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National Instruments 370755B-01 User's Manual

1. Figure 3 1 Controller C s Designed for Multiplicative Error Reduction The full order plant is G J A G and the reduced order model is G Since G G G A this means that A is the multiplicative error Another way one could measure the multiplicative error would be as G e In the matrix plant case interchange of the order of the product gives two more possibilities again The following multiplicative robustness result can be found in Vid85 National Instruments Corporation 3 1 Xmath Model Reduction Module Chapter 3 Multiplicative Error Reduction Multiplicative Robustness Result Xmath Model Reduction Module Suppose C stabilizes G that A G AG has no j axis poles and that G has the same number of poles in Re s 2 0 as G If for all IAAI GAC Guyer lt 1 3 1 then C stabilizes G This result indicates that if a controller C is designed to stabilize a nominal or reduced order model G satisfaction of Equation 3 1 ensures that the controller also will stabilize the true plant G In reducing a model of the plant there will be concern not just to have this type of stability property but also concern to have as little error as possible between the designed system based on G and the true system based on G Extrapolation of the stability result then suggests that the goal should be not just to have Equation 3 1 but to minimize the quantity on the left side of Equation 3 1 or its grea
2. 0 For continuous systems there is generally some error at because the D matrix is normally changed This means that normally the approximation of a strictly proper system through mreduce will not be strictly proper in contrast to the situation with balmoore For discrete systems the D matrix is also normally changed so that for example a system which was strictly causal or guaranteed to contain a delay that is D 0 will be approximated by a system SysR without this property The presentation of the Hankel singular values may suggest a logical dimension for the reduced order system thus if 6 O it may be sensible to choose nsr k With mreduce and a continuous system the reduced order system SysR is internally balanced with the grammian diag 0 Oz 0 SO that 1ts Hankel Singular Values are a subset of those of the original system Sys Provided O gt 0 p SysR also is controllable observable and stable This is not guaranteed if Ops O p SO it is highly advisable to avoid this situation Refer to the balmoore section for more on the balmoore algorithm With mreduce and discrete systems the reduced order system SysR is not in general balanced in contrast to balmoore _ and its Hankel singular values are not in general a subset of those of Sys Provided Onsr gt Osrn 1 the reduced order system SysR also is controllable observable and stable This is not guarant
3. C sI Ay B are computed by calling hankelsv The value of nsr is obtained if not prespecified either by prompting the user or by the error bound formula GrA89 Gre88 Glo86 varai UG Gpl lt T a yp 1 3 3 j nsr 1 with v 2 V 12 being assumed If V Vk 1 Vk 4 r for some k that is v has multiplicity greater than unity then v appears once only in the previous error bound formula In other words the number of terms in the product is equal to the number of distinct v less than Vasr There are restrictions on nsr nsr cannot exceed the dimension of a minimal realization of G s although v 2 1 nsr must obey Mnsr gt Nnsr 1 and While 1 gt v for all i itis necessary that 1 gt V 1 The number of v equal to 1 is the number of right half plane zeros of G s They must be retained in G s so the order of G s nsr must at least be equal to the number of v equal to 1 The software checks all these conditions The minimum order permitted is the number of Hankel 3 16 ni com National Instruments Corporation Chapter 3 Multiplicative Error Reduction singular values of F s larger than 1 e refer to steps 1 through 3 of the Restrictions section The maximum order permitted is the number of nonzero eigenvalues of W W larger than e Let r be the multiplicity of v The algorithm approximates F s C s1 A B by a transfer function matrix F s of order ns r using Ha
4. computed from Equation 1 1 and from Equation 1 2 should be similar e If Equation 1 1 and Equation 1 2 are excited with the same u have the same x t and Equation 1 1 has arbitrary x2 then x and y computed from Equation 1 1 and Equation 1 2 should be similar after a possible initial transient As far as the transfer function matrices are concerned it can be verified that they are actually equal at DC 1 12 ni com Chapter 1 Introduction Similar considerations govern the discrete time problem where x k 1 a dia x k B u x k 1 Ay 422 x k B x 4 12 4 LT wk c c m can be approximated by 1 xi k 1 4 4 4 4 1x 4 B 4 I 43 B lu k 1 Yk Ci Ca 42 Az x k D C I A3 B u k mreduce can carry out singular perturbation For further discussion refer to Chapter 2 Additive Error Reduction If Equation 1 1 is balanced singular perturbation is provably attractive Spectral Factorization Let W s be a stable transfer function matrix and suppose a system S with transfer function matrix W s is excited by zero mean unit intensity white noise Then the output of is a stationary process with a spectrum s related to W s by s W s W s 1 3 Evidently DUA WoW jo so that j is nonnegative hermitian for all when W j is a scalar so is D j with B ja IW j In the matrix case O is singular for some only if W
5. of states in G of states in G This number is always the multiplicity of a Hankel singular value Thus when the order of G is n _ the number of eliminated states is n n _ or the multiplicity of O 1 Oni For each order n _ of G s it is possible to find G and G so that EGO G 0 G 7 S On Choosing i 1 causes G to be of order zero identify n 0 Actually among all approximations of G s with stable part restricted to having degree n _ and with no restriction on the degree of the unstable part one can never obtain a lower bound on the approximation error than O in the scalar or SISO G s case the G s which achieves the previous bound is unique while in the matrix or MIMO G s case the G s which achieves the previous bound may not be unique Glo84 The algorithm we use to find G s and G s however allows no user choice and delivers a single pair of transfer function matrices The transfer function matrix G j alone can be regarded as a stable approximation of G j If the D matrix in G j is approximately chosen and the algorithm ensures that it is then ICID CGD S 5 4 6 2 3 1 ns ni com Chapter 2 Additive Error Reduction Thus the penalty for not being allowed to include G in the approximation is an increase in the error bound by 6 O A number of theoretical developments hinge on bounding the Hankel singular
6. An error bound is available In the continuous time case it is as follows Let G j and Gr j be the transfer function matrices of Sys and SysR respectively For the continuous case G j CRD lt 2 0 O 6 49 O nsr National Instruments Corporation 2 13 Xmath Model Reduction Module Chapter 2 Additive Error Reduction Related Functions ophank For the discrete time case jo 10 Gte Gale DI 2 sr 1FOnsra2 t Ons 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 is smaller than 26 5 no reduction is made In the continuous time case the error depends on frequency but is always zero at ce If the reduction in dimension is 1 or the system Sys is single input single output with alternating poles and zeros on the real axis the bound is tight It is far from tight when the poles and zeros approximately alternate along the jw axis It is not normally tight in the discrete time case and for both continuous time and discrete time cases it is not tight if there are repeated singular values The presentation of the Hankel singular values may suggest a logical dimension for the reduced order system thus if 0 O p it may be sensible to choose nsr k ophank balmoore Restriction Xmath Model Reduction Module SysR SysU HSV
7. Next the following definitions are made An Ato _ An Az Bi _ u B U i Co CP V V3 2 J 02A OAP V V QB Xmath Model Reduction Module 2 18 ni com Chapter 2 Additive Error Reduction and finally A Si Gi Avil Asi B S B SAAB C 43431 1 D D C AB These four matrices are the constituents of the system matrix of G s where G s G s G s Digression This choice is related to the ideas of Glo84 in the following way in Glo84 the complete set is identified of G s satisfying GGo Ga 6 with G having a stable part of order n _ The set is parameterized in terms of a stable transfer function matrix K s which has to satisfy C K s B 0 I K j K j lt 0 for all with C B2 being two matrices appearing in the course of the algorithm of Glo84 and enjoying the property C C B B3 The particular choice K s SCC Ci Bs in the algorithm of Glo84 and flagged in corollary 7 3 of Glo84 is equivalent to the previous construction in the sense of yielding the same G though the actual formulas used here and in Glo84 for the construction procedure are quite different In a number of situations including the case of scalar SISO G s this is the only choice The next step of the algorithm is to call stable _ to separate G s into its stable and unstable parts call them G s and G s stable will always assign the m
8. ophank Sys nsr onepass The ophank function calculates an optimal Hankel norm reduction of Sys This function has the following restriction e Only continuous systems are accepted for discrete systems use makecontinuous before calling bst then discretize the result Sys ophank makecontinuous SysD SysD discretize Sys 2 14 ni com Chapter 2 Additive Error Reduction 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 gt 0 If the transfer function matrices are G s G s and G s then e G s is a stable approximation of G s e G s G s is a more accurate but not stable approximation of G s and optimal in a certain sense Of course the algorithm works with state space descriptions that of G s can be minimal while that of G s cannot be These statements are explained in the Behaviors section If onepass is specified reduction is calculated in one pass If onepass is not called or is set to 0 onepass 0 reduction is calculated in number of states of Sys nsr passes There seems to be no general rule to suggest which setting produces the more accurate approximation G Therefore if accuracy of approximation for a given order is critical both should be tried As noted previously if an approximation involving an unstable system is desired the default onepass 1 is specified Be
9. Vol AC 31 pp 734 743 August 1986 B Francis A Course in L Control Theory Springer Verlag Berlin New York 1987 D S Flamm S Boyd G Stein and S K Mitter Tutorial Workshop on Lee Control Theory pre conference workshop Proceedings 26th IEEE Conference on Decision and Control December 1988 K Glover and J C Doyle State space formulae for all stabilizing controllers that satisfy an Loo norm bound and relations to risk sensitivity Systems and Control Letters Vol 11 pp 167 172 1988 J C Doyle K Glover P K Khargonekar and B Francis State space solutions to standard H and L control problems IEEE Transactions on Automatic Control Vol AC 34 No 8 pp 831 847 August 1989 N K Gupta Frequency Shaping of Cost Functionals An extension of LQG Design Methods AIAA Journal of Guidance and Control Vol 3 No 6 December 1980 ONR Honeywell Workshop on Advances in Multivariable Control Lecture Notes Minneapolis MN 1984 E E Osborne On Preconditioning of Matrices JACM 7 338 345 1960 M G Safonov Stability Margins of Diagonally Perturbed Multivariable Feedback Systems IEEE Proceedings 129 D 251 256 November 1982 M G Safonov and J C Doyle Optimal Scaling for Multivariable Stability Margin Singular Value Computation Proceedings of MECO EES 1983 Symposium 1983 M G Safonov and J C Doyle Minimizing Conservativeness of Robust Sin
10. to1 1 to1 in discrete time then X is likely to be ill conditioned and consequently Syss and SysU will also be ill conditioned For example the B matrix of Syss could contain very small values while the C matrix could contain large values In this case SysS would be very weakly controllable and very strongly observable This will cause problems when gramians and Hankel singular values are calculated To avoid this problem change tol to a value that is not close to the majority of eigenvalues A further transformation of A is constructed using X National Instruments Corporation E A 0 0 Ay 5 3 Xmath Model Reduction Module Chapter 5 Utilities Related Functions compare After this last transformation and with Bs By B C C Cy it follows that SysS A3 A5 C D and SysU A By Cy9 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 eigenvalues of A are stable or all are unstable this is flagged and T I stable can be combined with a reduction algorithm such as redschur orbalmoore toreduce the order of a system with some unstable and some stable poles One uses stable to separate the stable and unstable parts and then for example reduces the stable part with redschur the reduced stable part is added to the original unstable part to provide the d
11. with type output stab Xmath Model Reduction Module SysCR SysCLR HSV wtbalance Sys SysC type nscr SysvV The wtbalance function calculates a frequency weighted balanced truncation of a system wtbalance has two separate uses Reduce the order of a controller C s located in a stable closed loop with the plant P s known Frequency weighted balanced truncation is used with the weights involving P s and being calculated in a predominantly standard way 4 10 ni com Chapter 4 Frequency Weighted Error Reduction 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 towards the first use For the second use the user should set S 0 NS 0 This results in automatically SCLR NSCLR 0 The user will also select the type input spec Let C s be the reduced order approximation of C s which is being sought Its order is either specified in advance or the user responds to a prompt after presentation of the weighted Hankel singular values Then the different types concentrate on approximately minimizing certain error measures through frequency weighted balanced truncation These are shown in Table 4 1 Table 4 1 Types versus Error Measures Type Error Measure input stab I C C PU CPT output stab 7 Pcy Ptc c matc
12. E s K sI A K C E K 4 11 which is formed from the numerator and denominator of the MFD in Equation 4 5 The grammian equations Equation 4 8 and Equation 4 9 are replaced by P A K C 4 K C P BB K K Q A K C A K C O KpKp redschur type calculations are used to reduce E s and Equation 4 10 again yields the reduced order controller Notice that the HSVs obtained from Equation 4 10 or the left MFD Equation 4 5 of C s will in general be quite different from those coming from the right MFD Equation 4 6 It may be possible to reduce much more with the left MFD than with the right MED or vice versa before closed loop stability is lost As noted in the fracred input listing type left stab and right stab focus on a stability robustness measure in conjunction with Equation 4 5 and Equation 4 6 respectively Leaving aside for the moment the explanation the key differences in the algorithm computations lie solely in the calculation of the grammians P and Q For type left stab these are given by P A BK 4 BK P BB O A K C 4 K CYQ KKz and for right stab P A BKp 4 BKp P KpKg 4 12 O A K C 4 K C 0 C C 4 13 National Instruments Corporation 4 19 Xmath Model Reduction Module Chapter 4 Frequency Weighted Error Reduction Additional Background Xmath Model Reduction Module A discussion of the stability robustness measure can be found in A
13. The smaller the error measure is the greater the stability robustness output stab A similar stability robustness argument but based on breaking the loop at the controller input indicates that if C is stabilizing for P and the error measure is less that 1 then C is stabilizing for P The smaller the error measure is the greater the stability robustness match If T PC PC and T PC 1 PC are the two closed loop transfer function matrices then T T to first order in C C is given by I PC P C C U PCT so that the error measure looks at matching of the closed loop transfer function matrix match spec It may be important to match closed loop transfer function matrices more at certain frequencies than others frequency weighting is achieved by introducing V s Frequencies corresponding to larger values of V jq l or V j V jo will be the frequencies at which T j and T 0 should have smaller error input spec This is the one error measure that is not associated with a plant or closed loop of some kind It simply allows the user to emphasize certain frequencies in the reduction procedures Algorithm The major steps of the algorithm are as follows 1 Check dimension syntax stability of SysV closed loop stability and decomposition of C s into the sum of a stable part poles in Re s lt 0 and unstable part poles in Re s gt 0 stable is used fo
14. does not have full rank there and in the scalar case only if W has a zero there Spectral factorization as shown in Example 1 1 seeks a Wj given j In the rational case a W jw exists if and only if P j is National Instruments Corporation 1 13 Xmath Model Reduction Module Chapter 1 Introduction nonnegative hermitian for all If O is scalar then jw 0 for all Normally one restricts attention to with lim D 0 lt oo A key result is that given a rational nonnegative hermitian D j with lim DP 0 lt oo there exists a rational W s where e W e lt o e W s is stable e Ws is minimum phase that is the rank of W s is constant in Re s gt 0 In the scalar case all zeros of W s lie in Re s lt 0 or in Re s lt 0 if Gw gt 0 for all In the matrix case and if P 0 is nonsingular for some o it means that W s is square and W s has all its poles in Re s lt 0 or in Re s lt 0 if B j is nonsingular for all Moreover the particular W s previously defined is unique to within right multiplication by a constant orthogonal matrix In the scalar case this means that W s is determined to within a 1 multiplier Example 1 1 Example of Spectral Factorization Suppose o 1 O jo o 4 Then Equation 1 3 is satisfied by W s E which is stable and minimum phase 1 s 3 s l sTS 1 Also Equation 1 3 is satisfied bias st2 and 72542 and oy 5 an so forth b
15. is known as Nehari approximation The second special case arises when nsr n _ or NS 1 if the smallest Hankel singular value has multiplicity 1 In this case G s becomes a constant which can then be lumped in with G s so that G s of degree NS is then National Instruments Corporation 2 17 Xmath Model Reduction Module Chapter 2 Additive Error Reduction being approximated by a stable G s with the actual error as opposed to just the error bound satisfying G s T GD Ons 3 Note G is optimal that is there is no other G achieving a lower bound Onepass Algorithm The first steps of the algorithm are to obtain the Hankel singular values of G s by using hankelsv and identify their multiplicities Stability of G s is checked in this process If the user has specified nsr and this does not coincide with one of 0 n n an error message is obtained generally all the are different so the occurrence of error messages will be rare The next step of the algorithm is to calculate the sum G s G s G s following SCL90 A separate function ophred is called for this purpose The controllability and observability grammians P and Q are found in the usual way AP PA BB QA AQ C C and then a singular value decomposition is obtained of the matrix QP 071 E Sg OIV ly A B J OP 0 1 0 ojn There are precisely n n _ zero singular values this being the multiplicity of o
16. is the same as the number of zeros of G s in Re s gt 0 State variable realizations of W G and the stable part of W s G s can be connected in a nice way The algorithm computes an additive Hankel norm reduction of the stable part of W s G s to cause a degree reduction equal to the multiplicity of the smallest o The matrices defining the reduced order object are then combined in a new way to define a multiplicative approximation to G s as it turns out there is a close connection between additive reduction of the stable part of W s G s and multiplicative reduction of G s The reduction procedure then can be repeated on the new phase function of the just found approximation to obtain a further reduction again in G s A description of the algorithm for the keyword right follows It is based on ideas of Glo86 in part developed in GrA86 and further developed in SaC88 The procedure is almost the same when left is specified except the transpose of G s is used the following algorithm finds an approximation then transposes it to yield the desired G s 1 The algorithm checks that G s is square stable and that the transfer function is nonsingular at infinity 2 With G s D C sI A B square and stable with D nonsingular rank d must equal number of rows in d and G j nonsingular for all finite this step determines a state variable realization of a minimum phase stable W s such that W s W s G s
