Changeset 1469

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Timestamp:
12/02/13 13:54:59 (10 years ago)
Author:
smidl
Message:

upravy rand

Files:
5 modified

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Unmodified
Added
Removed

  • applications/pmsm/simulator_zdenek/ekf_example/CMakeLists.txt

    r1468 r1469  
    88 
    99include_directories(../../bdm) 
    10 add_library (ekf_obj ekf_obj.cpp fixed.cpp matrix.cpp matrix_vs.cpp mpf_double.cpp ekf_mm.cpp qmath.cpp) 
     10add_library (ekf_obj ekf_obj.cpp fixed.cpp matrix.cpp matrix_vs.cpp mpf_double.cpp ekf_mm.cpp qmath.cpp RArandn.cpp) 

  • applications/pmsm/simulator_zdenek/ekf_example/RArandn.h

    r1468 r1469  
    1 #define MAXrandn 997 
    2 static const floatx RArandn[MAXrandn]={ 
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-6.6462217e-01, 5.5817658e-01, -1.1885421e+00, -7.7548150e-01, 2.7104207e-01, 1.5349757e+00, -1.0522827e+00, 6.2555857e-01, -7.9762607e-01, -3.1352199e-01, -6.0220975e-01, 1.2590603e+00, 8.5848425e-01, -2.1052919e+00, -3.6093718e-01, 5.5355672e-01, -1.5563838e+00, -2.0666572e-01, -4.2556755e-01, 4.9377825e-01, -8.7090761e-01, 7.9827658e-02, -5.2161914e-01, -1.4138609e+00, -3.8429318e-01, -4.5792222e-01, -2.9147069e-01, -3.0122401e-01, -1.5885938e+00, 1.0942867e+00, 1.3241669e+00, -1.2647963e-01, -7.3716382e-01, 2.1371884e-01, -4.0052913e-01, 6.4937747e-02};//, -1.7579956e+00, 1.6867480e+00, 3.2740021e-01}; 
    4 static const floatx *randn_i=&RArandn[0]; 
    5 const static floatx *randn_last=&RArandn[MAXrandn-1]; 
     1#define MAXrandn 997    //1000 
     2static const float RArandn[MAXrandn]={-0.2895823932,-1.6226019832,0.1683147268,0.3306588368,-1.1034889832,1.2338979168,1.2321466168,0.0053491398,0.3702747768,0.2176215568,-0.1437261632,0.7687729668,-0.5453341232,2.2261682168,-0.0934134632,0.1569137268,1.1097506168,0.1022638778,-0.0526659882,-0.7893670432,0.3373932368,-1.2931994832,0.7573069668,1.6665445168,-0.6487932832,0.9009790868,1.2969838168,-1.5507471832,-1.3979819832,0.6141300368,-0.3569031632,0.7329797968,0.8586047068,0.7548907368,1.3332322168,0.7115829268,1.2338205168,-1.1594746832,0.0231928588,-0.1137348832,-1.5611031832,0.3002866468,-1.0134904832,1.4581239168,-0.7621079832,0.5717254268,0.2623030868,-0.8789192032,-2.1276920832,-0.0162054082,-0.9676512832,0.6574454668,0.5507232068,1.7354123168,0.6342650068,-0.6006127832,0.4233196668,-0.9661330832,0.0234717468,-0.0052383722,0.0430256086,-0.2748770332,1.1379861168,-1.8310078832,0.4711656868,0.9386208868,0.7739397568,0.6208397668,0.0832964488,0.7200716068,0.6118826268,-0.2126630032,-0.3344865432,-0.2529046932,-1.4321520832,-0.1910216332,0.1614272568,0.3577914568,1.4864906168,-0.3079923232,0.6662162668,0.8420310368,0.9838723568,-0.9491093232,0.2550175668,0.2808644868,-0.9647809832,-0.6990623332,1.1252774168,-0.0885172832,0.4328629068,0.1309695238,-0.5924828132,-0.5165908832,0.4866359068,-0.9069213832,0.8241640368,0.6119430668,-0.7787318732,-0.2226244332,-1.1447945832,-2.1593382832,1.0293198068,-0.4756526532,0.3703499768,0.2770394268,0.0644485558,-0.9609620832,-0.9041636432,-0.3314467832,-1.1429037832,-1.0129204832,1.5154623168,0.0987262488,-1.1743350832,0.0017552828,-1.0853614832,-1.3062950832,-0.2181192032,0.9964478668,0.1716268468,0.6994499268,-1.1248369832,-0.4176227632,-0.2194575332,-1.1701696832,-1.2764545832,0.9741999268,0.0542273128,-0.6021634032,0.8487112068,0.2746084268,-0.9467772532,1.3825681168,0.3324844468,1.5218995168,1.1810104168,-0.6411561732,-1.2489535832,-0.0299438592,-0.2876164632,-0.8006452232,0.5407520768,1.5314729168,-0.5034934732,-0.8037757432,-0.2033541132,0.7060065668,-0.8112149532,-1.1583323832,-0.0768870132,-0.0223115982,0.5282779768,-0.5525084832,-0.1066853232,-0.3917695132,-0.0363478062,1.5781347168,-0.5635004432,-1.3043802832,0.5123655368,-0.8605845232,0.0788620558,-0.5845488032,0.5783803668,0.5958659368,-0.1607080632,-2.0113422832,0.1755431468