ. Ofthesetwooptimizations,(14.14)ismoreusefulfor etermining the H 2 guarantee cost of a given system because its matrix inequalities are smaller in imension an the optimization problem has fewer ual variables. However, as we will see in Sects an 14.5, (14.15) will be more suitable for control esign. Relaxing the strict inequalities in either of these optimization problems to non-strict inequalities results in a SDP. Thus, a reasonable way to solve the H 2 guarantee cost analysis problem is to relax (14.14) to a SDP then solve the SDP using an appropriate solver. From the erivation, it is obvious that the H 2 guarantee cost is an upper boun on the worst-case H 2 performance of the interconnection in Fig when is a real matrix satisfying 1. What is not immeiately apparent, however, is that the H 2 guarantee cost is also an upper boun on the worst-case H 2 performance of the interconnection in Fig when is only known to be a causal LTI system satisfying 1.

11 14 H2 Guarantee Cost Control in Track-Following Servos Riccati Equation Approach We begin this subsection by noting that (14.14) can equivalently be expresse as where J τ (Ḡ) := inf P 0,W,V inf τ>0 J τ(ḡ) (14.16) tr{w} s.t. M (τ,p,w,v) 0. (14.17) It can be shown that J τ (Ḡ) is well-efine, i.e. for a fixe value of τ > 0, J τ (Ḡ) is inepenent of the realization of Ḡ. Whenever the optimization problem (14.17) is infeasible for a particular value of τ, wewillsaythatj τ (Ḡ)=. Defining [ Q S ] := C 2 T [ ] C 2 D 21 + τ C 1 T [ ] C 1 D 11 [ ] Q W S W := D T [ ] 22 D 22 D 21 + τ D T [ ] 12 D 12 D 11 R := D T 21 D 21 + τ( D T 11 D 11 I) φ := (Ā, B 1, Q, R, S) ψ := ( B 2, B 1, Q W, R, S W ) (14.18a) (14.18b) (14.18c) (14.18) (14.18e) it was shown (2) that J τ (Ḡ) if an only if the DARE R φ (P)=Phas a stabilizing solution, P 0,suchthat B T 1 P 0 B 1 + R 0. In this case, J τ (Ḡ)=tr{R ψ (P 0 )}. (14.19) It was also shown that if J τ (Ḡ) for τ = τ 0,thenJ τ for all τ > τ 0.Moreover, there exists a value of τ such that J τ (Ḡ) if an only if the interconnection in Fig is robustly stable over 1, i.e. the H norm of Ḡ from to q is less than 1. This conition is in turn equivalent to the DARE R ρ (P)=P having a stabilizing solution P 0 such that B T 1 P 0 B 1 + D T 11 D 11 I 0where ρ :=(Ā, B 1, C T 1 C 1, D T 11 D 11 I, C T 1 D 11 ). (14.20) Thus, once we have verifie that the optimization problem (14.16) is feasible by using the DARE R ρ (P)=P, wecanalwaysfinvaluesofτ for which J τ (Ḡ) simply by making τ increasingly large. With this in min, we woul like to know how J τ (Ḡ) varies as τ varies. First, since minimizing a convex function of several variables over a subset of those variables prouces a convex function of the remaining variables, we see that J τ (Ḡ) is aconvexnonlinearfunctionofτ. Secon,sincethestabilizingsolutionofaDARE is analytic in its parameters (5) an J τ (Ḡ) is an analytic function of the stabilizing DARE solution, J τ (Ḡ) is an analytic function of τ.wewillthereforefintheglobal optimal value of J τ (Ḡ) by fining a value of τ such that (/τ)(j τ (Ḡ)) = 0. Although the most straightforwar way to fin (/τ)(j τ (Ḡ)) is by irectly taking the erivative of the relevant equations with respect to τ, as was one in(2),

