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NI MA TRIXx TM Xmath TM Model Reduction Module Xmath Model Reduction Module April 2007 370755C-01.
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© National Instruments Corporatio n v Xmath Model Reducti on Module Contents Chapter 1 Introduction Using This Manual...................... .. .......................... ......................... .......................... .... 1-1 Document Organization.
Contents Xmath Model Reduction Module vi ni.com Onepass Algorithm .................. ......................... .......................... .................... 2-18 Multipass Algorithm . .............. .............. .......................... ........
Contents © National Instruments Corporatio n vii Xmath Model Reducti o n Module fracred( ) .................... ......................... ......................... .......................... ...................... .. 4-15 Restrictions ...............
© National Instruments Corporatio n 1-1 Xmath Model Reducti o n Module 1 Introduction This chapter starts with an outline of the manual an d some useful notes. It also provides an overview of the Model Reduction Module, describes the functions in this module, and introduces no mencl ature and concepts used throughout this manual.
Chapter 1 Introduction Xmath Model Reducti on Module 1-2 ni.com • Chapter 5, Utilities , describes three utilit y fun ctions: hankelsv( ) , stable( ) , and compare( ) . • Chapter 6, Tutorial , illustrates a number of the MRM functions and their underlying ideas.
Chapter 1 Introduction © National Instruments Corporatio n 1-3 Xmath Model Reducti o n Module Related Publications For a complete list of MATRIXx publications, refer to C hapter 2, MATRIXx Publications, Onlin e Help, and Customer Support , of the MATRIXx Getting Started Guide.
Chapter 1 Introduction Xmath Model Reducti on Module 1-4 ni.com As shown in Figure 1-1, functions are provided to handle four broad tasks: • Model reduction w ith additi v e errors • Model reducti.
Chapter 1 Introduction © National Instruments Corporatio n 1-5 Xmath Model Reducti o n Module Certain restrictions regarding minimality and stabil ity are required of the input data, and are summarized in T able 1-1. Documentation of the individual functions sometimes indicat es ho w the restrictions can be circumvented.
Chapter 1 Introduction Xmath Model Reducti on Module 1-6 ni.com • L 2 approximation, in which the L 2 norm of impulse response error (or, by P ar se val’ s theorem, the L 2 norm of the transfer -f.
Chapter 1 Introduction © National Instruments Corporatio n 1-7 Xmath Model Reducti o n Module • An inequality or bound is tight if it can be met in practice, for example is tight because the inequali ty becomes an equality for x =1 .
Chapter 1 Introduction Xmath Model Reducti on Module 1-8 ni.com • The controllability grammian is also E [ x ( t ) x ′ ( t )] when the system has been excited from time – ∞ b y zero mean white noise with . • The observability grammian can be thought of as measuring the information contained in the output concerning an initial state.
Chapter 1 Introduction © National Instruments Corporatio n 1-9 Xmath Model Reducti o n Module • Suppose the transfer-function matr ix corresponds to a di screte-time system, with state variable dimension n .
Chapter 1 Introduction Xmath Model Reduction Module 1-10 ni.com Internally Balanc ed Realizations Suppose that a reali zation of a transfer-function matrix has the controllability an d observability grammian p roperty that P = Q = Σ for some diagonal Σ .
Chapter 1 Introduction © National Instruments Corporatio n 1-11 Xmath Mod el Reduction Module This is almost the algorithm set out in Section II of [LHPW87]. Th e one difference (and it is minor) is that in [LHPW87], 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.
Chapter 1 Introduction Xmath Model Reduction Module 1-12 ni.com and also: Re λ i ( A 22 )<0 and . Usually , we expect that, in the sense that the intuitiv e argument hinges on this, but it is not necessary .
Chapter 1 Introduction © National Instruments Corporatio n 1-13 Xmath Mod el Reduction Module Similar considerations gov ern the discrete-time problem, where, can be approximated by: mreduce( ) can carry out singular perturbation. For further discussion, refer to Chapter 2, Additive Error Reduction .
Chapter 1 Introduction Xmath Model Reduction Module 1-14 ni.com nonnegati ve hermitian for all ω . If Φ is scalar , then Φ ( j ω ) ≥ 0 for all ω .
