{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Popup" -1 31 "" 0 0 0 128 128 1 1 0 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 256 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courie r" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Error" -1 8 1 {CSTYLE "" -1 -1 "Courier" 1 10 255 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output " -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE " " -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Author " -1 19 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 3 1 0 0 8 8 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Book Antiqua" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Title" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }1 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Auth or" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 8 8 1 0 1 0 2 2 0 1 }{PSTYLE "Author" -1 259 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 1 2 2 2 1 1 1 1 }3 1 0 0 8 8 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 260 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Title" -1 261 1 {CSTYLE "" -1 -1 "Times" 1 24 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Normal" -1 262 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 261 "" 0 "" {TEXT -1 77 "Identifiability of Linea r Time-Invariant Differential-Algebraic Systems - I: " }}{PARA 261 "" 0 "" {TEXT -1 41 "The Generalized Markov Parameter Approach" }}{PARA 19 "" 0 "" {TEXT -1 0 "" }}{PARA 18 "" 0 "" {TEXT 259 33 "Maple Sheet \+ for Selected Examples" }}{PARA 19 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 155 "******************* ********************************************************************** ******************************************************************" }} {PARA 257 "" 0 "" {TEXT -1 49 "Authors: Amos Ben-Zvi, Jim McLellan, K im McAuley" }}{PARA 257 "" 0 "" {TEXT -1 59 "Institution: Queen's Uni versity, Kingston, Ontario, Canada" }}{PARA 257 "" 0 "" {TEXT -1 26 "D ate: Janurary 29th, 2003" }}{PARA 257 "" 0 "" {TEXT -1 14 "Version: \+ 1.00" }{TEXT 258 1 " " }}{PARA 258 "" 0 "" {TEXT 257 34 "E-mail: mcle llanj@chee.queensu.ca" }}{PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 154 "*************************************************** ********************************************************************** *********************************" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT 256 432 "Sheet Abstract: This Maple sheet was generated to illustrate how the solutions to the examples found in Se ctions six and seven of the above paper were computed. It is intended as an illustration of the use of computer algebra software (in this c ase Maple Version 7) for generating the GMPs of an LTI DAE system. In addition, this sheet allows the reader to see in detail an implementa tion of the algorithm described in the paper." }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 205 "**** ********************************************************************** ********************************************************************** *************************************************************" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }{TEXT -1 0 "" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 31 31 "All \+ sheet variables are erased." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }{TEXT -1 0 "" }}{PARA 7 "" 1 " " {TEXT -1 80 "Warning, the protected names norm and trace have been r edefined and unprotected\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 31 124 "Pa ckage linalg and LinearAlgebra are loaded. These allow the defining o f the structures \"matrix\" and \"Matrix\" respectively." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with(LinearAlgebra ):" }}{PARA 7 "" 1 "" {TEXT -1 64 "Warning, the assigned name GramSchm idt now has a global binding\n" }}}{PARA 3 "" 0 "" {TEXT -1 18 "Proble m Definition" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 " " {TEXT 31 130 "****************************************************** ********************************************************************** ******" }}{PARA 0 "" 0 "" {TEXT 31 70 "The Matrices E, M, B and C from the example in Section 6 are entered. " }}{PARA 0 "" 0 "" {TEXT 31 131 "***************************************************************** ***************************************************************** " } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "E:=matr ix(3,3,[p1,0,0,0,1/p5,0,0,0,0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"EG-%'matrixG6#7%7%%#p1G\"\"!F+7%F+*&\"\"\"F.%#p5G!\"\"F+7%F+F+F+" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "M:=matrix(3,3,[0,1,0,p3,0,p 2,exp(p1),p4,0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG-%'matrixG6 #7%7%\"\"!\"\"\"F*7%%#p3GF*%#p2G7%-%$expG6#%#p1G%#p4GF*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "B:=matrix(3,1,[p1*p2,0,p6]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'matrixG6#7%7#*&%#p1G\"\"\"%#p 2GF,7#\"\"!7#%#p6G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "C:=ma trix(1,3,[0,0,p5]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CG-%'matrix G6#7#7%\"\"!F*%#p5G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 31 130 "********************************* ********************************************************************** ***************************" }}{PARA 0 "" 0 "" {TEXT 31 398 "The deter minant of the matrix pencil (lambda*E+M) is computed. Note that this \+ determinant is not identically zero as a function of the parameters. \+ As a result, the system is solvable for any value of the parameter vec tor [p1,p2,p3,p4,p5] such that the determinant of the matrix pencil i s nonsingular. This set of solvable parameter values forms an open an d dense subset of the parameter space. " }}{PARA 0 "" 0 "" {TEXT 31 131 "***************************************************************** ***************************************************************** " } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "eval(de t(evalm(lambda*E+M)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&**%'lambda G\"\"\"%#p1GF&%#p2GF&%#p4GF&!