MFNWtKUb<ob<R=MDLCdNVZZJ:tN>H:xXVErps:;BNSDOETlMXlgwgiW;mD[UUUWUsKitUf]Wfv_ivmixoYKEVcsIyuyvayvUIv_ioixoOWkgxwiywOveCHwgIxiIxmyqAYs]IwgYtUiuIXpCIFiSIaBAAsa;GbYyvcixqyxeYweyuYyuWdMWTuUYuyyyyA;:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::ZjifDqEtk]`N\\@Nd\\QgqxH`jwhSWDQVyPxPLAIXU`wyyySUun`r[DNZ]WmUjPuJZ]Y_lSLqqWioSxwwtLEQl@UNGiOC@XyQjXLYbIvN<xwaLnAt=uOZdQnAtE<SIdQnQJLYRIdq:`xJYryqJBhyNFvL?^^YoOA[yYelofiGbt?w[w[PhdK?gSO^DGpLYeJp]t?fjHo\\I_:yo;H]\\`\\:GoDF]`hqEht=w[F_alS=wUToTtOHPwCborY[w:=EpYdRYrYMChKdE?BDmidKG=QsC_YRmHnQBLYr?QeE_X_krige:[iBYcf_DDaGeSs\\eTPOb_wYrwsXirdIviGbNwG];TYeTKmgywvJGBsyCy]VlmFeyEQwcX=jjyx:`sQMP^\\YPho_Tk>xMsmtsIPMhKmYLwMXwIWXqMxqPIUkEQT?moDhtHEo_lY@mQHQpZDyLUrYHpn<yRutnHUv<lpxKYPWwIXR_p`I`pXfWOyy>eMy_JWu=qaR>ppVxO^funr?G`Hv^Qia]vuuocJpwUQdTgd`_mex]Tvf\\xfrhdbXvpIe_Hs[IiH>nUonv@bKpiZHtX`ibhfKO`JFdPPkIqvy^q<?m@vuvA[k`fDhbkYdNqxj_c>_fOfv_wdx^_E?uYXyQ@olFqYIf;_e]IyPVqnosfPyJA^=asuq[j`ZR?kE^yjHtHQgOHxSn\\wYoIh`TY\\Fg`Rx`Iq[Vwq:@]TyybQxv@]k>kivdaY\\ui\\dWirn[PqrTgpPYbx^tvFfkWZbihlYa>^bK@wTwsQhvOyb@?]gqhwomng_>og=>wpGarAc]hibAyX@eLogQnhlykD?s<_c\\>b@QuvA^kxm^ppAXvjVZsF^AFo^_nVVflixrifhaqi?bHI\\Jf_]O]s^`lyssAsp_b=IZ]akdPmJniAv^PnaNAw:Gi>VqmfvRIuyF_[NmpQjc?pIq^PWjiFdUYrc>glPqhP[B?jLNqKAwyxnVhq\\ajYQ^ZFVQxk?e;_f@UbISs??T<aBw=fK]UyYy[oRAMyR=HwiwEUHfmRPSty]TsStbAHxSuYMs^yGKUu=IB=QxemUA=rrwI;aIX=BJ?b^ss[_TXEYTCeEkuGgCNgeEKY:yxEKBLWbmuBHkvjOgvacI_W=_dGktRegYwr]WFQ?yTKBBUwI[HTYrByGjyF\\Wbwgvw]SxawaaWs;yAwTCAS^yxd?Xd=sBgyRaDW=DjsT:=h\\KgmMG[av\\Kd]sTJEcv[dV;fvch;wS:_DkYu]QwOCdO;sg=yoeytSG`kImsFyog^?xEOBLCFViDIgI@Gy]ot^irP;HK?hZOsjgS\\oH?EUSuDGMUAuFJIHi_FKSWUwRT[ho=Succ^;Is_VTUE=ICoSIswCWqRZQG<_iUacrCehOcaIRWuspqRfYT@ccfMuhsyCWrYmIPKIbQhdCehqx\\st?]DG]EqMIFYfW]rgUCbqvIGSgofLWg`aHJKdluEqEeu=ixkwQStrSWtWgcgwJSIGku^oxgKVyQWZEt^gBeKGZKxced=IdTOhJEfR[xrMBkKg^mGJ]Hc[trOT:_R?eFd_FVCXZCD?QCqSX]YetGF<EuQeUfcCLMhjGvVKs_STkUw^]CEUEl[f<?hNEwdoH?MWf]FbesPKU]kgH]bSES;QVV?hHqdT=ce?bp_h=GGuqGD[y@SU=IFKExEeUWAhNMX^wdRYFIMevKeHYWSsCl[HGau[AEZiR_iTJUDS]YckXsoV>]GBqb@;VM=DluVHgVuQeDqxLUE]]WSAR_oB?oxLgr==vqkR?McPAEG]WBKVP[HVOI>IrEuBkUcqSckCwpsFo_Rc?eB[hhCXYSrFChVASt[UUWWs]ceYBhyD>aUTMWZ;vDoR<MigQDu_TtCUuQeTqXLOI>QV_CiI]w_CruEHosRwoFf?EcQiJ?bh<rTuX[Xm>QN?YtNdpPQMSxUM<Lq=q@INBAKETJBhxStLsEq:\\VmYMcEJvLM`\\joAWKlvL`oTExbqR``uRqK;=PX<LAusChO?@mNEjeaP]ISWhp@yWl\\Wc=y<QlPXJQuSwlW=xtYyvJHOTtK;TW>lOIDODTJZyNoUPRLxHlwPelKxT;toREv\\Alc]kbppf`yolyvPvOMkxDK>]u\\EVC]NAanAYc=F^K_udgd[Q^Vi^Dr>[tR=H\\aG_?GT[rtSru[XBuGDsUKag?QUEGEKCigcGMeYoGB<URBIb[ebvYFAKbGyGK=CMQCQ]C^[UkUTFcXVEh=]g<[VDoBAIgOyXCgsQsd`CFc=ujQHK]Yc;xOOi@YxlOFXEbxOGeCs<khaIRVIgOms=eTOIyPyrfyBtqtVuyREy:orPce\\IgqkbVMUZAX>sHsUuOkYqAgC=syoYAsv`KChAX^WR]_xvcF_kRgAc^IcP[SI_D[Uf=MerofcQGoYBfcca[TiETvae;=HGctaqvuWHd;IbAiJEYdytG?hBordUTXWC;ebcisL[UxYDC[gkQrHMgHqebmrikvj?HrSiPyrckxkCwQADCoIeEvbUbGAboQhXEh[;d\\KHHmb_OFtWu_yb_UcROtnwbQUHjEuL=Up_Rb=UYAgUME>gCAgCiySEosEQUGqdWMWq=c?ErKMWIGFwOCeGw_?c@YBM=s`Qes?U`GvDIGu]Dh_U\\aECQCkig`KY^Id<UrFSGdidCQd?wvjsgjoc`av?ABUcCqkDbgUQmYdWTyUHIEI^?vO=xrIY@_IXKxyey:Wy]YRruxDiGiSv?uiHGbVQl:DK[HmrpPHPq^PlE\\kAMkvmLLylFAokljcev=lqi<YWtRLewqIQP]nuTjSqvo]xgtr:`TDeos`qsXoUts^<QVAKd=lHEwRQOVYqEyTo`Y:aYNPKh\\V>AsNQx;TxrdW^YJ^tja\\vHdnlUkRekoYJvXOVesOqlUMN@mPnUPoXmT@jtmUdpKoHxhmuD=QHewk\\nBlOhuqWXowys<\\VPdkZAJgERo`R@ev[evpTq`aSx<NUAvyUra]nvtRiHuBTQITs\\qV\\yLl]raXt\\@PCHS@tr;\\XmDS;XmFpVRyXuHjIMKB`mZivW=NHTSA\\srptgMmIANqeuY@qJMOFhrxELi]vomrP@kg]TEMSNEXrelmyroxkU\\YyMlm=K`AvvaXWmKQqmA<QTIU^IQhmw^IYHQq^\\sWllE]skls=QYwARtpUPHVWin>TKyeq`DLkYPD=VYxOUiu<QTo=u=PUcYXLykEMmBHYwuOSdsmuu_dRm]WlpLI\\xKlqy@K^AO:IJ\\ao;YsdHxRHpO@yD@L?IpLxrdUp_Hvcpvg]uEQVKXwvdnp`VNqVV@t[lL_io;qOIUNwLSfiJ:mt>yVTqNgMVoaoP]RNiVDQO`@VDisdHywtY;@VftLqYlstrE<vhmrBimUMr>EkJAuGxvYiYJmWxxYqdjGxKl]T@QPVYOY`LJ`m`ajN\\MBMVQmysLNDYsq<opYySDm`AvP@qBHlPiO\\Ax<qo\\@YeXrSHPR@VeYVGASrxQZYPGttsPk]eLEhWo<PGAP?QxZLXX<ucMS\\lJydSSMwG@kQLvjAMWTyUtoxULkUPXTu<PQ\\hsaPkdPKNhuHdkAtuCQPZQMKDSvQVPYypLRTxyMTPVMUUhqsmmDpncYlX=NqlqkxRpdPOekRxMp@kSlU]HW;xt?=S_Lm=Atn`LqUQEEVBAWBUnn=tBLXxptF`NSATdUNGHTE<WNINPxWNIRGewJTNwHu=MYV@uE@K<MM_eSGEk_DP@aV@ml@=L\\EuCPvcywSpka@u[tQhDp<eS@Avm@U\\Mv=AQZDW;MMwUkQ=m=aTMAY]LouuS:MN\\yPs\\QXmPVIwvqJoPMTIprAs^QRvlsS]tNdKCEl?xTwmn[prj]WMxKODNIIL^@sn<YfEkXHQNdQMtWLaPMQLqUT?MyZtWTaRCUlk@X;@mWdK?mnVDlF@xvtLVQQsIm^TWs<oX=KaEuEhYELt`]Qr@yTHXRtxBXuvDrZdt^MnVHXIaRxqLKLSGASMHw]@jdyrNTM==se]r`]oG<K\\=VP`YKDjXuTuIjE@wCQSxdM^@wPPS=Msb=k>LO[\\o;tsm]UCILdEVj<S;PTEiNUMVMmoPuJGLTHTNGpXKPKgDpJlUTHKsuo<PcJn_cxp@FwagZNY_WpeM`qAWg]h\\fIsA`bZ^atituw`>Aiayh[PrEQigpbMOwyvaJvx:HgbYg;Xm<OrMogdPw\\>^??kNFaVXqGP^dyZwFrWGxKn[kfgL>`GYnPYkdwbKqbYXpphhOGs>y[[FsuV]Av:GuVKGX?rtmbU=UAyBXQVIOwDqEKIEsoe:ad]kXJavRGdd[BwMcuEY\\eEDex][dxOe]AuRIdBKvS[D>CXgAVH]BwUUGsBYixByfVwvrkSa]BmGWgcfq_il;hgig<ARuarHuhNQdqkHkKWqAdpEcGoGCMI^iwaWcWyFSmSlqsI_WmgGcqeVismqboWhYWGdWSFGCnwvkOH?kFPCsjohaaDoygB;cesRR=HPAcA[ivarNgrGQyf;ce[cSab>oF<eXJ]bA?rgIUJKetmscqCqEULmU<MfeQwrytKsXtsvgwfR=ssee]eVr=iXgTusREKIrawVaesoD[]yqAgT=VE;XwgTX;CsicmshZIyh]tf;D>agYkSZ;cEeeOUbpgBK]C^cch=X[Wcw?TK]D_kwZcr^aIMaiv]gC?uR[iAeGn=gWIRDYxaAgLyiU[cosfTMs`WHECeWihZSi[SSH?vD]CCSyJSeLawp=dB_xOOxKqg@cy^?bb_wJSC`UYT[bNEGd]HZiIOAbVOrJeEcKIj?FnaWgAelSCA_WK]t\\AUAqSMMDhSVRgep?BFMu]kRDkvOKw`]iy]SsAxnUt>Sfi;IWuuuGev=rhWd@qggYFHucr=ffovLQB_[e>=ieAwjAd\\ihDUtOSHFeW^wiAeWnMCcsXsmh]oxC?g_?fYgrnsSvguuMb[If?Cf`mRPEvYusl;XQ=SqmDRAsuctPGvsgxHgBKui\\;Fv]GLAD^[CoME>?SOWUigTKUYTYDTcIVCGt;`FO]UfkWV^GwkaW`ufkKPZt?\\Dw^oO_\\^iw@bn_mvN^]?vKqc`v`tVs<@yfN_e`\\DvsFg`iYcLHtw_bKwZk^hKVnhVfmg\\sIwQPfAG`Pnhuikjww=atXPbdweN@jNWuG`agib^ViYfeaPir?g;_[=Or:x\\vqi=iqZiZTvk@afXvpdqlSW\\ipydarYG\\JFm=Y_xge?_ZBqcDH_NwrNXxkPn=W[lYn=a_=HZ[in?PnAayT?yxviixbhXr;igdHiooQ]eDWfbiF@Sx=ctvwu[QWIiI^sE]WrP[EbMugWRnsBd[yR[d?YU=MybchadqR=loMvwDSWYPWxR\\=LOdWj<LImuL]w<XNAEROhwKHLxmKftSxYUf<KEMxxtLNlYKxr>uQtpTVYUt\\mudTa\\t]enZDQp`YkqSI@o]qOp`lo]PCxxxAkrxx_DsTImQHSayThaRE\\JLtptqRAuQyXONdUyUN;ax[MWsxxN=t?