National Instruments 370755B-01 manuale d’uso

MA TRIXx TM Xmath TM Model Reduction Module Xmath Mo del Reducti on Modul e April 2004 Edit ion Part Numb er 37075 5B-01[...]

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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[...]

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

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

Contents © Nationa l Instrume nts Corpora tion vii Xmath Mod el Redu ction Modu le Algorithm ....... ............ ................ ............ ........... ................. ........... ................. . 4-18 Additional Backgr ound .................... ........... ................. ............ ........... ............ 4-20 Related Functions . .[...]

© Nationa l Instrume nts Corpora tion 1-1 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[...]

Chapter 1 Introduction Xmath Model R eduction M odule 1-2 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[...]

Chap ter 1 Int rodu cti on © Nationa l Instrume nts Corpora tion 1-3 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[...]

Chapter 1 Introduction Xmath Model R eduction M odule 1-4 ni.com As sho wn in Figure 1-1, 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[...]

Chap ter 1 Int rodu cti on © Nationa l Instrume nts Corpora tion 1-5 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 1-1. Documentation of the individual functions sometimes indicates how th e restrictions can be circu mvented. Th ere are a n umb[...]

Chapter 1 Introduction Xmath Model R eduction M odule 1-6 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 er-fun 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, [...]

Chap ter 1 Int rodu cti on © Nationa l Instrume nts Corpora tion 1-7 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[...]

Chapter 1 Introduction Xmath Model R eduction M odule 1-8 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[...]

Chap ter 1 Int rodu cti on © Nationa l Instrume nts Corpora tion 1-9 Xmath Mod el Redu ction Modu le • Suppose t he transfer -fu nction matri x corresponds to a discr ete-ti 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[...]

Chapter 1 Introduction Xmath Model R eduction M odule 1-10 ni.com Interna lly Ba lanced Re alizatio ns Suppose that a realization of a transfer-fu 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 [...]

Chap ter 1 Int rodu cti on © Natio nal Instrum ents Cor poration 1-11 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 [...]

Chapter 1 Introduction Xmath Model R eduction M odule 1-12 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, (1-2) Equation 1-2 may be an ap[...]

Chap ter 1 Int rodu cti on © Natio nal Instrum ents Cor poration 1-13 Xma th Model Re ductio n Module Similar considerations govern the discrete-time 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 1-1 is b alanced, [...]

Chapter 1 Introduction Xmath Model R eduction M odule 1-14 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 )<[...]

Chap ter 1 Int rodu cti on © Natio nal Instrum ents Cor poration 1-15 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 [...]

Chapter 1 Introduction Xmath Model R eduction M odule 1-16 ni.com multiplicati ve red uction, as described in Chapter 4, Frequ ency-Weighted 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 .[...]

© Nationa l Instrume nts Corpora tion 2-1 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 [...]

Chapter 2 Additive Error Red uction Xmath Model R eduction M odule 2-2 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 (2-1) and the realization is internally balanced, then a tru ncation is provided b y[...]

Chapte r 2 Additive Erro r Reduc tion © Nationa l Instrume nts Corpora tion 2-3 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 (2-2) with the diagonal entries of Σ in decre[...]

Chapter 2 Additive Error Red uction Xmath Model R eduction M odule 2-4 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[...]

Chapte r 2 Additive Erro r Reduc tion © Nationa l Instrume nts Corpora tion 2-5 Xmath Mod el Redu ction Modu le order model is not one in general obtainable by truncation of an internally-balanced realization of the full order model. Figure 2-1 sets o ut se veral rou tes to a reduced-o rder realization. In continuous time, a truncation of a bala n[...]

Chapter 2 Additive Error Red uction Xmath Model R eduction M odule 2-6 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[...]

Chapte r 2 Additive Erro r Reduc tion © Nationa l Instrume nts Corpora tion 2-7 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 (?[...]