,1.6359231168,1.0613942168,-1.5374200832,-0.0356795022,-0.6386744432,-0.9815706832,-1.1913710832,0.3317894368,-0.3863205832,0.0987836068,-0.3248911532,-0.4219909532,0.4139429968,0.7712653468,2.1551426168,-1.3143152832,-0.9796276832,1.0808166168,-0.3468171332,-1.3382831832,0.3585250468,1.5962250168,0.7508762968,2.0003672168,0.5475247668,1.9075114168,-0.2968293632,-1.0967969832,-0.1681410632,1.2332273168,-1.0732263832,0.6782565468,-0.5584297132,0.5941671268,-1.0568580832,0.1289730098,-1.9615808832,-0.4501055032,0.5050304268,-0.2780222732,1.2795381168,-0.5882972432,-2.2822286832,-1.1886540832,1.0986308168,-0.0702415732,0.4222060368,0.9871821468,-2.0774442832,-0.6016965032,-0.6613193132,-0.9751547832,-0.1390994532,1.5639956168,0.0045436528,1.2704304168,-0.6532223832,0.0505069033,-0.7399106232,0.6299209768,-0.2082249532,0.5231182368,0.7111374468,-0.0353387792,0.9321550368,2.3522699168,0.5676210968,0.0311950928,0.9561232368,0.0989230958,-1.0640874832,0.5284801268,0.0379773430,-0.2332354432,1.3194349168,1.9063830168,-0.4795768832,0.1464068668,-0.7646667132,0.7234209968,-2.3216073832,1.0330972868,0.2618815368,0.3046448768,1.2564269168,-0.2316845732,-0.0901520332,-1.2275177832,-1.6206240832,-0.6605718432,0.3238629068,-0.4982269132,-1.2905482832,1.1156687168,-0.6691030332,0.0316968558,0.0421653876,-0.2064538632,0.4395577368,-0.2210309332,-1.6210284832,-0.9859926832,0.2860771168,-1.2136076832,-0.3042007732,-0.8983897732,-1.1315778832,-0.9781592832,-0.3586843132,0.2166480868,-0.0731360732,1.1071015168,-0.2024038832,-1.4745566832,0.0527165759,0.1143552818,0.3595182268,0.5428080868,1.3210665168,-0.5048337332,0.3037908168,0.0298057448,-0.5372815832,2.1792908168,-0.2146347032,-1.3665460832,1.8130833168,0.3685283968,-1.0760571832,0.6633325568,1.3127642168,-0.8530600932,0.1781578568,-0.0960575932,-1.1204128832,1.2267019168,0.0275527548,0.5792011068,-0.6734462032,-0.6125769732,0.3573451768,0.1497964968,1.8911986168,-0.2321232632,2.2555365168,1.5515082168,-1.9020961832,-1.6375603832,-0.5305517132,-0.1428341132,0.0519165325,0.8799323068,-0.6792882532,-0.6785080632,-0.1581985832,0.0225182558,0.3218724168,1.1012772168,0.6646556968,-1.7076328832,0.7403299668,0.8544682768,0.6793273668,1.3530627168,0.3700799368,-0.6300107432,-0.1063450832,-2.4060353832,0.5162680268,0.1599280768,-0.5481214232,-0.6117252632,-1.0376794832,-0.0047484482,0.4223269568,-0.2873786332,-0.4569158332,0.0070038088,-0.1317779132,-0.9142826632,1.3355303168,0.4838920568,1.3239233168,-0.4547473932,-1.0757341832,0.8506320368,0.0841819958,-0.7132261932,-0.0461467312,-1.9658678832,1.1269004168,-0.9382081432,-0.6455062232,1.3824619168,-0.8662607432,-0.3698753132,-0.4631807732,1.6627302168,0.1238831278,-1.03807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     3 
     4static const float *randn_i=&RArandn[0]; 
     5const static float *randn_last=&RArandn[MAXrandn-1]; 
    66 
    77 
    88#define MAXrand 100 
    9 static const floatx RArandu[MAXrand]={0.8147, 0.9058, 0.1270, 0.9134, 0.6324, 0.0975, 0.2785, 0.5469, 0.9575, 0.9649, 
     9static const float RArandu[MAXrand]={0.8147, 0.9058, 0.1270, 0.9134, 0.6324, 0.0975, 0.2785, 0.5469, 0.9575, 0.9649, 
    1010                                      0.1576, 0.9706, 0.9572, 0.4854, 0.8003, 0.1419, 0.4218, 0.9157, 0.7922, 0.9595, 0.6557, 0.0357, 
    1111                                      0.8491, 0.9340, 0.6787, 0.7577, 0.7431, 0.3922, 0.6555, 0.1712, 0.7060, 0.0318, 0.2769, 0.0462, 
    … …  
    1717                                      0.9340, 0.1299, 0.5688, 0.4694, 0.0119, 0.3371 
    1818                                     }; 
    19 static const floatx *randu_i=&RArandu[0]; 
    20 const static floatx *randu_last=&RArandu[MAXrand-1]; 
     19static const float *randu_i=&RArandu[0]; 
     20const static float *randu_last=&RArandu[MAXrand-1]; 
     21 
     22float qrandu(); 
     23float qrandn(); 
     24void rng_init(); 
     25 

