The Hierarchical Control of ST-Finite State Machines Peter E. Caines
y
Vineet Gupta
z
Gang Shen
Abstract
This paper follows [3] where a new notion of state aggregation for nite machines was introduced via the concept of the dynamical consistency (DC) relation between the blocks of states in any given state space partition . This formulation results in a de nition of high level dynamics on the nite (partition) machine whose states correspond to the given partition elements. This paper treats the more general case of ST-systems where there is a preferred sense of ow from a set of source states (S ) to a set of target states (T ) which is to be achieved by (hierarchical) control. A generalisation of the theory of [3] to ST-systems is given which includes the generalisation of the notions of dynamical consistency, in block controllability and hierarchical feedback control on the associated hierarchical lattices.
M
Keywords: Controllability, Discrete Event Systems, Hierarchical Control, Lattices, Dynamical Consistency, Source-Target Systems, State Space Partitions.
1 Introduction Hierarchically structured information and control systems occur for at least two related reasons: rst, the great complexity of many natural and designed systems limits the ability of humans and machines to describe and comprehend them, and, second, the inherent limitations on the information processing capacity of feedback regulators leads to such regulators (and possibly the controlled systems themselves) being organised in special, in particular hierarchical, con gurations. Theoretical work on hierarchical control has a large literature and has connections to, among other subjects, game theory, mathematical programming and optimal resource allocation. These topics are presented together with their connections to control theory in [6]. More recently, formulations of hierarchical control have appeared in stochastic control [9], automated highway system studies [10] and within the supervisory Department of Electrical Engineering, McGill University, 3480 University Street, Montreal, Quebec, Canada H3A 2A7. Work partially supported by NSERC grant number OGP 0001329, NSERC-FCARNortel grant number CRD 180190 and NASA-Ames Research Center grant number NAG-2-1040. @cim.mcgill.edu. y Also aliated with the Canadian Institute for Advanced Research. z Xerox Palo Alto Research Center, 3333 Coyote Hill Road, Palo Alto, CA, 94304, USA.
[email protected].
1
control formulation of discrete event system theory (e.g. [7], [8], [13], [14]). The work in this paper follows that of Wei and Caines in [12], [3] and [11], where a new notion of state aggregation is introduced via the concept of the dynamical consistency (DC) relation between the sets of states constituting the members of any given partition of the state space. This formulation results in a de nition of high level dynamics on the nite (partition) machine M whose states correspond to the given partition elements. The DC relation is then similarly de ned on any further partition of the state set of M ; and so on. The theoretical development in [3] gives the lattice structure of the class of so-called in block controllable (IBC) partition machines. The notion of state aggregation given by the concept of the DC dynamics of a partition machine permits a natural construction of a large class of hierarchical control structures on any given nite state machine. It is to be noted that in the purely graph theoretic setting, without any controlled dynamics, an idea related to that of DC dynamics is to be found in [5]. We note that there is a natural parallel between the formulation of levels in hierarchical system theory and their de nition in hybrid system theory, where discrete systems play the roles of both models and controllers for ner continuous state systems. The reader is referred to [2] and [4], where the theory of [3] is generalised to the hybrid case, and to [1]. In the analysis and design of hierarchical control systems, one is often interested in the reachability of some set of terminal states from some initial (or start) set of states. Many examples of systems with such a preferred sense of ow are to be found among natural and designed systems. As a result, in this paper, we consider a generalisation of the theory of hierarchical control initiated in [3] to systems in which there are distinguished source and target sets for the controlled ow.