17. near to1 or 1 to1 then Syss and Sysu might be ill conditioned To avoid this problem choose tol to a value that is not close to the majority of poles The algorithm begins by transforming the A matrix to Schur form and counting the number of stable and unstable eigenvalues together with those for which classification is doubtful Stable eigenvalues are those in either of the following e Re s lt 0 continuous time e Izl lt 1 discrete time 5 2 ni com Chapter 5 Utilities 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 1 tol 1 tol A warning is given if doubtful eigenvalues exist The algorithm then computes a real ordered Schur decomposition of A so that after transformation As su 0 Ay A where the eigenvalues of As and Ay are respectively stable and unstable A matrix X satisfying A sx XA y Asy 0 is then determined by calling the algorithm sylvester The eigenvalue properties of Ag and Ay guarantee that X exists If doubtful eigenvalues are present they are assigned to the unstable part of Sys In this circumstance you get the message The system has poles near or upon the jw axis for continuous systems and the following for discrete systems The system has poles near the unit circle 3 Note If A has eigenvalues clustered near
18. programs tutorials application notes instrument drivers and so on Free Technical Support All registered users receive free Basic Service which includes access to hundreds of Application Engineers worldwide in the NI Developer Exchange at ni com exchange National Instruments Application Engineers make sure every question receives an answer Training and Certification Visit ni com training for self paced training eLearning virtual classrooms interactive CDs and Certification program information You also can register for instructor led hands on courses at locations around the world System Integration If you have time constraints limited in house technical resources or other project challenges NI Alliance Program members can help To learn more call your local NI office or visit ni com alliance If you searched ni com and could not find the answers you need contact your local office or NI corporate headquarters Phone numbers for our worldwide offices are listed at the front of this manual You also can visit the Worldwide Offices section of ni com niglobal to access the branch office Web sites which provide up to date contact information support phone numbers email addresses and current events National Instruments Corporation B 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
19. quantities v The Schur decompositions are V POV Dace VpPOV des where V Vp are orthogonal and Se Saes are upper triangular 4 Define submatrices as follows assuming the dimension of the reduced order system nsr is known 0 Ler V nig vd l V big Vo 0 nsr Determine a singular value decomposition VobigSebig Pobia V Ibig V big and then define transformation matrices Sibig OE S big A The reduced order system G is Ap SAS mis Br SipigB Ar CS hig Dp D where step 4 is identical with that used in redschur except the matrices P Q which determine V Vp and so forth are the controllability and observability grammians of Cy s A B rather than of C sI A B the controllability grammian of G s and the observability grammian of W s The error formula WaS90 is a G G any vi 3 2 1 v All v obey v lt 1 One can only eliminate v where v lt 1 Hence if nsr is chosen so that Vasr 1 the algorithm produces an error message The algorithm also checks that nsr does not exceed the dimension of a minimal National Instruments Corporation 3 7 Xmath Model Reduction Module Chapter 3 Multiplicative Error Reduction state variable representation of G In this case the user is effectively asking for G G When the phase matrix has repeated Hankel singular values they must all be included or all excluded from the model that is Vasr Vnsr 1 18 no
20. 1 0862 9638 5846 5646 3144 The most attractive candidate for reducing to second order is right stab This is because the HSVs being discarded columns 3 to 8 are smaller relative to those being retained columns 1 and 2 for right stab than for the other three candidates ay Note The relative values count not the absolute values Related Functions redschur wtbalance O National Instruments Corporation 4 21 Xmath Model Reduction Module Utilities This chapter describes three utility functions hankelsv stable and compare The background to hankelsv which calculates Hankel singular values was presented in Chapter 1 Introduction Hankel singular values are also calculated in other functions sometimes by other procedures A comparison of the procedures is given in the Hankel Singular Values section The function compare serves to facilitate the comparisons of an unreduced and a reduced system from various points of views The function stable is used to separate additively a system into its stable and unstable parts that is given G s the function determines G s and G s the first with all poles in Re s lt 0 the second with all poles in Re s gt 0 such that G s G s G s The function is used within some of the other functions of the Model Reduction Module It should also be used when reduction of an unstable G s is contemplated The norm
21. 79 DoS81 Appendix A Bibliography 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 M G Safonov Imaginary axis zeros in multivariable H optimal control Modeling Robustness and Sensitivity Reduction in Control Ed R F Curtain Springer Verlag Berlin 1987 M G Safonov R Y Chiang and DIN Limebeer Optimal Hankel model reduction for nonminimal systems JEEE Transactions on Automatic Control Vol 35 No 4 pp 496 502 1990 M Vidyasagar Control Systems Synthesis A Factorization Approach MIT Press Cambridge MA 1985 W Wang and M G Safonov A tighter relative error bound for balanced stochastic truncation Systems and Control Letters Vol 14 1990 pp 307 317 W Wang and M G Safonov Comparison between continuous and discrete time model truncation Proceedings for the 29th CDC 1990 S Boyd V Balakrishnan and P Kabamba A bisection method for computing the Leo norm of a transfer matrix and related problems Mathematical Controls Signals and Systems Vol 2 No 3 pp 207 219 1989 A Berman and R J Plemmons Nonnegative Matrices in the Mathematical Sciences Computer Science and Applied Mathematics series Academic Press 1979 S Boyd and V Balakrishnan A regularity result for the singular values of a transfer matrix and a quadra
22. A A lt 0 Then there exist symmetric matrices P Q defined by PA AP BB QA A Q C C These are termed the controllability and observability grammians of the realization defined by A B C D Sometimes in the code WC is used for P and WO for Q They have a number of properties National Instruments Corporation P 2 0 with P gt 0 if and only if A B is controllable Q 0 with Q gt 0 if and only if A C is observable P e BB e dt and O oca With ved P denoting the ea vector formed by stacking column 1 of P on column 2 on column 3 and so on and denoting Kronecker product 8 4 40 I vecP vec BB The controllability grammian can be thought of as measuring the difficulty of controlling a system More specifically 1f the system is in the zero state initially the minimum energy as measured by the L norm of u required to bring it to the state xp is x P xo so small eigenvalues of P correspond to systems that are difficult to control while zero eigenvalues correspond to uncontrollable systems 1 7 Xmath Model Reduction Module Chapter 1 Introduction The controllability grammian is also E x t x t when the system x Ax Bw has been excited from time o gt by zero mean white noise with E w t w s 6 t s The observability grammian can be thought of as measuring the information contained in the output concerning an initial state If x Ax y Cx with x 0
23. C and are computed reliably with the aid of schur If one is found you are warned that results may be unreliable Next a stabilizing solution Q is found for the following Riccati equation QA AQ C B yo DD Y C BO 0 The singriccati function is used failure of the nonsingularity condition on G 0 G O will normally result in an error message that no stabilizing solution exists To obtain the best numerical results singriccati isinvoked with the keyword method schur Although Dy Cw are not needed for the remainder of the algorithm they are simply determined in the square case by De D Cys DPD C 8 0 with minor modification in the nonsquare case The real point of the algorithm is to compute P and Q the matrix Q satisfies square or nonsquare case OA A0 CyCy 0 P Q are the controllability and observability grammians of the transfer function Cy sI A B This transfer function matrix it turns out is the strictly proper stable part of O s W 7 s G s which obeys the matrix all pass property 0 s 0 s J and is the phase matrix associated with G s Compute ordered Schur decompositions of PO with the eigenvalues of PQ is ascending and descending order Obtain the phase matrix Hankel singular values that is the Hankel singular values of the 3 6 ni com Chapter 3 Multiplicative Error Reduction strictly proper stable part of O s as the square roots of the eigenvalues of PQ Call these
24. G oooooccccnononoccnonnoanannnncnnnnananononno 3 9 Imaginary Axis Zeros Including Zeros at 09 eee eseeseeeeeeeeeeeeenaes 3 10 Related Funct onsesc2 ciate het ease ad eee einen 3 14 m lhank 9 22 A dt Seated 3 14 Restrictions Ha ia 3 14 Alcosa ia cesses EI EAA E ate aes aw entees 3 14 TIC ANG let o oia 3 15 Consequences of Step 5 and Justification of Step oononcnnnnonnnonconnonncnnnnonon 3 18 Error Bounds im ad 3 20 Imaginary Axis Zeros Including Zeros at o cooccnncnocnnoncocnonnnonnnnnonnconnnnnccnnos 3 21 Related Funcom lc aci 3 24 Chapter 4 Frequency Weighted Error Reduction Introduction rt AA A A A tddi 4 1 Controller Reduction 4 2 Controller Robustness Result ooooocononococncocononanococonononanncnonanononocononnnnnononannns 4 2 Fractional Representati0WS ooooconocnocnnonoonnnnncnncnnnonnccnnonn nono can conc eere i 4 5 Wibal it dt deis 4 10 ALS Orit in A iria dadas 4 12 Related Fun CtHOnS sic 4 15 TACTO A HAA le 4 15 RESTCUIONS seeders tichs tr a di Beets 4 15 Defining and Reducing a Controller ooooncnnonocincnnonanoncnnnonncnncnn conc cnn nonncnnccnnos 4 16 Xmath Model Reduction Module vi ni com Contents ALS Om SA a a r e eee 4 18 Additional Background insna ai ri non ncnn E E E 4 20 Related Functions id aen co a e id ea a a ai 4 21 Chapter 5 Utilities ha kelsy Josi nduru e ha Mae RM aoe ie E 5 1 Related F NGUONS its 5 2 Sable OD a E E as 5 2 ALO titi 5 2 Related Function
25. G s with W s Dy CASA Bs The various state variable matrices in W s are obtained as follows The controllability grammian P associated with G s is first found from AP PA BB 0 then Ay AB PC BDD D The algorithm checks to see if there is a zero or singularity of G s close to the ja axis The zeros are determined by calculating the National Instruments Corporation 3 15 Xmath Model Reduction Module Chapter 3 Multiplicative Error Reduction Xmath Model Reduction Module 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 O is found for the following Riccati equation OA A Q C B OY DD Y CB 0 0 The function singriccati is used failure of the nonsingularity condition of G jo will normally result in an error message To obtain the best numerical results singriccati is invoked with the keyword method schur The matrix C is given by C D C B OQ Notice that Q satisfies QA A Q C C 0 so that P and Q are the controllability and observability grammians of F s C sI A B This strictly proper stable transfer function matrix is the strictly proper stable part under additive decomposition of 0 s W s G s which obeys the matrix all pass property 050 s 1 It is the phase matrix associated with G s The Hankel singular values v of F s
26. It may well be that closed loop transfer function matrices should be better matched at some frequencies than others if this weighting on the error in the closed loop transfer function matrices is determined by the input spectrum Y V O then one really wants T T V to be small so that Equation 4 4 is replaced by Eys PCy PC CAE POI National Instruments Corporation 4 3 Xmath Model Reduction Module Chapter 4 Frequency Weighted Error Reduction Most of these ideas are discussed in Enn84 AnL89 and AnM89 The function wtbalance implements weighted reduction with five choices of error measure namely Ers Eos Em Ems and E with arbitrary V jo The first four are specifically for controller reduction whereas the last is not aimed specifically at this situation Several features of the algorithms are Only the stable part of C is really reduced the unstable part is copied exactly into C A modification of balanced realization truncation underpins the algorithms namely frequency weighted balanced truncation although to avoid numerical problems the actual construction of a frequency weighted balanced realization of C is avoided Frequency weighted Hankel singular values can be computed and although no error bound formula is available in contrast to the unweighted problem generally speaking there is little damage done in reducing by a number of states equal to the number of relatively small Hankel
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29. National Instruments Corporation 6 3 Xmath Model Reduction Module Chapter 6 Tutorial recovery at low frequencies there is consequently a faster roll off of the loop gain at high frequencies than for KC ol Ay Bl and this is desired Figure 6 2 displays the magnitudes of the plant transfer function the compensator transfer function and the loop gain as well as the constraints evidently the compensated plant meets the constraints You can enter the following commands to create a plot equivalent to Figure 6 2 sysol sys sysc svals svplot sys w radians svalsc svplot sysc w radians svalsol svplot sysol w radians plot svals x_log grid ylab line _width 2 hold plot svalsc keep plot svalsol keep f2 plot wc constr keep legend plant compensator compensated plant constraint plot hold 150 plant compensator compensated plant 100 constraint 50 50 100 150 20 E Pon 0 01 0 1 1 10 100 Frequency rad sec Figure 6 2 Frequency Response for Plant Compensator and Compensated Plant Xmath Model Reduction Module 6 4 ni com Chapter 6 Tutorial Controller Reduction This section contrasts the effect of unweighted and weighted controller reduction Unweighted reduction is at first examined through redschur using balance or balmoore will give similar results The Hankel singular values of the controller transfer funct
30. a for ophank is more attractive than that for redschur the error itself will be better for ophank O National Instruments Corporation 6 11 Xmath Model Reduction Module Chapter 6 Tutorial wtbalance Xmath Model Reduction Module 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 Hankel singular values are 1 486 4 513 x 107 8 420 x 102 5 869 x 1 2 1 999 x 10 1 382 x 10 7 198 x 10 3 6 336 x 103 The relative magnitudes suggest that reduction to order 2 will produce less of an approximation error here in the closed loop transfer function than a reduction to this order through redschur or ophank where the implicit criterion is the unweighted error in approximating the controller transfer function Examination of Figures 6 9 6 10 and 6 11 reveals that far better approximation is now obtained Violation of the specification is to be observed in the open loop gain Notice though that e The error measure for wtbalance does not reflect the open loop gain it reflects the closed loop gain e While the error in dB looks large as an absolute value it is not extremely so wtbalance works with additive not multiplicative error Hence it cannot be concluded that the algorithm is not working Use of the option match spec with wtbalance might be conject
31. a late step of the procedure if approximation is done early the subsequent steps of the design procedure may have unpredictable effects on the approximation errors Hence the scheme based on high order controller design followed by reduction is generally to be preferred Controller reduction should aim to preserve closed loop properties as far as possible Hence the controller reduction procedures advocated in this module reflect the plant in some way This leads to the frequency weighted reduction schemes of wtbalance and fracred as described in Chapter 4 Frequency Weighted Error Reduction Plant reduction logically should also seek to preserve closed loop properties and thus should involve the controller With the controller unknown however this is impossible Nevertheless it can be argued on the basis of the high loop gain property within the closed loop bandwidth that is typical of many systems that O National Instruments Corporation 1 15 Xmath Model Reduction Module Chapter 1 Introduction 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 fully Xmath Model Reduction Module 1 16 ni com Additive Error Reduction This chapter describes additive error reduction including discussions of truncation of reduction by and perturbation of balanced
32. after approximating the signal will no longer be white noise so that argument is questionable Simple appeal to duality motivates using no frequency dependent weighting for H s These are two of the options offered by fracred Two more fracred options depend on examining stability robustness the options are duals of one another From the stability point of view the set up of Figure 4 3 is identical to that of Figure 4 4 with P P 1 Xmath Model Reduction Module 4 8 ni com Chapter 4 Frequency Weighted Error Reduction te y leer BK K P s 1 La gt C A 7 Cs E Ps gt Figure 4 4 Redrawn Individual Signal Paths as Vector Paths It is possible to verify that PG P Gia OB I C sI A K C K and accordingly the output weight 7 PG P W can be used in an error measure W G G It turns out that the calculations for frequency weighted balanced truncation of G and subsequent construction of C s are exceptionally easy using this weight The second fracred option is the dual of this The error measure is 4 H V where y f Ke st A BK B C sI A BK B It is possible to argue heuristically the relevance of these error measures from a second point of view It turns out that Vi Npr 10 07 D N V D W W r National Instruments Corporation 4 9 Xmath Model Reduction M
33. ages arising out of or related to this document or the information contained in it EXCEPT AS SPECIFIED HEREIN NATIONAL INSTRUMENTS MAKES NO WARRANTIES EXPRESS OR IMPLIED AND SPECIFICALLY DISCLAIMS ANY WARRANTY OF MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE CUSTOMER S RIGHT TO RECOVER DAMAGES CAUSED BY FAULT OR NEGLIGENCE ON THE PART OF NATIONAL INSTRUMENTS SHALL BE LIMITED TO THE AMOUNT THERETOFORE PAID BY THE CUSTOMER NATIONAL INSTRUMENTS WILL NOT BE LIABLE FOR DAMAGES RESULTING FROM LOSS OF DATA PROFITS USE OF PRODUCTS OR INCIDENTAL OR CONSEQUENTIAL DAMAGES EVEN IF ADVISED OF THE POSSIBILITY THEREOF This limitation of the liability of National Instruments will apply regardless of the form of action whether in contract or tort including negligence Any action against National Instruments must be brought within one year after the cause of action accrues National Instruments shall not be liable for any delay in performance due to causes beyond its reasonable control The warranty provided herein does not cover damages defects malfunctions or service failures caused by owner s failure to follow the National Instruments installation operation or maintenance instructions owner s modification of the product owner s abuse misuse or negligent acts and power failure or surges fire flood accident actions of third parties or other events outside reasonable control Copyright Under the copyright laws this publication may not
