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The rules should oblige the seller to give the purchaser an operating instrucion of National Instruments 370755B01, along with an item. The lack of an instruction or false information given to customer shall constitute grounds to apply for a complaint because of nonconformity of goods with the contract. In accordance with the law, a customer can receive an instruction in nonpaper form; lately graphic and electronic forms of the manuals, as well as instructional videos have been majorly used. A necessary precondition for this is the unmistakable, legible character of an instruction.
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Table of contents for the manual

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MA TRIXx TM Xmath TM Model Reduction Module Xmath Mo del Reducti on Modul e April 2004 Edit ion Part Numb er 37075 5B01[...]

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Supp ort Worl dwide Tec hnical Supp ort and Pro duct Infor mation ni.com Nati onal In strum ents Corp orate Headqu arter s 11500 Nor th Mopac E xpressway Austin, T exas 78759 3504 USA Tel: 512 683 01 00 Worldwide Offices A u s t r a l i a1 8 0 03 0 08 0 0 , A u s t r i a4 306 6 2 4 57 99 00 , B e l g i u m3 2027 5 70 02 0 , B r a z i l 5 51 13 2 6[...]

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Important Information Warrant y The media on whi ch you receive National Instruments softw are are warranted n ot to fail to exec ute programming ins tructions , due to defects in mat erials an d workman ship, for a period of 9 0 days fr om date of sh ipment, a s evidence d by receipt s or othe r documenta tion. N ational Instruments will, at its o[...]

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Conventions The follo wing con ventions are used in this manual: [ ] Square brackets enclose optional items—for ex ample, [ response ]. Square brackets also cite bibliographic references. » The » symbol leads you throu gh nested menu items and dialog box options to a f inal action. The s equence File»P age Setup »Options di rects y ou to pull[...]

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© Nationa l Instrume nts Corpora tion v Xmath Mod el Redu ction Modu le Contents Chapter 1 Introdu ctio n Using This Manual.............. ........... ................. ........... ................. ............ ........... ............ 11 Document Organization... ........... ................. ............ ................. ........... ...........[...]

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Contents Xmath Model R eduction M odule vi ni.com Onepass Algorithm ................... ................. ........... ................. ........... ............ .. 2 18 Multipass Algorithm ................ ................. ........... ........... ................. ............ .. 2 20 DiscreteTime Systems .................. ............ .......[...]

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Contents © Nationa l Instrume nts Corpora tion vii Xmath Mod el Redu ction Modu le Algorithm ....... ............ ................ ............ ........... ................. ........... ................. . 418 Additional Backgr ound .................... ........... ................. ............ ........... ............ 420 Related Functions . .[...]

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© Nationa l Instrume nts Corpora tion 11 Xmath Mod el Redu ction Modu le 1 Introduction This chapter starts with an outlin e of the m anual and s ome usefu l notes. It also provides an ov erview of the Model Re duction Module, describes th e functions in this module, and introdu ces nomenclature and concepts used throughout this manual. Using Thi[...]

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Chapter 1 Introduction Xmath Model R eduction M odule 12 ni.com • Chapter 5, Utilit ies , describes three utility functio ns: hankelsv( ) , stable( ) , and compare( ) . • Chapter 6, Tu torial , illustrates a nu mber of the MRM func tions and their underlying ideas. Bibliograph ic Reference s Througho ut this document, bibliog r aphic reference[...]

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Chap ter 1 Int rodu cti on © Nationa l Instrume nts Corpora tion 13 Xmath Mod el Redu ction Modu le Relate d Publica tions For a complete list of MATRIXx publicat ions, refer to Chapter 2, MATRIXx Pu blication s, Onlin e Help, and Customer Support , o f the MATRIXx Getting Started Guide. The following do cuments are particularly useful for topics[...]

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Chapter 1 Introduction Xmath Model R eduction M odule 14 ni.com As sho wn in Figure 11, fu nctions are p rov ided to ha ndle four broad t asks: • Model reduction with additi ve errors • Model reduction with multiplicativ e errors • Model reduction with frequency weighting of an additive error , including controller r eduction • Utility fu[...]

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Chap ter 1 Int rodu cti on © Nationa l Instrume nts Corpora tion 15 Xmath Mod el Redu ction Modu le Certain restriction s re garding minimality and stability are required of the input data, and are summarized in T able 11. Documentation of the individual functions sometimes indicates how th e restrictions can be circu mvented. Th ere are a n umb[...]

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Chapter 1 Introduction Xmath Model R eduction M odule 16 ni.com • L 2 approximation, in which the L 2 norm of impuls e respon se erro r (or, by P arse v al’ s theorem, the L 2 norm of the transf erfun ction error along the imaginary axis ) serv es as the error measur e • Marko v param eter or i mpulse respo nse matchin g, moment m atching, [...]

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Chap ter 1 Int rodu cti on © Nationa l Instrume nts Corpora tion 17 Xmath Mod el Redu ction Modu le • An inequality or bound is tigh t if it can be met in practice, for e xample is tight because the inequality becomes an equality for x =1 . A g a i n , if F ( j ω ) denotes the Fourier transform of some , the Heisenber g inequa lity states, and[...]

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Chapter 1 Introduction Xmath Model R eduction M odule 18 ni.com • The controllabilit y grammian is also E [ x ( t ) x ′ ( t )] when the system has been e xcited from time – ∞ b y zero mean white noise with . • The observability grammian can be thought of as measuring the information contained in the outp ut concerning an initial state. I[...]

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Chap ter 1 Int rodu cti on © Nationa l Instrume nts Corpora tion 19 Xmath Mod el Redu ction Modu le • Suppose t he transfer fu nction matri x corresponds to a discr eteti me system, with state v ariable dimen sion n . Then the infinite Hank el matrix, has for its singular values the n nonzero Hankel s ingular v alues, together with an infinit[...]

