ROBUST LMI PARSER: A COMPUTATIONAL PACKAGE TO
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1. A A1 A2 A3 gt poly_A poly_struct A A 3 1 2300 Anais do XIX Congresso Brasileiro de Automatica CBA 2012 Example 2 A polynomial matrix A a with degree g 2 that represents a system with N 2 vertices is given by A a af Ago a102A11 03A02 a Ag 10 and can be defined using the following sequence of Matlab commands gt A A20 A11 A02 gt poly_A poly_struct A A 2 2 The proper order of the coefficients in a poly nomial expression can be verified on the structure exponent of the polynomial variable poly_A or by executing the command exponent generate_homogenous_exponents vertices degree Example 3 A 3 x 3 identity matrix can be defined using gt poly_I poly_struct eye 3 I Example 4 A symmetric polynomial variable P a R of degree 2 used in an uncertain system with N 3 vertices can be defined by poly_P P poly_struct 3 3 P symmetric 3 2 The command poly_struct does not require the outputs to be used since the variables are de fined internally in the parser However if one needs to access a variable already defined the command rolmip can be used as poly rolmip getvar label The procedure rolmip is basically the interface between the user and the ROLMIP variables but it will not be detailed in this paper for the sake of brevity However further details of the procedure can be accessed by typing help rolmip2. if a problem can be cast as a set of LMIs then it can be considered as solved Boyd et al 1994 Unfortunately this is not completely true for large scale systems since LMI solvers are limited to a few thousands of variables and LMI rows but progresses are being made Usually LMIs are solved in two steps first an interface for parsing the conditions is used for example the YALMIP parser L fberg 2004 or the LMI Control Toolbox from Matlab Gahinet et al 1995 and an LMI solver is then applied to find a solution if any for example SeDuMi ISBN 978 85 8001 069 5 Sturm 1999 or SDPTS3 Toh et al 1999 Some auxiliary toolboxes may also be used in addi tion to the parser and the solver for example the SOSTOOLS Prajna et al 2004 which is used to transform a sum of squares problem into a SDP formulation and Gloptipoly Henrion and Lasserre 2003 used to handle optimization prob lems over polynomials Consider for instance the problem of analyz ing the stability of a discrete time linear system given by x k 1 Az k 1 with x k R being the state vector of the sys tem and A R being the dynamic matrix Using Lyapunov stability theory such system is stable if and only if there exists a symmetric ma trix P R such that the LMI H ae 2 holds Such LMI can be easily programmed and solved using respectively any LMI parser and solver available Consider now that s
3. label param vertices degree The output variable M is also internally defined and it is declared as a cell array being each cell a matrix with dimension rows x cols The ar gument param is a string that indicates if the variable is symmetric symmetric rectangu lar full symmetric Toeplitz toeplitz symmetric Hankel hanke1 or Skew symmetric skew The matrices will be declared as sym metric if param is not informed Note that such command is similar to the sdpvar instruction used to define the variables in the YALMIP parser If the variable is a scalar one may use the syntax poly poly_struct M label scalar which returns the structure related to the scalar M with label given by label It is important to notice that the parameter scalar is a lower case string If the scalar is a variable the following syntax may be used poly M poly_struct label scalar which returns a scalar variable M NOTE A scalar may also be defined as a 1 x 1 matrix omitting the variables vertices and degree and without using the parameter scalar However this approach may cause problems when ROLMIP is used to generate an independent Matlab executable file as presented in Section 6 Example 1 A polynomial matrix A a with degree g 1 that represents a system with N 3 vertices is given by A a 0A a2A2 0343 a Az 9 and can be defined using the following sequence of Matlab commands gt
4. 2012 Campina Grande 2a setembro 2012 Emails Anais do XIX Congresso Brasileiro de Automatica CBA 2012 ROBUST LMI PARSER A COMPUTATIONAL PACKAGE TO CONSTRUCT LMI CONDITIONS FOR UNCERTAIN SYSTEMS CRISTIANO M AGULHARI RICARDO C L F OLIVEIRA PEDRO L D PERES School of Electrical and Computer Engineering University of Campinas UNICAMP 13083 852 Campinas SP Brazil Abstract A computational package to construct linear matrix inequality LMI finite dimensional conditions from parameter dependent infinite dimensional LMIs whose parameters lie in the unit simplex is proposed The package named Robust LMI Parser is developed for Matlab and works jointly with YALMIP returning the entire set of LMIs through simple commands that describe the structure of the matrices involved and the robust LMI conditions to be programmed The performance of the parser is compared through numerical computations with P lya filter available in the robust optimization framework of YALMIP Keywords Parser LMIs computational package uncertain systems Resumo Neste artigo apresentado um pacote computacional para a interpreta o e constru o de condi es na forma de desigualdades matriciais lineares Linear Matrix Inequalities LMIs de dimens o finita a partir de LMIs de dimens o infinita dependentes de par metros pertencentes ao simplex unit rio O pacote denominado Robust LMI Parser desenvolvido para o M