\"\"*&-%$expG6#F'F&F(F&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 31 130 "**************************************** ********************************************************************** ********************" }}{PARA 0 "" 0 "" {TEXT 31 128 "\"temp\" is a ma trix that left-multiplies the state equation of the system in order to get the new pencil (lambda[inv(M)*E] + I) " }}{PARA 0 "" 0 "" {TEXT 31 131 "************************************************************** ******************************************************************** \+ " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "temp:=evalm(inverse(M)&*E); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%tempG-%'matrixG6#7%7%,$*(%#p4G \"\"\"-%$expG6#%#p1G!\"\"F1F-F2\"\"!F37%F1F3F37%*,%#p3GF-F,F-F.F2%#p2G F2F1F-*&F-F-*&F8F-%#p5GF-F2F3" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 31 131 "********************************************************************* **************************************************************" }} {PARA 0 "" 0 "" {TEXT 31 298 "\"diag_mat\" a jordan block matrix that \+ is similar (by a non-singular matrix multiplication) to temp. The matr ix \"T\" used in the article is the inverse of the matrix \"Tin\" prod uced by the function \"jordan\". The fourth line in this section show s how the matrix \"T\", \"temp\" and \"diag_mat\" are related." } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 31 131 "************************ ********************************************************************** ************************************ " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "n:=rowdim(E);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "diag_mat:=jordan(tem p,'Tin');" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)diag_matG-%'matrixG6#7 %7%\"\"!\"\"\"F*7%F*F*F*7%F*F*,$*(%#p4GF+-%$expG6#%#p1G!\"\"F4F+F5" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "T:=expand(inverse(Tin));" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"TG-%'matrixG6#7%7%\"\"!,$*(%#p1G! \"\",&**)%#p4G\"\"#\"\"\"F-F4%#p3GF4%#p5GF4F4*$)-%$expG6#F-F3F4F.F4F9! \"#F.**F2F4F9F.%#p2GF4F6F47%F4*&F9F.F2F4F*7%F4F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "simplify(evalm((T)&*temp&*inverse(T)));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%\"\"!\"\"\"F(7%F(F(F(7 %F(F(,$*(%#p4GF)-%$expG6#,$%#p1G!\"\"F)F3F)F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{EXCHG {PARA 0 "" 0 "" {TEXT 31 130 "******************************* ********************************************************************** *****************************" }}{PARA 0 "" 0 "" {TEXT 31 183 "The mat rix \"T\" calculated above has the proerty of putting the matrix \"tem p\" into block-diagonal form. This is done by pre multipication by T and post-multiplication by inv(T). . " }}{PARA 0 "" 0 "" {TEXT 31 469 "To be consistant with the notation used in the paper, we wish the upper diagonal block of the matrix \"diag_mat\" to be the one that is full rank, and the lower diagonal block to be the nil-potent block. \+ This is achieved by pre-multiplying \"diag_mat\" by \"K\" and post mul tiplying by the inverse of \"K\". Note that the upper diagonal block (1-by-1) of \"H_J_diag\" is full rank on an open and dense subset of \+ the parameter space, while the lower left block is nil-potent. " }} {PARA 0 "" 0 "" {TEXT 31 131 "**************************************** ********************************************************************** ******************** " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "K:=matrix(3,3,[0,0,1,1,0,0,0,1,0]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "simplify(evalm(K&*T&*temp&*inverse(T)&*inverse(K )));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"KG-%'matrixG6#7%7%\"\"!F* \"\"\"7%F+F*F*7%F*F+F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6# 7%7%,$*(%#p4G\"\"\"-%$expG6#,$%#p1G!\"\"F+F0F+F1\"\"!F27%F2F2F+7%F2F2F 2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "H_J_diag:=evalm(K&*dia g_mat&*inverse(K));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)H_J_diagG-%' matrixG6#7%7%,$*(%#p4G\"\"\"-%$expG6#%#p1G!\"\"F1F-F2\"\"!F37%F3F3F-7% F3F3F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 31 130 "********************************************** ********************************************************************** **************" }}{PARA 0 "" 0 "" {TEXT 31 161 "The scalar \"r\" is si mply the size of the upper block-diagonal matrix of \"H_J_diag\". It \+ is used with the submatrix command to seperate the differential equati on " }}{PARA 0 "" 0 "" {TEXT 31 130 "part, corresponding to the upper \+ block diagonal matrix from the algebraic part corresponding to the bot tom block diagonal matrix. " }}{PARA 0 "" 0 "" {TEXT 31 131 "********* ********************************************************************** *************************************************** " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "r:=1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 31 130 "***************************************************************** *****************************************************************" }} {PARA 0 "" 0 "" {TEXT 31 161 "As shown in the paper, the matrix \"H\" \+ is nothing but the inverse of the upper block diagonal submatrix of \" H_J_diag\". In addition, the matrix \"J\" is simply the " }}{PARA 0 " " 0 "" {TEXT 31 108 "lower block diagnal submatrix of \"H_J_diag\". T hese matricies are used to generate the GMPs for this system." }} {PARA 0 "" 0 "" {TEXT 31 131 "**************************************** ********************************************************************** ******************** " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "H:=inverse(submatrix(H_J_diag,1..r,1..r));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"HG-%'matrixG6#7#7#,$*(%#p4G!