34. al reduction functions for example balmoore or redschur require stability of the transfer function matrix G s being reduced If G s is unstable stable should be used to generate G s and G s reduction of G s should be performed and then G s added to the outcome using the operator to yield the desired reduction of G s hankelsv HSV Wc Wo hankelsv Sys noplot The hankelsv function computes the Hankel Singular Values of a stable system continuous or discrete and displays them in a bar plot National Instruments Corporation 5 1 Xmath Model Reduction Module Chapter 5 Utilities Related Functions stable The gramian matrices are defined by solving the equations in continuous time AW W A BB W A A W CC and in discrete time W AW A BB W A W A C C The computations are effected with 1yapunov and stability is checked which is time consuming The Hankel singular values are the square roots of the eigenvalues of the product lyapunov dlyapunov Algorithm Xmath Model Reduction Module SysS SysU stable Sys tol The stable function decomposes Sys into its stable SysS and unstable SysU parts such that Sys SysS SysU Continuous systems have unstable poles if real parts gt tol Discrete systems have unstable poles if magnitudes gt 1 tol The direct term D matrix is included in Syss e If Sys has poles clustered
35. ansfer function matrix has the controllability and observability grammian property that P Q 2 for some diagonal 2 Then the realization is termed internally balanced Notice that the diagonal entries of 2 are square roots of the eigenvalues of PQ that is they are the Hankel singular values Often the entries of X are assumed ordered with O gt 6j41 As noted in the discussion of grammians systems with small eigenvalues of P are hard to control and those with small eigenvalues of O are hard to observe Now a state transformation T amp J will cause P Q to be replaced by 02P 020 implying that ease of control can be obtained at the expense of difficulty of observation and conversely Balanced realizations are those when ease of control has been balanced against ease of observation Given an arbitrary realization there are a number of ways of finding a state variable coordinate transformation bringing it to balanced form A good survey of the available algorithms for balancing is in LHPW87 One of these is implemented in the Xmath function balance The one implemented in balmoore _ as part of this module is more sophisticated but more time consuming It proceeds as follows 1 Singular value decompositions of P and Q are defined Because P and Q are symmetric this is equivalent to diagonalizing P and Q by orthogonal transformations P U S U Q US U 2 The matrix 1 2 H si 1 2 UnSuV yH is co
36. aphy AnJ AnL89 AnM89 BoD87 Enn84 Enn84a GCP88 Glo84 Glo86 GrA86 GrA89 BDO Anderson and B James Algorithm for multiplicative approximation of a stable linear system in preparation BDO Anderson and Y Liu Controller reduction Concepts and approaches IEEE Transactions on Automatic Control Vol 34 1989 pp 802 812 BDO Anderson and J B Moore Optimal Control Linear Quadratic Methods Prentice Hall Inc Englewood Cliffs NJ 1989 S P Boyd and J Doyle Comparison of peak and RMS gains for discrete time systems System Control Letters Vol 9 No 1 pp 1 6 1987 D F Enns Model reduction with balanced realizations an error bound and a frequency weighted generalization Proceedings for the 23rd IEEE Conference on Decision and Control Las Vegas November 1984 pp 127 132 D F Enns Model reduction for control systems design PhD Thesis Dept of Aeronautics and Astronautics Stanford University CA USA 1984 K Glover R F Curtain and J R Partington Realisation and approximation of linear infinite dimensional systems with error bounds SIAM J Controls and Optimization Vol 26 1988 pp 863 898 K Glover All optimal Hankel norm approximations of linear multivariable systems and their L error bounds Int J Controls Vol 39 1984 pp 1115 1193 K Glover Multiplicative approximation of linear multivariable syst
37. ates the realization to form an approximation balance Computes an internally balanced realization of a system e truncate This function truncates a system It allows examination of a sequence of different reduced order models formed from the one balanced realization e redschur These functions in theory function almost the same as the two features of balmoore That is they produce a state variable realization of a reduced order model such that the transfer function matrix of the model could have resulted by truncating a balanced realization of the original full order transfer function matrix However the initially given realization of the original transfer function matrix is never actually balanced which can be a numerically hazardous step Moreover the state variable realization of the reduced Xmath Model Reduction Module 2 4 ni com Chapter 2 Additive Error Reduction 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 continuous time a truncation of a balanced realization is again balanced This is not the case for discrete time but otherwise it looks the same Full Order Realization balmoore balmoore bal dsch with both steps with first step manes See truncate Balanced Real
38. ative feedback the second output of G to the input that is set n C s d e Nothing has been said as to how d should be chosen and the end result of the reduction C s depends on d Nor has the reduction procedure been specified When C s has been designed to combine a state estimator with a stabilizing feedback law it turns out that there is a natural choice for d s As for the reduction procedure one possibility is to use a weight based on the spectrum of the input signals to G and in case C s has been determined by an LQG optimal design this spectrum turns out to be white that is independent of frequency so that no weight apart perhaps from scaling is needed A second possibility is to use a weight based on a stability robustness measure These points are now discussed in more detail To understand the construction of a natural fractional representation for C s suppose that P s C sI A B and let Kp Kg be matrices such that A BKp and A Kc are stable The controller A x Ax Bu K Cx y U KRX generates an estimate K px of the feedback control K x The controller can be represented as a series compensator A x Ax BK x K Cx Kgy u K pX with compensator input y and output u Allowing for connection with negative feedback the compensator transfer function matrix is C s Kp sI A BKp Kgc Kg Xmath Model Reduction Module 4 6 ni com Chapter 4 Fre
39. atrix D to G s and the final step of the algorithm is National Instruments Corporation 2 19 Xmath Model Reduction Module Chapter 2 Additive Error Reduction to choose the D matrix of G s by splitting D between G s and G s This is done by using a separate function ophiter Suppose G s is the unstable output of stable and let K s G s By applying the multipass Hankel reduction algorithm described further below K s is reduced to the constant Kg the approximation which satisfies K s Kollo lt 01 K 0 06 n K E Op 16 0 G that is if it is larger than G9 Koa lt Y 940 k n 1 then one chooses G G Ko G Gu Ko This ensures satisfaction of the error bound for G G given previously because G 6G 6 G G G Ko l G G Gul K Kol 6 G 6 4 1 6 0 G Multipass Algorithm Xmath Model Reduction Module We now explain the multipass algorithm For simplicity in first explaining the idea suppose that the Hankel singular values at every stage or pass are distinct 1 Find a stable order ns 1 approximation G _ s of G s with IGU Gps Jol 6 5 G This can be achieved by the algorithm already given and there is no unstable part of the approximation 2 20 ni com Chapter 2 Additive Error Reduction 2 Finda stable order ns 2 approximation G _ 5 of G _ 5 with Gas 17 ag Gus JO On
40. be reproduced or transmitted in any form electronic or mechanical including photocopying recording storing in an information retrieval system or translating in whole or in part without the prior written consent of National Instruments Corporation Trademarks MATRIXx National Instruments NI ni ccom and Xmath are trademarks of National Instruments Corporation Product and company names mentioned herein are trademarks or trade names of their respective companies Patents For patents covering National Instruments products refer to the appropriate location Help Patents in your software the patents txt file on your CD or ni com patents WARNING REGARDING USE OF NATIONAL INSTRUMENTS PRODUCTS 1 NATIONAL INSTRUMENTS PRODUCTS ARE NOT DESIGNED WITH COMPONENTS AND TESTING FOR A LEVEL OF RELIABILITY SUITABLE FOR USE IN OR IN CONNECTION WITH SURGICAL IMPLANTS OR AS CRITICAL COMPONENTS IN ANY LIFE SUPPORT SYSTEMS WHOSE FAILURE TO PERFORM CAN REASONABLY BE EXPECTED TO CAUSE SIGNIFICANT INJURY TO A HUMAN 2 IN ANY APPLICATION INCLUDING THE ABOVE RELIABILITY OF OPERATION OF THE SOFTWARE PRODUCTS CAN BE IMPAIRED BY ADVERSE FACTORS INCLUDING BUT NOT LIMITED TO FLUCTUATIONS IN ELECTRICAL POWER SUPPLY COMPUTER HARDWARE MALFUNCTIONS COMPUTER OPERATING SYSTEM SOFTWARE FITNESS FITNESS OF COMPILERS AND DEVELOPMENT SOFTWARE USED TO DEVELOP AN APPLICATION INSTALLATION ERRORS SOFTWARE AND HARDWARE COMPATIBILITY PROBLEMS MALFU
41. ce variable G q is used when both continuous and discrete systems are allowed e H s is used to denote the frequency response over some range of frequencies of a system where s is the Laplace variable H q is used to indicate that the system can be continuous or discrete e A single apostrophe following a matrix variable for example x denotes the transpose of that variable An asterisk following a matrix variable for example A indicates the complex conjugate or Hermitian transpose of that variable Conventions This publication makes use of the following types of conventions font format symbol mouse and note These conventions are detailed in Chapter 2 MATRIXx Publications Online Help and Customer Support of the MATRIXx Getting Started Guide Xmath Model Reduction Module 1 2 ni com Chapter 1 Introduction 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 particularly useful for topics covered in this manual e MATRIXx Getting Started Guide e Xmath User Guide e Control Design Module e Interactive Control Design Module e Interactive System Identification Module Part 1 e Interactive System Identification Module Part 2 e Model Reduction Module e Optimization Module e Robust Control Module e Xu Module MATRIXx Help Model Reduction Module fu
42. ch by degree one Then Cies SG EE G GG GG GGG G63 Gs G3 Also leal 6 G 6 1 lt l v Similarly G Gi lt 1 v 1 67 Gs 1 v gt Then lc Ga Vas 1 Vas Yns 1 el t Vas 1 Y ns 2 1 EV C1 Vas V gee 1 The error bound Equation 3 3 is only exact when there is a single reduction step Normally this algorithm has a lower error bound than bst in particular if the v are all distinct and v 1 the error bounds are approximately nsr ns ns y Vi for mulhank 2 gt A Vi for bst i nsr 1 i nsr 1 3 20 ni com Chapter 3 Multiplicative Error Reduction For mulhank _ this translates for a scalar system into 869 Y vdB lt 20108 19 Grsr GI i nsrt 1 lt 8 69 Y v dB i nsr l and ns phase error lt y v radians i nsr 1 The bounds are double for bst The error as a function of frequency is always zero at for bst or at 0 if a transformation s gt s is used whereas no such particular property of the error holds for mulhank Imaginary Axis Zeros Including Zeros at co When G j is singular or zero on the j axis or at co reduction can be handled in the same manner as explained for bst The key is to use a bilinear transformation Saf87 Consider the bilinear map defined by _ z a bz 1 sta bs 1 where 0 lt a lt b and mapping G s into G s through awed AN Ge National Instrume
43. curring at DC The bound also is exact in the special case of a single input single output transfer function which has poles and zero alternating along the negative real axis It is far from exact when the poles and zeros approximately alternate along the imaginary axis with the poles stable National Instruments Corporation 2 9 Xmath Model Reduction Module Chapter 2 Additive Error Reduction Xmath Model Reduction Module The actual approximation error for discrete systems also depends on frequency and can be large at 0 The error bound is almost never tight that is the actual error magnitude as a function of almost never attains the error bound so that the bound can only be a guide to the selection of the reduced system dimension In principle the error bound formula for both continuous and discrete systems can be improved that is made tighter or less likely to overestimate the actual maximum error magnitude when singular values occur with multiplicity greater than one However because of errors arising in calculation it is safer to proceed conservatively that is work with the error bound above when using the error bound to select nsr and examine the actual error achieved If this is smaller than required a smaller dimension for the reduced order system can be selected mreduce provides an alternative reduction procedure for a balanced realization which achieves the same error bound but which has zero error at
44. d order model of a plant The procedure requires G again to be nonsingular at and to have no j axis poles It is as follows 1 Form H G If Gis described by state variable matrices A B C D then H is described by A BD 1C BD D 1C D 7 H is square stable and of full rank on the ja axis 2 Form H of the desired order to minimize approximately H H H oo 3 Set G H Observe that H H H I H H I GG G G G The reduced order G will have the same poles in Re s gt 0 as G and be minimum phase Imaginary Axis Zeros Including Zeros at oo We shall now explain how to handle the reduction of G s which has a rank drop at s or on the j axis The key is to use a bilinear transformation Saf87 Consider the bilinear map defined by Z gt bz 1 s a bs 1 where 0 lt a lt b and mapping G s into G s through CDC ay co a Xmath Model Reduction Module 3 10 ni com Chapter 3 Multiplicative Error Reduction The values of G s as shown in Figure 3 2 along the j axis are the same as the values of G s around a circle with diameter defined by a j0 b7 jO on the positive real axis CS Kr gt 1 G s G s values values Figure 3 2 Bilinear Mapping from G s to Gs Case 1 Also the values of G s as shown in Figure 3 3 along the jw axis are the same as the values of G s around a circle with diame
45. e X AXA BB X A NX A CC x diag o 65 03 0 with 0 lt 0 gt 0 with theo the Hankel Singular Values of Sys In the second part of balmoore a truncation 2 8 ni com Chapter 2 Additive Error Reduction of the balanced system occurs assuming nsr is less than the number of states Thus if the state space representation of the balanced system is Ay Aiz A A An B c jc c with A possessing dimension nsr X nsr B possessing nsr rows and C possessing nsr columns the reduced order system SysR is continuous discrete x A x 8Bj u xi k 1 A x k 8Bj u k y Cx Du y k Cixi k Du k The following error formula is relevant continuous Cgor 4 1 C1001 47 B Df S2 On ope 1 Onsr 2 Ons nsr discrete licee r 4y B D C I 4 B Dl pa Lo a 1 Onsr 2 Ons It is this error bound which is the basis of the determination of the order of the reduced system when the keyword bound is specified If the error bound sought is smaller than 20 then no reduction is possible which is consistent with the error bound If it is larger than 21rZ then the constant transfer function matrix D achieves the bound For continuous systems the actual approximation error depends on frequency but is always zero at oe In practice it is often greatest at 0 if the reduction of state dimension is 1 the error bound is exact with the maximum error oc
46. e first impulse response coefficient D in the discrete time case while mreduce ensures matching of DC gains for Sys and SysR in both the continuous time and discrete time case For a additional information about the truncate function refer to the Xmath Help balance balmoore redschur mreduce National Instruments Corporation 2 11 Xmath Model Reduction Module Chapter 2 Additive Error Reduction 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 Algorithm The objective of redschur is the same as that of balmoore when the latter is being used to reduce a system this means that if the same Sys and nsr are used for both algorithms the reduced order system should have the same transfer function matrix However in contrast to balmoore redschur do not initially transform Sys to an internally balanced realization and then truncate it nor is SysR in balanced form The fact that there is no balancing offers numerical advantages especially if Sys is nearly nonminimal Sys should be stable and this is checked by the algorithm In contrast to balmoore minimality of Sys that is controllability and observability is not required If the Hankel singular values of Sys are ordered as 0 2 2 2 Ops 20 then those o
47. eed if O O 4 gt SO it is highly advisable to avoid this situation For additional information about the balmoore function refer to the Xmath Help 2 10 ni com Related Functions truncate Chapter 2 Additive Error Reduction balance truncate redschur mreduce Related Functions SysR truncate Sys nsr VD VA The truncate function reduces a system Sys by retaining the first nsr States and throwing away the rest to form a system SysR If for Sys one has A Ap B B Az Az B A C C ca the reduced order system in both continuous time and discrete time cases is defined by A B C and D If Sys is balanced then SysR is an approximation of Sys achieving a certain error bound truncate may well be used after an initial application of balmoore to further reduce a system should a larger approximation error be tolerable Alternatively it may be used after an initial application of balance or redschur If Sys was calculated from redschur and VA VD were posed as arguments then SysR is calculated as in redschur refer to the redschur section truncate should be contrasted with mreduce which achieves a reduction through a singular perturbation calculation If Sys is balanced the same error bound formulas apply though not necessarily the same errors truncate always ensures exact matching at s in the continuous time case or exacting matching of th
48. een the full order and reduced order controller in the vicinity of 0 1 radians per second The step response has overshoot of 50 as opposed to 40 and the ripple persists for longer We use the compare function refer to the compare section of Chapter 5 Utilities to reproduce Figures 6 4 and 6 5 Calculate the full order closed loop system then the closed loop system with the reduced order compensator syscl feedback sysol sysolr sys syscr sysclr feedback sysolr 6 6 ni com Chapter 6 Tutorial Generate Figure 6 4 compare syscl sysclr w radians type 5 f4 plot keep legend original reduced Singular Value Magnitude dB original 50 fof HE a reduced 50 be 00 bs 150 b oor oo o 1 10 100 Frequency rad sec Figure 6 4 Closed Loop Gain with redschur National Instruments Corporation 6 7 Xmath Model Reduction Module Chapter 6 Tutorial Generate Figure 6 5 tvec 0 140 99 140 compare syscl sysclr tvec type 7 5 plot keep legend original reduced Step Response original reduced E 0 20 40 60 80 100 Time sec Figure 6 5 Step Response with redschur Xmath Model Reduction Module 6 8 ni com Chapter 6 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 svalsro
49. ems with Leo error bounds Proceedings for American Controls Conference Seattle 1986 pp 1705 1709 M Green and BDO Anderson The approximation of power spectra by phase matching Proceedings for 25th CDC 1986 pp 1085 1090 M Green and BDO Anderson Model reduction by phase matching Mathematics of Control Signals and Systems Vol 2 1989 pp 221 263 O National Instruments Corporation A 1 Xmath Model Reduction Module Appendix A GrA90 Gre88 Gre88a HiP90 LAL90 Lau80 LHPW87 LiA86 LiA89 LiA90 Moo81 NJB84 PeS82 Bibliography M Green and BDO Anderson Generalized balanced stochastic truncation Proceedings for 29th CDC 1990 M Green Balanced stochastic realization Linear Algebra and Applications Vol 98 1988 pp 211 247 M Green A relative error bound for balanced stochastic truncation JEEE Transactions on Automatic Control Vol 33 1988 pp 961 965 D Hinrichsen and A J Pritchard An improved error estimate for reduced order models of discrete time systems IEEE Transactions on Automatic Control Vol 35 1990 pp 317 320 Y Liu BDO Anderson and U L Ly Coprime factorization controller reduction with Bezout identity induced frequency weighting Automatica Vol 26 No 2 1990 pp 233 249 A J Laub On computing balancing transformations Proceedings on Joint American Controls Confere
