Jun 6, 1992 - (14) become algebraic ones. Efficient numerical methods for solving corresponding algebraic equations are discussed in [SI. Note that both (4) ...
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cally very efficient for the case of time-varying systems where the corresponding differential equations are stiff.
and differentiating, we get
Cr
= li
€PV - EPV,
-
80 1
) = -QiC-Qu+U. Substituting for li and we get
REFERENCES
from the original system, and simplifying,
U
Oi = B,oa - G , ( P ) u
where
(7)
B,o = B , - PB,
and
G , ( P ) = E P+ PB, - E B , P- B,
+ EPB,P.
(8)
Also =
where
B4oP - G , ( Q ) u
B40
=
(9)
B4 - EQB,
2. Gajic, D. Petkovski, and X. Shen, Singularly Perturbed and
Weakly Coupled Linear Control Systems-A Recursive Approach. New York: Springer-Verlag, 1990. P. Kokotovic and H. Khalil, Singular Perturbations in Systems and Control. New York: IEEE Press, 1986. K. W. Chang, “Singular perturbations of a general boundary value problem,” SIAM J. Math. Anal., vol. 3, pp. 520-526, Aug. 1972. D. R. Smith, “Decoupling and order reduction via the Riccati transformation,” SIAMRev., vol. 29, pp. 91-113, Mar. 1987. T. Grodt and Z. Gajic, “The recursive reduced-order numerical solution of the singularly perturbed matrix differential Riccati equation,” IEEE Trans. Automat. Contr., vol. 33, pp. 751-754, Aug. 1988. W. Miranker, Numerical Methods for StiffEquations. Amsterdam, The Netherlands; Reidel, 1981.
and
G 2 ( Q )= EQ
+ EQB,
-
B4Q - B,
+EQB~Q.
(10)
By setting G , ( P ) = 0, and G 2 ( Q )= 0, we get the decoupled system ci = B,,a =
€6 = B40P
( B , - PB3)a,
(11)
= (B4 - eQB,)P
(12)
where P and Q can be calculated from the following two stiff differential equations:
E P = -PB4 EQ
= B4Q
+ B, + E ( B , P- P B , P ) ,
(13)
+ B , + E ( Q B I+ Q B z Q ) .
(14)
The initial conditions for differential equations (13) and (14) are arbitrary [3], [4]. For time-invariant systems, equations (13) and (14) become algebraic ones. Efficient numerical methods for solving corresponding algebraic equations are discussed in [SI. Note that both (4) and (5) and (13) and (14) are stiff differential equations. They can be solved by using methods from [6]. It is known that due to a huge initial slope, solution of these equations requires a lot of time [6]. T.hus, the introduced decoupling transformation is
where
[
1
T-I= Z + E P N Q E P N N
with N = ( I - E Q P )- I , and P and Q are the solutions of ( 1 3) and (14), respectively. It is important to notice that in (4) and (5) one has to solve one Riccati and one Lyapunov equation sequentially. The total processing time in that case is greater than t,, where t , is the time for solving the Riccati equation. However, in (13) and (14) solutions of two Riccati equations are required, but due to parallelism the total processing time is t,.
Geometric Theory for the Singular Roesser Model A. Karamancioilu and F. L. Lewis Abstract-( A , E , B)-invariant and ( E , A , B)-invariant subspaces for the two-dimensional singular Roesser model are investigated. These subspaces are related to the existence of the solutions when the boundary conditions are in these subspaces. Also existence of a solution sequence in certain subspaces derived from the invariant subspaces is shown. The boundary conditions that appear in the solution when some semistates in the solution are restricted to zero are also investigated.
I. INTRODUCTION Two-dimensional (2-D) state-space models have been studied extensively during the past decade and a half. During this time many I-D state-space techniques [I], [2] have been generalized to their 2-D counterparts [3]-[6]. However, only a few publications have emerged considering the 2-D singular models, which are more general [7]-[IO]. In fact, 2-D singular models deserve better consideration due to the physical motivations and their richer structure. 2-D system models may assume spatial parameters as well as time, consequently, they do not have any natural notion of causality. The notion of recursibility is a commonly assumed property for 2-D state-space models, and allows their solution. The 2-D singular models, however, do not require recursibility. This allows them to model systems whose states at any value of the parameters depend on data from any direction in the 2-D plane. For instance, the heat conduction problem over a finite plane, and a nonrecursible mask can be modeled as a singular, but not state-space 2-D systems [7], [ l o ] . Also, the 2-D singular models allow algebraic constraints in addition to their dynamics, which is an improvement over state-space models. In this note we consider geometric notions for the 2-D Roesser model (SRM). The geometric approach classifies system constraints and dynamics with respect to subspaces. In singular systems, where
III. CONCLUSION A different viewpoint is taken in developing the decoupling transformations for singularly perturbed linear systems. The proposed transformations have the advantage over the previous ones since they also decouple the transformation equations (13) and (14), enabling us to perform the computations in parallel. This is numeri-
Manuscript received June 15, 1990; revised February 15, 1991. This work was supported by the National Science Foundation under Grant ECS-8805932. The authors are with the Automation and Robotics Research Institute, University of Texas at Arlington, Fort Worth, TX 76118. IEEE Log Number 9107242.
0018-9286/92$03,00 0 1992 IEEE
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there may be infinitely many system behaviors for a given input sequence, the geometric approach becomes essential for the analysis. In Section I1 we propose two alternative system representations for the 2-D SRM. Each representation is seen to be useful depending on the subdomain of interest that we seek the solutions in. We also analyze a Fornasini-Marchesini-like (F-M) model, which encompasses both representations. In Section 111 we define ( A , E , B) and ( E , A , B)-invariant subspaces for the SRM. We relate the subspaces derived from these to the solutions of SRM. In Section IV we restrict some boundary conditions of the SRM to zero, and investigate the behavior of the system. Due to the peculiarities associated with the splitting of the Roesser local semistate into horizontal and vertical components, we are forced to define some new geometric notions that have no 1 - D counterparts. 11. REPRESENTATIONS FOR THE ROBSERMODEL
A. System Representations Consider a system modeled by the singular 2-D Roesser model (SRM) given by
where _A and B are some constant banded matrices and &(_U)is a vector formed by stacking x(: j ' s and xy, j ' s (U;,j ' s ) columnwise. However, direct analysis using ( 2 ) is not feasible since the dimension of _A and _B increases as the domain of interest gets larger. This motivates us to search for other representations of (1) so that the analysis is independent of the size of the domain of interest. Let us partition E as [ E , Eb] such that E, and Eb have n l and n , columns, respectively. Also define
E,
+
+
+
(XI:
__ A X = B3
(2)
and
E2 = [0 E b ]
+
E l x i + , ,j
+ G x i ,j + l
=
A x ; ,j + B ~ i . 1
in the domain of interest, where x i , :=
[
(3)
This equation
introduces extra variables xf N ; i EM and x x , j ; j ENon the final boundary. However, due to the location of zero entries of E , and E,, these variables have zero contributions to (3). Therefore, we do not care about the values of these variables, which are merely for notational convenience. Let us define Z I , ] for i E M + 1 and j E N + 1 by
['"'1 [
,
:= x " J - ' ] , and let x t and xY on the initial 2:. J xY-l,J boundary be arbitrary to make the shifting well-defined. As in the preceding case, these variables are for notational convenience, and have zero contribution to the equations we shall introduce. The above definition amounts to regrouping the semistate vectors. That is, we shift the horizontal semistates up and the vertical ones right, and arrange the coefficient matrices accordingly. We call Z;, the shifted semistate. Using the shifted semistates, (1) can be expressed as