\"\"-%$expG6 #%#p1G\"\"\"F1F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "J:=su bmatrix(H_J_diag,r+1..n,r+1..n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"JG-%'matrixG6#7$7$\"\"!\"\"\"7$F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 31 130 "******************************************************** ********************************************************************** ****" }}{PARA 0 "" 0 "" {TEXT 31 116 "The matrix \"B_new\" is the resu lt of left-multiplying the original system matrix \"B\" by the matrix \+ \"K*T(p)*inv(M)\". " }}{PARA 0 "" 0 "" {TEXT 31 131 "**************** ********************************************************************** ******************************************** " }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "B_new:=simplify((evalm((K)&* (T)&*evalm(inverse(M)&*B))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&B_ newG-%'matrixG6#7%7#,&**%#p4G\"\"\"-%$expG6#,$%#p1G!\"\"F-F2F-%#p2GF-F 3*&F.F-%#p6GF-F-7#,$*&-F/6#,$F2!\"#F-,&*&F4F--F/6#,$F2\"\"#F-F3**F,F-% #p5GF-%#p3GF-F6F-F-F-F37#F5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 31 130 "* ********************************************************************** ***********************************************************" }}{PARA 0 "" 0 "" {TEXT 31 231 "The matrix \"C_new\" is the value of the outpu t gain matrix \"C\" in the \"z\" coordinates. Recall that y(t) = C(p) x(t) = C(p)*inv(T)*inv(K)z(t). Also, we explicitly compute the new \" z\" cooridnates in terms of the old \"x\" coordinates. " }}{PARA 0 "" 0 "" {TEXT 31 131 "*************************************************** ********************************************************************** ********* " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "C_new:=map(simplify,(evalm(C&*inverse(T)&*inverse(K))));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&C_newG-%'matrixG6#7#7%,$**%#p1G!\" \",&**)%#p4G\"\"#\"\"\"F,F3%#p3GF3%#p5GF3F3-%$expG6#,$F,F2F-F3%#p2GF-F 1!\"#F-*(-F76#F,F3F:F-F1F-F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "z:=(evalm(K&*T&*[x1,x2,x3]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%\"zG-%'vectorG6#7%%#x1G,&**%#p1G!\"\",&**)%#p4G\"\"#\"\"\"F,F3%#p3G F3%#p5GF3F3*$)-%$expG6#F,F2F3F-F3F8!\"#%#x2GF3F-*,F1F3F8F-%#p2GF3F5F3% #x3GF3F3,&F)F3*(F8F-F1F3F " 0 "" {MPLTEXT 1 0 13 "b:=coldim(B);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"bG\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "B2:=submatrix(B_new,r+1..n,1..b);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#B2G-%'matrixG6#7$7#,$*&-%$expG6#,$%#p1G!\"#\"\"\", &*&%#p2GF2-F-6#,$F0\"\"#F2!\"\"**%#p4GF2%#p5GF2%#p3GF2%#p6GF2F2F2F:7#* &-F-6#,$F0F:F2F?F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "B1:=s implify(evalm(inverse(H)&*submatrix(B_new,1..r,1..b)));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#B1G-%'matrixG6#7#7#,$**%#p4G\"\"\"-%$expG6#,$ %#p1G!\"#F-F2F-,&*(F,F-F2F-%#p2GF-!\"\"%#p6GF-F-F7" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 31 130 "************************************************ ********************************************************************** ************" }}{PARA 0 "" 0 "" {TEXT 31 155 "Similar to what is done \+ for the input gain matrix \"B\", we also split the output gain matrix \+ \"C\" into two parts. The matrix \"C1\" denotes the effect of the " } }{PARA 0 "" 0 "" {TEXT 31 107 "dynamical variables on the output, whi le \"C2\" denotes the effect of the algebraic variables on the output. " }}{PARA 0 "" 0 "" {TEXT 31 131 "************************************ ********************************************************************** ************************ " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "crd:=rowdim(C);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$crdG\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "C2:=sub matrix(C_new,1..crd,r+1..n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#C2G -%'matrixG6#7#7$*(-%$expG6#%#p1G\"\"\"%#p2G!\"\"%#p4GF1**F.F1,&**)F2\" \"#F/F.F/%#p3GF/%#p5GF/F/*$)F+F7F/F1F/F0F1F2!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "C1:=submatrix(C_new,1..crd,1..r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#C1G-%'matrixG6#7#7#,$**%#p1G!\"\",&**)%#p 4G\"\"#\"\"\"F,F3%#p3GF3%#p5GF3F3*$)-%$expG6#F,F2F3F-F3%#p2GF-F1!\"#F- " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 31 130 "**************************** ********************************************************************** ********************************" }}{PARA 0 "" 0 "" {TEXT 31 167 "We n ow compute the generalized Markov parameters (GMPs). First, note tha t since the nilpotency of the matrix \"J\" is two, (i.e., J^2=0, but J is not the zero matrix)," }}{PARA 0 "" 0 "" {TEXT 31 31 " there are t wo GMPs associated " }}{PARA 0 "" 0 "" {TEXT 31 158 "with the algebrai c part. In addition, since the dimension of the dynamical part of the system is one there are (1*2=2) two GMPs associated with the dynamica l" }}{PARA 0 "" 0 "" {TEXT 31 75 " part of the system. Thus there are four GMPs in total for this system. " }}{PARA 0 "" 0 "" {TEXT 31 130 "***************************************************************** *****************************************************************" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "GMP:=vector(rowdim(J)+2*rowdim(H));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$GMPG-%&arrayG6$;\"\"\"\"\"%7\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "GMP[1]:=simplify(evalm(C2&*B2))[1,1];" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$GMPG6#\"\"\",$*,-%$expG6#%#p1GF',& *(%#p4GF'F-F'%#p2GF'!\"\"%#p6GF'F'F-F2F1F2F0!\"#F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "for i from 2 to (n-r) do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "GMP[i]:=(-1)^((n-r)-i)*simplify(evalm(C2&*evalm(J^ (i-1))&*B2)[1,1]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$GM PG6#\"\"#*(%#p2G!\"\"%#p4GF*%#p6G\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "GMP[3]: =expand(evalm(C1&*B1)[1,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$GM PG6#\"\"$,**,%#p4G\"\"#-%$expG6#%#p1G!\"#F/F+%#p3G\"\"\"%#p5GF2!\"\"*0 %#p2GF4F*F2F,F0F/F2F1F2F3F2%#p6GF2F2F/F2*(F6F4F*F4F7F2F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "for i from 1 to r do" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "GMP[i+3] :=expand(evalm(C1&*inverse(H)^i&*B1)[1,1]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$GMPG6#\"\"% ,**,-%$expG6#%#p1G!\"$F-\"\"$%#p4GF/%#p3G\"\"\"%#p5GF2F2*0%#p2G!\"\"F* F.F-\"\"#F0F7F1F2F3F2%#p6GF2F6*(F*F6F-F7F0F2F6**F5F6F*F6F-F2F8F2F2" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "evalm(GMP[1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*,-%$expG6#%#p1G\"\"\",&*(%#p4GF)F(F)%#p2GF)! \"\"%#p6GF)F)F(F.F-F.F,!\"#F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 31 130 "*********************** ********************************************************************** *************************************" }}{PARA 0 "" 0 "" {TEXT 31 211 "We now test the rank of the jacobian of the GMPs with respect to the \+ parameters. As expected this rank is four which is less then the tota l number of parameters, and therefore the system is not identifiable. \+ " }}{PARA 0 "" 0 "" {TEXT 31 130 "********************************** ********************************************************************** **************************" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "Jcob:=jacobian([GMP[1],GMP[2],GMP[3 ],GMP[4]],[p1,p2,p3,p4,p5,p6]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "rank(Jcob);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "#### ###################################################################### #####################" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 31 130 "******* ********************************************************************** *****************************************************" }}{PARA 0 "" 0 "" {TEXT 31 164 "This part of the sheet deals with the idealized gas-p hase reactor used in Section 7. This problem is more complex then the above problem. In the following lines, " }}{PARA 0 "" 0 "" {TEXT 31 160 "the matricies \"E\", \"M\", \"C\" and \"B\" are entered as before . Note that these entries replace the entries used by the above examp le in the memory of the computer." }}{PARA 0 "" 0 "" {TEXT 31 130 "*** ********************************************************************** *********************************************************" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "E:=matrix(9,9,0) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "for i from 1 to 4 do E [i,i]:=-V; od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "M:=matrix (9,9,0):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "M[1,1]:=-k1*V: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "M[2,1]:=k1*V:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "M[2,2]:=-k2*V:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "M[3,2]:=k2*V:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "for i from 6 to 8 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "M[i,i-5]:=-V/n_total;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "M[i,i-1]:=1;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "for i from 5 to 8 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "M[i-4 ,i]:=-F;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "M[5,i]:=1;" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 3 "od:" }}{PARA 8 "" 1 "" {TEXT -1 53 "Error, 2n d index, 5, larger than upper array bound 3\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "M[9,9]:=1: M[4,9]:=F:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalm(E);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'mat rixG6#7+7+,$%\"VG!\"\"\"\"!F+F+F+F+F+F+F+7+F+F(F+F+F+F+F+F+F+7+F+F+F(F +F+F+F+F+F+7+F+F+F+F(F+F+F+F+F+7+F+F+F+F+F+F+F+F+F+F/F/F/F/" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalm(M);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%'matrixG6#7+7+,$*&%#k1G\"\"\"%\"VGF+!\"\"\"\"!F.F., $%\"FGF-F.F.F.F.7+F),$*&%#k2GF+F,F+F-F.F.F.F/F.F.F.7+F.F3F.F.F.F.F/F.F .7+F.F.F.F.F.F.F.F/F07+F.F.F.F.F+F+F+F+F.7+,$*&F,F+%(n_totalGF-F-F.F.F .F+F.F.F.F.7+F.F9F.F.F.F+F.F.F.7+F.F.F9F.F.F.F+F.F.7+F.F.F.F.F.F.F.F.F +" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "rank(M);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#\"\")" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 31 130 "* ********************************************************************** ***********************************************************" }}{PARA 0 "" 0 "" {TEXT 31 246 "Once again, we check the determinant of the ma trix pencil (lambda*E+M). Note that for this example, the matrix penc il may be factored into monomials. Also, note that the determinant is nonzero on an open and dense subset of the parameter space." }}{PARA 0 "" 0 "" {TEXT 31 130 "********************************************** ********************************************************************** **************" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "factor(det(lambda*E+M));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*.%\"VG\"\"%%'lambdaG\"\"\",&*&%(n_totalGF(F'F(F(%\"F GF(F(,(F*F(*&F+F(%#k2GF(F(F,F(F(,(F,F(F*F(*&F+F(%#k1GF(F(F(F+!\"$!\"\" " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "B:=matrix(9,1,[F,0,0,0, 0,0,0,0,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'matrixG6#7+7# %\"FG7#\"\"!F+F+F+F+F+F+7#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "C:=matrix(1,9,[0,0,1,0,0,0,0,0,0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CG-%'matrixG6#7#7+\"\"!F*\"\"\"F*F*F*F*F*F*" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 31 130 "********************************* ********************************************************************** ***************************" }}{PARA 0 "" 0 "" {TEXT 31 102 "This exam ple is solvable. As a result, the matrix (E+c*M) is nonsingular for a lmost any value of c. " }}{PARA 0 "" 0 "" {TEXT 31 130 "************* ********************************************************************** ***********************************************" }{MPLTEXT 1 0 0 "" }} }{EXCHG {PARA 0 "" 0 "" {TEXT 31 130 "******************************** ********************************************************************** ****************************" }}{PARA 0 "" 0 "" {TEXT 31 114 "The matr ix (E+cM) is nonsingular, and is used to tranform the matrix pencil to the form (lambda*I+(1-lambda*c*M1)." }}{PARA 0 "" 0 "" {TEXT 31 130 " ********************************************************************** ************************************************************" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "M1:=eva lm(inverse(E*c+M)&*E);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#M1G-%'mat rixG6#7+7+*&%(n_totalG\"\"\",(*&%\"cGF,F+F,F,*&F+F,%#k1GF,F,%\"FGF,!\" \"\"\"!F4F4F4F4F4F4F47+**F1F,F+\"\"#,(F.F,*&F+F,%#k2GF,F,F2F,F3F-F3*&F +F,F8F3F4F4F4F4F4F4F47+*.F1F,F:F,,&F.F,F2F,F3F8F3F-F3F+\"\"$**F:F,F+F7 F>F3F8F3*&F+F,F>F3F4F4F4F4F4F47+*(F2F,F/F3F>F3FCFC*&F,F,F/F3F4F4F4F4F4 7+*&F-F3%\"VGF,F4F4F4F4F4F4F4F47+*,F1F,F+F,F8F3F-F3FGF,*&F8F3FGF,F4F4F 4F4F4F4F47+*0F1F,F:F,F+F7F>F3F8F3F-F3FGF,*,F:F,F+F,F>F3F8F3FGF,*&F>F3F GF,F4F4F4F4F4F47+,$FNF3FPFPF4F4F4F4F4F47+F4F4F4F4F4F4F4F4F4" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 31 130 "********************************* ********************************************************************** ***************************" }}{PARA 0 "" 0 "" {TEXT 31 178 "This line below uses a function called \"jordan\" to compute the Jordan form of the matrix \"M1\". In addition, the subroutine returns the matrix \" T\" such that T*M1_jord*inv(T)=M1. " }}{PARA 0 "" 0 "" {TEXT 31 167 " \+ Note that at this point, this matrix \"T\" is the inverse of the matri x \"T\" used in the paper. As a result, we set \"T=inv(T)\" in order \+ to be consistant with the paper." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 262 "" 0 "" {TEXT -1 531 "It is very important to note that Mapl e does not always return the matrix M1_jord using the same Jordan bloc k order. As a result, the when running this worksheet sometimes the m atrix \"K\" (discussed below) must be re-typed so that the \"1/c\" eig envalues make up the bottom diagonal elements of \"K*M1_jord*inverse(K )\". Alternatively, the command \"jordan(evalm(M1),T)\" can be run se veral times until Maple chooses a Jordan form that is compatible with \+ the tranfomration matrix \"K\" below. All of this has no effect on th e final GMPs." }}{PARA 0 "" 0 "" {TEXT 31 4 " " }}{PARA 0 "" 0 "" {TEXT 31 130 "******************************************************** ********************************************************************** ****" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 " M1_jord:=jordan(evalm(M1),T);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(M1 _jordG-%'matrixG6#7+7+\"\"!F*F*F*F*F*F*F*F*7+F**&%(n_totalG\"\"\",(*&% \"cGF.F-F.F.*&F-F.%#k1GF.F.%\"FGF.!\"\"F*F*F*F*F*F*F*7+F*F**&F-F.,&F0F .F4F.F5F*F*F*F*F*F*7+F*F*F**&F-F.,(F0F.*&F-F.%#k2GF.F.F4F.F5F*F*F*F*F* 7+F*F*F*F**&F.F.F1F5F*F*F*F*F)F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "T:=inverse(T);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"TG-%'matrixG6#7+7+\"\"\"\"\"!F+F+,$*&%(n_totalGF*%\"VG!\"\"F0F+F+F+F +7+F*F+F+F+F+F+F+F+F+7+F*F*F*F+F+F+F+F+F+7+F**&%#k1GF0,&%#k2GF0F5F*F*F +F+F+F+F+F+F+7+F*F*F*F*F+F+F+F+F+7+F0,$*&F/F*F.F0F0F+F+F-F*F+F+F+7+F+F ;F:F+F+F0F*F+F+7+F+F;,$F;\"\"#F+F*F+F0F*F+7+F+F:F:F+F0F+F+F0F*" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 31 130 "********************************* ********************************************************************** ***************************" }}{PARA 0 "" 0 "" {TEXT 31 154 "The matri x \"M1_jord\" which is the Jordan form of the matrix \"M1\" is unique \+ up to a permuation of the Jordan blocks. We use the matrix \"K\" to p ermute the " }}{PARA 0 "" 0 "" {TEXT 31 160 "jordan blocks so that the diagonal elements corresponding to \"1/c\" are all in the bottom left block diagonal submatrix of \"diag_h_j\". The matrix \"diag_h_j\" is " }}{PARA 0 "" 0 "" {TEXT 31 161 "the one appearing in Equation 7.4 o f the paper. It corresponds to begin a block diagonal matrix whose up per diagonal submatrix is called bar(H) in the paper and" }}{PARA 0 " " 0 "" {TEXT 31 60 " the lower diagonal submatrix is called bar(J) in \+ the paper." }}{PARA 0 "" 0 "" {TEXT 31 130 "************************** ********************************************************************** **********************************" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 197 "K:=matrix(9,9,[[0,1,0,0,0,0,0,0,0] ,[0,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[1,0,0,0, 0,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0] ,[0,0,0,0,0,0,0,0,1]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"KG-%'ma trixG6#7+7+\"\"!\"\"\"F*F*F*F*F*F*F*7+F*F*F+F*F*F*F*F*F*7+F*F*F*F+F*F* F*F*F*7+F*F*F*F*F+F*F*F*F*7+F+F*F*F*F*F*F*F*F*7+F*F*F*F*F*F+F*F*F*7+F* F*F*F*F*F*F+F*F*7+F*F*F*F*F*F*F*F+F*7+F*F*F*F*F*F*F*F*F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalm(T);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7+7+\"\"\"\"\"!F)F),$*&%(n_totalGF(%\"VG!\" \"F.F)F)F)F)7+F(F)F)F)F)F)F)F)F)7+F(F(F(F)F)F)F)F)F)7+F(*&%#k1GF.,&%#k 2GF.F3F(F(F)F)F)F)F)F)F)7+F(F(F(F(F)F)F)F)F)7+F.,$*&F-F(F,F.F.F)F)F+F( F)F)F)7+F)F9F8F)F)F.F(F)F)7+F)F9,$F9\"\"#F)F(F)F.F(F)7+F)F8F8F)F.F)F)F .F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "diag_h_j:=evalm(K&*M1_ jord&*inverse(K));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)diag_h_jG-%'m atrixG6#7+7+*&%(n_totalG\"\"\",(*&%\"cGF,F+F,F,*&F+F,%#k1GF,F,%\"FGF,! \"\"\"\"!F4F4F4F4F4F4F47+F4*&F+F,,&F.F,F2F,F3F4F4F4F4F4F4F47+F4F4*&F+F ,,(F.F,*&F+F,%#k2GF,F,F2F,F3F4F4F4F4F4F47+F4F4F4*&F,F,F/F3F4F4F4F4F47+ F4F4F4F4F4F4F4F4F4F?F?F?F?" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 31 130 "** ********************************************************************** **********************************************************" }}{PARA 0 "" 0 "" {TEXT 31 140 "We now compute the matrices \"H_bar\", and \"J_b ar\" which are the upper and lower digaonal block diagonal matricies o f \"diag_h_j\" respectively." }}{PARA 0 "" 0 "" {TEXT 31 130 "******** ********************************************************************** ****************************************************" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "H_bar:=submatrix(diag_ h_j,1..rank(E),1..rank(E));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&H_ba rG-%'matrixG6#7&7&*&%(n_totalG\"\"\",(*&%\"cGF,F+F,F,*&F+F,%#k1GF,F,% \"FGF,!\"\"\"\"!F4F47&F4*&F+F,,&F.F,F2F,F3F4F47&F4F4*&F+F,,(F.F,*&F+F, %#k2GF,F,F2F,F3F47&F4F4F4*&F,F,F/F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "J_bar:=submatrix(diag_h_j,rank(E)+1..rowdim(diag_h_j) ,rank(E)+1..rowdim(diag_h_j));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&J _barG-%'matrixG6#7'7'\"\"!F*F*F*F*F)F)F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 31 130 "******************************************************** ********************************************************************** ****" }}{PARA 0 "" 0 "" {TEXT 31 94 "We now compute the matrix \"H\" \+ using the matrix [I-c*H_bar] called \"H_temp\" in this worksheet." }} {PARA 0 "" 0 "" {TEXT 31 130 "**************************************** ********************************************************************** ********************" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "H_temp:=evalm(Matrix(rank(E),rank(E),shape=identity)- c*H_bar);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'H_tempG-%'matrixG6#7&7 &,&*(%\"cG\"\"\"%(n_totalGF-,(*&F,F-F.F-F-*&F.F-%#k1GF-F-%\"FGF-!\"\"F 4F-F-\"\"!F5F57&F5,&*(F,F-F.F-,&F0F-F3F-F4F4F-F-F5F57&F5F5,&*(F,F-F.F- ,(F0F-*&F.F-%#k2GF-F-F3F-F4F4F-F-F57&F5F5F5F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "H:=simplify(evalm(inverse(H_bar)&*H_temp));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"HG-%'matrixG6#7&7&*&,&*&%(n_totalG \"\"\"%#k1GF.F.%\"FGF.F.F-!\"\"\"\"!F2F27&F2*&F0F.F-F1F2F27&F2F2*&,&*& F-F.%#k2GF.F.F0F.F.F-F1F27&F2F2F2F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 31 130 "************ ********************************************************************** ************************************************" }}{PARA 0 "" 0 "" {TEXT 31 95 "We now compute the matrix \"J\" using the matrix [I-c*J_ bar] called \"J_temp\" in this worksheet. " }}{PARA 0 "" 0 "" {TEXT 31 130 "************************************************************** ********************************************************************" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "J_temp :=(evalm(Matrix(rowdim(diag_h_j)-rank(E),rowdim(diag_h_j)-rank(E),shap e=identity)-J_bar*c));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'J_tempG-% 'matrixG6#7'7'\"\"\"\"\"!F+F+F+7'F+F*F+F+F+7'F+F+F*F+F+7'F+F+F+F*F+7'F +F+F+F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "J:=evalm(inver se(J_temp)&*J_bar);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"JG-%'matrix G6#7'7'\"\"!F*F*F*F*F)F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 31 130 "*********************** ********************************************************************** *************************************" }}{PARA 0 "" 0 "" {TEXT 31 153 "The matrix \"B_new\" is computed using \"B_new\" = \"K*T*inv(E+M)*B\" . It should be noted that this matrix is related to the matricies \"B _1\" and \"B_2\" by pre-" }}{PARA 0 "" 0 "" {TEXT 31 117 "multiplicati on by the block diagonal matrix whose diagonal elements are \"inv(H_te mp)\" and \"inv(bar(J))\" respectively." }}{PARA 0 "" 0 "" {TEXT 31 130 "***************************************************************** *****************************************************************" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "B_new:=simplify(evalm(K&*sim plify(evalm((T)))&*inverse(E+M)&*B));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&B_newG-%'matrixG6#7+7#,$**%\"VG!\"\"%\"FG\"\"\"%(n_totalGF/,(F0F /*&F0F/%#k1GF/F/F.F/F-F-7#,$**F,F-F.F/F0F/,&F0F/F.F/F-F-7#,$**F,F-F.F/ F0F/,(F0F/*&F0F/%#k2GF/F/F.F/F-F-7#\"\"!F>F>F>F>7#F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "B1:=subs(c=1,simplify(evalm(inverse(H_bar )&*submatrix(B_new,1..4,1..1))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#B1G-%'matrixG6#7&7#,$*&%\"VG!\"\"%\"FG\"\"\"F-F)F)7#\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "B2:=subs(c=1,simplify(evalm( inverse(J_temp)&*submatrix(B_new,5..9,1..1))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#B2G-%'matrixG6#7'7#\"\"!F)F)F)7#\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 31 130 "**************************************** ********************************************************************** ********************" }}{PARA 0 "" 0 "" {TEXT 31 150 "The matrix \"C_n ew\" is computed using \"C_new\" = \"C*inv(T)*inv(K)\". The submatri ces \"C1\" and \"C2\" are the output gains corresponding to the dynami cal " }}{PARA 0 "" 0 "" {TEXT 31 35 "and algebraic states, repsectivel y." }}{PARA 0 "" 0 "" {TEXT 31 128 "********************************** ********************************************************************** ************************" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "C_new:=evalm(C&*(inverse(T))&*inverse(K));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&C_newG-%'matrixG6#7#7+*&%#k2G\"\"\",&F+!\"\"%#k1GF,F .F,,$*&F/F,F-F.F.\"\"!F2F2F2F2F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "C1:=submatrix(C_new,1..1,1..rowdim(H));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#C1G-%'matrixG6#7#7&*&%#k2G\"\"\",&F+!\"\"%#k1 GF,F.F,,$*&F/F,F-F.F.\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "C2:=submatrix(C_new,1..1,rowdim(H)+1..rowdim(E));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#C2G-%'matrixG6#7#7'\"\"!F*F*F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 31 130 "**************************************** ********************************************************************** ********************" }}{PARA 0 "" 0 "" {TEXT 31 156 "We now compute t he generalized Markov parameters (GMPs). First, note that since the \+ nilpotency of the matrix \"J\" is one, there is only one GMP associate d " }}{PARA 0 "" 0 "" {TEXT 31 160 "with the algebraic part. In addit ion, since the dimension of the dynamical part of the system is four t here are (4*2=8) four GMPs associated with the dynamical" }}{PARA 0 " " 0 "" {TEXT 31 160 " part of the system. Thus there are nine GMPs in total for this system. The scalar \"j\" is an index used to test whe ther the matrix \"J\" to the power of \"j\" is " }}{PARA 0 "" 0 "" {TEXT 31 161 "identically zero. If the rank of \"J\" to the \"j\"th p ower is zero, then the matrix has zero for every entry and any GMPs ge nerated by this or higher powers of \"J\"" }}{PARA 0 "" 0 "" {TEXT 31 15 " will be zero. " }}{PARA 0 "" 0 "" {TEXT 31 130 "***************** ********************************************************************** *******************************************" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 31 130 "********************** ********************************************************************** **************************************" }}{PARA 0 "" 0 "" {TEXT 31 158 "The scaler \"theta\" is used to remember how many algebraic GMP w ere generated for this system. In this case the value of theta is one . The algorithm now goes" }}{PARA 0 "" 0 "" {TEXT 31 157 " and places the remaining GMPs in the \"theta+i\"th entry of the vector \"GMP\". \+ The scalar \"i\" is used to denote the element of the vector \"GMP\" \+ which is being" }}{PARA 0 "" 0 "" {TEXT 31 170 " currently computed. \+ If the value of \"i\" is theta (as it must be for the first time thr ough the loop) the matrix \"Hi\" (which is the identity matrix) is use d to compute" }}{PARA 0 "" 0 "" {TEXT 31 163 " the GMP. If \"i\" is s trictly greater then theta then succesivley higher powers of \"H\" are used to compute the GMP. Finally after all entries of the vector \"G MP\" " }}{PARA 0 "" 0 "" {TEXT 31 169 "are computed, the vector is pri nted to the screen. This vector is used for identifiability analysis \+ in the paper. Note that we substituted a value of 1 for the scalar " }}{PARA 0 "" 0 "" {TEXT 31 153 "\"c\". We could have used any scalar \+ such that the matrix pencil E+cM is non singular. The interested rea der is reffered to the book \"Matrix Theory\" by " }}{PARA 0 "" 0 "" {TEXT 31 79 "Gantmacher, Volume II, Chapter XII (1960, Chelsa Publishi ng Company, New York)." }}{PARA 0 "" 0 "" {TEXT 31 131 "************** ********************************************************************** ********************************************** " }{MPLTEXT 1 0 0 "" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "GMP:=vector(9); GMP[1]:=evalm(C2&*B2)[1,1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$GMPG-%&arrayG6$;\"\"\"\"\"*7\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$GMPG6#\"\"\"\"\"!" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 " j:=1: test:=evalm(J):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "while(ra nk(test)>0 and j < n) do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "test:=e valm(test&*J);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "j:=j+1;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "od; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "theta:=evalm(j);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%&thetaG\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "for i f rom theta to 2*4 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "Hi:=ev alm(Matrix(4,4,shape=identity));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "if (i>theta) then " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "for j from 2 to i do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Hi:=evalm(Hi&*H);" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "end if;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "GMP[i+1]:=(evalm(C1 &*Hi&*B1)[1,1]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "od: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "GMP:=map(simplify,map(evalm,subs(c=1,evalm( GMP))));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$GMPG-%'vectorG6#7+\"\"!F)F),$**%#k1G\"\"\"%#k2GF-%\" FGF-%\"VG!\"\"F1,$*.,(*&%(n_totalGF-F,F-F-*&\"\"$F-F/F-F-*&F6F-F.F-F-F -F,F-F6F1F.F-F/F-F0F1F1,$*.,.*&)F6\"\"#F-)F,F?F-F-**\"\"%F-F6F-F,F-F/F -F-*(F.F-F>F-F,F-F-*&\"\"'F-)F/F?F-F-**FBF-F6F-F.F-F/F-F-*&F>F-)F.F?F- F-F-F,F-F6!\"#F.F-F/F-F0F1F1,$*.,6*&)F6F8F-)F,F8F-F-**\"\"&F-F>F-F@F-F /F-F-*(F@F-FOF-F.F-F-**\"#5F-F6F-F,F-FFF-F-*,FRF-F,F-F>F-F.F-F/F-F-*(F ,F-FOF-FIF-F-*&FUF-)F/F8F-F-**FUF-F6F-F.F-FFF-F-**FRF-F>F-FIF-F/F-F-*& FOF-)F.F8F-F-F-F,F-F6!\"$F.F-F/F-F0F1F1,$*.,@*&)F6FBF-)F,FBF-F-**FEF-F OF-FPF-F/F-F-*(FPF-F]oF-F.F-F-**\"#:F-F>F-F@F-FFF-F-*,FEF-F@F-F.F-FOF- F/F-F-*(F@F-F]oF-FIF-F-**\"#?F-F6F-F,F-FYF-F-*,FboF-F,F-F.F-F>F-FFF-F- *,FEF-F,F-FIF-FOF-F/F-F-*(F,F-F]oF-FgnF-F-*&FboF-)F/FBF-F-**FfoF-F6F-F .F-FYF-F-**FboF-F>F-FIF-FFF-F-**FEF-FOF-FgnF-F/F-F-*&F]oF-)F.FBF-F-F-F ,F-F6!\"%F.F-F/F-F0F1F1,$*.,L*&)F6FRF-)F,FRF-F-**\"\"(F-F]oF-F^oF-F/F- F-*(F^oF-FfpF-F.F-F-**\"#@F-FOF-FPF-FFF-F-*,FipF-FPF-F.F-F]oF-F/F-F-*( FPF-FfpF-FIF-F-**\"#NF-F>F-F@F-FYF-F-*,F\\qF-F@F-F.F-FOF-FFF-F-*,FipF- F@F-FIF-F]oF-F/F-F-*(F@F-FfpF-FgnF-F-**F`qF-F6F-F,F-F[pF-F-*,F`qF-F,F- F.F-F>F-FYF-F-*,F\\qF-F,F-FIF-FOF-FFF-F-*,FipF-F,F-FgnF-F]oF-F/F-F-*(F ,F-FfpF-F`pF-F-*&F\\qF-)F/FRF-F-**F`qF-F6F-F.F-F[pF-F-**F`qF-F>F-FIF-F YF-F-**F\\qF-FOF-FgnF-FFF-F-**FipF-F]oF-F`pF-F/F-F-*&FfpF-)F.FRF-F-F-F ,F-F6!\"&F.F-F/F-F0F1F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 31 130 "***** ********************************************************************** *******************************************************" }}{PARA 0 "" 0 "" {TEXT 31 155 "This section of the worksheet is used to show that \+ when the parameters \"k1\" and \"k2\" are interchanged, the values of \+ the GMPs do not change. The function " }}{PARA 0 "" 0 "" {TEXT 31 155 "\"test\" is used to show what we intend to do with every element \+ of the vector \"GMP\". We simply substitute \"k_temp\" for \"k2\", wh ere \"k_temp\" is a temporary " }}{PARA 0 "" 0 "" {TEXT 31 151 "place \+ holder. We then replace \"k2\" by \"k1\" and finally, we replace \"k_ temp\" by \"k1\". The vector \"GMP_2\" has the elements of \"GMP\" wi th \"k1\" and \"k2\" " }}{PARA 0 "" 0 "" {TEXT 31 153 "switched. We t hen subtract \"GMP2\" from \"GMP\" and use the command \"simplify\" on the result. As can be seen, the result is the zero vector, we theref ore " }}{PARA 0 "" 0 "" {TEXT 31 30 " conclude that \"GMP\"=\"GMP_2\" ." }}{PARA 0 "" 0 "" {TEXT 31 131 "*********************************** ********************************************************************** ************************* " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "te st:=k1/k2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%testG*&%#k1G\"\"\"%#k 2G!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "test := k1/k2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%testG*&%#k1G\"\"\"%#k2G!\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "test:=subs(k_temp=k1,subs(\{ k1=k2\},subs(\{k2=k_temp\},evalm(test))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%testG*&%#k2G\"\"\"%#k1G!\"\"" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 66 "GMP_2:=subs(k_temp=k1,subs(\{k1=k2\},subs(\{k2 =k_temp\},evalm(GMP))));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&GMP_2G- %'vectorG6#7+\"\"!F)F),$**%#k1G\"\"\"%#k2GF-%\"FGF-%\"VG!\"\"F1,$*.,(* &%(n_totalGF-F,F-F-*&\"\"$F-F/F-F-*&F6F-F.F-F-F-F,F-F6F1F.F-F/F-F0F1F1 ,$*.,.*&)F6\"\"#F-)F,F?F-F-**\"\"%F-F6F-F,F-F/F-F-*(F.F-F>F-F,F-F-*&\" \"'F-)F/F?F-F-**FBF-F6F-F.F-F/F-F-*&F>F-)F.F?F-F-F-F,F-F6!\"#F.F-F/F-F 0F1F1,$*.,6*&)F6F8F-)F,F8F-F-**\"\"&F-F>F-F@F-F/F-F-*(F@F-FOF-F.F-F-** \"#5F-F6F-F,F-FFF-F-*,FRF-F,F-F>F-F.F-F/F-F-*(F,F-FOF-FIF-F-*&FUF-)F/F 8F-F-**FUF-F6F-F.F-FFF-F-**FRF-F>F-FIF-F/F-F-*&FOF-)F.F8F-F-F-F,F-F6! \"$F.F-F/F-F0F1F1,$*.,@*&)F6FBF-)F,FBF-F-**FEF-FOF-FPF-F/F-F-*(FPF-F]o F-F.F-F-**\"#:F-F>F-F@F-FFF-F-*,FEF-F@F-F.F-FOF-F/F-F-*(F@F-F]oF-FIF-F -**\"#?F-F6F-F,F-FYF-F-*,FboF-F,F-F.F-F>F-FFF-F-*,FEF-F,F-FIF-FOF-F/F- F-*(F,F-F]oF-FgnF-F-*&FboF-)F/FBF-F-**FfoF-F6F-F.F-FYF-F-**FboF-F>F-FI F-FFF-F-**FEF-FOF-FgnF-F/F-F-*&F]oF-)F.FBF-F-F-F,F-F6!\"%F.F-F/F-F0F1F 1,$*.,L*&)F6FRF-)F,FRF-F-**\"\"(F-F]oF-F^oF-F/F-F-*(F^oF-FfpF-F.F-F-** \"#@F-FOF-FPF-FFF-F-*,FipF-FPF-F.F-F]oF-F/F-F-*(FPF-FfpF-FIF-F-**\"#NF -F>F-F@F-FYF-F-*,F\\qF-F@F-F.F-FOF-FFF-F-*,FipF-F@F-FIF-F]oF-F/F-F-*(F @F-FfpF-FgnF-F-**F`qF-F6F-F,F-F[pF-F-*,F`qF-F,F-F.F-F>F-FYF-F-*,F\\qF- F,F-FIF-FOF-FFF-F-*,FipF-F,F-FgnF-F]oF-F/F-F-*(F,F-FfpF-F`pF-F-*&F\\qF -)F/FRF-F-**F`qF-F6F-F.F-F[pF-F-**F`qF-F>F-FIF-FYF-F-**F\\qF-FOF-FgnF- FFF-F-**FipF-F]oF-F`pF-F/F-F-*&FfpF-)F.FRF-F-F-F,F-F6!\"&F.F-F/F-F0F1F 1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "simplify(evalm(GMP-GMP _2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7+\"\"!F'F'F'F'F' F'F'F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 31 130 "********************** ********************************************************************** **************************************" }}{PARA 260 "" 0 "" {TEXT 31 103 "***********************************This is the end of the workshe et.***********************************" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 31 131 "***************************************************** ********************************************************************** ******* " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "60 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }