National Instruments 370755B-01 - Manual

© National Instruments Corporation v Xmath Model Reduction Module Contents Chapter 1Introduction Using This Manual......................................................................................................... 1-1 Document Organization..........................................................

Contents © National Instruments Corporation vii Xmath Model Reduction Module Algorithm ........................................................................................................ 4-18Additional Background .....................................................................................

© National Instruments Corporation 1-1 Xmath Model Reduction Module 1 Introduction This chapter starts with an outline of the manual and some useful notes. It also provides an overview of the Model Reduction Module, describes the functions in this module, and introduces nomenclature and concepts use...

Chapter 1 Introduction Xmath Model Reduction Module 1-2 ni.com • Chapter 5, Utilities , describes three utility functions: hankelsv( ) , stable( ) , and compare( ) . • Chapter 6, Tutorial , illustrates a number of the MRM functions and their underlying ideas. Bibliographic References Throughout this...

Chapter 1 Introduction © National Instruments Corporation 1-3 Xmath Model Reduction Module Related Publications For a complete list of MATRIXx publications, refer to Chapter 2, MATRIXx Publications, Online Help, and Customer Support , of the MATRIXx Getting Started Guide. The following documents are...

Chapter 1 Introduction Xmath Model Reduction Module 1-4 ni.com As shown in Figure 1-1, functions are provided to handle four broad tasks: • Model reduction with additive errors • Model reduction with multiplicative errors • Model reduction with frequency weighting of an additive error, including con...

Chapter 1 Introduction Xmath Model Reduction Module 1-6 ni.com • L 2 approximation, in which the L 2 norm of impulse response error (or, by Parseval’s theorem, the L 2 norm of the transfer-function error along the imaginary axis) serves as the error measure • Markov parameter or impulse response mat...

Chapter 1 Introduction © National Instruments Corporation 1-7 Xmath Model Reduction Module • An inequality or bound is tight if it can be met in practice, for example is tight because the inequality becomes an equality for x = 1. Again, if F ( j ω ) denotes the Fourier transform of some , the Heisen...

Chapter 1 Introduction Xmath Model Reduction Module 1-8 ni.com • The controllability grammian is also E [ x ( t ) x ′ ( t )] when the system has been excited from time – ∞ by zero mean white noise with . • The observability grammian can be thought of as measuring the information contained in the out...

Chapter 1 Introduction © National Instruments Corporation 1-9 Xmath Model Reduction Module • Suppose the transfer-function matrix corresponds to a discrete-time system, with state variable dimension n . Then the infinite Hankel matrix, has for its singular values the n nonzero Hankel singular values...

Chapter 1 Introduction Xmath Model Reduction Module 1-10 ni.com Internally Balanced Realizations Suppose that a realization of a transfer-function matrix has the controllability and observability grammian property that P = Q = Σ for some diagonal Σ . Then the realization is termed internally balance...

Chapter 1 Introduction © National Instruments Corporation 1-11 Xmath Model Reduction Module This is almost the algorithm set out in Section II of [LHPW87]. The 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...

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 intuitive argument hinges on this, but it is not necessary. Then a singular perturbation is obtained by replacing by zero; this means that: Accordingly,...

Chapter 1 Introduction © National Instruments Corporation 1-13 Xmath Model Reduction Module Similar considerations govern 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 . I...

Chapter 1 Introduction Xmath Model Reduction Module 1-14 ni.com nonnegative hermitian 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, given a rational, nonnegative hermitian Φ ( j ω ) with lim ω...

Chapter 1 Introduction © National Instruments Corporation 1-15 Xmath Model Reduction Module Low Order Controller Design Through Order Reduction The Model Reduction Module is particularly suitable for achieving low order controller design for a high order plant. This section explains some of the broa...

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. Chapter 3, Multiplicative Error Reduction , and Chapter 4, Frequency-Weighted Error Reduction , develop these arguments more ...

© National Instruments Corporation 2-1 Xmath Model Reduction Module 2 Additive Error Reduction This chapter describes additive error reduction including discussions of truncation of, reduction by, and perturbation of balanced realizations. Introduction Additive error reduction focuses on errors of t...

Chapter 2 Additive Error Reduction Xmath Model Reduction Module 2-2 ni.com Truncation of Balanced Realizations A group of functions can be used to achieve a reduction through truncation of a balanced realization. This means that if the original system is (2-1) and the realization is internally balan...

Chapter 2 Additive Error Reduction © National Instruments Corporation 2-3 Xmath Model Reduction Module A very attractive feature of the truncation procedure is the availability of an error bound. More precisely, suppose that the controllability and observability grammians for [Enn84] are (2-2) with ...

Chapter 2 Additive Error Reduction Xmath Model Reduction Module 2-4 ni.com proper. So, even if all zeros are unstable, the maximum phase shift when ω moves from 0 to ∞ is (2n – 3) π /2. It follows that if G ( j ω ) remains large in magnitude at frequencies when the phase shift has moved past (2n – 3...

Chapter 2 Additive Error Reduction © National Instruments Corporation 2-5 Xmath Model Reduction Module order model is not one in general obtainable by truncation of an internally-balanced realization of the full order model. Figure 2-1 sets out several routes to a reduced-order realization. In conti...

Chapter 2 Additive Error Reduction Xmath Model Reduction Module 2-6 ni.com with controllability and observability grammians given by, in which the diagonal entries 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...

Chapter 2 Additive Error Reduction © National Instruments Corporation 2-7 Xmath Model Reduction Module function matrix. Consider the way the associated impulse response maps inputs defined over (– ∞ ,0] in L 2 into outputs, and focus on the output over [0, ∞ ). Define the input as u ( t ) for t <...

Chapter 2 Additive Error Reduction Xmath Model Reduction Module 2-8 ni.com Further, the which is optimal for Hankel norm approximation also is optimal for this second type of approximation. In Xmath Hankel norm approximation is achieved with ophank( ) . The most comprehensive reference is [Glo84]. b...

Chapter 2 Additive Error Reduction Xmath Model Reduction Module 2-10 ni.com The actual approximation error for discrete systems also depends on frequency, and can be large at ω = 0. The error bound is almost never tight, that is, the actual error magnitude as a function of ω almost never attains the...

Chapter 2 Additive Error Reduction © National Instruments Corporation 2-11 Xmath Model Reduction Module Related Functions balance() , truncate() , redschur() , mreduce() truncate( ) SysR = truncate(Sys,nsr,{VD,VA}) The truncate( ) function reduces a system Sys by retaining the first nsr states and t...

Chapter 2 Additive Error 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 Corporation 2-13 Xmath Model Reduction Module Next, Schur decompositions of W c W o are formed with the eigenvalues of W c W o in ascending and descending order. These eigenvalues are the square of the Hankel singular values of Sys , and if S...

Chapter 2 Additive Error Reduction Xmath Model Reduction Module 2-14 ni.com For the discrete-time case: When {bound} is specified, the error bound just enunciated is used to choose the number of states in SysR so that the bound is satisfied and nsr is as small as possible. If the desired error bound...

Chapter 2 Additive Error Reduction © National Instruments Corporation 2-15 Xmath Model Reduction Module Algorithm The algorithm does the following. The system Sys and the reduced order system SysR are stable; the system SysU has all its poles in Re [ s ] > 0. If the transfer function matrices are...

Chapter 2 Additive Error Reduction © National Instruments Corporation 2-17 Xmath Model Reduction Module Thus, the penalty for not being allowed to include G u in the approximation is an increase in the error bound, by σ n i + 1 + ... + σ ns . A number of theoretical developments hinge on bounding th...

Chapter 2 Additive Error 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 bound) satisfying: Note G r is optimal, that is, there is no other G r achieving a lower bound. Onepass Algorithm The first steps o...

Chapter 2 Additive Error Reduction © National Instruments Corporation 2-19 Xmath Model Reduction Module and finally: These four matrices are the constituents of the system matrix of , where: Digression: This choice is related to the ideas of [Glo84] in the following way; in [Glo84], the complete set...

Chapter 2 Additive Error 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 Corporation 2-21 Xmath Model 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 , ..., this being a...

Chapter 2 Additive Error Reduction Xmath Model Reduction Module 2-22 ni.com We use sysZ to denote G(z) and define: bilinsys=makepoly([-1,a]/makepoly([1,a]) as the mapping from the z-domain to the s-domain. The specification is reversed because this function uses backward polynomial rotation. Hankel ...

Chapter 2 Additive Error Reduction © National Instruments Corporation 2-23 Xmath Model Reduction Module It follows by a result of [BoD87] that the impulse response error for t > 0 satisfies: Evidently, Hankel norm approximation ensures some form of approximation of the impulse response too. Unsta...

© National Instruments Corporation 3-1 Xmath Model Reduction Module 3 Multiplicative Error Reduction This chapter describes multiplicative error reduction presenting two reasons to consider multiplicative rather than additive error reduction, one general and one specific. Selecting Multiplicative Er...

Chapter 3 Multiplicative Error Reduction Xmath Model Reduction 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 . This result indicates that if a...

Chapter 3 Multiplicative Error Reduction Xmath Model Reduction Module 3-4 ni.com The objective of the algorithm 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...

Chapter 3 Multiplicative Error Reduction © National Instruments Corporation 3-5 Xmath Model Reduction Module These cases are secured with the keywords right and left , respectively. If the wrong option is requested for a nonsquare G ( s ), an error message will result. The algorithm has the property...

Chapter 3 Multiplicative Error Reduction Xmath Model Reduction Module 3-6 ni.com 2. With G ( s ) = D + C ( sI – A ) –1 B and stable, with DD ´ nonsingular and G ( j ω ) G '(– j ω ) nonsingular for all ω , part of a state variable realization of a minimum phase stable W ( s ) is determined such that ...

Chapter 3 Multiplicative Error Reduction Xmath Model Reduction Module 3-8 ni.com state-variable representation of G . In this case, the user is effectively asking for G r = G . When the phase matrix has repeated Hankel singular values, they must all be included or all excluded from the model, that i...

Chapter 3 Multiplicative Error Reduction © National Instruments Corporation 3-9 Xmath Model Reduction Module Hankel Singular Values 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 . The corresponding quantities ...

Chapter 3 Multiplicative Error Reduction Xmath Model Reduction Module 3-10 ni.com which also can be relevant in finding a reduced order model of a plant. The procedure requires G again to be nonsingular at ω = ∞ , and to have no j ω -axis poles. It is as follows: 1. Form H = G –1 . If G is described...

Chapter 3 Multiplicative Error Reduction © National Instruments Corporation 3-11 Xmath Model Reduction Module The values of G ( s ), as shown in Figure 3-2, along the j ω -axis are the same as the values of around a circle with diameter defined by [ a – j 0, b –1 + j 0] on the positive real axis. Fi...

Chapter 3 Multiplicative Error Reduction © National Instruments Corporation 3-15 Xmath Model Reduction Module The conceptual basis of the algorithm can best be grasped by considering the case of scalar G ( s ) of degree n . Then one can form a minimum phase, stable W ( s ) with | W ( j ω )| 2 = | G ...

Chapter 3 Multiplicative Error Reduction Xmath Model Reduction Module 3-16 ni.com eigenvalues 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. Next, a stabilizing solution Q is found for the following Riccati equation: The funct...

Chapter 3 Multiplicative Error Reduction © National Instruments Corporation 3-17 Xmath Model Reduction Module singular values of F ( s ) larger than 1– ε (refer to steps 1 through 3 of the Restrictions section). The maximum order permitted is the number of nonzero eigenvalues of W c W o larger than ...

Chapter 3 Multiplicative Error Reduction 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 multivariable case when ophank( ) is used to find a Hankel norm approximation of F ( s ). 5. The a...

Chapter 3 Multiplicative Error Reduction © National Instruments Corporation 3-19 Xmath Model Reduction Module • and stand in the same relation as W ( s ) and G ( s ), that is: – – With , there holds or – With there holds or – – is the stable strictly proper part of . • The Hankel singular values of ...

Chapter 3 Multiplicative Error Reduction Xmath Model Reduction Module 3-20 ni.com Error Bounds The error bound formula (Equation 3-3) is a simple consequence of iterating (Equation 3-5). To illustrate, suppose there are three reductions → → → , each by degree one. Then, Also, Similarly, Then: The er...

Chapter 3 Multiplicative Error Reduction © National Instruments Corporation 3-21 Xmath Model Reduction Module For mulhank( ) , this translates for a scalar system into and The bounds are double for bst( ) . The error as a function of frequency is always zero at ω = ∞ for bst( ) (or at ω = 0 if a tra...

Chapter 4 Frequency-Weighted Error Reduction Xmath Model Reduction Module 4-2 ni.com (so that ) is logical. However, a major use of weighting is in controller reduction, which is now described. Controller Reduction Frequency weighted error reduction becomes particularly important in reducing control...

Chapter 4 Frequency-Weighted Error Reduction © National Instruments Corporation 4-3 Xmath Model Reduction Module is minimized (and of course is less than 1). Notice that these two error measures are like those of Equation 4-1 and Equation 4-2. The fact that the plant ought to show up in a good formu...

Chapter 4 Frequency-Weighted Error Reduction Xmath Model Reduction Module 4-4 ni.com Most of these ideas are discussed in [Enn84], [AnL89], and [AnM89]. The function wtbalance( ) implements weighted reduction, with five choices of error measure, namely E IS , E OS , E M , E MS , and E 1 with arbitra...

Chapter 4 Frequency-Weighted Error Reduction © National Instruments Corporation 4-5 Xmath Model Reduction 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 the reduced order controller. A different approa...

Chapter 4 Frequency-Weighted Error Reduction Xmath Model Reduction Module 4-6 ni.com • Form the reduced controller by interconnecting using negative feedback the second output of G r to the input, that is, set Nothing has been said as to how should be chosen—and the end result of the reduction, C r ...

Chapter 4 Frequency-Weighted Error Reduction © National Instruments Corporation 4-7 Xmath Model Reduction Module Matrix algebra shows that C ( s ) can be described through a left or right matrix fraction description with D L , and related values, all stable transfer function matrices. In particular:...

Chapter 4 Frequency-Weighted Error Reduction Xmath Model Reduction Module 4-8 ni.com The left MFD corresponds to the setup of Figure 4-3. Figure 4-3. C(s) Implemented to Display Left MFD Representation The setup of Figure 4-2 suggests approximation of: whereas that of Figure 4-3 suggests approximati...

Chapter 4 Frequency-Weighted Error Reduction 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 . For...

Chapter 4 Frequency-Weighted Error Reduction © National Instruments Corporation 4-11 Xmath Model Reduction Module • Reduce the order of a transfer function matrix C ( s ) through frequency-weighted balanced truncation, a stable frequency weight V ( s ) being prescribed. The syntax is more accented t...

Chapter 4 Frequency-Weighted Error Reduction © National Instruments Corporation 4-13 Xmath Model Reduction Module 3. Compute weighted Hankel Singular Values σ i (described in more detail later). If the order of C r ( s ) is not specified a priori , it must be input at this time. Certain values may b...

Chapter 4 Frequency-Weighted Error Reduction Xmath Model Reduction Module 4-14 ni.com and the observability grammian Q , defined in the obvious way, is written as It is trivial to verify that so that Q cc is the observability gramian of C s ( s ) alone, as well as a submatrix of Q . The weighted Han...

Chapter 4 Frequency-Weighted Error Reduction © National Instruments Corporation 4-15 Xmath Model Reduction Module From these quantities the transformation matrices used for calculating C sr ( s ), the stable part of C r ( s ), are defined and then Just as in unweighted balanced truncation, the reduc...

Chapter 4 Frequency-Weighted Error Reduction Xmath Model Reduction Module 4-16 ni.com 3. Only continuous systems are accepted; for discrete systems use makecontinuous( ) before calling bst( ) , then discretize the result. Sys=fracred(makecontinuous(SysD)); SysD=discretize(Sys); Defining and Reducing...

Chapter 4 Frequency-Weighted Error Reduction © National Instruments Corporation 4-17 Xmath Model Reduction Module to, for example, through, for example, balanced truncation, and then defining: For the second rationale, consider Figure 4-5. Figure 4-5. Internal Structure of Controller Recognize that ...

Chapter 4 Frequency-Weighted Error Reduction Xmath Model Reduction Module 4-20 ni.com Additional Background A discussion of the stability robustness measure can be found in [AnM89] and [LAL90]. The idea can be understood with reference to the transfer functions E ( s ) and E r ( s ) used in discussi...

Chapter 5 Utilities © National Instruments Corporation 5-3 Xmath Model Reduction Module Doubtful ones are those for which the real part of the eigenvalue has magnitude less than or equal to tol for continuous-time, or eigenvalue magnitude within the following range for discrete time: A warning is gi...

Chapter 5 Utilities Xmath Model Reduction Module 5-4 ni.com After this last transformation, and with it follows that and By combining the transformation yielding the real ordered Schur form for A with the transformation defined using X, the overall transformation T is readily identified. In case all...

© National Instruments Corporation 6-1 Xmath Model Reduction Module 6 Tutorial This chapter illustrates a number of the MRM functions and their underlying ideas. A plant and full-order controller are defined, and then the effects of various reduction algorithms are examined. The data for this exampl...

Chapter 6 Tutorial Xmath Model Reduction Module 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 frequencies (for performance) and low loop gain at high frequencies (to guarantee stability in the presence of unstructured ...

