National Instruments 370760B-01 manual

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Table of contents for the manual

  • Page 1

    MA TRIXx TM Xmath TM X µ Manual MA TRIXx Xmath Basics The MA TRIXx products and related items ha ve[...]

  • Page 2

    Support Worldwide Technical Support and Product Info rmation ni.com National Instruments Corporate H[...]

  • Page 3

    Important Information Warranty The media on which you receive National In struments software are war[...]

  • Page 4

    Con ten ts 1 In tro duction 1 1 . 1 N o t a t i o n ....... ....... .......... ......... .... 1 1.2 [...]

  • Page 5

    iv CONTENTS 2.2.5 Obtaining Robust Control Mo dels for Ph ysica l Systems . . . . . . 28 2.3 H ∞ a[...]

  • Page 6

    CONTENTS v 2 . 6 M o d e l R e d u c t i o n .. ....... .......... ......... .... 6 4 2.6.1 T runcat[...]

  • Page 7

    vi CONTENTS 3 . 5 S y s t e m I n t e r c o n n e c t i o n ..... .......... ......... .... 9 1 3.6 [...]

  • Page 8

    CONTENTS vii 4.1.5 µ An alysis of the H ∞ C o n t r o l l e r ........ ......... .. 1 3 8 4.1.6 F[...]

  • Page 9

    viii CONTENTS A . 3 S y s t e m R e s p o n s e F u n c t i o n s .. ......... ....... .... 3 9 8 A [...]

  • Page 10

    Chapter 1 In tro duction X µ is a suite of Xmath functions for the mo deling , ana lysis and syn th[...]

  • Page 11

    2 CHAPTER 1. INTR ODUCTION Notatio n Meaning pdm Xmath parameter dependen t matrix data ob ject Dyna[...]

  • Page 12

    1.3. HO W TO A VOID RE ALL Y READING THIS MANUAL 3 1.3 Ho w to a v oid really reading this M an ual [...]

  • Page 13

    Chapter 2 Ov erview of the Underlying Theory 2.1 In tro duction The material cov ered here is taken [...]

  • Page 14

    6 CHAPTER 2. OVER VIE W OF THE UNDE RL YING THEO R Y several studies in volving pro cess control app[...]

  • Page 15

    2.1. INTRODUCTION 7 Figure 2.1: The generic rob ust cont rol model stru cture T race( M ) trace of M[...]

  • Page 16

    8 CHAPTER 2. OVER VIE W OF THE UNDE RL YING THEO R Y is interpreted to mean that the signa l y is th[...]

  • Page 17

    2.1. INTRODUCTION 9 Euclidean norm. Given, x =    x 1 . . . x n    , the E uclidea[...]

  • Page 18

    10 CHAPTER 2. OVER VIE W OF THE UNDE RL YING THEO R Y F or p ersistent signals, where the ab ov e no[...]

  • Page 19

    2.1. INTRODUCTION 11 signals. Strictly sp eaking, sig nals in H 2 or H ⊥ 2 are not defined on the[...]

  • Page 20

    12 CHAPTER 2. OVER VIE W OF THE UNDE RL YING THEO R Y where σ max denotes the maxim um sing ular v [...]

  • Page 21

    2.2. MOD ELING UNCER T AIN SYSTEM S 13 The set o f all s ystems with b ounded ∞ -nor m is denoted [...]

  • Page 22

    14 CHAPTER 2. OVER VIE W OF THE UNDE RL YING THEO R Y robust control model is therefo re a set descr[...]

  • Page 23

    2.2. MOD ELING UNCER T AIN SYSTEM S 15 as sp ecifying a maximum percen tage error b etw een P nom an[...]

  • Page 24

    16 CHAPTER 2. OVER VIE W OF THE UNDE RL YING THEO R Y Real 0 0.5 1 -0.5 1.5 Imaginary -0.5 0 0.5 -1 [...]

  • Page 25

    2.2. MOD ELING UNCER T AIN SYSTEM S 17 Figure 2.4: Unit y gain negativ e feedbac k for the example s[...]

  • Page 26

    18 CHAPTER 2. OVER VIE W OF THE UNDE RL YING THEO R Y Figure 2.5 : Generic LFT mo del s tructure inc[...]

  • Page 27

    2.2. MOD ELING UNCER T AIN SYSTEM S 19 in an LFT format. The open-lo op system is described by , y =[...]

  • Page 28

    20 CHAPTER 2. OVER VIE W OF THE UNDE RL YING THEO R Y The issue o f the inv er tibilit y o f ( I −[...]

  • Page 29

    2.2. MOD ELING UNCER T AIN SYSTEM S 21 Figure 2.6 : Example mo del: multiplicativ e output p er turb[...]

  • Page 30

    22 CHAPTER 2. OVER VIE W OF THE UNDE RL YING THEO R Y Robust control models a re therefor e set desc[...]

  • Page 31

    2.2. MOD ELING UNCER T AIN SYSTEM S 23 Pac k ar d [19] discuss the implications of this a ssumption [...]

  • Page 32

    24 CHAPTER 2. OVER VIE W OF THE UNDE RL YING THEO R Y systems. W e will now lo ok at other p ossible[...]

  • Page 33

    2.2. MOD ELING UNCER T AIN SYSTEM S 25 = Cz − 1 ( I − z − 1 A ) − 1 B + D = F u ( P ss ,z ?[...]

  • Page 34

    26 CHAPTER 2. OVER VIE W OF THE UNDE RL YING THEO R Y where the transport dela y , T d , and th e co[...]

  • Page 35

    2.2. MOD ELING UNCER T AIN SYSTEM S 27 Putting all the pieces together gives an engine mo del in the[...]

  • Page 36

    28 CHAPTER 2. OVER VIE W OF THE UNDE RL YING THEO R Y and ∆ ∈ B∆ , with the structure defined[...]

  • Page 37

    2.3. H ∞ AND H 2 DESI GN METHODOLOGIES 29 An area of work, kno wn as iden tificat ion in H ∞ , [...]

  • Page 38

    30 CHAPTER 2. OVER VIE W OF THE UNDE RL YING THEO R Y Figure 2.7: LFT confi guration for control le[...]

  • Page 39

    2.3. H ∞ AND H 2 DESI GN METHODOLOGIES 31 extend these approac hes t o the case where P ( s ) i s [...]

  • Page 40

    32 CHAPTER 2. OVER VIE W OF THE UNDE RL YING THEO R Y 2.3.2 Assumptions for the H ∞ Design Problem[...]

  • Page 41

    2.3. H ∞ AND H 2 DESI GN METHODOLOGIES 33 that the eff ect of all disturban ces, w , at ev ery fr[...]

  • Page 42

    34 CHAPTER 2. OVER VIE W OF THE UNDE RL YING THEO R Y W e hav e partitioned the matr ix in to tw o n[...]

  • Page 43

    2.3. H ∞ AND H 2 DESI GN METHODOLOGIES 35 Cho ose γ> 0 and fo rm the following Ha miltonian ma[...]

  • Page 44

    36 CHAPTER 2. OVER VIE W OF THE UNDE RL YING THEO R Y an y with a zero real part. In practice w e mu[...]

  • Page 45

    2.3. H ∞ AND H 2 DESI GN METHODOLOGIES 37 and the component s, C 1 x and D 12 u are orthogonal . D[...]

  • Page 46

    38 CHAPTER 2. OVER VIE W OF THE UNDE RL YING THEO R Y L ∞ = − Y ∞ C T 2 Z ∞ =( I − γ − [...]

  • Page 47

    2.3. H ∞ AND H 2 DESI GN METHODOLOGIES 39 a) Choose γ ≥ γ opt b) F orm H ∞ and J ∞ c) Chec[...]

  • Page 48

    40 CHAPTER 2. OVER VIE W OF THE UNDE RL YING THEO R Y eigen v alue of X ∞ (and Y ∞ ) is displaye[...]

  • Page 49

    2.3. H ∞ AND H 2 DESI GN METHODOLOGIES 41 framew ork. W e again assume the simplif ying assumption[...]

  • Page 50

    42 CHAPTER 2. OVER VIE W OF THE UNDE RL YING THEO R Y Figure 2.8: Closed loop system, G ( s ), for p[...]

  • Page 51

    2.4. µ ANAL YSIS 43 descriptions are considered, where B again denotes the unit ball. Power : BP = [...]