  • applications/pmsm/simulator_zdenek/ekf_example/mpf_double.cpp

    r1468 r1469  
    1 #include "math.h" 
     1#include <math.h> 
    22#include <stdio.h> 
    33#include "fast_exp.h" 
    44#include "fastexp.h" 
    55#include "mpf_double.h" 
     6 
     7//Parameters of PMSM 
     8#define Lsd (3.465e-3*0.9)      //      *0.9)//H        ... zajimave - chodi to i pri Lsd = Lsq 
     9#define Lsq 3.465e-3//H 
     10//#define Ls  3.465e-3//H 
     11#define Rs 0.28//Ohm 
     12#define Fpm (0.1989*1.1)        //*1.15)        //V.s           // !!xx!! 
     13#define pp 4 
     14#define Ism 35.//40. 
    615 
    716static floatx om[N]; 
    … …  
    1423static floatx w[N]; 
    1524static floatx lwi[N]; 
     25static int i_best; 
    1626 
    1727static floatx qth; 
    … …  
    2030static floatx rinv; 
    2131 
    22 #if 1 
    23 #include "RArandn.h" 
    24 floatx qrandu() { 
    25         if ( randu_i==randu_last ) 
    26                 randu_i =&RArandu[0]; 
    27         else 
    28                 randu_i++; 
    29  
    30         return *randu_i; 
    31 } 
    32  
    33 floatx qrandn() { 
    34         if ( randn_i==randn_last ) 
    35                 randn_i =&RArandn[0]; 
    36         else 
    37                 randn_i++; 
    38  
    39         return *randn_i; 
    40 } 
    41  
    42 void rng_init() 
    43 { 
    44 randu_i=&RArandu[0]; 
    45 randn_i=&RArandn[0]; 
    46 } 
    47  
    48 #else 
    49  
    50 #include "rnor.h" 
    51 floatx qrandu() { 
    52         return UNI; 
    53 } 
    54  
    55 floatx qrandn() { 
    56         return PolarRNG(); 
    57 } 
    58  
    59 void rng_init() 
    60 { 
    61 zigset(86947731 ); 
    62 } 
    63  
    64  
    65 #endif 
     32// staticke promenne pro model 
     33static float _ad; 
     34static float _aq; 
     35static float _b; 
     36static float _cd; 
     37static float _cq; 
     38static float _dd; 
     39static float _dq; 
    6640 
    6741 
    … …  
    9569 
    9670        while ( i_from<N ) { 
     71                w[i_from]=1.0/N; 
    9772                while ( N_babies[i_from]>1 ) { // 1 baby stays where it is 
    9873                         
    … …  
    139114        floatx ydC; 
    140115 
    141         floatx maxlwi, sumlwi; 
     116        floatx maxlwi, sumlwi,sumw2,neff_th; 
    142117        int i; 
    143118         
    … …  
    145120        floatx *w_i, *lw_i; 
    146121        floatx *sth_i, *cth_i; 
     122        floatx sumw_inv; 
    147123// implementation starts here 
    148124        th_i = &th[0]; 
    … …  
    155131        for ( i=0; i<N; i++ ) { 
    156132 
     133          /////////////////////////// HACK!!!!!!!!!!! 
     134          //om[i] = 0; 
     135          /////////////////////////// HACK!!!!!!!!!!! 
     136           
    157137                // time update 
    158138                *Pt_i = 1.0*1.0*(*Pt_i)+qom; // Pt is now predictive variance 
    159                 *th_i += *om_i*_dt+qth *  qrandn() ; 
     139                floatx ff =qrandn() ; 
     140//              printf("%f\n", ff); 
     141                *th_i += *om_i*_dt+qth * ff;  
    160142                while ( *th_i>M_PI ) *th_i -=2*M_PI; 
    161143                while ( *th_i<-M_PI ) *th_i+=2*M_PI; 
    … …  
    170152                usq = -(*sth_i)*usa+(*cth_i)*usb; 
    171153 
    172                 Cd = isq*_dt; 
