Uniform Sample Generations from Contractive Block Toeplitz Matrices

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1 IEEE TRASACTIOS O AUTOMATIC COTROL, VOL 5, O 9, SEPTEMBER Uniform Samle Generations from Contractive Bloc Toelitz Matrices Tong Zhou and Chao Feng Abstract This note deals with generating a series of random matrices uniformly and indeendently from contractive lower triangular bloc Toelitz (LTBT) matrices All the contractive LTBT matrices are arameterized nonlinearly but recursively in a closed-form by a sequence of contractive unstructured matrices Moreover, the conditional robability density function (PDF) is obtained in a closed-form for all the matri arameters, as well as the squares of their singular values It has been made clear that the conditional PDFs of the matri arameters are unitarily invariant A rocedure is develoed to accurately roduce the required random matri samles Inde Terms Model set validation, Monte Carlo method, random matri, robust control, Toelitz matri, uniform distribution I ITRODUCTIO Robustness is one of the major required erformances in control system synthesis Recent studies, however, show that when modelling errors are structured and norm-bounded, the analysis and synthesis of a controller is P (nondeterministic olynomial time)-hard with the number of uncertainty blocs, as well as the consistency verification of a model set with eerimental data [], [5] To overcome these difficulties, a Monte Carlo simulation based framewor is suggested In alying this aroach, an essential issue is the generation of uniform and indeendent random samles from a bounded modelling error set [], [4], [6], [], [], [3] [6] Recently, several advances have been made on this subject Secifically, when a matri is of bloc diagonal structure, the corresonding random matri samle generation roblem has been successfully solved in [3] and [4] In [5], random samle generation from contractive LTBT matrices has been converted to a sequence of uniform samling over unstructured contractive matrices Samle reuse has been suggested in [6] to imrove the comutational efficiency in robustness analysis, and so on When the algorithm of [5] is adoted, however, several roblems still arise At first, the generated samles are only asymtotically uniformly distributed Second, at every ste, square roots of matrices and solutions to matri equations are required which generally have a high dimension Finally, the number of the totally generated samles is much greater than the required one, and it increases almost eonentially with increasing the data length In this note, the conditional robability density function (PDF)- based aroach is adoted for uniform samling over contractive lower triangular bloc Toelitz (LTBT) matrices A contractive LTBT matri is nonlinearly but recursively arameterized in a closed-form by a sequence of unstructured contractive matrices, which is an imrovement of the results in [5] Under the condition that a LTBT matri is Manuscrit received October 4, 5; revised May 5, 6 Recommended by Associate Editor L Xie This wor was suorted in art by the ational atural Science Foundation of China under Grant 65748, the Trans-Century Training Program Foundation for the Talents of MOE, and the Secialized Research Fund for the Doctoral Program of Higher Education, PRC, under Grant 5396 The authors are with Deartment of Automation, Tsinghua University, Beijing, 84, China ( tzhou@mailtsinghuaeducn; fengchao@tsinghua educn) Digital Object Identifier 9/TAC6888 uniformly distributed, it is roved that the matri arameters are indeendent of each other and have unitarily invariant PDFs These roerties mae it ossible to reduce their random samle generation into three indeendent ones Based on Jacobians of matri transformations and matri integrations [3], [], [], a closed-form has been derived for the conditional PDFs of all the matri arameters, as well as the squares of their singular values From these conditional PDFs, an