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Chapter 1 Introduction Xmath Model R eduction M odule 110 ni.com Interna lly Ba lanced Re alizatio ns Suppose that a realization of a transferfu nction matrix has the controllability and observ ability grammian property that P = Q = Σ for some diagonal Σ . Then the realization is termed internally balanced. Notice that the diagona l entries σ [...]

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Chap ter 1 Int rodu cti on © Natio nal Instrum ents Cor poration 111 Xma th Model Re ductio n Module This is almost the algorith m set out in Section II of [LHPW87]. The one dif ference (and it is minor) is that in [LH PW87], lower triangular Cholesky factors of P and Q are used, in place of U c S c 1/2 and U O S O 1/2 in forming H in step 2. Th [...]

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Chapter 1 Introduction Xmath Model R eduction M odule 112 ni.com and also: Re λ i ( A 22 )< 0 and . Usually , we expect that, in the sense that th e intuiti ve ar gument h inges on this, b ut it is n ot necessary . Then a sing ular perturbation is ob tained by r eplacing by zero ; this means that: Acco rdin gly, (12) Equation 12 may be an ap[...]

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Chap ter 1 Int rodu cti on © Natio nal Instrum ents Cor poration 113 Xma th Model Re ductio n Module Similar considerations govern the discretetime problem, where, can be appro ximated by: mreduce( ) can carry out si ngular pe rturbatio n. For fu rther di scussion , refer to Chapter 2, Addi tive Erro r Reduct ion . If Equation 11 is b alanced, [...]

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Chapter 1 Introduction Xmath Model R eduction M odule 114 ni.com nonnegati ve h ermitian for all ω . If Φ is scalar , then Φ ( j ω ) ≥ 0 for all ω . Normally one restricts attention to Φ (·) with lim ω→∞ Φ ( j ω )< ∞ . A key result is that, gi ven a rational, nonnegati ve hermitian Φ ( j ω ) with lim ω→∞ Φ ( j ω )<[...]

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Chap ter 1 Int rodu cti on © Natio nal Instrum ents Cor poration 115 Xma th Model Re ductio n Module Low Order Controller De sign Through Order Reduction The Model Reduction Mo dule is par ticularly sui table for ach ieving l ow order cont roller des ign for a high or der plant. Th is section ex plains s ome of the bro ad issu es involved . Most [...]

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Chapter 1 Introduction Xmath Model R eduction M odule 116 ni.com multiplicati ve red uction, as described in Chapter 4, Frequ encyWeighted Error Redu ction , is a sound appro ach. Chapter 3, Multiplicative Error Reduction , and Ch apter 4, Frequency Weighted Error Reduction , de v elop these ar guments mor e fully .[...]

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© Nationa l Instrume nts Corpora tion 21 Xmath Mod el Redu ction Modu le 2 Additive Error Reduction This chapter describes additive error reduction includin g discussions of truncation of, redu ction by, and p erturbatio n of balanced real izations. Introduction Additive erro r reduction focuses o n errors of t he form, where G is the originally [...]

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Chapter 2 Additive Error Red uction Xmath Model R eduction M odule 22 ni.com T runcation of Bal anced Realizations A group of funct ions can be used to achieve a reduct ion through tru ncation of a balanced r ealization. This m eans that if the original s ystem is (21) and the realization is internally balanced, then a tru ncation is provided b y[...]

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Chapte r 2 Additive Erro r Reduc tion © Nationa l Instrume nts Corpora tion 23 Xmath Mod el Redu ction Modu le A very attracti ve f eature of the truncation procedure is the av ailability of an error bound. More precisely , suppose that the controllability and observability grammians for [E nn84] are (22) with the diagonal entries of Σ in decre[...]

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Chapter 2 Additive Error Red uction Xmath Model R eduction M odule 24 ni.com proper . So, ev en if all zeros are un stable, the max imum phase shift when ω mov es from 0 to ∞ is (2 n – 3) π /2 . It follows that if G ( j ω ) rem ains large in magnitude at frequen cies when the phase s hift has mo ved pas t (2n – 3) π /2, approximat ion of[...]

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Chapte r 2 Additive Erro r Reduc tion © Nationa l Instrume nts Corpora tion 25 Xmath Mod el Redu ction Modu le order model is not one in general obtainable by truncation of an internallybalanced realization of the full order model. Figure 21 sets o ut se veral rou tes to a reducedo rder realization. In continuous time, a truncation of a bala n[...]

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Chapter 2 Additive Error Red uction Xmath Model R eduction M odule 26 ni.com with controllability and obser vab ility grammians given b y , in which the diagonal entries o f Σ are in decreasing order , that is, σ 1 ≥σ 2 ≥ ···, and su ch that the last diagonal entry of Σ 1 ex ceeds the fir st diagonal entry of Σ 2 . It turns out that Re[...]

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Chapte r 2 Additive Erro r Reduc tion © Nationa l Instrume nts Corpora tion 27 Xmath Mod el Redu ction Modu le function matrix. Consider th e way th e associated impulse resp onse maps inputs defined over (– ∞ ,0] in L 2 into output s, and focus on the output over [0, ∞ ). Define the input as u ( t ) for t <0 , a n d s e t v ( t )= u (?[...]

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Chapter 2 Additive Error Red uction Xmath Model R eduction M odule 28 ni.com Further , the which is optimal for Hankel norm approxim ation also is optimal for this seco nd type o f approximat ion. In Xmath Han kel no rm approximat ion is achie ved with ophank( ) . The most com prehensive ref erence is [Glo84]. balmoore( ) [SysR,HSV,T] = balmoore(S[...]

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Chapte r 2 Additive Erro r Reduc tion © Nationa l Instrume nts Corpora tion 29 Xmath Mod el Redu ction Modu le of the balanced system occurs, ( assuming nsr is less than the number of states). Thus, if the statespace repres e ntation of the balanced system is with A 11 p ossessing dimensi on nsr × nsr , B 1 posses sing nsr ro ws and C 1 posse s[...]