    173                 Cq = -_b - isd* _dt; 
     154                Cd = isq*_dd; 
     155                Cq = -_b - isd* _dq; 
    174156 
    175157                difid=isd- _ad  * (*isdm_i) - _cd *(usd); 
    … …  
    211193 
    212194        } 
    213 /*      maxlwi=-1e10; 
     195        maxlwi=-1e10; 
    214196        lw_i = &lwi[0]; 
     197        i_best = 0; 
    215198        for ( i=0;i<N;i++,lw_i++ ) { 
    216                 if ( *lw_i>maxlwi ) maxlwi=*lw_i; 
    217         }*/ 
     199                if ( *lw_i>maxlwi ) {maxlwi=*lw_i;i_best=i;} 
     200        } 
     201/*!!*/ 
    218202        lw_i = &lwi[0]; 
    219203        w_i = &w[0]; 
    220204        for ( i=0;i<N;i++,lw_i++,w_i++ ) { 
    221 //              *lw_i-=maxlwi; 
    222                 (*w_i)=fasterexp ( *lw_i ); // always resample -> wmin=1/n; 
     205                *lw_i-=maxlwi; 
     206                (*w_i)*=fasterexp ( *lw_i ); // do not always resample  
    223207        } 
    224208 
    225209        sumlwi=0.0; 
     210        sumw2=0.0; 
    226211        w_i = &w[0]; 
    227         for ( i=0;i<N;i++,w_i++ ) sumlwi+=*w_i; 
     212        for ( i=0;i<N;i++,w_i++ ) {sumlwi+=*w_i;sumw2+=*w_i**w_i;} 
     213        sumw_inv = 1.0/sumlwi; // multiplication is faster than division! 
    228214        w_i = &w[0]; 
    229         for ( i=0;i<N;i++,w_i++ ) *w_i/=sumlwi; 
    230          
    231         // NEFF 
    232         floatx sw2=0.0; 
    233         for ( i=0;i<N;i++ ) sw2+=w[i]*w[i]; 
    234         printf("%f\n",1./sw2*N); 
    235          
    236         resample(); 
     215        for ( i=0;i<N;i++,w_i++ ) *w_i*=sumw_inv; 
     216         
     217        neff_th = 0.8*N*sumw2*sumw_inv*sumw_inv; 
     218        if (1<neff_th) 
     219          resample(); 
    237220} 
    238221 
    … …  
    240223void mpf_init(floatx qom0, floatx qth0, floatx r0) { 
    241224         
    242         rng_init(); 
     225//      rng_init(); 
    243226         
    244227        int i; 
    … …  
    259242                w[i]=1.0/N; 
    260243        } 
     244 
     245   // inicializace promennych 
     246   #if (VYBER_MODELU==0) 
     247    // plny model s respektovanim Lsd/Lsq 
     248    _ad = (1.-Rs*_dt/(Lsd)); 
     249    _aq = (1.-Rs*_dt/(Lsq)); 
     250    _b = Fpm/Lsq*_dt; 
     251    _cd = _dt/(Lsd); 
     252    _cq = _dt/(Lsq); 
     253    _dd = _dt*Lsq/Lsd; 
     254    _dq = _dt*Lsd/Lsq; /**/ 
     255  #else 
     256  // parametry modelu identifikovane z namerenych dat z prototypu v EL204 
     257    _ad = 0.9880; 
     258    _aq = 0.9817; 
     259    _b = 0.0076; 
     260    _cd = 0.0289; 
     261    _cq = 0.0292; 
     262    _dd = _dt; 
     263    _dq = _dt; 
     264  #endif    
    261265 
    262266} 
    … …  
    277281 
    278282} 
     283 
     284void mpf_best( floatx *Ecosth, floatx *Esinth, floatx *Eome )  
     285{ 
     286        *Ecosth=cth[i_best]; 
     287        *Esinth=sth[i_best]; 
     288        *Eome=om[i_best]; 
     289} 
     290 
     291 
    279292void mpf_th(floatx th1[N]){ 
    280293        int i; 
    … …  
    330343                usq = -sth*usa+cth*usb; 
    331344 
    332                 Cd = isq*_dt; 
    333                 Cq = -_b - isd* _dt; 
     345                Cd = isq*_dd; 
     346                Cq = -_b - isd* _dq; 
    334347 
    335348                difid=isd- _ad  *isdm - _cd *usd; 

  • applications/pmsm/simulator_zdenek/ekf_example/mpf_double.h

    r1468 r1469  
    11// Bayesian Filtering for PMSM using Marginalized Particle Filter  
    22 
    3 #include "parametry_motoru.h" 
    4 #define Fpm 0.1989 
    5 //#define pi 3.141596 
     3//#define Fpm 0.1989 
     4#ifndef M_PI 
     5#define M_PI 3.141596 
     6#endif 
    67 
    78#define N 4 
    89 
    9 #define Lsd Ls*0.9 
    10 #define Lsq Ls 
    11 #define _dt 0.000125 
    12 #define _ad (1.-Rs*_dt/(Lsd)) 
    13 #define _aq (1.-Rs*_dt/(Lsq)) 
    14 #define _b  Fpm/Lsq*_dt 
    15 #define _cd  _dt/(Lsd) 
    16 #define _cq  _dt/(Lsq) 
     10//#define Lsd Ls*0.9 
     11//#define Lsq Ls 
    1712 
     13#define VYBER_MODELU    0       // 0 = plny model s respektovanim Lsd/Lsq ... implicitne 
     14                                                        // 1 = parametry modelu z identifikace 
     15#define _dt     0.000125 
    1816 
    19 # define floatx float 
     17#define floatx float 
    2018 
    2119floatx qrandn(); 
    … …  
    2826void mpf_th(floatx th1[N]); 
    2927void mpf_om(floatx om1[N]); 
     28void mpf_best( floatx *Ecosth, floatx *Esinth, floatx *Eome); 
    3029 
    3130floatx kalman_om ( const floatx isa, const floatx isb , const floatx usa, const floatx usb, const floatx th ); 