  • Page 52

    44 CHAPTER 2. OVER VIE W OF THE UNDE RL YING THEO R Y Figure 2.9: Perturb ed closed lo op s ystem fo[...]

  • Page 53

    2.4. µ ANAL YSIS 45 Consider the case where the model ha s only one full ∆ block ( m =1 a n d q =[...]

  • Page 54

    46 CHAPTER 2. OVER VIE W OF THE UNDE RL YING THEO R Y F u ( G ( s ) , ∆) stable fo r al l ∆ ∈ [...]

  • Page 55

    2.4. µ ANAL YSIS 47 if and on ly if k µ ( G ( s )) k ∞ < 1 , wher e µ is taken wi th r esp e[...]

  • Page 56

    48 CHAPTER 2. OVER VIE W OF THE UNDE RL YING THEO R Y F or the o ther extreme consider a single full[...]

  • Page 57

    2.4. µ ANAL YSIS 49 Actually , the lower bound is always equal to µ but the implied optimization h[...]

  • Page 58

    50 CHAPTER 2. OVER VIE W OF THE UNDE RL YING THEO R Y and tw o blo ck structures, ∆ 1 (compa tible[...]

  • Page 59

    2.4. µ ANAL YSIS 51 2.4.6 State-sp ace Robustness Analysis T ests W e will lo ok at some more adv a[...]

  • Page 60

    52 CHAPTER 2. OVER VIE W OF THE UNDE RL YING THEO R Y Therefore µ 1 ( A ) < 1 is equiv alen t to[...]

  • Page 61

    2.4. µ ANAL YSIS 53 Figure 2.10: P erturb ed system for st ate-space robust ness tests[...]

  • Page 62

    54 CHAPTER 2. OVER VIE W OF THE UNDE RL YING THEO R Y Note that the nominal system is given by , G n[...]

  • Page 63

    2.4. µ ANAL YSIS 55 iii ) There exists a c onstant β ∈ [0 , 1] s uch that for e ach fixe d ∆ [...]

  • Page 64

    56 CHAPTER 2. OVER VIE W OF THE UNDE RL YING THEO R Y i ) State-spa ce upp er bound: inf D s ∈D s [...]

  • Page 65

    2.4. µ ANAL YSIS 57 The gap betw een the state- space (or constant D ) upp er b o und and the frequ[...]

  • Page 66

    58 CHAPTER 2. OVER VIE W OF THE UNDE RL YING THEO R Y 2.4.7 Analysis with b oth Real and Complex Per[...]

  • Page 67

    2.5. µ SYNTHESIS AND D - K ITERA TION 59 Figure 2.11: The generic in terconnecti o n structu re for[...]

  • Page 68

    60 CHAPTER 2. OVER VIE W OF THE UNDE RL YING THEO R Y Recall that this is an upp er bo und for the ?[...]

  • Page 69

    2.5. µ SYNTHESIS AND D - K ITERA TION 61 dynamic system. This requires fitting an appro x imation [...]

  • Page 70

    62 CHAPTER 2. OVER VIE W OF THE UNDE RL YING THEO R Y Figure 2. 12: D - K iterat ion pro cedure: a) [...]

  • Page 71

    2.5. µ SYNTHESIS AND D - K ITERA TION 63 frequency we w ould hav e, D =      d 1 I 1[...]

  • Page 72

    64 CHAPTER 2. OVER VIE W OF THE UNDE RL YING THEO R Y Sev eral asp ects of t his pro cedure are wort[...]

  • Page 73

    2.6. MODEL REDUCTION 65 2.6.1 T runcation and Residualization The simplest fo rm of mo del r eductio[...]

  • Page 74

    66 CHAPTER 2. OVER VIE W OF THE UNDE RL YING THEO R Y The controllabilit y grammian, Y is defined a[...]

  • Page 75

    2.6. MODEL REDUCTION 67 W e will now lo ok at a particular choice of transfor mation. F or a minimal[...]

  • Page 76

    68 CHAPTER 2. OVER VIE W OF THE UNDE RL YING THEO R Y and Glov er [79] independently obtained the fo[...]

  • Page 77

    2.6. MODEL REDUCTION 69 Consider the problem of finding the stable, order k realization w hich mini[...]

  • Page 78

    Chapter 3 F unctional Description of X µ 3.1 In tro duction This chapter describes the X µ functio[...]