Chapter 1 Introduction © National Instruments Corporatio n 1-15 Xmath Mod el Reduction Module Low Order Controller Design Through Order Reduction The Model Reduction Module is particularly suitab le for achieving low order controller design for a high order plant.
Chapter 1 Introduction Xmath Model Reduction Module 1-16 ni.com multiplicative reduction, as described in Chapter 4, Frequency-Weighted Error Reduction , is a sound approach. Chapt er 3, Multiplicative Error Reduction , and Chapter 4, Frequency-Weighted Error Reduction , develop these arguments more fully .
© National Instruments Corporatio n 2-1 Xmath Model Reducti o n Module 2 Additive Error Reduction This chapter describes additive error reductio n including discussions of truncation of, reduction by, and pert urbation of balanced realizations.
Chapter 2 Additive Er ror Reduction Xmath Model Reducti on Module 2-2 ni.com T runcation of Balanced Realizations A group of functions can be used to achieve a reduction through truncation of a balanced realization.
Chapter 2 Additive Error Reduction © National Instruments Corporatio n 2-3 Xmath Model Reducti o n Module A very attracti ve feature of the truncation procedure is the av ail ability of an erro r bou nd.
Chapter 2 Additive Er ror Reduction Xmath Model Reducti on Module 2-4 ni.com proper . So, e v en if all zeros are uns table, the maximum phase shift when ω mov es from 0 to ∞ i s ( 2 n–3 ) π /2.
Chapter 2 Additive Error Reduction © National Instruments Corporatio n 2-5 Xmath Model Reducti o n Module order model is not one in general obtainable by truncation of an internally-balanced realizatio n of the full order model. Figure 2-1 sets out sev eral routes to a reduced-order realization.
Chapter 2 Additive Er ror Reduction Xmath Model Reducti on Module 2-6 ni.com with controllability and observability grammians given b y , in which the diagonal entri es of Σ are in decreasing order , that is, σ 1 ≥σ 2 ≥ ···, and such that the last diagonal entry of Σ 1 exceeds the first diagonal entry of Σ 2 .
Chapter 2 Additive Error Reduction © National Instruments Corporatio n 2-7 Xmath Model Reducti o n Module function matrix. Con si der th e way th e associated impulse response maps inputs d efin ed ov er (– ∞ ,0] in L 2 into outputs, and focus on the output ov er [0, ∞ ).
Chapter 2 Additive Er ror Reduction Xmath Model Reducti on Module 2-8 ni.com Further , the which is optimal for Hankel norm approximation also is optimal for this second type of approximat ion . In Xmath Hankel norm approximation is achieved with ophank( ) .
Chapter 2 Additive Error Reduction © National Instruments Corporatio n 2-9 Xmath Model Reducti o n Module of the balanced system occurs, (assuming nsr is less than the number of states).
Chapter 2 Additive Er ror Reduction Xmath Model Reduction Module 2-10 ni.com The actual approximation error for discrete systems als o depends on frequency , and can be large at ω = 0.
Chapter 2 Additive Error Reduction © National Instruments Corporatio n 2-11 Xmath Mod el Reduction Module Related Functions balance() , truncate() , redschur() , mreduce() truncate( ) SysR = truncate.
Chapter 2 Additive Er ror Reduction Xmath Model Reduction Module 2-12 ni.com redschur( ) [SysR,HSV,slbig,srbig,VD,VA] = redschur(Sys,{nsr,bound}) The redschur( ) function uses a Schur method (from Safonov and Chiang) to calculate a reduced version of a continuous or discrete system without balancing.
Chapter 2 Additive Error Reduction © National Instruments Corporatio n 2-13 Xmath Mod el Reduction Module Next, Schur decompositions of W c W o are formed with the eigen v alues of W c W o in ascending and descending order . These eigen values are the square of the Hankel singular v alues of Sys , and if Sys is nonminimal, some can be zero.
Chapter 2 Additive Er ror Reduction Xmath Model Reduction Module 2-14 ni.com For the discrete-time case: When {bound} is specif ied, the error bound ju st enunciated is used to choose the number of states in SysR so that the bound is satisfied and nsr is as small as possible.