2 The ST-Dynamical Consistency Relation and Partition Machines
Consider a nite state machine, M = (X; U; ), where X is a nite set of states, U is a nite set of inputs , and : X U ! X is the state (partial) transition function of the system. We shall use the standard notion U for the set of all nite sequences (including the empty string ) of elements of U and if v 2 U is a pre x (respectively, proper pre x) of u 2 U , we write v u (respectively, v < u). Denote a distinguished subset of states called source states and a distinguished subset called terminal states with S X and T X respectively. We shall term nite systems with such distinguished subsets ST-systems. We note that in the case where the sets S , T and X are identical all the de nitions and results below become consistent with their counterparts in [3]. In this sense this paper generalises [3]. 2
De nition 2.1 (ST-Controllability) (i) (Strong ST-Controllable) M is said to be strongly ST-controllable if and only if for every x 2 S and for every y 2 T , there exists a u 2 U such that (x; u) = y. (ii) (Weak ST-Controllability) M is said to be weakly ST-controllable if and only if for every x 2 S there exists a y 2 T and there exists a u 2 U such that (x; u) = y: 2 Clearly, strong ST-controllability (and hence weak ST- controllability) is a signi cantly weaker property than standard controllability because the latter requires the accessibility of every state from every other state.
Example 1 In the following 8-state machine M8, shown in Fig. 1, S = f1g and T = f8g. Clearly, this ST-system is both strongly ST-Controllable and weakly ST-Controllable.
2
We recall that a partition of a nite set X is a collection of pairwise disjoint subsets called blocks, Xi X , such that Xi \ Xj = ; for i 6= j and X = [ni=1 Xi, where n = jj. A partial order relation (less than) on partitions of X is de ned such that for two partitions, 1 = (X11; X21; :::; Xm1 ) and 2 = (X12 ; X22; :::; Xn2), 1 2 if and only if for each block Xi1 2 1, there is an Xj2 2 2 such that Xi1 Xj2, where 1 i m; 1 j n, i.e. 1 is a re nement of 2. In the example above, for the two partitions 1 = (1; f2; 4g; 3; f5; 7g; 6; 8); 2 = (f1; 2; 3; 4g, f5; 6; 7; 8g), we see that 1 2.
De nition 2.2 (I (Xi; S; T ), O(Xi; S; T )) Consider a partition = fX1; X2; :::; Xng, n = jj, of the state set X of a nite state machine M . In each block Xi 2 , 1 i n, we specify two subsets, respectively I (Xi; S; T ) and
O(Xi; S; T ), which are termed the local entries (or input states) and local exits (or output states); these are de ned respectively as follows: x 2 I (Xi; S; T ) , x 2 S or there exists x0 2 (X ? Xi ), i.e. the complement of Xi in X , and there exists u 2 U such that (x0 ; u) = x; y 2 O(Xi; S; T ) , y 2 T or there exists y0 2 (X ? Xi) and there exists u 2 U such that (y; u) = y0. 2 In the partition = (f1; 2; 3; 4g; f5; 6; 7; 8g) of Example 1, for X1 = f1; 2; 3; 4g, I (X1; S; T ) = f1; 3; 4g and O(X1; S; T ) = f2; 3g; for X2 = f5; 6; 7; 8g, I (X2; S; T ) = f5; 6g and O(X2; S; T ) = f6; 7; 8g. We now shall de ne the appropriate generalisation of the notions of dynamical consistency and partition machines to ST-systems. 3
4
2
6
S
T 1
8 3
7
5
Figure 1: An 8-state machine
De nition 2.3 (ST-Dynamical Consistency (ST-DC))
The relation of ST-dynamical consistency for an ordered pair of blocks hXi; Xj i in a partition is de ned as follows: hXi; Xj i 2 is called ST-dynamically consistent (ST-DC) if one of the following cases holds: ( ) i 6= j . For each x 2 I (Xi; S; T ) (i) there exists y 2 O(Xi; S; T ) and there exists ui;i 2 U such that (x; u0i;i) 2 Xi for all u0i;i ui;i, and (x; ui;i) = y; and (ii) for at least one such y, there exists z 2 I (Xj ; S; T ) and there exists ui;j 2 U such that (y; ui;j ) = z. We write ui;i ui;j as uji , where denotes concatenation. ( ) i = j For every x 2 I (Xi; S; T ) there exists y 2 I (Xi; S; T ), and there exists some non-null ui;i 2 U such that (x; u0i;i) 2 Xi for all u0i;i ui;i and (x; ui;i) = y. 2 In the case i = j we simply require that each input state of Xi has a non-empty controlled step which keeps it within the input set so as to form a pseudo-cycle. This is postulated essentially to obtain desirable properties for the formal language of high level transitions as de ned below.