12 246 R. Conway, J. Nie, an R. Horowitz we will pursue a slightly ifferent approach here which is more computationally efficient. First, we efine ε := τ 1 an ˆφ := (Ā, B 1, ε Q, ε R, ε S) ˆψ := ( B 2, B 1, ε Q W, ε R, ε S W ). (14.21a) (14.21b) Multiplying the DARE by ε yiels R ˆφ (εp 0)=εP 0.AlsonotethatεR ψ (P 0 )= R ˆψ (εp 0 ).Takingerivativesofthesetwoequations yiels, after some algebra, that ) T ε (εp 0)= (Ā + B 1 K φ (P 0) ε (εp 0) (Ā + B 1 K φ 0)) (P T ( ) +( C 2 + D 21 K φ 0)) (P C 2 + D 21 K φ (P 0) ( εr ψ (P 0 ) ) = ( B 2 + B 1 K ψ (P 0 ) ) T ε ε (εp 0) ( B 2 + B 1 K ψ (P 0 ) ) (14.22a) + ( D 22 + D 21 K ψ (P 0 ) ) T ( D 22 + D 21 K ψ (P 0 ) ). (14.22b) The first of these equations is a iscrete Lyapunov equation for (/ε)(εp 0 ).Since Ā + B 1 K φ (P 0) is stable (by the efinition of a stabilizing solution of a DARE), we see that there exists upper triangular Ū such that Ū T Ū =(/ε)(εp 0 ) an we can irectly solve for Ū using the lyapchol function in MATLAB. Using this, we express { } tr ε (εr ψ(p 0 )) = Ū( B 2 + B 1 K ψ (P 0 )) 2 F + D 22 + D 21 K ψ (P 0 ) 2 F. (14.23) Using the chain rule, we see that ε (εr ψ(p 0 )) = R ψ (P 0 )+ε ε (R ψ(p 0 )). (14.24) Thus, taking the erivative of (14.19) an applying the chain rule to the right-han sie yiels { } { τ (J τ(ḡ)) = τ 2 tr ε (R ψ(p 0 )) = τ 1 tr R ψ (P 0 ) } ε (εr ψ(p 0 )). Therefore, we have that (14.25) τ (J τ(ḡ)) = 1 τ ( Jτ (Ḡ) Ū( B 2 + B 1 K ψ (P 0 )) 2 F D 22 + D 21 K ψ (P 0 ) 2 F ). (14.26) In comparison to formulas given in (2), using (14.26) to compute the cost erivative is more computationally efficient because it requires fewer matrix multiplications an aitions. The value an erivative of J τ (Ḡ) is also useful for generating a lower boun on the optimal value of (14.16). Consier Fig , which shows a representative

13 14 H2 Guarantee Cost Control in Track-Following Servos 247 J ( ¹ G) ^J 2 Fig Illustration of ^J lower boun computation in 1 H 2 guarantee cost analysis graph of J τ (Ḡ) in which τ 0 is known to be an upper boun on the minimizing value of τ. Byconvexity,ifτ 1 is known to be a lower boun on the minimizing value of τ,thevalueanerivativeofj τ (Ḡ) at τ 0 gives us the lower boun Jˆ 1.Ifinstea,the value an erivative of J τ (Ḡ) at τ 0 an τ 2 is known, we have the lower boun Jˆ 2. These lower bouns are respectively given by ˆ J 1 = J τ0 m 0 (τ 0 τ 1 ) (14.27a) Jˆ 2 = m 2[m 0 (τ 0 τ 2 ) (J τ0 J τ2 )] + J τ2 (14.27b) m 0 m 2 where J τi is interprete as J τ (Ḡ) evaluate at τ = τ i an m i is (/τ)(j τ (Ḡ)) evaluate at τ = τ i.itshoulbenotethattheseconoftheselowerbounsisless conservative when it is applicable. With these results in place, we can easily solve (14.16) using the following methoology: 1. Check Feasibility: Check that the DARE R ρ (P)=P has a stabilizing solution P 0 such that B T 1 P 0 B 1 + D T D I Fin Initial Interval: Choose α > 1. As previously note, the DARE R φ (P)= P will have a stabilizing solution with the require properties for τ = α k,for large enough k. Startingfromk = 0, iterate over increasing k until a value of τ = α k is foun such that the DARE has a stabilizing solution P 0 that satisfies B T 1 P 0 B 1 + R 0an(/τ)(J τ (Ḡ)) > 0. Denote this value of τ by τ u.notethat this correspons to an upper boun on the optimal value of τ.ifτ u = 1, then 0 is alowerbounontheoptimalvalueofτ, otherwiseτ u /α is a lower boun. 3. Bisection: Solve the equation (/τ)(j τ (Ḡ)) = 0overτ using bisection. Use (14.26) for evaluating (/τ)(j τ (Ḡ)).WhenevertheDARER φ (P)=P oes not have a stabilizing solution with the require properties for a given value of τ,this correspons to a lower boun on the optimal value of τ. In our implementation, we use α = 100. Also, except when the lower boun on the optimal value of τ is 0, we use the geometric mean instea of the arithmetic mean to better eal with large intervals in which the optimal value of τ coul lie. We use two stopping criteria in our implementation. If we efine the relative error as ν := 1 J/J τ (Ḡ) where J is the lower boun compute by (14.27), we terminate the algorithm when either ν < or 30 iterations have been execute in steps 2 3.