Chapter 2 Additive Error Reduction © National Instruments Corporatio n 2-15 Xmath Mod el Reduction Module Algorithm The algorithm does the fo llowing. The system Sys and the reduced order system SysR are stable; the system SysU has all its poles in Re [ s ] > 0.
Chapter 2 Additive Er ror Reduction Xmath Model Reduction Module 2-16 ni.com By abuse of notation, when we say that G is reduced to a certain order , this corresponds to the order of G r ( s ) alone; the unstable part of G u ( s ) of the approximation is most frequent ly thro wn a w ay .
Chapter 2 Additive Error Reduction © National Instruments Corporatio n 2-17 Xmath Mod el Reduction Module Thus, the penalty for not being allowed to include G u in the approximati on is an increase in the error bound, by σ n i + 1 + .
Chapter 2 Additive Er ror Reduction Xmath Model Reduction Module 2-18 ni.com being approximated by a stable G r ( s ) with the actual error (as opposed to just the error bou nd) satisfying: Note G r is optimal, that is, there is no other G r achieving a lower boun d.
Chapter 2 Additive Error Reduction © National Instruments Corporatio n 2-19 Xmath Mod el Reduction Module and finally: These four matrices are the constitu ents of the system matr ix of , where: Digr.
Chapter 2 Additive Er ror Reduction Xmath Model Reduction Module 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( ) . Suppose G u ( s ) is the unstable output of stable( ) , and let K ( s )= G u (– s ).
Chapter 2 Additive Error Reduction © National Instruments Corporatio n 2-21 Xmath Mod el Reduction Module 2. Find a stable order ns – 2 approximation G ns –2 of G ns –1 ( s ), with 3. (Step ns–nr) : Find a stable order nsr approximation of G nsr +1 , with Then, because for , for , .
Chapter 2 Additive Er ror Reduction Xmath Model Reduction Module 2-22 ni.com We u s e sysZ to denote G(z) and def ine: bilinsys=makepoly([-1,a]/mak epoly([1,a]) as the mapping from the z-domain to th e s-domain. The specification is rev ersed because this function uses backward polynomial rotation.
Chapter 2 Additive Error Reduction © National Instruments Corporatio n 2-23 Xmath Mod el Reduction Module It follows b y a result of [ BoD87 ] that the impulse response error for t >0 satisfies: Evidently , Hankel norm approximation ensures som e form of approximation of the impulse response too.
© National Instruments Corporatio n 3-1 Xmath Model Reducti o n Module 3 Multiplicative Error Reduction This chapter describes multipl i cative error reduction presenting two reasons to consider m ultiplicative rather th an additive error reduction, one general and one specific.
Chapter 3 Multiplicative Error Re duction Xmath Model Reducti on Module 3-2 ni.com Multiplicative Robustness Result Suppose C stabilizes , that has no j ω -axis poles, and that G has the same number of poles in Re [ s ] ≥ 0 as . If for all ω, (3-1) then C stabilizes G .
Chapter 3 Multiplicative Error R eduction © National Instruments Corporatio n 3-3 Xmath Model Reducti o n Module bandwidth at the expense of being larger outside this bandwidth, which would be preferable. Second, the previously used mu ltiplicati ve error is .
Chapter 3 Multiplicative Error Re duction Xmath Model Reducti on Module 3-4 ni.com The objecti ve of the algorit hm is to approximate a high-order stable transfer function matrix G ( s ) by a lower -order G r ( s ) with either inv(g)(g -gr) or (g-gr)inv(g) minimized, under the condition that G r is stable and of the prescribed order .
Chapter 3 Multiplicative Error R eduction © National Instruments Corporatio n 3-5 Xmath Model Reducti o n Module These cases are secured with the ke ywords right and left , respecti vely . If the wrong option is req uested for a nonsquare G ( s ), an error message will result.
Chapter 3 Multiplicative Error Re duction Xmath Model Reducti on Module 3-6 ni.com 2. W ith G ( s )= D + C ( sI – A ) –1 B and stable, with DD ´ nonsingular and G ( j ω ) G '(– j ω ) non.