Chapter 6 Tutorial © National Instruments Corporation 6-3 Xmath Model Reduction Module With a state weighting matrix, Q = 1e-3*diag([2,2,80,80,8,8,3,3]); R = 1; (and unity control weighting), a state-feedback control-gain is determined through a linear-quadratic performance index minimization as: [K...

Chapter 6 Tutorial Xmath Model Reduction Module 6-4 ni.com recovery at low frequencies; there is consequently a faster roll-off of the loop gain at high frequencies than for , and this is desired. Figure 6-2 displays the (magnitudes of the) plant transfer function, the compensator transfer function ...

Chapter 6 Tutorial Xmath Model Reduction Module 6-6 ni.com Figures 6-3, 6-4, and 6-5 display the outcome of the reduction. The loop gain is shown in Figure 6-3. The error near the unity gain crossover frequency may not look large, but it is considerably larger than that obtained through frequency we...

Chapter 6 Tutorial © National Instruments Corporation 6-9 Xmath Model Reduction Module ophank( ) ophank( ) is next used to reduce the controller with the results shown in Figures 6-6, 6-7, and 6-8. Generate Figure 6-6: [syscr,sysu,hsv]=ophank(sysc,2); svalsrol = svplot(sys*syscr,w,{radians}); plot(s...

Chapter 6 Tutorial © National Instruments Corporation 6-11 Xmath Model Reduction Module Generate Figure 6-8: tvec=0:(140/99):140; compare(syscl,sysclr,tvec,{type=7}) f8=plot({keep,legend=["original","reduced"]}) Figure 6-8. Step Response with ophank The open-loop gain, closed-loop ga...

Chapter 6 Tutorial Xmath Model Reduction Module 6-12 ni.com wtbalance The next command examined is wtbalance with the option "match" . [syscr,ysclr,hsv] = wtbalance(sys,sysc,"match",2) Recall that this command should promote matching of closed-loop transfer functions. The weighted Ha...

Chapter 6 Tutorial © National Instruments Corporation 6-13 Xmath Model Reduction Module The following function calls produce Figure 6-9: svalsrol = svplot(sys*syscr,w,{radians}) plot(svalsol, {keep}) f9=plot(wc, constr, {keep,!grid, legend=["reduced","original","constrained"]...

Chapter 6 Tutorial © National Instruments Corporation 6-15 Xmath Model Reduction Module Generate Figure 6-11: tvec=0:(140/99):140; compare(syscl,sysclr,tvec,{type=7}) f11=plot({keep,legend=["original","reduced"]}) Figure 6-11. Step Response with wtbalance Figures 6-9, 6-10, and 6-11 ...

Chapter 6 Tutorial Xmath Model Reduction Module 6-20 ni.com fracred fracred , the next command examined, has four options— "right stab" , "left stab" , "right perf" , and "left perf" . The options "left stab" , "right perf" , and "left perf" al...

Chapter 6 Tutorial Xmath Model Reduction Module 6-22 ni.com Generate Figure 6-18: tvec=0:(140/99):140; compare(syscl,sysclr,tvec,{type=7}) f18=plot({keep,legend=["original","reduced"]}) Figure 6-18. Step Response with fracred The end result is comparable to that from wtbalance( ) wit...

Appendix A Bibliography © National Instruments Corporation A-3 Xmath Model Reduction Module [SaC88] M. G. Safonov and R. Y. Chiang, “Model reduction for robust control: a Schur relative-error method,” Proceedings for the American Controls Conference , 1988, pp. 1685–1690. [Saf87] M. G. Safonov, “Ima...

Appendix A Bibliography Xmath Model Reduction Module A-4 ni.com [Doy82] J. C. Doyle. “Analysis of Feedback Systems with Structured Uncertainties.” IEEE Proceedings , November 1982. [DWS82] J. C. Doyle, J. E. Wall, and G. Stein. “Performance and Robustness Analysis for Structure Uncertainties,” Proce...

© National Instruments Corporation I-1 Xmath Model Reduction Module Index Symbols *, 1-6´, 1-6 A additive error, reduction, 2-1algorithm balanced stochastic truncation (bst), 3-4fractional representation reduction, 4-18Hankel multi-pass, 2-20optimal Hankel norm reduction, 2-15stable, 5-2weighted bal...

Index Xmath Model Reduction Module I-2 ni.com G grammians controllability, 1-7description of, 1-7observability, 1-7 H Hankel matrix, 1-9Hankel norm approximation, 2-6Hankel singular values, 1-8, 3-9, 5-1hankelsv, 1-5, 5-1 algorithm, multipass, 2-20 help, technical support, B-1 I instrument drivers (...