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Chapter 2 Additive Error Red uction Xmath Model R eduction M odule 210 ni.com The actual ap proximati on error for discrete systems als o depends on frequenc y , and can be lar ge at ω = 0. The error bou nd is almos t ne v er tight, that is, the actual erro r magnitude as a function o f ω alm ost never at ta ins the error boun d, so that the bou[...]

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Chapte r 2 Additive Erro r Reduc tion © Natio nal Instrum ents Cor poration 211 Xma th Model Re ductio n Module Related F unctions balance() , truncate() , redschur () , mreduce() truncate( ) SysR = truncate(Sys,nsr,{VD,VA}) The truncate( ) function r educes a system Sys b y retaining the first nsr states and throwing aw ay the rest to form a s y[...]

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Chapter 2 Additive Error Red uction Xmath Model R eduction M odule 212 ni.com redschur( ) [SysR,HSV,slbig,srbig,VD,VA] = redschur(Sys,{nsr,bound}) The redschur( ) functi on us es a Sc hur m ethod (from Safonov a nd Chiang) t o calculate a reduced v ersion of a continu ous or dis crete system withou t balancin g. Algorith m The objective of redschu[...]

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Chapte r 2 Additive Erro r Reduc tion © Natio nal Instrum ents Cor poration 213 Xma th Model Re ductio n Module Nex t, Schur decomposi tions of W c W o are formed with th e eigen v alues of W c W o in ascending and descending order . These eigenv alues are the square of the Hankel singular v alues of Sys , and if Sys is non minimal, some can be z[...]

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Chapter 2 Additive Error Red uction Xmath Model R eduction M odule 214 ni.com For the d iscretetime case: When {bound} is specif ied, the error bound jus t enunciated is use d to choose the nu mber of states in SysR so that the bound is satisfied and nsr is as small as possible. If the desired error bound is smaller than 2 σ ns , no reduction is[...]

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Chapte r 2 Additive Erro r Reduc tion © Natio nal Instrum ents Cor poration 215 Xma th Model Re ductio n Module Algorith m The algorithm do es the following. T he system Sys an d the reduced or der system SysR are stable; the system SysU h as all its poles in Re [ s ] > 0. If the transfer f unction matrices are G ( s ), G r ( s ) and G u ( s )[...]

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Chapter 2 Additive Error Red uction Xmath Model R eduction M odule 216 ni.com By ab use of notatio n, when we say that G is r educed to a certain order, this correspon ds to the order of G r ( s ) alone; the uns table part of G u ( s ) of the approximation is mo st frequently thrown away . The number of eliminated states (retaining G u ) refer s t[...]

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Chapte r 2 Additive Erro r Reduc tion © Natio nal Instrum ents Cor poration 217 Xma th Model Re ductio n Module Thus, the penalty for not being allo wed to include G u in the approximation is an increase in the er ror bound , b y σ n i + 1 + ... + σ ns . A num ber of theoretical dev el opment s hinge on boundi ng the Hank el singular val ues of[...]

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Chapter 2 Additive Error Red uction Xmath Model R eduction M odule 218 ni.com being appro ximated by a stable G r ( s ) with the actual error (as opposed to just the er ror bou nd) satis fying: Note G r is optimal, that is, there is no other G r achie ving a lower b ound. Onepass Algorithm The first steps of the algorithm are to obtain the Hankel [...]

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Chapte r 2 Additive Erro r Reduc tion © Natio nal Instrum ents Cor poration 219 Xma th Model Re ductio n Module and f inally: These four matrices are the constituents o f the system matrix of , where: Digress ion: This choice is related to the ideas of [Glo84] in the foll o wing way; in [Glo84], the complete set is id entified of satisfying with [...]

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Chapter 2 Additive Error Red uction Xmath Model R eduction M odule 220 ni.com to choose the D matrix of G r ( s ), by splitting between G r ( s ) and G u ( s ). This is done by using a separate function ophiter( ) . Suppos e G u ( s ) is the un sta ble ou tput of stable( ) , a nd le t K ( s )= G u (– s ). By ap plyi ng t he multipass Hankel redu[...]

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Chapte r 2 Additive Erro r Reduc tion © Natio nal Instrum ents Cor poration 221 Xma th Model Re ductio n Module 2. Find a stab le order ns – 2 approxi mation G ns –2 of G ns –1 ( s ), with 3. (S tep ns–nr ) : Find a stable orde r nsr approximation of G nsr +1 , with Then, becaus e for , for , ..., this being a property of the algor ithm, [...]

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Chapter 2 Additive Error Red uction Xmath Model R eduction M odule 222 ni.com We u s e sysZ to den ote G(z) and def ine: bilinsys=makepoly([1,a]/makepoly([1,a ]) as the mappin g from the zdomain to the sdomain. The specif ication is re vers ed because this fu nction uses b ackward poly nomial rotation. Ha nkel norm reduction is then applied to [...]

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Chapte r 2 Additive Erro r Reduc tion © Natio nal Instrum ents Cor poration 223 Xma th Model Re ductio n Module It follows by a result of [BoD87] th at the impulse response error for t >0 satisfies: Evidently , Hankel norm approx imation ensures some fo rm of approximat ion of th e impulse res ponse too. Unstable System Approximation A transf [...]

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© Nationa l Instrume nts Corpora tion 31 Xmath Mod el Redu ction Modu le 3 Multiplicative Error Reduction This chapter describes multiplicative error reductio n presenting two reasons to consider multiplicativ e rather than additive error reduction, one general and on e specific. Selecting Multiplicative Error Reduction The general reason to use [...]

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Chapter 3 Multiplicative Error Reduction Xmath Model R eduction M odule 32 ni.com Multipl icative Robustness Re sult Suppose C stab ilizes , that has no j ω axis pol es, and that G has the s ame number of poles in Re [ s ] ≥ 0 as . If for a ll ω, (31) then C stabilizes G . This result indicates that if a controller C is designed to stabili z[...]