Chapter 3 Multiplicative Error R eduction © National Instruments Corporatio n 3-7 Xmath Model Reducti o n Module strictly proper stable part of θ ( s ), as the square roots of the eigen v alues of PQ . Call these quantities ν i . The Schur decompositions are, where V A , V D are orthogonal and S as c , S des are upper triangular .
Chapter 3 Multiplicative Error Re duction Xmath Model Reducti on Module 3-8 ni.com state-v ariable representation of G . In this case, the user is ef fectively asking for G r = G .
Chapter 3 Multiplicative Error R eduction © National Instruments Corporatio n 3-9 Xmath Model Reducti o n Module Hankel Singular V alues of Phase Matrix of G r The ν i , i = 1,2,..., ns have been termed above the Hankel singular values of the phase matrix associated with G .
Chapter 3 Multiplicative Error Re duction Xmath Model Reduction Module 3-10 ni.com which also can be relev ant in finding a reduced order model of a plant. The procedure requires G again to be nonsingu lar at ω = ∞ , and to have no j ω -axis poles.
Chapter 3 Multiplicative Error R eduction © National Instruments Corporatio n 3-11 Xmath Mod el Reduction Module The values of G ( s ), as shown in Figure 3-2, along the j ω -axis are the same as the v alues of around a circle with diameter defi ned by [ a – j 0, b –1 + j 0] on the positi ve real axis.
Chapter 3 Multiplicative Error Re duction Xmath Model Reduction Module 3-12 ni.com Any zero (or rank reduction) on the j ω -axis o f G ( s ) becomes a zero (or rank reduction) in Re [ s ] > 0 of ,.
Chapter 3 Multiplicative Error R eduction © National Instruments Corporatio n 3-13 Xmath Mod el Reduction Module again with a bilinear transf ormation to s ecure multiplicativ e approximations over a limited frequency band.
Chapter 3 Multiplicative Error Re duction Xmath Model Reduction Module 3-14 ni.com There is one potential source of fa ilure of the algorithm. Because G ( s ) is stable, certainly will be, as its pol es will be in the left half plane circle on diameter .
Chapter 3 Multiplicative Error R eduction © National Instruments Corporatio n 3-15 Xmath Mod el Reduction Module The conceptual basis of the algorithm can best be grasped by considering the case of scalar G ( s ) of degree n .
Chapter 3 Multiplicative Error Re duction Xmath Model Reduction Module 3-16 ni.com eigen v alues of A – B/D * C with the aid of schur( ) . If any real part of the eigenvalues is less than eps , a warning is displayed.
Chapter 3 Multiplicative Error R eduction © National Instruments Corporatio n 3-17 Xmath Mod el Reduction Module singular values of F ( s ) lar ger than 1– ε (refer to steps 1 through 3 of the Restrictions sect ion). The maximum order permitted is the number of nonzero eigen values of W c W o larger than ε .
Chapter 3 Multiplicative Error Re duction Xmath Model Reduction Module 3-18 ni.com Note The expression is the strictly proper part of . The matrix is all pass; this property is not always secured in the multiv ariable case when ophank( ) is used to find a Hank el norm approximation of F ( s ).
Chapter 3 Multiplicative Error R eduction © National Instruments Corporatio n 3-19 Xmath Mod el Reduction Module • and stand in the same relation as W ( s ) and G ( s ), that is: – – W ith , there holds or – W ith there holds or – – is the stable strictly proper part of .
Chapter 3 Multiplicative Error Re duction Xmath Model Reduction Module 3-20 ni.com Error Bounds The error bound form ula (Equation 3-3) is a simple consequence of iterating (Equation 3 -5). To illustrate, suppose there ar e three reductions →→ → , each by degree one.
Chapter 3 Multiplicative Error R eduction © National Instruments Corporatio n 3-21 Xmath Mod el Reduction Module For mulhank( ) , this translates for a scalar system into and The bounds are double for bst( ) .
Chapter 3 Multiplicative Error Re duction Xmath Model Reduction Module 3-22 ni.com The values of G ( s ) along the j ω -axis are the same as the v alues of around a circle with diameter defined by [ a – j 0, b –1 + j 0] on the positi ve real axis (refer to Figure 3-2 ).