=J_yU?PK?Qx:Hn_YNk<SLIy\\UUDYOhhv:MufmMkIOrUJqXslDrU@wYqQEhLKAk[Xnkqv_]xcUKl=TaEnUENPmvHipVpSHYtcPQKxTIUw`XXeAq[iXyhqXdW]quSlkuUNTQw[hyWalRESQhKudMPuRNHQVTpZ<VAHnsHlYxv^mmVHJtDYoTw]at\\XX^Yy<LjJ`kV@o]tS=XJqMQ]Epq`sVIJ^pWtPKcLmkdNIhuZ=WYyvhTNrpTPiMwTQaqqZERnXje=nI`Ux=pPutt]X^@wGUKLxj[=qJQnY]VMLUFMrq`wIDsXhPc@Q\\AyrtTuHN=IkfdUeqO[lLqQpHtv=qJ>qNNmRUUL^]wRTLD@q]`sWhxkTK<hN^drxQmkARFQrVApJXYrlLWhVKqs`pxRLTwuQjdqf<wR=lOmu_]N\\aXCisgmJtDXsdUtUL<Pk`Xp^<URPuE\\Ty@UGDKXMKlQM:pt<`nHyvg<kEyn^lVFIQ>qPnuwBeyruTmHYmXJ_urDpKqIRpPLLxRV]JtLSkujxmokElxMuxAXNYWchP=hRXxUjpvqqnGEnv=YjQrZTP^epTqja\\PFDYnucSPdiflVg[jw[Hw\\j^_owuVPdfg_CVgdXnHhhkQwMVshgZTxolYbh`ojHqw_`eXZ\\>wXOne`m?goL_wOn`Bw`a_vfyyXGuJGugfso`mgivtHmX^cSpmQaf\\^]nyh=oZx_wPXnOitrib:XwYOhpWy]qdlWvu`cX>]Jgsm>tdqssn_F?anfyNhZKgg\\NgDyp;Ah:_lhAs[vtDF\\Mwh\\gwBAl[ybMX^?We]YdEnZwhy:Qw>aut@_lOl:>hfgaoxuFQbKnbYHpHQobw^C_nW@qDnpcQqEGawV\\`@rnpclhck>^XGdN@qdAu[FfUI^u_\\:`qfvq?_soG\\UguAA\\An]kPlFNdB@sKVpdNtH^gAfoipdaGdEGlPwbJPt[OsQn^UN`mFZZvlnob>ygL^wWYm\\VheVeMGjPhrJHenIbp@x\\we]Xoc``hpe`xp:vuXweMYg[PqTpniH`oo[Jg]t?si@`pvofItsn\\^Id`ovVagAqlaIxV@]jV]dvaQFal_hbowAOxD`_aYjJhloqkWYlJ^fAfbi>lMP`QNf[grX>r@_nH_j_a^TNvoxiJVrs^euPco@\\QO[O_pE>gYPm@_moP^UQ_BpfENcH`jMnZiYtmx`VOgxv\\fOqhod@yoWAoHNk^WbCYdsOhrygJndKvqVXbR@]i>jAHyW^]h?]fxgCIcNn]Io^lNwHFf>@gYAkQVcD@iB?\\UGrTV^hfjDifg^ytAyIv\\Q`myVx`v_DQZ\\Hxt`^Qq_sQm@hdCntT@c=xfg@`UYo\\YxxfpgYjHI`dggYo_q`thI^W`a=GrBheUVoPwkhxydGZS^np?yF^mGhhvh]TI^<qhwq^HF\\sQpVGtoo[GabIV\\f@fBywC?jOwoGF\\cFyqnmmNhewn:wkfxoaOipho:w_^w]GXi@^xiQeqFiOn_gA^oVpUYn<NxEgl?Iigi`ZQhlGuWovA_xna\\XNs:yb_PprX^Giv;Fhqxg<Ite^dDFajHfSvoQYi?WxZPdcI_NGm=iZ^Iv`>dY?p=qhmPp=>]O`bIQwNgelQd?VbY@i_O`\\IbDIeZfrmpblvlZfZy^svnsnIhmNh[apjVbmVfUfZ[At=`fBgvKfgWxkb?cfojdGvrhiLfv`Y_C_dipgXwoCXtsHl\\n]NPZmO`yW\\e^hT_xDFlh`[PI^ZnpWpmDgZ=_cGfccVvZnnJYkVofg^hlWw>pa\\`lMpfHPjCPj>GfnO`T>icv][fj@vktPronymPTgdbPp>yNdTnpdorQmTay:DoCxr`iP>QLchN@DXDTryXyI]jG<uEhL@TuA=leyOf\\XrPtZpOimYiqQgQrqXvrlWquqtLPvQyp]r?QYNur`uNWeR`xq^HjFipgUYXDsAYxNTrm@OJqPIYn:eoFMXYEtcLPRqLGHwKlnaUMpHocMwN=yZVfE^^_Iq\\FnC_cTHhnWsW?oN@nbP]]hrvGbs?oqnmB_a[xvn>fc@_EOi>XhNfpuVa@xhNIc^ormIqffoF_mfHcgydgN]__v:H[p@_R^^K`]eaoMijRW\\ZOy`hhApeUpmBh`@@mQ`n[fhMqbT_oOfhEphtAl`?lnidb@vhwbq^xVGcmo`nhuJavkNZI^`<gghHoD?`h_mIQyIHqFVg[q_T_n]Xb^H\\Sf^]nt`wfWo_VnvLHfSnbsYyui`AQgcq]D>sK@fqf[ChiS_Zvff[HtBRlgF`?sOSSrqvdAIH?xtmU]uW]GGDaRiSF?MTpKvokhbYF>QhhKUN?Dy=Vfwv=EVGWivoeK_uLagD]rVCtM?d?=s=GRQoYfIY;=UqoerMDsgILoS>EX<mCk_GFAY\\iEvIDggEAqehqgdWhSaSOUdAUBqeyMMco;gf]wO_slsGEaIlUtoegg?EaITkSu`QrwovxKy^keL]r]Gvj[WJshO_igADBwg]`vSmLLIsRHoCdLlHMFPxfpkLPr`]jmPOFQxvIkdLn>xVQysDtsEMptxvKaN``thlRl`qI`XiimSeJ:\\SiLU<mR]Hrb]xAmudTkWePQiuQ<lourv]tMtWUajiyo:pnW=PE=oLDQ^yL>`oG`jXtLplxs`mj`t[HODPnkDOtettHqVHKU=TglNadR^xUKYR;Xn;<YG`PtQYXTOlPtSDQNEliMw^dsvIpftPUikHal>LVqxkXxRShUC`TipvaAN;pssPOEQptlL^mrK=MKyTC`uc=mEdpR`u=Aqexjr@RkUPqXUGALn]l>HvcXKD@ycIv;QWKtSUmU_URB`kR`MBDlXTnbLnq=YXUnVPtutyO`Qx`JfmUbmQGxKlQmTQlLUMi=RZXmFeN`lXU<uQatxDYXqQLaVUXrdyUKMyAhkMQqTDjuqSTxpJTKBqv^QynlK]dXl@sXpogxR^qOXdvGxmYYWIMnf\\OYuueLM?iRneRwywcQRrPnFLPG=VRPO^iK]<RqpMYlquAxWpYwqKmufyfiq?^c@_DNnnYw@o]k>[^iuBfhQIyrNu@inlp[_Wfu^_Yp]EAal_y^ve_`bP^agga=AeRwaHAjQoeDOyV^e]oqUG`y?chw]=NxxO\\wVwZndk?bV>pPVjPYjDng@xc=qpQ_cOH\\Q>\\_f`Gfm^odnQ`>XZdWe\\GmFFvsPuapp=`lFVthi[Bya=NqqFxYYcmq\\pfsaGx;A\\DQldPwBPqMf\\UonsVuhHhX^fcNoQoceOkhIsti^;qtZGtOnxX?^vqdYPpjo]hPl<qc@VkTheGOwEokn@[Naiy^mBQj\\IrcOmOnk<ok]qt=qr=VoIV_d_jXvuqiowXw[ncxH_LY]tgpundPNeGpik>v[ivRFt_GuKnqFIw\\n[Dpht`vgIkZav_Q]NqvbAkmHqAqgUOq_H^ZXjE>c@ObIXjtnofGo?qq^gek>ut@iu`tp?l\\IpOhwCas\\aonfiIvm?PdqQgMNhVQjFgeDAdSFf\\VeDAghqlRG\\UFqMYepp^xvlPPv^G^e@dF?^L@mKIurQ^^yxKFiKnmXGj:IvRymv>lgOlr?q<htJI]lw[^quL@_\\Gfugp<?xGw[l_w?ieg_ijn\\D^\\eInBw[<gZnH\\oQ_FAbVnaFaaR`_>G`rHna^skP[@Fjhih>`bBpbgvvrAaBxp=Aqoqb@w_j^q;qa\\VuFq[@x[V@[?AgJ^[kxerPenhqfWafWy@qfmqvcwZkng[awq@ona^EG]KgrbY]_`]Nx^fGZrOxeA`RFh:@wFNjfhZ]Ncgiq=YddW\\b^m@@erFcgq[LQmRApeA[qAxA>rTGrBHmJpxoxayqqiagVO`cIn[>[^Xr@IhK>b:Ywm^koNxqg[X`g<QyC_gM`[x_fy>t\\@fbxb;A`h^hdQiKXufFnFH]Lhqchj^fyOeC[fcowYsvRqI;kW[;teGV^?TIUwW[etGDOmhlUf[ubfcyi]IMogiwH^Uh:owAwfWaRpmtCCu=IFdAe^OhsKrueeZESZUB^cinMIfwE_mfskxJsEn?fOac@?GJ=uXkDIiB:qYOWBUKR?=u?qgNath?vdGFP?u`Uyw]fCccUOtvUGYci?Cu=gXp]xIcIyOrAmyxYxIcUNKFpQDYYR;CwB=sPqiF?eF?C>UTYef>=RJMT\\EBCGUDihhKUR]vgAenUvZiCeibWkCcUBoqdoocwyCC_r@if;Ssqksw_DVoBuyY;yrdqfuSchAYAUgqSySkgZuWmkD=QcvYwY[bksrJsEaKWiWuM=u_gc:Eg_YfMYtnqHogRisEi=xpsXq]grIhrWFiECySU:mRe_xgEBs[C\\yRo?c:sifCgn[XHeDUgWKoiicEPYIqqHxkWwAgYchqgDjgI;aFpKDWQv_OBZYt>WwdAUYuRL;coyqNuVh]Uqptdik;pXdykxTQGlTQtvX`YV=q:@Vm@rfTrITlTyjYtxo@jnUSSHSs]q_uyq]sCeWNMuHmK>\\jW=lwdNZdoB`WspoPDLoImBUJaaXXmXPaLSeon=UuamQqmpEUWHsLTyvEXTeOiiYyiqy=qq]XHARH`M_IQc<YAenHyQTInyMlRYKKxkfEXyAmYyQT=pthy[duxAmCAmAyYwUosdyA`PkQkUtOyxr`IR^tmMLvePL;DYZenryvu<YGuNr`xkPqPxMrmmC=S@\\vayJkMxnutcav>@jVDv`lqdmL;iwfTkBeLqlUWQPfUXQXyZuLgmsOyQ=HV@uNxeQuumr@QOMykDMcLms=OH@sQIU=HQm]rxujluyGqpxYuNqUD]Yxar;avC`mH\\U;tP;lWkuQsIJmxMGMLvMNaYPS`mrYOhqYJ@SreQx]TbHOrHlcIm?EypIjnhlkEwgYnFtXohJSUslLrw<kq`XcXuHmS:IPPMW^\\neLNrEYTELkhyWDW:pv@PRNASAqoq`YLxUytXZMvr`TjlmkUq=hQPQRvdtSEXTYUuULq@YxAT?ySLmUVdNoQNV\\oKQwoUX=myJ@v^yrfIsJdOR=X>=WX@URHjyIVIuXILLS`nk\\LsdLJxVXpmjPrpyJ:YqgMSUurFXqeaWDxpW`Y:=RpPLLao?TQMDNl`PdYp<mX>yXXELE`wQmT>]QetyZEUExkr=R?yraIXTIYjuufinriK\\@ySqKExX^]x]`rX=R:LVhpPBlPeYY\\dLbUNT@PXeS^awZiwAukjXW<<medjwP`qweXNrkoh_Yo=hbkW]w`wE@eNWblwkhX\\h^rNobMQrywiKYkYy`yWt]ifhylt^hyAx\\x\\yHkTYkBxsk_yjAh=hv[_^yfua_d\\?ktxar>jowbYyZlQaPVmj^iwPq?`[AH^^Q`pAy`wdZ`cMpyvi^;FrmxqywtPpwWN`qqa[QdlXxJ_hwgZL_rHxkiGaZWq\\xqMHw`xvNAhI_pjwsL@pMWrx^ohqiZ?ohyw^xx<OtQW[cqlrFcZoa\\N]SWbKwq?hykwj=y]`_feNsf_Z[@i^xvxX`iv^w^fXWdQv`t?bqYvfV^qphkgnm_hOIlfn\\fynmpxP?yfvpuiepnhyfruyi?>qYo^rxuxAu;hxeia_wyYf[Iv]`Hiq?nHWxDy^IPZyhm?QaVojWapPgmnI_x`n_yim@`jyr?OyyW\\`xqA@uXNaHW\\?pdAykuveI@c;AqmIyCw\\eAf?OoWapPGxIHpxyjRfjyakN`gUFxaOcSvyYAl?qefnu;fvI`xtowHHrpWh^icYOitqy<pdyOmS>yin`\\ya=hntvo[gtmpyYVjkyiO?btFfuaxe_etyhlXZyyZ?xpAxy\\>kYFv;P[to`bIvl@n_ifMIa]P`Iyy`Pcw^xjyqEnks_b:ojdIi<fxpVyqQtKYryyabHsKv\\E?lYgvjouu>dw^etHyjX`s?yTqZUYuR^^X`Z?ykxoit?qn?sL`_;hlj?]^xdshgsyqmAoNw[w^yyN[Bv_<AgOftih`SIa=h_fpbx@uIAdIvlHV^f?bdYf@hwS__:vh^Iw_IoyPoVYbIv\\=V_J>p;FmhYeJ>xan]bYoxoitCV[bBKeqqroeGBCI=MHlQycIw[Qv;?cAaxJ=Rxay_EfXKYryy:oxvcdr=TVYCCuCw?X:IX;CIrIiAoEhSEtiWkqEt=w?tKw\\x<vjXniu^yAv]AYcNiedPgjD:;j^PNaLNQENjD5BAnalysis of a Dual-Input, Multi-Loop Servo\251 2004 J.M. Redwood