14 248 R. Conway, J. Nie, an R. Horowitz Application to Har Disk Drives So far in this section, we have evelope two methoologies for analyzing the robust performance of a system with ynamic uncertainty one base on the solution of a SDP an another base on nonlinear convex optimization involving Riccati equation solutions. We will now use these two methoologies to examine the performance of aclose-loophddsystemforaspecifiecontroller. For the controller, we choose the controller returne by the MATLAB Robust Control Toolbox function hinfsyn applie to the optimization problem inf K Fl (Ĝ H,K) (14.28) where Ĝ H := iag[0.001,0.001,1,1,1]g H.Thecontrollerreturnebyhinfsyn, K o,containe19statesanachieve F l (Ĝ H,K o ) = For this controller, the nominal H 2 norm of the interconnection in Fig (i.e. with = 0) was We use three approaches to fin the H 2 guarantee cost of the system G cl := F l (G H,K o ):solvingthesdp(14.14)usingthemincx comman, solving (14.14) using SeDuMi an YALMIP, an solving (14.16) using the methoology at the en of Sect We will refer to the latter of these approaches as the DARE approach. Using these three approaches to analyze the performance of this 38 th -orer system yiele the results liste in Table Although all three approaches yiele similar values of the H 2 guarantee cost an the corresponing value of τ, the DARE approach was more than 100 times faster for this system than the other two approaches. We now look more closely at the results of applying the DARE approach to this problem. In particular, we are intereste in the values of J τ (G cl ) an (/τ)(j τ (G cl )) as functions of τ. Figure14.11showsaplotofthesetwoquantities for 50 linearly space points in the interval [2,6] along with an estimate of (/τ)(j τ (G cl )) obtaine by applying the central ifference approximation to J τ (G cl ).Aswewoulexpect,thecurvesaresmooth,(/τ)(J τ (G cl )) is monotonic non-ecreasing, an the compute values of (/τ)(j τ (G cl )) agree with the central ifference approximations. It is interesting to note that as τ becomes large, (/τ)(j τ (G cl )) becomes constant an J τ varies linearly with τ. The H 2 guarantee costs compute so far reflect a combination of the sizes of the signals y h, u v,anu p.however,itismoremeaningfultolookatthe sizes of Table 14.3 Analysis of close-loop HDD performance using three approaches Approach Optimization Time (s) H 2 Guarantee Cost Optimal τ mincx SeDuMi DARE

15 14 H2 Guarantee Cost Control in Track-Following Servos J τ (G cl ) (/τ) (J τ (G cl )) Compute erivative Central ifference approximation τ Fig J τ (G cl ),thecomputevalueof(/τ)(j τ (G cl )), anthecentralifferenceapproximation to (/τ)(j τ (G cl )) these three signals separately. To o so, we remove the outputs we are not intereste in an compute the H 2 guarantee cost. For example, when we were intereste in the H 2 guarantee cost associate with y h,weremovetheoutputsu v an u p from G cl an then compute the H 2 guarantee cost of the resulting system using the DARE approach. Doing this for y h, u v,anu p yiele the H 2 guarantee costs given in Table So, for the controller K o,thereverylittlecontroleffortisuse to achieve this level of robust position error performance. Table 14.4 Close-loop HDD performance Signal y h u v u m H 2 Guarantee Cost nm V V 14.4 Full Information H2 Guarantee Cost Control In this section, we show how to use the analysis results of the previous section to esign controllers which optimize the H 2 guarantee cost. The particular control esign problem we consier is the full information control problem in which the controller has access to the state of the system an the isturbances acting on the system. We first present an SDP for etermining an optimal controller an then show that, as in the previous section, this convex optimization can be efficiently solve using nonlinear convex optimization involving DARE solutions. The results in this section are taken from (3). Again, for the sake of brevity an clarity of presentation, we o not present the proofs here; intereste reaers shoul rea the paper cite above.

16 250 R. Conway, J. Nie, an R. Horowitz Semi-efinite Programming Approach We begin by letting G fi in Fig have the known state-space realization This correspons to letting A B 1 B 2 B 3 C 1 D 11 D 12 D 13 G fi C 2 D 21 D 22 D 23 I 0 0 I (14.29) I 0 y fi = x fi w (14.30) in Fig where x fi is the state variable of G fi.inthiscontext,weareintereste in fining a controller K fi that achieves the best possible H 2 guarantee cost using this information. We will refer to this as the full information control problem. It has been shown for the full information control problem that, given any statespace controller, it is always possible to construct a static gain controller which achieves the same H 2 guarantee cost. This allows us, without any loss of closeloop performance, to consier only controllers of the form u = K x x + K + K w w (14.31) where K x, K,anK w are static gains. Equivalently, we make the restriction K fi = [K x K K w ].Thus,weareinteresteinclose-loopsystemsoftheform A + B 3 K x B 1 + B 3 K B 2 + B 3 K w F l (G fi,k fi ) C 1 + D 13 K x D 11 + D 13 K D 12 + D 13 K w. (14.32) C 2 + D 23 K x D 21 + D 23 K D 22 + D 23 K w In particular, we woul like to fin K x, K,anK w such that F l (G fi,k fi ) achieves the best possible H 2 guarantee cost. Fig LFT representation of H 2 guarantee cost control structure q p y fi G fi K fi w u