Chapter 3 Multiplicative Error R eduction © National Instruments Corporatio n 3-23 Xmath Mod el Reduction Module The error will be ov erbounded by the error , and G r will contain the same zeros in Re [ s ] ≥ 0 as G .
Chapter 3 Multiplicative Error Re duction Xmath Model Reduction Module 3-24 ni.com Multiplicativ e approxi mation of (along the j ω -axis) corresponds to multiplicative approximation of G ( s ) around a circle in the right half plane, touching t he j ω -axis at the origin.
© National Instruments Corporatio n 4-1 Xmath Model Reducti o n Module 4 Frequency-W eighted Error Reduction This chapter describes frequency-weighted error reduction problems. This includes a discussion of controller red uction and fractional representations.
Chapter 4 F requency-Weighted Error R eduction Xmath Model Reducti on Module 4-2 ni.com (so that ) is logical. Howe ver , a major use of weighting is in controller reduction, which i s no w described. Controller Reduction Frequency weighted error reduction becomes particularly importan t in reducing controller dim ension.
Chapter 4 Frequency-Weig hted Error Re duction © National Instruments Corporatio n 4-3 Xmath Model Reducti o n Module is minimized (and of course is less than 1).
Chapter 4 F requency-Weighted Error R eduction Xmath Model Reducti on Module 4-4 ni.com Most of these ideas are discussed in [Enn 84], [A nL89], and [AnM89]. The function wtbalance( ) implements weighted red ucti on, w ith f ive choices of error measure, namely E IS , E OS , E M , E MS , and E 1 w ith arbitrary V( j ω ).
Chapter 4 Frequency-Weig hted Error Re duction © National Instruments Corporatio n 4-5 Xmath Model Reducti o n Module Fractional Representations The treatment of j ω -axis or right half plane poles in the above schemes is crude: they are simply copied into th e reduced order controller.
Chapter 4 F requency-Weighted Error R eduction Xmath Model Reducti on Module 4-6 ni.com • Form the reduced controller by interconnecting using negativ e feedback the second output of G r to the input , th at is, set Nothing has been said as to how should be cho sen—and the end resul t of the reduction, C r ( s ), depends on .
Chapter 4 Frequency-Weig hted Error Re duction © National Instruments Corporatio n 4-7 Xmath Model Reducti o n Module Matrix algebra sho ws that C ( s ) can be described through a left or right matrix fraction descript ion with D L , and related values, all stable transfer function matrices.
Chapter 4 F requency-Weighted Error R eduction Xmath Model Reducti on Module 4-8 ni.com The left MFD corresponds to the setup of Figure 4-3. Figure 4-3.
Chapter 4 Frequency-Weig hted Error Re duction © National Instruments Corporatio n 4-9 Xmath Model Reducti o n Module Figure 4-4. Redrawn; Individual Signal Paths as Vector Paths It is possible to verify that and accordingly the output weight can be used in an error measure .
Chapter 4 F requency-Weighted Error R eduction Xmath Model Reduction Module 4-10 ni.com (Here, the W i and V i are submatrices of W ,V .) Evidently , Some manipulation shows that trying to preserve these identities after approximation of D L , N L or N R , D R suggests use of the error measures and .
Chapter 4 Frequency-Weig hted Error Re duction © National Instruments Corporatio n 4-11 Xmath Mod el Reduction Module • Reduce the order of a tr ansfer function matrix C ( s ) through frequency-weighted balanced truncation, a stable frequency weight V ( s ) being prescribed.
Chapter 4 F requency-Weighted Error R eduction Xmath Model Reduction Module 4-12 ni.com This rather crude approach to the handling of the unstable part of a controller is avoided in fracred( ) , which provides an alternati ve to wtbalance( ) for controller reduction, at least for an important family of controllers.
Chapter 4 Frequency-Weig hted Error Re duction © National Instruments Corporatio n 4-13 Xmath Mod el Reduction Module 3. Compute weighted Hankel Singular V alues σ i (described in more detail later). If the order o f C r ( s ) is not specif ied a pr iori , it must be input at this time.