This worksheet demonstrates the use of Maple to develop transfer functions for a naval gunmounting.The following Maple techniques are highlighted:Developing the modelPlotting results from the transfer functionsCreating a Nyquist diagram

Transfer functions are developed in this worksheet for a dual-input, multi-loop servo representing the training motion of a naval gunmounting and are analyzed by classical methods using Bode attenuation and phase diagrams. The closed loop responses are analyzed in Part II.

The purpose of the servo analyzed in this worksheet was to train a naval gunmounting so that its twin guns were aimed continuously at the target as directed by the ship's gun fire control system. (Training is the angular motion in a plane parallel to the ship's deck.) When a gun fires, the propellant gases acting on the gun's breech produce a force about the gunmounting's training axis, so causing a load torque tending to throw the guns off aim. Thus, the training servo had two inputs - a training signal from the gun fire control system, and a load torque. The training signal was transmitted by a synchro transmitter in the gun fire control system and the gunmounting's actual training was measured by a synchro control transformer located on the gunmounting. The difference between the two provided the error signal input to the servo. The servo had three feedback loops - the gunmounting's angular position, angular velocity and angular acceleration. The block diagram shown below illustrates the inputs, loops and main components of the servo. The ship is assumed to be in flat calm water, so that it does not suffer from sea-induced linear, or angular, motions. (This avoids the complications that arise when such motions have to be taken into account.) The details of the servo have been adapted from those given in the reference, which are believed to have been used on a gunmounting similar to the one discussed in this worksheet. The servo's transfer functions are derived in this worksheet, and the open loop performance is analyzed using Bode attenuation and phase diagrams. Closed loop performance is analyzed separately in Part II.

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Return to Filter Return to H1 Return to H2

R is the reference input (angular position in training required by the gun fire control system). e is the error between the controlled output and the reference input. L is an external disturbing torque applied to the motor shaft that drives the load (gunmounting). C is the controlled output (angular position of the gunmounting in training). C/R is the closed loop response of the controlled output to the reference input. C/L is the closed loop response of the controlled output to an external disturbing torque applied to the motor shaft that drives the load. N.B. The servo is assumed to be linear throughout the operating region and the principle of super-position therefore applies.

The "Bode2" package will be required for plotting the Bode attenuation and phase margin diagrams. It is therefore necessary to copy bode2.lib, bode2.ind and bode2.hdb to Maple's lib directory, for example, "C:\Program Files\Maple 9.5\LIB". It is also necessary to call the "plots" package and to change the imaginary unit to j, instead of I. A reduction to 4 decimal places shown makes the display easier to read, so "displayprecision" is set to 4, (computation, however, is at the precision set by "Digits").restart; interface(imaginaryunit=j,displayprecision=4): with(plots): with(Bode2);a. Pre-amplifier (including gains of synchros and error-phase-discriminating circuits).K1; K1 has units of volts/radian.b. Power Amplifier and Amplidyne.K2G2 := K2/((Tf*s + 1)*(Tq*s + 1)); K2 is gain of the power amplifier and amplidyne generator (volt/volt). Tf and Tq are the time constants (sec) of the amplidyne generator field and armature quadrature currents respectively.c. Combined Motor and Load.K3G3 := K3/(s*(Tm*s + 1)); K3 is the combined motor and load velocity constant (radian/sec/volt). Tm is the motor time constant = the combined motor and load inertias divided by the combined friction and armature current losses (sec).d. Tacho-generator (gunmounting's angular velocity feedback).K4G4 := K4*s; K4 is the tacho-generator constant (volt/radian/sec).e. Accelerometer (gunmounting's angular acceleration feedback).K5G5 := K5*s^2; K5 is the accelerometer constant (volt/radian/sec squared).f. Filter. Block DiagramK6G6 := K6*s^2*(Tc*s + 1)/((Ta*s + 1)*(Tb*s + 1)*(Td*s + 1)); K6 is the filter constant (sec squared). Ta, Tb, Tc and Td are time constants (sec) of the filter network.g. Applied load torque to motor voltage.K7; K7 is the reciprocal of the DC motor torque-per-volt constant, and its units are volt/Nm. It transforms the external load torque applied at the motor shaft into the equivalent voltage that would be required at the terminals of the motor's armature to produce that torque. (Note that it is voltage that is summed at the input to the DC motor.)h. G7. Let G7 be the transfer function of the forward loop in the block diagram from the input to the power amplifier and amplidyne to the output. Block Diagram Return to Observations G7 := K2G2*K3G3;i. K54G54. Let K54G54 be the ratio of accelerometer transfer function to that of the tacho-generator (sec).K54G54 := K5G5/K4G4;j. H1 Let H1 be the transfer function of the inner feedback loop (comprising the tacho-generator, accelerometer and filter). Block Diagram H1 := K4G4*K6G6*(1 + K54G54);