  • Page 79

    72 CHAPTER 3. FUNCTIONAL DESCRIPTION OF X µ 3.2.1 D ynamic System s Xmath has a dynamic system data[...]

  • Page 80

    3.2. D A T A OBJECTS 73 As above, these p olynomials can be sp ecified b y their ro ots or their co[...]

  • Page 81

    74 CHAPTER 3. FUNCTIONAL DESCRIPTION OF X µ Because the dynamic system is a built-in data ob ject, [...]

  • Page 82

    3.2. D A T A OBJECTS 75 App ending and Merging Data Time functions, for cre ating simulation inputs [...]

  • Page 83

    76 CHAPTER 3. FUNCTIONAL DESCRIPTION OF X µ Data ca n also b e extracted b y indep endent v ariable[...]

  • Page 84

    3.2. D A T A OBJECTS 77 # index of the pdm the value 100. idxlst = indexlist([4,1,2] ) pdm1(idxlst) [...]

  • Page 85

    78 CHAPTER 3. FUNCTIONAL DESCRIPTION OF X µ # Select columns 1, 3 ,4&5a n d rows 2 & 7 from[...]

  • Page 86

    3.2. D A T A OBJECTS 79 and B . Augmentation for pdm s simp ly p erforms the augmen tation at eac h [...]

  • Page 87

    80 CHAPTER 3. FUNCTIONAL DESCRIPTION OF X µ poles, or a zero D term. Generating rando m systems is [...]

  • Page 88

    3.3. MA TRIX INFORMA TION, DISPLA Y AND PLOTTING 81 # N is the decimation ratio. smallpdm = bigpdm([[...]

  • Page 89

    82 CHAPTER 3. FUNCTIONAL DESCRIPTION OF X µ 3.3.2 F ormatted Displa y F unctions It is often useful[...]

  • Page 90

    3.3. MA TRIX INFORMA TION, DISPLA Y AND PLOTTING 83 g1 = ctrlplot(sys1g, { bode } ); g1 = ctrlplot(s[...]

  • Page 91

    84 CHAPTER 3. FUNCTIONAL DESCRIPTION OF X µ # Nyquist plots g2 = ctrlplot(sys1g, { nyqui st } ); g2[...]

  • Page 92

    3.4. SYSTEM RESP ONSE FUNCTIONS 85 3.4 System Resp onse F unctions 3.4.1 Creating Time Domain Signal[...]

  • Page 93

    86 CHAPTER 3. FUNCTIONAL DESCRIPTION OF X µ spacing in the input signa l pdm . This mea ns that the[...]

  • Page 94

    3.4. SYSTEM RESP ONSE FUNCTIONS 87 [a,b,c,d] = abcd(sys) sys = system(a,b,c,d) y1 = system(sys, { X0[...]

  • Page 95

    88 CHAPTER 3. FUNCTIONAL DESCRIPTION OF X µ and intstep are the in terp olation order an d integrat[...]

  • Page 96

    3.4. SYSTEM RESP ONSE FUNCTIONS 89 maximum. This can b e ha ndy when first examining a high order s[...]

  • Page 97

    90 CHAPTER 3. FUNCTIONAL DESCRIPTION OF X µ Frequency 0.1 1 10 0.01 100 Magnitude 1e-05 0.0001 0.00[...]

  • Page 98

    3.5. SYSTEM INTERCONNECTION 91 Figure 3.2: Generic Red heffer in terconnect ion structure 3.5 Syste[...]

  • Page 99

    92 CHAPTER 3. FUNCTIONAL DESCRIPTION OF X µ Figure 3.3: Example in terconnecti on of subsystem s Fi[...]

  • Page 100

    3.5. SYSTEM INTERCONNECTION 93 Using sysic to fo rm this interconnection can be considered a s four [...]

  • Page 101

    94 CHAPTER 3. FUNCTIONAL DESCRIPTION OF X µ Specify the o utputs in terms of the subsystem names an[...]

  • Page 102

    3.6. H 2 AND H ∞ ANAL YSIS AND SYNTHESIS 95 (rad/sec) ratio -1.0000e+00 0.0000e+00 1.0000e+00 1.00[...]

  • Page 103

    96 CHAPTER 3. FUNCTIONAL DESCRIPTION OF X µ Figure 3.5: In terconnection structu re for control ler[...]

  • Page 104

    3.6. H 2 AND H ∞ ANAL YSIS AND SYNTHESIS 97 equation solutio n pro cedure o ften b ecomes po o rly[...]

  • Page 105

    98 CHAPTER 3. FUNCTIONAL DESCRIPTION OF X µ Frequency 0.01 0.1 1 10 0.001 100 Magnitude 0.1 1 10 0.[...]

  • Page 106

    3.6. H 2 AND H ∞ ANAL YSIS AND SYNTHESIS 99 Figure 3.7: W eight ed design in terconnect ion struct[...]

  • Page 107

    100 CHAPTER 3. FUNCTIONAL DESCRIPTION OF X µ Frequency 0.01 0.1 1 10 0.001 100 Magnitude 0.01 0.1 1[...]

  • Page 108

    3.6. H 2 AND H ∞ ANAL YSIS AND SYNTHESIS 101 12.500 5.2e-01 1.7e-03 1.0e-02 0.0e+00 0.0000 p 6.250[...]

  • Page 109

    102 CHAPTER 3. FUNCTIONAL DESCRIPTION OF X µ (rad/sec) ratio -1.4046e-01 -2.3161e-01 2.7087e-01 0.5[...]

  • Page 110

    3.6. H 2 AND H ∞ ANAL YSIS AND SYNTHESIS 103 Frequency 0.01 0.1 1 10 0.001 100 Magnitude 0.001 0.0[...]

  • Page 111

    104 CHAPTER 3. FUNCTIONAL DESCRIPTION OF X µ Frequency 0.01 0.1 1 10 0.001 100 Magnitude 0.1 1 0.01[...]

  • Page 112

    3.6. H 2 AND H ∞ ANAL YSIS AND SYNTHESIS 105 g3 = ctrlplot(step,g3, { line style=2 } ); g3 = plot([...]

  • Page 113

    106 CHAPTER 3. FUNCTIONAL DESCRIPTION OF X µ Bounds on the H ∞ norm are ret urned as out . An est[...]

  • Page 114

    3.7. STRUCTURED SINGULAR V ALUE ( µ ) ANAL YSIS AND SYNTHE SIS 107 3.7 Structured Singular V alue ([...]

  • Page 115

    108 CHAPTER 3. FUNCTIONAL DESCRIPTION OF X µ The outputs of the µ functio n a re: the upper and lo[...]

  • Page 116

    3.7. STRUCTURED SINGULAR V ALUE ( µ ) ANAL YSIS AND SYNTHE SIS 109 c = 1/sqrt(gamma); d = -sqrt(bet[...]

  • Page 117

    110 CHAPTER 3. FUNCTIONAL DESCRIPTION OF X µ mubnds2? mubnds2 (a column vector) = 3.17155 3.17155 d[...]

  • Page 118

    3.7. STRUCTURED SINGULAR V ALUE ( µ ) ANAL YSIS AND SYNTHE SIS 111 (a) Cont roller and closed loop [...]

  • Page 119

    112 CHAPTER 3. FUNCTIONAL DESCRIPTION OF X µ Figure 3. 9: H ∞ controller design. Step 9 in the en[...]

  • Page 120

    3.7. STRUCTURED SINGULAR V ALUE ( µ ) ANAL YSIS AND SYNTHE SIS 113 Both the D and D − 1 systems ([...]

  • Page 121

    114 CHAPTER 3. FUNCTIONAL DESCRIPTION OF X µ creating weigh ts fro m data and simple system identi?[...]

  • Page 122

    3.7. STRUCTURED SINGULAR V ALUE ( µ ) ANAL YSIS AND SYNTHE SIS 115 plant = 1/makepoly([1,0,-0.01], [...]

  • Page 123

    116 CHAPTER 3. FUNCTIONAL DESCRIPTION OF X µ Wperf = makepoly([0.01,1],"s")/ makepoly([ 1[...]

  • Page 124

    3.7. STRUCTURED SINGULAR V ALUE ( µ ) ANAL YSIS AND SYNTHE SIS 117 nms = ["plant";"W[...]

  • Page 125

    118 CHAPTER 3. FUNCTIONAL DESCRIPTION OF X µ blk = [1,1; 2,2] [rpbnds1,D1,Dinv1,Delt a1,sens1] = mu[...]

  • Page 126

    3.7. STRUCTURED SINGULAR V ALUE ( µ ) ANAL YSIS AND SYNTHE SIS 119 Frequency (Hz) 0.1 1 10 0.01 100[...]

  • Page 127

    120 CHAPTER 3. FUNCTIONAL DESCRIPTION OF X µ 0.1 1 10 0.01 100 Magnitude 0.5 1 1.5 2 2.5 3 3.5 0 4 [...]

  • Page 128

    3.8. MODEL REDUCTION 121 pert = randpert(blk, { sys,sfreq, complex,pn orm } ) The user ca n specify [...]

  • Page 129

    122 CHAPTER 3. FUNCTIONAL DESCRIPTION OF X µ A g reater r ange of mo del reductio n functio ns is a[...]

  • Page 130

    3.8. MODEL REDUCTION 123 g1 = ctrlplot(sysout2g,g1, { logmagplo t,line style=4 } ); g1 = ctrlplot(re[...]

  • Page 131

    124 CHAPTER 3. FUNCTIONAL DESCRIPTION OF X µ # Displaying the Hankel singular values shows which # [...]

  • Page 132

    3.8. MODEL REDUCTION 125 3.8.3 Hank el Singular V alue Approximation The function ophank (also cr os[...]

  • Page 133

    126 CHAPTER 3. FUNCTIONAL DESCRIPTION OF X µ Frequency 0.1 1 10 0.01 100 Magnitude 1e-05 0.0001 0.0[...]

  • Page 134

    Chapter 4 Demonstration Examples 4.1 The Himat Example The following demo can be run by executing th[...]