Chapter 4 F requency-Weighted Error R eduction Xmath Model Reduction Module 4-14 ni.com and the observ ability grammian Q , defined in the obvious way , is written as It is trivial to v erify that so that Q cc is the observability gramian of C s ( s ) alone, as well as a submatrix of Q .
Chapter 4 Frequency-Weig hted Error Re duction © National Instruments Corporatio n 4-15 Xmath Mod el Reduction Module From these quantities the transforma tion matrices used for calculating C sr ( s .
Chapter 4 F requency-Weighted Error R eduction Xmath Model Reduction Module 4-16 ni.com 3. Only continuous systems are accep ted; for discrete systems use makecontinuous( ) before calling bst( ) , then discre tize the result.
Chapter 4 Frequency-Weig hted Error Re duction © National Instruments Corporatio n 4-17 Xmath Mod el Reduction Module to, for example, through, for example, balanced truncation, and then def ining: For the second rationale, consider Figure 4-5. Figure 4-5.
Chapter 4 F requency-Weighted Error R eduction Xmath Model Reduction Module 4-18 ni.com Controller reduction proceeds by imple menting the same connection rule but on reduced v e rsi ons of th e two transfer function matrices.
Chapter 4 Frequency-Weig hted Error Re duction © National Instruments Corporatio n 4-19 Xmath Mod el Reduction Module 6. Check the stability of th e closed-loop system with C r ( s ). When the type="left perf" is specif ied, one works with (4-11) which is formed from the numerator and denominator of the MFD in Equation 4-5.
Chapter 4 F requency-Weighted Error R eduction Xmath Model Reduction Module 4-20 ni.com Additional Background A discussion of the stabil ity robustness measure can b e found in [AnM89] and [LAL90]. The idea can be unders t ood w ith reference to the transfer functions E ( s ) and E r ( s ) used in discussing type="right perf" .
Chapter 4 Frequency-Weig hted Error Re duction © National Instruments Corporatio n 4-21 Xmath Mod el Reduction Module The four schemes all produce different HSVs; it follows that it may be prudent to try all four schemes for a particular con troller reduction.
© National Instruments Corporatio n 5-1 Xmath Model Reducti o n Module 5 Utilities This chapter describes three utilit y functions: hankelsv( ) , stable( ) , and compare( ) . The background to hankelsv( ) , which calculates Hankel singular v alues, was presented in Chapter 1, Introduction .
Chapter 5 Utilities Xmath Model Reducti on Module 5-2 ni.com The gramian matrices are defined by solving the equations (in continuous time) and, in discrete time The computations are ef fected with lyapunov( ) and stability is check ed, which is time-consu ming.
Chapter 5 Utilities © National Instruments Corporatio n 5-3 Xmath Model Reducti o n Module Doubtful ones are those for which th e real part of the eigen value has magnitude less than or equal to tol for contin uous-time, or eigen value magnitude within the following range for discrete time: A warning is gi ven if doubtful eigenv alues exist.
Chapter 5 Utilities Xmath Model Reducti on Module 5-4 ni.com After this last transformation, and wit h it follows that and By combining the transformati on yieldi ng the real ordered Schur form for A with the transformation defined using X, the ov erall transformati on T is readily identified.
© National Instruments Corporatio n 6-1 Xmath Model Reducti o n Module 6 T utorial This chapter illustrates a number of the MRM functions and their underlying ideas. A plant and fu ll-order controller are de fined, an d then the effects of various reduction algorith ms are examined.
Chapter 6 T utorial Xmath Model Reducti on Module 6-2 ni.com A minimal realization in mo dal coor dinates is C ( sI – A ) –1 B where: The specifications seek high loop gain at low frequencies (for performance) and low loop ga in at high frequencies (to gu arantee stability in the presence of unstructured uncertainty).
Chapter 6 T utorial © National Instruments Corporatio n 6-3 Xmath Model Reducti o n Module With a state weighting matrix, Q = 1e-3*diag([2,2,80,80,8,8 ,3,3]); R = 1; (and unity control wei ghting), a.