algorithm is constructed for a recursive and accurate generation of the required random matri samles The rest of this note is organized as follows In the net section, a mathematical descrition is given for the samling roblem, as well as the derivation of the recursive structure of a contractive LTBT matri The involved conditional PDFs are obtained in Section III In this section, the random samle generation algorithm has also been rovided This note is concluded by Section IV, in which some further research toics have also been discussed An aendi is included which gives roofs of the theoretical results The following notation is adoted in this note Vec(X) denotes the oeration of stacing the columns of matri X from left to right, and A B the Kronecer roduct of matrices A and B () reresents the comlete Gamma function, while Vol(X ) the volume of a set X f () and f ( j y) stand, resectively, for the PDF and the conditional PDF of a random variable U(X ) reresents the uniform distribution over the set X To reduce the number of symbols, no difference is made in this note between a random variable and its s js realizations Moreover, with a little abuse of symbols, f (j) is defined to be whenever s <s The other notation is fairly standard Its recise definition is therefore omitted, but can be found, for eamle, in [3] [5], and [4] II PROBLEM STATEMET AD STRUCTURE OF A COTRACTIVE LTBT MATRIX For a given matri sequence i j i, let denote the LTBT matri with its last bloc row being [ ] When unmodelled uncertainties are eistent in a lant model and Monte Carlo simulations are alied to a control-related roblem, a basic issue is to generate a required number of random matri samles [i] that are uniformly and indeendently distributed over a matri set defined as follows: fj ; i R q ; i ; ; ;;() < g: () Here, q and reresent, resectively, the dimensions of the outut and the inut of an uncertain matri-valued transfer function, while the length of time-domain eerimental data or the data length to be simulated The imortance of this roblem results from the facts that H-norm is often used in the measurement of model uncertainties, the eistence of a H-norm bounded matri-valued transfer function is equivalent to the contractiveness of the LTBT matri constructed from its first +unit imulse resonse matrices [5], [5], [7] For conciseness, it is assumed afterwards that q The case in which q < can be simly dealt with through transforming i to its transose i T On the other hand, with a little abuse of notation, in the remaining of this note, instead of, eressions lie i j i and i j i U() are usually used, in order to emhasize that the generation of a random real contractive LTBT matri samle [i] is in fact the generation of a sequence of random matri samles i j i To deal with the revious uniform samling roblem, we at first investigate the structure of the set For notational simlicity, ma /$ 6 IEEE

2 56 IEEE TRASACTIOS O AUTOMATIC COTROL, VOL 5, O 9, SEPTEMBER 6 trices t ; ~ and r, ; ; ; are, resectively, defined as t [ ] and ~ r : Using these symbols, the following results can be derived from the well-nown Parrott matri dilation theorem Theorem : Assume that Then, there eists a + such that T + + < I (+) if and only if T < I (+) Moreover, when this condition is satisfied, all the desirable + can be eressed as follows + [] + ij i + R l;+ ij i w + R r;+ ij i in which w + R q and (w + ) < Furthermore R l;+ i j i R r;+ i j i j I q w j w T j Here, [] + ( ij i) t [I (+) ~ T ~ ] ~ T r ; R l;+ ( ij i) [I q t (I (+) ~ T ~ ) t T ], and R r;+ ( i j i) [I r T (I (+)q ~ ~ T ) r ] Define set W as W fw ij i j (w i) < ;i ; ; ; ;g ote that [ ] is still a LTBT matri and it shares the same maimum singular value with This means that through some aro- riate augmentations, we can get a nonlinear but recursive arameterization for all the sequences j that constructs a contractive real LTBT matri through a sequence of matrices w j belonging to W It is worthwhile to mention that + of () taes a similar form as that of H + given immediately after (7) in [5], ecet that analytic eressions have been obtained in Theorem for [] + ( ij i); R l;+ ( i j i) and R r;+ ( i j i) Moreover, a recursive rocedure can be easily develoed for comuting r T (I (+)q ~ ~ T ) r ; t (I (+) ~ T ~ ) t T and t (I (+) ~ T ~ ) ~ T r Furthermore, the comutational burden is indeendent of both and for obtaining the square roots of I r T (I (+)q ~ ~ T ) r and I q t (I (+) ~ T ~ ) t T [8] These are different from the arameterization of [5], in which at the ( +)th ste, two matri equations must be solved which have a norm constraint and a dimension roortional to +, and the square roots must be comuted for two matrices that also have dimensions roortional to + As both R l;+ ( i j i) and R r;+ ( i j i) are invertible whenever ( ) <, it is aarent that for each ; ; ;, the maing defined by () is bijective between the first matrices of () (3) ij i and the first matrices of w ij i W With a little abuse of notation, in the rest of this note, [] + ij i ; R l;+ ( i j i) and R r;+ ( ij i) are, resectively, written as [] + (wij i); R l;+ (w i ji) and R r;+ (w i ji) III UIFORM SAMPLIG OVER COTRACTIVE LTBT MATRICES As has been noticed in [5], an interesting roerty of the arameterization in Theorem is that every element in the sequence w j can be selected indeendently ote that many control-related roblems can be formulated as an otimization roblem over the set [], [3], [5], [7] It may be argued that uniform samles of w j are more aroriate in control engineering In fact, samling over the set is converted in [5] to uniform and indeendent samling over the set W However, it can be roved that although both uniform samles of and random samles constructed from uniform samles of W have the same concentration tendency to the boundary of with increasing the data length +, the concentration is generally more raid if [i] is generated from an uniform samle of W ote that the faster the concentration rate is, the smaller the robability for generating a consistent model in model set validation or a nearly worst erturbation in robust erformance analysis It can, therefore, be declared that uniform samles from the set are usually more areciative in control engineering [8][9] In order to accurately generate a required samle of a multivariate random variable, a basic aroach is based on conditional PDFs [4], [], [], [4] When this aroach is utilized in our samle generation roblem, uniform samling over the set is equivalent to sequentially generate a random matri samle of w according to its conditional PDF f w (w j(w j j j )); ; ; ; ; In the following two subsections, we, resectively, discuss the derivation of this conditional PDF and the generation of w A Conditional PDF for w To derive the conditional PDF for w with rescribed w jj j, vectors and w are defined as follows Vec( ) Vec( ) w Vec(w ) Vec(w ) : (4) As the oeration of stacing the columns of a matri into a column vector is a bijective maing, it is aarent that when j and w j W, the comosite maing between and w is also bijective which is defined by () and (4) On the other hand, from the recursive structure in constructing a contractive LTBT matri, it is easy to see is a lower triangular bloc matri with its first diagonal bloc to be the identity matri I q and the remaining diagonal blocs as Rr;(w T i j i ) R l;(w i j i );; ; j j R T j j r;j w i i R l;j w i i R r;j w i j i q R l;j w i j i : (5) Recall that when j is uniformly distributed over the set, its joint PDF can be eressed as f ( j ) Vol () whenever j From this eression and (3) and (5), it can be directly

3 IEEE TRASACTIOS O AUTOMATIC COTROL, VOL 5, O 9, SEPTEMBER 6 56 roved that when j U(), the joint PDF of w j taes a nonzero value only on the set W which can be eressed as follows: f w (w f ( j ) Vol() j Vol() j R r;j w i j i j I q w w T q Rl;j w i j i (+q) I q w w T ()(+q) : (6) Vol() ote that w is an arbitrary contractive real q -dimensional matri From the above joint PDF, it is obvious that w j are indeendent of each other This imlies that for every [;], the conditional PDF f w (w j (w jj j )) can be written as follows: f w w w j j f w (w ) K ji q w w T j ()(+q) : (7) Here, K is a normalization factor to guarantee (w f )< w (w )dw, which can be derived using matri integrations Corollary : Define n(m) as n(m) n(n)4 (m)((m )) ((m n + )) Then, for every integer between and ; see (8), as shown at the bottom of the age We are now ready to discuss the generation of the required random matri samle of w j When the lant is of single-inut, w is a q real vector Let w j; reresent the jth row element of w ; j ; ; ;q Then, the conditional PDF f w (w j(w ij i )) is nonzero only on the set fw j; jjj q q j w j; < g that can be rewritten as f w (w j(w ij i )) K ( q j w j;) ()(q+) From this eression, it is not very difficult to obtain a closed-form for every ; ; ;; j ; ; ;q, of the conditional PDF for w j; when w ij and i w l;j j l are given As the derivations are quite direct and standard, the details are not included Therefore, when, the required samles can be generated through a standard conditional PDF-based aroach However, the random samle generation for w becomes much more comlicated when Based on some results about the Jacobians of matri transformations and the roerties of a Vandermonde matri [4], [7], [], [], [4], an algorithm can be develoed for random w generations This is investigated in the net section B Random Generation for w When In this section, we discuss how to generate a random samle for w, on the basis of the well nown conditional PDF based aroach For this urose, let j; j j reresent the first singular values of matri w arranged in a descending order Recall that q is assumed in the roblem descrition It is obvious that ji q w w T j j ( j;) From Corollary and (7), it is aarent that the conditional PDF f w (w j(w i j i )) is only a function of the singular values of w A PDF of random matrices with this roerty is called an unitarily invariant density, and its samle generation can be converted into three indeendent random matri samle generations [3], [4], [7], [] The following theorem is an immediate result of [, Th 5] and [4, Th ] The roof is omitted due to its obviousness To mae the statements concise, a diagonal matri set D is defined as D fd j D diagf ij ig; < < < < < g Moreover, a symbol [X] ;i is adoted to reresent the first nonvanishing element of a q -dimensional real matri X in its ith column, i ; ; ; Theorem : Assume that w U D V T, in which D diagf i; j i g;ut U I ;U R q ;V T V V V T I, and i; j i are the square roots of the eigenvalues of matri w T w If matri w is of full-column ran and [U ] ;i j are i restricted to be ositive, then, when q, the maing defined by w U D V T between w and (U ;D ;V ) is bijective Moreover, the joint PDF of (U ;D ;V ) can be eressed as f U ;D ;V (U ;D ;V ) f U (U )f D (D )f V (V ) Here, f U (U ) U(fUjU T U I ; [U ] ;i > ;i [;]g); f V (V ) U(fV jv T V VV T I g), and f D (D ) taes a nonzero value only when D Dthat has an eression as f D (D ) Here j q j; j; K ((8) (q) )( (q+) ) ((( )))(( )) q q+ (q+)() ij+ j; i; : (9) ((( )))(( )): ote that in the set of q -dimensional real matrices, the measure of matrices with a ran smaller than is equal to zero whenever q, as well as that of matrices with at least two identical nonzero singular values [][7] From Theorem, it is clear that when q, the generation of a random matri samle of w can be divided into three indeendent generations of random matri samles The generation of a random matri samle for U or V has been widely studied and is well settled [3], [4], [], [] For eamle, [4, Alg 4] is generally regarded to be a simle yet efficient rocedure However, it is not so simle to roduce a samle of the random diagonal matri D To solve this roblem, define a set X as X f[ ; ; ; ] T j < < < < < g Denote j; by j; ; j ; ; ;, and define vector as K () (q) (+q) l (3)(((q + )( ) +q + l +)) ( + l)((q + l))(((q + )( ) +l +)) (8)

4 56 IEEE TRASACTIOS O AUTOMATIC COTROL, VOL 5, O 9, SEPTEMBER 6 [ ; ; ; ; ; ; ] T As all the diagonal entries of an element in D are restricted to be ositive, this definition leads to a bijective maing between sets X and D Therefore, random samle generation for D is equivalent to that for ote that d j; ( j; )d j; ;j ; ; ; It is obvious that f ( ) is nonzero only when X In this case f ( )f D (D ) j ij+ j j; (q) j; ( j; ) (q+)() ( j; i; ) : () To roduce a random vector samle for, the conditional PDF-based aroach is adoted In this rocedure, the most imortant ste is the comutation of the marginal PDF ( i; j j [j] i ); j, which is defined as f ( i; j j i ) f ( )d ; d ; d j+; With this marginal PDF available, a desirable random samle for j; can be generated according to its conditional PDF f ( j; j( i; j j i )) given by f [j] f j; j i; i j i; i i; j j i : () Define q -dimensional real vector-valued functions g () and h (), resectively, as g () (q) ( ) (q+)() [; ; ; ; ] T ;h () g ()d Then, based on the roerties of a Vandermonde matri and the method of integration over alternate variables, the following conclusions can be established Theorem 3: Let A ( i; j j i ) denote the (j )-dimensional real matri-valued function [g ( ; ) g ( ; ) g ( j;)]; S ( j; ) the -dimensional real matri-valued function [g ()h T () h ()g T ()]d Then, f [j] ( i; j j i ) jm ( j; )j if j ; and ( i; j j i ) if j Here M ( j; ) A ( i; j j i M ( j; ) ) A T ( i; j j i ) () S ( j; ) g ( j; ) g T ( j; ) ; j even S ( j; ) g ( j; ) h ( j; ) g T ( j; ) ; j odd: h T ( j; ) From Theorem 3, it can be declared that when a random samle for i; j j i has been generated, the conditional PDF f ( j; j( i; j j i )) can be calculated at an arbitrarily rescribed value of j; According to this value, a random samle for j; can be generated On the other hand, note that ecet the comutation of the elements of matri S ( j; ), () taes comletely the same form as that of () in [3] Therefore, the recursive comutation algorithm given in [3, Sec ] can be directly adoted ow, we investigate the comutation of matri S ( j; ) Let g i; () and h i; () denote, resectively, the ith row entry of g () and h () Then, it is obvious that g i; () (q)+i ( ) (q+)() and h i; () g i;()d On the other hand, it is also aarent that no matter what integral values are taen by and q, one and only one of (q ) and (q + ) is an integer From these observations, an analytic eression can be derived by direct algebraic oerations for the ith row lth column entry of S ( j; ), which is equivalent to [g i; ()h l; () h i; ()g l; ()]d It is worthwhile to oint out that this eression is quite lengthy and comlicated, but it can be well aroimated by a simle function This is because g i; () is nonnegative whenever (; ) Moreover, this function has an unique maimum and vanishes very fast from its maimum when is large The details are omitted due to their obviousness and sace considerations C Random Samle Generation From the Set Summarizing the above arguments, a rocedure can be simly develoed for uniform samling over the set When q, a candidate algorithm can be stated as follows Initialize the samling rocedure by setting [] (w ij i ) q; R l; (w i j )I i q and R r; (w i j )I i Generate an uniform random samle V and an uniform random samle U, resectively, from fv j V T V VV T I g and fu j U T U I ; [U ] ;i > ; i ; ; ;g Generate a random series j; j j according to the conditional PDF given in () Let D diagf j; j jg; w U D V T Comute by [] (w ij i ) + R l; (w i j )w i R r; (w i j ), as well as R i l;+(w i ji); R r;+ (w iji), and [] + (wij i) Reeat the revious stes until has been generated Construct a samle [i] of according to its structure using the generated ij i When, the revious rocedure is still alicable with some minor modifications The above algorithm taes a similar structure as that of [5, Algs and ] However, some differences eist between their realizations Secifically, in the algorithms of [5], b jc uniform and indeendent samles of w are required at the th ste Here, bc reresents the oeration of taing the largest integer that does not eceed ; a regulation factor which is a ositive function of ; s the required number of indeendently and identically distributed (iid) random LTBT matri samles From Theorem, it can be easily shown that j ji i q w i wi T j (+q) This imlies that in order to revent the samle generation rocedure from being terminated intermediately, a large is usually areciative But this will mae the number of the totally generated samles increase in a larger eonential seed with increasing, and much greater than the required number of the iid samles In the revious algorithm, these roblems do not occur It is because the required samles for can be searately generated, and in generating each samle [i], one and only one random samle for w is required at the th ste umerical simulations show that uniform and indeendent samles from the setreally outerform those from the set W in some controlrelated roblems The details are reorted in [8]

5 IEEE TRASACTIOS O AUTOMATIC COTROL, VOL 5, O 9, SEPTEMBER IV COCLUDIG REMARKS In this note, uniform random samling over singular value-bounded LTBT matrices has been discussed All the contractive LTBT matrices have been nonlinearly but recursively arameterized by a sequence of norm-bounded but unstructured real matrices w j Owing to the recursive structure of the arameterization, an analytic eression has been derived for the joint PDF of w j, as well as the conditional one It becomes clear from the structure of these PDFs that w j are indeendent of each other, and their conditional PDFs deend only on their singular values and are therefore unitarily invariant Based on these roerties, it has been roved that the generation of a random samle for w is equivalent to the indeendent random generations of the factors in its singular value decomosition Moreover, a closedform has been derived for the conditional PDF of the squares of w s singular values An algorithm has also been established for generating a series of random j and, therefore, a uniform random samle of contractive LTBT matrices The results resented in this note are alicable to many control-related roblems, such as robust control system analysis and synthesis, robust control oriented system identification, etc [], [3], [5], [6] As further research, it remains interesting to investigate the alicability of Theorem to unfalsified robability comutations for a given model set utilizing time-domain eerimental data In this case, in order to get a good evaluation for the unfalsified robability, it aears that random samles from contractive LTBT matrices with a matri-variate Gaussian distribution are more areciative [8] As both I (+) ^ T ^ and I (+)q ^ ^ T are ositive definite whenever ( ) <, the arameterization of () can now be roved through direct algebraic oerations On the other hand, from the structure of matri ~, it is obvious that ~ [ ~ ] Based on this relation and the well nown t matri inversion formula [7][], it is not very difficult to rove that I r T [I (+)q ~ ~ T ] r I r I q I q ~ ~ T I q T diag I q ~ ~ T I q t t ~ T I ~ T ~ ~t T ; t T I q I q ~ ~ T I q I r T I q ~ ~ T +[ + t ~ T r I q ~ ~ T I q t I ~ T ~ t T r r ] T A Proof of Theorem APPEDIX THEOREM PROOFS I ote that ~ It is obvious that I ( )( ~ ) This imlies that the conclusions of Theorem can be roved through relacing by ~ On the other hand, note that [ ~ ][ ~ ] and r ~ t ~ ~ + ~ r : t + + t ~ T I q ~ ~ T [I r T I w T w I q ~ ~ T I r T I q ~ ~ T We, therefore, have that I r T I (+)q ~ ~ T r ] r r r : (4) From the well-nown Parrott matri dilation theorem [5][5], it can be directly roved that there eists a + such that ( ~ + ) <, if and only if ( ~ ) < Moreover, let ^ denote the matri [ ~ ] Then, all the + satisfying ( ~ + ) < can be eressed by + Y ^ T Z +(I q YY T ) w + (I Z T Z) : (3) Here, w + is an arbitrary q -dimensional contractive matri, while Y and Z are contractive matrices, resectively, satisfying I r T I q ~ ~ T r I w T w : (5) ote that w Moreover, through defining q; r and ~ can, resectively, be modified to r [ q T ] T and q, while I q to I q From these modifications, it can be easily roved that when ; ji r T (I q ~ ~ T ) r j ji w T w j It can now be concluded by induction from the revious equation that I (+)q ^ ^ T r [ t ]Y I (+) ^ T ^ : I r T I (+)q ~ ~ T r j I w T j w j : (6)

6 564 IEEE TRASACTIOS O AUTOMATIC COTROL, VOL 5, O 9, SEPTEMBER 6 From the definitions of matrices ~ ;r and t, it is aarent that through a relacement of i by T i ;i; ; ;, the following relation can be directly obtained from (6): I q t I (+) ~ T ~ t T j I q w jw T j : (7) The roof can now be comleted by noting that ji q w j w T j j ji w T j w j j ] B Proof of Corollary To rove Corollary, the following two lemmas (Lemmas A and A) are introduced, which are well nown in multivariate analysis [7][] Here, functions f i()j s i defined on a set with measure is said to be functionally indeendent, if (fjf i () < i ;i ; ; ;ng) n i (fjf i() < i g) is satisfied for an arbitrary ositive integer n not greater than s and arbitrary real numbers i j n i Moreover, to simlify eressions, matri sets fu j U T U I ;U R q g and fv j V T V I ;V R g are, resectively, denoted by U ~ and V ~ Lemma A: Assume that X is a real q matri-valued function with full-column ran and q functionally indeendent entries Moreover, assume that all the nonzero singular values of matri X, reresent them by ; ; ;, are distinct Then, there eist functions U and V taing values, resectively, in U ~ and V, ~ such that X UDV T,in which D diagf ij ig Moreover, let dx; dd; du and dv denote resectively the differentials of matrices X; D; U, and V Then, dd i d i and dx i i q i ji+ i j dddudv: Lemma A: Assume that q Then ~U du q (q) Proof: Define sets D ~ and W ~ resectively as D ~ f i j i j < ; ; ; < ; i 6 j if i 6 jg and W ~ fxjx UDV T ;D diagf ij ig; U U;V ~ V; ~ ij i Dg ~ Let j; j denote the square roots of the eigenvalues of matri j wt w, and U D V T a singular value decomosition of w with U U ~ and V V ~ ote that K is a finite ositive number Moreover, ji q w w T j (q+)() is also finite when (w ) < and Furthermore, recall that in the set of finite q -dimensional real matrices with q, the measure of matrices with a ran smaller than, as well as that of matrices with at least two identical nonzero singular values, are equal to zero [7][] From these observations and Lemmas A and A, we have (8), as shown at the bottom of the age The roof can now be comleted by noting that (w )< f w (w )dw from the roerties of a PDF ] C Proof of Theorem 3 To derive a closed-form for f [j] ( i; j j i ), the following lemma is introduced, which is roved in [3] using the Pfaffian of a sew-symmetric matri and multile integrations of a matri determinant obtained by de Bruijn [] Lemma A3: Assume that both () and () are real n -dimensional vector-valued functions of the variable with all their entries integrable over the interval [;] Moreover, assume that A() is a real n r-dimensional matri-valued function indeendent of i j l i Furthermore, assume that n l+r Denote [() T () () T ()]d by S() Then j[a() ( ) ( ) ( ) ( ) ( l ) ( l )]j d l d l! S() A() A T () : (9) Proof: For notational simlicity, denote [ i; i; i; ]T and (q) i; ( i; ) (q+)(), resectively, by i and a i ; i ; ; ; Then, it is easy to see that j[a a a ] j j[ ]j i a i Moreover, from the roerties of a Vandermonde matri, we have that j[ ] j i ji+ ( j; i; ) ote that g ( i; )a i i by definitions From () and these relations, it can be directly roved that when X f ( ) i i ji+ ( j; i; ) (q) i; ( i; ) (q+)() j[g ( ; ) g ( ; ) g ( ; )]j () An attractive characteristic of the above eression is that each column of matri [g ( ; ) g ( ; ) g ( ; )] is the same function with an indeendent variable This imlies that the method of integration over alternate variables [3], [7], [], []can be alied More recisely, when j and j is even, the following equation f w (w )< (w )dw K I q w w T (q+)() dw ~W K ( j;) (+q)() ~U V ~ D ~ j K du dv ~U ~V (+q) K () (q) l j j q j; i ji+ ( j;) (+q)() j j i; j;j q j; i ji+ ( + l)((q + l))(((q + )( ) +l +)) (3)(((q + )( ) +q + l +)) d ; d ; du dv i; j; d ; d ; (8)