17 14 H2 Guarantee Cost Control in Track-Following Servos 251 Using the change of variables ˆP := P 1 ˆV := VP 1 ε := τ 1 ˆK x := K x P 1 ˆK := τ 1 K (14.33a) (14.33b) (14.33c) (14.33) (14.33e) with the H 2 guarantee cost characterization (14.15) applie to F l (G fi,k fi ) yiels, after multiplying M on the left an right by iag[p 1,τ 1 I,I,P 1,τ 1 I,I], theoptimization problem inf tr{w} s.t. M fi (ε, ˆP,W, ˆV, ˆK x, ˆK,K w ) 0 (14.34) ε, ˆP,W, ˆV, ˆK x, ˆK,K w where ˆP 0 εi M fi := ˆV 0 W A ˆP + B 3 ˆK x εb 1 + B 3 ˆK B 2 + B 3 K w ˆP. (14.35) C 1 ˆP + D 13 ˆK x εd 11 + D 13 ˆK D 12 + D 13 K w 0 εi C 2 ˆP + D 23 ˆK x εd 21 + D 23 ˆK D 22 + D 23 K w 00 I For any feasible ε, ˆP,W, ˆV, ˆK x, ˆK,K w,acontrollerwhichachievesthesquareh 2 guarantee cost tr{w} (or better) is given by K fi = [ ˆK x ˆP 1 ε 1 ˆK K w ]. (14.36) If the strict inequality in (14.34) is relaxe to a non-strict inequality, the optimization becomes a SDP. Thus, a reasonable way to solve the optimal full information control problem is to relax (14.34) to a SDP, solve the SDP using an appropriate solver, an then reconstruct the controller using (14.36) Riccati Equation Approach In this subsection, we examine the optimal full information control problem when the following so-calle regularity conitions hol: D T 13 D 13 + D T 23 D 23 is invertible (A,B 3 ) is -stabilizable (A fi,c fi ) is -etectable for all ε > 0where A fi := A B 3 (D T 13 D 13 + εd T 23 D 23) 1 (D T 13 C 1 + εd T 23 C 2) (14.37a) [ ] [ ] C1 D13 C fi := (D C 2 D T 13 D 13 + εd T 23 D 23) 1 (D T 13 C 1 + εd T 23 C 2). (14.37b) 23

18 252 R. Conway, J. Nie, an R. Horowitz As in the previous section, we begin by writing the relevant convex optimization problem as a neste optimization problem. In particular, we express (14.34) as where J fi,ε := inf J fi,ε (14.38) ε>0 inf tr{w} s.t. M fi (ε, ˆP,W, ˆV, ˆK x, ˆK,K w ) 0. (14.39) ˆP 0,W, ˆV, ˆK x, ˆK,K w Whenever (14.39) is infeasible for a particular value of ε, wewillsaythatj fi,ε =. Notethat,sinceε = τ 1,performingtheoptimization(14.39)isthesameas optimizing J τ (F l (G fi,k fi )) via choice of K fi for τ = ε 1. Defining [ ] [ ] [ ] Q S := C T 1 C1 D 11 D 13 + εc T 2 C2 D 21 D 23 (14.40a) [ ] [ ] [ ] QW S W := D T 12 D12 D 11 D 13 + εd T 22 D22 D 21 D 23 (14.40b) R := [ ] [ ] [ R11 D T := 11 D 11 I D T R 21 R 22 D T 13 D 11 D T 13 D + ε 21 D D T 23 D 21 D T 23 D 23 φ := (A,[B 1 B 3 ],Q,R,S) ψ := (B 2,[B 1 B 3 ],Q W,R,S W ) ] (14.40c) (14.40) (14.40e) it was shown (3) that J fi,ε if an only if the DARE R φ (P)=P has a stabilizing solution P 0 0suchthatthefactorization [ ] [ ] [ B T 1 [ ] R11 T T B T P 0 B1 B 3 + = 21 T 21 T11 T T ] 11 3 R 21 R 22 T22 T T 21 T22 T T (14.41) 22 exists with T 11 an T 22 invertible. In this case, J fi,ε = ε 1 tr{r ψ (P 0 )} an the optimal values of the controller parameters are [ K o x K o ] Ko w := (B T 3 P 0 B 3 + R 22 ) 1 [ B T 3 P 0 D T ] 13 εdt A B 1 B 2 23 C 1 D 11 D 12. (14.42) C 2 D 21 D 22 If the factorization (14.41) exists, then it can be forme using the following steps: 1. Perform the Cholesky factorization T22 T T 22 = B T 3 P 0B 3 + R Choose T 21 = T22 T (BT 3 P 0B 1 + R 21 ). 3. Perform the Cholesky factorization T11 T T 11 = T21 T T 21 B T 1 P 0B 1 R 11. One of the key steps in eriving this result from (14.38) was using matrix variable elimination to eliminate the controller parameters from the optimization problem. This woul not have been possible if we ha use the H 2 norm characterization (14.7) instea of the characterization (14.8). It was also shown (3) that if J fi,ε0,thenj fi,ε for all ε (0,ε 0 ).Notethat this correspons to J fi,ε for all ε 1 > ε 1 0.Unliketheprevioussection,thereis