The servo components had the following parameters (expressed in the SI units mentioned above). Params := {K1 = 14.8, K2 = 2300, K3 = 0.526, K4 = 0.98, K5 = 0.01368, K6 = 0.193, Tf = 1/31.5, Tq = 1/17.6, Tm = 1/9.1, Ta = 1/0.986, Tb = 1/5.26, Tc = 0.016, Td = 0.0016, K7 = 0.63062};

When determining the transfer function from the reference input, R, to the controlled output, C, the externally applied torque is set to zero. Let the loop transfer function from the pre-amplifier output, to the controlled output, C, be G8. ThenG8 := G7/(1 + G7*H1); Define K1G8 as the loop transfer function from the pre-amplifier input, e, to the controlled output, C. ThusK1G8 := K1*G8; Hence, the closed loop transfer function from the reference input, R, to the controlled output, C, is`C/R` := simplify(K1*G8/(1+K1*G8)):

When determining the transfer function from the externally applied disturbing torque, L, to the controlled output, C, the reference input R is set to zero.The loop is then opened at the output from K2 G2, the power amplifier and amplidyne. Let H2 be the transfer function of K1 in parallel with the inner feedback loop, H1, determined above. Block DiagramH2 := K1+H1; Let H3 be the transfer function of the feedback loop, thus Return to Observations H3 := K2G2*H2; Define G9 as the open loop transfer function from the external torque, L, to the controlled output, C, thusG9 := K7*K3G3; Hence, the closed loop transfer function between the externally applied torque, L, and the controlled output, C, is`C/L` := G9/(1+K3G3*H3); The dimensions of C/L = NiMqKDcjJSNLN0ciIiI3IyomJiUiS0c2IyIiJEYmJiUiR0dGK0YmRiYsJkYmRiYqKEYnRiY3IyomJSNLMkdGJiZGLjYjIiIjRiZGJjcjJiUiSEdGNUYmRiYhIiI= are NiMqJiUldm9sdEciIiIqJiUiTkdGJSUibUdGJSEiIg== \327 NiMqJiUncmFkaWFuRyIiIiUldm9sdEchIiI=. This equation is normalized by dividing both sides by 1 NiMqJiUncmFkaWFuRyIiIiomJSJOR0YlJSJtR0YlISIi.

Return to Observations The transfer function, G7H1, for the inner loop from the power amplifier's input isG7H1 := G7*H1; Inserting the parameter values in G7H1 to yield G7H1_attenG7H1_atten := eval(G7H1,Params); G7H1_atten is now plotted on the Bode attenuation diagram together with the straight-line asymptotic approximations thus,k := 228.8215720:num := [.016,.01395918367]:den := [.03174603175,.05681818182,.1098901099,1.014198783,.1901140684,.0016]:a := linelog(k,2,num,den,.01,5000,top):b := attenlog(G7H1_atten,.01,5000):display(a,b); Since the normalizing time constant, T = 1 second, the values on the abscissa may be read as frequencies in 1/sec (= radians/second). The gain of G7H1 increases at 12 dB/octave from very low frequencies, breaks twice before reaching a maximum of 57 dB at a frequency of w = 6/sec. The gain then declines at an average rate of about 11.5 dB/octave becoming 12 dB/octave when the frequency is greater than w = 625/sec. The break points are located at w = 0.986/sec, 5.26/sec, 9.1/sec, 17.6/sec, 31.5/sec, 62.5/sec, 71.6/sec and 625/sec. These correspond to the reciprocals of the time constants Ta, Tb, Tm, Tq, Tf, Tc, K54, and Td, where K54 is K5/K4. Ta and Tb are used to reduce the gain rate at frequencies between 1/sec and 10/sec, while Tc and K54 are used to reduce the attenuation rate at frequencies of 62.5/sec and 71.6/sec until 625/sec, when the attenuation rate again increases to 12 dB/octave. Since NiMvJSVLMUc4RyomJSNLMUciIiIsJkYnRiclJ0sxRzdIMUdGJyEiIg==, this is approximately NiMqJiIiIkYkJSVHN0gxRyEiIg== when K1G7H1 is large. Thus the shape of the main forward loop, K1G8, is the same as that of the reciprocal of G7H1. This is very convenient, since the latter is easier to use when choosing time constants to produce good performance.

Inserting the parameters in K1G8, the open loop gain for the reference input can be plotted. However, to plot the straight-line approximations, K1G8 must be factored as follows.K1G8_atten := eval(K1G8,Params):K1G8_atten := factor(K1G8_atten); If a line segment of the plot starts and ends at the same frequency, w = 1/T, the "linelog" function in the Bode package produces an error message "plot ranges must be non-empty". That would occur at each of the double break points, unless they are separated very slightly as follows: a. "Digits" is increased, and b. The second of each approximate factor of the quadratic's approximate factors in the denominator is decreased very slightly from that of the first. That is the second approximate w = 1/T for each quadratic is reduced slightly. As a result, "linelog" plots an extremely short line segment between the two very slightly separated frequencies. To the eye, this appears as a double break point. "Digits" is restored to 10 after plotting the Bode attenuation diagram.Digits := 13:k := 90331642.97:num := [625.0000001, 5.259999996, .9860000007]:den := [79.07244591, 59.53034908, sqrt(796354.8919), sqrt(796354.89189), sqrt(.4362516033e-2), sqrt(.43625160329e-2)]:a := linelog2(k,1,num,den,.001,5000):b := attenlog(K1G8_atten,.001,5000):display(a,b);Digits := 10: Using the "apparent gain" of 90331642.97, "linelog2" calculated the true gain as 85 dB (17905.04) by multiplying the apparent gain of K1G8_atten by all its numerator time constants and dividing it by all its denominator time constants. This is the gain of K1G8 when expressed in canonical form, and matches the gain indicated by extrapolating the low frequency asymptote of K1G8 to the gain axis. It is known as the velocity error coefficient, Kv, and has units of 1/sec. Kv determines the closed loop system's error when following a constant velocity input, which is an important operational requirement for a gunmounting. If the constant velocity input is W radian/sec, the error after all the transients have decayed, is NiMvKiYlJk9tZWdhRyIiIiYlIktHNiMlInZHISIiKiZGJUYmLSUmRmxvYXRHNiQiKC8weiIhIiNGKw== radians = .5585e-4 W radians. This will be examined further in Part II. Since the normalizing time constant, T, is 1 second, the values on the abscissa may be read as frequencies in 1/sec (= radians/second). The break points of K1G8_atten occur at w = 0.986/sec, 5.26/sec, 59/sec, 79/sec, 625/sec. The double break points at NiMvJSZvbWVnYUctJSVzcXJ0RzYjLSUmRmxvYXRHNiQiK0xnXmlWISM3 = 0.066/sec and NiMtJSVzcXJ0RzYjLSUmRmxvYXRHNiQiKz4qW04neiEiJQ=== 892/sec. Break point frequencies that can be adjusted are 1/Ta = 0.986/sec and 1/Tb = 5.26/sec; the others are affected by the values of 1/Tc = 62.5/sec, 1/K54 = 71.6/sec and K6 = 0.193/NiMqJCUkc2VjRyIiIw==. The very high gain at low frequencies is reduced rapidly by the -18 dB/octave segment between w = 0.066/sec and 0.98/sec, and by the -12 dB/octave segment that follows it from w = 1/Ta = 0.98/sec to w = 1/Tb = 5.26/sec. Together with the overall gain setting of 85 dB, Ta and Tb produce gain crossover over (also known as feedback cutoff) about 1/3 down the 3.5 dB long, -6 dB/octave segment from w = 5.26/sec to 59/sec. This indicates a very stable system that should follow the control signal well, while rejecting much of the noise in it. After a -18 dB/octave section from w = 59/sec to 625/sec, the loop gain then declines at -24 dB/octave, ensuring that unwanted high frequency signals are well attenuated. There is a second very short, -18 dB/octave section between w = 625/sec and 892/sec, before the loop gain decreases at the final rate of -24 dB/octave. It will be seen that at frequencies less than 59/sec, the attenuation is essentially the same as the reciprocal of the inner loop's transfer function, G7H1, plotted previously. As noted above this can be helpful when choosing the time constants Ta, Tb and Tc. The frequency at which gain crossover over occurs is obtained by substituting ja = s, where a = wT, setting K1G8 = 1, and solving for a. omega*T = abs(fsolve(subs(s=j*alpha,K1G8_atten)=1,alpha)); Alternatively, the cursor may be used to read the frequency at gain crossover over, thus log10(wT) = 1.16, whence wT = 14.4, which is sufficiently close for all practical purposes.

The loop phase margin is shown byphaselog(K1G8_atten,.001,5000); The sudden loss of phase margin seen at about log10(wT) = -1.4 occurs when the open loop attenuation changes slope from -6dB/octave to -18 dB/octave. At gain crossover, log10(wT) = 1.17, the phase margin is 43 degrees. Although, this indicates a very stable system, it appears to be only conditionally stable. If the gain were to reduce, and gain crossover-over occurred at about log10(wT) = 0.4, i.e. w = 2.4/sec, the phase margin would fall to zero. For this to happen, the loop gain would need to reduce by about 28 dB i.e. by a factor of 25. This is unlikely since it is greater than the pre-amplifier gain (K1 = 14.8) and because the power amplifier and amplidyne gains are most unlikely to halve at the same time as the pre-amplifier loses all of its gain. Nevertheless, a reduction of phase margin to, say, 12 degrees caused by the pre-amplifier's gain halving, would cause sluggish, oscillatory performance. This is illustrated in Part II, when the transient responses are examined.

When considering the feedback loop from the (load) output to the motor, the reference input is set to zero. Before examining this loop, it is worth looking briefly at K3G3_atten, the motor and load transfer function, which is shown below.K3G3_atten := eval(K3G3,Params);a := linelog(0.526,1,[ ], [.1098901099],.01,100):b := attenlog(K3G3_atten,.01,100):display(a,b); K3G3_atten demonstrates the usual characteristics of a motor and load - small gain at low frequencies, and attenuation at 12 dB/octave at frequencies above the single break-point at w = 1/Tm = 9.1/sec (again, the normalizing time constant, T = 1 second). The transfer function for the complete open loop feedback from the (load) output to the motor is H3,H3; Inserting the component parameters, this becomesH3_atten := factor(eval(H3,Params)); The attenuation diagram for the open loop feedback transfer function from the (load) output is plotted below.Digits := 13:k := 174605.1797:n := [91.70572349,15.84188672,.9721856834,sqrt(248.0537289),sqrt(248.05372889)]:d := [624.9999999,31.49999999,17.60000001,5.260000000,.9859999999]:a := linelog2(k,0,n,d,0.01,5000):b := attenlog(H3_atten,0.01,5000):display(a,b);Digits := 10: The very large low frequency gain of H3_atten remains virtually constant and never dips below 81 dB, thus ensuring that even a small movement of the load shaft produces a very large opposing torque at all frequencies. This is caused by the feedbacks from the tacho-generator and accelerometer offsetting the small low frequency gain and attenuation of K3G3, the motor and load transfer function.