  • library/bdm/design/arx2statespace_notes.lyx

    r723 r1469  
    1 #LyX 1.6.4 created this file. For more info see http://www.lyx.org/ 
    2 \lyxformat 345 
     1#LyX 2.0 created this file. For more info see http://www.lyx.org/ 
     2\lyxformat 413 
    33\begin_document 
    44\begin_header 
    55\textclass article 
    66\use_default_options true 
     7\maintain_unincluded_children false 
    78\language english 
     9\language_package default 
    810\inputencoding auto 
     11\fontencoding global 
    912\font_roman default 
    1013\font_sans default 
    1114\font_typewriter default 
    1215\font_default_family default 
     16\use_non_tex_fonts false 
    1317\font_sc false 
    1418\font_osf false 
    … …  
    1721 
    1822\graphics default 
     23\default_output_format default 
     24\output_sync 0 
     25\bibtex_command default 
     26\index_command default 
    1927\paperfontsize default 
    2028\use_hyperref false 
    … …  
    2331\use_amsmath 1 
    2432\use_esint 1 
     33\use_mhchem 1 
     34\use_mathdots 1 
    2535\cite_engine basic 
    2636\use_bibtopic false 
     37\use_indices false 
    2738\paperorientation portrait 
     39\suppress_date false 
     40\use_refstyle 0 
     41\index Index 
     42\shortcut idx 
     43\color #008000 
     44\end_index 
    2845\secnumdepth 3 
    2946\tocdepth 3 
    3047\paragraph_separation indent 
    31 \defskip medskip 
     48\paragraph_indentation default 
    3249\quotes_language english 
    3350\papercolumns 1 
    … …  
    3653\tracking_changes false 
    3754\output_changes false 
    38 \author ""  
     55\html_math_output 0 
     56\html_css_as_file 0 
     57\html_be_strict false 
    3958\end_header 
    4059 
    … …  
    4362\begin_layout Standard 
    4463System 
    45 \begin_inset Formula \[ 
    46 y_{t}=y_{t-1}+bu_{t}+e_{t}\] 
     64\begin_inset Formula  
     65\[ 
     66y_{t}=y_{t-1}+bu_{t}+e_{t} 
     67\] 
    4768 
    4869\end_inset 
    … …  
    6384 
    6485 the same system is:  
    65 \begin_inset Formula \[ 
    66 y_{t}=y_{t-1}+bu_{t-1}+e_{t}\] 
     86\begin_inset Formula  
     87\[ 
     88y_{t}=y_{t-1}+bu_{t-1}+e_{t} 
     89\] 
    6790 
    6891\end_inset 
    6992 
    7093and can be transformed as 
    71 \begin_inset Formula \begin{eqnarray*} 
     94\begin_inset Formula  
     95\begin{eqnarray*} 
    7296x_{t+1} & = & x_{t}+bu_{t}+e_{t}\\ 
    73 y_{t} & = & x_{t}\end{eqnarray*} 
     97y_{t} & = & x_{t} 
     98\end{eqnarray*} 
    7499 
    75100\end_inset 
    … …  
    93118\begin_layout Standard 
    94119System 
    95 \begin_inset Formula \[ 
    96 y_{t}=y_{t-1}+bu_{t}+cu_{t-1}+e_{t}\] 
     120\begin_inset Formula  
     121\[ 
     122y_{t}=y_{t-1}+bu_{t}+cu_{t-1}+e_{t} 
     123\] 
    97124 
    98125\end_inset 
    99126 
    100127is 