5. 4 Operating on polynomials The basic mathematical operations sum subtrac tion and multiplication as well as the transpose operation can be performed in polynomial vari ables by using the procedure parser_poly whose syntax is poly_res parser_poly expr newlabel ISBN 978 85 8001 069 5 The variable expr is a string that describes the op eration to be performed in the polynomials The names of the variables used in expr must be equiv alent to the labels given to the polynomials in their definition The label of the resulting polynomial will be equal to newlabel if such parameter is in formed otherwise the label will be equal to expr Example 5 Consider the polynomial matrices A a a Ai az Bla ai By a2Bo Cla aC azC 2 The polynomial R a A a C a B a C a can be calculated by performing the following sequence of Matlab commands gt A A1 A2 gt B B1 B2 gt C1 C2 gt poly_struct A A 2 1 gt poly_struct B B 2 1 gt poly_struct C C 2 1 gt poly_R parser_poly A C B C In this example since A a C a B a C a A a B a C a the procedure parser_poly may also be used as gt poly_R parser_poly A B C Example 6 Consider the variables A a aA az Bla ai By a2Bo Cla aC a2C The polynomial R a A a B a C a can be calculated by performing the following se quence of Matlab commands gt poly_stru
6. NAME1 VALUE1 NAME2 VALUE2 as done in the sdpsettings procedure from YALMIP Finally the file needs to be closed by using the command lmifiles close fid The first argument may be replaced by c The new Matlab executable file is stored at the same directory of the program used to create it The input parameters of the main function are the sys tem matrices and other user defined variables and the output is a structure whose fields contain the values of the resulting variables if the LMI is fea sible Example 10 The creation of a Matlab m file to solve the LMI presented in 3 considering a system with N 2 vertices and polynomial variables with degree g 1 can be done through the following sequence of Matlab commands ISBN 978 85 8001 069 5 gt fid lmifiles o0 discr_stab gt poly_struct A A 2 1 gt poly_struct P P 2 1 gt Term 1 1 parser_poly P gt Term 1 2 parser_poly A P gt Term 2 2 parser_poly P gt lmifiles i fid Term gt gt lmifiles c fid The resulting function can then be called through the command gt Output discr_stab A 7 Numerical Experiments The objective of the experiments is to compare the complexity of ROLMIP presented in this paper with the Robust Toolbox from YALMIP which is the state of the art and a more general parser The complexity of the parsers are measured us ing the memory usage and the time s
7. atlab funciona em conjunto com o YALMIP e capaz de construir todo o conjunto de LMIs desejado a partir de simples comandos que descrevem a estrutura das matrizes envolvidas e as condi es LMIs robustas a serem programadas O desempenho do interpretador comparado por meio de agulhari dt fee unicamp br ricfow dt fee unicamp br peres dt fee unicamp br experimentos num ricos com o filtro de P lya dispon vel no pacote Robust Optimization do YALMIP Palavras chave Interpretador LMIs pacote computacional sistemas incertos 1 Introduction In the last decades problems formulated in terms of Linear Matrix Inequality LMI conditions and solved by Semidefinite Programming SPD tech niques became more and more common in several fields related to engineering and applied mathe matics Specifically in control theory the grow ing usage of such tools have led to important re sults on the analysis of systems stability synthe sis of stabilizing robust controllers for uncertain systems and synthesis of optimal control mod els just to name a few Boyd et al 1994 Chesi et al 2009 Accompanying the growth of the usage of LMI conditions a large number of solvers based on in terior point methods were developed as well as interfaces for parsing the LMIs most of them free and easily accessible Thanks to such remark able advance in the computational tools to define manipulate and solve LMIs in many cases one can say that
8. ct A A 2 1 gt poly_struct B B 2 1 gt poly_struct C C 2 1 gt poly_R parser_poly A B C Mathematically such operation results on A a B a C a a A1 Bi Q1Q2 A1 B2 Ao B a5 A2Ba a101 a2C2 11 Note that the polynomial A a B a has degree g 2 and C a has degree g 1 According to the assumptions on Section 2 all the polynomi als are considered to be homogeneous In order to return a homogeneous polynomial the follow ing operation is automatically performed by the procedure parser_poly on matrix C a C a a1 a2 C a af Ch aya2 Cy C2 0302 12 2301 Anais do XIX Congresso Brasileiro de Automatica CBA 2012 The resulting homogeneous polynomial R qa is then given by R a a A B C1 a1Q2 A Bp AgB C C2 02 A2B2 C2 13 Example 7 Consider the variables A q a A a2 Ag B a amp B amp 2 B2 Qe Ao 14 The polynomial R a B a A a can be cal culated by performing the following sequence of Matlab commands gt poly_struct A A 2 1 gt poly_struct B B 2 1 gt poly_R parser_poly B A Since B a A a A a B a the proce dure parser_ poly may also be used as gt poly_R parser_poly A B 5 Composing matrices and LMIs Consider again the parameter dependent LMI 3 Each entry i e block of the matrix can be cal culated by using the procedure parser_poly as present
9. d with a lower memory usage Table 1 Values of y min y execution time t in seconds and the number K of allocated sdpvar objects resultant from using ROLMIP and the Ro bust Toolbox of YALMIP when varying the de gree of the polynomial variable P a deer yi ROLMIP YALMIP t K t K 0 1 0848 3 iil 8 125 iL 1 0016 5 41 17 455 2 100015 EmO 3 1 0000 29 201 179 2215 ISBN 978 85 8001 069 5 7 2 Experiment 2 Consider now the robust stability analysis condi tion presented in 3 for uncertain discrete time systems The number of sdpvar objects allocated to assess the stability of a system of order n n 2 3 4 whose uncertainties are represented by a polytope of N vertices N 1 2 8 is shown in Figure 1 Note that the number of vari ables generated by ROLMIP is considerably lower than the number of variables generated by the Ro bust Toolbox from YALMIP which implies on a lower memory usage Concerning the execution times ROLMIP spent 1 second on average while YALMIP spent 9 seconds on average to oy Number of scalar variables id i i i i i i 3 4 5 6 Number N of vertices Figure 1 Number of sdpvar objects created when programming condition 3 using ROLMIP dashed line and the Robust Toolbox from YALMIP solid line 8 Conclusion A computational package named Robust LMI Parser is presented in this paper The main pur p
10. ed in Section 4 In order to construct a matrix of polynomials as well as the set of LMIs generated by a parameter dependent condition as the one given by 3 one can use the procedure named construct_lmi whose syntax is poly_matr construct_1mi Term param The variable Term is a cell array in which Term i j corresponds to the element i 7 of the matrix to be defined If the matrix to be defined is square and symmetric which is the case when working with LMIs it suffices to inform only the upper or lower triangular elements The input param is a string either containing an inequality symbol gt lt gt lt meaning that the desired result is a set of LMIs defined using the command set from YALMIP L fberg 2004 or representing the label of the new matrix The procedure construct_1mi also applies an homog enization operation assuring that all the elements in the matrix are homogeneous polynomials with the same degree NOTE If an LMI is to be defined then the matrix represented by the variable Term must be symmetric and therefore square If the matrix is rectangular and the variable param is an inequal ity symbol an error will occur ISBN 978 85 8001 069 5 Example 8 The LMI presented in 3 considering a system with N 2 vertices and polynomial variables with degree g 1 can be implemented through the following sequence of Matlab commands gt poly_struct A A 2 1 gt poly_str
11. er Verlag Berlin Germany Gahinet P Nemirovskii A Laub A J and Chi lali M 1995 LMI Control Toolbox User s Guide The Math Works Natick MA Henrion D and Lasserre J B 2003 Glop tiPoly global optimization over polynomi als with Matlab and SeDuMi ACM Transac tions on Mathematical Software 29 2 165 194 L fberg J 2004 YALMIP A toolbox for modeling and optimization in MAT LAB Proceedings of the 2004 IEEE In ternational Symposium on Computer Aided Control Systems Design Taipei Taiwan pp 284 289 http control ee ethz ch joloef yalmip php L fberg J 2012 Automatic robust convex pro gramming Optimization Methods and Soft ware 27 1 115 129 Oliveira R C L F and Peres P L D 2006 LMI conditions for robust stability analysis based on polynomially parameter dependent Lyapunov functions Systems amp Control Let ters 55 1 52 61 Oliveira R C L F and Peres P L D 2007 Parameter dependent LMIs in robust analy sis Characterization of homogeneous poly nomially parameter dependent solutions via LMI relaxations IEEE Transactions on Au tomatic Control 52 7 1334 1340 ISBN 978 85 8001 069 5 Oliveira R C L F and Peres P L D 2008 A convex optimization procedure to compute H z and H norms for uncertain linear sys tems in polytopic domains Optimal Control Applications and Methods 29 4 295 312 Prajna S Papachristodoulou A Se
12. for using ROLMIP is to define all the variables and constants used in the LMIs through the procedure poly_struct which has basically three possible syntaxes If the coefficients of the monomials are already defined the related poly nomial structure is obtained by poly poly_struct M label vertices degree The output poly whose detailed description can be found on the ROLMIP user manual is a structured variable that fully describes the poly nomial variable All the terms used in the LMIs must be defined through the poly_struct proce dure even the scalars and the precisely known ma trices Since the variables defined by poly_struct 1 Available at http www dt fee unicamp br agulhari softwares robust_lmi_parser zip 2299 Anais do XIX Congresso Brasileiro de Automatica CBA 2012 3 a Pa a are automatically stored by ROLMIP the output poly is not mandatory The variable M corresponds to the coefficients of the monomials of the homogeneous polynomial to be defined If the polynomial has a number r of monomials all the r components must be given in M For example consider that matrix A a has the following structure A q a1 A a2A2 a a2 1 a gt 0 a2 gt 0 7 i e A a is characterized by a polytope of N 2 vertices and is modeled as a homogeneous poly nomial of degree g 1 There are two ways of in putting the coefficients A and Ag using the vari able M e Concatenating the matrices A a
13. function gama calc_hinf A B C D degP n length A 1 vert length A size B1i 2 poly_struct A A vert 1 T poly_struct B B vert 1 poly_struct C C vert 1 poly_struct D D vert 1 poly_struct n n P vert degP poly_struct eye r Ir polymu mu poly_struct mu scalar LMIs construct_lmi parser_poly P gt Term 1 1 parser_poly A P P A C C Term 2 1 parser_poly B P D C Term 2 2 parser_poly D D mu Ir LMIs LMIs construct_lmi Term lt solvesdp LMIs mu gama sqrt double mu As illustrated the implementation of LMI conditions using ROLMIP is simple and straight forward Moreover the different sets of LMIs for larger degrees of the homogeneous polynomial variable P a can be readily obtained by simply changing degP differently from the robust tool box of YALMIP where the polynomial strucutre of the variables must be manually set The re sults of executing the function calc_hinf to cal culate the bound y min y of the fourth order mass spring system borrowed from Oliveira and Peres 2008 is shown in Table 1 along with ex ecution times spent to solve the problem and the number of scalar sdpvar objects created by YALMIP information obtained from the com mand getvariables The values of y are the same for both parser as expected but it is clear that ROLMIP allows the resolution of the prob lem spending less time than YALMIP an
14. iler P and Parrilo P A 2004 SOSTOOLS Sum of squares optimization toolbox for MATLAB Ramos D C W and Peres P L D 2001 A less conservative LMI condition for the ro bust stability of discrete time uncertain sys tems Systems amp Control Letters 43 5 371 378 Ramos D C W and Peres P L D 2002 An LMI condition for the robust stability of uncertain continuous time linear systems IEEE Transactions on Automatic Control 47 4 675 678 Sturm J F 1999 Using SeDuMi 1 02 a MAT LAB toolbox for optimization over symmet ric cones Optimization Methods and Soft ware 11 1 4 625 653 http sedumi mcmaster ca Toh K C Todd M J and T t nc R 1999 SDPT3 A Matlab software package for semidefinite programming Version 1 3 Op timization Methods and Software 11 1 545 581 2305
15. nd Ag gt M A1 A2 e Using M as a cell array gt M 1 A1 M 2 The number r of monomials that must be provided in M depends on the number N of vertices and on the degree g of the polynomial Oliveira and Peres 2007 being calculated by D N g 1 g N 1 8 Note that the number of vertices and the degree are also input parameters of the poly_struct procedure If the variable to be defined is not parameter dependent the parameters vertices and degree may be omitted The variable label is a string and consists of a name used to refer the variable when defining the polynomial operations described later The string is case sensitive and must correspond to a valid variable of Matlab NOTE A constant matrix may be defined by setting vertices with the number of vertices of the considered system and degree 0 However the structure will be more complex and may result on more expensive computations NOTE According to the assumptions made in Section 2 all the matrix variables must have the same number of vertices The only exception is for constant matrices where the number of vertices as well as the degree may be set to zero or does not need to be informed ISBN 978 85 8001 069 5 N 1 N ai 1D L ALD i SET dee P P R ASP sa 1 a i 1 j i If the coefficients of the monomials are deci sion variables to be calculated one may use the following syntax poly M poly_struct rows cols