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Chapte r 3 Mult iplicative Erro r Reduction © Nationa l Instrume nts Corpora tion 33 Xmath Mod el Redu ction Modu le bandwidth at the e xpense o f being lar ger out side this bandwidth, which would be preferable. Second, the previously used multiplicati ve error is . In the algorithms that follow , the er ror appears. It is easy to check that: an[...]

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Chapter 3 Multiplicative Error Reduction Xmath Model R eduction M odule 34 ni.com The objecti ve of the algorithm is to ap proximate a highorder stable transfer function matri x G ( s ) b y a lower order G r ( s ) with either in v(g)(ggr) or (ggr)inv(g) minimi zed, under the condition t hat G r is stable and of the prescribed order . Restricti[...]

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Chapte r 3 Mult iplicative Erro r Reduction © Nationa l Instrume nts Corpora tion 35 Xmath Mod el Redu ction Modu le These cases are secured with the keyw ords right and left , respecti v ely . If the wrong opt ion is req uested for a nonsq uare G ( s ) , an error message will result. The algorithm has the prop erty that right half plane zeros of[...]

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Chapter 3 Multiplicative Error Reduction Xmath Model R eduction M odule 36 ni.com 2. W ith G ( s )= D + C ( sI – A ) –1 B and stable, with DD ´ nonsingul ar and G ( j ω ) G '(– j ω ) nonsin gular for al l ω , part of a state va riable realization of a minimum phase stab le W ( s ) is determined such that W´(–s) W(s) = G(s)G´ (?[...]

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Chapte r 3 Mult iplicative Erro r Reduction © Nationa l Instrume nts Corpora tion 37 Xmath Mod el Redu ction Modu le strictly proper s table par t of θ ( s ) , as the square r oots of the eigen v alues of PQ . Call these q uantities ν i . The Schur decompositions are, where V A , V D are orthogonal and S asc , S des are upper tri angular . 4. D[...]

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Chapter 3 Multiplicative Error Reduction Xmath Model R eduction M odule 38 ni.com statev ariable representation of G . In this case, the user is ef fectiv ely asking for G r = G . When the phase matrix h as repeated Hankel s ingular v alues, they must all be included or all excluded from the model, that is, ν nsr = ν nsr + 1 is not permitted; t[...]

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Chapte r 3 Mult iplicative Erro r Reduction © Nationa l Instrume nts Corpora tion 39 Xmath Mod el Redu ction Modu le Hankel Singul ar Values of Phase Matrix of G r The ν i , i = 1 ,2,..., ns have been termed above the Hank el singular v alues of the phase matrix associated with G . The c orrespondin g quant ities for G r are ν i , i = 1,..., ns[...]

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Chapter 3 Multiplicative Error Reduction Xmath Model R eduction M odule 310 ni.com which also can be rele vant i n find ing a redu ced order model of a plant. The procedure requires G again to be n onsingul ar at ω = ∞ , an d to ha v e no j ω axis poles. It is as follows: 1. Form H = G –1 . If G is des cribed b y stat ev ariable matrice s [...]

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Chapte r 3 Mult iplicative Erro r Reduction © Natio nal Instrum ents Cor poration 311 Xma th Model Re ductio n Module The v alues of G ( s ), as sho wn in Figure 32, al ong the j ω axis are the same as the v alues o f around a circle with diameter def ined b y [ a – j 0, b –1 + j 0] on the positi ve real axis. Figu re 3 2. Biline ar Mapp [...]

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Chapter 3 Multiplicative Error Reduction Xmath Model R eduction M odule 312 ni.com An y zero (or rank r eduction) o n the j ω axis of G ( s ) becomes a zero (or rank reduction) in Re [ s ] > 0 of , and if G ( s ) has a zero ( or rank reduction) at infinity , thi s is shifted to a zero (or rank reduction) of at the point b –1 , (in Re [ s ] [...]

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Chapte r 3 Mult iplicative Erro r Reduction © Natio nal Instrum ents Cor poration 313 Xma th Model Re ductio n Module again with a bilinear transformation to secure multip licati ve approximat ions o ver a limited fr equenc y band. S uppose that Create a system that co rresponds to with: gtildesys=subs(gsys,(makep([eps,1])/m akep([1,])) bilinsy[...]

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Chapter 3 Multiplicative Error Reduction Xmath Model R eduction M odule 314 ni.com There is one potential source of f ailure of th e algorithm. Because G ( s ) i s stable, certainly will be, as i ts poles wil l be in th e left half p lane circle on diameter . If acq uires a pole outside this circle (but still in the left half plane of course)—an[...]

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Chapte r 3 Mult iplicative Erro r Reduction © Natio nal Instrum ents Cor poration 315 Xma th Model Re ductio n Module The conceptual b asis of the algo rithm can best be g rasped b y considering the case of scalar G ( s ) o f de gree n . Then one can form a minimum phas e, stable W ( s ) with  W ( j ω ) 2 =  G ( j ω ) 2 and then an allpas [...]

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Chapter 3 Multiplicative Error Reduction Xmath Model R eduction M odule 316 ni.com eigen v al ues of A – B/D * C with the aid of schur( ) . If an y real part of th e eigenvalues is less than eps , a war ning is displayed. Next, a stabilizing solution Q is found for the fo llo wing Riccati equation: The funct ion singriccati( ) is used; f ailure [...]

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Chapte r 3 Mult iplicative Erro r Reduction © Natio nal Instrum ents Cor poration 317 Xma th Model Re ductio n Module singu lar v alues of F ( s ) lar ger than 1– ε (ref er to step s 1 thro ugh 3 of the Restrictions section). The ma ximum order per mitted is t he number o f nonzero eigen values o f W c W o larger than ε . 4. Le t r be the mul[...]

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Chapter 3 Multiplicative Error Reduction Xmath Model R eduction M odule 318 ni.com Note The e xpression is the strictly proper part of . The matrix is all pass; this property is not alway s secured in the multiv ariable case when ophank( ) is used to f ind a Hank el norm app roximation o f F ( s ). 5. The algorithm constructs and , which satisfy ,[...]