Before using the Nyquist stability criterion for a multi-loop system such as this, it is necessary to determine the number of poles, P, of the open loop transfer function that lie in the right half of the s plane. This is given by,solve(K1G8,{s}); Since Ta, Tb and Td are all positive real constants, K1G8 has no poles in the right-half plane, i.e. P = 0. Moreover, this result is independent of all gain settings. (Maple may not be able to find the roots of the open loop transfer function symbolically. However, it should be able to find numerical solutions after the servo's parameters have been inserted in K1G8. In such cases, it is necessary to check that gain changes do not affect the number of poles, or their location in the left, or right-half planes.) Nyquist's stability criterion derives from the theorem that N = Z - P, where Z is the number of zeros in the right-half plane of the denominator of the closed loop transfer function (1 + K1G8), P is the number of poles in the right half plane of the open loop transfer function (K1G8), and N is the number of net clockwise encirclements of the [-1, j0] point described by a representative point as it travels from w = -\245 to +\245 around the locus of the open loop transfer function in the complex plane. (A negative value of N signifies the number of net anti-clockwise encirclements of the [-1, j0] point.) For stability, Z must be identically zero. Otherwise, the closed loop transfer function would contain poles in the right-half plane, and the controlled output, C, would increase without bound. Thus, the system is stable if, and only if, N \272 -P. In this case P = 0, so N must be zero for stability. It is now convenient to obtain a function f (jw) for the loop transfer function in the complex plane thus,K1G8_jw := eval(K1G8,Params):K1G8_jw := subs(s=j*omega,K1G8_jw):f := unapply(K1G8_jw,omega): The Nyquist diagram is plotted below with some representative points shown on the locus of K1G8_jw.a := complexplot([seq(evalc(f(i)),i={12,15,20,30,65,100,-12,-15,-20,-30,-65,-100})],x=-1..0, style=point,symbol=circle,colour=black):b := complexplot(f,10..1000,colour=blue):c := complexplot(f,-10..-1000,colour=red,linestyle=3):d := plot([[-1,0]],style=POINT,symbol=cross,symbolsize=10,color=black):e := coordplot(polar,[0..1,Pi/2..3*Pi/2],grid=[4,13],color=[black,black],linestyle=[2,2]): f := textplot([-1,-0.05,"[-1, j0]"],font=[TIMES,ROMAN,11],colour=black):display(a,b,c,d,e,f,scaling=constrained,title="Nyquist Diagram\nPositive frequencies (blue), Negative frequencies (red)",titlefont=[TIMES,ROMAN,12]); The Nyquist diagram above shows the locus, in blue, of K1G8_jw between w = 11/sec and w \256 +\245, which is located at the origin of the diagram. It also shows the locus, in red, for the portion between w = -11/sec and w \256 -\245, also located at the origin. Reading from the grid, the blue locus shows that the phase margin of K1G8_jw is slightly less than 225 - 180 = 45 degrees. Using the cursor, it is`phase margin` = evalf(arctan(-0.68/(-0.73))*180/Pi*degrees); The remainder of the Nyquist diagram can be inferred from the Bode diagrams. There, it was seen that the phase margin becomes negative when log10(wT) equals -1.18, 0.38 and 1.81 (i.e. w = 0.07/sec, 2.4/sec and 64.5/sec). Hence, the loci of K1G8_jw must cross the negative real axis three times before becoming asymptotic to the imaginary axis. The highest frequency of crossing has been plotted in the diagram above, while the other two are located far to the left on the negative real axis. The complete diagram is illustrated in the sketch below, which shows zero encirclements of the Nyquist point. (The area enclosed always lies to the right of a representative point as it traces the locus from w = -\245 to +\245. Since the Nyquist point is not enclosed within this area, there are zero net encirclements.) However, if the scale of the axes is either increased, or decreased, while leaving the locus fixed, the locus would encircle the Nyquist point twice in the clockwise sense. This would violate the stability criterion, and so the system would be unstable. The change of scale corresponds to a change of the servo's gain, and indicates that the servo is conditionally stable depending upon the gain. As already noted in the discussion about the Bode diagrams, the case of the gain being too small is unlikely to occur.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 The difficulty with the low frequency portion of the Nyquist diagram can be partly overcome by plotting the inverse diagram, i.e. plotting the reciprocal of K1G8_jw. Unfortunately, such plots can be difficult to interpret and serve to illustrate the superiority of Bode diagrams in the low frequency region.

1. The choice of compensation and time constants has not been elaborated in this worksheet. It is, however, worth mentioning that acceleration feedback is sometimes used to increase the load's apparent inertia in a manner analogous to using velocity feedback to increase the load's apparent viscous friction. In this case, the increased apparent inertia would decrease the throw-off caused by the guns firing. Having included acceleration feedback, velocity feedback is the best choice for the servo's compensation. The accelerometer and tacho-generator gains and the filter constants were chosen to cause rapid attenuation of the gain at frequencies above about 70/sec and so provide the high "apparent inertia" to sudden load torques. They were also chosen to attenuate the large low frequency gain rapidly at frequencies greater than about 0.07/sec, and to produce a long 6 dB/octave attenuation at gain crossover. These are important for achieving good, constant-velocity performance needed for aiming accurately at moving targets, and for good servo stability with noise rejection. These choices are usually made with the help of Bode diagrams for the inner feedback loops involved, in this case, the G7H1 loop and the H3 loop.G7H1 H3 2. The modern design process includes the classical, open loop, frequency analysis shown in this worksheet together with the closed loop analysis shown in Part II. However, specialized software for servos is normally used to try out different approaches and to optimize the parameter settings, taking account of non-linearities, backlash, etc.3. Much painstaking work with slide-rule, log-tables, and semi-log charts were needed in the past to perform the calculations and draw the Bode diagrams shown in this worksheet. Great care was necessary when developing the transfer functions - a minor slip with the algebra could cause immense problems. Maple does it all quickly, easily and without errors. Not only that, it can feed optimized code to certain software used for designing servos, thus reducing setting up times and errors.

In Part I, transfer functions were developed and open loop responses were analyzed for a multi-input, multi-loop servo representing the training motion of a naval gunmounting. This worksheet examines the closed loop responses of the servo to standard inputs and the responses to load torques caused by a gun firing.

The purpose of the servo analyzed in this worksheet was described in the introduction to Part I. Briefly, it was to train a naval gunmounting so that its twin guns were aimed continuously at the target by the ship's gun fire control system, and to resist the load torque created when a gun fired. Part I, developed the servo's transfer functions and analyzed the open loops. The closed loop responses to standard signals at the reference input and to approximations for load torques are examined in this worksheet. The servo had three feedback loops - the gunmounting's angular position (i.e. training), angular velocity and angular acceleration. The block diagram shown below illustrates the inputs, loops and main components of the servo. The ship is assumed to be in flat calm water, so that it does not suffer from sea-induced linear, or angular, motions. (This avoids the complications that arise when such motions have to be taken into account.) The details of the servo have been adapted from those given in the reference, which are believed to have been used on a gunmounting similar to the one discussed in this worksheet.

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 Return to H1

R is the reference input (angular position in training required by the gun fire control system). e is the error between the controlled output and the reference input. L is an external disturbing torque applied to the motor shaft that drives the load. C is the controlled output (angular position of the gunmounting in training). C/R is the closed loop response of the controlled output to the reference input. C/L is the closed loop response of the controlled output to an external disturbing torque applied to the motor shaft that drives the load. N.B. The servo is assumed to be linear throughout the operating region and the principle of super-position therefore applies.

The "Bode2" package will be required for plotting the Bode attenuation and phase margin diagrams. It is therefore necessary to copy bode2.lib, bode2.ind and bode2.hdb to Maple's lib directory, for example, "C:\Program Files\Maple 9.5\LIB". It is also necessary to call the "plots" package and to change the imaginary unit to j, instead of I. A reduction to 4 decimal places shown makes the display easier to read, so "displayprecision" is set to 4, (computation, however, is at the precision set by "Digits").restart; interface(imaginaryunit=j,displayprecision=4): with(plots): with(Bode2); with(inttrans):a. Pre-amplifier (including gains of synchros and error-phase-discriminating circuits).K1; K1 has units of volts/radian.b. Power Amplifier and Amplidyne.K2G2 := K2/((Tf*s + 1)*(Tq*s + 1)); K2 is gain of the power amplifier and amplidyne generator (volt/volt). Tf and Tq are the time constants (sec) of the amplidyne generator field and armature quadrature currents respectively.c. Combined Motor and Load.K3G3 := K3/(s*(Tm*s + 1)); K3 is the combined motor and load velocity constant (radian/sec/volt). Tm is the motor time constant = the combined motor and load inertias divided by the combined friction and armature current losses (sec).d. Tacho-generator (gunmounting's angular velocity feedback).K4G4 := K4*s; K4 is the tacho-generator constant (volt/radian/sec).e. Accelerometer (gunmounting's angular acceleration feedback).K5G5 := K5*s^2; K5 is the accelerometer constant (volt/radian/sec squared).f. Filter.K6G6 := K6*s^2*(Tc*s + 1)/((Ta*s + 1)*(Tb*s + 1)*(Td*s + 1)); K6 is the filter constant (sec squared). Ta, Tb, Tc and Td are time constants (sec) of the filter network.g. Applied load torque to motor voltage.K7; K7 is the reciprocal of the DC motor torque-per-volt constant, and its units are volt/Nm. It transforms the external load torque applied at the motor shaft into the equivalent voltage that would be required at the terminals of the motor's armature to produce that torque. (Note that it is voltage that is summed at the input to the DC motor.)h. G7. Let G7 be the transfer function of the forward loop in the block diagram from the input to the power amplifier and amplidyne to the output. G7 := K2G2*K3G3;i. K54G54. Let K54G54 be the ratio of accelerometer transfer function to that of the tacho-generator (sec).K54G54 := K5G5/K4G4;j. H1. Let H1 be the transfer function of the inner feedback loop (comprising the tacho-generator, accelerometer and filter). Block Diagram H1 := K4G4*K6G6*(1 + K54G54);

The servo components had the following parameters (expressed in the SI units mentioned above). Params := {K1 = 14.8, K2 = 2300, K3 = 0.526, K4 = 0.98, K5 = 0.01368, K6 = 0.193, Tf = 1/31.5,Tq = 1/17.6, Tm = 1/9.1, Ta = 1/0.986, Tb = 1/5.26, Tc = 0.016,Td = 0.0016, K7 = 0.63062};