  • Page 135

    128 CHAPTER 4. DEMONSTRA TION EXAMPLES δv Perturbations along the velo city v ector. α Angle of at[...]

  • Page 136

    4.1. TH E HIMA T EXAMPLE 129 Figure 4.2: In terconnection st ructure for the himat design exam ple T[...]

  • Page 137

    130 CHAPTER 4. DEMONSTRA TION EXAMPLES himat (a state space system) = A -0.0226 -36.6 -18.9 -32.1 0 [...]

  • Page 138

    4.1. TH E HIMA T EXAMPLE 131 angle-of-attack pitch angle System is continuous 4.1.3 Creating a W eig[...]

  • Page 139

    132 CHAPTER 4. DEMONSTRA TION EXAMPLES Frequency 0.01 0.1 1 10 100 1000 0.001 10000 Magnitude 1 10 0[...]

  • Page 140

    4.1. TH E HIMA T EXAMPLE 133 sysn = ["himat";"wdel";"wp"] in = ["[...]

  • Page 141

    134 CHAPTER 4. DEMONSTRA TION EXAMPLES gamma Hx eig X eig Hy eig Y eig nrho xy p/f 6.000 2.3e-02 5.6[...]

  • Page 142

    4.1. TH E HIMA T EXAMPLE 135 Zeros: real imaginary frequency damping (rad/sec) ratio -2.2516e-02 0.0[...]

  • Page 143

    136 CHAPTER 4. DEMONSTRA TION EXAMPLES Frequency 1 10 100 1000 0.1 10000 Magnitude 0.01 0.1 0.001 1 [...]

  • Page 144

    4.1. TH E HIMA T EXAMPLE 137 -2.2517e-02 0.0000e+00 2.2517e-02 1.0000 -2.2600e-02 0.0000e+00 2.2600e[...]

  • Page 145

    138 CHAPTER 4. DEMONSTRA TION EXAMPLES 1 10 100 1000 0.1 10000 0.5 1 1.5 0 2 Singular value plot of [...]

  • Page 146

    4.1. TH E HIMA T EXAMPLE 139 complex v alued blo cks. The upp er and lower bounds of the µ function[...]

  • Page 147

    140 CHAPTER 4. DEMONSTRA TION EXAMPLES 1 10 100 1000 0.1 10000 0.6 0.8 1 1.2 1.4 1.6 0.4 1.8 Max. si[...]

  • Page 148

    4.1. TH E HIMA T EXAMPLE 141 The second thing to note is that the in ter connection structure has th[...]

  • Page 149

    142 CHAPTER 4. DEMONSTRA TION EXAMPLES Frequency (Hz) 1 10 100 1000 0.1 10000 Magnitude 0.1 1 0.01 1[...]

  • Page 150

    4.1. TH E HIMA T EXAMPLE 143 comment D1sys "system approx. to D1" comment D1invsys "s[...]

  • Page 151

    144 CHAPTER 4. DEMONSTRA TION EXAMPLES g2 = starp(himat ic,k2) comment g2 "closed loop: iterati[...]

  • Page 152

    4.1. TH E HIMA T EXAMPLE 145 1 10 100 1000 0.1 10000 0.6 0.8 1 1.2 1.4 1.6 0.4 1.8 Robustness analys[...]

  • Page 153

    146 CHAPTER 4. DEMONSTRA TION EXAMPLES g3 = starp(himat ic,k3) g3g = freq(g3,omega) [bnds3,D3,D3inv,[...]

  • Page 154

    4.1. TH E HIMA T EXAMPLE 147 d = zeros(2,2) klp = system(a,b,c,d) comment klp "loop shape contr[...]

  • Page 155

    148 CHAPTER 4. DEMONSTRA TION EXAMPLES comment gsim mu nom "nominal closed loop sys: mu ctrl&qu[...]

  • Page 156

    4.1. TH E HIMA T EXAMPLE 149 time 0.5 1 1.5 0 2 -0.5 0 0.5 1 -1 1.5 Kmu step dist. response (nominal[...]

  • Page 157

    150 CHAPTER 4. DEMONSTRA TION EXAMPLES The lo opsha ping design gives a decoupled resp onse. Both th[...]

  • Page 158

    4.1. TH E HIMA T EXAMPLE 151 time 0.5 1 1.5 0 2 -0.5 0 0.5 1 -1 1.5 Kmu step dist. response (perturb[...]

  • Page 159

    152 CHAPTER 4. DEMONSTRA TION EXAMPLES The lo opsha ping controller had goo d nominal perfo rmance a[...]

  • Page 160

    4.2. A SIMPLE FLEX IBLE STR UCTURE EXAMPLE 153 4.2 A Simple Flexible Structure Example The demonstra[...]

  • Page 161

    154 CHAPTER 4. DEMONSTRA TION EXAMPLES Figure 4.3: Sc hematic diagram of t he JPL Phase B opti cal t[...]

  • Page 162

    4.2. A SIMPLE FLEX IBLE STR UCTURE EXAMPLE 155 static gain. W e will include some dynamic uncertaint[...]

  • Page 163

    156 CHAPTER 4. DEMONSTRA TION EXAMPLES The additive weigh t clearly provides for significant high f[...]

  • Page 164

    4.2. A SIMPLE FLEX IBLE STR UCTURE EXAMPLE 157 F = 100 Wactp = makepoly([1/(2*pi*F),1] ,"s"[...]

  • Page 165

    158 CHAPTER 4. DEMONSTRA TION EXAMPLES Wdist,Wnoise,Wperf,W actp,Wactv ,piezo) size(P)? ans (a row v[...]

  • Page 166

    4.2. A SIMPLE FLEX IBLE STR UCTURE EXAMPLE 159 Frequency 10 100 1 1000 Magnitude 0.01 0.1 1 10 100 1[...]

  • Page 167

    160 CHAPTER 4. DEMONSTRA TION EXAMPLES And examine the design weigh ts. weightsg = freq(weights,omeg[...]

  • Page 168

    4.2. A SIMPLE FLEX IBLE STR UCTURE EXAMPLE 161 Frequency 10 100 1 1000 Magnitude 0.01 0.1 1 10 0.001[...]

  • Page 169

    162 CHAPTER 4. DEMONSTRA TION EXAMPLES 4.2.3 Design of an H ∞ Con troller An H ∞ design is now p[...]

  • Page 170

    4.2. A SIMPLE FLEX IBLE STR UCTURE EXAMPLE 163 -3.5543e+02 -3.5543e+02 5.0265e+02 0.7071 -3.5543e+02[...]

  • Page 171

    164 CHAPTER 4. DEMONSTRA TION EXAMPLES Frequency 10 100 1 1000 Magnitude 0.001 0.01 0.1 0.0001 1 Con[...]

  • Page 172

    4.2. A SIMPLE FLEX IBLE STR UCTURE EXAMPLE 165 4.2.4 Robustness Analysis The blo ck str ucture has t[...]

  • Page 173

    166 CHAPTER 4. DEMONSTRA TION EXAMPLES gph4 = ctrlplot(npbnds, { log,lin e style=4 } ); gph4 = ctrlp[...]

  • Page 174

    4.2. A SIMPLE FLEX IBLE STR UCTURE EXAMPLE 167 0.5 1 1.5 0 2 -0.01 -0.005 0 0.005 0.01 -0.015 0.015 [...]

  • Page 175

    168 CHAPTER 4. DEMONSTRA TION EXAMPLES 4.2.5 D - K Iteration W e will now perfo rm one D - K itera t[...]

  • Page 176

    4.2. A SIMPLE FLEX IBLE STR UCTURE EXAMPLE 169 Kmug = freq(Kmu,omega) gph5 = ctrlplot(Kmug, { bode }[...]

  • Page 177

    170 CHAPTER 4. DEMONSTRA TION EXAMPLES Frequency 10 100 1 1000 Magnitude 0.001 0.01 0.1 0.0001 1 Con[...]

  • Page 178

    4.2. A SIMPLE FLEX IBLE STR UCTURE EXAMPLE 171 W e now examine the robustness p roperties of the new[...]

  • Page 179

    172 CHAPTER 4. DEMONSTRA TION EXAMPLES 10 100 1 1000 0.05 0.1 0.15 0.2 0.25 0 0.3 mu analysis of Gmu[...]

  • Page 180

    4.2. A SIMPLE FLEX IBLE STR UCTURE EXAMPLE 173 4.2.6 A Sim ulation Study No w the t w o con trollers[...]

  • Page 181

    174 CHAPTER 4. DEMONSTRA TION EXAMPLES 0.5 1 1.5 0 2 -0.004 -0.002 0 0.002 0.004 -0.006 0.006 Simula[...]

  • Page 182