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Chapte r 3 Mult iplicative Erro r Reduction © Natio nal Instrum ents Cor poration 319 Xma th Model Re ductio n Module • and stand in the s ame relation as W ( s ) and G ( s ), that is: – – W ith , there holds or – W ith there holds or – – is the stab le strictly proper part of . • The Hankel s ingular v alues of (and ) ar e the firs[...]

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Chapter 3 Multiplicative Error Reduction Xmath Model R eduction M odule 320 ni.com Error Bounds The error b ound formu la (Equation 33) is a simple consequence o f iterating (Equation 35). To illustrate, suppose there ar e three reductions →→ → , each by degree one. Th en, Also, Similarly , Then: The error b ound (Equation 3 3) is on ly e[...]

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Chapte r 3 Mult iplicative Erro r Reduction © Natio nal Instrum ents Cor poration 321 Xma th Model Re ductio n Module For mulhank( ) , this translates for a scalar system into and The bound s are do uble for bst( ) . The error as a fu nction of frequency is always zero at ω = ∞ for bst( ) (or at ω = 0 if a tran sformation s → s –1 is used[...]

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Chapter 3 Multiplicative Error Reduction Xmath Model R eduction M odule 322 ni.com The v alues of G ( s ) along t he j ω axis are the same as the v alues of around a circle with diameter d efined b y [ a – j 0, b –1 + j 0] on th e positi ve real axis (refer to Figure 32). Also, th e valu es of along the j ω axis are the same as the values[...]

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Chapte r 3 Mult iplicative Erro r Reduction © Natio nal Instrum ents Cor poration 323 Xma th Model Re ductio n Module The error will be ov erbounded by the error , and G r will contain the same zeros in Re [ s ] ≥ 0 a s G . If th ere i s no zer o (or rank r educ tion) of G ( s ) at the origin , one can take a =0 a n d b –1 = ban dwidth ov er [...]

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Chapter 3 Multiplicative Error Reduction Xmath Model R eduction M odule 324 ni.com Multiplicativ e approximation of (along the j ω axis) corresp onds to multiplicat i ve appr oximatio n of G ( s ) around a circle in the r ight half plane, touching the j ω axis at the origin. For those points on the j ω axis near the circle, there will be goo[...]

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© Nationa l Instrume nts Corpora tion 41 Xmath Mod el Redu ction Modu le 4 FrequencyW eighted Error Reduction This chapter descr ibes frequencyweighted error reduction problems. This includes a discuss ion of controller reduction and fractional repres entations. Introduction Frequencyw eighted erro r reduction m eans that the erro r is measure[...]

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Chapter 4 Frequenc yWeighted Error R eduction Xmath Model R eduction M odule 42 ni.com (so that ) is logical. Howe ver , a major use of weighting is in controll er reductio n, which i s no w descr ibed. Controller Red uction Frequency weighted error reducti on become s particul arly impor tant in reducing controller dimen sion. The LQG and desi g[...]

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Chapte r 4 Frequency Weighted Erro r Reduc tion © Nationa l Instrume nts Corpora tion 43 Xmath Mod el Redu ction Modu le is minimized (and of course is less th an 1). Notice that these two error measures are like thos e of Equation 41 and Equation 42. The f act that the plant ought to show up in a goo d formulation of a controller reductio n p[...]

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Chapter 4 Frequenc yWeighted Error R eduction Xmath Model R eduction M odule 44 ni.com Most of thes e ideas are discuss ed in [Enn84 ], [AnL89], and [AnM89]. The fun ction wt balance( ) implements weighted reduct ion, with fi v e choices of error measure, namely E IS , E OS , E M , E MS , and E 1 with arbitrary V( j ω ). The first fo ur are spec[...]

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Chapte r 4 Frequency Weighted Erro r Reduc tion © Nationa l Instrume nts Corpora tion 45 Xmath Mod el Redu ction Modu le Fractional R epresentatio ns The treatment of j ω axis or right half plane poles in the above schemes is crude: they are simply co pied into the reduced order contro ller. A differen t approach com es when one uses a so cal[...]

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Chapter 4 Frequenc yWeighted Error R eduction Xmath Model R eduction M odule 46 ni.com • For m the redu ced controll er by intercon necting us ing ne gati v e feedback the secon d output of G r to the input, that is, set Nothing h as been sa id as to ho w shou ld be chosen— and the en d result of the reduction, C r ( s ), depends on . Nor has[...]

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Chapte r 4 Frequency Weighted Erro r Reduc tion © Nationa l Instrume nts Corpora tion 47 Xmath Mod el Redu ction Modu le Matrix algebra shows that C ( s ) can be desc ribe d thr ough a left or righ t matrix fraction descr iption with D L , and related values, all stable transfer function matrices. In particul ar: For matrix C(s), the left and ri[...]

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Chapter 4 Frequenc yWeighted Error R eduction Xmath Model R eduction M odule 48 ni.com The left MF D corresp onds to the setup of Figu re 43. Figu re 43 . C(s) Implement ed to Display L eft MFD Repr esentatio n The setu p of Figur e 42 su ggests approxim ation of : whereas that of Figu re 43 s uggests appro ximation of: In the LQG optimal cas[...]

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Chapte r 4 Frequency Weighted Erro r Reduc tion © Nationa l Instrume nts Corpora tion 49 Xmath Mod el Redu ction Modu le Figu re 4 4. Redrawn; Indivi dual Sign al Paths as Vector Path s It is possible to verify that and according ly the output weig ht can be us ed in an error measure . It turns out that the calculations for frequenc y weighted [...]

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Chapter 4 Frequenc yWeighted Error R eduction Xmath Model R eduction M odule 410 ni.com (Here, the W i and V i are submatr ices of W ,V .) Evidently , Some manipulation sh o ws that trying to preserve these identities after approximat ion of D L , N L or N R , D R suggests use of the err or measures and . For fu rther details , refer to [AnM89 ] [...]

