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1. click LOAD and load the string called demolmi 15 s You can save this description in a MATLAB string of your choice SAVE Click SAVE and type demolmi2 as the name of the string who demolmi2 generate the internal representation lmisys of this LMI System by typing lmisys2 as the name of the LMI system string and clicking on CREATE gt gt who Your variables are A S ans lmisys2 B X demolmi C ZZZ ehdl demolmi2 visualize the LMIVAR and LMITERM commands that create Imisys click on VIEW COMMANDS 16 E Figure 1 LMI Editor J describe the matrix var setlist M lmivar 1 6 1 Selmivart1 2 02 1 fe describe the Lills az MATLAB exe view commands A X X amp CPSC NPB BPM S lt 0 MeO S21 ee ete ee setlmis X lmivar 1 6 11 S lmivar 1 2 0 2 11 write in a file this series of commands click on WRITE Click on CLOSE to exit LMIEDIT 6 17 Example 8 2 EXAMPLE 8 2 IN THE Old LMI USER S MANUAL or IN Chapter 9 of the Robust Control Toolbox Manual oe oe oe A 1 2 1 3 2 1 1 2 1 Bs 1 0 11 Q 1 1 0 1 3 12 0 12 36 Consider the optimization problem Minimize Trace X subject to A X XA XBB X Q lt 0 9 9 It can be shown that the minimizer X is simply the stabilizing solution of the algebraic Riccati equation A X XA XBB X Q 0 This solution can be computed directly with the Riccati solv2. E MEE if and only if 4 3b Q gt 0 and R S Q S 0 Proof LMI Examples Ex 1 Z x e R depends affinely on x and Z x G Z x Then Z lt 1 ZWZ lt I ie I Z x Z x gt 0 I pu 20 Z x I Proof Schur complement a Ex 2 c x e R P x P x e R depends affinely on x c G P x e x 1 P x gt 0 P x b 20 c x 1 Proof Schur complement b Then Ex 3 P x P x e R and S x e R depend affinely on x Tr S OP Q Q 1 PG 20 X o Tr X l1 S x P x ba eR Proof Then Ex 4 Convert the quadratic matrix inequality Riccati inequality into an LMI The Riccati inequality A P PA PBR B P Q lt 0 where A B Q Q R R 0 are given matrices and P P is the variable is equivalent to the following LMI A P PA PB s Q gt 0 BP R Proof Linear Matrix Inequalities ov LMI LAB DEMO EXAMPLE 8 1 IN THE Old LMI USER S MANUAL or IN Chapter 9 of the new Robust Control Toolbox Manual oe oe oe Author P Gahinet Copyright 1995 2004 The MathWorks Inc SRevision 1 1 6 1 o ol load lmidem gt gt who gt gt A B C s disp LMI CONTROL TOOLBOX disp kkkkkkkkkkkkkkkkkkkkxk DE disp DEMO OF LMI LAB disp Specification and manipulation of LMI systems disp Example 8 1 of the Tutorial Section sj Given G s C sl A B Minimize the H infinity norm of DG s D Over a set of scaling
3. 1 eig lhsl rhs1 the first LMI is indeed satisfied o 3 get the values of the left and right hand sides of the second LMI with SHOWLMI lhs2 rhs2 showlmi evlmi 2 gt gt eig rhs2 4 get the values of the left and right hand sides of the third LMI with SHOWLMI lhs3 rhs3 showlmi evlmi 3 eig rhs3 3 4 3 4 o amp 13 Finally let us check that the H infinity norm of G s was not less than one from the start To do this we can remove the scaling D by setting S 2 I and solve the resulting feasibility problem Find X such that A X XA C C XB amp 0 B X I This new LMI system is derived from the previous one by setting S 2 I with SETMVAR j newsys setmvar lmisys S 2 lmiinfo newsys This is a system of 3 LMI s with 1 matrix variables Do you want information on v matrix variables 1 LMIs q quit q It has been a pleasure serving you oe Now call FEASP to solve the modified LMI problem tmin xfeas feasp newsys These LMI constraints were found infeasible Infeasible The H infinity norm of G s was larger than one ove 14 s You can also specify this system with the LMI editor gt gt lmiedit who clear who load lmidem who demolmi lmiedit Here you specify the variables in the upper half of the window and type the LMIs as MATLAB expressions in the lower half To see how this should look like
4. MEMS800 007 Chapter 4a Linear Matrix Inequality Approach Reference Linear matrix inequalities in system and control theory Stephen Boyd et al SIAM 1994 Linear Matrix Inequality LMI approach have become a powerful design tool in almost all areas of control system engineering The LMI approach has the following advantages Many control system design specifications and constraints can be expressed as LMIs The LMI problems can be solved numerically very efficiently using interior point methods e For those problems that analytical solutions are impossible the LMI approach often can provide solutions numerically LMI A linear matrix inequality LMI has the form F x F xF gt 0 4 1 i l where xe R is the variable the symmetric matrices F e R i 0 1 m are given Positive definite matrix F x 0 means that F x is positive definite i e u F x u gt 0 for all nonzero ue R Affine function X X X Xa t X40 x a b 1 2 m 1 1 Dod mom Ex 0 Lyapunov inequality A P PA 0 4 2 where Ae R is given and P P is the variable Eq 4 2 can be rewritten in the form of 4 1 Let P P P be a basis for the symmetric nxn matrices m n n 1 2 then take F 20 and F 2 A P PA 3 Nonlinear convex inequalities can be converted to LMI form using Schur complements Schur Complenment Q S a gt 0 s E if and only if 4 3a R 0 and Q SR S s 0 Q S b
5. er care and compared to the minimizer returned by mincx From an LMI optimization standpoint problem 9 9 is equivalent to the following linear objective minimization problem Minimize Tr X subject to A X XA Q XB lt 0 B X I Since Trace X is a linear function of the entries of X this problem falls within the scope of the mincx solver and can be numerically solved as follows o sj o 1 1 Define the LMI constraint 9 9 by the sequence of commands o i setlmis l 18 X lmivar 1 3 11 variable X full symmetric lImiterm 1 1 1 X 1 A s lmiterm 1 1 1 01 9 lmiterm 1 2 2 0 1 lmiterm 1 2 1 X B 1 A X XA Q XB B X I o ge LMIs getlmis lmiinfo LMIs This is a system of 1 LMI s with 1 matrix variables Do you want information on v matrix variables 1 LMIs q quit gt q It has been a pleasure serving you 1 2 Write the objective Trace X as c x where x is the vector of free entries of X Since c should select the diagonal entries of X it is obtained as the decision vector corresponding to X I that is j c mat2dec LMIs eye 3 1 Note that the function defcx provides a more systematic way of specifying such objectives see Specifying c x Objectives for mincx on page 9 37 for details o sj help defcx 19 o 5 3 Call mincx to compute the minimizer xopt and the global minimum copt c xopt of the objective j
6. matrices D with some given structure This problem arises in Mu theory robust stability analysis The system of LMIs is A X XA C SC XB B X S where X is symmetric S D D is symmetric block diagonal with prescribed structure 0 X50 S gt I To specify this LMI system with LMIVAR and LMITERM 1 resets the internal varibales used for creating LMIs so that a new system of LMIs can be created setlmis 2 define the 2 matrix variables X S lmivar 1 6 1 X is a 6x6 full symmetric matrix variable S lmivar 1 2 0 2 1 S is diag 2x2 diagonal block 2x2 full symmetric block o9 bd oe oe oe help lmivar s 3 specify the terms appearing in each LMI For convenience you can give a name to each LMI with NEWLMI optional o di help limterm A X XA C SC XB 0 B X S 6 lst LMI BRL newlmi lmiterm BRL 1 1 X 1 A s Imiterm BRL 1 1 S C C lmiterm BRL 1 2 X 1 B lmiterm BRL 2 2 S 1 1 2nd LMI X gt 0 Xpos newlmi lmiterm Xpos 1 1 X 1 1 3rd LMI Sol Slmi newlmi Imiterm Slmi 1 1 S 1 1 lmiterm Slmi 1 1 01 1 10 oe 4 get the internal description of this LMI system with GETLMIS sj lmisys getlmis Done A full description of this LMI system is now stored in the MATLAB variable LMISYS i s You can retrieve various information about the LMI system you just defined sj Oo number of LMIs lminbr lmisys Oo number of ma
7. options 1e 5 0 0 0 0 copt xopt mincx LMIs c options Here 1e 5 specifies the desired relative accuracy on copt The following trace of the iterative optimization performed by mincx appears on the screen o sj c xopt Upon termination mincx reports that the global minimum for the objective Trace X c x is 18 716695 with relative accuracy of at least 9 5 by 10 6 This is the value copt returned by mincx o sj Oo s 4 mincx also returns the optimizing vector of decision variables xopt The corresponding optimal value of the matrix variable X is given by j Xopt dec2mat LMIs xopt X This result can be compared with the stabilizing Riccati solution computed by care Xst care A B Q 1 Xst 6 3542e 000 5 8895e 000 2 2046e 000 5 8895e 000 6 2855e 000 2 2046e 000 2 2201e 000 norm Xopt Xst oe help norm 2 2201e 000 26077167000 20
8. trix variables matnbr lmisys Oo variables and terms in each LMI type q to exit lmiinfo lmiinfo lmisys This is a system of 3 LMI s with 2 matrix variables Do you want information on v matrix variables 1 LMIs q quit V Which variable matrix enter its index k between 1 and 2 1 X1 is a 6x6 symmetric block diagonal matrix its 1 1 block is a full block of size 6 This is a system of 3 LMI with 2 variable matrices Do you want information on v matrix variables 1 LMIs q quit 11 gt q It has been a pleasure serving you s We now call FEASP to solve our system of LMIs A X XA C SC XB m B X S X gt 0 S gt I oe tmin xfeas feasp lmisys sl tmin 1 839011 lt 0 the problem is feasible gt there exists a scaling D such that DG s D 1 The output XFEAS is a feasible value of the vector of decision variables the free entries of X and S j xfeas Use DEC2MAT to get the corresponding values of the matrix variables X and S s Xf dec2mat lmisys xfeas X Sf dec2mat lmisys xfeas S eig Xf eig Sf oe the constraints X gt 0 and S gt I are satisfied 12 To verify that the first LMI is satisfied 1 evaluate the LMI system for the computed decision vector XFEAS tj evlmi evallmi lmisys xfeas 2 get the values of the left and right hand sides of the first LMI with SHOWLMI sj lhs1 rhs1 showlmi evlmi
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