    4.2. A SIMPLE FLEX IBLE STR UCTURE EXAMPLE 175 gph8 = ctrlplot(u(2,1)); gph8 = plot(gph8, { title=&q[...]

  • Page 183

    176 CHAPTER 4. DEMONSTRA TION EXAMPLES 0.5 1 1.5 0 2 micrometers -0.002 -0.001 0 0.001 0.002 -0.003 [...]

  • Page 184

    4.2. A SIMPLE FLEX IBLE STR UCTURE EXAMPLE 177 A nominal r esp onse is calcula ted by setting ∆ = [...]

  • Page 185

    178 CHAPTER 4. DEMONSTRA TION EXAMPLES 0.5 1 1.5 0 2 micrometers -5 0 5 -10 10 open loop beam length[...]

  • Page 186

    4.2. A SIMPLE FLEX IBLE STR UCTURE EXAMPLE 179 Now w e consider the closed-lo op nominal resp onse w[...]

  • Page 187

    180 CHAPTER 4. DEMONSTRA TION EXAMPLES 0.5 1 1.5 0 2 micrometers -0.2 -0.1 0 0.1 0.2 0.3 -0.3 0.4 cl[...]

  • Page 188

    4.2. A SIMPLE FLEX IBLE STR UCTURE EXAMPLE 181 gph11 = ctrlplot(yclphinf(2,1)) ; gph11 = ctrlplot(yc[...]

  • Page 189

    182 CHAPTER 4. DEMONSTRA TION EXAMPLES 0.5 1 1.5 0 2 -0.01 -0.005 0 0.005 0.01 -0.015 0.015 closed l[...]

  • Page 190

    4.2. A SIMPLE FLEX IBLE STR UCTURE EXAMPLE 183 gph12 = ctrlplot(yclphinf(3,1)) ; gph12 = ctrlplot(yc[...]

  • Page 191

    184 CHAPTER 4. DEMONSTRA TION EXAMPLES 0.5 1 1.5 0 2 -0.01 0 0.01 0.02 -0.02 0.03 closed loop: piezo[...]

  • Page 192

    4.2. A SIMPLE FLEX IBLE STR UCTURE EXAMPLE 185 Note that Kmu achiev es better performance at the exp[...]

  • Page 193

    186 CHAPTER 4. DEMONSTRA TION EXAMPLES gph13 = ctrlplot(ybclpmu(1,1),g ph13); gph13 = plot(gph13, { [...]

  • Page 194

    4.2. A SIMPLE FLEX IBLE STR UCTURE EXAMPLE 187 Frequency 10 100 1 1000 Magnitude 0.001 0.01 0.1 0.00[...]

  • Page 195

    188 CHAPTER 4. DEMONSTRA TION EXAMPLES gph14 = ctrlplot(ybclphinf(2,1) ); gph14 = ctrlplot(ybclpmu(2[...]

  • Page 196

    4.2. A SIMPLE FLEX IBLE STR UCTURE EXAMPLE 189 0.5 1 1.5 0 2 -0.01 -0.005 0 0.005 0.01 -0.015 0.015 [...]

  • Page 197

    190 CHAPTER 4. DEMONSTRA TION EXAMPLES gph15 = ctrlplot(ybclphinf(3,1) ); gph15 = ctrlplot(ybclpmu(3[...]

  • Page 198

    4.2. A SIMPLE FLEX IBLE STR UCTURE EXAMPLE 191 0.5 1 1.5 0 2 -0.02 -0.01 0 0.01 0.02 -0.03 0.03 pert[...]

  • Page 199

    Bibliograph y [1] J. C. Doyle, “Lecture notes on adv ances in multiv ariable control.” ONR/Honey[...]

  • Page 200

    194 BIBLIOGRAPHY [11] G. J. Ba las and J. C. Doyle, “Robust con tr ol of flexible modes in the co[...]

  • Page 201

    BIBLIOGRAPHY 195 [24] D. L. Laughlin, K. G. Jordan, and M. Morari, “In ternal mo del con trol and [...]

  • Page 202

    196 BIBLIOGRAPHY [37] R. Kosut, M . Lau, and S. Bo yd, “Param eter set ident ification of systems[...]

  • Page 203

    BIBLIOGRAPHY 197 [50] J. M. Krause, P . P . Khargonek ar, and G. Stein , “Robust param eter adjust[...]

  • Page 204

    198 BIBLIOGRAPHY [63] A. J. La ub, “A Sch ur method for solving algebr aic Riccati equatio ns,” [...]

  • Page 205

    BIBLIOGRAPHY 199 [76] M. Dahleh, A . T esi , and A. V icino, “Extremal prop erties for t he parame[...]

  • Page 206

    Chapter 6 F unction Reference 6.1 X µ F unctions The following pages contain descriptions of the X [...]

  • Page 207

    202 CHAPTER 6. FUNCTION RE FERENCE rifd ....... ........ ......... ........ ....... ........ .......[...]

  • Page 208

    6.1. X µ FUNCTIONS 203 Mo del r e duction and state-sp ac e functions balmoore ...... ........ ....[...]

  • Page 209

    204 CHAPTER 6. FUNCTION RE FERENCE Misc el lane ous functions conpdm ...... ........ ......... .....[...]

  • Page 210

    balmoore 205 balmo ore Syn t ax [SysR,HSV,T] = balmoore(Sys, { nsr,bound } ) Pa r a m e t e r Li s t[...]

  • Page 211

    206 Chapter 6. F unctio n Reference Reference B.C. Mo ore, “Prin cip al Comp onen t Analysis in Li[...]

  • Page 212

    balmoore 207 See Also: minimal , ophank .[...]

  • Page 213

    blknorm 209 blknorm Syn t ax normM = blknorm(M,blk,p,Frobeni us) Pa r a m e t e r Li s t Inputs: M M[...]

  • Page 214

    210 Chapter 6. F unctio n Reference Examples A = random(3,3)-0.5*ones(3 ,3)? A (a square matrix) = 0[...]

  • Page 215

    blknorm 211 2.23607 3 4 7.81025 7 8 13.4536 11 12 # and compare to norm(B(1,1:2)) ans (a scalar) = 2[...]

  • Page 216

    conpdm 213 conp dm Syn t ax outpdm = conpdm(mat,domain, { skipChk s } ) Pa r a m e t e r Li s t Inpu[...]

  • Page 217

    consys 215 consys Syn t ax outsys = consys(mat, { skipChks } ) Pa r a m e t e r Li s t Inputs: mat c[...]

  • Page 218

    csum 217 csum Syn t ax [outpdm] = csum(inpdm, { channels } ) Pa r a m e t e r Li s t Inputs: inpdm r[...]

  • Page 219

    218 Chapter 6. F unctio n Reference ans (a rectangular matrix) = 1 0.608453 2 1.46287 3 1.52714 4 2.[...]

  • Page 220

    csum 219 ans (a pdm) = domain | Col 1 Col 2 -------+-------------- --------- 1 | Row 1 1 0.608453 | [...]

  • Page 221

    ctrlplot 221 ctrlplot Syn t ax graph = ctrlplot(pdm,old graph, { keywords } ) Pa r a m e t e r Li s [...]

  • Page 222

    222 Chapter 6. F unctio n Reference linear linear domain. Default = 1 for timeresp k eyword. Defa ul[...]

  • Page 223

    ctrlplot 223 Description This function per forms some co mmon control system related plotting. The u[...]

  • Page 224

    224 Chapter 6. F unctio n Reference legend = ["sys1","sys2"] } )?[...]

  • Page 225

    ctrlplot 225 Frequency 0.1 1 0.01 10 Magnitude 0.0001 0.001 0.01 0.1 1 1e-05 10 Bode plots sys1 sys2[...]

  • Page 226

    226 Chapter 6. F unctio n Reference # Nyquist plots g2 = ctrlplot(sys1g, { nyqui st } ); g2 = ctrlpl[...]

  • Page 227

    ctrlplot 227 Real -1 0 1 -2 2 Imaginary -1.5 -1 -0.5 0 -2 0.5 Nyquist plots sys1 sys3 critical point[...]

  • Page 228

    228 Chapter 6. F unctio n Reference # Create a second order lightly damped system to illustrate # ti[...]

  • Page 229

    ctrlplot 229 2468 0 10 -2 -1 0 1 -3 2 input x0 = zero non-zero x0[...]