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Chapte r 4 Frequency Weighted Erro r Reduc tion © Natio nal Instrum ents Cor poration 411 Xma th Model Re ductio n Module • Reduce the or der of a transfer fu nction matrix C ( s ) t hrough frequenc ywe ighted bala nced truncatio n, a stable frequenc y wei ght V ( s ) being prescribed. The syntax is more accented to wards the fi rst use. For [...]

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Chapter 4 Frequenc yWeighted Error R eduction Xmath Model R eduction M odule 412 ni.com This rather crude ap proach to the han dling of the un stable part of a controll er is a v oided i n fracred( ) , whi ch provi des an alter nat iv e to wtbalance( ) for controller reduction, at least for an important family of contr oll ers. Algorith m The maj[...]

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Chapte r 4 Frequency Weighted Erro r Reduc tion © Natio nal Instrum ents Cor poration 413 Xma th Model Re ductio n Module 3. Comput e weighted Hankel Si ngular V alues σ i (d escribed in more detail later). If the order of C r ( s ) is not specif ied a p riori , it must be input at this time. Certain v alues may be flagged as unacceptab le for [...]

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Chapter 4 Frequenc yWeighted Error R eduction Xmath Model R eduction M odule 414 ni.com and the observ ability grammian Q , defined in the obvious way , is written as It is trivial to ver ify that so that Q cc is the observability gramian of C s ( s ) alone, as well as a s ubmatrix of Q . The weight ed Hankel singul ar val ues of C s ( s ) ar e t[...]

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Chapte r 4 Frequency Weighted Erro r Reduc tion © Natio nal Instrum ents Cor poration 415 Xma th Model Re ductio n Module From these quantities th e transformation matrices used for calculatin g C sr ( s ), the stable part of C r ( s ), are d efined and then Just as in un weighted balanced truncation, the reduced order transfer function matrix i[...]

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Chapter 4 Frequenc yWeighted Error R eduction Xmath Model R eduction M odule 416 ni.com 3. Only continuou s systems are accep ted; for discrete syst ems use makecontinuous( ) before call ing bst( ) , then discretize the result. Sys=fracred(makecontinuous(SysD)); SysD=discretize(Sys); Defining and Reducing a Controller Suppose P ( s ) = C ( sI –[...]

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Chapte r 4 Frequency Weighted Erro r Reduc tion © Natio nal Instrum ents Cor poration 417 Xma th Model Re ductio n Module to, for e xample, throug h, for e xample, bala nced tru ncation, and then def ining: For the second rationale, consider Figure 45. Figur e 45. Inte rnal Struc ture of Contro ller Recognize that the contro ller C ( s ) (show[...]

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Chapter 4 Frequenc yWeighted Error R eduction Xmath Model R eduction M odule 418 ni.com Controller reduction proceeds b y implem enting the same connection rule bu t on reduce d v ersions of the tw o transfer function matrices. When K E has been d efined through Kalman f iltering con siderations, the spectrum o f the si gnal dri ving K E in Figur[...]

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Chapte r 4 Frequency Weighted Erro r Reduc tion © Natio nal Instrum ents Cor poration 419 Xma th Model Re ductio n Module 6. Check the stability of th e closedloop syst em with C r ( s ). When the type="left perf" is specif ied, one w orks with (41 1) which is f ormed fr om the numerat or and deno minator of the MFD in Equation 45. [...]

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Chapter 4 Frequenc yWeighted Error R eduction Xmath Model R eduction M odule 420 ni.com Additional Background A discussion of the stability robu stness measure can be found in [AnM89] and [LAL90]. The idea can be un derstood with reference to the transfer functions E ( s ) a nd E r ( s ) used in d iscussi ng type="right perf" . It is po[...]

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Chapte r 4 Frequency Weighted Erro r Reduc tion © Natio nal Instrum ents Cor poration 421 Xma th Model Re ductio n Module The four schemes all produce dif ferent HSVs; it follows that it may be prudent to try all four schemes for a particular controller reductio n. Recall again that their relati ve s izes are only a guide as to what can b e thro[...]

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© Nationa l Instrume nts Corpora tion 51 Xmath Mod el Redu ction Modu le 5 Utilities This chapter describes three utility fun ctions: hankelsv( ) , stable( ) , and compare( ) . The backgro und to hankelsv( ) , which calculates Hankel singular v alues, was pres ented in Chapter 1, Introd uction . Hankel s ingula r v alues are also calculated in ot[...]

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Chapte r 5 Utiliti es Xmath Model R eduction M odule 52 ni.com The grami an matrices are defi ned by solving the equat ions (in cont inuous time) and, in discrete time The computations are ef fected with lyapunov( ) and stability is checked, which is t imeconsumi ng. The Hank el sing ular v alues are t he square r oots of the eigen values of the [...]

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Chapte r 5 Utili ties © Nationa l Instrume nts Corpora tion 53 Xmath Mod el Redu ction Modu le Doubtful ones are those for which the real part of the eigen v al ue has magnitude less than or equal to tol fo r continuo ustime, or eigen value magnitude within the following range for discrete time: A warn ing is gi ven if dou btful eigen v alues ex[...]

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Chapte r 5 Utiliti es Xmath Model R eduction M odule 54 ni.com After this last transform ation, and with it follows that and By combini ng the transf ormation yi elding the real ordered Schur form for A with th e transfor mation def ined using X, th e ov erall tra nsform ation T is readily identif ied. In case all eigen v alue s of A are stable or[...]

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© Nationa l Instrume nts Corpora tion 61 Xmath Mod el Redu ction Modu le 6 T utorial This chapter illustrates a number of the MRM functio ns and their underly ing ideas. A plant and fu llord er controll er are defi ned, and th en the effects of various red uction algo rith ms ar e examined. Th e data for this example is stored in the file mr_dis[...]

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Chapter 6 T utoria l Xmath Model R eduction M odule 62 ni.com A minimal realization in modal coordinates is C ( sI – A ) –1 B where: The specifications seek high loop gain at low frequ encies (f or perfor mance) and low loop gain at high f requencies (to guar antee st ability in the presen ce of unstructured uncertainty). More sp ecifically, t[...]