x,, := where the horizontal (local) semistate x t E R"1, vertical (local) semistate x:, E R n 2 (with n, n , = n ) , (local) input U,, E R", and the coefficient matrices are constant with E and A possibly rectangular. In the sequel we denote { H , H l ; . . , K - l } by [ H , K ) , where the integer H satisfies H < K . For the special case [0, K ) we use K. Using this notation, the domain of interest for (1) is the rectangular set of ( i , j ) ' s such that i E Mand j €8. Consider the mapping X h :M + 1 x &'+ R"1, X u :_M x N 1+ R"2, and U : _M x U+ R m . We call any evaluation of X " , X u , and U a horizontal semistate, a vertical semistate, and an J ) , (x,",J ) , and input sequence, respectively. We denote them by u l , respectively. A horizontal and a vertical semistate sequence pair is called a semistate sequence. We say a semistate sequence ( x l ,J ) is a solution for (1) if it satisfies (1) for some input sequence ( u ~ , ~Given ) . a subspace W, we denote the space of W-valued sequences by Y (W). Also we define the ( K , L ) t h forward boundary by { ( i , L ) l i E [ K , M ) } U { ( K , j j)E( [ L , N ) } , and the ( K , L ) t h backward boundary by { ( i , L ) 1 i E K } U {( K , j ) I j EL}. In the sequel the (0,O)th forward and ( M , N)th backward boundaries will be called the initial boundary and final boundary, respectively. Also a projection of a semistate sequence to a boundary or its subset is called the boundary conditions for the SRM. We assume neither existence or uniqueness of the solutions of (1). Our fundamental concern is existence of solutions that satisfy certain properties. To investigate this we consider problems of the following sort. Are the boundary conditions a projection of a solution in Y ( W ) , for some subspace W? We shall consider boundary conditions on various boundaries to highlight different properties of (1). This will amount to considering the existence problem in a subdomain and in its complement separately to show the existence of solutions in the domain of interest. Writing (1) for each ( i , j ) in the domain of interest, we obtain a single set of equations
[ E , 01
=
such that E = E, E,. We define A , , A , , A , , and A, likewise. Equation (1) can be expressed equivalently as
Ezi+l,j+l
~
=AIZi,j+l +A22i+,,j+Bui,j
(4)
for i E _M and j EN.The convenience of this equation for geometric analysis will be discussed in the following sections.
B. Subspace Representations In this subsection we present tools to obtain the subspaces of R" which contain solutions of (1). The solutions of (1) are, by ( 2 ) , contained in the subspace Ker [ _A E]. The dimension of this subspace depends on the size of the domain of interest, which is not desirable in the computational viewpoint. Therefore, we consider local semistates and restrict the geometric analysis to subspaces of R". Consider a Fornasini-Marchesini-like (F-M) model [ 111 given by
Fxi+l,j + l = G l x i , j + l + ' , x i + , , ,
+ Hui, j
(5)
where semistate vectors x i , E R " , input vectors U;, E R", and the coefficient matrices are constant. We let the domain of interest be the same as for (1). In this section we shall define some geometric notions for ( 5 ) . In the next section, we shall use these general notions to analyze the special cases of formulations (3) and (4) of the SRM, both of which have forms similar to (5). Definition 2.1: A subspace V C R" is said to be a (2 - D ) ( G I , G, , F , H)-invariant subspace for (5) if it satisfies
G,V
+ G,V C FV + Im H.
(6)
Theorem 2.2: Let V be a subspace of R". Equation (5) has a
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solution in Y (V ) for any Y-valued initial boundary conditions if and only if Y is a ( G I , G,, F,H)-invariant subspace for (5). Proof: (Sufficiency) Assume that (6) is satisfied for some Y. Then existence of a solution for (5) can be seen considering the problem in the order of (i, j ) = (0, 0), (0, l), (0,2), . . . , (0, N l), (1, 0), (2,0), * * . , (M - 1,O). Clearly there exist semistates in Y satisfying (5) on the first forward boundary. The existence on the second forward boundary can be seen similarly. Therefore, inductively, there exists a solution in Y (V). (Necessity) Obvious. 0 Our objective is to find the maximum Y satisfying (6). We shall use an equivalent definition for these subspaces for mathematical simplicity. Let us define the sets of subspaces K and L by
K := { Y C R"IG,V
+ G,YC
FV
+ Im H }
803
To show that V * is maximal, let f be a ( G I ,G,,-F, H)-invariant-subspace fo_r (5). We will show tht V * 3 V. Clearly Yo 3 Y. If V, 3 Y , then
",I
[ 0' F ']I[ V,
= [::]-I{
",+I
3
t
CL
This implies Y* 3 V 0 At this point we would like to comment on the ( F , G,,,, H I ,,)invariant subspace defined for the second model of Fornasini an& Marchesini (F-MI1 model) in [SI. In [8] V is said to be an (F,G,,,, HI,,)-invariant subspace of the system
and
if it satisfies Theorem 2.3: The sets K and L are equal. Proof: We first show that K C L. Let VI be a member of K. Suppose that VI# L, then there exists x E Y , such that
This implies at least
G,x#FV,
+ Im H
(7)
G,x$FV,
+ Im H
(8)
or holds true. If (7) holds true then
GIx
The definition of invariance in [SI is different from ours in the following sense. There exists a subspace, say W, not satisfying ( l l ) , however, if the semistates on the initial boundary are in this subspace then there exists a solution in Y ( W ) .That is, (11) may not be used to give both necessary and sufficient conditions for existence of a solution to (10). Example: Consider the system, modeled by the F-MI1 model, given by
+ G,O # F Y I + Im h
which contradicts with the hypothesis that Y , E K. Similarly, contradiction is obtained if (8) holds true. L c K is obvious. U (GI, G,, F,H)-invariant subspaces are closed under subspace addition, that is, if VI and V, are (GI, G,, F,H)-invariant subso is Y , V,. This implies existence of a maximal spaces for (3, ( G I , G,, F, H)-invariant subspace for (5). Theorem 2.4: There exists a maximal ( G I , G,, F,H)-invariant subspace Y * for (5) given by the limit of the ( G I G,, , F,H ) recursion
+
with Yo = R".The limit of the recursion is reached in no more than
n steps. Proof: This proof utilizes the equivalence of the sets K and L introduced previously. Clearly V, 3 VI. Let V,- 3 V,, then
,
= VK+I.
This shows that V, is a nonincreasing sequence. Since V, has a finite dimension n, this algorithm terminates in n steps. Y * is a member of L . By Theorem 2.3, it is also a member of K.
X i ,j + I
,
for N = M = 1. Y --s pan(
[ A]}is not an ( F ,G,,,,
in-
variant subspace for the given system, however, for any initial. conditions contained in this subspace one can find a solution contained in it. 0 In fact, there is no known way to define the invariant subspace for the F-MI1 model to obtain a counterpart to Theorem 2.2. This is due to the appearance of two inputs in (10). In the next two sections we use two special cases of the F-M-like equation given by (5) to derive geometric properties of (3) and (4), and hence of the SRM.
III. INVARIANTSUBSPACES A. (A, E, B)-Invariant Subspaces In this subsection we investigate existence of the solutions for (1) within a given set, when the initial boundary conditions are in certain subspaces related to this set. For this we make use of the representation (4) of (1). For the sake of compactness we denote ( A I , A , , E , B ) by (-4, E, B). Due to the peculiarities associated with the two parts x t and x:, of x,, in (l), the definition of ( A , E, B)-invariant subspaces is not sufficient for our purposes. We, therefore, define the next two subspaces V h and Y " by
z:]
Y h := ( x h e R n ~ ~ E[ Y
forsome x u ]
(13)
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and
Using this procedure, one can find the inputs and the semistates satisfying (1) in the remaining part of the domain of interest. The last statement of the theorem is straightforward to observe. 0 Next we change the initial boundary in the hypotheses of Theorem 3.1 to some other boundaries. In Corollary 3.2, we consider the boundary conditions given on the ( K , L)th forward boundary, which amounts to considering Theorem 3.1 in transilated coordinates. In Corollary 3.3 the boundary conditions are given on the ( K , L)th backward boundary. The corollary states the existence of a solution for (1) towards the final boundary. Since the proofs of the following corollaries are similar to that of Theorem 3.1, we shall omit them. Corollary 3.2: Let K E M and L E H be fixed numbers, and x i , E V h ; j E [ L , N) and xy, E V "; i E [ K , M ) be given. Then there exists a solution for (1) in the subdomain { ( i , j ) I i E [ K , M ) and j E [ L, N)} with these boundary conditions. Moreover, this solution satisfies x: E V h and xy. E V " in this subdomain. 0 Corollary 3.3: Let KE_M and L E N be fixed numbers, and E V h ; j E & and xy, E V "; i E K be given. Then there exists a solution for (1) in the subdomain obtained by subtracting { ( i , j ) I i E .K and j E &} from the original domain of interest. Moreover, this solution satisfies x: ,E V h and xy, E V " in this subdomain. 0
I[ ]I:
{
V " := x U ~ R n 2
E
V
for some x h )
(14)
where V is the maximal ( A , E , B)-invariant subspace for (1). These notions are needed for the geometric analysis of the SRM, and have no 1-D counterparts. The next result is our main result on existence of solutions given the initial boundary conditions. Theorem 3. I: Let the initial boundary conditions x i , E V h ; j E H and x : , E~ V "; i E M be given. Then there exists a solution for (1) with these boundary conditions. Moreover, this solution satisfies x t E V and xy, E V u for all i and j in the domain of * .interest. Proof: Let us use the shifted representation (4) of (1). We first show the existence in the shifted domain, then we complete the proof by shifting back to the original domain. We first show the existence of uo,o and Z,, satisfying the shifted equation (4), then u ~ ,uo, ~ , Z,, ,, and Z2, It will be seen that the remaining ones can be shown inductively. such that Z,,o E V . Select a Zi, such that Zo, E V , and Then
,
,.
,
xi^
B. (E, A , B)-Invariant Subspaces
,, ,,
hold for some xf, xf, x ; , x,b E V and u : , ~u, : , ~ U,:, U$. Adding the first block row (BR) of (15) and the second BR of (16), we obtain
AIZ0.l
,+
+ A22l.O = E%,, + Bu0.0
+
u ; , ~ Notice . that with Zl,, = xf, xf, and uo,o = E V , since xf, , xf, E V . Also notice that in equation space
,
,
(17)
Z,, ,
and (17) are equivalent. It is obvious that (18) is the shifted representation, of (1) at ( i , j ) = (0,O). Next, we do the same things for ( i , j ) = (0, 1) and ( i , j ) = (1,O). The following holds true for some values of the variables given on the right-hand side:
[
:;]'0.2
=
[ E0
' :I;]: : : [
O E][
(21)
-
Addition of the first BR of (19) and the second BR of (20), and the second BR of (19) and the first BR of (21) give rise to
A I % , , + A252.0 = E?,,, + Bu1.0 and
AlPO,,
+ A2Zl.l
= EZI.2
+ Bu0.1
In this subsection we relate ( E , , E,, A , B)-invariant subspaces of (1) to its solutions when the jinal boundary conditions are given. In the sequel we denote ( E , , E,, A , B) compactly by ( E , A , B ) . Let us modify V h and V " to define S h and S". To do this, we replace V by S in the definition of V h and V u , given by (13) and (14), where S is the maximal ( E , A , B)-invariant subspace for (1). Due to the structural similarity between (3) and (4), proofs of the following theorem and its corollary are similar to the previous ones and are therefore omitted. Given the boundary conditions on the final boundary (resp., the ( K , L)th boundary) are in an appropriate set, Theorem 3.4 (resp., Corollary 3.5) states the existence of a solution for (1) towards the initial boundary. Theorem 3.4: Let the final boundary conditions x h , E Sh;J EH and x:, , E S " ; iE_M be given. Then there exists a solution for (1) in Y ( S ) . 0 Corollary 3.5: Let K E M and L E&' be fixed numbers, and x i , i ~ Sj ~E &; and xy, E S " ;i E K be given. Then there exists a solution for (1) in the subdomain { ( i , j ) I i E K and j EL}. Moreover, there exists an S-valued solution in this subdomain. 0 Note that ( E , A , B)-invariance does not rely on a shifted representation. Therefore, in Theorem 3.4 we have imposed a stronger condition on a solution, namely we have been able to keep it in Y ( S ) .In Theorem 3.1, however, we have kept the shifted semistates Zi,,in V , but due to shifting back to the original representation, we could not guarantee that the semistates x i , are in V .
C . 2 - 0 Concatenation of Solutions In this subsection we introduce two subspaces that can be related to the existence of solutions when any forward or backward boundary conditions are given. We define the subspaces ( S V ) hand ( S V ) " by
( S v ) h : ={.h€..Ij[
;:]
E
V
,,
with the obvious definitions of Z,, ZI,,, u , , ~and , u0,,. Shifting back to the original representation, it is not difficult to see that (1) is satisfied for ( i , j ) = (0, 1) and ( i , j ) = (1,O).