  • Page 230

    230 Chapter 6. F unctio n Reference See Also: plot .[...]

  • Page 231

    daug 231 daug Syn t ax out = daug (sys1,sys2,...) Pa r a m e t e r Li s t Inputs: sys1 Input systems[...]

  • Page 232

    232 Chapter 6. F unctio n Reference ans (a square matrix) = 110 0 002 0 002 0 000I n f sys1 = randsy[...]

  • Page 233

    daug 233 -1.67106 B 0.579502 C 0.262815 0.436099 D 0.911055 0.808267 X0 1 State Names ----------- sy[...]

  • Page 234

    234 Chapter 6. F unctio n Reference D 0.492058 0.748961 0 0 001 0 0 0 0 0 0.911055 0 0 0 0.808267 X0[...]

  • Page 235

    daug 235 0| 1 0 -------+----- 1| 1 0 -------+----- 2| 1 0 -------+----- daug(pdm1,pdm2) ans (a pdm) [...]

  • Page 236

    delsubstr 237 delsubstr Syn t ax [outstr] = delsubstr(str,charstr) Pa r a m e t e r Li s t Inputs: s[...]

  • Page 237

    238 Chapter 6. F unctio n Reference out2 (a column vector of strings) = string one aaa xy # If execu[...]

  • Page 238

    fitsys 239 fitsys Syn t ax [sys] = fitsys(data,npoles ,nzeros,we ight, { skipchks,Hertz } ) Pa r a [...]

  • Page 239

    240 Chapter 6. F unctio n Reference The primary use of this routine is the fitting of D scal e w ei[...]

  • Page 240

    fitsys 241 real imaginary frequency damping (rad/sec) ratio -1.0000e+00 0.0000e+00 1.0000e+00 1.0000[...]

  • Page 241

    242 Chapter 6. F unctio n Reference Frequency 0.01 0.1 1 10 0.001 100 Phase (degrees) -300 -200 -100[...]

  • Page 242

    fitsys 243 # Create fitting weight. 1/s works well for logspaced # data. wght = 1/makepoly([1,0],&qu[...]

  • Page 243

    244 Chapter 6. F unctio n Reference Frequency 0.01 0.1 1 10 0.001 100 Magnitude 0.1 1 10 0.01 100 Da[...]

  • Page 244

    fitsys 245 Limita tions Limited to SISO systems . See Also tfid[...]

  • Page 245

    gstep 247 gstep Syn t ax gPdm = gstep (ytime,timespec,valspec , { skipChks } ) Pa r a m e t e r Li s[...]

  • Page 246

    248 Chapter 6. F unctio n Reference Frequency 0.01 0.1 1 10 0.001 100 Magnitude 0.1 1 10 0.01 100 Da[...]

  • Page 247

    gstep 249 See Also randpdm , gcos , gsin , gpulse , gsawtooth , gsquarewave[...]

  • Page 248

    hinfnorm 251 hinfnorm Syn t ax [out,omega] = hinfnorm(sys,tol, { imag eps,max it } ) Pa r a m e t e [...]

  • Page 249

    252 Chapter 6. F unctio n Reference Stable Dynamic System norms a re calculated by an itera tive Ham[...]

  • Page 250

    hinfnorm 253 25.000 5.2e-01 1.7e-03 1.0e-02 0.0e+00 0.0000 p 12.500 5.2e-01 1.7e-03 1.0e-02 0.0e+00 [...]

  • Page 251

    h2norm 255 h2norm Syn t ax out = h2norm(sys) Pa r a m e t e r Li s t Inputs: sys Co n tin uous time [...]

  • Page 252

    256 Chapter 6. F unctio n Reference Wperf = 100/makepoly([100,1],"s ") Wact = makepoly([0.[...]

  • Page 253

    hinfsyn 257 hinfsyn Syn t ax [k,gfin,stat] = hinfsyn(p,nmeas,ncon,gamma, { keywords } ) Pa r a m e t[...]

  • Page 254

    258 Chapter 6. F unctio n Reference Description The H ∞ (sub)optimal controller for the interconne[...]

  • Page 255

    hinfsyn 259 1. ( a, b 2 ,c 2 ) is stabiliza ble and detectable 2. d 12 and d 21 ha ve full rank 3. T[...]

  • Page 256

    260 Chapter 6. F unctio n Reference wghtic = sysic(sysnames,sy sinp,sysou t,syscnx,p lant,... Wperf,[...]

  • Page 257

    hinfsyn 261 real imaginary frequency damping (rad/sec) ratio -5.0000e-02 -3.1225e-01 3.1623e-01 0.15[...]

  • Page 258

    262 Chapter 6. F unctio n Reference Frequency 0.01 0.1 1 10 0.001 100 Magnitude 0.01 0.1 1 0.001 10 [...]

  • Page 259

    hinfsyn 263 # Use sysic to create unweighted interconnection ic = sysic("plant",["ref[...]

  • Page 260

    264 Chapter 6. F unctio n Reference gph2 = ctrlplot(sensg, { logmagpl ot } ); gph2 = plot(gph2, { ti[...]

  • Page 261

    hinfsyn 265 Frequency 0.01 0.1 1 10 0.001 100 Magnitude 0.1 1 0.01 10 Kinf controller: sensitivity f[...]

  • Page 262

    266 Chapter 6. F unctio n Reference # Examine step response step = gstep([0:0.1:10],0,1) y = clpinf*[...]

  • Page 263

    hinfsyn 267 2468 0 10 0.2 0.4 0.6 0.8 1 0 1.2 Kinf controller: step response[...]

  • Page 264

    268 Chapter 6. F unctio n Reference See also hinfsyn , hinfnorm , h2norm[...]

  • Page 265

    h2syn 269 h2syn Syn t ax k = h2syn(p,nmeas,ncon, { ke ywords } ) Pa r a m e t e r Li s t Inputs: p G[...]

  • Page 266

    270 Chapter 6. F unctio n Reference p k - zw y u The v ariab les ncon and nmeas are used to sp ecify[...]

  • Page 267

    h2syn 271 Reference This function uses the state-spa ce formulae given in: “Stat e-sp ace formu la[...]

  • Page 268

    272 Chapter 6. F unctio n Reference K2 = h2syn(wghtic,nmeas,nc trls) rifd(K2) Poles: real imaginary [...]

  • Page 269

    h2syn 273 Frequency 0.01 0.1 1 10 0.001 100 Magnitude 0.001 0.01 0.1 0.0001 1 K2 Frequency 0.01 0.1 [...]

  • Page 270

    274 Chapter 6. F unctio n Reference # Use sysic to create unweighted interconnection ic = sysic(&quo[...]

  • Page 271

    h2syn 275 sensg = freq(sens,omega) gph2 = ctrlplot(sensg, { logmagpl ot } ); gph2 = plot(gph2, { tit[...]

  • Page 272

    276 Chapter 6. F unctio n Reference Frequency 0.01 0.1 1 10 0.001 100 Magnitude 1 0.1 10 K2 controll[...]

  • Page 273

    h2syn 277 # Examine step response step = gstep([0:0.1:10],0,1) y = clp2*step gph3 = ctrlplot([y,step[...]

  • Page 274

    278 Chapter 6. F unctio n Reference 2468 0 10 0.2 0.4 0.6 0.8 0 1 K2 controller: step response[...]

  • Page 275

    h2syn 279 See also hinfsyn , h2norm , hinfnorm[...]

  • Page 276

    interp 281 in terp Syn t ax outpdm = interp(inpdm,stepsize,final { keywords } ) outpdm = interp(inpd[...]

  • Page 277

    282 Chapter 6. F unctio n Reference This function differs from the interpolate function in tha t it[...]

  • Page 278

    interp 283 1234 0 5 1 2 3 4 0 5 1st order interp. 0 order interp. original pdm[...]