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Chap ter 6 T utoria l © Nationa l Instrume nts Corpora tion 63 Xmath Mod el Redu ction Modu le With a state weightin g matrix, Q = 1e3*diag([2,2,80,80,8,8,3,3]); R = 1; (and unity control weighting), a statef eedback controlgain is de termined throug h a linearqu adratic p erformance index minimi zation as: [Kr,ev] = regulator(sys,Q,R); A –[...]

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Chapter 6 T utoria l Xmath Model R eduction M odule 64 ni.com recovery at low f requencies; there is consequ ently a faster rollo ff of the loop gain at high f requencies than for , and this is desi red. Figure 6 2 displays the (magn itudes of t he) plant tran sfer fun ction, th e compensator transfer functio n and the loop gain, as well as the [...]

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Chap ter 6 T utoria l © Nationa l Instrume nts Corpora tion 65 Xmath Mod el Redu ction Modu le Controller Reduction This section contras ts the effect of u nweighted and weighted controller reduction. U nweighted reduct ion is at f irst examined, through redschur( ) (usi ng balance( ) or balmoore( ) will give similar results). The Hank el singula[...]

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Chapter 6 T utoria l Xmath Model R eduction M odule 66 ni.com Figures 63, 64, and 65 displ ay the outcome o f the redu ction. The loo p gain is shown in Figure 63. The error near the unity gain cros sover frequency m ay not look large, but it is con siderably larger than that obtained throu gh frequ ency weight ed reductio n methods, as descri[...]

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Chap ter 6 T utoria l © Nationa l Instrume nts Corpora tion 67 Xmath Mod el Redu ction Modu le Gen erate Figu re 6 4: compare(syscl,sysclr,w,{radians,type=5 }) f4=plot({keep,legend=["original","redu ced"]}) Figu re 64 . Closed Loo p Gain with redsch ur[...]

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Chapter 6 T utoria l Xmath Model R eduction M odule 68 ni.com Gen erate Figu re 6 5: tvec=0:(140/99):140; compare(syscl,sysclr,tvec,{type=7}) f5=plot({keep,legend=["original","redu ced"]}) Figu re 65 . Step Response with redschur[...]

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Chap ter 6 T utoria l © Nationa l Instrume nts Corpora tion 69 Xmath Mod el Redu ction Modu le ophank( ) ophank( ) is next used to reduce the controller with the re sults shown in Figures 6 6, 67, an d 68. Gen erate Figu re 6 6: [syscr,sysu,hsv]=ophank(sysc,2); svalsrol = svplot(sys*syscr,w,{radians }); plot(svalsol, {keep}) f6=plot(wc, const[...]

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Chapter 6 T utoria l Xmath Model R eduction M odule 610 ni.com Gen erate Figu re 6 7: syscl = feedback(sysol); sysolr=sys*syscr; sysclr=feedback(sysolr); compare(syscl,sysclr,w,{radians,type=5 }) f7=plot({keep,legend=["original","redu ced"]}) Figur e 67. ClosedL oop Ga in with ophank[...]

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Chap ter 6 T utoria l © Natio nal Instrum ents Cor poration 611 Xma th Model Re ductio n Module Gen erate Figu re 6 8: tvec=0:(140/99):140; compare(syscl,sysclr,tvec,{type=7}) f8=plot({keep,legend=["original","redu ced"]}) Figur e 68. St ep Re spons e with o phank The openl oop gai n, closedloop gain and s tep resp onse ar[...]

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Chapter 6 T utoria l Xmath Model R eduction M odule 612 ni.com wtbalan ce The next comman d examined is wtbal ance with the option "match" . [syscr,ysclr,hsv] = wtbalance(sys,sysc ,"match",2) Recall that this command should p romote matching of clo sedloop transfer functions. The weigh ted Hankel sing ular values are: 1.486 4.[...]

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Chap ter 6 T utoria l © Natio nal Instrum ents Cor poration 613 Xma th Model Re ductio n Module The following fu nction calls produce Figur e 69: svalsrol = svplot(sys*syscr,w,{radians }) plot(svalsol, {keep}) f9=plot(wc, constr, {keep,!grid, legend=["reduced","original","constrai ned"], title="OpenLoop Gain U[...]

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Chapter 6 T utoria l Xmath Model R eduction M odule 614 ni.com Gen erate Figu re 6 10: syscl = feedback(sysol); sysolr=sys*syscr; sysclr=feedback(sysolr); compare(syscl,sysclr,w,{radians,type=5 }) f10=plot({keep,legend=["original","red uced"]}) Figu re 61 0. Clos edL oop G ain with wtb alance[...]

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Chap ter 6 T utoria l © Natio nal Instrum ents Cor poration 615 Xma th Model Re ductio n Module Gen erate Figu re 6 11: tvec=0:(140/99):140; compare(syscl,sysclr,tvec,{type=7}) f11=plot({keep,legend=["original","red uced"]}) Figu re 611. Ste p Response wit h wtbalance Figures 69, 610, and 611 are obtained for wtbalance wi[...]

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Chapter 6 T utoria l Xmath Model R eduction M odule 616 ni.com Gen erate Figu re 6 12: vtf=poly([0.1,10])/poly([1,1.4]) [,sysv]=check(vtf,{ss,convert}); svalsv = svplot(sysv,w,{radians}); Figu re 612 . Frequency Re sponse of th e Weight V( j ω )[...]

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Chap ter 6 T utoria l © Natio nal Instrum ents Cor poration 617 Xma th Model Re ductio n Module Gen erate Figu re 6 13: [syscr,sysclr,hsv] = wtbalance(sys,sys c, "input spec",2,sysv) svalsrol = svplot(sys*syscr,w,{radians }) plot(svalsol, {keep}) f13=plot(wc,constr,{keep, !grid, legend=["reduced","original","co[...]