50. er b j0 a j0 for almost all choices of a b If a and b are chosen small enough G s will have all its poles inside this circle and no zero or rank reduction on it while G s then will have all poles in Re s lt 0 and no zero or rank reduction on the jo axis including s co The steps of the algorithm when G s has a zero on the j axis or at s are as follows 1 Forsmalla b withO lt a lt b form G s Gl gtildesys s a Ree as shown for 2 Reduce G s to G s this being possible because G s is stable and has full rank on s jo including os sta 3 FormG s a G G will be overbounded by the error oo and G will contain the same zeros in Re s gt 0 as G as shown for gsys 1 The error la a G 6 Tf there is no zero or rank reduction of G s at the origin one can take a 0 and b bandwidth over which a good approximation of G s is needed and at the very least b sufficiently large that the poles of G s lie in the circle of diameter b j0 a j0 If there is a zero or rank reduction at the origin one can replace a 0 by a b It is possible to take b too small or if there is a zero at the origin to take a too small The user will be presented with an error message that there is a j axis zero and or that the Riccati equation solution may be in error The basic explanation is that as b 0 and thus a gt 0 the zeros of G s a
51. er function which is nonminimum phase and has a double pole at the origin is as follows 2 Be a 2 s2 0 5 Oo SC 0 5 0 sta 2 2 a 1 Wo Gs a ot a Ia 4s s2L 0 5 05 s 260s 03 s 2L 045 0 2 2 f 2 gt 03 0 with E 70 02 0m 1 j 0 4 0m 5 65 6 70 02 0 0 765 E _ 0 02 1 41 E 70 02 1 85 a 4 84 O National Instruments Corporation 6 1 Xmath Model Reduction Module Chapter 6 Tutorial A minimal realization in modal coordinates is C sJ AFB where A dia 0 0 LE 0 015 ne En 1 410 0 04 1 85 0 765 0 015 1 410 0 028 1 85 0 04 0 026 0 251 0 033 0 886 4 017 0 145 3 604 0 280 0 996 0 105 0 261 0 009 0 001 0 043 0 002 0 026 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 uncertainty More specifically the loop gain has to lie outside the shaded region shown in Figure 6 1 En 2 E lt 4 40 dB decade O a S 03 Frequency rad sec gt 0 07 40 dB decade gt Xmath Model Reduction Module Figure 6 1 Loop Gain Constraints 6 2 ni com Chapter 6 Tutorial With a state weighting matrix Q le 3 diag 2 2 80 80 8 8 3 31 R 1 and unity control weighting a state feedback control gain is determined through a linear quadratic performance index minimization as Kr ev
52. es ReA A and IA A I for a square matrix A denote an arbitrary real part and an arbitrary magnitude of an eigenvalue of A IXGo for a transfer function X denotes sup o lt WM lt oo Ana GOXO An all pass transfer function W s is one where X j 1 for all to each pole there corresponds a zero which is the reflection through the ja axis of the pole and there are no j axis poles An all pass transfer function matrix W s is a square matrix where W jo W jo 1 P gt Oand P 2 0 for a symmetric or hermitian matrix denote positive and nonnegative definiteness P gt P and P P for symmetric or hermitian P and P denote P P is positive definite and nonnegative definite A superscripted number sign for a square matrix A denotes the Moore Penrose pseudo inverse of A 1 6 ni com Chapter 1 Introduction An inequality or bound is tight if it can be met in practice for example l logx x lt 0 is tight because the inequality becomes an equality for x 1 Again if F jw denotes the Fourier transform of some f t e L the Heisenberg inequality states Jlcoad 2 2 1 2 2 2 73 541 f ho de fo IF jo do and the bound is tight since it is attained for f t exp k Commonly Used Concepts This section outlines some frequently used standard concepts Controllability and Observability Grammians Suppose that G s D C sI A B is a transfer function matrix with Re
53. esired system reduction sylvester schur redschur balmoore respdiff Xmath Model Reduction Module compare Sys SysRed FTvec Fmin Fmax npts radians type The compare function provides a number of different graphical tests which can be used to compare two state space system implementations compare can be used as a tool for evaluating a reduced order system by comparing it with the original full order system from which it was obtained However it can be used for more general comparisons as well such as examining the results of different discretization or identification techniques 5 4 ni com 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 example is stored in the file mr_disc xmd in the Xmath demos directory To follow the example start Xmath and then select File Load from the Xmath Commands menu or enter the load command with the file specification appropriate to your operating system from the Xmath Commands area For example load XMATH demos mr_disc Plant and Full Order Controller The plant in question comprises four spinning disks connected by a flexible shaft A motor applies torque to the third disk and the output variable of interest is the angular displacement of the first disk The plant transf
54. f SysR in the continuous time case are O 20 2 26 gt 0 A restriction of the algorithm is that 6 gt O sr 1 18 required for both continuous time and discrete time cases Under this restriction SysR is guaranteed to be stable and minimal The algorithms depend on the same algorithm apart from the calculation of the controllability and observability grammians W and W of the original system These are obtained as follows continuous W A AW BB WA A W CC C discrete W AW_A BB W AW A The maximum order permitted is the number of nonzero eigenvalues of W W that are larger than e Xmath Model Reduction Module 2 12 ni com Chapter 2 Additive Error Reduction Next Schur decompositions of W W are formed with the eigenvalues of W W in ascending and descending order These eigenvalues are the square of the Hankel singular values of Sys and if Sys is nonminimal some can be zero S V a W W Va asc Vn W W Vp Dies The matrices Va Vp are orthogonal and S Saes are upper triangular Next submatrices are obtained as follows 0 List V big E vd bie al 0 and then a singular value decomposition is found aid 4 Unie opis Vetig V lbig V big From these quantities the transformation matrices used for calculating SysR are defined 1 2 Sibig VibigUebigSebig 1 2 S big V big ebighebig and the reduced order system is Ap SmigAS big By SipigB Ap CS rig D
55. f the Weight V ja Xmath Model Reduction Module 6 16 ni com Chapter 6 Tutorial Generate Figure 6 13 syscr sysclr hsv wtbalance sys sysc input spec 2 sysv svalsrol svplot sys syscr w radians plot svalsol keep 13 plot wc constr keep grid legend reduced original constrained title Open Loop Gain with wtbal input spec Open Loop Gain with wtbal input spec 100 50 ea a a E 0 T D gt E 50 0 gt 3 100 5 150 2 Pon 0 01 0 1 1 10 100 Frequency rad sec Figure 6 13 Open Loop Gain from wtbalance with input spec National Instruments Corporation 6 17 Xmath Model Reduction Module Chapter 6 Tutorial Generate Figure 6 14 syscl feedback sysol sysolr sys syscr sysclr feedback sysolr compare syscl sysclr w radians type 5 f14 plot keep legend original reduced Singular Value Magnitude dB original 50 T T H reduced 50 bss 100 H 150 fp eon aon on 1o oo Frequency rad sec Figure 6 14 System Singular Values of wtbalance with input spec Xmath Model Reduction Module 6 18 ni com Chapter 6 Generate Figure 6 15 tvec 0 140 99 140 compare syscl sysclr tvec type 7 f15 plot keep legend original reduced Tutorial Step Response original reduced 0 20 40 60 80 100 Time sec Figure 6 15 Step Respo
56. fferent scalar inputs it is advisable to introduce at some stage a weight e 2 into the reduction process After preliminary checks the algorithm steps are 1 Form the observability and weighted through Q controllability grammians of E s in Equation 4 7 by P A BK A BK P Kg0 Kg 4 8 O A BKy A BKp O KpKp C C 4 9 2 Compute the square roots of the eigenvalues of PQ Hankel singular values of the fractional representation of Equation 4 5 The maximum order permitted is the number of nonzero eigenvalues of PQ that are larger than e 3 Introduce the order of the reduced order controller possibly by displaying the Hankel singular values HSVs to the user Broadly speaking one can throw away small HSVs but not large ones 4 Using redschur type calculations find a state variable description of E s This means that E s is the transfer function matrix of a truncation of a balanced realization of E s but the redschur type calculations avoid the possibly numerically difficult step of balancing the initially known realization of E s Suppose that A Sipigl A BKR S pigs Kg SibigKk 5 Define the reduced order controller C s by Acr SipigA BKp KeO S pi 4 10 so that aj Cs Cop sl Acr Ber 4 18 ni com Chapter 4 Frequency Weighted Error Reduction 6 Check the stability of the closed loop system with C s When the type left perf is specified one works with
57. full order state space system wtbalance A state space system must be linear and continuous Interconnection of controller and plant must be stable and or weight must be stable Documentation of the individual functions sometimes indicates how the restrictions can be circumvented There are a number of model reduction methods not covered here These include e Pad Approximation e Methods based on interpolating or matching at discrete frequencies National Instruments Corporation 1 5 Xmath Model Reduction Module Chapter 1 Introduction Nomenclature Xmath Model Reduction Module L approximation in which the L norm of impulse response error or by Parseval s theorem the L norm of the transfer function error along the imaginary axis serves as the error measure Markov parameter or impulse response matching moment matching covariance matching and combinations of these for example q COVER approximation Controller reduction using canonical interactions balanced Riccati equations and certain balanced controller reduction algorithms This manual uses standard nomenclature The user should be familiar with the following sup denotes supremum the least upper bound The acute accent denotes matrix transposition A superscripted asterisk denotes matrix transposition and complex conjugation Amax A for a square matrix A denotes the maximum eigenvalue presuming there are no complex eigenvalu
58. gular Values Multivariable Control pp 197 207 S G Tzafestas ed D Reidel Publishing Company 1984 Xmath Model Reduction Module A 4 ni com Appendix A Bibliography SLH81 M G Safonov A J Laub and G L Hartmann Feedback Properties of Multivariable Systems The Role and Use of the Return Difference Matrix IEEE Transactions on Automatic Control Vol AC 26 February 1981 SA88 G Stein and M Athans The LQG LTR Procedure for Multivariable Control Design IEEE Transactions on Automatic Control Vol AC 32 No 2 February 1987 pp 105 114 Za81 G Zames Feedback and optimal sensitivity model reference transformations multiplicative semi norms and approximate inverses IEEE Transactions on Automatic Control Vol AC 26 pp 301 320 1981 KS72 H Kwakernaak and R Sivan Linear Optimal Control Systems Wiley 1972 O National Instruments Corporation A 5 Xmath Model Reduction Module Technical Support and Professional Services Visit the following sections of the National Instruments Web site at ni com for technical support and professional services Support Online technical support resources at ni com support include the following Self Help Resources For immediate answers and solutions visit the award winning National Instruments Web site for software drivers and updates a searchable KnowledgeBase product manuals step by step troubleshooting wizards thousands of example
59. gular values 1 8 3 9 5 1 hankelsv 1 5 5 1 algorithm multipass 2 20 help technical support B 1 instrument drivers NI resources B 1 internal balancing 1 10 K KnowledgeBase B 1 L Imax A 1 6 lyapunov 1 8 MATRIXx Help 1 3 minimality requirements 1 5 model reduction schur 2 5 Moore Penrose pseudo inverse 1 6 mreduce 1 5 1 13 2 10 mulhank 1 5 1 14 3 14 multiplicative error 1 1 3 1 Xmath Model Reduction Module National Instruments support and services B 1 nomenclature 1 2 nomenclature for MRM 1 6 numerical conditioning 2 8 0 ophank 1 5 2 14 discrete time systems 2 21 error formulas 2 22 impulse response error 2 22 multipass 2 20 onepass 2 18 unstable system approximation 2 23 P Pad approximation 1 5 perturbation of balanced realization 2 5 singular 1 11 2 6 phase function 3 5 3 15 phase matrix 3 6 3 16 pole zero pairs in reduction 2 4 programming examples NI resources B 1 R redschur 1 5 2 4 2 12 reduced order system 2 3 Reli A 1 6 S singular perturbation 1 11 skipChks 6 3 software NI resources B 1 spectral factorization 1 13 stability requirements 1 5 ni com Index stable 1 5 5 2 U sup 1 6 support technical B 1 unstable zeros 2 3 W T Web resources B 1 technical support B 1 wtbalance 1 5 4 10 tight equality bounds 1 7 training and certification NI resources B 1 transfer function allpass 1 6 t
60. h Z PCT PLC C 111 PCT match spec PCT PIC CAU PCI V input spec C C Y These error measures have certain interpretations as shown in Table 4 2 In case C s is not a compensator in a closed loop and the error measure VODCA C Go is of interest you can work with type input spec and C V in lieu of C and V There is no restriction on the stability of C s or indeed of P s in the algorithm though if C s is a controller the closed loop must be stabilizing Also V s must be stable Hence all weights on the left or right of Co C 0 in the error measures will be stable The algorithm however treats unstable C s in a special way by reducing only the stable part of C s under additive decomposition and copying the unstable part into C s National Instruments Corporation 4 11 Xmath Model Reduction Module Chapter 4 Frequency Weighted Error Reduction This rather crude approach to the handling of the unstable part of a controller is avoided in fracred which provides an alternative to wtbalance for controller reduction at least for an important family of controllers Table 4 2 Error Measure Interpretation for wtbalance Type Error Measure Interpretations input stab A stability robustness argument based on breaking the loop at the controller output indicates that if C is stabilizing for P and the error measure is less than 1 then Cr is stabilizing for P
61. haviors The following explanation deals first with the keyword onepass Suppose that 01 69 O are the Hankel Singular values of S which has transfer function matrix G s Suppose that the singular values are ordered so that 0 0 0 gt On 410 On qq On gt On 41 gt On oS On FIS 6 On5 20 Thus there are n equal values followed by n n equal values followed by n m equal values and so forth The order nsr of G s cannot be arbitrary when there are equal Hankel singular values In fact the orders shown in Table 2 1 for the strictly stable G all poles in Re s lt 0 and strictly unstable G all poles Re s gt 0 are possible and there are no other possibilities National Instruments Corporation 2 15 Xmath Model Reduction Module Chapter 2 Xmath Model Reduction Module Additive Error Reduction Table 2 1 Orders of G Number of Number of Order of Order of Eliminated States Eliminated States G nsr G nsu Retaining G Discarding G 0 ns ny n ns n ns m m n ns ny Ny ns 13 a n ns m y y y y Nm 1 0 NS Ny_ NS My 1 By abuse of notation when we say that G is reduced to a certain order this corresponds to the order of G s alone the unstable part of G s of the approximation is most frequently thrown away The number of eliminated states retaining G refers to of states in G
62. he desired degree nsr is obtained For example in the second iteration G s is given by Gs G s Pao FO 3 4 Consequences of Step 5 and Justification of Step 6 A number of properties are true G s is of order ns r with lee Ole v 3 5 Xmath Model Reduction Module 3 18 ni com Chapter 3 Multiplicative Error Reduction W s and Gs stand in the same relation as W s and G s that is W s W s G s G s With P4 AP BpB p there holds By PCs BaD or BpD V C P DCp B yU 2 y Br v TD With Q4p 4p0 C pCp there holds 1 A Cy Dg C B yO or Ud v T U v T Cr DU va DI DCF B yU 2 BrD V C O Dy DS E i F is the stable strictly proper part of W s G s e The Hankel singular values of F p and F are the first as r Hankel singular values of F x Ipa 7 P UQV Y QU 2 O V PU E X U PV Gs has the same zeros in Re s gt 0 as G s These properties mean that one is immediately positioned to repeat the reduction procedure on Gs with almost all needed quantities being on hand National Instruments Corporation 3 19 Xmath Model Reduction Module Chapter 3 Multiplicative Error Reduction Error Bounds Xmath Model Reduction Module The error bound formula Equation 3 3 is a simple consequence of iterating Equation 3 5 To illustrate suppose there are three reductions G gt G gt G gt G3 ea
63. he functions in question are balmoore balance refer to the Xmath Help truncate redschur One only can speak of internally balanced realizations for systems which are stable if the aim is to reduce a transfer function matrix G s which contains unstable poles one must additively decompose it into a stable part and unstable part reduce the stable part and then add the unstable part back in The function stable described in Chapter 5 Utilities can be used to decompose G s Thus G s G s G s G s stable G s unstable Gas found by algorithm reduction of G s G s G s G s reduction of G s Xmath Model Reduction Module 2 2 ni com Chapter 2 Additive Error Reduction 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 poto j 2 2 0 E with the diagonal entries of in decreasing order that is 0 2 62 2 Then the key result is IGU G jo _ lt 2trE with G G the transfer function matrices of Equation 2 1 and Equation 2 2 respectively This formula shows that small singular values can without great cost be thrown away It also is valid in discrete time and can be improved upon if there are repeated Hankel singular values Provided that the smallest diagonal entry of 2 strictly exceeds the largest diagonal entry of 2 the red
64. he phase of G somehow ensures a good approximation albeit in a multiplicative sense of G itself Algorithm with the Keywords right and left The following description of the algorithm with the keyword right is based on ideas of GrA86 developed in SaC88 The procedure is almost the same when left is specified except the transpose of G s is used the algorithm finds an approximation in the same manner as for right but transposes the approximation to yield the desired G s 1 The algorithm checks e That the system is state space continuous and stable e That a correct option has been specified if the plant is nonsquare e That Dis nonsingular if the plant is nonsquare DD must be nonsingular National Instruments Corporation 3 5 Xmath Model Reduction Module Chapter 3 Multiplicative Error Reduction Xmath Model Reduction Module With G s D C sI Ay UB and stable with DD nonsingular and G j G jo nonsingular for all part of a state variable realization of a minimum phase stable W s is determined such that WY s W s G s G s with W s Dy Cy sl A By The state variable matrices in W s are obtained as follows The controllability grammian P associated with G s is first found from AP PA BB 0 then Aw A By PC BD When G s is square the algorithm checks to see if there is a zero or singularity of G s close to the j axis the zeros are given by the eigenvalues of A BD