19 14 H2 Guarantee Cost Control in Track-Following Servos 253 no Riccati equation result that allows us to check whether or not there exists a value of ε such that J fi,ε. Thisisthecasebecausetheexistenceofsuchavalueof ε is equivalent to the solvability of a full information H control problem which might not be solvable via a Riccati equation. Despite this, if (14.34) is feasible, we can always fin values of ε > 0forwhichtheinneroptimizationproblemin(14.38) is feasible simply by ecreasing ε > 0untilJ fi,ε. With this in min, we woul like to know how J fi,ε varies as ε varies. Using the same reasoning as in the previous section, we see that J fi,ε is an analytic function of ε. WewillthereforefintheglobaloptimalvalueofJ fi,ε by fining a value of ε such that (/ε)(j fi,ε )=0. Implicitly ifferentiating the DARE R φ (P 0 )=P 0 an ifferentiating the expression for R ψ (P 0 ) with respect to ε yiels ε (P 0) = ( A + [ B 1 B 3 ] Kφ (P 0 ) ) T ε (P 0) ( A + [ ] B 1 B 3 Kφ (P 0 ) ) + ( C 2 + [ ] D 21 D 23 Kφ (P 0 ) ) T ( C2 + [ ] D 21 D 23 Kφ (P 0 ) ) (14.43a) ε (R ψ(p 0 )) = ( B 2 + [ ] B 1 B 3 Kψ (P 0 ) ) T ε (P 0) ( B 2 + [ ] B 1 B 3 Kψ (P 0 ) ) + ( D 22 + [ D 21 D 23 ] Kψ (P 0 ) ) T ( D 22 + [ D 21 D 23 ] Kψ (P 0 ) ) (14.43b) The first of these equations is a iscrete Lyapunov equation for (/ε)(p 0 ).Since A +[B 1 B 3 ]K φ (P 0 ) is stable (by the efinition of a stabilizing solution of a DARE), we see that there exists upper triangular U such that U T U =(/ε)(εp 0 ) an we can irectly solve for U using the lyapchol function in MATLAB. Using this, we express { } ε (J fi,ε)= ε 2 tr{r ψ (P 0 )} + ε 1 tr ε (R ψ(p 0 )) (14.44) which implies that ( ε (J fi,ε) =ε 1 ( U B2 + [ ] B 1 B 3 Kφ (P 0 ) ) 2 F + D22 + [ ] D 21 D 23 Kφ (P 0 ) ) 2 F J fi,ε. (14.45) J fi; Fig Illustration of lower boun computation in optimal full information H 2 guarantee cost control ^J 2 ^J

20 254 R. Conway, J. Nie, an R. Horowitz The value an erivative of J fi,ε is also useful for generating a lower boun on the optimal value of (14.38). Consier Fig , which shows a representative graph of J fi,ε in which ε 0 is known to be a lower boun on the minimizing value of ε. By convexity, if ε 1 is known to be an upper boun on the minimizing value of ε, the value an erivative of J fi,ε at ε 0 gives us the lower boun Jˆ 1.Ifinstea,thevalue an erivative of J fi,ε at ε 0 an ε 2 is known, we have the lower boun Jˆ 2.These lower bouns are respectively given by ˆ J 1 = J fi,ε0 + m 0 (ε 1 ε 2 ) (14.46a) Jˆ 2 = m 2[ m 0 (ε 2 ε 0 ) (J fi,ε2 J fi,ε0 )] + J fi,ε2 (14.46b) m 2 m 0 where J fi,εi is interprete as J fi,ε evaluate at ε = ε i an m i is (/ε)(j fi,ε ) evaluate at ε = ε i.itshoulbenotethattheseconoftheselowerbounsislessconservative when it is applicable. With these results in place, we can easily solve (14.38) using the following methoology: 1. Fin Initial Interval: Choose α > 1. Check if J fi,ε an (/ε)(j fi,ε ) < 0 when ε = 1. If so, start from k = 1anincrementk until either of these conitions fails to be met when ε = α k.denotingthecorresponingvalueofε as ε u,there exists an optimal value of ε in the interval (α 1 ε u,ε u ). If instea either J fi,ε = or (/ε)(j fi,ε ) > 0whenε = 1, start from k = 1an increment k until J fi,ε an (/ε)(j fi,ε ) < 0whenε = α k.denotingthe corresponing value of ε as ε l,thereexistsanoptimalvalueofε in the interval (ε l,αε l ). 2. Bisection: Solve the equation (/ε)(j fi,ε )=0overε using bisection. Use (14.45) for evaluating (/ε)(j fi,ε ).Whenever,foraparticularvalueofε, the DARE R φ (P)=P oes not have a stabilizing solution P 0 0suchthatthefactorization (14.41) exists for invertible T 11 an T 22,thisvalueofε is an upper boun on the optimal value of ε. 3. Controller Construction: For the value of ε which yiele the smallest cost, reconstruct the corresponing controller using (14.42). In our implementation, we use α = 100 an, in the bisection step, we use the geometric mean instea of the arithmetic mean. We use two stopping criteria in our implementation. If we efine the relative error as ν fi := 1 J fi /J fi,ε where J fi is the lower boun compute by (14.46), we terminate the algorithm when either ν fi < or 30 iterations have been execute in steps Application to Har Disk Drives So far in this section, we have evelope two methoologies for esigning an optimal full information controller in terms of its H 2 guarantee cost one base on the solution of a SDP an another base on nonlinear convex optimization involving Riccati equation solutions.

21 14 H2 Guarantee Cost Control in Track-Following Servos 255 Although we woul like to apply these methoologies to the moel G H introuce in Sect. 14.2, it is not in the form of a full information control problem. However, we can create a full information control problem by replacing the measurements generate by the moel, y, bythevector[x T T w T a w T r w T n ] T where x is the state of G H.Themoelwasnotbalanceafterreefiningthemeasurement vector. We use three approaches to fin optimal full information H 2 guarantee cost controllers for this system: solving the SDP (14.34) using the mincx comman, solving (14.34) using SeDuMi an YALMIP, an solving (14.38) using the methoology at the en of Sect We will refer to the latter of these approaches as the DARE approach. Using these three approaches on the moel of G H with the reefine measurement vector yiele the results liste in Table Although the DARE approach an the mincx approach yiele similar accuracy, the SeDuMi approach seeme to yiel better accuracy. However, analyzing the close-loop systems using the DARE approach in Sect showe that the actual H 2 guarantee cost performance achieve by controller esigne using the SeDuMi approach was only , whereas the other two controllers achieve the costs state in Table Thus, for this system, SeDuMi actually ha the worst numerical accuracy of the three approaches. In terms of efficiency, the DARE approach was more than 10 times faster than the other two approaches.theifference in computation time is less ramatic than the ifference in Sect becausetheclose-loopsystems here have 19 states whereas the system analyze in Sect ha 38 states. We now look more closely at the DARE approach to this problem. In particular, we are intereste in the values of J fi,ε an (/ε)(j fi,ε ) as functions of ε. Figure shows a plot of these two quantities for 50 linearly space points in the interval [0,1.2] along with an estimate of (/ε)(j fi,ε ) obtaine by applying the central ifference approximation to J fi,ε.aswewoulexpect,thecurvesare smooth, (/ε)(j fi,ε ) is monotonic non-ecreasing, an the compute values of (/ε)(j fi,ε ) agree with the central ifference approximations. Also, for this particular example, J fi,ε is monotonic ecreasing over the chosen values of ε. The H 2 guarantee costs compute so far reflect a combination of the sizes of the signals y h, u v,anu p.however,itismoremeaningfultolookatthe sizes of these three signals separately. As in Sect , we remove the outputs we are not intereste in an compute the relevant H 2 guarantee costs associate with each output signal. Doing this for the close-loop system esigne using the DARE approach yiels the H 2 guarantee cost given in Table Thus, we see that the Table 14.5 Analysis of close-loop HDD performance using three approaches Approach Optimization Time (s) H 2 Guarantee Cost Optimal ε DARE SeDuMi mincx

22 256 R. Conway, J. Nie, an R. Horowitz J fi,ε (/τ) (J fi,ε ) x Compute erivative Central ifference approximation ε Fig J fi,ε,thecomputevalueof(/ε)(j fi,ε ),anthecentralifferenceapproximation to (/ε)(j fi,ε ) Table 14.6 Analysis of close-loop HDD performance for each close-loop signal Signal y h u v u p H 2 Guarantee Cost nm V V control effort is rather small for this controller. This suggests that we shoul eemphasize the control effort in the cost function in orer to esign controllers which achieve a higher level of performance Output Feeback H2 Guarantee Cost Control In this section, we consier the optimal output feeback H 2 guarantee cost control problem. We first present a non-convex optimization problem for etermining an optimal controller an a solution heuristic which is base on the solution of SDPs. We then give an algorithm which exploits Riccati equation structure to give a more computationally efficient heuristic for fining an optimal controller. For the sake of brevity an clarity of presentation, we o not give full proofs here; these will be presente in future papers Sequential Semi-efinite Programming Approach We now consier an optimal H 2 guarantee cost control problem of the form shown in Fig (a) on p. 262 in which G has the known 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. (14.47) C 3 D 31 D 32 0