Example 2 In Example 1, hf1; 2; 3g; f4; 6gi is ST-DC. Here the rst partition block is such that its input set is equal to the whole block and the output set is the pair of elements f2; 3g. The second block is such that its input set is the element f6g, which is accessible in one step from f2g, and the output set of f4; 6g is the whole set. But hf1; 2; 3g; f4gi, where f4g is both an input and and output set, is not ST-DC since f4g is not accessible in one step from
the rst set. A general ST-DC relation is represented in Fig. 2, in which we see displayed I -unreachable states (i.e those not reachable from the input states of Xi) and O-inaccessible states (i.e. those from which the output states of Xi are not reachable) in the block Xi. 2 4
Xi
I
O
I Xj
Figure 2: hXi; Xj i is ST-DC but not DC A high level transition (input) event Uji is de ned, and denoted by Uji , if and only if hXi; Xj i is ST-DC; in other words, for any pair i; j , Uji is de ned if and only if the conditions of De nition 2.3 hold. The term high level transition is to be taken relative to the base machine M, but no mention shall be made of this whenever the context is clear. In order to de ne the partition machine M = (; U ; ) (based upon the partition of M), we de ne the state transition function : U ! by (Xi; Uji) = Xj , Xi; Xj 2 ,1 i; j jj, whenever hXi; Xj i is ST-DC. We may now de ne : (U ) ! recursively as follows: rst set (Xi; ) = Xi; then, for strings in (U ) of length one, the de nition of gives (Xi; Uij ) = Xj whenever hXi; Xj i is ST-DC; nally, for strings U in (U ) of length greater than one we set (Xi; Uij U ) = (Xj ; U ) if (Xi; Uij ) = Xj and if (Xj ; U ) is de ned. Here indicates the concatenation of two strings. From this recursive de nition we immediately obtain the following fact as a special case.
Lemma 2.1 (Semi-Group Property) (Xi; U1 U2 ) = ( (Xi; U1 ); U2) as long as (Xi; U1) and ( (Xi; U1 ); U2) are de ned, where means the concatenation of two sets. 2 It may be veri ed that when a chain of high level transitions is de ned it is the case that the appropriate generalisation of De nition 2.3 holds; that is to say, for every state in the I -set of the initial block there exists a path through a chain of O and I state sets in the successive blocks which terminates in the I states of the nal block. It is to be noted that the de nition of one step pseudo-cycles in De nition 2.3 permits well de ned chains of high level transition event strings to contain sets of pseudo-cycles, i.e. high level identity elements. For a given state space X and partition , let S denote the elements of containing states lying in S , i.e., Xi 2 S if Xi \ S 6= ;, and similarly let T denote the elements of containing states in T , i.e., Yj 2 T if Yj \ T 6= ;. Then we may give the following de nition 5
of the controllability of the partition machine M = (; U ; ).
De nition 2.4 (ST-Between Block Controllability) A partition machine M = (; U ; ) is
(i) strongly ST between block controllable (strongly ST-BBC) if it is the case that for every Xi 2 S and Yj 2 T , there exists a U 2 (U ) such that (Xi; U ) = Yj ; and (ii) weakly ST between block controllable (weakly ST-BBC) if it is the case that for every Xi 2 S , there exists a Yj 2 T and there exists a U 2 (U ) such that (Xi; U ) = Yj . 2 We note that we may obtain weak ST-BBC from strong ST-BBC by simply exchanging a universal quanti er for an existential one. As this applies to the results below we henceforth only discuss strong properties in detail and leave the development of its analogous weak properties to the reader.
De nition 2.5 (ST-In Block Controllability) A block Xi; 1 i jj is ST-in block controllable (ST-IBC) if and only if either I (Xi; S; T ) = ; or O(Xi; S; T ) = ; or the following holds: (i) For every x 2 I (Xi; S; T ) there exists y 2 O(Xi; S; T ), and there exists u 2 U such that (x; u0) 2 Xi for each u0 < u, and (x; u) = y. (ii) For every y 2 O(Xi; S; T ) and for every z 2 O(Xi; S; T ) with z = 6 y, there exists v 2 U such that (y; v0) 2 Xi for each v0 < v, and (y; v) = z, i.e., the states in O(Xi; S; T ) are mutually accessible with respect to Xi. A partition is said to be ST-in block controllable (ST-IBC) if every block of is ST-IBC.
2
In other words, any input state of an ST-IBC block Xi 2 must have an internal trajectory going to an exit state of Xi, and the exit states of Xi must be mutually accessible, i.e., Xi is weakly ST-controllable and O(Xi; S; T ) is mutually accessible with respect to Xi. In the 8-state machine of Example 1, the block f1; 2; 3; 4g is ST-IBC, although it is not IBC in the sense de ned in [3]. We observe that in the standard case where S = T = X , ST-in block controllability specialises to the standard IBC property because in this case all elements of any block Xj are mutually accessible. Further, in case S = T = X , ST-between block controllability clearly implies that the standard BBC property holds. IBC We let IBC ST (X ) represent the collection of all ST-IBC partitions of X and MST (M) denote the collection of all partition machines of M corresponding to partitions in IBC ST (X ).
Theorem 2.1 If M is an ST-in block controllable partition machine of M, then M is strongly (respectively, weakly) ST-between block controllable if and only if M is strongly (respectively, weakly) ST-controllable. 6
Proof:
We only prove the implications concerning strong ST-controllability as the weak case follows by an analogous argument. =): Given M 2 MIBC ST (M). Assume M is strongly ST-controllable and let us look at arbitrary blocks X1 in S and Xk 2 T . Now consider any x 2 X1 \ S and y 2 Xk \ T ; then by De nition 2.1, we know there is a trajectory from x to y. Suppose this trajectory traverses a chain of blocks in , in the order X1 ) X2 ) ) Xk . Because all of the blocks are ST-IBC it follows that the block pairs hX1; X2i; hX2; X3 i; ::; ; hXk?1; Xk i satisfy the conditions to be ST-DC. Thus, we have well-de ned high level transitions, (X1; U21 ) = X2 ; (X2; U32 ) = X3; :::; (Xk?1; Ukk?1 ) = Xk , and hence (X1; U21 U32 ::: Ukk?1) = Xk , i.e., M is strongly ST-between block controllable. (=: Let M be strongly ST-between block controllable. Consider an x 2 S and y 2 T . Then there must be some X1 2 S and Xk 2 T such that x 2 X1 and y 2 Xk . By De nition 2.4, we know, there is a sequence of partition machine transitions U21 U32 ::: Ukk?1 from X1 to Xk . This concatenation gives a trajectory from x to some input state of Xk . Because of the ST-in block controllability of Xk , this input state must have an in-block trajectory leading to y 2 O(Xk ; S; T ). Thus, we get a complete trajectory from x to y. Hence, M is strongly ST-controllable. 2
3 The T-Accessible Realisation of an ST-Machine and Its Associated ST-IBC Lattice The S-unreachable and T-inaccessible states (i.e. the states from which T is not reachable) of a machine M are irrelevant to the ST-control problem; this is because any path from a state in S to a state in T cannot pass through either of the S-unreachable and Tinaccessible states. From the point of view of ST-controllability, that all the S-unreachable or T-inaccessible states may be deleted before we investigate the ST-control problem. This would yield a minimal realisation M0, by which we mean every state of M0 should be Sreachable and T-accessible, and hence there are no redundant states with respect to the ST-reachability problem for the resulting ST-system. In this paper, however, it is sucient to eliminate the T-inaccessible states of a nite state machine M, we denote the nite state machine obtained after this thinning process by Mt = (Xt; Ut ; t ).