65. hen a choice such as 7 5 Q or 10 Q needs to be made Reduction then proceeds as follows 1 Form G s 2 Reduce G s through bst 3 Form G s GAs s with gsys subsys gtildesys gtildesys makep eps 1 makep 1 0 Notice that the number of zeros of G s in the circle of diameter 0 j0 s sets a lower bound on the degree of G s for such zeros become right half plane zeros of G s and must be preserved by bst Obviously zeros at s are never in this circle so a procedure for reducing G s 1 d s is available National Instruments Corporation 3 13 Xmath Model Reduction Module Chapter 3 Multiplicative Error Reduction Related Functions mulhank There is one potential source of failure of the algorithm Because G s is stable G s certainly will be as its poles will be in the left half plane circle on diameter e j0 0 If G s acquires a pole outside this circle but still in the left half plane of course and this appears possible in principle G s will then acquire a pole in Re s gt 0 Should this difficulty be encountered a smaller value of should be used redschur mulhank Restrictions Algorithm Xmath Model Reduction Module SysR HSV mulhank Sys nsr left right bound method The mulhank _ function calculates an optimal Hankel norm reduction of Sys for the multiplicative case This function has the following restrictio
66. ing if and only if G has the same property right half plane zeros of G are still preserved by the algorithm The error G G G though now zero at 0 is in general nonzero at 3 8 ni com Chapter 3 Multiplicative Error Reduction Hankel Singular Values of Phase Matrix of G The v i 1 2 ns have been termed above the Hankel singular values of the phase matrix associated with G The corresponding quantities for G are Vi i 1 NST Further Error Bounds The introduction to this chapter emphasized the importance of the error measures G G G or 6 e e co for plant reduction as opposed to G eje L or e G G The BST algorithm ensures that in addition to Equation 3 2 there holds WaS90a Io e a 52 Y r r i v i nsrt 1 which also means that for a scalar system G l v 201 lt 8 69 2 dB F 10 G X l y i i nsr l and if the bound is small ns phase G phase G lt Dy i nsrt 1 v radians y Reduction of Minimum Phase Unstable G For square minimum phase but not necessarily stable G it also is possible to use this algorithm with minor modification to try to minimize for G of a certain order the error bound ke 66 co National Instruments Corporation 3 9 Xmath Model Reduction Module Chapter 3 Multiplicative Error Reduction which also can be relevant in finding a reduce
67. ion on the j axis including oo If G s is nonsingular for almost all values of s it will be nonsingular or have no zero or rank reduction on the circle of diameter b j0 a j0 for almost all choices of a b If a and b are chosen small enough G s will have all its poles inside this circle and no zero or rank reduction on it while G s then will have all poles in Re s lt O and no zero or rank reduction on the jo axis including s The steps of the algorithm when G s has a zero on the j axis or at s are as follows 1 Forsmalla b with 0 lt a lt b form G s af gtildesys 2 Reduce G s to G s this being possible because G s is stable and has full rank on s jo including oo sta 3 FormG s Gn s a are as shown for as shown for gsys 3 22 ni com Chapter 3 Multiplicative Error Reduction The error eG G will be overbounded by the error lee G and G will contain the same zeros in Re s gt 0 as G If there is no zero or rank reduction of G s at the origin one can take a 0 and b bandwidth over which a good approximation of G s is needed and at the very least b sufficiently large that the poles of G s lie in the circle of diameter b j0 a j0 If there is a zero or rank reduction at the origin one can replace a 0 by a b It is possible to take b too small or if there is a zero at the origin to take a too s
68. ion are 6 26410 4 901x10 2 58110 2 474x102 1 545x10 1 335x107 9 467x103 9 466x10 3 A reduction to order 2 is attempted The ending Hankel singular values that iS 03 O4 Og have a sum that is not particularly small with respect to and 60 this is an indication that problems may arise in the reduction syscr hsv redschur sysc 2 svalsRol svplot sys syscr w radians plot svalsol keep f3 plot wc constr keep grid legend reduced original constrained title Open Loop Gain Using redschur 100 50 aa CG a 8 0 E T gt g 50 o gt 3 100 Do E mn 150 01 Open Loop Gain Using redschur reduced AT TT mr Y original constrained 0 01 0 1 1 10 100 Frequency rad sec Figure 6 3 Open Loop Gain Using redschur O National Instruments Corporation 6 5 Xmath Model Reduction Module Chapter 6 Tutorial Xmath Model Reduction Module 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 weighted reduction methods as described later Figure 6 3 also shows the inability to suppress all three plant resonances in contrast to the full order controller Two are such as to cause violation of the specifications The closed loop gains differ by some 4 to 5 dB betw
69. ions are used in this manual Square brackets enclose optional items for example response Square brackets also cite bibliographic references The symbol leads you through nested menu items and dialog box options to a final action The sequence File Page Setup Options directs you to pull down the File menu select the Page Setup item and select Options from the last dialog box This icon denotes a note which alerts you to important information Bold text denotes items that you must select or click in the software such as menu items and dialog box options Bold text also denotes parameter names Italic text denotes variables emphasis a cross reference or an introduction to a key concept This font also denotes text that is a placeholder for a word or value that you must supply Text in this font denotes text or characters that you should enter from the keyboard sections of code programming examples and syntax examples This font is also used for the proper names of disk drives paths directories programs subprograms subroutines device names functions operations variables filenames and extensions Bold text in this font denotes the messages and responses that the computer automatically prints to the screen This font also emphasizes lines of code that are different from the other examples Italic text in this font denotes text that is a placeholder for a word or value that you must supply Contents Cha
70. ization of Nonbalanced Reduced Order Model Realization of in continuous time Reduced Order Model Reduced Order Model Transfer Function Figure 2 1 Different Approaches for Obtaining the Same Reduced Order Model Singular Perturbation of Balanced Realization Singular perturbation of a balanced realization is an attractive way to produce a reduced order model Suppose G s is defined by ares ae Ay An 2 B y le C x Du National Instruments Corporation 2 5 Xmath Model Reduction Module Chapter 2 Additive Error Reduction with controllability and observability grammians given by p o z a NO in which the diagonal entries of 2 are in decreasing order that is 2 2 and such that the last diagonal entry of 2 exceeds the first diagonal entry of 2 It turns out that ReA 4 gt lt 0 and Reh A A e 0 and a reduced order model G s can be defined by El al x Ay 4 474 gt x 8 4 42 B u 1 1 y Ci C4431 x D C 4 B gt u The attractive feature LiA89 is that the same error bound holds as for balanced truncation For example GUD GGA lt 2trE Although the error bounds are the same the actual frequency pattern of the errors and the actual maximum modulus need not be the same for reduction to the same order One crucial difference is that balanced truncation provides exact matching at co but does not match at DC while sing
71. l svplot sys syscr w radians plot svalsol keep f6 plot wc constr keep grid title Open loop gain using ophank Open loop gain using ophank 100 ar reduced original constrained 50 50 100 Singular Value Magnitude dB 150 0 0 01 0 1 1 10 100 Frequency rad sec Figure 6 6 Open Loop Gain Using ophank O National Instruments Corporation 6 9 Xmath Model Reduction Module Chapter 6 Tutorial Generate Figure 6 7 syscl feedback sysol sysolr sys syscr sysclr feedback sysolr compare syscl sysclr w radians type 5 7 plot keep legend original reduced Singular Value Magnitude dB 50 4 50 be 100 Lb 150 20 EE ioi Ered FRIO i Poi A toot 0 01 0 1 1 10 100 Frequency rad sec original reduced Figure 6 7 Closed Loop Gain with ophank Xmath Model Reduction Module 6 10 ni com Chapter 6 Tutorial Generate Figure 6 8 tvec 0 140 99 140 compare syscl sysclr tvec type 7 f8 plot keep legend original reduced Step Response original reduced 0 20 40 60 80 100 Time sec Figure 6 8 Step Response with ophank The open loop gain closed loop gain and step response are all inferior to those obtained with redschur This emphasizes the point that one cannot automatically assume that because the error bound formul
72. long the j axis corresponds to multiplicative approximation of G s around a circle in the right half plane touching the j axis at the origin For those points on the jo axis near the circle there will be good multiplicative approximation of G j If a good approximation of G s over an interval jQ jQ it is desired then 5 Q or 10 Q are good choices Reduction then proceeds as follows 1 FormG s 2 Reduce G s through bst 3 Form G s G s 1 s with gsys subsys gtildesys gtildesys makep eps 1 makep 1 0 Notice that the number of zeros of G s in the circle of diameter 0 e j0 sets a lower bound on the degree of G s for such zeros become right half plane zeros of G s and must be preserved by bst Zeros at s are never in this circle so a procedure for reducing G s 1 d s is available There is one potential source of failure of the algorithm Because G s is stable G s certainly will be as its poles will be in the left half plane circle on diameter ce j0 0 If G s acquires a pole outside this circle but still in the left half plane of course and this appears possible in principle G s will then acquire a pole in Re s gt 0 Should this difficulty be encountered a smaller value of should be used singriccati ophank bst hankelsv 3 24 ni com Frequency Weighted Error Reduction This chapter describes frequency weighted erro
73. lot keep legend original reduced Singular Value Magnitude dB 50 E d original reduced 100 b 150 p tio sor of 4 ie 400 Frequency rad sec Figure 6 10 Closed Loop Gain with wtbalance Xmath Model Reduction Module 6 14 ni com Chapter 6 Tutorial Generate Figure 6 11 tvec 0 140 99 140 compare syscl sysclr tvec type 7 11 plot keep legend original reduced Step Response original 15 reduced I SS A a TE rias ra eee S A ria A 0 0 20 40 60 80 100 Time sec Figure 6 11 Step Response with wtbalance Figures 6 9 6 10 and 6 11 are obtained for wtbalance with the option input spec Evidently there is little difference between this and the result with the option match One notices marginally better matching in the region of interest 0 1 to 5 rad per second at the expense of matching at other frequencies The weighted Hankel singular values again indicate that it is reasonable to seek a second order controller National Instruments Corporation 6 15 Xmath Model Reduction Module Chapter 6 Tutorial Generate Figure 6 12 vtf poly 0 1 10 poly 1 1 4 sysv check vtf ss convert svalsv svplot sysv w radians System Singular Values 15 o Singular Value Magnitude dB on tonm ao or 1 10 100 Frequency rad sec Figure 6 12 Frequency Response o
74. m order permitted is the number of nonzero eigenvalues of P Q that are larger than e The matrices Va Vp are orthogonal and Sase and Sges are upper triangular Next submatrices are obtained as follows 0 Likor V ibig a k iig Vo 0 and then a singular value decomposition is formed UsbigSebig V hig T Vivig Kobe 4 14 ni com Related Functions fracred Chapter 4 Frequency Weighted Error Reduction From these quantities the transformation matrices used for calculating C s the stable part of C s are defined 1 2 Sibig Piet big 1 2 S big Pl biota and then ACR SipigA cS big Bor SipigBc ACR CCS pig Bcer De Just as in unweighted balanced truncation the reduced order transfer function matrix is guaranteed stable the same is guaranteed to be true in weighted balanced truncation when either a left output weight or a right input weight is used It is suspected to be true when both input and output weights are present The overall algorithm is not however at risk in this case since it is stability of the closed loop system which is the key issue of concern except for type input spec but here there is only a single weight and so the theory guarantees preservation of stability balance redschur stable fracred Restrictions SysCR HSV fracred Sys Kr Ke type nscr Qyy The fracred function uses fractional representations to calculate a reduction of a continuous ti
75. mall In these cases an error message results saying that there is a j axis zero and or that the Riccati equation solution may be in error The basic explanation is that as b 0 and thus a gt 0 the zeros of G s approach those of G s Thus for sufficiently small b one or more zeros of G s may be identified as lying on the imaginary axis The remedy is to increase a and or b above the desirable values The previous procedure for handling j axis zeros or zeros at infinity will be deficient if the number of such zeros is the same as the order of G s for example if G s 1 d s for some stable d s In this case it is possible again with a bilinear transformation to secure multiplicative approximations over a limited frequency band Suppose that G s Gl 2 es 1 Create a system that corresponds to G s with gtildesys subs gsys makep eps 1 makep 1 bilinsys makep eps 1 makep 1 0 sys subsys sys bilinsys Under this transformation e Values of G s along the j axis correspond to values of G s around a circle in the left half plane on diameter e jO 0 e Values of G s along the jw axis correspond to values of G s around a circle in the right half plane on diameter 0 e j0 National Instruments Corporation 3 23 Xmath Model Reduction Module Chapter 3 Multiplicative Error Reduction Related Functions Xmath Model Reduction Module Multiplicative approximation of G s a
76. me compensator comprising a state estimator with state feedback law 1 The closed loop system SCLR NSCLR is calculated from sysol scr sys open loop system syscl feedback sysol closed loop system 2 Initial state values state names and input and output names are not considered by fracred National Instruments Corporation 4 15 Xmath Model Reduction Module Chapter 4 Frequency Weighted Error Reduction 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 a Controller Suppose P s C sI A B and A BKp and A K C are stable where Kp is a stabilizing state feedback gain and K a stabilizing observer gain A controller for the plant P s can be defined by a x Ax Bu K Cx y K px u with u the plant input and y the plant output The associated series compensator under unity negative feedback is C s Kg sI A BKp K C K and this may be written as a left or right MFD as follows C s I Kg sI A K C BI Kp sI A K C K 4 5 C s Kp sI A BKp Kell C sI A BK Kel 4 6 The reduction procedures right perf and left perf have similar rationales We shall describe right perf refer to AnM89 and LiA86 The first rationale involves observing that to reduce C s one might as well reduce its numerator a
77. nM89 and LAL90 The idea can be understood with reference to the transfer functions E s and E s used in discussing type right perf Itis possible to argue through block diagram manipulation that e C s stabilizes P s when E s stabilizes as a series compensator with unity negative feedback P s Pc s I e E s also will stabilize P s and then C s will stabilize P s provided Cora K C B 1 CGwl A K Cy K siid EGo E 0 g Accordingly it makes sense to try to reduce E by frequency weighted balanced truncation When this is done the controllability grammian for E s remains unaltered while the observability grammian is altered Hence Equation 4 5 at least with Q Z and Equation 4 12 are the same while Equation 4 6 and Equation 4 13 are quite different The calculations leading to Equation 4 13 are set out in LAL90 The argument for type left perf is dual Another insight into Equation 4 14 is provided by relations set out in NJB84 There it is established in a somewhat broader context that C jol A K C B I C jol A K C K K s1 A BK Kz x I 1 C jol A BKp Kg The left matrix is the weighting matrix in Equation 4 14 the right matrix is the numerator of C j stacked on the denominator or alternatively E jo P I This formula then suggests the desirability of retaining the weight in the approximation of E j by E j 4 20 ni com Chapter 4 Frequency Weighted Erro
78. nce San Francisco CA 1980 Section FA8 E A J Laub M T Heath C C Paige and R C Ward Computation of system balancing transformations and other applications of simultaneous diagonalizing algorithms JEEE Transactions on Automatic Control Vol AC 32 1987 pp 115 122 Y Liu and BDO Anderson Controller reduction via stable factorization and balancing Int J Control Vol 44 1986 pp 507 531 Y Liu and BDO Anderson Singular perturbation approximation of balanced systems International Journal of Control Vol 50 1989 pp 1379 1405 Y Liu and BDO Anderson Frequency weighted controller reduction methods and loop transfer recovery Automatica Vol 26 No 3 pp 487 489 B C Moore Principal component analysis in linear systems Controllability observability and model reduction IEEE Transactions on Automatic Control Vol AC 26 No 1 1981 pp 17 32 C N Nett C A Jacobson and M J Balas A connection between state space and doubly coprime fractional representations IEEE Transactions on Automatic Control Vol AC 29 1984 pp 831 832 L Pernebo and L M Silverman Model reduction via balanced state space representations JEEE Transactions on Automatic Control Vol AC 27 No 2 1982 pp 382 387 Xmath Model Reduction Module A 2 ni com SaC88 Saf87 SCL90 Vid85 WaS90 WaS90a BBK88 BeP79 BoB90 BoB91 BH69 DoS
79. nction reference information is available in the MATRIXx Help The MATRIXx Help includes all Model Reduction functions Each topic explains a function s inputs outputs and keywords in detail Refer to Chapter 2 MATRIXx Publications Online Help and Customer Support of the MATRIXx Getting Started Guide for complete instructions on using the help feature Overview The Xmath Model Reduction Module MRM provides a collection of tools for reducing the order of systems Many of the functions are based on state of the art algorithms in conjunction with researchers at the Australian National University who were responsible for the original development of some of the algorithms A common theme throughout the module is the use of Hankel singular values and balanced realizations although considerations of numerical accuracy often dictates whether these tools are used implicitly rather than explicitly The tools are particularly suitable when as generally here quality of approximation is measured by closeness of frequency domain behavior National Instruments Corporation 1 3 Xmath Model Reduction Module Chapter 1 Introduction Functions Xmath Model Reduction Module 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 controller reduction Utility func
80. nd denominator simultaneously and then form a new fraction C s of lower order than C s This amounts to reducing E s Ejea BK Kg 4 7 Xmath Model Reduction Module 4 16 ni com Chapter 4 Frequency Weighted Error Reduction to for example q C sI A K through for example balanced truncation and then defining A 1 A af 1 C s Kp st A Kelf C sI A Kel Kp sI A REC Ke For the second rationale consider Figure 4 5 P s ye Figure 4 5 Internal Structure of Controller Recognize that the controller C s shown within the hazy rectangle in Figure 4 5 can be constructed by implementing K sI A BK Kg and C sI A BK K and then applying an interconnection rule connect the output of the second transfer function matrix back to the input at point X in Figure 4 5 National Instruments Corporation 4 17 Xmath Model Reduction Module Chapter 4 Frequency Weighted Error Reduction Algorithm Xmath Model Reduction Module Controller reduction proceeds by implementing the same connection rule but on reduced versions of the two transfer function matrices When K has been defined through Kalman filtering considerations the spectrum of the signal driving K in Figure 4 5 is white with intensity Q It follows that to reflect in the multiple input case the different intensities on the di