In Part I, the loop transfer function from the pre-amplifier input, e, to the controlled output, C, was found to be,K1G8 := K1*K2/(Tf*s+1)/(Tq*s+1)*K3/s/(Tm*s+1)/(1+K2/(Tf*s+1)/(Tq*s+1)*K3*s^2/(Tm*s+1)*K4*K6*(Tc*s+1)/(Ta*s+1)/(Tb*s+1)/(Td*s+1)*(1+K5*s/K4)); Hence the closed loop transfer function between the reference input, R, and the controlled output, C, is`C/R` := simplify(K1G8/(1+K1G8)): In Part I, the closed loop transfer function between the externally applied load torque, L, and the controlled output, C, was found to be,`C/L` := K7*K3/s/(Tm*s+1)/(1+K3/s/(Tm*s+1)*K2/(Tf*s+1)/(Tq*s+1)*(K1+K4*s^3*K6*(Tc*s+1)/(Ta*s+1)/(Tb*s+1)/(Td*s+1)*(1+K5*s/K4)));

Inserting the servo's parameters, the closed loop-frequency response is`C/R freq` := eval(`C/R`,Params):a := attensemilog(`C/R freq`,.01,100):b := pointplot([[-3,-3],[2,-3]],connect=true,colour=black,linestyle=2): display(a,b,title="C/R Closed Loop gain dB"); The resonance peak of the closed loop response, C/R, is about 3.5 dB = 1.5 at the damped natural frequency of about w = 10/sec (since the normalizing time constant T is 1 sec). The bandwidth of the servo, i.e. the frequency range in which the magnitude of the closed loop response, C/R, is greater than -3 dB, is approximately 27/sec. Both can be obtained more accurately by using the "linelog" command to plot C/R instead of "linesemilog" and then using the cursor. Or, by substituting ja = s (where a = wT) in C/R freq, a function, C/R f,can be obtained, from which the maximum value of C/R freq can then be found thus, `C/R f` := map(abs, subs(s=j*alpha,`C/R freq`) ):`C/R max` = maximize(`C/R f`,location,alpha=10..20); The maximize function yields not only the maximum value of C/R freq = 1.5, but also its location, a = wT = 10.72, hence w = 10.7/sec Setting C/R f = -3 dB and solving for a yields the cutoff frequency, NiMmJSZvbWVnYUc2IyUiY0c=. In this case, the bandwidth extends from w = 0/sec (direct current) to the cutoff frequency, so it has the same value as the cutoff frequency.omega[c]*T = fsolve( `C/R f`=10^(-3/20), alpha,0..40); The bandwidth is an important operational parameter, since fast response requires a wide bandwidth. However, too wide a bandwidth allows noise to appear in the response. The bandwidth of 28/sec found here is a satisfactory compromise between these conflicting requirements.

At this point, it is convenient to define some options that will be used in the transient response plots.txt := [TIMES,ROMAN,12]:opt1 := axesfont=txt,labelfont=txt,titlefont=txt,colour=blue:opt2 := style=point,symbol=diamond,axesfont=txt,titlefont=txt,labelfont=[SYMBOL], symbolsize=18,colour=red: While considering the open loop attenuation and phase margin in Part I, it was noted that a decrease of gain, K1, (the gain of the synchro chain, phase-discrimination circuits and pre-amplifier) could cause a sluggish, oscillatory response of the servo. It was also suggested that the gain could be increased slightly without affecting stability. This is examined below, by plotting the closed loop response to a unit step input of position with four different settings of the gain K1, namely 20.92, 14.8, 7.4, and 3.7 volt/radian. These represent an increase of 3 dB on the proper setting, the proper setting, and reductions of 6 dB and 12 dB from it. First, K1 is removed from the set of parameters so that it can be given four different settings, each associated with a colour.Params := Params minus {K1=14.8};K1s := array([[20.92,14.8,7.4,3.7],[green,blue,magenta,maroon]]); Next the closed loop transfer functions are obtained for each of the four settings of K1, `C/R 4` := eval(`C/R`,Params):`C/R 4` := [seq( subs(K1=K1s[1,i],`C/R 4`) ,i=1..4)]: The Laplace transforms of the four closed loop responses, C, to a unit step of (angular) position input areposn_steps := [seq( `C/R 4`[i]/s ,i=1..4)]: The inverse Laplace transforms of these four responses are normalized by dividing them by the input (R = 1 radian) and plotted, thusposn_step_resp := seq( invlaplace(posn_steps[i],s,t) ,i=1..4): a := seq(plot(posn_step_resp[i],t=0..2, title="Responses to Unit Step of Input Position for\nK1 = 20.92 (green), 14.8 (blue), 7.4 (magenta) & 3.7(maroon)",labels=["t sec","C/R"],opt1,colour=K1s[2,i]),i=1..4): b:= pointplot([[0,1],[2,1]],connect=true,colour=black,linestyle=2):display(a,b); The plot shows how the response becomes sluggish and oscillatory as the pre-amplifier gain reduces by 6 dB (halves) from its proper setting and then halves again. With a setting of K1 = 7.4 volts/radian, the settling time more than doubles to t = 1.15 sec, which is probably unacceptable. On the other hand, a 3 dB increase in K1 causes only a slight increase of the overshoot and settling time. The sensitivity of the closed loop response to gain reductions in the pre-amplifier and power amplifier would need careful evaluation during breadboard and prototype trials. The normalized error (i.e. Error/R = 1 - C/R) is more easily seen in the following plot,plot([seq( (1-posn_step_resp[i]) ,i=1..4)], t=0..3/2, colour = [seq(K1s[2,i],i=1..4)],title="Error in Response to Unit Step of Input Position for\nK1 = 20.92 (green), 14.8 (blue), 7.4 (magenta) & 3.7(maroon)",labels=["t sec","Error/R"],opt1); With the gain setting of K1 = 14.8 volts/radian, the error of the response to a step displacement has a single overshoot of 38% and the transient is virtually complete after t = 0.44 sec. This should be satisfactory for a gun-aiming servo.

The Laplace of transforms of the responses, C, to a unit step of (angular) velocity input, or ramp input, for each of the 4 settings of K1 arevel_steps := [seq( `C/R 4`[i]/s^2 ,i=1..4)]: The inverse Laplace transforms of these (angular) position responses, C, are then normalized by dividing them by the input's (angular) position, R. That is, each C is divided by R = velocity \327 t = 1 radian/sec \327 t sec = t radian, thus, vel_step_resp := seq( (invlaplace(vel_steps[i],s,t)) / (1*t), i=1..4): aa := seq(plot(vel_step_resp[i], t=0..3/2,title="Responses to Unit Step of Input Velocity for\nK1 = 20.92 (green), 14.8 (blue), 7.4 (magenta) & 3.7(maroon)",labels=["t sec","C/R"], opt1,colour=K1s[2,i]),i=1..4): display(aa,b); With K1 = 14.8 volts/radian, C/R reaches unity by t = 0.3 sec and the transient has a very small overshoot of about 2%. Although the transient has a rather long tail (caused by zeros in the numerator of the closed loop transfer function that are associated with the tacho-generator and accelerometer feedbacks) it is substantially complete by t = 1 sec. These are important characteristics for gun aiming servos, since they have to respond quickly and accurately to changing input velocities. (In passing, it should be noted that the accelerations involved are limited by the "g forces" that the target can withstand.) As K1 decreases, the response becomes sluggish and oscillatory, as already seen with a step input of position. A 3 dB increase of K1 to 20.92 volts/radian produces a near ideal response. However, it would increase the servo's damped resonant frequency and its bandwidth, making it more responsive to noise in the input. For this reason, K1 = 14.8 volts/radian, is probably the best compromise. It would, though, be worth investigating the response to a noisy signal with the higher setting during breadboard and prototype testing. The response, C, to a unit step of velocity input is simply the normalized response shown above, multiplied by the normalizing factor R = 1 radian/sec \327 t sec, and is shown below with K1 = 14.8 volts/radian.aaa := plot( vel_step_resp[2]*(1*t), t=0..1, title="Response (blue) to Unit Step Input of Velocity (red)",labels=["t sec","C and R"],opt1): bbb := plot(1*t,t=0..1,colour=red):display([aaa,bbb]); The plot shows that after a transient, the response, C, leads the input, R = 1 radian/sec \327 t sec. The lead is caused by the velocity feedback and the values chosen for the feedback gain and filter constants. The very small difference between response and input results from the fairly large value of the velocity error coefficient, Kv, noted in Part I. The difference between the input and the response, R - C, is a position error and is plotted below.plot( 1000*(1*t - vel_step_resp[2]*t), t=0..3/2,title="Error of Response to Unit Step of Velocity", labels=["t sec","1000 \327 Error"],opt1); The plot shows that while following a constant velocity input of 1 radian/sec, the error is less than 5/1000 radians after t = 1 sec. A 500 knot (257 m/sec) target crossing the line of fire perpendicularly at range r would generate an angular rate of NiMqKCIkZCMiIiIlIm1HRiUqJiUickdGJSUkc2VjR0YlISIi, so the gun aiming error would be`aiming error` := (257*m/(r*sec))*((5*radian/1000.)/(1*radian/sec))*r; The gun aiming error of 1.29 m should be acceptable. The plot shows that the velocity error approaches a very small value asymptotically. This can be found from the Laplace transform of the velocity by means of the final value theorem. First, it is necessary to reset K1 = 14.8, re-insert it in the servo's parameters, and evaluate K1G8 with these parameters.Params := Params union {K1=14.8}:K1G8 := eval(K1G8,Params): Assuming that the input velocity is W, the transform of the velocity error is given by epsilon(s) := 'Omega/s^2/(1 + K1G8)'; The final value theorem states that NiMvLSUmbGltaXRHNiQtJSJ5RzYjJSJ4Ry9GKiUpaW5maW5pdHlHLUYlNiQqJiUic0ciIiItRig2I0YwRjEvRjAiIiE=, hence the steady state error resulting from an input velocity of W is ss_error := limit(s*eval(epsilon(s)),s=0); At full display precision, this is 0.5585019637e-4 \327 W, which is identical to that found in Part I as NiMvKiYlJk9tZWdhRyIiIiUjS3ZHISIiKiZGJUYmLSUmRmxvYXRHNiQiKC8weiIhIiNGKA==. With W = 1 radian/sec, the steady state velocity error is about 1% of the velocity error (5/1000) when t = 1 sec.

The poles of C/R are the roots of the characteristic equation of the C/R transfer function. This is obtained by removing any common factors from its numerator and denominator and equating the latter to zero.`char C/R` := denom(normal(eval(`C/R`,Params))) = 0:`C/R poles` := [solve(`char C/R`)]; These are now plotted, thusplot(map([Re, Im],`C/R poles`),-16.. 0,-16..16, style=point, symbol=diamond, title="Poles of C/R", labels=[\302,\301], opt2 ); The plot does not show the distant poles since their effect should be small. The damping ratio indicated by the complex poles nearest to the imaginary axis issolve( sqrt(1-zeta^2)/zeta = abs( Im(`C/R poles`[6])/Re(`C/R poles`[6]) ),{zeta}); This value of damping ratio suggests a more sluggish response than that obtained for the unit step displacement. However, the servo is not a simple, second-order system, to which the standard response curves apply.

The closed loop response to a load torque input is given by`C/L` := eval(`C/L`,Params):display(attenlog(`C/L`,.01,100),title="C/L Closed Loop Gain dB"); The dimensions of C/L = NiMqKDcjJSNLN0ciIiI3IyomJiUiS0c2IyIiJEYmJiUiR0dGK0YmRiYsJkYmRiYqKEYnRiY3IyomJSNLMkdGJiZGLjYjIiIjRiZGJjcjJiUiSEdGNUYmRiYhIiI= are NiMqJiUldm9sdEciIiIqJiUiTkdGJSUibUdGJSEiIg== \327 NiMqJiUncmFkaWFuRyIiIiUldm9sdEchIiI=. This equation is normalized by dividing both sides by 1 NiMqJiUncmFkaWFuRyIiIiomJSJOR0YlJSJtR0YlISIi, so that the result is non-dimensional and can be expressed in decibels (dB). C/L is the response to torques applied to the motor shaft. If the torque is applied to the load, i.e. the gunmounting's structure, instead of the motor shaft, it will be reduced by the gearing. The firing load torque can be visualized as an approximately triangular pulse with a duration of about 3/4 second, and can be represented by a Fourier series with frequencies of NiMqKCIiJSIiIiUjUGlHRiUqJiIiJEYlJSRzZWNHRiUhIiI=, NiMqKCIiKSIiIiUjUGlHRiUqJiIiJEYlJSRzZWNHRiUhIiI=, etc. In this frequency range, the attenuation of C/L is at least -88 dB. The gearing ratio of the train drive is about 200:1, so the throw off in the training motion would be less than `throw off` := evalf( (75000*N*m) * (1*radian/(N*m)) * (1/200) * (10^(-88/20)) * (180*degrees/(Pi*radian)) );

The firing load could be considered as a torque that increases steadily for 1/4 second up to a maximum value, and then decreases steadily to zero for 1/2 second. (The actual shape and duration of the pulse is determined by the design of the gun's recoil and recuperator systems that attach it to the gunmounting.) The (angular) position of the output, C, in response to a unit step input of torque rate, i.e. a suddenly applied ramp of 1 Nm/sec of torque input, is given bytorque_rate_resp := `C/L`/s^2:torque_rate_resp := invlaplace(torque_rate_resp,s,t): At this point, it is convenient to obtain a function, f, for the response, C, to the 1 Nm/sec ramp of torque input thus,f := unapply(torque_rate_resp,t): The response is plotted below.plot(10^6*f, 0..0.4, labels=["t sec","C \327 10^6"], title="Response to Unit Ramp of Load Torque", opt1); The response, C, to a torque rate of 1 Nm/sec for 1/4 sec, followed by one of -0.58 Nm/sec, lasting for 1/2 sec is shown in the plot below,plot( piecewise(t<1/4, 10^6*f(t),t>=1/4, 10^6*( f(1/4) - 0.58* f(t- 1/4))), t=0..3/4,labels=["t sec","C \327 10^6"], title="Response to Triangular Pulse of Torque",opt1); Assuming that the firing-load torque increases linearly, it would produce a torque rate of 75 kNm / (0.25 sec) = 300 kNm/sec. Since the gearing is 200:1, the maximum throw off is given by`throw off` := evalf( (300*10^3*N*m/sec) * (1/200) * (simplify(f(1/4)) * radian/((N*m)/sec)) * (180*degrees/(Pi*radian)) ); Thus, the throw-off produced by one gun firing at zero elevation is between 0.6 and 0.9 degrees. This should be within the amplifier's linear region when controlled by a 36-speed synchro, and so the assumptions of linearity and super-position hold good. However, this would need to be checked using actual firing pulse data. The extent of the throw-off and the time taken to recover from it are important operational parameters, and would have to be measured during prototype trials.

The (angular) position response, C, of the gunmounting's training servo to a unit step of torque input is shown below for interest.torque_step_resp := ( `C/L`/1 )/s:torque_step_resp := invlaplace(torque_step_resp,s,t):plot(10^6*torque_step_resp,t=0..0.5,labels=["t sec","C \327 10^6"], title="Response to Unit Step Input of Load Torque",opt1); After t = 0.36 sec, the transient is complete and, in the steady state, the load torque applied at the motor shaft keeps the gunmounting trained away from its correct position by about 18.2 m radians. It is analogous to a torque twisting a shaft, or a force compressing a spring. Thus, the steady state torsional "stiffness" of the training motion from motor shaft to gunmounting position is 18.2 m radians/Nm (3.75 seconds of arc per Nm).

The poles of C/L are the roots of the characteristic equation of the C/L transfer function. This is obtained by removing any common factors from its numerator and denominator and equating the latter to zero.`char C/L` := denom(normal(`C/L`)) = 0:`C/L poles` := [solve(`char C/L`)];plot(map([Re, Im],`C/L poles`),-16.. 0,-16..16, title="Poles of C/L", labels=[\302,\301], opt2 ); The poles are identical to those of C/R.

1. The choice of compensation, gains and time constants was discussed in Part I. The closed loop frequency and transient response analyses in this worksheet demonstrate that the values chosen should have satisfied the operational requirements for a gunmounting's training motion. The effect of sea-induced ship motion would need to be examined before leaving the theoretical design stages, but that is outside the scope of this analysis.2. The classical analyses shown in this worksheet are part of the modern design process and are often included in specialized software for designing servos. This is also used to try out different approaches and to optimize the parameter settings, taking account of non-linearities, backlash, etc. Despite the efficacy of such software, the designer's skill and experience is still the most important factor in achieving a successful design.3. The final hardware will depart from the theoretical design because of the assumptions that have to be made - for example, about the rigidity of the gunmounting's structure and the ship's supporting structure. Extensive experimental work with breadboard models and a prototype almost invariably follow the theoretical work. In the case of a naval gunmounting, this work would have included live firings in heavy seas and against drone aircraft.4. Long, painstaking work with slide-rule, tables of Laplace transforms, log-tables, and semi-log charts (and eraser!) were needed in the past to perform the calculations and draw the plots shown in this worksheet. The transient responses were particularly arduous. Maple does it all quickly, easily and without errors. Not only that, it can feed optimized code to certain software used for designing servos, thus reducing setting up times and errors.

Page 378 excerpted from Servomechanisms & Regulating System Design, Vol I, by Harold Chestnut & Robert W. Mayer, originally published by John Wiley & Sons, 1951 - 1980, copyright by The General Electric Company 1951. Used with permission of The General Electric Company.