23 14 H2 Guarantee Cost Control in Track-Following Servos 257 Note that we are choosing many of the state-space entries to be the same as in the previous section. The only ifference is the choice of information that is available to the controller. In this context, we are intereste in fining a controller K which achieves the best possible H 2 guarantee cost using only the measurements y. We will refer to this as the output feeback control problem. Using techniques similar to the one use in (3), it can be shown that, given any state-space controller, it is always possible to construct a state-space controller with the same number of states as G that achieves the same H 2 guarantee cost. This allows us, without any loss of close-loop performance, to consier only state-space controllers of the form [ ] Ac B K c (14.48) C c D c where A c has the same imensions as A Coorinate Descent Approach We are intereste in solving the optimization problem inf τ>0,k J τ(f l (G,K)) (14.49) where K has the realization (14.48). Applying the Lyapunov shaping paraigm (15) to the characterization (14.15) for the system F l (G,K) yiels the optimization problem where inf tr{w} s.t. M ˇ(τ,X,Y,W, ˆV 1, ˆV 2,Â, ˆB,Ĉ, ˆD) 0(14.50) τ,x,y,w, ˆV 1, ˆV 2,Â, ˆB,Ĉ, ˆD ˇ M := X I Y 0 0 τi ˆV 1 ˆV 2 0 W ˇM 11 ˇM 12 ˇM 13 ˇM 14 X ˇM 21 ˇM 22 ˇM 23 ˇM 24 IY τ ˇM 31 τ ˇM 32 τ ˇM 33 τ ˇM 34 00τI ˇM 41 ˇM 42 ˇM 43 ˇM I (14.51a) ˇM 11 ˇM 14 AX + B 3 Ĉ A+ B 3 ˆDC 3 B 1 + B 3 ˆDD 31 B 2 + B 3 ˆDD := Â YA+ ˆBC 3 YB 1 + ˆBD 31 YB 2 + ˆBD 32 C 1 X + D 13 ĈC 1 + D 13 ˆDC 3 D 11 + D 13 ˆDD 31 D 12 + D 13 ˆDD 32 ˇM 41 ˇM 44 C 2 X + D 23 ĈC 2 + D 23 ˆDC 3 D 21 + D 23 ˆDD 31 D 22 + D 23 ˆDD 32. (14.51b)

24 258 R. Conway, J. Nie, an R. Horowitz For any feasible τ,x,y,w, ˆV 1, ˆV 2,Â, ˆB,Ĉ, ˆD, acontrollerthatachievesthesquare H 2 guarantee cost tr{w} (or better) is given by [ N 1 [Â YAX ˆBC 3 X YB 3 (Ĉ ˆDC 3 X)]M T N 1 ] ( ˆB YB 3 ˆD) K (Ĉ ˆDC 3 X)M T (14.52) ˆD where M an N are chosen so that MN T = I XY. (14.53) Although we use the QR ecomposition to factorize I XY in our implementation, apivoteluecompositionwoulbeequallysuitable. The optimization (14.50) is a non-convex optimization because the matrix inequality is nonlinear in the optimization parameters; the proucts τx, τĉ an τ ˆD appear in the matrix inequality. Thus, the matrix inequality is a bilinear matrix inequality (BMI) an the optimization (14.50) is a BMI optimization problem. If the value of τ is fixe or the values of X, Ĉ, an ˆD are fixe, then the BMI becomes an LMI. In either of these cases, if the strict inequalities in (14.50) are relaxe to non-strict inequalities, the optimization becomesasdp.thus,foragiveninitial guess for τ, areasonableheuristicforsolving(14.50)istoalternatebetweensolving (14.50) for fixe τ an solving (14.50) for fixe X, Ĉ,an ˆD. There are two challenges in using this approach. The first ifficultly we encounter is the ifficulty of selecting the initial value of τ; sincebmioptimizationproblems are non-convex, the selection of a goo initial iterate is especially critical. The secon ifficulty that we encounter is in reconstructing the controller using (14.52). In particular, since the controller reconstruction epens on both M 1 an N 1, we see that the controller reconstruction will be ill-conitione if I XY is illconitione with respect to inversion. We will show in the next two sections that it is possible to eal with both of these problem using semi-efinite programming Initial Controller Design We now examine the problem of fining an initial value of τ. Toealwiththis problem, we follow the approach use in (10) in which the solution of two SDPs yiel initial values of all optimization parameters. Since we are only intereste in the initial value of τ,wewillnotexplicitlyconstructinitialvaluesfortheremaining optimization parameters in this paper. The first convex optimization is a state feeback control esign. This is one by relaxing (14.34) to an SDP, an then solving it with the aitional constraints that ˆK = 0anK w = 0. This yiels a state feeback control law u = K x x,wherek x is a static gain an x is the state variable of G. In the secon convex optimization, we first restrict the class of controllers to ones which have C c = K x an D c = 0. If the state variable of the controller is interprete as an estimate of the state of G, thisrestrictioncanbeinterpreteasimposinga separation structure on the controller. We then form a realization of F l (G,K) whose state is given by [x T (x x c ) T ] T where x c is the state variable of K. Usingthe