81. nkel norm approximation The procedure is slightly different from that used in ophank Construct an SVD of OP vI E aI suas 00 OP visl U l 00 V with 2 of dimension ns r x ns r and nonsingular Also obtain an orthogonal matrix T satisfying Bot C T 0 where B and C are the last r rows of B and C the state variable matrices appearing in a balanced realization of C s 1 A 1B It is possible to calculate T without evaluating B B C as it turns out refer to AnJ and the algorithm does this Now with Fis De Grlsl Ap Br F s Cr sI Ar BF there holds Ap E U A QAP v C TB V Br X U LOB v Cy Cr C P v TB V Dr Vast 3 17 Xmath Model Reduction Module Chapter 3 Multiplicative Error Reduction B Note The expression F p s is the strictly proper part of F s The matrix Y F s F s 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 algorithm constructs G and W which satisfy G s G s W lt s LF s F s and W s i Viel JU va W s LF s F s G s through the state variable formulas a 5 x JA G s D 1 v sT DCr ByUZ sl Ar Br and Ws v 17 D U v T d v ea ns Cr sI Ar BrD V C A Continue the reduction procedure starting with G W F and repeating the process till G of t
82. nonnonncnnncnnonnnconnnnnonn corno anno non ron ncnn nin ncnncinnos 2 6 Dalimoore io e ec tthe E E a A 2 8 Related Functions ci A 2 11 trunca aa daa at lito oda ata 2 11 Related Functions iii nia 2 11 Tedschur dio oda lia 2 12 ATEON e ea iaaii i 2 12 Related M AOTEA ELOI aE ETE E EE 2 14 OPM eri IT 2 14 RES tri A eet n se a tac eae E ES 2 14 Algorithm ici diia 2 15 Behaviors cd tists lata 2 15 National Instruments Corporation v Xmath Model Reduction Module Contents Chapter 3 On pass Alporithin ans ae ae eee tes ae cee 2 18 Multipass Algoritmo adria ei 2 20 Discrete Time SYSTEMS erorita ro e EE EE ARRE S 2 21 Impulse Response ETOT i a E E E E E E Ei 2 22 Unstable System Approximation ooccoocnnonconnnoncononancnnnonnconcnnonn non nan ccnncnnncn conan 2 23 Related Functions vicios 2 23 Multiplicative Error Reduction Selecting Multiplicative Error ReductiON oooonncnnncnocnnnnnononnconcnnnonncnnnonccon crono nn conan crono 3 1 Multiplicative Robustness Result oonoonicnncnnccnnnnonncnnncononanonncanorancnn con cconcrnnonnos 3 2 DIA a 3 3 Restrictions id acia 3 4 BUSOT Mt AS A ltd 3 4 Algorithm with the Keywords right and left oononnnnnnnnnnnnnnnnonnocnnnncannnnonnos 3 5 Securing Zero Erroriat DC vas saver aces diia 3 8 Hankel Singular Values of Phase Matrix Of Gioccoconncnocnnonnonnonnconnnanonncrnncnnccnnos 3 9 Further Error Bounds dey vey ieee cten ee EA aves svete eee 3 9 Reduction of Minimum Phase Unstable
83. ns The user must ensure that the input system is stable and nonsingular at s infinity e The algorithm may be problematic if the input system has a zero on the JO axis e Only continuous systems are accepted for discrete systems use makecontinuous before calling mulhank then discretize the result Sys mulhank makecontinuous SysD SysD discretize Sys The objective of the algorithm like bst is to approximate a high order square stable transfer function matrix G s by a lower order G s with either ke G Ga or lG A approximately minimized under the constraint that G is stable and of prescribed order The algorithm has the property that right half plane zeros of G s are retained as zeros of G s This means that if G s has order NS with N zeros in Re s gt 0 G s must have degree at least N else given that it has N zeros in Re s gt 0 it would not be proper GrA89 3 14 ni com right and left Chapter 3 Multiplicative Error Reduction 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 jw 2 G jw I and then an all pass function the phase function W s G s This all pass function has a mixture of stable and unstable poles and it encodes the phase of G j Its stable part has n Hankel singular values 6 with lt 1 and the number of equal to 1
84. nse of wtbalance with input spec National Instruments Corporation 6 19 Xmath Model Reduction Module Chapter 6 Tutorial 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 all produce instability Given the relative magnitudes of the Hankel singular values this is perhaps not surprising Figures 6 16 6 17 and 6 18 illustrate the results using right stab Generate Figure 6 16 svalsrol svplot sys syscr w radians plot svalsol keep f16 plot wc constr keep grid legend reduced original constrained title Open Loop Gain Using fracred Open Loop Gain Using fracred 100 E ee reduced original constrained 50 co a a O a re o gt y 700 2 E gt 100 Do lt Ww 150 2 Toot 0 01 0 1 1 10 100 Frequency rad sec Figure 6 16 Open Loop Gain Using fracred Xmath Model Reduction Module 6 20 ni com Chapter 6 Tutorial Generate Figure 6 17 syscl feedback sysol sysolr sys syscr sysclr feedback sysolr compare syscl sysclr w radians type 5 17 plot keep legend original reduced Singular Value Magnitude dB original 50 7 i Tt F reduced 100 Hs 150 b oo 0 01 ih a 10 100 Frequency rad sec Figure 6 17 Closed Loop Response with fracred National Instruments Co
85. nstructed and from it a singular value decomposition is obtained H UySyV y 3 The balancing transformation is given by T US ass The balanced realization is T AT TB CT 1 10 ni com Chapter 1 Introduction 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 S 2 and U S in forming H in step 2 The grammians themselves need never be formed as these Cholesky factors can be computed directly from A B and C in both continuous and discrete time this however is not done in balmoore The algorithm has the property that TOT T P T Y Sy Thus the diagonal entries of Sy are the Hankel singular values The algorithm implemented in balance is older refer to Lau80 A lower triangular Cholesky factor L of P is found so that L L P Then the symmetric matrix L QL is diagonalized through a singular value decomposition thus L QL VRU with actually V U Finally the coordinate basis transformation is given by T L VR7 resulting in T QT T P TY R Singular Perturbation A common procedure for approximating systems is the technique of Singular Perturbation The underlying heuristic reasoning is as follows Suppose there is a system with the property that the unforced responses of the system contain some modes which decay to zero extremely fast Then an appr
86. nts Corporation 3 21 Xmath Model Reduction Module Chapter 3 Multiplicative Error Reduction The values of G s along the j axis are the same as the values of G s around a circle with diameter defined by a j0 bs JO on the positive real axis refer to Figure 3 2 Also the values of G s along the j axis are the same as the values of G s around a circle with diameter defined by b j0 a j0 We can implement an arbitrary bilinear transform using the subsys _ function which substitutes a given transfer function for the s or z domain operator as previously shown i s a To implement G s c bsa use gtildesys subsys gsys makep b 1 makep 1 al a Sta To implement G s dE T use gsys subsys gtildesys makep b 1 makep 1 a 3 Note The systems substituted in the previous calls to subsys invert the function specification because these functions use backward polynomial rotation Xmath Model Reduction Module Any zero or rank reduction on the j axis of G s becomes a zero or rank reduction in Re s gt 0 of G s and if G s has a zero or rank reduction at infinity this is shifted to a zero or rank reduction of G s at the point b again in Re s gt 0 If all poles of G s are inside the circle of diameter b7 jO a 50 all poles of G s will be in Re s lt 0 and if G s has no zero or rank reduction on this circle G s will have no zero or rank reduct
87. odule Chapter 4 Frequency Weighted Error Reduction wtbalance Here the W and V are submatrices of W V Evidently Nr Dp D N V I and W I Some manipulation shows that trying to preserve these identities after approximation of D Ni or Ng Dg suggests use of the error measures W G G _ and H H V For further details refer to AnM89 and LAL90 In all four fracred options it is possible to construct weighted Hankel singular values and to use them as a guide to the likely quality of approximation The patterns tend to be different for the four options The fracred options are normally different in outcome from the wtbalance options However if the controller has been designed by the loop transfer recovery method and is stable then one of the fracred options is essentially the same as one of the wtbalance options refer to LiA90 More precisely if the LTR design is performed with input noise or process noise weighting tending to infinity reduction with fracred and type left stab which uses the error measure H H V leads to effectively the same reduction as wtbalance with the type input stab If the LTR design is performed with state or output weighting tending to infinity in the index determining the state feedback law reduction with fracred and type right stab using the error measure W G G leads to effectively the same reduction as wtbalance
88. onds to a discrete time system with state variable dimension n Then the infinite Hankel matrix CB CAB CAB H CAB CA B CAB has for its singular values the n nonzero Hankel singular values together with an infinite number of zero singular values The Hankel singular values of a stable all pass system or all pass matrix are all 1 Slightly different procedures are used for calculating the Hankel singular values and so called weighted Hankel singular values in the various functions These procedures are summarized in Table 1 2 Table 1 2 Calculating Hankel Singular Values balance For a discussion of the balancing algorithm refer to the Internally Balanced Realizations section the Hankel singular values are given by diag R HSV balmoore For a discussion of the balancing algorithm refer to the Internally Balanced Realizations section the matrix Sy yields the Hankel singular values through diag SH hankelsv real sqrt eig p a ophank Calls hankelsv redschur Computes a Schur decomposition of P Q and then takes the square roots of the diagonal entries bst Same as redschur except either P or O can be mulhank a weighted grammian wtbalance fracred National Instruments Corporation 1 9 Xmath Model Reduction Module Chapter 1 Introduction Internally Balanced Realizations Xmath Model Reduction Module Suppose that a realization of a tr
89. ons The treatment of j axis or right half plane poles in the above schemes is crude they are simply copied into the reduced order controller A different approach comes when one uses a so called matrix fraction description MFD to represent the controller and controller reduction procedures based on these representations only for continuous time are found in fracred Consider first a scalar controller C s n s d s One can take a stable polynomial d s of the same degree as d and then represent the controller as a ratio of two stable transfer functions namely hal ae Now n ig d is the numerator andd d the denominator Write d d as 1 e d Then we have the equivalence shown in Figure 4 1 ly Figure 4 1 Controller Representation Through Stable Fractions gt gt C s Quis Evidently C s can be formed by completing the following steps 1 Construction of the one input two output stable transfer function matrix d e d which has order equal to that of d or d 2 Interconnection through negative feedback of the second output to the single input These observations motivate the reduction procedure e Reduce Gto G notice that G is stable Let G be n dy G _ e d National Instruments Corporation 4 5 Xmath Model Reduction Module Chapter 4 Frequency Weighted Error Reduction Form the reduced controller by interconnecting using neg
90. oximation to the system behavior may be attained by setting state variable derivatives associated with these modes to zero even in the forced case The phrase associated with the modes is loose exactly what occurs is shown below The phrase even in the forced case captures a logical flaw in the argument smallness in the unforced case associated with initial conditions is not identical with smallness in the forced case Suppose the system is defined by P j Ta H J al z Ay Ayy 2 18 y e C HEZ 2 1 1 National Instruments Corporation 1 11 Xmath Model Reduction Module Chapter 1 Introduction Xmath Model Reduction Module and also Reh A lt 0 and ReA A 41243421 lt 0 Usually we expect that Reh Ay Red A A ADA in the sense that the intuitive argument hinges on this but it is not necessary Then a singular perturbation is obtained by replacing x by zero this means that 1 A x X Bou 0 or X A74 x z Bou Accordingly i 1 Xx 411 41 424 1 x By 4 4 B u 1 1 y Ci C 4y7491 x1 D C2422B2 U 1 2 Equation 1 2 may be an approximation for Equation 1 1 This means that e The transfer function matrices may be similar e IfEquation 1 2 is excited by some u with initial condition x and if Equation 1 1 is excited by the same u with initial condition given by K1 t and xa t A 22A91X1 t0 A27 Bonlto then x and y
91. oy ISO A lt _ _ lagal T Tigo This means that if either bound is small so is the other with the bounds approximately equal Two algorithms for multiplicative reduction are presented bst a mnemonic for balanced stochastic truncation and mulhank Roughly they relate to one another in the same way that redschur and ophank _ relate that is one focuses on balanced realization truncation and the other on Hankel norm approximation Some of the similarities and differences are as follows When the errors are small the error bound formula for bst is about one half of that for bst e Withbst the actual multiplicative error as a function of frequency goes to zero as Ys or after using an optional transformation given in the algorithm description to zero as 0 with mulhank the error tends to be more uniform as a function of frequency e bst can handle nonsquare reduction while mulhank cannot e Both algorithms are restricted to stable G s both preserve right half plane zeros that is these zeros are copied into the reduced order object both have difficulties with j axis zeros of G s but these difficulties are not insuperable bst SysR HSV bst Sys nsr left right bound method The bst function calculates a balanced stochastic truncation of Sys for the multiplicative case National Instruments Corporation 3 3 Xmath Model Reduction Module Chapter 3 Mul
92. pproach those of G s Thus for sufficiently small b one or more zeros of G s may be identified as lying on the imaginary axis The remedy is to increase a and or b above the desirable values The procedure for handling j axis zeros or zeros at infinity will be deficient if the number of such zeros is the same as the order of G s for example if G s 1 d s for some stable d s In this case it is possible 3 12 ni com Chapter 3 Multiplicative Error Reduction again with a bilinear transformation to secure multiplicative approximations over a limited frequency band Suppose that G s c es 1 Create a system that corresponds to G s with gtildesys subs gsys makep eps 1 makep 1 bilinsys makep eps 1 makep 1 0 sys subsys sys bilinsys Under this transformation e Values of G s along the j axis correspond to values of G s around a circle in the left half plane on diameter e j0 0 e Values of G s along the jw axis correspond to values of G s around a circle in the right half plane on diameter 0 e 0 Multiplicative approximation of G s along the j axis corresponds to multiplicative approximation of G s around a circle in the right half plane touching the j axis at the origin For those points on the jo axis near the circle there will be good multiplicative approximation of G j If it is desired to have a good approximation of G s over an interval jQ jQ t
93. pter 1 Introduction Using This Manual ct dit 1 1 Document Organization eee eeeceseeseeeseeseeeseceeseseeseceaeeaeseeesseseaeseseaees 1 1 Bibliographic References iecerei a E E E EEE 1 2 Commonly Used Nomenclature cooccnoccnonconnnoncnnnnoncnncnnnonnnonccononnnonncnnncnncrnncnnnos 1 2 COMVENMONS ss ti A e lid idee A aa 1 2 Related Publications vaina ta 1 3 MATRIX He Petra aani ea Seles eect EEN E a E AE RE E aSa 1 3 OVET e A na 1 3 PUNCH ONS rita AAA A Mevetdaad eeu eess 1 4 Nomenclature iia ia cd 1 6 Commonly UsediConceptsitciiiinaaa vi iia 1 7 Controllability and Observability Grammians oooococcnicnonnnonconnnnnnoncnancnnccnncnnon 1 7 Hankel Singular Values irere anors he cesngs teuecsein eee E ENERE Ros 1 8 Internally Balanced RealizatiODS oooncnncninnnncnncnnonconononcnnnnncconono conc cronica conncnn 1 10 Singular PerturbatiOn iii o 1 11 Spectral FactorizdtOM ceci 1 13 Low Order Controller Design Through Order Reduction ooconcnonnnonnonnnoncnnnconcnnccnonnos 1 15 Chapter 2 Additive Error Reduction TEO UC ii a has Bevin Pn wie een hed 2 1 Truncation of Balanced Realizations ccccccccessccessnceessseceseeeeeeeeeseneeeeseeeeessneeeneneeens 2 2 Reduction Through Balanced Realization Truncation cccesceessecesceeseeceneeceeeeeneeeeaee 2 4 Singular Perturbation of Balanced Realizati0M oononncnnncnonnnnnnonnonnnonconncnn cnn nonn cnn ccnncnnnos 2 5 Hankel Norm Approximation oocoococononocconcnnnon
94. quency Weighted Error Reduction Matrix algebra shows that C s can be described through a left or right matrix fraction description C s D7 s N s Ng s D 5 with D and related values all stable transfer function matrices In particular D I Kg sI A K Cy B N Kg sI A K C K Na Kg sI A BKp Ke Dg I C sI A BK K For matrix C s the left and right matrix fraction descriptions are distinct entities It is the right MFD which corresponds to Figure 4 1 refer to Figure 4 2 P s yr gt C s gt P s Figure 4 2 C s Implemented to Display Right MFD Representation National Instruments Corporation 4 7 Xmath Model Reduction Module Chapter 4 Frequency Weighted Error Reduction The left MFD corresponds to the setup of Figure 4 3 P s gt K sI A K C l4 y w A Kp 4 Figure 4 3 C s Implemented to Display Left MFD Representation The setup of Figure 4 2 suggests approximation of G s MES BK K whereas that of Figure 4 3 suggests approximation of H s Kp s1 A K C B Kg In the LQG optimal case the signal driving Kg in Figure 4 2 is white noise the innovations process this motivates the possibility of using no frequency dependent weighting in approximating G s but observe that
95. r Reduction The four schemes all produce different HSVs it follows that it may be prudent to try all four schemes for a particular controller reduction Recall again that their relative sizes are only a guide as to what can be thrown away without incurring much error There is no simple rule to indicate which of the four schemes will be the most effective at controller reduction Two rough rules can however be formulated e Problems with instability through reduction to too low a controller order are more likely with left perf and right perf than left stab or right stab e Ifthe controller has been designed using the loop transfer recovery idea left stab will probably be attractive if the input noise covariance is very large and right stab will probably be attractive if the output weighting in the performance index is very large LiA90 The reduced controllers will then actually be very similar to those obtained using wtbalance with the option input stab in the first case and output stab in the second case One example gives the HSVs summarized in Table 4 3 for an eighth order controller Table 4 3 HSVs for an Eighth Order Controller 1 2 3 4 5 6 7 8 right perf 0339 0164 0128 0102 0040 0037 0000 0000 left perf 4 9075 4 8742 3 8457 3 7813 1 2255 1 1750 5055 0413 right stab 3 3081 7278 1123 0783 0242 0181 0107 0099 left stab 1 3914 1 317 1 1269
96. r reduction problems This includes a discussion of controller reduction and fractional representations Introduction Frequency weighted error reduction means that the error is measured not as previously by E GGo G Go but rather by E GGo G 0 0 4 1 or E W Go GGo G 0 1 4 2 or E W Go GGo G G0 1 0 4 3 where W V are certain weighting matrices Their presence reflects a desire that the approximation process be more accurate at certain frequencies where V or W have large singular values than at others where they have small singular values For scalar G 0 all the indices above are effectively the same with the effective weight just IV j l 1W 0 l or Wo Vo When the system G is processing signals which do not have a flat spectrum and is to be approximated there is considerable logic in using a weight If the signal spectrum is ja then taking V jq as a stable spectral factor O National Instruments Corporation 4 1 Xmath Model Reduction Module Chapter 4 Frequency Weighted Error Reduction so that VW D 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 controller dimension The LQG and A design procedures lead to controllers which have order equal to or roughly equal to the order of the plant Very of