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Chapter 6 T utoria l Xmath Model R eduction M odule 618 ni.com Gen erate Figu re 6 14: syscl = feedback(sysol); sysolr=sys*syscr; sysclr=feedback(sysolr); compare(syscl,sysclr,w,{radians,type=5 }) f14=plot({keep,legend=["original","red uced"]}) Figu re 614 . System Sing ular Values of wtbalanc e with "i nput spec "[...]

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Chap ter 6 T utoria l © Natio nal Instrum ents Cor poration 619 Xma th Model Re ductio n Module Gen erate Figu re 6 15: tvec=0:(140/99):140; compare(syscl,sysclr,tvec,{type=7}) f15=plot({keep,legend=["original","red uced"]}) Figur e 615. Step Respon se of wtba lanc e with "inp ut spec "[...]

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Chapter 6 T utoria l Xmath Model R eduction M odule 620 ni.com fracred fracred , the next command examined, has f our options — "right stab" , "left stab" , "right perf " , and "left perf" . The optio ns "left stab" , "right perf" , and "left perf" all produce instability. Giv[...]

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Chap ter 6 T utoria l © Natio nal Instrum ents Cor poration 621 Xma th Model Re ductio n Module Gen erate Figu re 6 17: syscl = feedback(sysol); sysolr=sys*syscr; sysclr=feedback(sysolr); compare(syscl,sysclr,w,{radians,type=5 }) f17=plot({keep,legend=["original","red uced"]}) Figu re 617. Closed Loop Respon se with fracred[...]

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Chapter 6 T utoria l Xmath Model R eduction M odule 622 ni.com Gen erate Figu re 6 18: tvec=0:(140/99):140; compare(syscl,sysclr,tvec,{type=7}) f18=plot({keep,legend=["original","red uced"]}) Figur e 618. Step Response with fracre d The end result is comparable to that from wtbalance( ) with option "match" . We can [...]

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Chap ter 6 T utoria l © Natio nal Instrum ents Cor poration 623 Xma th Model Re ductio n Module hsvtable = [... "right stab:", string(hsvrs'); "left stab:", string(hsvls'); "right perf:", string(hsvrp'); "left perf:", string(hsvlp')]? hsvtable (a rectangular matrix of stri ngs) = right s[...]

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© Nationa l Instrume nts Corpora tion A1 Xmath Mod el Redu ction Modu le A Bibliography [AnJ] BDO Anderson and B. James, “ Algorithm for multiplicati v e approximation of a s table linear system, ” in preparation. [AnL89] BDO Anderson and Y . Liu, “C ontroll er reduction : Concepts and approaches , ” IEEE T r ansa ction s on Auto matic Co[...]

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Appendix A Bibliogr aphy Xmath Model R eduction M odule A2 ni.com [GrA90] M. Green an d BDO Anderson, “Gener alized balanced stochas tic truncation, ” Pr oceedings for 29th CDC , 1990. [Gre88] M. Green, “Balanced stochastic realization, ” Linear Alg ebra and Ap plications , V ol. 98, 1988, pp . 211–2 47. [Gre88a] M. Green , “ A relati [...]

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Appendix A Bibliography © Nationa l Instrume nts Corpora tion A3 Xmath Mod el Redu ction Modu le [SaC88] M. G. Safono v and R. Y . Chiang , “Model redu ction for r ob ust control : a Schur relati v eerror m ethod, ” Pr oceedings for the American Contr ols Confer ence , 1988, pp. 1685 –1690. [Saf87] M. G. Saf onov , “Imaginaryaxi s zeros[...]

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Appendix A Bibliogr aphy Xmath Model R eduction M odule A4 ni.com [Do y82] J. C. Doyle. “ Analysis of Feedback Systems with Struct ured Uncertainties. ” IEEE Pr oceedings , Nove mber 1982. [ D W S 8 2 ] J . C . D o y l e , J . E . Wa l l , a n d G . S t e i n . “Performance and Robu stness Analysis for Structu re Uncertainties, ” Proceedin[...]

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Appendix A Bibliography © Nationa l Instrume nts Corpora tion A5 Xmath Mod el Redu ction Modu le [SLH81] M. G. Safono v , A. J. Laub, and G. L. Hartman n, “Feedback Prop erties of Multi variable Systems: The R ole and Use of the Return Dif ference Matrix, ” IEEE T ransac tions on Automatic Contr ol , V ol. A C26, Februar y 1981. [SA88] G. S [...]

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© Nationa l Instrume nts Corpora tion B1 Xmath Mod el Redu ction Modu le B T echnical Support and Professional Ser vices Visit the followin g sections of the Nationa l Instruments Web site at ni.com for technical suppor t and prof essional services: • Support — Online technical s upport resources at ni.c om/support include the following: – [...]

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© Nationa l Instrume nts Corpora tion I1 Xmath Mod el Redu ction Modu le Index Symbols *, 16 ´, 16 A additive error, reduction, 21 algorithm bala nced stocha stic trunc atio n (bst) , 34 fractional re presentat ion redu ction, 418 Hankel multipass, 220 optimal Ha nkel norm redu ction, 215 stable, 5 2 weighted bal ance, 412 allpass tra[...]

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Index Xmath Model R eduction M odule I2 ni.com G grammians controllability, 1 7 desc ripti on of , 17 observability, 1 7 H Hankel matrix, 19 Hankel no rm approxi mation, 26 Hankel si ngular values, 18, 39 , 51 hankels v, 15, 51 algorithm, m ultipass, 2 20 help, technical support, B1 I instrum ent drivers (NI resources ), B1 internal b[...]

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Index © Nationa l Instrume nts Corpora tion I3 Xmath Mod el Redu ction Modu le stable, 1 5, 52 sup, 16 suppor t, technical, B 1 T technical support, B 1 tight equality bounds, 17 training and certification (NI r esources), B1 transfer function, allpass, 16 troubles hooting (NI resou rces), B1 truncate, 1 5, 24, 2 11 U unstable zer os,[...]