  • Page 279

    284 Chapter 6. F unctio n Reference See Also interpolate[...]

  • Page 280

    mergeseg 285 mergeseg Syn t ax outpdm = mergeseg(pdm1,pdm2, { keywords } ) Pa r a m e t e r Li s t I[...]

  • Page 281

    286 Chapter 6. F unctio n Reference Exam ple time1 = [0:0.025:1] pdm1 = gsin(time1, { frequency= 2 }[...]

  • Page 282

    mergeseg 287 0.5 1 0 1.5 -0.5 0 0.5 -1 1 outpdm pdm1 pdm2[...]

  • Page 283

    mkpert 289 mkp ert Syn t ax [pertsys] = mkpert(Delta,blk,mubnds, { f select,pn orm,Hertz } ) Pa r a [...]

  • Page 284

    290 Chapter 6. F unctio n Reference The bo unds, m ubnds, ar e used to determine the ”worst-case?[...]

  • Page 285

    mkphase 291 mkphase Syn t ax [cdata] = mkphase(magdata, { skipchks,Hertz } ) Pa r a m e t e r Li s t[...]

  • Page 286

    292 Chapter 6. F unctio n Reference Limita tions Limited to SISO systems . See Also fitsys , ccepstr[...]

  • Page 287

    modalstate 293 mo dalstate Syn t ax outsys = modalstate(sys, { keywords } ) Pa r a m e t e r Li s t [...]

  • Page 288

    294 Chapter 6. F unctio n Reference -8.1602e-01 1.3353e+00 1.5649e+00 0.5215 -8.1602e-01 -1.3353e+00[...]

  • Page 289

    mu 295 mu Syn t ax [mubnds,D,Dinv,Delta, sens] = mu(M,blk) Pa r a m e t e r Li s t Inputs: M Matrix [...]

  • Page 290

    296 Chapter 6. F unctio n Reference # that mu is not equal to its upper bound for # more than three [...]

  • Page 291

    mu 297 ans (a scalar) = 2.81264 mubnds2? mubnds2 (a column vector) = 2.81264 2.81264 det(eye(4,4) - [...]

  • Page 292

    musynfit 299 m usynfit Syn t ax [Dsys,Dinvsys] = musynfit(Dmag,blk,nmeas,nc trls,.. weight,M,order,[...]

  • Page 293

    300 Chapter 6. F unctio n Reference Keyw ords: He rtz Boolean. This ke yword i s mandatory as th e f[...]

  • Page 294

    musynfit 301 P = 1/makepoly([1,0,-0.01] ,"s") W = makepoly([1,20],"s")/m akepoly[...]

  • Page 295

    302 Chapter 6. F unctio n Reference 1.562 6.8e-01 8.0e-04 7.7e-03 0.0e+00 0.6811 p 1.406 6.7e-01 8.1[...]

  • Page 296

    musynfit 303 0.1 1 10 0.01 100 0.2 0.4 0.6 0.8 1 0 1.2 mu analysis nominal perf. robust stab. robust[...]

  • Page 297

    304 Chapter 6. F unctio n Reference # Fit transfer functions to D1 & Dinv1 for a mu # synthesis [...]

  • Page 298

    musynfit 305 Frequency (Hz) 0.1 1 10 0.01 100 Magnitude 10 100 1000 1 10000 D scale fit, block: 1 Ma[...]

  • Page 299

    306 Chapter 6. F unctio n Reference # Apply the D scales to another H infinity design Kmu = hinfsyn([...]

  • Page 300

    musynfit 307 0.1 1 10 0.01 100 0.2 0.4 0.6 0.8 1 0 1.2 Kmu & Kinf mu analysis Kmu: robust perf. [...]

  • Page 301

    308 Chapter 6. F unctio n Reference # Look at the worst case perturbations for each of # the Kinf an[...]

  • Page 302

    musynfit 309 ymunom(1,1),ymupert(1 ,1)]); gph4 = plot(gph4, { legend=["inpu t step";"[...]

  • Page 303

    310 Chapter 6. F unctio n Reference 123 0 4 0.2 0.4 0.6 0.8 1 1.2 0 1.4 input step Kinf nominal Kinf[...]

  • Page 304

    musynfit 311 # Compare with a random perturbation yinfrandp = infrpert*step ymurandp = murpert*step [...]

  • Page 305

    312 Chapter 6. F unctio n Reference 123 0 4 0.2 0.4 0.6 0.8 1 1.2 0 1.4 input step Kinf nominal Kinf[...]

  • Page 306

    musynfit 313 See Also: mu , hinfsyn , mkpert , hinfnorm .[...]

  • Page 307

    ophank 315 ophank Syn t ax [SysR,SysU,HSV] = ophank(Sys, { nsr,o nepass } ) Pa r a m e t e r Li s t [...]

  • Page 308

    316 Chapter 6. F unctio n Reference Uses additional s ubroutines ophiter , ophred , ophmult and stab[...]

  • Page 309

    ophank 317 Frequency 0.1 1 10 0.01 100 Magnitude 0.001 0.01 0.0001 0.1 original system reduced: opha[...]

  • Page 310

    318 Chapter 6. F unctio n Reference See Also minimal , balmoore .[...]

  • Page 311

    orderstate 319 orderstate Syn t ax outsys = orderstate(sys,indx) Pa r a m e t e r Li s t Inputs: sys[...]

  • Page 312

    320 Chapter 6. F unctio n Reference 0.710595 0.688873 C 0.659532 0.181512 0.390497 D 0.15869 X0 0.1 [...]

  • Page 313

    orderstate 321 0.2 0.1 0.3 State Names ----------- s2 s1 s3 Input Names ----------- Input 1 Output N[...]

  • Page 314

    randpdm 323 randp dm Syn t ax pdmout = randpdm (ndomain,nrows,ncolumns, { keywords } ) Pa r a m e t [...]

  • Page 315

    324 Chapter 6. F unctio n Reference Examples pdm0 = randpdm(3,1,2, { zeromean } )? pdm0 (a pdm) = do[...]

  • Page 316

    randpdm 325 10.0323 | 0.922528 ---------+------------ 18.3191 | 0.81113 ---------+------------ pdm3 [...]

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    randpert 327 randp ert Syn t ax [pert] = randpert(blk, { sys,sfreq,complex,pnorm } ) Pa r a m e t e [...]

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    328 Chapter 6. F unctio n Reference Exam ple The use of randpert is studied in co ntext in the on-li[...]

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    randsys 329 randsys Syn t ax sys = randsys (nstates,noutputs,ninput s, { keywor ds } ) Pa r a m e t [...]

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    330 Chapter 6. F unctio n Reference Description A rando m system, with user sp ecified state, input[...]

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    randsys 331 real imaginary frequency damping (rad/sec) ratio -1.0408e+00 0.0000e+00 1.0408e+00 1.000[...]

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    332 Chapter 6. F unctio n Reference Frequency 0.1 1 10 0.01 100 Magnitude 0.01 0.1 1 0.001 10 sys1 ([...]

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    randsys 333 sys3 = randsys(6,1,1, { discrete, dt=5 } ) rifd(sys3)? Poles: radius angle (radians) 0.1[...]

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    rifd 335 rifd Syn t ax [stat] = rifd(vec, { discrete,Hertz,d egrees } ) Pa r a m e t e r Li s t Inpu[...]

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    336 Chapter 6. F unctio n Reference Examples sys1 = randsys(4,3,2, { stable } ) rifd(sys1) Poles: re[...]

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    rifd 337 sys2 = randsys(3,3,2, { stable,di screte } ) rifd(sys2) Poles: radius angle (radians) 0.964[...]

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    sdtrsp 339 sdtrsp Syn t ax [v,y,u] = sdtrsp(Sys,dSys,w,tfinal,. .. { ord,intstep,cdelay } ) Pa r a m[...]

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    340 Chapter 6. F unctio n Reference Description Time doma in simulation of a sa mpled data interconn[...]

