We propose an abstract block diagram formalism based on the notions of a signal as a time-varying quantity, a block as a signal transformer, a connection between blocks as a signal equality constraint, and a block diagram as a collection of interconnected blocks. It does not enforce implementation details (like internal state-space) or particular kinds of dynamic behavior (like alternation of discrete steps and continuous evolutions) on blocks and can be considered as an abstraction of block diagram languages used by engineering system designers. We study its properties and give general conditions for well-definedness of the operation of a system specified by a block diagram for each admissible input signal(s).
Many software tools for developing control, signal processing, and communication systems are based on block diagram notations familiar to control engineers. Examples include system design software Simulink [1], Scicos [2], Dymola [3], SCADE [4], declarative synchronous languages [5], some embedded programming languages [6, 7].
In such notations, a diagram consists of blocks (components) connected by links. Typically, blocks have input and output ports, and (directed) links connect output ports of one block with input ports of the same or another block. A block is interpreted as an operation which transforms input signals (i.e. time-varying quantities) flowing though its input ports into output signals. ? Part of this research has been supported by the project Verisync (ANR-10-BLAN0310), France.
Wide applicability of block diagram notations makes them an interesting object of study from a theoretical perspective. Classical control theory and signal processing already provide some degree of formal treatment of block diagrams [8, 9], but this is normally not suficient to handle such aspects of modern system design languages as mixing of analog and discrete-time blocks, partially defined block operations, non-numeric data processing, etc.
To take these issues into account, researches developed formal semantics for various block diagram languages [10–13]. Some efort has been made to unify approaches taken by diferent engineering system modeling and analysis tools and make them interoperable by the use of exchange languages with well-defined semantics [14, 15] such as Hybrid System Interchange Format (HSIF) which gives semantics of hybrids systems in terms of dynamic networks of hybrid automata [16, 17].
Although hybrid automata-based approaches like HSIF can be used to give semantics to block diagram languages [18] and have many advantages (e.g. availability of verification theory for hybrid automata), we consider them not entirely satisfactory from a theoretical standpoint for the following reasons: – Semantics of a system component (block) is based on the notion of a dynamical system with an internal state, and so the components with the same externally observable behavior can be semantically distinguishable. In our opinion, this is not needed for a high-level semantics which does not intend to describe details of the component’s physical/logical implementation. – Semantics of a system has a computational nature: it describes a sequence of discrete steps, where a step may involve function computations, solving initial-value problems for diferential equations (continuous evolution), etc. It may be adequate for certain classes of discrete-continuous systems, but it does not always capture the behavior of a physical realization of a system (and thus may conflict which the view of a system designer).
For example, a Zeno execution [17] of a hybrid automaton can be described as an infinite sequence of discrete steps which takes a bounded total time (but each step takes a non-zero time). This normally does not correspond to the behavior of a physical system described by the automaton. In many cases this is caused by system modeling simplifications. The conflict is usually resolved by applying a certain method of continuation of an execution beyond Zeno time (regularization, Fillipov solution, etc.) [19]. But an extended execution is not, in fact, a sequence of discrete steps, as it resumes after the accumulation point.
In our opinion, in the general case, the dynamic behavior of a system should not be restricted to a particular scheme like a sequence of discrete steps and continuous evolutions.
The goal of this paper is to introduce abstract formal models for blocks and block diagrams which overcomes limitations of the classical control/signaltheoretic approach to them and does not enforce implementation details (like internal state-space) or particular kinds of dynamic behavior (like alternation of discrete steps and continuous evolutions) on blocks.
These models can be used to identify the most general properties of block diagram languages which are valid regardless of implementation details. In particular, in the paper we will give a general formulation and conditions for welldefinedness of the operation of a system specified by a block diagram for each admissible input signal(s).
To achieve our goal, we will use a composition-nominative approach [20]. The main idea of this approach is that semantics of a system is constructed from semantics of components using special operations called compositions, and the syntactic representation of a system reflects this construction.