Chapter 2 Additive Error Red uction Xmath Model R eduction M odule 2-8 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[...]

Chapte r 2 Additive Erro r Reduc tion © Nationa l Instrume nts Corpora tion 2-9 Xmath Mod el Redu ction Modu le of the balanced system occurs, ( assuming nsr is less than the number of states). Thus, if the state-space 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[...]

Chapter 2 Additive Error Red uction Xmath Model R eduction M odule 2-10 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[...]

Chapte r 2 Additive Erro r Reduc tion © Natio nal Instrum ents Cor poration 2-11 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[...]

Chapter 2 Additive Error Red uction Xmath Model R eduction M odule 2-12 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[...]

Chapte r 2 Additive Erro r Reduc tion © Natio nal Instrum ents Cor poration 2-13 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[...]

Chapter 2 Additive Error Red uction Xmath Model R eduction M odule 2-14 ni.com For the d iscrete-time 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[...]

Chapte r 2 Additive Erro r Reduc tion © Natio nal Instrum ents Cor poration 2-15 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 )[...]

Chapter 2 Additive Error Red uction Xmath Model R eduction M odule 2-16 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[...]

Chapte r 2 Additive Erro r Reduc tion © Natio nal Instrum ents Cor poration 2-17 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[...]

Chapter 2 Additive Error Red uction Xmath Model R eduction M odule 2-18 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 [...]

Chapte r 2 Additive Erro r Reduc tion © Natio nal Instrum ents Cor poration 2-19 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 [...]

Chapter 2 Additive Error Red uction Xmath Model R eduction M odule 2-20 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[...]

Chapte r 2 Additive Erro r Reduc tion © Natio nal Instrum ents Cor poration 2-21 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, [...]

Chapter 2 Additive Error Red uction Xmath Model R eduction M odule 2-22 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 z-domain to the s-domain. 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 [...]

Chapte r 2 Additive Erro r Reduc tion © Natio nal Instrum ents Cor poration 2-23 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 [...]

© Nationa l Instrume nts Corpora tion 3-1 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 [...]

Chapter 3 Multiplicative Error Reduction Xmath Model R eduction M odule 3-2 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, (3-1) then C stabilizes G . This result indicates that if a controller C is designed to stabili z[...]

Chapte r 3 Mult iplicative Erro r Reduction © Nationa l Instrume nts Corpora tion 3-3 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[...]

Chapter 3 Multiplicative Error Reduction Xmath Model R eduction M odule 3-4 ni.com The objecti ve of the algorithm is to ap proximate a high-order stable transfer function matri x G ( s ) b y a lower -order G r ( s ) with either in v(g)(g-gr) or (g-gr)inv(g) minimi zed, under the condition t hat G r is stable and of the prescribed order . Restricti[...]

Chapte r 3 Mult iplicative Erro r Reduction © Nationa l Instrume nts Corpora tion 3-5 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[...]

Chapter 3 Multiplicative Error Reduction Xmath Model R eduction M odule 3-6 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´ (?[...]

Chapte r 3 Mult iplicative Erro r Reduction © Nationa l Instrume nts Corpora tion 3-7 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[...]

Chapter 3 Multiplicative Error Reduction Xmath Model R eduction M odule 3-8 ni.com state-v 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[...]

Chapte r 3 Mult iplicative Erro r Reduction © Nationa l Instrume nts Corpora tion 3-9 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[...]

Chapter 3 Multiplicative Error Reduction Xmath Model R eduction M odule 3-10 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 e-v ariable matrice s [...]

Chapte r 3 Mult iplicative Erro r Reduction © Natio nal Instrum ents Cor poration 3-11 Xma th Model Re ductio n Module The v alues of G ( s ), as sho wn in Figure 3-2, 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 [...]

Chapter 3 Multiplicative Error Reduction Xmath Model R eduction M odule 3-12 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 ] [...]