De nition 3.1 (Chain Union [C ) The chain union of two partitions 1 and 2 of X , denoted 1 [C 2 , is the least upper bound of 1 and 2 with respect to in the set of partitions of X . 2 The following algorithm may be used to calculate the chain union of two partitions. Each of the distinct blocks Zi ofS the partition 1 [C 2 ; 1; 2 2 IBC ST (X ), can be constructed recursively by setting Zi = Nn=1 Zi;n; N = maxfj1 j; j2jg, where Zi;n is given by the following 7
algorithm: Set Zi;1 = Xi for some Xi 2 i , then for all n; 1 n < N ,
Zi;n+1 =
Zi;n [ fX 0 2 1 : Zi;n \ X 0 6= g n odd, Zi;n [ fY 0 2 2 : Zi;n \ Y 0 6= g n even
Alternatively, let us de ne x x0 if either x; x0 2 Xi1 2 1, 1 i j1 j, or x; x0 2 Xj2 2 2 , 1 j j2j. Then the equivalence classes of the transitive closure of this relation are the blocks of 1 [C 2 . The property of ST-in block controllability which we have de ned above restricts the set of partitions in such a way that it is preserved under chain union.
Theorem 3.1 For 1 ; 2 2 IBC ST (Mt ), the chain union of 1 and 2 is ST-IBC, i.e., C IBC 1 [ 2 2 ST (Mt). Proof: Suppose 1 and 2 are two ST-IBC partitions of Xt , let us look at their chain union 1 [C 2 = fZ1; Z2; :::; Zr g. First, we prove that if A and B are ST-IBC, A [ B is ST-IBC whenever A \ B = 6 ;. It is clear that I (A [ B; S; T ) I (A; S; T ) [ I (B; S; T ) and O(A [ B; S; T ) O(A; S; T ) [ O(B; S; T ) so that we need only consider the nontrivial cases when O(A; S; T ) [ O(B; S; T ) = 6 ;. After eliminating all states which are not accessible to the output states of ST-IBC blocks, at least one of the output states of A or B is in A \ B . This is shown as follows. Since we may suppose O(A; S; T ) 6= ; (recall O(A; S; T ) [ O(B; S; T ) 6= ;), without loss of generality, take x 2 A \ B , y 2 O(A; S; T ) then there is an internal path in A from x to y (since Xt has been thinned, i.e., all T-inaccessible states in X have been eliminated, such that the states left in Xd must have a trajectory to T via an output state of its block; and all output states of A in an ST-IBC partition, are mutually reachable). If (i) y 2 A \ B , the above claim holds; otherwise if (ii) y 2 A ? (A \ B ), then the path within A from x to y of the form x ! x1 ! ::: ! xk ! y must have a one step transition leaving B of the form b ! a, where a 2 A ? (A \ B ); b 2 A \ B and b; a 2 fx; x1 ; :::xk ; yg. Thus, b 2 O(B; S; T ), so the conclusion follows again. Now consider the case I (A [ B; S; T ) 6= ; and O(A [ B; S; T ) 6= ; (otherwise, A [ B is trivially ST-IBC), therefore, we have two cases to analyse as follows (see Fig. 3): Case (i) (O(A; S; T ) \ O(B; S; T )) 6= ;. Let x 2 A \ B be such a common output state of both A and B (see Fig. 3(i)). Then, by the mutual accessibility of the output states in ST-IBC blocks, all output states of O(A; S; T ) and O(B; S; T ) can communicate with each other through x. Hence, the second condition of De nition 2.5 holds. Since I (A [ B; S; T ) I (A; S; T ) [ I (B; S; T ), every state in I (A [ B; S; T ) can be driven to some state in O(A; S; T ) or O(B; S; T ) because A and B are ST-IBC; moreover, it may be then driven to some state in O(A [ B; S; T ) 6= ;. Hence, it follows that A [ B is ST-IBC. Case (ii) (O(A; S; T ) \ O(B; S; T )) = ;. Without loss of generality, by symmetry we may assume there exists x 2 O(A; S; T ) \ (A \ B ) (we have shown above (O(A; S; T ) [ O(B; S; T )) \ (A \ B ) 6= ;) such that x 2= O(B; S; T ) (see Fig. 3(ii)). Now x 2= O(A [ B; S; T ) since the 8
y
A
A
s
x
x z
B
B CASE (i)
CASE (ii)