7 IEEE TRASACTIOS O AUTOMATIC COTROL, VOL 5, O 9, SEPTEMBER is immediately obtained through integrating j+; ; j+4; ; ; ; over their resective intervals i; j i j[g ( ; ) g ( ; ) g ( ; )]j d ; d ; d j+; A i; j i g ( j; ) g ( j+; ) h ( j+; ) h ( j+3; ) g ( j+3; ) h ( j+3; ) h ( j+5; ) g ( ;) h ( ;) d ;d 3; d j+; : () Recall that the determinant of a matri remains unchanged when one of its columns is added to its another column We can, therefore, further have that ( i; j j i ) A i; j j i g ( j; ) g ( j+; ) h ( j+; ) g ( j+3; ) h ( j+3; ) g ( ;) h ( ;) d ;d 3; d j+; j! A i; j j i g ( j; ) g ( j+; ) h ( j+; ) g ( j+3; ) h ( j+3; ) g ( ;) h ( ;) d ;d 3; d j+; : () In obtaining the last equality, the roerty has been utilized that the integrand in the first equality remains unchanged when any two of variables j+; ; j+3; ; ; ;, are interchanged [3][] ote that the integration at the right hand side of the aforementioned equation taes comletely the same form as that in Lemma A3 We, therefore, have that Obviously, the revious derivations remain correct when j and j is even, ecet that the term A ( i; j j i ) is no longer available On the other hand, when j is odd, the corresonding conclusions can be established similarly This comletes the roof ] ACKOWLEDGMET The authors would lie to than the reviewers for their constructive suggestions and for drawing attention to [3] REFERECES [] B Barmish and C Lagoa, The uniform distribution: A rigorous justification for its use in robustness analysis, Math Control, Signals, Syst, vol, 3, 997 [] V Blondel and J Tsitsilis, A survey of comutational comleity results in systems and control, Automatica, vol 36, 49 74, [3] G Calafiore and F Dabbene, A robabilistic framewor for roblems with real structured uncertainty in systems and control, Automatica, vol 38, 65 76, [4] G Calafiore, F Dabbene, and R Temo, Randomized algorithms for robabilistic robustness with real and comle structured uncertainty, IEEE Trans Autom Control, vol 45, no, 8 35, Dec [5] J Chen and G Gu, Control-Oriented System Identification: An H Aroach ew Yor: Wiley, [6] X Chen, K Zhou, and J Aravene, Fast construction of robustness degradation function, SIAM J Control Otim, vol 4, no 6, 96 97, 4 [7] K Fang and Y Zhang, Generalized Multivariate Analysis Berlin, Germany: Sringer-Verlag, 99 [8] C Feng, Random tranfer function generation and its alications, MS thesiis, Tsinghua Univ, Beijing, China, 6 [9] C Feng and T Zhou, Remars on Monte-Carlo methods in model set validation submitted for ublication [] J Gentle, Random umber Generation and Monte Carlo Methods, nd ed ew Yor: Sringer-Verlag, 3 [] A Mathai, Jacobians of Matri Transformations and Functions of Matri Argument Singaore: World Scientific, 997 [] M Mehta, Random Matrices, nd ed Boston, MA: Academic, 99 [3] L Ray and R Stengel, A Monte Carlo aroach to the analysis of control system robustness, Automatica, vol 9, 9 36, 993 [4] C Robert and G Casella, Monte Carlo Statistical Methods Berlin, Germany: Sringer-Verlag, 999 [5] M Sznaier, C Lagoa, and M Mazzaro, An algorithm for samling subsets of H with alications to ris-adjusted erformance analysis and model (in)validation, IEEE Trans Autom Control, vol 5, no 3, 4 46, Mar 5 [6] R Temo, E Bai, and F Dabbene, Probabilistic robustness analysis: Elicit bounds for the minimum number of samles, Syst Control Lett, vol 3, 37 4, 997 [7] T Zhou and H Kimura, Time domain identification for robust control, Syst Control Lett, vol, 67 78, 993 [8] T Zhou, L Wang, and Z Sun, Model set validation under a stochastic framewor, Automatica, vol 38, no 9, , ( i; j j i ) S ( j; ) A ( i; j j i ) g ( j; ) A T ( i; j j i ) g T ( j;) A ( i; j j i M ( ) j;) : A T ( i; j j i ) (3)