16. ng a consid erable amount of time This is performed through the command lmifiles The syntax used for opening the file is fid lmifiles open filename 2302 Anais do XIX Congresso Brasileiro de Automatica CBA 2012 The first argument may be replaced by o The parameter filename is both the name of the file and the name of the main function being declared without the extension m The file identifier fid associated is returned The file must be opened before the definition of the first LMI To insert an LMI in the file one can use the 1lmifiles procedure with the following syntax lmifiles insert fid Term ineq The first argument may be replaced by i The parameter fid is the file identifier returned when opening the file The structure Term is the cell array that defines the LMI in which Termfi j corresponds to the element i j of the matrix and ineq is the signal of the inequality If the inserted LMIs are constraints of an opti mization problem the command lmifiles is also used to define the objective function of a mini mization problem Considering that the objective is the minimization of the variable whose label is given by label_obj one may use the command lmifiles insert fid label_obj In the final m file the problem is solved by ap plying the function solvesdp The desired op tions for the solvesdp can be defined using the command lmifiles as lmifiles insert fid sdpsettings
17. onservative set of condi tions may be obtained by modeling the variable P a in 6 as a homogeneous polynomial with generic integer degree g gt 1 and then imposing the positivity of all matrix coefficients Oliveira and Peres 2006 Bliman et al 2006 Program ming these LMIs requires an a priori knowledge on the formation law of the monomials which de pends on the number N of uncertain parameters and on the degree of the polynomial variable P a In Oliveira and Peres 2007 a systematic way to deal with such cases has been developed but in the context of robust LMIs presenting at most products between two parameter dependent ma trices When the LMIs to be solved are more com plex and have products involving three or more parameter dependent matrices the rules to com pose the monomials become more complicated Moreover each new case demands the manipula tion of different polynomials Such task as well as programming the resulting LMIs is tedious time demanding and can be a source of programming errors Such problems can be partially mitigated by the Polya filter provided in the robust opti mization framework of YALMIP L fberg 2012 affine parameter dependent LMIs can be easily programmed using YALMIP and even P lya re laxations can be automatically performed but the ISBN 978 85 8001 069 5 definition of variables that are modeled as ho mogeneous polynomials of degree g gt 1 is not straightforward The comp
18. ose of the parser is to facilitate the task of pro gramming LMIs that are sufficient conditions for robust LMIs i e for parameter dependent LMI conditions whose entries are algebraic manipula tions of homogeneous polynomials of generic de gree with parameters lying in the unit simplex Some examples illustrate the advantages of using such a specific purpose parser over a general one for this kind of problem The parser is under con stant evolution and some new features are to be implemented such as code optimizations and the capacity of handling uncertain parameters in the multi simplex domain 2304 Anais do XIX Congresso Brasileiro de Automatica CBA 2012 Acknowledgments This work is supported by CNPq FAPESP and CAPES References Bliman P A Oliveira R C L F Montag ner V F and Peres P L D 2006 Ex istence of homogeneous polynomial solutions for parameter dependent linear matrix in equalities with parameters in the simplex Proceedings of the 45th IEEE Conference on Decision and Control San Diego CA USA pp 1486 1491 Boyd S El Ghaoui L Feron E and Balakrish nan V 1994 Linear Matrix Inequalities in System and Control Theory SIAM Studies in Applied Mathematics Philadelphia PA Chesi G Garulli A Tesi A and Vicino A 2009 Homogeneous Polynomial Forms for Robustness Analysis of Uncertain Systems Vol 390 of Lecture Notes in Control and In formation Sciences Spring
19. pent to solve a specific problem The computer used to per form the simulations is an Intel Core 2 T5500 1 66 GHz 1GB RAM 981 MHz Ubuntu 11 10 us ing YALMIP and SeDuMi Sturm 1999 within the Matlab environment 7 1 Experiment 1 Consider the continuous time uncertain time invariant system given by r t A a x y t Cla a with A a R B a R C a RI and D a R4 The transfer function from the disturbance input w to the output y for a fixed a is given by t B a w t t Diayw t 97 1 H s a C a sI A a Ba D a The bounded real lemma assures the Hurwitz sta bility of A a i e all eigenvalues have negative real part for all a Ay and a bound y to the Ho norm of the transfer function from w to y It can be formulated as follows Boyd et al 1994 Lemma 1 Matriz A a is Hurwitz and H s a lo lt y for alla E An if and only if there exists a positive definite symmetric matriz P a P a gt 0 such that A a P a P a A a C a C a B a P a D a C a Dla Dla 921 lt 0 Aw 18 2The x means a symmetric block 2303 Anais do XIX Congresso Brasileiro de Automatica CBA 2012 The implementation of convergent LMI re laxations as the degree of the Lyapunov matrix grows that search for a feasible solution while minimizing u 7 is given in the following being the input matrices defined as cell arrays
20. uct P P 2 1 gt Term 1 1 parser_poly P gt Term 1 2 gt Term 2 2 parser_poly P gt LMIs construct_lmi Term gt parser_poly A P Example 9 Multiplying inequality 3 on the left by I A o 15 and on the right by its transpose yields P a A a P a A a gt 0 16 which is along with the inequality P a gt 0 an equivalent robust stability condition The above multiplication can be performed by the following sequence of Matlab commands gt poly_struct A A 2 1 gt poly_P poly_struct P P 2 1 gt LMIs construct_lmi poly_P gt gt Term 1 1 parser_poly P gt Term 1 2 parser_poly A P gt Term 2 2 parser_poly P gt construct_lmi Term M gt T 1 1 poly_struct eye 2 I gt T 1 2 parser_poly A gt construct_lmi T T gt LMIs LMIs construct_1lmi parser_poly T M T gt 6 Creating Matlab m files Although ROLMIP is a very practical pack age and allows the programming of complex parameter dependent LMIs without spending much effort it is true that the computational time required for the parser to compose the LMIs is considerably higher than setting the LMIs mono mial by monomial To circumvent this drawback it is possible to use ROLMIP to create a m Mat lab file that contains all the LMIs already properly constructed Therefore the parsing is performed only once in the file generation savi
21. utational package pre sented in this paper called Robust LMI Parser ROLMIP was developed to deal specifically with operations concerning parameter dependent variables whose parameters are contained in the unit simplex The parser is developed for Mat lab and works jointly with YALMIP returning the entire set of LMIs through a few simple com mands that describe the structure of the known matrices and variables involved in the parameter dependent LMIs to be investigated Since the parser is a specific purpose application it is considerably faster than the P lya filter from YALMIP The paper is organized as follows Section 2 presents the assumptions considered in the parser and the notation used throughout the paper The details on the syntaxes of the commands are pre sented in Sections 3 4 5 and 6 A comparison between the parser and YALMIP is performed through some illustrative examples in Section 7 and Section 8 concludes the paper 2 Preliminaries The parser named ROLMIP currently considers the following assumptions e All the matrices are described as homoge neous polynomials on the uncertain param eter vector a e The vector of parameters a belong to the unit simplex Ay defined in 5 Several examples illustrating the usage of the commands are presented throughout the pa per Such commands are displayed using the Typewriter font and start with gt 3 Defining the polynomial structure The first step
22. ystem 1 is affected by uncertainties i e the system matrix is parameter dependent and denoted by A a The robust sta bility of such system can be assessed by rewriting the LMI condition 2 as A a P a P a P a P a A a _ but in this case 3 must hold for all admissible a In order to transform the parameter dependent 2298 Anais do XIX Congresso Brasileiro de Automatica CBA 2012 LMI into a finite set of standard LMIs some in formation about A a must be added as well as some structure must be imposed to the unknown variable P a For instance consider that A a and P a have a polytopic structure N N Ala X ai Ai P a X aP a An i 1 i 1 4 being A and P the vertices of the respective poly topes N the number of vertices and Ay the set known as unit simplex given by s N 5 Using the chosen structure for A a and P a given by 4 to the robust stability condition ex pressed in the parameter dependent LMI 3 and multipliying the entries on the main diagonal by Da Qi one gets the homogeneous polynomial matrix inequality of degree 2 on a shown in 6 top of next page A sufficient but not necessary way to guar antee that 3 holds is to impose that all the matrix coefficients of the monomials are posi tive definite This has been done for instance in Ramos and Peres 2001 discrete time sys tems and Ramos and Peres 2002 continuous time systems A less c
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