97. r this purpose 2 Compute input right weight and or output left weight as appropriate for the specified type Xmath Model Reduction Module 4 12 ni com Chapter 4 Frequency Weighted Error Reduction 3 Compute weighted Hankel Singular Values described in more detail later If the order of C s is not specified a priori it must be input at this time Certain values may be flagged as unacceptable for various reasons In particular nscr cannot be chosen so that Onser Onser 1 4 Execute reduction step on stable part of C s based on a modification of redschur to accommodate frequency weighting and yielding stable part of C s 5 Compute C s by adding unstable part of C s to stable part of C s 6 Check closed loop stability with C s introduced in place of C s at least in case C s is a compensator More details of steps 3 and 4 will be given for the case when there is an input weight only The case when there is an output weight only is almost the same and the case when both weights are present is very similar refer to Enn84a for a treatment Let C s D C sI A B W s D C sI A B be a stable transfer function matrix to be reduced and its stable weight Thus W s may be PU CP corresponding to input stab and will thus have been calculated in step 2 or it maybe an independently specified stable V s Then 1 I A BC B D A peT pe SIA y w The controllability grammian P sa
98. realizations Introduction Additive error reduction focuses on errors of the form EGO GC jo where G is the originally given transfer function or model and G is the reduced one Of course in discrete time one works instead with ec Ge As is argued in later chapters if one is reducing a plant that will sit inside a closed loop or if one is reducing a controller that again is sitting in a closed loop focus on additive error model reduction may not be appropriate It is however extremely appropriate in considering reducing the transfer function of a filter One pertinent application comes specifically from digital filtering a great many design algorithms lead to a finite impulse response FIR filter which can have a very large number of coefficients when poles are close to the unit circle Model reduction provides a means to replace an FIR design by a much lower order infinite impulse response IIR design with close matching of the transfer function at all frequencies National Instruments Corporation 2 1 Xmath Model Reduction Module Chapter 2 Additive Error Reduction 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 X Az A 2 122 y C C x D c c os and the realization is internally balanced then a truncation is provided by X1 Aj X1 Byu y C x Du T
99. regulator sys Q R A BxK is stable Next with an input noise variance matrix O W BBW where W DIAG 0 346 0 346 0 024 0 0240 042 0 0420 042 0 042 and measurement noise covariance matrix R 1 an estimation gain K so that A K C is stable is determined Qhat Wt b b wt Rhat 1 Ke ev estimator sys Qhat Rhat skipChks The keyword skipChks circumvents syntax checking in most functions It is used here because we know that Qhat does not fulfill positive semidefiniteness due to numerics sysc lqgcomp sys Kr Ke poles sysc ans a column vector 0 296674 0 292246 j 0 296674 0 292246 j 0 15095 0 765357 j 0 15095 0 765357 j 0 239151 1 415 j 0 239151 1 415 j 0 129808 1 84093 j 0 129808 1 84093 j The compensator itself is open loop stable A brief explanation of how Q and wt are chosen is as follows First Q is chosen to ensure that the loop gain K Jol A B which would be relevant were the state measurable meets the constraints as far as possible However it is not possible to obtain a 40 dB per decade roll off at high frequencies as LQ design virtually always yields a 20 dB per decade roll off Second a loop transfer recovery approach to the choice of as pBB for some large p is modified through the introduction of the diagonal matrix wt The larger entries of Wt because of the modal coordinate system in effect promote better loop transfer
100. rnally balanced realization of a continuous system and then optionally truncates it to provide a balance reduced order system using B C Moore s algorithm When balmoore _ is being used to reduce a system its objective mirrors that of redschur therefore if the same Sys and nsr are used for both algorithms the reduced order system should have the same transfer function though in general the state variable realizations will be different When balmoore is being used to balance a system its objective like that of balance is to generate an internally balanced state variable realization The implementations are not identical balmoore _ only can be applied on systems that have a stable A matrix and are controllable and observable that is minimal Checks which are rather time consuming are included The computation is intrinsically not well conditioned if Sys is nearly nonminimal The first part of balmoore serves to find a transformation matrix T such that the controllability and observability grammians after transformation are equal and diagonal with decreasing entries down the diagonal that is the system representation is internally balanced The condition number of T is a measure of the ill conditioning of the algorithm If there is a problem with ill conditioning redschur can be used as an alternative If this common grammian is 2 then after transformation continuous ZA H AX BB YA AD CC discret
101. roubleshooting NI resources B 1 truncate 1 5 2 4 2 11 National Instruments Corporation 1 3 Xmath Model Reduction Module
102. roximation A transfer function G s with stable and unstable poles can be reduced by applying stable to separate G s into a stable and unstable part The former is reduced and then the unstable part can be added back on For additional information about the ophank function refer to the Xmath Help Related Functions stable redschur balmoore National Instruments Corporation 2 23 Xmath Model Reduction Module 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 Error Reduction The general reason to use multiplicative error reduction is that many specifications are given using decibels 1 db corresponds to a multiplicative error of about 12 Specifications regarding phase shift also can be regarded as multiplicative error statements 0 05 radians of phase shift is like 5 multiplicative error also The more specific reason arises in considering the problem of plant approximation with a high order possibly very high order plant being initially prescribed with no controller having been designed and with a requirement to provide a simpler model of the plant possibly to allow controller design Consider the arrangement of Figure 3 1 controller C s designed for G s j with G s I A G s
103. rporation 6 21 Xmath Model Reduction Module Chapter 6 Tutorial Generate Figure 6 18 tvec 0 140 99 140 compare syscl sysclr tvec type 7 18 plot keep legend original reduced Step Response original T reduced 20 40 60 80 100 Time sec Xmath Model Reduction Module Figure 6 18 Step Response with fracred The end result is comparable to that from wtbalance with option match We can create a table to examine the values of the Hankel singular values based on different decompositions approaches set precision 3 Optional set format fixed we set a smaller precision here so we could fit the table in the manual syscr hsvrs fracred sys Kr Ke right stab 2 syscr hsvls fracred sys Kr Ke left stab 2 syscr hsvrp fracred sys Kr Ke right perf 2 syscr hsvlp fracred sys Kr Ke left perf 2 6 22 ni com Chapter 6 Tutorial hsvtable right stab string hsvrs left stab string hsvls right perf string hsvrp left perf string hsvlp hsvtable a rectangular matrix of strings right stab 3 308 0 728 0 112 0 078 0 024 0 018 0 011 0 010 left stab 1 403 1 331 1 133 1 092 0 965 0 549 0 526 0 313 right perf 0 034 0 016 0 013 0 010 0 004 0 004 0 000 0 000 left perf 4 907 4 874 3 846 3 781 1 225 1 175 0 505 0 041 National Instruments Corporation 6 23 Xmath Model Reduction Module Bibliogr
104. s 1 Gus 1 3 Step ns nr Find a stable order nsr approximation of Gs 1 with Gasp 10 gt Gus JO Onsr Gays 1 Then because 6 G _ 0 G for i lt ns 6 G _ O Gys_ 1 for lt s i this being a property of the algorithm there follows KAEO Gis JO Onsr 1 Gnsr 1 Sy s G lt Y 016 i nsrt 1 The only difference that arises when singular values have multiplicity in excess of is that the degree reduction at a given step is greater Thus if 5 G has multiplicity k then G s is approximated by G _ S of degree ns k No separate optimization of the D matrix of G is required The approximation error bound is the same as for the first algorithm The actual approximation error may be different Notice that this second algorithm does not calculate an unstable G s such that EGo Grs 10 iS G Go Onsr41 Discrete Time Systems No special algorithm is presented for discrete time systems Any stable discrete time transfer function matrix G z can be used to define a stable continuous time transfer function matrix by a bilinear transformation thus when 0 is a positive constant National Instruments Corporation 2 21 Xmath Model Reduction Module Chapter 2 Additive Error Reduction 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 rever
105. s ese i en eaves 5 4 compare O AN 5 4 Chapter 6 Tutorial Plant and Full Order Controller oonnnnnnnnuconacuouaoaonncnonononononononanananannnnnnnnnnncccnonononinonon 6 1 Controller REGUCHOM sii dt dou ss dd da de 6 5 ophank cccitiiits Sieve A A 6 9 WIDE A ads 6 12 e Fo EEE EE Ses Se eR a aN a ee 6 20 Appendix A Bibliography Appendix B Technical Support and Professional Services Index National Instruments Corporation vil Xmath Model Reduction Module 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 used throughout this manual Using This Manual This manual describes the Model Reduction Module MRM which provides a collection of tools for reducing the order of systems Readers who are not familiar with Parameter Dependent Matrices PDMs should consult the Xmath User Guide before using MRM functions and tools Although several MRM functions accept both PDMs and matrices as input parameters PDMs are preferable because they can include additional information that is useful for simulation plotting and signal labeling Document Organization This manual includes the following chapters e Chapter 1 Introduction starts with an outline of the manual and some useful notes It also provides an overview of the Model Reduction Module de
106. scribes the functions in this module and introduces nomenclature and concepts used throughout this manual e Chapter 2 Additive Error Reduction describes additive error reduction including discussions of truncation of reduction by and perturbation of balanced realizations e Chapter 3 Multiplicative Error Reduction describes multiplicative error reduction presenting considerations for using multiplicative rather than additive error reduction e Chapter 4 Frequency Weighted Error Reduction describes frequency weighted error reduction problems including controller reduction and fractional representations National Instruments Corporation 1 1 Xmath Model Reduction Module Chapter 1 Introduction e Chapter 5 Utilities describes three utility functions hankelsv stable and compare e Chapter 6 Tutorial illustrates a number of the MRM functions and their underlying ideas Bibliographic References Throughout this document bibliographic references are cited with bracketed entries For example a reference to VODM1 corresponds to a paper published by Van Overschee and De Moor For a table of bibliographic references refer to Appendix A Bibliography Commonly Used Nomenclature This manual uses the following general nomenclature e Matrix variables are generally denoted with capital letters vectors are represented in lowercase e G s is used to denote a transfer function of a system where s is the Lapla
107. sed because this function uses backward polynomial rotation Hankel norm reduction is then applied to H s to generate a stable reduced order approximation H s and unstable H s such that 7 H A O 1 H H 0 0 1 0 ns Here the s are the Hankel singular values of both G z and H s they are the same oe note ouo Ho We then implement the s domain equivalent with sysS subsys sysZ bilinsys There is no simple rule for choosing the choice amp 1 is probably as good as any The orders of G and G are the same as those of H and H respectively The error formulas are as follows lace 6 2 CDa 6 Jace 6 2 S 6 5 21 Ons Impulse Response Error Xmath Model Reduction Module If G is determined by the first single pass algorithm the impulse response error for t gt 0 between the impulse responses of G and G can be bounded As shown in Corollary 9 9 of Glo84 if G is of degree 1 and the multiplicity of the ith larger singular value 0 of G is r then o G G lt 0 G forj 1 2 2i 2 r lt 0 1 G forj 2i 1 r ns i 1 2 22 ni com Chapter 2 Additive Error Reduction It follows by a result of BoD87 that the impulse response error for t gt 0 satisfies g t g t 1 2 21 2 r 0 G 96 Evidently Hankel norm approximation ensures some form of approximation of the impulse response too Unstable System App
108. seni com 2000 2004 National Instruments Corporation All rights reserved Important Information Warranty The media on which you receive National Instruments software are warranted not to fail to execute programming instructions due to defects in materials and workmanship for a period of 90 days from date of shipment as evidenced by receipts or other documentation National Instruments will at its option repair or replace software media that do not execute programming instructions if National Instruments receives notice of such defects during the warranty period National Instruments does not warrant that the operation of the software shall be uninterrupted or error free A Return Material Authorization RMA number must be obtained from the factory and clearly marked on the outside of the package before any equipment will be accepted for warranty work National Instruments will pay the shipping costs of returning to the owner parts which are covered by warranty National Instruments believes that the information in this document is accurate The document has been carefully reviewed for technical accuracy In the event that technical or typographical errors exist National Instruments reserves the right to make changes to subsequent editions of this document without prior notice to holders of this edition The reader should consult National Instruments if errors are suspected In no event shall National Instruments be liable for any dam
109. sentation reduction 4 18 Hankel multi pass 2 20 optimal Hankel norm reduction 2 15 stable 5 2 weighted balance 4 12 all pass transfer function 1 6 2 4 B balance 1 5 2 4 algorithm 1 11 balanced realization definition 1 10 internally balanced 3 9 singular perturbation 2 5 truncation 2 2 2 4 balanced stochastic truncation 3 3 See also bst balmoore 1 5 2 4 algorithm 1 10 bst 1 5 1 14 3 3 c compare 1 5 5 4 controller reduction 4 2 with fractional representations 4 5 National Instruments Corporation controller robustness 4 2 conventions used in the manual iv D diagnostic tools NI resources B 1 documentation conventions used in the manual iv NI resources B 1 drivers NI resources B 1 E equality bounds tight 1 7 error bound 2 7 for balanced stochastic truncation 3 8 for balanced truncation 2 2 for impulse responses 2 3 for multiplicative Hankel reduction 3 16 for stochastic truncation 3 9 error formulas ophank 2 22 error reduction additive 2 1 frequency weighted 1 1 4 1 multiplicative 1 1 3 1 examples NI resources B 1 F for Hankel norm approximation 2 7 fracred 1 5 4 15 reduction 4 18 frequency weighted error reduction 1 1 4 1 controller reduction 4 2 Xmath Model Reduction Module Index G grammians controllability 1 7 description of 1 7 observability 1 7 H Hankel matrix 1 9 Hankel norm approximation 2 6 Hankel sin
110. singular values The error measures themselves deserve certain comments Xmath Model Reduction Module The two stability based measures Es and Eggs are derived from a sufficiency condition for stability rather than a necessity and sufficiency condition and so capture stability a little crudely For any constant nonsingular N the error measure 7 can be replaced by lace C PU CIN ls and the robustness result remains valid Use of an N may improve or worsen the quality of the approximation Having T T small normally ensures closeness of the closed loop impulse and step responses In classical control especially constraints on the loop gain can be imposed Minimum value of gain in one band maximum value of gain in another band for example None of the methods presented directly addresses the task of retaining satisfaction of these constraints after reduction of a high order acceptable controller However judicious use of a weight V can assist Suppose that above the closed loop bandwidth there is an overbound constraint on the loop gain which is violated when a controller reduction is performed but not with the original controller At these frequencies roughly PC and PC are small so that T T P C C Introduction of a weight V in Eys penalizing frequencies in the region in question will evidently encourage PC to better track PC 4 4 ni com Chapter 4 Frequency Weighted Error Reduction Fractional Representati
111. t permitted the algorithm checks for this The number of v equal to 1 is the number of zeros in Re s gt 0 of G s and as mentioned already these zeros remain as zeros of G s If error is specified then the error bound formula Equation 3 2 in conjunction with the v values from step 3 is used to define nsr for step 4 For nonsquare G with more columns than rows the error formula is 1 2 ie v aS 2 l v i nsr 1 IG G G G G G If the user is presented with the v the error formula provides a basis for intelligently choosing nsr However the error bound is not guaranteed to be tight except when nsr ns 1 Securing Zero Error at DC Xmath Model Reduction Module The error G G G as a function of frequency is always zero at When the algorithm is being used to approximate a high order plant by a low order plant it may be preferable to secure zero error at 0 A method for doing this is discussed in GrA90 for our purposes 1 We need a bilinear transformation of sys 1 z Given G s we generate H s through bilinsys makepoly b3 b4 makepoly b1 b2 sys subsys sys bilinsys 2 Reduce with the previous algorithm sr nsr hsv bst sys 3 Use the bilinear transformation s 1 z again sr1 nsr1 bilinear sr nsr 0 1 1 0 The v are the same for G s and H s G s The error bound formula is the same H is stable and H 0 H jo of full rank for all includ
112. t that info of degree k Ic E Gla Ok i G and there is an algorithm available for obtaining G Further the optimum G which is minimizing IG 2 G y does a reasonable job of minimizing ie Gl because it can be shown that IG Gl lt Y o j k 1 where n deg G with this bound subject to the proviso that G and G are allowed to be nonzero and different at s The bound on G Gl is one half that applying for balanced truncation However e It is actual error that is important in practice not bounds e The Hankel norm approximation does not give zero error at co or at 0 Balanced realization truncation gives zero error at and singular perturbation of a balanced realization gives zero error at 0 There is one further connection between optimum Hankel norm approximation and L error If one seeks to approximate G by a sum G F with G stable and of degree k and with F unstable then G G Fl 0 1 G info of degree k and F unstable O National Instruments Corporation 2 7 Xmath Model Reduction Module Chapter 2 Additive Error Reduction balmoore Further the G 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 Xmath Model Reduction Module SysR HSV T balmoore Sys nsr bound Thebalmoore function computes an inte
113. ten controllers of much lower order will result in acceptable performance and will be desired on account of their greater simplicity It is almost immediately evident that unweighted additive approximation of a controller will not necessarily ensure closeness of the behavior of the two closed loop systems formed from the original and reduced order controller together with the plant This behavior is dependent in part on the plant and so one would expect that a procedure for approximating controllers ought in some way to reflect the plant This can be done several ways as described in the Controller Robustness Result section The following result is a trivial variant of one in Vid85 dealing with robustness in the face of plant variations Controller Robustness Result Xmath Model Reduction Module Suppose that a controller C stabilizes a plant P and that C is a reduced order approximation to C with the same number of unstable poles Then C stabilizes P also provided CUa C Go PU U CU PUOT lt 1 or r PG CU PG LCUMC Go lt 1 An extrapolation to this thinking AnM89 suggests that C will be a good approximation to C from the viewpoint of some form of stability robustness if Ers CC PU CPy or E 9 I c c pu cpPy 4 2 ni com Chapter 4 Frequency Weighted Error Reduction is minimized and of course is less than 1 Notice that these two error measures are like those of Equation 4 1 and Eq