Figure 3: ST-IBC is closed under chain union states in O(A [ B; S; T ) are output states of both A and B , otherwise, it would violate the current hypothesis. Because we have already assumed that all T-inaccessible states have been eliminated, x 2 B is T-accessible implies that there is a trajectory in B from x to some output state of B , say z. Now we need to consider the following alternative two situations: (ii)(1) If all output states of A are inside A\B , then the current hypothesis of case (ii) implies that none of the output states of A can be an output state of A [ B , since these are common output states of both A and B . Because O(A [ B ) 6= ;, O(A [ B; S; T ) O(B; S; T ). But by the ST-IBC property of A, every input state of A has a trajectory to x 2 O(A; S; T ) \ A \ B and hence to z 2 O(B; S; T ). Then by the ST-IBC property of B , z has a trajectory to an element of O(A [ B; S; T ). Hence, A [ B is ST-IBC. (ii)(2) If (ii)(1) does not hold, there exist y 2 O(A; S; T ) \ (A ? B ). Clearly, all output states of A are accessible to the elements of O(B; S; T ) through x and z, since all output states of A, including x, are mutually reachable by De nition 2.5(ii) (see Fig. 3(ii)). But, by the mutual accessibility of O(A; S; T ), it is evident that x 2 O(A; S; T ) must have a trajectory connecting it to y 2 O(A; S; T ) \ (A ? B ) through some s 2 O(B; S; T ) \ (A \ B ). Thus, all output states of B are accessible to y through s 2 O(B; S; T ) by the mutual accessibility of the output states of a ST-IBC block. Hence, all output states of A and output states of B are mutually accessible with respect to A [ B . Furthermore, since A and B all are ST-IBC, their input states have in-block trajectories leading to their output states and hence to a state in O(A [ B; S; T ) 6= ;. It follows again that A [ B is ST-IBC. By induction on n, the chain union 1 [C 2 de ned in De nition 3.1 is ST-IBC. Furthermore, 1 [C 2 is the least upper bound of 1 and 2 in the collection of ordinary partitions ordered by , and, since it is ST-IBC, it is also the least upper bound of 1 and 2 in the IBC 2 ST (X ). In the light of De nition 2.5, a block containing only one state must be ST-IBC, since that state is either an input and output state, or one (or both) of the input or output (singleton) sets of this (singleton) block is the empty set. Thus, the partition id = X must be an ST-IBC partition. This partition id acts as a lower bound of all ST-IBC partitions. 9
Theorem 3.2 For two ST-IBC partitions 1 ; 2 2 IBC ST (Mt ) of an ST nite state machine, the greatest lower bound 1 u 2 2 IBC ( M ) exists. 2 t ST The proof is readily obtained by use of a standard lattice theoretic argumant.
We extend the partial order of partitions to their corresponding partition machines by de ning M1 M2 if and only if 1 2 . The following theorem is a straightforward result of theorem 3.1 and Theorem 3.2.
Theorem 3.3 All ST-IBC partition machines of an ST nite state machine Mt, ordered C by , form a lattice hMIBC ST (Mt ); ; [ ; ui, denoted by HIBCST , which takes the machine id Mt = Mt as its bottom element. In case Mt is ST-IBC, then HIBCST has as top element the trivial partition Mtrt . 2
4 Hierarchical Control for ST-Systems As stated in the introduction, a feature of the ST-IBC lattice structure for any ST-system M is that it permits the construction of all possible sets of hierarchical feedback control systems (for ST-reachability control problems) for the given machine. To be speci c, parallel to the standard IBC hierarchical control problem (see [4]), once the underlying ST-IBC lattice of the thinned machine Mt has been constructed, one may select any chain C from the base element Midt to the top element in the ST-IBC lattice HIBCST . (This may or may not be the trivial partition machine depending on whether T is a mutually accessible set.). Then any set of partition machines lying along such a chain is called an ST-hierarchical control structure. Concerning such a control structure we have the following theorem which is readily veri ed using the results established above.