The paper is organized in the following way: – In Section 2 we give definitions of the auxiliary notions which are used in the rest of the paper. – In Section 3 we introduce abstract notions of a block, a connection, a block diagram, and a block composition. We show how they fit into a system design process and give conditions of well-definedness of the operation of a system specified by a block diagram for each admissible input signal(s). 2 2.1
We will use the following notation: N = {1, 2, 3, ...}, N0 = N ∪ {0}, R+ is the set of nonnegative real numbers, f : A → B is a total function from A to B, f : A →˜B is a partial function from A to B, 2A is the power set of a set A, f |X is the restriction of a function f to a set X. If A, B are sets, then BA denotes the set of all total functions from A to B. For a function f : A→˜ B the symbol f (x) ↓ (f (x) ↑) means that f (x) is defined (respectively undefined) on the argument x.
We denote the domain and range of a function as dom(f ) = {x | f (x) ↓} and range(f ) = {y | ∃x f (x) ↓ ∧ y = f (x)} respectively. We will use the the same notation for the domain and range of a binary relation: if R ⊆ A × B, then dom(R) = {x | ∃ y (x, y) ∈ R} and range(R) = {y | ∃ x (x, y) ∈ R}.
We will use the notation f (x) ∼= g(x) for the strong equality (where f and g are partial functions): f (x) ↓ if g(x) ↓ and f (x) ↓ implies f (x) = g(x).
The symbol ◦ denotes a functional composition: (f ◦ g)(x) ∼= g(f (x)).
By T we denote the (positive real) time scale [0, +∞). We assume that T is equipped with a topology induced by the standard topology on R.
Additionally, we define the following class of sets:
T0 = {∅, T } ∪ {[0, x) | x ∈ T \{0}} ∪ {[0, x] | x ∈ T } i.e. the set of (possibly empty, bounded or unbounded) intervals with left end 0. 2.2
A multi-valued function [20] assigns one or more resulting values to each argument value. An application of a multi-valued function to an argument is interpreted as a nondeterministic choice of a result. Definition 1 ([20]).
tm (denoted as f : A−→ B) is a function f : A → 2B\{∅}.
Thus the inclusion y ∈ f (x) means that y is a possible value of f on x. 2.3
We will use a simple notion of a named set to formalize an assignment of values to variable names in program and system semantics.
Definition 2. ([20]) A named set is a partial function f : V →˜ W from a nonempty set of names V to a set of values W .
In this definition both names and values are unstructured. A named set can be considered as a partial (”flat”) case of a more general notion of nominative data [20] which reflects hierarchical data organizations and naming schemes.
We will use a special notation for the set of named sets: V W denotes the set of all named sets f : V →˜W (this notation just emphasises that V is interpreted as a set of names). We consider named sets equal, if their graphs are equal.
An expression of the form [n1 7→ a1, n2 7→ a2, ...] (where n1, n2, ... are distinct names) denotes a named set d such that the graph of d is {(n1, a1), (n2, a2), ...}. A nowhere-defined named set is called an empty named set and is denoted as [].
For any named sets d1, d2 we write d1 ⊆ d2 (named set inclusion ), if the graph of a function d1 is a subset of the graph of d2. We extend set-theoretical operations of union ∪, intersection ∩ and diference \ to the partial operations on named sets in the following way: the result of a union (intersection, diference) of named sets (operation’s arguments) is a named set d such that the graph of d is the union (intersection, diference) of graphs of the arguments (if such d exists). 3 3.1
Let us introduce abstract notions of a signal as a time-varying quantity and a block as a signal transformer. We will use a real time scale for signals, but we will not require them to be continuous or real-valued. So the signals can be piecewise-constant as well and can be used to represent discrete evolutions.
Informally, a block (see Fig. 1) is a device which receives input signals and produces output signals. We call a collection of input (output) signals an input (resp. output) signal bunch. At each time moment (the value of) a given signal may be present or absent. In the general case, the presence of an input signal at a given time does not imply the presence of an output signal at the same or any other time moment.
A block can operate nondeterministically, i.e. for one input signal bunch it may choose an output signal bunch from a set of possible variants. However, for any input signal bunch there exists at least one corresponding output signal bunch (although the values of all signals in it may be absent at all times, which means that the block does not produce any output values).