Chapter 6 T utorial Xmath Model Reducti on Module 6-4 ni.com recovery at low frequencie s; there is consequently a faster roll-off of the loop gain at high frequen cies than for , and this is desired.
Chapter 6 T utorial © National Instruments Corporatio n 6-5 Xmath Model Reducti o n Module Controller Reduction This section contrasts the effect of unweighted and weighted cont roller reduction. Unweighted reduction is at first exam ined, throug h redschur( ) (using balance( ) or balmoore( ) will give similar results).
Chapter 6 T utorial Xmath Model Reducti on Module 6-6 ni.com Figures 6-3, 6-4, and 6-5 display the outcome of the reducti on. The loop gain is shown in Figure 6-3.
Chapter 6 T utorial © National Instruments Corporatio n 6-7 Xmath Model Reducti o n Module Generate Figure 6-4: compare(syscl,sysclr,w,{radi ans,type=5}) f4=plot({keep,legend=["origi nal","reduced"]}) Figure 6-4.
Chapter 6 T utorial Xmath Model Reducti on Module 6-8 ni.com Generate Figure 6-5: tvec=0:(140/99):140; compare(syscl,sysclr,tvec,{t ype=7}) f5=plot({keep,legend=["origi nal","reduced"]}) Figure 6-5.
Chapter 6 T utorial © National Instruments Corporatio n 6-9 Xmath Model Reducti o n Module ophank( ) ophank( ) is next used to reduce the cont roller with the results shown in Figures 6-6, 6-7, and 6-8.
Chapter 6 T utorial Xmath Model Reduction Module 6-10 ni.com Generate Figure 6-7: syscl = feedback(sysol); sysolr=sys*syscr; sysclr=feedback(sysolr); compare(syscl,sysclr,w,{radi ans,type=5}) f7=plot({keep,legend=["origi nal","reduced"]}) Figure 6-7.
Chapter 6 T utorial © National Instruments Corporatio n 6-11 Xmath Mod el Reduction Module Generate Figure 6-8: tvec=0:(140/99):140; compare(syscl,sysclr,tvec,{t ype=7}) f8=plot({keep,legend=["origi nal","reduced"]}) Figure 6-8.
Chapter 6 T utorial Xmath Model Reduction Module 6-12 ni.com wtbalance The next command examined is wtbalance with the option "match" . [syscr,ysclr,hsv] = wtbalanc e(sys,sysc,"match",2) Recall that this command should prom ote matching o f closed-loop transfer functions.
Chapter 6 T utorial © National Instruments Corporatio n 6-13 Xmath Mod el Reduction Module The following function calls produ ce Figu re 6-9: svalsrol = svplot(sys*syscr, w,{radians}) plot(svalsol, {.
Chapter 6 T utorial Xmath Model Reduction Module 6-14 ni.com Generate Figure 6-10: syscl = feedback(sysol); sysolr=sys*syscr; sysclr=feedback(sysolr); compare(syscl,sysclr,w,{radi ans,type=5}) f10=plot({keep,legend=["orig inal","reduced"]}) Figure 6-10.
Chapter 6 T utorial © National Instruments Corporatio n 6-15 Xmath Mod el Reduction Module Generate Figure 6-11: tvec=0:(140/99):140; compare(syscl,sysclr,tvec,{t ype=7}) f11=plot({keep,legend=["orig inal","reduced"]}) Figure 6-11.
Chapter 6 T utorial Xmath Model Reduction Module 6-16 ni.com Generate Figure 6-12: vtf=poly([-0.1,-10])/poly([- 1,-1.4]) [,sysv]=check(vtf,{ss,conver t}); svalsv = svplot(sysv,w,{radi ans}); Figure 6-12.
Chapter 6 T utorial © National Instruments Corporatio n 6-17 Xmath Mod el Reduction Module Generate Figure 6-13: [syscr,sysclr,hsv] = wtbalan ce(sys,sysc, "input spec",2,sysv) svalsrol = sv.
Chapter 6 T utorial Xmath Model Reduction Module 6-18 ni.com Generate Figure 6-14: syscl = feedback(sysol); sysolr=sys*syscr; sysclr=feedback(sysolr); compare(syscl,sysclr,w,{radi ans,type=5}) f14=plot({keep,legend=["orig inal","reduced"]}) Figure 6-14.
Chapter 6 T utorial © National Instruments Corporatio n 6-19 Xmath Mod el Reduction Module Generate Figure 6-15: tvec=0:(140/99):140; compare(syscl,sysclr,tvec,{t ype=7}) f15=plot({keep,legend=["orig inal","reduced"]}) Figure 6-15.
Chapter 6 T utorial Xmath Model Reduction Module 6-20 ni.com fracred fracred , the next command exami ned, has four options— "right stab" , "left stab" , "right perf" , and "left perf" . The options "left stab" , "right perf" , a nd "left perf" all produce instability.
Chapter 6 T utorial © National Instruments Corporatio n 6-21 Xmath Mod el Reduction Module Generate Figure 6-17: syscl = feedback(sysol); sysolr=sys*syscr; sysclr=feedback(sysolr); compare(syscl,sysclr,w,{radi ans,type=5}) f17=plot({keep,legend=["orig inal","reduced"]}) Figure 6-17.
Chapter 6 T utorial Xmath Model Reduction Module 6-22 ni.com Generate Figure 6-18: tvec=0:(140/99):140; compare(syscl,sysclr,tvec,{t ype=7}) f18=plot({keep,legend=["orig inal","reduced"]}) Figure 6-18. Step Response with fracred The end result is comp arable to that from wtbalance( ) with option "match" .
Chapter 6 T utorial © National Instruments Corporatio n 6-23 Xmath Mod el Reduction Module hsvtable = [... "right stab:", string(hsvrs' ); "left stab:", string(hsvls'); .
© National Instruments Corporatio n A- 1 Xmath Mod el Reduction Module A Bibliography [AnJ] BDO Anderson and B. James, “ Algorith m fo r multiplicativ e appro ximation of a stable linear system, ” in preparation.
Appendix A B ibliography Xmath Model Reducti on Module A-2 ni.com [GrA90] M. Green and BDO Anderson, “General ized balanced stochastic truncation, ” Pr oceedings for 29th CDC , 1990. [Gre88] M. Green, “Balanced stochastic realization, ” Linear Algebra and Applicati ons , V ol.
Appendix A B ibliography © National Instruments Corporatio n A- 3 Xmath Mod el Reduction Module [SaC88] M. G. Safonov and R. Y . Chiang, “Mod el reductio n for rob ust control: a Schur relativ e -er ror meth od, ” Pr oceedings for the Americ an Contr ols Conferenc e , 1988, pp.
Appendix A B ibliography Xmath Model Reducti on Module A-4 ni.com [Doy82] J. C. Doyle. “ Analysis of Feedback Systems with Structured Uncertainties. ” IEEE Pr oceedings , November 1982.
Appendix A B ibliography © National Instruments Corporatio n A- 5 Xmath Mod el Reduction Module [SLH81] M. G. Safonov , A. J. Laub, and G. L. Hartmann, “Feedback Prope rties of Multiv ariabl e Systems: The Role and Use of the Return Difference Matrix, ” IEEE T ransactions on Automatic Contr ol , V ol.
© National Instruments Corporatio n B- 1 Xmath Mod el Reduction Module B T echnical Support and Professional Ser vices Visit the following sections of the National Instruments Web site at ni.com for technical support an d professional services: • Support —Online technical support resources at ni.
© National Instruments Corporatio n I-1 Xmath Model Reduction Module Index Symbols *, 1-6 ´, 1-6 A additive error, reduction, 2-1 algorithm balanced stochastic truncation (bst), 3-4 fractional repre.
Index Xmath Model Reducti on Module I-2 ni.com G grammians controllability, 1-7 description of, 1-7 observability, 1-7 H Hankel matrix, 1-9 Hankel norm approximation, 2-6 Hankel singular values, 1-8, .
Index © National Instruments Corporatio n I-3 Xmath Model Reduction Module spectral factorization, 1-13 stability requirements, 1 -5 stable, 1-5, 5-2 sup, 1-6 support, technical, B-1 T technical supp.
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