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    sdtrsp 341 triangle ho ld equiv alent is used. This is the same a linea rly connecting the samples a[...]

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    342 Chapter 6. F unctio n Reference inps = "ref" outs = "digP" cnx = ["digC[...]

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    sdtrsp 343 0.2 0.4 0.6 0.8 0 1 -1 0 1 2 3 -2 4 sampled data calc. discrete calc.[...]

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    344 Chapter 6. F unctio n Reference See Also trsp[...]

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    simtransform 345 sim transform Syn t ax out = simtransform(sys,X) Pa r a m e t e r Li s t Inputs: sy[...]

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    346 Chapter 6. F unctio n Reference -1.9737e+01 0.0000e+00 1.9737e+01 1.0000 -9.7569e+00 -2.9129e+01[...]

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    simtransform 347 X0 0 0 0 Input Names ----------- Input 1 Output Names ------------ Output 1 System [...]

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    spectrad 349 sp ectrad Syn t ax out = spectrad(mat) Pa r a m e t e r Li s t Inputs: ma t Square matr[...]

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    350 Chapter 6. F unctio n Reference eig(pdm1)? ans (a pdm) = domain | -------+-------------- -------[...]

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    sresidualize 351 sresidualize Syn t ax sysout = sresidualize(sysin,ord) Pa r a m e t e r Li s t Inpu[...]

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    352 Chapter 6. F unctio n Reference sysout1 = sresidualize(sys1,3) sysout2 = truncate(sys1,3) fHz = [...]

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    sresidualize 353 Frequency 0.1 1 10 0.01 100 Magnitude 0.0001 0.001 0.01 1e-05 0.1 original system r[...]

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    354 Chapter 6. F unctio n Reference See also rifd , simtransform , orderstate modalstate , truncate [...]

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    starp 355 starp Syn t ax out = starp (upper,lower,dim1,dim 2,skipChks ) Pa r a m e t e r Li s t Inpu[...]

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    356 Chapter 6. F unctio n Reference specified. This is equiv alent to: dim1 = min ( upper output di[...]

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    starp 357 # structure for closing control loops. The following is # the structure for a simple unity[...]

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    358 Chapter 6. F unctio n Reference -1.0000e+01 0.0000e+00 1.0000e+01 1.0000 # Now consider the syst[...]

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    starp 359 Poles: real imaginary frequency damping (rad/sec) ratio -5.0000e+00 -8.0623e+00 9.4868e+00[...]

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    substr 361 substr Syn t ax littlestring = substr(bigstring,charinde x, { skipChk s } ) Pa r a m e t [...]

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    sysic 363 sysic Syn t ax [sys] = sysic(sysNames,sysInput s,sysOutp uts,sysCon nects,... subsys1,subs[...]

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    364 Chapter 6. F unctio n Reference Matrices can a lso be included in the interconnection and are co[...]

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    trsp 365 trsp Syn t ax [y,uint] = trsp(Sys,u,tfinal, ord,intstep) Pa r a m e t e r Li s t Inputs: Sy[...]

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    366 Chapter 6. F unctio n Reference necessary . If int step is not specified the in tegration steps[...]

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    trsp 367 0.5 1 1.5 0 2 -1 -0.5 0 0.5 1 -1.5 1.5 output input[...]

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    368 Chapter 6. F unctio n Reference # Compare trsp calculation to standard [ytrsp,uint] = trsp(P,u) [...]

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    trsp 369 0.5 1 1.5 0 2 -1 -0.5 0 0.5 1 -1.5 1.5 Time response calculation comparisons trsp calc. * c[...]

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    370 Chapter 6. F unctio n Reference # Now look at 1st order interpolation [y1trsp,u1int] = trsp(P,u,[...]

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    trsp 371 0.5 1 1.5 0 2 -1 -0.5 0 0.5 1 -1.5 1.5 Time response calculation comparisons 1st order inte[...]

  • Page 357

    sresidualize 373 truncate Syn t ax sysout = truncate(sysin,ord) Pa r a m e t e r Li s t Inputs: sysi[...]

  • Page 358

    374 Chapter 6. F unctio n Reference Exam ple This example is iden tica l to tha t describ ed for sre[...]

  • Page 359

    sresidualize 375 Frequency 0.1 1 10 0.01 100 Magnitude 0.0001 0.001 0.01 1e-05 0.1 original system r[...]

  • Page 360

    376 Chapter 6. F unctio n Reference See also rifd , simtransform , sresidualize , orderstate modalst[...]

  • Page 361

    6.2. X µ SUBR OUTINES AND UTILITIES 377 6.2 X µ Subroutines and Utilities Several subroutines ma y[...]

  • Page 362

    blkbal 379 blkbal Syn t ax d = blkbal(M) Description Balances a square ma trix assuming only scalar [...]

  • Page 363

    hinfcalc 381 hinfcalc Syn t ax [X,Y,f,h,Ric fail,HX,HY,HXmin,HYmin ] = ... hinfcalc(p,nmeas,nco n,g,[...]

  • Page 364

    382 Chapter 6. F unctio n Reference Pa r a m e t e r Li s t Inputs: p Generalized interconnection st[...]

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    hinfcalc 383 Description F orm and solve th e Riccati equations for the H ∞ con trol problem. X an[...]

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    powermu 385 po w e r m u Syn t ax [lbnd,delta,errstat] = powermu(M,blk,rp,cp) Description Low er bou[...]

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    riccati eig 387 riccati eig Syn t ax [x1,x2,stat,Heig min] = riccati eig(H,epp) Pa r a m e t e r Li [...]

  • Page 368

    388 Chapter 6. F unctio n Reference v ariab le, X = x 2 x − 1 1 , is the stabilizing solution to t[...]

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    riccati schur 389 riccati sc h ur Syn t ax [x1,x2,stat,Heig min] = riccati schur(H,epp) Pa r a m e t[...]

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    390 Chapter 6. F unctio n Reference v ariab le, X = x 2 x − 1 1 , is the stabilizing solution to t[...]

  • Page 371

    App endices A T ranslation Bet ween Ma tlab µ -T o ols and X µ This app endix outlines the functio[...]

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    392 APPEND ICES The functionally equiv alen t commands will b e listed, in each sub-section, for con[...]

  • Page 373

    A. TRANSLA TION BETWE EN MA TLA B µ -TOOLS AND X µ 393 Description µ -T o ols F unction Xmath/X ?[...]

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    394 APPEND ICES Subblo cks: s electing input & outputs In µ -T o ols the function sel selects r[...]

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    A. TRANSLA TION BETWE EN MA TLA B µ -TOOLS AND X µ 395 Note that the transpose and conjugate tran [...]

  • Page 376

    396 APPEND ICES Description µ -T o ols F unction Xmath/X µ equiv alen t peak norm pkvnorm norm , m[...]

  • Page 377

    A. TRANSLA TION BETWE EN MA TLA B µ -TOOLS AND X µ 397 Miscellaneou s Utilities Several utilities [...]

  • Page 378

    398 APPEND ICES A.3 System R esp onse F unctions Creating Ti me Domain Signal s The Xmath pdm data o[...]

  • Page 379

    A. TRANSLA TION BETWE EN MA TLA B µ -TOOLS AND X µ 399 information. A.4 System In terconnection Si[...]

  • Page 380

    400 APPEND ICES Description µ -T o ols function Xma th/X µ equiv alen t H 2 norm calcula tion h2no[...]

  • Page 381

    A. TRANSLA TION BETWE EN MA TLA B µ -TOOLS AND X µ 401 Description µ -T o ols function Xma th/X ?[...]

  • Page 382

    402 APPEND ICES g1g = freq(g1,omega) [mubnds,Dmagdata] = mu(g1g,blk) F rom this, frequency do main D[...]

  • Page 383

    A. TRANSLA TION BETWE EN MA TLA B µ -TOOLS AND X µ 403 The adv an tage of this is that in order to[...]

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    T echnical Support and Professional Ser vices Visit the following sections of the National Instrumen[...]