Legal Notice: The copyright for this application is owned by the author(s). Neither Maplesoft nor the author are responsible for any errors contained within and are not liable for any damages resulting from the use of this material.. This application is intended for non-commercial, non-profit use only. Contact the author for permission if you wish to use this application in for-profit activities. MFNWtKUb<ob<R=MDLCdNVZZJ:@L>H:TKGxMkJ:<O`Lo\\lQxlQWdMWpsHqShmWhYoeXOPmTPmV`mvqyxq=Xj=xXquXaxnaXcEWc=UR=UweYwELKDLqtPq<R:=r^av^uRAurZ@nZtVauVb=WbMYtMyvayvYyuYYxmYxqyxqYyuYyEYsEYpmXpyyyyypqxp=J:>::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::dy<TypC>qULCTJcDXoXusT<aupkcfWMX@JCeU`dNuTmWxyyyppuPCDSSuLClu><xTpQmlsb]MihUO`qTeXSQO;@JxV]wOl:@syFv<w\\t@tsNnQn\\V?w<w\\?FqJijXynZVvnyHErmiB__tWit[MyxYRIIXvWgtSS=;gQMwAIC]IYrGXRogc[EpqYtsxn=BVSUGuEA[WxKrWaSHssoYBPkynKctqgmyUKAYQYUw_rs=wboYTWXI?IQKyo[X@wydqytYRGAy`ixs[SlyXaSyquy:mel=dXqydIfvgRIeSUkUmUBGwuZitS;eQ?S>AdMasnkySGbDSuimbSabjytNAyMuXlaTWaCp;y?at;_txaTwath?cj=GbgYVGCA[eAkh^ihyaIGoVdGxyWeQatamVHYx:SEIewyacmcSBAvgOyyssEyBVWCwQFtYWxYdMgcY_y^Uy?gce[WXQCDcwGuwHMw?qwx[gacscGrwOtuKFXKsc[FZIBOqIrII]kuICfRosM_yTSEWWcKQs_qGHeIiaWBsvaAXWoFsYTyuIYSdWCet[fZpOYtv[\\XSMvN=Xhluxel]ylvUn;PYsqvkmmCxSEQPsMOeUpQEKN`yVAqcqRQpYxHr[xU\\AtgPVexmHHQYDXptL;ey_\\XHxyTpLQ=qJhJklqA=wPxqOtpPmwQ=kWdSSYjxhQt=li<X=Pr\\HoxMKxppdUPGxl`<RadWsEMUhnMinaqvy\\t]pJw\\Pttt:lw_hy;PxuElWpfypiQyg<IbgHqQ?wRwvFgcQnmtI]lXZoauvw\\]Vi\\?yuIjGqyA_]j^cia\\^vaYfmXYvV_foyd_wZa?yIPfNXpOimbInwiieQyZ@[jf[p_`s?\\N@qaw[<a_=qpdIu]>gnHpUi\\^a[AGcS_y]pnHg_oIi=XkM`bK^yUWjFhhCpif?llhelhkKqk=qgCqqIokJadZ@]IOspHjgQgUv^Mp^[akXNokxcFaxMX>Efx=GJyY]=uKWXuefcYCV_DO;X]oeDwI]UrhIXhKdtYgv=sYMxyMhEAbdKdFED;MBimUYgvNsfBuDgqw^sRZoieyiYEfEAsYOcU;uf_C^;g>EIUmWy]xZ[H?UTiwhayb<EWUAhmghUee]ODLyfkYdOQDNMsleg]mHGkynUrrUhjgbvstrICsOiU?upUhtME_cVUeywWrSeSvIwHqsEUvwaS`mv_kCEgDEEVOoyfSFYGXh[xe;wfsya?Hbcu_SiHUfrStqsgICUKmR;IEGGiEUxSSewkBRcic?f[GHs]WBCeFSXMec@qwQYiOCFi;bd_epghCcrSIbrUFfKXpOE>CdGUVH_ss=GaEF\\Mh_uDJcXeWGSkIA=T`[uhOiKOy;Ido_sBQgPGbiMxZIx[=RNQHCUwlIhVAs>Mxv=t;Iekec[iToeB]YSVsI]UGkMgC=xM_cv]rCkGlOyE=wVsymoRPERGUWoKs>?dNGcqOvL=DcgUUid=SdBYtacBcyT;sC??sXsBFEIPKdwUibUUuowtCxLERxGUPOc=eeWWDJ_tBIFj[RMWXoaIniFDYyvIfFYH;EifaWAAdkQgSuIoYHS?s\\aYnkYcCRXAy;=urSsUEGXovmkdU?bIkuvIhf;hHKRmsIqkGkCIEGSQiUy?r[chy]DW?UJweo_HI;I[iRPuYCce]yIQGSR=SFcY@IHNabEyhT;H\\gC[iiEubXIY[?FhkfAaRyccQ;D<MBLksUGvM]FOSWZaFnmUVOB]Mh`gu]ew:CSX[VU[d^iWCITMkingVmcY;EuIkFZgetaSlkeD_SlUd?SU[Wh`_IHkuNaIBEY@KhQ[IbSfl_CpgV]IBgcf:CrOWWliVPSDMuEkwBYQbgKxGiWfcdg_cCoXDyFoAF<CYd_fZSUKOXmUErmvpWgaQIeWGyMiuOfheFY[UWgdGwe[;X@Yh<owskTwUgjYdvEhnTP`LJatUmyo]xlkUpgPSHmSOiSXtM?HsHhWglnu=ypMosmPWQtXmlLDR^erappAPq@Twu\\mf<ytMo_tNQDmwuUBal[TKM]UZ\\VsUPg\\OhXU]iw>lT>TtolYUeM\\`q:iNFQkMeuB<Y^yq[TqwLxyYk^mPDhUTEL[mxdYTrUwHYpp`R]tsyhm<\\rdhN\\]VGejEyTBLlXhUidSklVcImkuJA\\OFAJxXTJ\\oRpUr\\qnEUf<POaocioXxYUTRxhmKHnoUuBavvxt]@ordyqIl`tycEyg=St<V;LY`DoDElChWYdkpIkSMophnhqkeMW<QX^dogEmM<kxAYM=mpPKmTTMmXeQLnuK?HMeIU``TqMSdeNqmxHeLK=OUpx^@kiYp`xXVdoU@L=PprAPIuR[Qp@YlvPWwQToMpG`jOXyFhxAETieRADKgioVPOyXUlXT:Iwc<NgeMNup\\XWrdQFPQvlP=Toseo>qXbiWO\\yE=PUiPAASgLtxXLG=STASAxj=@WixwX`XOAtHloIeoHiLvyuouMtLtTyJsAxBXr@TqWXOsEKopuAEU<uyO\\LTyPAXm=tOUQneaND]KOYyLyXbtxuhmcYrXMkh\\ylLo_eq`tSeAOH]lqUwiPnkPwlHPgHrehY^pKhPwGPJ;<O<`qU=tMxUUEPW@RdITfYjjaowTqMQjXHJS\\M<EvappT@mWMJ@iOVhyLQKq]T=Eyc=UhqNa]PJ\\X\\Lu[DsQ@O[XRw<Rb`P`tSuejceYX@UN=rFexuHmDmk]XRLaYElRmIP]Pech`rxma?araaCxvWQ[\\aZ`yiFAj?gvVVd^@mGy[hhjxQvjIwMVwPGyXW_EpjDNnsy^EhvE_d:PnkOaDA^CnxEAoCh_ewc;pb[I[ZwcU?kpGwxvcVV\\OWaYGZWqbGG^jVkAQ]mXckfwTVfovZVnZLwfoIeS>e@HtcvsgPn<YqDOxcqbdNmPxtqwhsfag>myOedhqCFkNWqspy]@_VQrIIu]ncLIb>_xdQ^[yw^`^YqbSxeyga>OkV@fpVfeNhmxeSwn^?_GOklf`QqgK_yK?yj@pxvwbHtI`yYai?HvJ^wvQvYngAVo=XhwcReBIMflKTU_b`qrFQC<UGRWY=kVWAiv]X<CSyMycyweoE>?ttksVgBTmtGIXvKDT;D`atpaGQEVA=efoH@]TgswsCfWGEbCCLIYtSwG;tRaC?]hi[TfwSPUcSQYZCuloE[KTnOSTuDPqfpQU_Yx[?UZ=b`yCuETUectcrsaWIGhPUVdCXo[Dn;GTof=AVBcYRGgaaYbsvt=UBuVIOeZKgGmhHQr]]umsifyTPWtneyZKydmHjoWRAsSQHewDS=Hj]C>qdH[XHIgkwTGuvI_sgYDgabSsiLYrb]Ic[uZUuCeGN]InyyjiVnMuJibq]E>=sH[thQDXgT\\qhNwTVmGdoSiKsD]DD]UOksO=fX;XvIdbUwRiisCEv?tEAS?eH[EHiOy[mcE?hY;ewKCr[x;ECpUEaItRMUeMI@wF=GuqIdriXmAiHouB]UEkvboD`]bDeu^UHOsxwKSogVE_GNQbBAduMYQ;Y_]XbqBe[FFYGF=tXgxryYpAFDoidIRHgUf?uXGg]WguGig]URQrp;u=MHYIXxcIamsqEl<uR<PMwtwNMqNYMB?\\aIiqvboxhknwDOv]^r:a\\[WhExsn_cdQo@Ng]orLPnCptE?wJqi:ad`?gjX\\Bol:@dJis[vel^pK>]TpcIHhoSZoXJOhw[WgsesuBfEg]=uuUY=qXZWVYMSZECHWHqeX<Su^EuvYX;AFQQC]]Fl]SNqIO=ILQwhIwZoeqEoOqVY@TTprWANqYsuxNA@WjlpuaXytmXMRkdpI]K\\LT@=Pd\\SxHJSXNhulFYQmtwJhWI<QsuRUpwm\\rQDLyuMgMv>@pS@pftRiUniTV:uRRil<lRY<wltSViLhHKD@vViS`DOfaTvAsyMuKmQUhvqlQuLW@qlr`RddRKIm^QYAaXxdP\\TuVlktMYmyPA`xRivRUoLxKmANalL`qV`eTDIO;MY\\HoQiYnMkHLNqhylUJ\\tS^uKJIMKAY[qufMrxAXfxJyXxe`RPqxOiorlJW]XEHXw\\lJqr=XwN<T>`nFPklHv^LTd]kviu:YwlhWkTyDpLSUVUqQCAuTTliPopuoTHNSQyRts>IqKYKhTNQMseAjoalrQvbIslMp=\\ojLUMDuDQymaoiQulmPMELwhpuplnIvypP`XlCDM>LY@`rdqtoyn@MLFTUUPo\\UWR\\WMetOAoEewLIUctRw@t]ERG@XtqKuHQWqjWLqZ`LTUOTusmHPcYk?DN=uT\\aXSeLNuKrttf@kIunUTXCMtYyRUQplXw`Xv=iXppuLmRUqwTMm[]qxhLElt>lNi@qQ=Q_lRL<NgerhhXwAryAL=iw]IxYTUyhj;poqXPmUgHG\\ganfWfF>hrAwtwy[Ys<VuGXhSGxePjM^exn\\vabHNjTffFYwDNre@qoheHWmoW`]P\\gfq]Ikxx\\?vknnc\\giupovIhMaZOIkjIdVqtv?efnhe`i=OixVueVopxjJOuNY`[W\\jX\\SNkeqrQ_pUghjNiNQtpG\\CIe_IabYs@wwBw\\L`xO?r`qZi?c@WsW`^@fjogeppjkIpnXkKPndGadGidocE>m?Fjf_bYf\\\\?p]HieNqWggeIuCAnhiZwaepYnkgeFyjvOhu_[GQkpioSNa?ndiprUFjcV\\pQngw]R?]WFeWx`>i_H@tAwdbny<x__O`FyggqujAtJhaiAnSAs=xwtp^aYnloln?eYQtA^mJvwD?k\\Ql]xqMPc`_sjV]gvreOsIOkpP^Vy^[Vw`O[gwmLqi]NmZ@hBAriP]O>[@HdmYZyir[Nn<YpeNfonso^]dnfIYuXwkEAcUyn^A`]VeyYulPogAn;?\\K?mt^gp^jXGxf>ysfZsgu=`seb_aIESSJcWewtmCrECfgERaqENChB;f^IvxYL=PS]=yKXmGeMYLmrTSBpL_`UAlmXmXlUTXEn^EsSmmfyREXsDEwelvQqlQaX@@tj<pkTYkDSNqxPQjlusiTJELXQ\\Rw`sPaSUYJwPjdes_QsK`j@Ij_DuFmJmPLmllh<SSPKV<W[eOaaTN@wLltv=qd@OOHrc<K>huhPP=ApSURP]mbIVSurlDLqpKuaVliV>IoOxJxLyGXOhqt=QPBQVItRjdV?]PFPPCyvs]YB]RXAsPLysQT^MuLUODMueDP=UPpHsFUx:XJ`hNlEYKykqQLQHSEur^aX_XJH]UyxtgMRCXtjuo?EQWML[aRSikidoeLsUduWEMthYZyQ[qwxHT[tOu<VGxqb`qp<OQAWOeYIIw^Tv`HrNyP;EKhDLiTqcXLq<NXejsEKseT;MYA<osmuf@U@txUMJYaMFuvVajUelv]xX`ncuThTxB\\wxtvCiu@HsQUQ:msJyUVXLOeUALmdaY]TMouqEExW`xK=QQLyGAyiHP\\xOf]tG>cJw`gxw^f]mIdJwgXiybX]_^\\]x]wXoovfJ`vgQklWrhq`sxqThd_AuXHotauxqvVPs>fXQEG_YGyujGWqaCOyE>WX[wuEwysMHsACawYfsIiqvWiWpWGoGYmqwAeh;_XqGSy[YQUW<kFaUGmuhqeYE;xdwbDUDdWV<OYjmwc]rL?TpuwF_snWumiiaAInyB[aUbyx\\yy`cSLmHxsInwYLwf=ob_ktxgUJWTB]TtIvKkDDMICMVZCH<WWF;vXeuOGe^QeLwik]HkCfrUXu_DgoC[OIyuh_Iyb[eEhqryQ?MwTexIuNbumv<sOiwy]uO>ie?oNXpnFb]iykyv@pnM?^bQbcOp]@pM_wOIZ\\i]tVpGIu=PdbHfMxcxXat?aWPZsww>xaDvv<wqQvyk^piAr_@fdYyfoxsactW_uvgBPmqvmK_ZMArZWZyAvCPmuYd\\AbZp]ZNgXwryXaxva>wfYpcZgem>uxiu[GiYnuwQu<aiJns?\\UNpqHgjfwhq[bahb@xCGbHVkk_nTPeiobfycUf`XnaxidlwiTHjmheF?sw>qWXxTWygQbupZtYpgqpkwwfWvcHZcAw[iuMiyb^mEfyh_yyXsIIosXdJfxvq]>yaR_ZVxy\\bS?EbAws]w]wvcOFoMhwSURagyCYdiTwABuAEGWFuSIGoEkKYIGFYUY]uw`uwXoGuAFVWkGwqyfb@qrrifj?sYpu=@_]on=g[Q@ltQbQNZDf\\FWe\\yquw[<pu^>lvQx\\Yw<w\\<VxRPn=yxiN[CNgB^irOpwGnEfyyWntqw:gwEfZSpi_G\\<?`QnxV?wygm<NZ^qyaGpxxiMpk_OhqYrWx\\t@t?@vAA\\eq_rQqv>uy@tya`Wyy:xvmysXwyYf[MWxoWmIgvoE:;B:MTKWDKWgJ;eZ1: