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Beneath-Beyond Method and Construction of Lyapunov Functions

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The purpose of this paper is to pro- pose an improved method of the beneath{beyond method to solve dynamic convex hull problem in d- dimensional space. The traditional beneath{beyond method requires the whole data of its faces and their inclusion
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  Beneath-Beyond Method and Construction of Lyapunov Functions Yuzo Ohta Yasushi Nagai and Lei GongKobe University, Nada, Kobe 657, Japan, ohta@seg.kobe-u.ac.jp Abstract —  The purpose of this paper is to pro-pose an improved method of the beneath–beyondmethod to solve dynamic convex hull problem in  d -dimensional space. The traditional beneath–beyondmethod requires the whole data of its faces and theirinclusion relations. However, in the application of thedynamic convex hull problem we usually need the re-sultant facets but do not need the data of faces of lower dimensions. When the dimension  d  becomeslarge, then the computing time to maintain these un-necessary data becomes very large. In this respect,we propose a method which need data of facets andsubfacets and their inclusion relations. This is usefulin saving not only the storage but also the computingtime. I. INTRODUCTION The convex hull problem is one of the most fundamen-tal problems in the computational geometry. In partic-ular, in stability analysis of dynamical systems usingcomputer, the dynamic convex hull algorithm playsa crucial role [1] -[7]. The beneath–beyond method[8]–[10] is one of the most powerful method to solvedynamic convex hull problem in  d -dimensional spaces.In the srcinal beneath–beyond method, we need tomaintain the whole data of its faces and their inclu-sion relations, and, hence, it requires huge storage andcomputing time. However, in many applications, weusually need the resultant facets but do not need thedata of faces of lower dimensions.In this paper, we propose an improved methodwhich need data of facets, subfacets and their inclusionrelations. This is useful to save not only the storagebut also the computing time . Notation : In this paper,  R  denotes the real numbersystem, and  R d is the usual vector space of real d–dimensional vectors  x  = [ x 1 ,  x 2 , ...,  x d ] T  . All vectorsare to be regarded as column vectors for the purposeof matrix multiplications. The inner product of twovectors  x  and  y  in  R d is expressed by  < x,y >  =  di =1 x i y i . For a set  P   in  R d , the interior, the affinehull and the convex hull of   P   are denoted by int  P  ,aff   P   and conv  P  , respectively. Let  V   =  { v 1 ,  v 2 , ..., v n }  be a set of vectors in  R d and let dim aff   V   = m , then the convex hull  P   = conv  V   is called an  m -polytope. Let  P   be a  d -polytope. The set of all facesof   P   and the set of all  k -faces of a polytope  P   aredenoted by  F  ( P  ) and  F  k ( P  ), respectively. A  d  −  1face  F   ∈ F  d − 1 ) P  ) and a  d − 2 face  G  ∈ F  d − 2 ) P  ) arecalled a facet and subfacet, respectively. The set of all extreme points ( i.e. , 0-faces) of a polytope  P   isdenoted by ext  P  . For a set  V   having a finite numberof elements  | V  |  denotes the number of elements of   V  . II. POLYTOPE LYAPUNOV FUNCTION Let us consider the following systems described by˙ x  ∈  conv ( A 1 x,A 2 x,...,A m x ) ,  (1)where  x  ∈  R d and  A i  ∈  R d × d .Let  P   be a  d -polytope such that 0  ∈  int  P  , then thepolytope Lyapunov function is defined by V  ( x ) = inf  { ρ >  0 |  x  ∈  ρP  } ,  (2)where  ρP   =  { ρx |  x  ∈  P  } .Stability of the system (1) is examined by the fol-lowing [2]: Theorem 1  The equilibrium point 0 of (1) is expo-nentially stable (in the large) if there exists positiveconstants ∆ and  γ   and a  d -polytope  P   such that1) 0  ∈  int  P  2) for each  x  ∈  ext  P   and  A i x + ∆[ γx + A i x ]  ∈  P.  (3)A polytope  P   satisfying the condition 2) of Theorem1 is constructed through the following algorithm: Algorithm PLF 2 1) for each  x  ∈  ext  P   append  x  to  Q ;2) while  Q   =  ∅  begin3)  x  := top( Q );4) for each  i  ∈ { 1 , 2 ,...,m }  begin5)  y  :=  x + ∆[ γx + A i x ];6) if   y  |∈  P   then begin7) append  y  to  Q ;8)  P   := conv ( P   ∪{ y } );9) end;10) end;11) end.  In Step 8), we need to computeˆ P   = conv ( P   ∪{  p } ) ,  (4)which is called the convex hull problem. We also notethat Step 6) is solved in the initial stage of solving theconvex hull problem.This method has been extended so that we can ex-amine stability of nonlinear systems [2] – [6]. III. BENEATH-BEYOND THEOREM Suppose that a  d -polytope  P   ∈  R n and a point  p  ∈  R d are given. The problem we consider here is compute(4). If   p  ∈  P  , then we immediately have ˆ P   =  P  , and,hence, in the following, we assume that  p  |∈  P  .For a facet  F   ∈ F  d − 1 ( P  ), the corresponding sup-porting hyperplane  H  ( a,α ) of   P   is given by aff   F  .Wedefine an open half spaces  S  − ( F  ) and  S  + ( F  ) by S  − ( F  ) =  { x  ∈  R d |  < x,a > < α } , , and  S  + ( F  ) = { x  ∈  R d |  < x,a > > α } . Definition 1  Coloring of faces. Let  p  ∈  R d ,  F   ∈F  d − 1 ( P  ). The color of facet  F   is classified as follows:1)  F   is yellow if   p  ∈  aff   F  ,2)  F   is blue if   p  ∈  S  − ( F  ), and3)  F   is red if   p  ∈  S  + ( F  ).Let  F   be a facet and let ext  F   =  { x 1 ,  x 2 , ...,  x m } . If 0  ∈  int  P  , then the normalized normal vector  a  ∈  R d of   F   is a vector satisfying X  T  a  =  e  = [11 ... 1] T  ∈  R d (5)where  X   is the  d × m  matrix whose  i -th column is  x i and rank  X   =  d . The normalized normal vector  a  maybe given by a  = ( XX  T  ) − 1 Xe.  (6)Suppose that  p  ∈  R d is given. Then, F   is  red ,  if   < a,p > >  1blue ,  if   < a,p > <  1yellow ,  if   < a,p >  = 1(7)We code colors by using 3-bit data: We correspondyellow, blue and red to 001, 010 and 100, respectively.We define the logical OR operation  ∨  to these 3-bitdata as follows: ( b 1  b 2  b 3 ) = ( b 11  b 12  b 13 )  ∨  ( b 21  b 22  b 23 )is defined by  b j  =  b 1 j  ∨  b 2 j ,  i  = 1, 2, 3. Moreover, wedefine ( b 1  b 2  b 3 ) =   mk =1  ( b k 1  b k 2  b k 3 ) by  b j  =   mk =1  b kj ,  j  = 1, 2, 3, and m  k =1 b kj  = ( ··· (( b 1 j  ∨ b 2 j ) ∨ b 3 j ) ··· ) ∨ b mj  ) Definition 2  Coloring of   k -faces.A  k -face  E   of   P   with  k <  dim P   − 1, and let  { F  k } mk =1 be the set of all superfaces of   E  . Suppose that  F  k  hasbeen classified as ( b k 1 ,  b k 2 ,  b k 3 ). Let ( b 1  b 2  b 3 ) =  mk =1 ( b k 1 b k 2  b k 3 ). Then,  E   is classified as follows:1)  E   is yellow if ( b 1  b 2  b 3 ) = (001),2)  E   is blue if ( b 1  b 2  b 3 ) = (010),3)  E   is red if ( b 1  b 2  b 3 ) = (100),4)  E   is orange if ( b 1  b 2  b 3 ) = (101),5)  E   is green if ( b 1  b 2  b 3 ) = (011),6)  E   is