Normally, a block processes the whole input signal bunch, and does or does not produce output values. However, in certain cases a block may not process the whole input signal bunch and may terminate at some time moment before its end. This situation is interpreted as an abnormal termination of a block (e.g. caused by an invalid input).
Definition 3. (
Note that on V -signal bunches is a partial order (for an arbitrary V ). Later we will need generalized versions of the prefix relation for pairs and indexed families of pairs of signal bunches.
For any signal bunches s1, s2, s01, s02 let us denote (s1, s2) 2 (s01, s02) if there exists A ∈ T0 such that s1 = s01|A and s2 = s02|A.
For any indexed families of pairs of signal bunches (sj , s0j )j∈J and (s0j0, s0j00)j∈J of signal bunches let us denote (sj , s0j )j∈J J,2 (s0j0, s0j00)j∈J if there exists A ∈ T0 such that sj = s0j0|A and s0j = s0j00|A for all j ∈ J .
It is easy to check that 2 is a partial order on pairs of signal bunches and J,2 is a partial order on J -indexed families of pairs of signal bunches.
A block has a syntactic aspect (e.g. a description in a specification language) and a semantic aspect – a partial multi-valued function on signal bunches.
sociated set of input names In(B), a set of output names Out(B), and a total
multi-valued function Op(B) : Sb(In(B), W )−tm→ Sb(Out(B), W ) (operation,
semantic aspect) such that o ∈ Op(B)(i) implies dom(o) ⊆ dom(i).
(
Out(B2), and Op(B1) = Op(B2).
(
An inclusion o ∈ Op(B)(i) means that o is a possible output of a block B on the input i. For each input i there is some output o. The domain of o is a subset of the domain of i. If o becomes undefined at some time t, but i is still defined at t, we interpret this as an error during the operation of the block B (the block cannot resume its operation after t).
We interpret the operation of a block as a (possibly nondeterministic) choice of an output signal bunch corresponding to a given input signal bunch. However, we would also like to describe this choice as dynamic, i.e. that a block chooses the output signal values at each time t, and in doing so it cannot rely on the future values of the input signals (i.e. values of the input signals at times t0 > t).
If a block is deterministic, this requirement can be formalized in the same way as the notion of a causal (or nonanticipative) input-output system [26]. Definition 6. A deterministic block B is causal if for all signal bunches i1, i2 and A ∈ T0, o1 ∈ Op(B)(i1), o2 ∈ Op(B)(i2), the equality i1|A = i2|A implies o1|A = o2|A. This means that the value of the output signal bunch at time t can depend only on the values of the input signal at times ≤ t.
Some works in the domain of systems theory extend the notion of a causal (deterministic) system to nondeterministic systems. However, there is no unified approach to an extension of this kind. For example, in the work [21], a system, considered as a binary relation on (total) signals S ⊆ AT × BT , where T is a time domain (Mesarovic time system, [22]) is “non-anticipatory”, if it is a union of (graphs of) causal (non-anticipatory) selections from S, i.e, S = S{f : dom(S) → range(S) | f ⊆ S, f is causal}. In the work [23] the authors define another notion of a “non-anticipatory” or “causal” system in nondeterministic case. In the theory developed in the work [24], the authors use a similar notion of a “precausal” system, which is also defined in [22], as a generalization of the notion of a causal system to the nondeterministic case.
In this work, we generalize the notion of a non-anticipatory system in sense of [23] to blocks and call such blocks nonanticipative, and generalize the notion of a non-anticipatory system in sense of [21] to blocks, but call such blocks strongly nonanticipative. We will show that strongly nonanticipative block is nonanticipative. We will consider the words “causal” and “nonanticipative” as synonyms when they are used informally, but we will distinguish them in the context of formal definitions to avoid a conflict with Definition 6.