Chapte r 3 Mult iplicative Erro r Reduction © Natio nal Instrum ents Cor poration 3-13 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[...]

Chapter 3 Multiplicative Error Reduction Xmath Model R eduction M odule 3-14 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[...]

Chapte r 3 Mult iplicative Erro r Reduction © Natio nal Instrum ents Cor poration 3-15 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 all-pas [...]

Chapter 3 Multiplicative Error Reduction Xmath Model R eduction M odule 3-16 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 [...]

Chapte r 3 Mult iplicative Erro r Reduction © Natio nal Instrum ents Cor poration 3-17 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[...]

Chapter 3 Multiplicative Error Reduction Xmath Model R eduction M odule 3-18 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 ,[...]

Chapte r 3 Mult iplicative Erro r Reduction © Natio nal Instrum ents Cor poration 3-19 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[...]

Chapter 3 Multiplicative Error Reduction Xmath Model R eduction M odule 3-20 ni.com Error Bounds The error b ound formu la (Equation 3-3) is a simple consequence o f iterating (Equation 3-5). 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[...]

Chapte r 3 Mult iplicative Erro r Reduction © Natio nal Instrum ents Cor poration 3-21 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[...]

Chapter 3 Multiplicative Error Reduction Xmath Model R eduction M odule 3-22 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 3-2). Also, th e valu es of along the j ω -axis are the same as the values[...]

Chapte r 3 Mult iplicative Erro r Reduction © Natio nal Instrum ents Cor poration 3-23 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 [...]

Chapter 3 Multiplicative Error Reduction Xmath Model R eduction M odule 3-24 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[...]

© Nationa l Instrume nts Corpora tion 4-1 Xmath Mod el Redu ction Modu le 4 Frequency-W eighted Error Reduction This chapter descr ibes frequency-weighted error reduction problems. This includes a discuss ion of controller reduction and fractional repres entations. Introduction Frequency-w eighted erro r reduction m eans that the erro r is measure[...]

Chapter 4 Frequenc y-Weighted Error R eduction Xmath Model R eduction M odule 4-2 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[...]

Chapte r 4 Frequency -Weighted Erro r Reduc tion © Nationa l Instrume nts Corpora tion 4-3 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 4-1 and Equation 4-2. The f act that the plant ought to show up in a goo d formulation of a controller reductio n p[...]

Chapter 4 Frequenc y-Weighted Error R eduction Xmath Model R eduction M odule 4-4 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[...]

Chapte r 4 Frequency -Weighted Erro r Reduc tion © Nationa l Instrume nts Corpora tion 4-5 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[...]

Chapter 4 Frequenc y-Weighted Error R eduction Xmath Model R eduction M odule 4-6 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[...]

Chapte r 4 Frequency -Weighted Erro r Reduc tion © Nationa l Instrume nts Corpora tion 4-7 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[...]

Chapter 4 Frequenc y-Weighted Error R eduction Xmath Model R eduction M odule 4-8 ni.com The left MF D corresp onds to the setup of Figu re 4-3. Figu re 4-3 . C(s) Implement ed to Display L eft MFD Repr esentatio n The setu p of Figur e 4-2 su ggests approxim ation of : whereas that of Figu re 4-3 s uggests appro ximation of: In the LQG optimal cas[...]

Chapte r 4 Frequency -Weighted Erro r Reduc tion © Nationa l Instrume nts Corpora tion 4-9 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 [...]

Chapter 4 Frequenc y-Weighted Error R eduction Xmath Model R eduction M odule 4-10 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 ] [...]

Chapte r 4 Frequency -Weighted Erro r Reduc tion © Natio nal Instrum ents Cor poration 4-11 Xma th Model Re ductio n Module • Reduce the or der of a transfer fu nction matrix C ( s ) t hrough frequenc y-we 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 [...]

Chapter 4 Frequenc y-Weighted Error R eduction Xmath Model R eduction M odule 4-12 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[...]