114. ter defined by b j0 a j0 pr gt G s G s values values Figure 3 3 Bilinear Mapping from G s to Gs Case 2 We can implement an arbitrary bilinear transform using the subsys _ function which substitutes a given transfer function for the s or z domain operator a s a To implement G s G ees use gtildesys subsys gsys makep b 1 makep 1 al i Sta To implement G s Gli F 2 use gsys subsys gtildesys makep b 1 makep 1 a 3 Note The systems substituted in the previous calls to subsys invert the function specification because these functions use backward polynomial rotation National Instruments Corporation 3 11 Xmath Model Reduction Module Chapter 3 Multiplicative Error Reduction Xmath Model Reduction Module Any zero or rank reduction on the j axis of G s becomes a zero or rank reduction in Re s gt 0 of G s and if G s has a zero or rank reduction at infinity this is shifted to a zero or rank reduction of G s at the point b in Re s gt 0 If all poles of G s are inside the circle of diameter b jO a j0 all poles of G s will be in Re s lt 0 and if G s has no zero or rank reduction on this circle G s will have no zero or rank reduction on the ja axis including oo If G s is nonsingular for almost all values of s it will be nonsingular or have no zero or rank reduction on the circle of diamet
115. test value mala lGjo coa ucu h However there are difficulties The principal one is that if we are reducing the plant without knowledge of the controller we cannot calculate the measure because we do not know C jq Nevertheless one could presume that for a well designed system GC Gc will be close to J over the operating bandwidth of the system and have smaller norm than 1 tending to zero as Wes in fact outside the operating bandwidth of the system This suggests that in the absence of knowledge of C one should carry out multiplicative approximation by seeking to minimize max A jo Ago 0 This is the prime rationale for unweighted multiplicative reduction of a plant Two other points should be noted First because frequencies well beyond the closed loop bandwidth GC GC will be small it is in a sense wasteful to seek to have A j small at very high frequencies The choice of maxa A j as the index is convenient because it removes a requirement to make assumptions about the controller but at the same time it does not allow A j to be made even smaller in the closed loop 3 2 ni com Chapter 3 Multiplicative Error Reduction bandwidth at the expense of being larger outside this bandwidth which would be preferable Second the previously used multiplicative error is G GG In the algorithms that follow the error G G G appears It is easy to check that o AGO Po and
116. tically convergent algorithm for computing its Leo norm Systems Control Letters Vol 15 pp 1 7 1990 S Boyd and C Barratt Linear Controller Design Limits of Performance Prentice Hall 1991 A E Bryson and Y C Ho Applied Optimal Control p 149 Blaisdell Publishing Co 1969 J C Doyle and G Stein Robustness with Observers IEEE Transactions on Automatic Control August 1979 J C Doyle and G Stein Multivariable Feedback Design Concepts for a Classical Modern Synthesis IEEE Transactions on Automatic Control Vol AC 26 February 1981 pp 4 16 National Instruments Corporation A 3 Xmath Model Reduction Module Appendix A Doy82 DWS82 FaT88 FaT86 Fr87 FPGM87 GD88 DGKF89 Gus0 ONR84 Osb60 Saf82 SD83 SD84 Bibliography J C Doyle Analysis of Feedback Systems with Structured Uncertainties IEEE Proceedings November 1982 J C Doyle J E Wall and G Stein Performance and Robustness Analysis for Structure Uncertainties Proceedings IEEE Conference on Decision and Control pp 629 636 1982 M K Fan and A L Tits m form Numerical Range and the Computation of the Structured Singular Value IEEE Transactions on Automatic Control Vol 33 pp 284 289 March 1988 M K Fan and A L Tits Characterization and Efficient Computation of the Structured Singular Value IEEE Transactions on Automatic Control
117. tions Additive Error Model Reduction Multiplicative Model Reduction Frequency Weighted Model Reduction balmoore redschur ophank truncate balance mreduce bst mulhank wtbalance fracred Utility Functions hankelsv stable compare Figure 1 1 MRM Function ni com Chapter 1 Introduction Certain restrictions regarding minimality and stability are required of the input data and are summarized in Table 1 1 Table 1 1 MRM Restrictions balance A stable minimal system balmoore A state space system must be stable and minimal having at least one input output and state bst A state space system must be linear continuous time and stable with full rank along the jo axis including infinity compare Must be a state space system fracred A state space system must be linear and continuous hankelsv A system must be linear and stable mreduce A submatrix of a matrix must be nonsingular for continuous systems and variant for discrete systems mulhank A state space system must be linear continuous time stable and square with full rank along the jo axis including infinity ophank A state space system must be linear continuous time and stable but can be nonminimal redschur A state space system must be stable and linear but can be nonminimal stable No restriction truncate Any
118. tiplicative Error Reduction Restrictions Algorithm Xmath Model Reduction Module The objective of the algorithm is to approximate a high order stable transfer function matrix G s by a lower order G s with either inv g g gr or g gr inv g minimized under the condition that G is stable and of the prescribed order This function has the following restrictions The user must ensure that the input system is stable and nonsingular at s infinity e The algorithm may be problematic if the input system has a zero on the JO axis e Only continuous systems are accepted for discrete systems use makecontinuous before calling bst then discretize the result Sys bst makecontinuous SysD SysD discretize Sys The modifications described in this section allow you to circumvent the previous restrictions The objective of the algorithm is to approximate a high order stable transfer function matrix G s by a lower order G s with in the square G s case either G G Gl or G G Gol approximately minimized under the constraint that G is stable and of prescribed order nsr In case G is not square but has full row rank the algorithm seeks to minimize co G G GG G G Recall that X s X s so that when s jo XJA X jo When G is not square but has full column rank the algorithm seeks to minimize G 6 G G G G co 3 4 ni com Chapter 3 Multiplicati
119. tisfying A B py eo eles Cul py i ID B B 0 C Be A 0 A B wc w is written as National Instruments Corporation 4 13 Xmath Model Reduction Module Chapter 4 Frequency Weighted Error Reduction Xmath Model Reduction Module and the observability grammian O defined in the obvious way is written as Occ Der It is trivial to verify that O A 4 0 C C so that Q is the observability gramian of C s alone as well as a submatrix of Q The weighted Hankel singular values of C s are the square roots of the eigenvalues of P O They differ from the usual or unweighted Hankel singular values because P is not the controllability gramian of C s but rather a weighted controllability gramian The usual controllability gramian can be regarded as E x x when C s is excited by white noise The weighted controllability gramian is still E xx but now C s is excited by colored noise that is the output of the shaping filter W s which is excited by white noise Small weighted Hankel singular values are a pointer to the possibility of eliminating states from C s without incurring a large error in CGo C G W No error bound formula is known however The actual reduction procedure is virtually the same as that of redschur except that P is used Thus Schur decompositions of P Q are formed with the eigenvalues in ascending and descending order VP Que Va Sic VP ocQec Vp Sdes The maximu
120. uation 4 2 The fact that the plant ought to show up in a good formulation of a controller reduction problem is evidenced by the appearance of P in the two weights It is instructive to consider the shape of the weighting matrix or function P 1 CP Consider the scalar plant case In the pass band IPCI is likely to be large and if this is achieved by having ICI large then IP CP will be approximately small Also outside the plant bandwidth IP CPY I will be small This means that it will be most likely to take its biggest values at frequencies near the unity gain cross over frequency This means that the approximation C is being forced to be more accurate near this frequency than well away from it a fact very much in accord with classical control where one learns the importance of good loop shaping round this frequency The above measures FE and Eos are advanced after a consideration of stability and the need for its preservation in approximating C by C If one takes the viewpoint that the important thing to preserve is the closed loop transfer function matrix a different error measure arises With T T denoting the closed loop transfer function matrices T T PCU PC PCI PC Then to a first order approximation in C C there holds T T I PO TPC C I PO The natural error measure is then Ey lU PCY PC CU P 4 4 and this error measure parallels Ez in Equation 4 3 Further refinement again is possible
121. uced order system is guaranteed to be stable Several other points concerning the error can be made e The error G j G jo as a function of frequency is not flat it is zero at and may take its largest value at 0 so that there is in general no matching of DC gains of the original and reduced system e The actual error may be considerably less than the error bound at all frequencies so that the error bound formula can be no more than an advance guide However the bound is tight when the dimension reduction is 1 and the reduction is of a continuous time transfer function matrix e With g and g denoting the impulse responses for impulse responses of G and G and with G of degree k the following L bound holds GCP88 g g l lt 442k 1 tr2 This bound also will apply for the L error on the step response It is helpful to note one situation where reduction is likely to be difficult so that 2 will contain few diagonal entries which are relatively very small Suppose G s strictly proper has degree n and has n 1 unstable zeros Then as runs from zero to infinity the phase of G s will change by 2n 1 n 2 Much of this change may occur in the passband Suppose G s has degree n 1 it can have no more than n 2 zeros since it is strictly National Instruments Corporation 2 3 Xmath Model Reduction Module Chapter 2 Additive Error Reduction proper So even if all zeros are
122. ular perturbation is exactly the other way round Perfect matching at DC can be a substantial advantage especially if input signals are known to be band limited Singular perturbation can be achieved with mreduce _ Figure 2 1 shows the two alternative approaches For both continuous time and discrete time reductions the end result is a balanced realization Hankel Norm Approximation Xmath Model Reduction Module In Hankel norm approximation one relies on the fact that if one chooses an approximation to exactly minimize one norm the Hankel norm then the infinity norm will be approximately minimized The Hankel norm is defined in the following way Let G s be a rational stable transfer 2 6 ni com Chapter 2 Additive Error Reduction function matrix Consider the way the associated impulse response maps inputs defined over c 0 in L into outputs and focus on the output over 0 c Define the input as u t for t lt 0 and set v t u Define the output as y t for t gt 0 Then the mapping is y t Cexpa e Bv r dr 0 if G s C sI A B The norm of the associated operator is the Hankel norm Gl of G A key result is that if o gt 0 gt are the Hankel singular values of G s then Gl To avoid minor confusion suppose that all Hankel singular values of G are distinct Then consider approximating G by some stable G of prescribed degree k much that lG a El is minimized It turns ou
123. unstable the maximum phase shift when moves from 0 to is 2n 3 x 2 It follows that if G w remains large in magnitude at frequencies when the phase shift has moved past 2n 3 n 2 approximation of G by G will necessarily be poor Put another way good approximation may depend somehow on removing roughly cancelling pole zeros pairs when there are no left half plane zeros there can be no rough cancellation and so approximation is unsatisfactory As a working rule of thumb if there are p right half plane zeros in the passband of a strictly proper G s reduction to a G s of order less than p 1 is likely to involve substantial errors For non strictly proper G s having p right half plane zeros means that reduction to a G s of order less than p is likely to involve substantial errors An all pass function exemplifies the problem there are n stable poles and n unstable zeros Since all singular values are 1 the error bound formula indicates for a reduction to order n 1 when it is not just a bound but exact a maximum error of 2 Another situation where poor approximation can arise is when a highly oscillatory system is to be replaced by a system with a real pole Reduction Through Balanced Realization Truncation This section briefly describes functions that reduce balance and truncate to achieve reduction balmoore Computes an internally balanced realization of a system and optionally trunc
124. ured as a device for reducing the violation of the specification one could introduce a weight V jw emphasizing frequencies from 0 1 radians per second to 5 radians per second For example Via s 0 1 s 10 s 1 s 1 4 This would tend to force the closed loop transfer functions derived from the full order and reduced controller to match better over this range because their absolute value is small there they are approximately equal to the open loop gains which accordingly may be close The flaw in this reasoning is that a second order controller with four independent parameters only can only do so much and the totality of designer demands cannot be fully met 6 12 ni com Chapter 6 Tutorial 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 title Open Loop Gain Using wtbalance Open Loop Gain Using wtbalance reduced original constrained 100 on o 100 Singular Value Magnitude dB an o 150 eoo 0 01 0 1 1 10 100 Frequency rad sec Figure 6 9 Open Loop Gain with wtbalance National Instruments Corporation 6 13 Xmath Model Reduction Module Chapter 6 Tutorial Generate Figure 6 10 syscl feedback sysol sysolr sys syscr sysclr feedback sysolr compare syscl sysclr w radians type 5 10 p
125. ut none of these is minimum phase bst and mulhank both require execution within the program of a spectral factorization the actual algorithm for achieving the spectral factorization depends on a Riccati equation The concepts of a spectrum and spectral factor also underpin aspects of wtbalance Xmath Model Reduction Module 1 14 ni com Chapter 1 Introduction 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 broad issues involved Most modern controller design methods for example LOG and H yield controllers of order roughly comparable with that of the plant It follows that to obtain a low order controller using such methods one must either follow a high order controller design by a controller reduction step or reduce an initially given high order plant model and then design a controller using the resulting low order plant with the understanding that the controller will actually be used on the high order plant Refer to Figure 1 2 High Order Plant m High Order Controller Plant Controller Reduction Reduction Low Order Plant Low Order Controller Figure 1 2 Low Order Controller Design for a High Order Plant Generally speaking in any design procedure it is better to postpone approximation to
126. values of G s and G s in terms of those of G s With G s of order n _ there holds 6 G S6 G k 1 2 2 The transfer function matrix G s being unstable does not have Hankel singular values however G s which is stable does have Hankel singular values They satisfy o 1G s lt op G In most cases the Hankel singular values of G s are distinct If accordingly G Gr Gyl o then G has degree i 1 G has degree ns i and G Glo 0 0 1 Ons 2 4 Observe that the bound Equation 2 3 or Equation 2 4 which is not necessarily obtained is one half that applying for both balanced truncation as implemented by balmoore or effectively by redschur it also is one half that obtained when applying mreduce to a balanced realization In general the D matrices of G and G are different that is G cc G ce in contrast to balmoore andredschur Similarly G 0 G 0 in general in contrast to the result when mreduce is applied to a balanced realization The price paid for obtaining a smaller error bound overall through Hankel norm reduction is that one no longer normally secures zero error at or 0 Two special cases should be noted If nsr 0 then G s is a constant only This constant can be added onto G s so that G s is then being approximated by a totally unstable transfer function matrix with error this type of approximation
127. ve Error Reduction 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 that right half plane zeros of G s remain as right half plane zeros of G s This means that if G s has order nsr with n zeros in Re s gt 0 G s must have degree at least n else given that it has n zeros in Re s gt 0 it would not be proper Gre88 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 I G j and then an all pass function the phase function W s G s This all pass function has a mixture of stable and unstable poles and it encodes the phase of G j Its stable part has n Hankel singular values 6 with lt 1 and the number of 6 equal to is the same as the number of zeros of G s in Re s gt 0 State variable realizations of W G and the stable part of W s G s can be connected in a nice way and when the stable part of W s G s has a balanced realization we say that the realization of G is stochastically balanced Truncating the balanced realization of the stable part of W s G s induces a corresponding truncation in the realization of G s and the truncated realization defines an approximation of G Further a good approximation of a transfer function encoding t
128. xy then Jy Or dt x40x 0 Systems that are easy to observe correspond to Q with large eigenvalues and thus large output energy when unforced lyapunov A B B produces P and lyapunov A C C produces O For a discrete time G z D C zI A B with l1 4 I lt 1 P and Q are P APA BB Q A QA CC The first dot point above remains valid Also P Y 4 BB A and Q YN 4 c ca k 0 k 0 with the sums being finite in case A is nilpotent which is the case if the transfer function matrix has a finite impulse response LAS A vec P vec BB lyapunov can be used to evaluate P and O Hankel Singular Values If P O are the controllability and observability grammians of a transfer function matrix in continuous or discrete time the Hankel Singular Values are the quantities PQ Notice the following Xmath Model Reduction Module All eigenvalues of PO are nonnegative and so are the Hankel singular values The Hankel singular values are independent of the realization used to calculate them when A B C D are replaced by TAT TB CT and D then P and Q are replaced by TPT and TY OT then PO is replaced by TPOT and the eigenvalues are unaltered The number of nonzero Hankel singular values is the order or McMillan degree of the transfer function matrix or the state dimension in a minimal realization 1 8 ni com Chapter 1 Introduction e Suppose the transfer function matrix corresp

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