Note, however, that the notion of a strongly nonanticipative block defined below is very diferent from the notion of a ”strictly causal“ system, defined in some works [25] as a system which uses only past (but not current or future) values of the input signal(s) to produce a current value of the output signal(s). Definition 7. A block B is nonanticipative, if for each A ∈ T0 and i1, i2 ∈ Sb(In(B), W ), if i1|A = i2|A, then
{o|A | o ∈ Op(B)(i1)} = {o|A | o ∈ Op(B)(i2)}.
Definition 8. A block B is a sub-block of a block B0 (denoted as B E B0), if In(B) = In(B0), Out(B) = Out(B0), and IO(B) ⊆ IO(B0).
Informally, a sub-block narrows nondeterminism of a block.
Definition 9. A block B is strongly nonanticipative, if for each (i, o) ∈ IO(B) there exists a deterministic causal sub-block B0 E B such that (i, o) ∈ IO(B0).
Informally, the operation of a strongly nonanticipative block B can be interpreted as a two-step process: 1. before receiving the input signals, the block B (nondeterministically) chooses a deterministic causal sub-block B0 E B (response strategy); 2. the block B0 receives input signals of B and produces the corresponding output signals (response) which become the output signals of B. Intuitively, it is clear that in this scheme at any time the block B does not need a knowledge of the future of its input signals in order produce the corresponding output signals.
tive.
Proof. Follows immediately from Definition 6.
The following theorem gives a characterization of a nonanticipative block which does not rely on comparison of sets of signal bunches.
(
o0 = o|A ∈ {o00|A | o00 ∈ Op(B)(i)} = {o00|A | o00 ∈ Op(B)(i0)} by Definition 7. Then o0 = o00|A for some o00 ∈ Op(B)(i0). Moreover, dom(o00) ⊆ dom(i0) ⊆ A. Thus o0 = o00 and (i0, o0) ∈ IO(B).
Let us prove (
o|A ∈ {o00|A | o00 ∈ Op(B)(i)} = {o00|A | o00 ∈ Op(B)(i0)} by Definition 7. Then o|A = o0|A for some o0 ∈ Op(B)(i0). Moveover, dom(o) ⊆ dom(i) ⊆ A, whence o = o|A = o0|A. Thus (i, o) 2 (i0, o0).
(
(
lim o[y](t) = f t→+∞
lim i[x](t) ; t→+∞ • otherwise, Op(B)(i) is the set of all {y}-signal bunches o such that dom(o) = dom(o[y]) = S{A ∈ T0 | A ⊆ dom(i[x])}.
Obviously, in this definition Op(B)(i) 6= ∅ (because T0 is closed under unions) and dom(o) ⊆ dom(i) for each o ∈ Op(B)(i). So B is indeed a block. Using Theorem 1 it is not dificult to show that B is nonanticipative. Suppose that B has a a deterministic causal sub-block B0. Let a ∈ R and ak ∈ R, k = 1, 2, ... be a sequence such that limk→∞ ak = a. Let us show that limk→∞ f (ak) = f (a). Let us define sequences ik ∈ Sb({y}, R), ok ∈ Sb({y}, R), and tk ∈ T , k = 1, 2, ... by induction as follows. Let i1(t) = [x 7→ a1] for all t ∈ T , o1 be a unique member of Op(B0)(i1), and t1 = 0. If i1, i2, ..., ik are already defined, let ik+1(t) = ik(t), if t ∈ [0, tk] and ik+1(t) = [x 7→ ak+1], if t ∈ T \[0, tk]. Let ok+1 be a unique member of Op(B0)(ik+1). Because B0 E B, dom(ok+1) = dom(ok+1[y]) = T and limt→+∞ ok+1[y](t) = f (limt→+∞ ik+1[x](t)) = f (ak+1). Then let tk+1 = 1 + max{tk, inf{τ ∈ T | We have defined sequences ik, ok, tk. The sequence tk, k = 1, 2, ... is a strictly increasing and unbounded from above and t1 = 0.