Theorem 4.1 For an ST-Controllable nite state machine Mt, consider any pair of distinct 1 elements Mt and Mt 2 , Mt 1 Mt 2 , in a hierarchical control structure hMIBC ST (Mt ); C 1 2 ; [ ; ui; then Mt is ST-BBC and ST-IBC with respect to Mt . Further, any (necessarily solvable) state to state ST-reachability problem for Mt has a decomposition into a set
of recursively de ned, solvable, block to block ST-reachability problems for a sequence of machine pairs Mt n and Mt n+1 , 1 n N ? 1, corresponding to the elements of the N-level C hierarchical control structure hMIBC ST (Mt ); ; [ ; ui. A (hierarchical family of) solution (trajectories) to this set of problems gives a solution to the original ST-reachability problem.
2
Theorem 4.1 shows that any ST-state reachability control problem may be decomposed into a sequence of hierarchical control problems; these are such that the feedback controller at any level steers the level-n aggregated state (i.e. level-n partition machine state, containing the base level system state) along a trajectory solving the level-n partition T-reachability control problem. 10
Mπ1 {1,2,4} S
{3}
{6}
Mπ2
{8} T
Mπ3
{5,7}
Mπ4
(i)
(ii)
Figure 4: The 5-state partition machine M5 of M8 and its ST-IBC lattice
Example 3 Let us examine the 5-state machine shown in Fig. 4.(i); this is a partition machine of the model M8, in Example 1, and we choose S = f1; 2; 4g 2 M8; T = f8g. We notice that M8 is already thinned with respect to T and that its corresponding ST-IBC lattice is given in Fig. 4.(ii).
In this lattice the rightmost chain of partitions from the top to the bottom of the lattice is given by: 1 = Xtr = (ff1; 2; 4g; 3; f5; 7g; 6; 8g); 2 = (ff1; 2; 4g; 3g; ff5; 7g; 6; 8g); 3 = (f1; 2; 4; g; f3g; ff5; 7g; 6; 8g); 4 = X5 = (f1; 2; 4g; f3g; f5; 7g; f6g; f8g). In this chain will shall choose the sub-chain 1 ; 2; 4 as the hierarchical control structure. In 1 , the trivial partition, we know that S 1 = T 1 = 1 . Noting the containment relations, we have S = S 4 = f1; 2; 4g S 2 = ff1; 2; 4g; 3g S 1 = 1 and T = T 4 = f8g T 2 T 1 = 1 . Since the partition 1 is ST-IBC, there is an internal trajectory from S 1 to T 1 . For the partition machine of 2 , we know hff1; 2; 4g; 3g; ff5; 7g; 6; 8gi is DC (relative to 4 ), ff1; 2; 4g; 3g ) ff5; 7g; 6; 8g. Now the M2 controller chooses the unique one step control event to drive S 2 to T 2 . This is realized (at the next level of the control hierarchy) by the M4 controller which has the choice of driving S 2 = ff1; 2; 4gg to f6g in one step, or S 2 to ff5; 7gg via f3g. As far as the realization of this hierarchical control law is concerned, the choice is arbitrary and may be determined by any well de ned rule. At the nest (4 ) level, the controller terminates the path to T 4 by nding a path from f5; 7g to T = f8g. Finally, expressing the corresponding ST-DC relations as the state to state one step transitions at the nest (4) level, we obtain the particular trajectory f1; 2; 4g ! f3g ! f5; 7g ! f6g ! f8g solving the ST-reachability problem. 2
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