for some x u and
[ f:]
E S for some
x"
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and
for some x h and
[
We first use the shifted representation (4) of (1). Considering the limit is also an ( A , E , &invariant subspace for (l), we compute the variables on the (1, 1)th forward boundary as in the proof of Theorem 3. I. However, on this boundary there exist shifted semisES
for some g h ]
(23)
where V and S are as defined previously. These new quantities are needed to extend the 1-D notion of subspace intersection to the 2-D Roesser case. The next theorem can be viewed as a concatenation property for the 2-D singular Roesser model. In 1-D systems two trajectories of a system can be concatenated if the initial condition of one trajectory coincides with the final condition of the other. This makes the intersection of admissible initial conditions and final conditions subspaces particularly important in 1-D singular models [ 121. Each of the subspaces we defined above is composed of two subspaces with similar functions, so that there exists analogy between the 1-D and 2-D cases which can also be displayed by the following theorem. Theorem 3.6: Let KE_M and LE_N be fixed numbers, and X ; , , E ( S V ) ~j ;E L and X;,,E(SY)"; i E K begiven. Then there exists a solution for (1) such that the semistates between the ( K , L)th backward boundary and the initial boundary are members of S, and the rest of the domain of interest, the horizontal semistates are in V h and the vertical ones are in V u . Proof: Existence of the semistates satisfying (1) between the ( K , L)th backward boundary and the initial boundary is a direct result of Corollary 3.5. Note that, by applying this corollary we have imposed no constraints on x;,,; j E _ L and x,lfL; i E K . Therefore, using Corollary 3.3, existence of a solution for (1) between the ( K , L)th backward boundary and the final boundary can be shown. Note that in the process of using this corollary, we use the shifted representation (4) of (1). The shifting back to the original representation determines x;, ,;j EL and x,lf L ; i E K . 0 Another version of Theorem 3.6 where ( K , L)th forward boundary conditions are given can be treated similarly, and IS omiaed for the sake of compact presentation of the material.
IV. ATTAINABLE SUBPACES In this section we propose a subspace of semistates that can be reached from zero boundary conditions, or equivalently, that can be driven to zero boundary conditions. These subspaces are relevant in connection with finite-memory filters. Attainable subspaces, which are introduced next, play a major role in constructing these subspaces. We call V, an attainable ( A , E , B)-invariant subspace of (1) if it is contained in limit of the ( A , E , B ) recursion [defined by (9)] with the modified initial condition V, = 0. We define an attainable ( E , A , B)-invariant subspace of (1) similarly, and denote it by So. Attainable subspaces extend the usual notion of "controllable subspace" to singular 2-D systems. Let us define V," by replacing V by Vo in the definition of V h , given by (13). We define V i , S:, and Si by modifying V ", S h , and S" likewise. Theorem 4.1: Let M L n and N L n . Let the initial boundary conditions xt, E V,"; J E& and x;.~E V i ; i EM be given. Then there exists a solution for (1) such that all semistates on the ( r , r)th forward boundary is zero for some r 5 n. Proof: ( A , E , B ) recursion with the initial condition V , = 0 produces nondecreasing V,. Since V, can have maximum dimension n , the recursion has a limit reached in no more than n steps. Let it be the rth step ( r In).
,
805
,
tates contained in V,- . Notice that multiplying (9) by employing the identity A A - ' X C X ( A and operator and subspace) imply
[ :;]
and
X are any linear
that is, existence of the shifted semistates in V , - , is guaranteed. Then existence of the shifted semistates contained in Yr-2 can be shown for (2,2)th forward boundary, and so on. Finally, at the rth step, the semistates on the ( r , r)th boundary are in Yo = 0. Shifting back to the original representation completes the proof. 0 This theorem shows the existence of inputs between the initial boundary and the ( r , r)th boundary (rE_n) so that if the initial boundary conditions are in an appropriate set, then there exists a solution with zero semistates on the ( r , r)th boundary. The next theorem is the attainable ( E , A , @-invariant version of Theorem 4.1, that is, the final boundary conditions in an appropriate set are given, and we show the existence of a solution that can be zero on the ( r , r)th boundary ( r E E ) . Due to its similarity to the preceding proof, we omit it. Theorem 4.2: Let M 5 n and N 2 n, and let the final boundary conditions x h , , E S , ~j ;E & and x , U N e S i ;i E M be given. Then there exists a solution for (1) with these boundary conditions such that all semistates on the (N- r , M - r)th backward bound0 ary are zero for some r i n . In Theorem 4.1, it is possible to replace the initial boundary by the ( K , L)th forward boundary provided that K + n IM and L n IN . Also, a similar comment holds true for Theorem 4.2. Let us define the subspaces ( S V ) ; and ( S Y ) ; by modifying ( S V ) hand ( S V ) " ,respectively. To do this, we change Y to Yo and S to S, in the definitions of ( S Y ) hand ( S V ) " ,given by (22) and (23). The next theorem generalizes Theorems 4.1 and 4.2 to the ( K , L)th forward boundary provided that ( K , L ) is sufficiently far away from the initial and final boundary. Theorem 4.3: Let M 2 2 n and N 2 2 n be satisfied. Also let K E [ n , M - n) and L E [ n , N - n ) be fixed numbers, and x i , , E ( S V ) : ; j E [ L , N ) and x , U L ~ ( S V ) i ;E [ K , M )be given. Then there exists a solution for (1) such that all the semistates on the ( K + r , L r)th and ( K - s, L - s)th forward boundaries are zero for some r , s In . Proof: Theorem 3.6 together with Theorems 4.1 and 4.2 lead 0 to the conclusion. This theorem states that if the ( K , L)th forward boundary conditions are in an appropriate set, then there exists an input sequence so that the solution is zero outside the band whose thickness is less than or equal to 2 n. This conclusion holds true if the ( K , L)th forward boundary is replaced by the ( K , L)th backward boundary.
+
+
V . CONCLUSIONS
We have investigated ( A , E , B)-invariant and ( E , A , B)-invariant subspaces for the two-dimensional (2-D) singular Roesser model. Also, their attainable invariant versions have been investigated. Particularly for (attainable) ( A , E , B)-invariance we have made use of the shifted representation of the 2-D singular Roesser model. We have shown the relationship between (attainable) ( A , E , B ) invariant subspaces of the singular Roesser model and its solutions
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TRANSACTIONSON AUTOMATIC CONTROL,VOL. 37, NO. 6, JUNE
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towards the final boundary. Similarly, the (attainable) ( E , A , B)-invariant subspaces are related to the solutions towards the initial boundary. Also we have presented a subspace such that if the boundary conditions on any boundary are in it, then there exists a solution with these boundary conditions. Due to the peculiarities associated with the splitting of the Roesser local semistate into horizontal and vertical components, we have been forced to define some new geometric notions that have no 1-D counterparts. ACKNOWLEDGMENT The authors would like to thank Prof. A. Banaszuk and Prof. M. Kociecki of Warsaw Tech. for their valuable conversation with them on this topic. REFERENCES T. Kailath, Linear Systems. Englewood Cliffs, NJ: Prentice-Hall, 1980. W. M. Wonham, Linear Multivariable Control: A Geometric Approach. New York: Springer-Verlag, 1979. M. Morf, B. C. Levy, and S. Kung, “New results in 2-D systems theory-Part 1: 2-D polynomial matrices, factorization, coprimeness,” Proc. IEEE, pp. 861-872, June 1977. S . Kung, B. C. Levy, M. Morf, and T. Kailath, “New results in 2-D systems theory-Part 2: 2-D state-space models, realization and the notions of controllability, observability and minimality,” Proc. ZEEE, pp. 945-961, June 1977. T. Kaczorek, 2 - 0 Linear Systems. New York: Springer-Verlag, 1985. G. Conte and A. Perdon, “A geometric approach to the theory of 2-D systems,” IEEE Trans. Automat. Contr., vol. 33, pp. 946-950, Oct. 1988. G. Beauchamp, “Algorithms in singular systems,” Ph.D. dissertation, School Elect. Eng., Georgia Inst. Tech., Atlanta, GA, Jan. 1990.
G. Conte, A. Perdon, and T. Kaczorek, “A geometric approach to
singular 2D linear systems,” IFAC Workshop, Prague, Czechoslovakia, Sept. 1989. T. Kaczorek, “Singular general model of 2-D systems and its solutions,” IEEE Trans. Automat. Contr., vol. 33, pp. 1060-1091, 1988. F. L. Lewis, “A survey of 2-D implicit systems,” presented at the Imacs Int. Symp. Math. Intelligent Models in Syst. Simulation. Brussels, Belgium, Sept. 1990. E. Fornasini and G. Marchesini, “Doubly indexed dynamical systems: State-space models and structural properties,” Math. Syst. Theory, vol. 12, pp. 59-72, 1978. K. Ozcaldiran and F. Lewis, “Generalized reachability subspaces for singular systems,” SIAM J. Contr. Optimiz., vol. 27, pp. 495-510, May 1989.
Control of Interconnected Nonlinear Dynamical Systems: The Platoon Problem Shahab Sheikholeslam and Charles A. Desoer
1992
platoon); each vehicle is driven by the same input U and the state of the kth vehicle affects the dynamics of the ( k + 1)th vehicles; furthermore, the dynamics of each vehicle is affected by its (local) state-feedback controller. Under very general conditions, it is shown that for sufficiently slowly varying inputs, decentralized controllers can be designed so that the platoon maintains its cohesion.
I. INTRODUCTION The congestion of highways is a universal problem. One possible remedy is to form “platoons”: N vehicles follow, under automatic control, a lead vehicle. The tight control allows a larger number of vehicles per kilometer, hence, produces a greater traffic flow per lane. The platoon concept has been studied by several groups over the years. A recent overview of the platoon concept, the associated communication requirements, and some proposed control laws are described in [2], [14], and references therein. In [2], [13], and [15], using a simplified model of vehicle dynamics we studied the platoon control problem: a decentralized control law has been developed for this model and the platoon performance evaluated. Certain specific properties of the model greatly simplified the decentralized controller design. The platoon concept with the assumed pattern leads to a special interconnection of dynamical subsystems, each one representing a vehicle. The purpose of this note is to demonstrate that under general qualitative conditions imposed on the nonlinear dynamical subsystems interconnected as above, it is possible to obtain appropriate dynamical behavior for the overall system using only decentralized control. The study of interconnections of dynamical systems has a long history usually under the heading of “Large Scale Systems.” Some of the main results are to be found in [ 101 and [ 111. The treatise [ 121 on singular perturbations is an excellent reference on the concepts and techniques associated with the notions of slow and fast dynamics. From a system design point of view these studies show that two aspects are very important: 1) the graph of the interconnection [ l 11; and 2) the time-scale separation of dynamics [ 121. The system under study has a special interconnection which is dictated by the platoon concept: the system consists of N nonlinear subsystems, each one representing a vehicle. To maintain the cohesion of the platoon, the lead vehicle’s velocity and acceleration is transmitted to each vehicle of the platoon, and vehicle k measures the distance A k between it and the preceding vehicle. As an approximation we may view the dynamics of the sensors and actuators and that of the engine as fast with respect to that of the vehicle. We show that by suitable design of each controller in each vehicle it is possible to achieve the following: given that the platoon is operating in the steady state at const ant velocity, U , at t = t o , and that the lead vehicle accelerates to reach a constant velocity U , at some later time T , decentralized control laws can be designed so that for all k 2 1, A , ( t ) is bounded on [ t 0, 03) and, for some < 1, llAA-)llm5 a I I A k - , ( * ) I I m + II@k(.)llmwhere @ J t ) - + O exponentially as t 03 at a rate controlled by the choice of the control laws; here 11 . denotes the sup norm over [ T , 00). --$
,Abstract-The problem in this note is motivated by a highway automation project [Z]. The overall system consists of N vehicle (the
Manuscript received July 16, 1990; revised December 21, 1990. This work was supported in part by the PATH Project, under Grant RTA-74H221, and by the National Science Foundation under Grant ECS-88-05767. The authors are with the Department of Electrical Engineering and Computer Science, University of California at Berkeley, Berkeley, CA 94720. IEEE Log Number 9107244. 0018-9286/92$03.00
11. PROBLEM MOTIVATION
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1 vehicles traveling in the same Consider a “platoon” of N lane of a straight stretch of highway and closely following one another. Initially, all vehicles travel at the same constant velocity, say U. The lead vehicle is labeled “ I , ” the next one is labeled “1,” and the last one “ N ” : x k denotes the abscissa of the rear bumper of the kth vehicle and x, that of the lead vehicle; each vehicle is
0 1992 IEEE