Chapte r 4 Frequency -Weighted Erro r Reduc tion © Natio nal Instrum ents Cor poration 4-13 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 [...]

Chapter 4 Frequenc y-Weighted Error R eduction Xmath Model R eduction M odule 4-14 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[...]

Chapte r 4 Frequency -Weighted Erro r Reduc tion © Natio nal Instrum ents Cor poration 4-15 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[...]

Chapter 4 Frequenc y-Weighted Error R eduction Xmath Model R eduction M odule 4-16 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 –[...]

Chapte r 4 Frequency -Weighted Erro r Reduc tion © Natio nal Instrum ents Cor poration 4-17 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 4-5. Figur e 4-5. Inte rnal Struc ture of Contro ller Recognize that the contro ller C ( s ) (show[...]

Chapter 4 Frequenc y-Weighted Error R eduction Xmath Model R eduction M odule 4-18 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[...]

Chapte r 4 Frequency -Weighted Erro r Reduc tion © Natio nal Instrum ents Cor poration 4-19 Xma th Model Re ductio n Module 6. Check the stability of th e closed-loop syst em with C r ( s ). When the type="left perf" is specif ied, one w orks with (4-1 1) which is f ormed fr om the numerat or and deno minator of the MFD in Equation 4-5. [...]

Chapter 4 Frequenc y-Weighted Error R eduction Xmath Model R eduction M odule 4-20 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[...]

Chapte r 4 Frequency -Weighted Erro r Reduc tion © Natio nal Instrum ents Cor poration 4-21 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[...]

© Nationa l Instrume nts Corpora tion 5-1 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[...]

Chapte r 5 Utiliti es Xmath Model R eduction M odule 5-2 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 ime-consumi ng. The Hank el sing ular v alues are t he square r oots of the eigen values of the [...]

Chapte r 5 Utili ties © Nationa l Instrume nts Corpora tion 5-3 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 us-time, or eigen value magnitude within the following range for discrete time: A warn ing is gi ven if dou btful eigen v alues ex[...]

Chapte r 5 Utiliti es Xmath Model R eduction M odule 5-4 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[...]

© Nationa l Instrume nts Corpora tion 6-1 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 ll-ord 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[...]

Chapter 6 T utoria l Xmath Model R eduction M odule 6-2 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[...]

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

Chapter 6 T utoria l Xmath Model R eduction M odule 6-4 ni.com recovery at low f requencies; there is consequ ently a faster roll-o 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 [...]

Chap ter 6 T utoria l © Nationa l Instrume nts Corpora tion 6-5 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[...]

Chapter 6 T utoria l Xmath Model R eduction M odule 6-6 ni.com Figures 6-3, 6-4, and 6-5 displ ay the outcome o f the redu ction. The loo p gain is shown in Figure 6-3. 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[...]

Chap ter 6 T utoria l © Nationa l Instrume nts Corpora tion 6-7 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 6-4 . Closed -Loo p Gain with redsch ur[...]

Chapter 6 T utoria l Xmath Model R eduction M odule 6-8 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 6-5 . Step Response with redschur[...]

Chap ter 6 T utoria l © Nationa l Instrume nts Corpora tion 6-9 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, 6-7, an d 6-8. 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[...]

Chapter 6 T utoria l Xmath Model R eduction M odule 6-10 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 6-7. Closed-L oop Ga in with ophank[...]

Chap ter 6 T utoria l © Natio nal Instrum ents Cor poration 6-11 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 6-8. St ep Re spons e with o phank The open-l oop gai n, closed-loop gain and s tep resp onse ar[...]

Chapter 6 T utoria l Xmath Model R eduction M odule 6-12 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 sed-loop transfer functions. The weigh ted Hankel sing ular values are: 1.486 4.[...]