purple if ( b 1  b 2  b 3 ) = (110), and7)  E   is brown if ( b 1  b 2  b 3 ) = (111).We denote the set of all yellow (blue, red, orange,green, purple, brown, respectively)  k -faces of   P   by F  Y k  ( P  ) ( F  Blk  ( P  ),  F  Rk  ( P  ),  F  Ok  ( P  ),  F  Gk  ( P  ),  F  P k  ( P  ), F  Brk  ( P  ), respectively).The following is the precise representation of thebeneath–beyond theorem [9], [10]. Lemma 1  Beneath–Beyond TheoremLet us consider  d -polytope  P   ∈  R n and a point  p  ∈  R d such that  p  |∈  P  , and let ˆ P   = conv ( P   ∪ {  p } ). Let k  ∈ { 0 , 1 ,...,d − 1 }  and let F  Bk  ( P  ) =  F  Blk  ( P  ) ∪F  Gk  ( P  ) ∪F  P k  ( P  ) ∪F  Brk  ( P  ) (8) F  PBrk − 1  ( P  ) = ( F  P k − 1  ∪F  Brk − 1 ) ,  (9)ˆ F  PBrk  ( ˆ P  ) =  { conv ( F   ∪{  p } ) |  F   ∈ F  PBrk − 1  ( P  ) } ,  (10)and F  Y k  ( ˆ P  ) =  { conv ( F   ∪{  p } ) |  F   ∈ F  Y k  ( P  ) }  (11)Then, the set  F  k ( ˆ P  ) of all  k -faces of  ˆ P   is given by F  k ( ˆ P  ) =  F  Bk  ( P  ) ∪  ˆ F  PBrk  ( ˆ P  ) ∪ F  Y k  ( ˆ P  ) (12)and F  Bk  ( P  ) ∩  ˆ F  PBrk  ( ˆ P  ) =  F  Bk  ( P  ) ∩F  Y k  ( ˆ P  ) =  ∅ .  (13)ˆ F  PBrk  ( ˆ P  ) ∩F  Y k  ( ˆ P  ) =  ∅ .  (14)In [9], [10], it is not mentioned about the dimension of faces, but it is easy to prove Lemma 1. IV. DYNAMIC CONVEX HULLALGORITHMA. DYNAMIC CONVEX HULL PROBLEM In this section, we consider a dynamic convex hullproblem that is, for a given  d -polytope  P  0 ∈  R d anda set of points  {  p n ∈  R d } N n =1 , we want to compute { ext ( P  n ) } N n =1 , where P  n = conv ( P  n − 1 ∩{  p n } ) , n  = 1 , 2 ,...,N   (15)ext  P  n is easily obtained from ( ∪ F  ∈F  d − 1 ( P  n )  ext  F  ).Lemma 1 means that we can compute  F  k ( P  n ) if weknow  F  k ( P  n − 1 ),  F  k − 1 ( P  n − 1 ) and color of them. In  other word, if we want to solve the dynamic convexhull problem by applying Lemma 1 directly, then weneed to update all  F  ( P  n ) and their color. However,when we consider dynamic hull problem, we need notto enumerate all faces of   P  n . All we need is to computeext  P  n and to update data to construct  P  n +1 .The idea is the following: Suppose that we are given F  d − 1 ( P  n − 1 ),  F  d − 2 ( P  n − 1 ) and the set of the normal-ized normal vector of   F   ∈ F  d − 1 ( P  n − 1 ). Then colorof   F   is assigned by (7) when  p n is given, and we cancompute  F  d − 1 ( P  n ) by applying Lemma 1. Here wenote that any facet  F   ∈ F  d − 1 ( P  n − 1 ) is blue, red oryellow and that there is no subfacet  E   ∈ F  d − 2 ( P  n − 1 )which is brown.  F  d − 1 ( P  n ) is computed by the follow-ing formula: F  d − 1 ( P  n ) =  F  Bd − 1 ( P  n − 1 ) ∪  ˆ F  P d − 1 ( P  n ) ∪F  Y d − 1 ( P  n ) (16)whereˆ F  P d − 1 ( P  n )=  { conv ( E   ∪{  p n } ) |  E   ∈ F  P d − 2 ( P  n − 1 ) }  (17)and F  Y d − 1 ( P  n ) =  { conv ( F   ∪{  p n } ) | F   ∈ F  Y d − 1 ( P  n − 1 ) }  (18)To compute  F  d − 1 ( P  n +1 ) by the same procedure, weneed to know  F  d − 1 ( P  n ),  F  d − 2 ( P  n ) and the set of thenormalized normal vector of   F   ∈ F  d − 1 ( P  n ).At this point we note the following:1) The normalized normal vector of   F   ∈ F  Bd − 1 ( P  n − 1 )is already computed and need not to compute again. If we know ext  F  ,  F   ∈  ˆ F  P d − 1 ( P  n ) ∪F  Y d − 1 ( P  n ), then thenormalized normal vector of   F   is computed by usingthe formula (6).2) Let  F   ∈ F  PBrk − 1  ( P  ) and ˆ F   = conv ( F   ∪{  p } ). Then  p  |∈  aff   F  , and, hence, we have ext ˆ F   = { ext  F  ∪{  p }} .Therefore, in this case, ext ˆ F   is easily computed.3) Let  F   ∈ F  Y k  ( P  ) and  F   = conv ( F  ∪{  p } ). Then  p  ∈ aff   F  , and, in general, we have ext  F    = { ext  F  ∪{  p }} .Therefore, we need additional effort to compute ext  F  .We will propose an efficient method in Subsection B..4) Since we do not have data of   F  PBrd − 3  ( P  n − 1 ), we cannot apply Lemma 1 to compute  F  d − 2 ( P  n ). Therefore,we need an alternative method to compute it. We willshow the method in Subsection B.. B. COMPUTATION OF  F  Y k  ( ˆ P  )  and  F  d − 2 ( P  n ) F  Y k  ( ˆ P  ) is computed by using the following result. Theorem 2  Let  F   ∈ F  Y k  ( P  ) and let  F  Gk − 1 ( P  ; F  ) = { E   ∈ F  Gk − 1 ( P  ) |  E   ⊆  F  } . Then we haveconv ( F   ∪{  p } )= conv ( {  p }∪ (  E  ∈F  Gk − 1 ( P  ; F  ) ext  E  )) (19)andext [conv ( {  p }∪ (  E  ∈F  Gk − 1 ( P  ; F  ) ext  E  )]=  {  p }∪ (  E  ∈F  Gk − 1 ( P  ; F  ) ext  E  ) (20)In (19), all the elements of   E  ∈F  Gk − 1 ( P  ; F  )  ext  E   aredifferent each other, that is, we need to eliminate du-plicate elements and leave exactly one of them. Wecan do this by a similar method with the merge sort.Moreover, we note because of (20), we need not com-pute the convex hull in (19). This is the point of The-orem 2.From Lemma 1 and Theorem 2, we immediatelyhave the following: Corollary 1 F  d − 1 ( P  n ) =  F  Bd − 1 ( P  n − 1 )  ∪  ˆ F  P d − 1 ( P  n ) ∪ F  Y d − 1 ( P  n ) (21)where ˆ F  P d − 1 ( P  n ) is given by (17) and F  Y d − 1 ( P  n ) =  { conv [(  E  ∈F  Gd − 2 ( P  n − 1 ; F  )) E  ) ∪{  p } ] | F   ∈ F  Y d − 1 ( P  n − 1 ) }  (22)To compute  F  d − 2 ( P  n ) we use the following: Theorem 3  Let ˜ F  PY d − 1 ( P  n ) = ˆ F  P d − 1 ( P  n )  ∪ F  Y d − 1 ( P  n ),where ˆ F  P d − 1 ( P  n ) and F  Y d − 1 ( P  n ) are given by (17), (22).Then,  F  d − 2 ( P  n ) is given by F  d − 2 ( P  n ) =  F  Bd − 2 ( P  n − 1 ) ∪  ˜ F  PY d − 2 ( P  n ) (23)where˜ F  PY d − 2 ( P  n ) =  F  1 ,F  2  ∈  ˜ F  PY d − 1 ( P  n )dim( F  1  ∩ F  2 ) =  d − 2 F  1  ∩ F  2  (24) C. Outline of Algorithm We note that the color of facets are red, yellow, orblue, and, hence, it suffices to determine the color of red or yellow facets and assume that the color of theremaining facets is blue. The color of subfacets are  also determined by a similar idea, that is, we need todetermine the color of subfacets which are facets of these red or yellow facets. An efficient method of thiscoloring process are shown in [10].Now we are ready to state an algorithm to solvedynamic convex hull problem (15), where we assumethat 1)  P  0 is a  d -polytope such that 0  ∈  int  P  0 2) F  d − 1 ( P  0 ) and  F  d − 2 ( P  0 ) are given and 3) each facet F   ∈ F  d − 1 ( P  0 ) has a data of its normal vector. Algorithm 1)  k  := 1;  P   :=  P  0 ;2) if   k > N   then stop;3) Determine  F  RY d − 1 ( P  ) :=  F  Rd − 1 ( P  ) ∪F  Y d − 1 ( P  );4) if   F  Rd − 1 ( P  ) =  ∅  then goto Step 18);5) Determine color of all subfacets  F   ⊆ F  RY d − 1 ( P  );6) for  F   ∈ F  RY d − 1 ( P  ) begin7) if   F   ∈ F  R ( P  ) then begin8) for ( E   ∈ F  P d − 2 ( P  )  ∧  E   ⊆  F  )make a new facet conv ( E   ∪{  p k } );9) end;10) else begin11)  G  := conv (  E  ∈F  Gd − 2 ( F  ) ext  E  );12) make a new facet conv ( G ∪{  p k } );13) end;14) remove  F  ;15) end;16) for  E   ∈ F  ROY d − 2  ( P  ) remove  E  ;17) generate ˜ F  PY d − 2 ( P  );18)  k  :=  k  + 1;19) goto Step 2);In Step 15),  F  ROY d − 2  ( P  k ) denote the set of all ( d − 2)-faces whose color is red, orange or yellow.At this point we note the following:1) Steps except 3), 5), 8) 11), 16), and 17) can beexecuted in constant time.2) The computing of Steps 3) and 5) is  T  R  +  O ( d | N  RY  | ), where  T  R  is the time for finding a red facet,and  N  RY  is the number of subfacets of   F  RY d − 1 ( P  ). Inour applications, most of   p k are very close to some of extreme points of   P   and  F  Y d − 1 ( P  ) =  ∅ . In this case, T  R  =  O ( | ext  P  | d 2 ) if there exists a red facet.2) The computing conv ( G ∪{  p k } ) can be done  O ( d 3 ),which is needed to compute the normalized normalvector by using (6).3) The computing time for Step 11) is  O ( N  G  log N  G ),where  N  G  =  E  ∈F  Gd − 2 ( F  ) | ext  E  | .5) The computing time for Steps 6) – 15) is O ( d 3 |F  P d − 2 ( P  ) | + |F  RY d − 1 ( P  ) | +  F  ∈F  Y  d − 1 ( P  ) N  G  log N  G ) . 6) The computing time for Step 16) is  O ( |F  ROY d − 2  ( P  ) | ).7) The computing time for Step 17) is  O ( N  2 N   N  N  N  ),where  N  N   =  |F  new | ,  F  new  is the set of new facets gen-erated in Steps 8) and 12), and  N  N  N   is the maximalnumber of   | ext  F  | ,  F   ∈ F  new .8) In theoretical point of view,  |F  d − 1 ( P  ) |  = O ( | ext  P  | ⌊ d/ 2 ⌋ ), and the computing time to solve dy-namic convex hull problem is  O ( N  ⌊ ( d +1) / 2 ⌋ ) However,our experience says that, in most of in our applications, |F  d − 1 ( P  ) |  depends on  d  but is proportional to  | ext  P  | and he computing time to solve dynamic convex hullproblem is  O ( n 2 )  ∼  O ( n 3 ). V. Conclusion In this paper, we proposed an improved method of thebeneath–beyond method, which can save both com-puting time and storage very much. The computingtime of the proposed method is 1 / 10  ∼  1 / 2 of that of the srcinal beneath–beyond method. References [1] H. H. Rosenbrock: A method of investigating sta-bility, Proc. 2nd IFAC World Congress, Basel,Switzerland, pp.590-594, 1963.[2] R. K. Brayton and C. H. Tong: Stability of dy-namical systems: a constructive approach, IEEETrans. Circuits and Systems, Vol.CAS-26, No.4,pp.224-234, 1979.[3] R. K. Brayton and C. H. Tong: Constructivestability and asymptotic stability of dynami-cal systems, IEEE Trans. Circuits and Systems,Vol.CAS-27, No.11, pp.1121-1130, 1980.[4] A. N. Michel, N. R. Sarabudla, and R. K. Miller:Stability analysis of complex dynamical systems:some computational methods, Circuits SystemsSignal Process, vol.1, no.2, pp.171-202, 1982.[5] A. P. Molchanov and E. S. Pyatnitskii: Stabilitycriteria for sector-linear differential inclusions, So-viet Math. Dokl., Vol. 36, No.3, pp.421-424, 1988.[6] Y. Ohta, H. Imanishi, L. Gong and H. Haneda:Computer generalized Lyapunov functions for aclass of nonlinear systems, IEEE Trans. on Cir-cuits and Systems-I, Vol.40, No.5, pp.343-354,1993.[7] F. Blanchini: Nonquadratic Lyapunov functionsfor robust control, Automatica, Vol. 31, No. 3,pp.451-461, 1995.[8] F. P. Preparata and M. I. Shamos, Computa-tional Geometry - An Introduction, Springer-Verlag New York, 1985.[9] L. Gong, Y. Ohta and H. Haneda, ”Complementof beneath–beyond theorem and dynamic con-vex hull maintenance,” COMP 90-33, 1990 (inJapanese).[10] H. Edelsbrunner, Algorithms in CombinatorialGeometry, Springer-Verlag Berlin Heidelberg,1987.
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