    101 \begin_inset Formula \begin{eqnarray*} 
     128\begin_inset Formula  
     129\begin{eqnarray*} 
    102130\left[\begin{array}{c} 
    103131x_{t+1}\\ 
    104 u_{t}\end{array}\right] & =\left[\begin{array}{cc} 
     132u_{t} 
     133\end{array}\right] & =\left[\begin{array}{cc} 
    1051341 & c\\ 
    106 0 & 0\end{array}\right] & \left[\begin{array}{c} 
    107 x_{t}\\ 
    108 u_{t-1}\end{array}\right]+\left[\begin{array}{c} 
     1350 & 0 
     136\end{array}\right] & \left[\begin{array}{c} 
     137x_{t}\\ 
     138u_{t-1} 
     139\end{array}\right]+\left[\begin{array}{c} 
    109140b\\ 
    110 1\end{array}\right]u_{t}+e_{t}\\ 
    111 y_{t} & = & x_{t}\end{eqnarray*} 
    112  
    113 \end_inset 
    114  
     1411 
     142\end{array}\right]u_{t}+e_{t}\\ 
     143y_{t} & = & x_{t} 
     144\end{eqnarray*} 
     145 
     146\end_inset 
     147 
     148 
     149\end_layout 
     150 
     151\begin_layout Standard 
     152System #2: 
     153\begin_inset Formula  
     154\[ 
     155y_{t}=ay_{t-1}+a_{2}y_{t-2}+bu_{t-1}+cu_{t-2}+e_{t} 
     156\] 
     157 
     158\end_inset 
     159 
     160is 
     161\begin_inset Formula  
     162\begin{eqnarray*} 
     163\left[\begin{array}{c} 
     164x_{t+1}\\ 
     165x_{t}\\ 
     166u_{t} 
     167\end{array}\right] & =\left[\begin{array}{ccc} 
     168a_{1} & a_{2} & c\\ 
     1691 & 0 & 0\\ 
     1700 & 0 & 0 
     171\end{array}\right] & \left[\begin{array}{c} 
     172x_{t}\\ 
     173x_{t-1}\\ 
     174u_{t-1} 
     175\end{array}\right]+\left[\begin{array}{c} 
     176b\\ 
     177\\ 
     1781 
     179\end{array}\right]u_{t}+e_{t}\\ 
     180y_{t} & = & x_{t} 
     181\end{eqnarray*} 
     182 
     183\end_inset 
     184 
     185 
     186\end_layout 
     187 
     188\begin_layout Standard 
    115189Notation in BDM: 
    116190\end_layout 
    … …  
    126200\begin_layout Standard 
    127201System 
    128 \begin_inset Formula \[ 
    129 y_{t}=y_{t-1}+bu_{t-1}+g+e_{t}\] 
     202\begin_inset Formula  
     203\[ 
     204y_{t}=y_{t-1}+bu_{t-1}+g+e_{t} 
     205\] 
    130206 
    131207\end_inset 
    132208 
    133209must be modelled as  
    134 \begin_inset Formula \[ 
     210\begin_inset Formula  
     211\[ 
    135212\left[\begin{array}{c} 
    136213x_{t+1}\\ 
    137 one\end{array}\right]=\left[\begin{array}{cc} 
     214one 
     215\end{array}\right]=\left[\begin{array}{cc} 
    1382161 & c\\ 
    139 0 & 0\end{array}\right]\left[\begin{array}{c} 
    140 x_{t}\\ 
    141 one\end{array}\right]+\left[\begin{array}{c} 
     2170 & 0 
     218\end{array}\right]\left[\begin{array}{c} 
     219x_{t}\\ 
     220one 
     221\end{array}\right]+\left[\begin{array}{c} 
    142222b\\ 
    143 1\end{array}\right]u_{t}+e_{t}\] 
     2231 
     224\end{array}\right]u_{t}+e_{t} 
     225\] 
    144226 
    145227\end_inset