25 14 H2 Guarantee Cost Control in Track-Following Servos 259 characterization (14.15) for this realization with the restriction P = iag[ X,Ỹ] an the change of variables [ à B ] := Ỹ [ A c B c ] (14.54) yiels the optimization problem inf tr{w} τ, X,Ỹ,W,V 1,V 2,Ã, B s.t. X 0 Ỹ 0 0 τi V 1 V 2 0 W M 11 M 12 M 13 M 14 X 0 (14.55) M 21 M 22 M 23 M 24 0 Ỹ M 31 M 32 M 33 M 34 00τI M 41 M 42 M 43 M I where M 11 M 14 XA cl XB 3 K x XB 1 XB := ỸA cl à BC 3 à ỸB 3 K x ỸB 1 BD 31 ỸB 2 BD 32 τ(c 1 + D 13 K x ) τd 13 K x τd 11 τd 12 M 41 M 44 C 2 + D 23 K x D 23 K x D 21 D 22 (14.56) an A cl := A + B 3 K x.althoughitisnotusebyouralgorithmevelopehere,it is worth mentioning that, for any feasible τ, X,Ỹ,W,V 1,V 2,Ã, B, acontrollerwhich achieves the square H 2 guarantee cost tr{w} or better is given by [Ỹ K 1 à Ỹ 1 ] B. (14.57) K x 0 If the strict inequalities in (14.55) are relaxe to non-strict inequalities, the optimization becomes a SDP. The value of τ which results from solving this optimization problem is a suitable initial value of τ for the BMI optimization problem (14.50) Conitioning of the Controller Reconstruction Step As mentione in Sect , we woul like to make the matrix I XY wellconitione with respect to inversion. Define [ ] X I S :=. (14.58) IY Since S 0foranyfeasibleiterateof(14.50),weseebySchurcomplementsthat Y 0anX Y 1 0foranyfeasibleiterateof(14.50).Inparticular,Y an X Y 1 are invertible. Exploiting the invertibility of these matrices yiels

26 260 R. Conway, J. Nie, an R. Horowitz [ S 1 (X Y = 1 ) 1 (I YX) 1 ] (I XY) 1 Y 1 +Y 1 (X Y 1 ) 1 Y 1. (14.59) Since (I XY) 1 explicitly appears in the expression for S 1,weseethatifS is easy to invert then I XY must also be easy to invert. Since S is a positive efinite matrix for any feasible iterate of (14.50), we woul therefore like to make the ratio of its largest eigenvalue to its smallest eigenvalue (i.e. its conition number) as small as possible. Since the conition number of S 0islessthanκ if an only if there exists t > 0 such that I ts κi, weseethatareasonablewaytoimprovetheconitioningof the controller reconstruction is to solve inf κ s.t. tmˇ 0, I ts κi, tr{tw} < tγ (14.60) t,κ,tx,ty,tw,t ˆV 1,t ˆV 2,tÂ,t ˆB,tĈ,t ˆD where γ is some acceptable level of H 2 guarantee cost performance for the closeloop system. Note that we have fixe the value of τ an the level of H 2 guarantee cost performance in this optimization. Also note that tmˇ 0impliesthatt > 0. Thus, this optimization problem minimizes the conition number of S subject to a constraint on the close-loop H 2 guarantee cost an a fixe value of τ. When the strict inequalities in (14.60) are relaxe to non-strict inequalities, it becomes an SDP in the variables t, κ, tx, ty, tw, t ˆV 1, t ˆV 2, tâ, t ˆB, tĉ, ant ˆD.Thus, improving the conitioning of the controller reconstruction process for a fixe value of τ an a fixe close-loop H 2 guarantee cost can be solve using this SDP Solution Methoology With the results of the previous sections in place, a reasonable methoology for trying to solve (14.49) is: 1. Fin Initial Value of τ a. State Feeback Controller Design: Solve (14.34) with the aitional constraints that ˆK = 0anK w = 0usinganSDPsolver.Fromtheresulting set of optimization parameter values, reconstruct the state feeback gain K x := ˆK x ˆP 1. b. Separation Principle Controller Design: Solve (14.55) using an SDP solver. 2. Controller Design a. Controller Design (Fixe τ): Fix τ to be the value obtaine in the previous optimization. Solve (14.50) using an SDP solver. b. Controller Design (Fixe X,Ĉ, ˆD): Fix X, Ĉ,an ˆD to be the values obtaine in the previous optimization. Solve (14.50) using an SDP solver. If the stopping criteria have not been met (see below), return to step 2a.