Chap ter 6 T utoria l © Natio nal Instrum ents Cor poration 6-13 Xma th Model Re ductio n Module The following fu nction calls produce Figur e 6-9: svalsrol = svplot(sys*syscr,w,{radians }) plot(svalsol, {keep}) f9=plot(wc, constr, {keep,!grid, legend=["reduced","original","constrai ned"], title="Open-Loop Gain U[...]

Chapter 6 T utoria l Xmath Model R eduction M odule 6-14 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 6-1 0. Clos ed-L oop G ain with wtb alance[...]

Chap ter 6 T utoria l © Natio nal Instrum ents Cor poration 6-15 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 6-11. Ste p Response wit h wtbalance Figures 6-9, 6-10, and 6-11 are obtained for wtbalance wi[...]

Chapter 6 T utoria l Xmath Model R eduction M odule 6-16 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 6-12 . Frequency Re sponse of th e Weight V( j ω )[...]

Chap ter 6 T utoria l © Natio nal Instrum ents Cor poration 6-17 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[...]

Chapter 6 T utoria l Xmath Model R eduction M odule 6-18 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 6-14 . System Sing ular Values of wtbalanc e with "i nput spec "[...]

Chap ter 6 T utoria l © Natio nal Instrum ents Cor poration 6-19 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 6-15. Step Respon se of wtba lanc e with "inp ut spec "[...]

Chapter 6 T utoria l Xmath Model R eduction M odule 6-20 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[...]

Chap ter 6 T utoria l © Natio nal Instrum ents Cor poration 6-21 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 6-17. Closed -Loop Respon se with fracred[...]

Chapter 6 T utoria l Xmath Model R eduction M odule 6-22 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 6-18. Step Response with fracre d The end result is comparable to that from wtbalance( ) with option "match" . We can [...]

Chap ter 6 T utoria l © Natio nal Instrum ents Cor poration 6-23 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[...]

© Nationa l Instrume nts Corpora tion A-1 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[...]

Appendix A Bibliogr aphy Xmath Model R eduction M odule A-2 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 [...]

Appendix A Bibliography © Nationa l Instrume nts Corpora tion A-3 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 e-error m ethod, ” Pr oceedings for the American Contr ols Confer ence , 1988, pp. 1685 –1690. [Saf87] M. G. Saf onov , “Imaginary-axi s zeros[...]

Appendix A Bibliogr aphy Xmath Model R eduction M odule A-4 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[...]

Appendix A Bibliography © Nationa l Instrume nts Corpora tion A-5 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 C-26, Februar y 1981. [SA88] G. S [...]

© Nationa l Instrume nts Corpora tion B-1 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: – [...]

© Nationa l Instrume nts Corpora tion I-1 Xmath Mod el Redu ction Modu le Index Symbols *, 1-6 ´, 1-6 A additive error, reduction, 2-1 algorithm bala nced stocha stic trunc atio n (bst) , 3-4 fractional re presentat ion redu ction, 4-18 Hankel multi-pass, 2-20 optimal Ha nkel norm redu ction, 2-15 stable, 5- 2 weighted bal ance, 4-12 all-pass tra[...]

Index Xmath Model R eduction M odule I-2 ni.com G grammians controllability, 1- 7 desc ripti on of , 1-7 observability, 1- 7 H Hankel matrix, 1-9 Hankel no rm approxi mation, 2-6 Hankel si ngular values, 1-8, 3-9 , 5-1 hankels v, 1-5, 5-1 algorithm, m ultipass, 2 -20 help, technical support, B-1 I instrum ent drivers (NI resources ), B-1 internal b[...]

Index © Nationa l Instrume nts Corpora tion I-3 Xmath Mod el Redu ction Modu le stable, 1 -5, 5-2 sup, 1-6 suppor t, technical, B -1 T technical support, B -1 tight equality bounds, 1-7 training and certification (NI r esources), B-1 transfer function, allpass, 1-6 troubles hooting (NI resou rces), B-1 truncate, 1 -5, 2-4, 2- 11 U unstable zer os,[...]