Let i be a {x}-signal bunch such that dom(i) = T , i(t1) = i1(t1), and i(t) = ik+1(t), if t ∈ (tk, tk+1], k ∈ N, and o be a (unique) member of Op(B0)(i). We have ik+1[x](t) = ak+1 for all k = 1, 2, ... and t > tk. Then i[x](t) ∈ {ak+1, ak+2, ...} for all k ∈ N and t > tk. For each > 0 there exists k ∈ N such that |ak0 − a| < for all k0 ≥ k, whence |i[x](t) − a| < for all t > tk. Thus limt→+∞ i[x](t) = a. Then dom(o) = dom(o[y]) = T and limt→+∞ o[y](t) = f (a), because B0 E B.
On the other hand, ik+1|[0,tk] = ik|[0,tk] for all k ∈ N. Because tk is an increasing sequence, we have ik0 |[0,tk] = ik|[0,tk] for all k and k0 ≥ k. Besides, i|(tk,tk+1] = ik+1|(tk,tk+1] for all k ∈ N, whence i|(tk,tk+1] = ik0 |(tk,tk+1] for all k0 ≥ k + 1. Also, ik(t1) = i1(t1) for all k ∈ N. Then i|[0,tk] = i|{t1}∪(t1,t2]∪...∪(tk−1,tk] = ik|[0,tk] for all k = 2, 3, ..., whence o|[0,tk] = ok|[0,tk], because B0 is causal. Then o(tk) = ok(tk) for all k = 2, 3, ..., and from the definition of tk we have |o[y](tk) − f (ak)| = |ok[y](tk) − f (ak)| ≤ k1 for all k = 2, 3, .... This implies that limk→∞ f (ak) = f (a), because limt→+∞ o[y](t) = f (a). We conclude that f is sequentially continuous and thus is continuous. This contradicts our choice of f as a discontinuous function. Thus B has not deterministic causal sub-blocks. Consequently, B is not strongly nonanticipative, though it is nonanticipative.
The proof of this theorem gives a reason of why Definition 9 better captures
a intuitive idea of causality than Definition 7. Consider, for example, the block
B constructed in the proof of the item (
In the rest of the paper we will focus on strongly nonanticipative blocks, as more adequate models of (real-time) information processing systems.
Consider an example of a strongly nonanticipative block.
Let u, y be names. Assume that W = R.
Example 1. Let B be a block such that In(B) = { } u , Out(B) = {y}, and for each i, Op(B)(i) = {o1(i), o2(i)}, where o1(i), o2(i) ∈ Sb(Out(B), W ) are signal bunches such that – dom(o1(i)) = dom(o2(i)) = dom(i); – o1(i)(t) = [y 7→ i[u](t)] for all t ∈ dom(i); – o2(i)(t) = [y 7→ 2i[u](t)] for all t ∈ dom(i).
Informally, this means that B is a gain block with a slope which is either 1 or 2 during the whole duration of the block’s operation.
Obviously, B satisfies Definition 4(
The blocks B1, B2 are deterministic and it is easy to see that they are causal. Obviously, each I/O pair (i, o) ∈ IO(B) belongs either to IO(B1), or to IO(B2), so B is strongly nonanticipative.
Now let us consider an example of a block which is not nonanticipative. Example 2. Let B0 be a block such that In(B0) = {u}, Out(B0) = {y}, and – Op(B0)(i) = {o1}, where dom(o1) = dom(i) and o1(t) = [y 7→ 1] for all t ∈ dom(i), if dom(i[u]) = T ; – Op(B0)(i) = {o2}, where dom(o2) = dom(i) and o2(t) = [y 7→ 0] for all t ∈ dom(i), otherwise.
Informally, the block B0 decides whether its input signal u is total. It is easy to
see that B0 indeed satisfies Definition 4(
Definition 10. (
j∈J
j∈J
) is called regular, if each Bj , j ∈ J is strongly Note that a diagram may consist of an infinite set of blocks. A relation ( j, x) (j0, x0) means that the output x of the j-th block is connected to the input x0 of the j0-th block.
A block diagram is only a syntactic aspect of a block composition. We define semantics of a block composition only for strongly nonanticipative blocks.
To describe it informally consider Fig. 3. The connection between y2 and x2 means a signal equality constraint. The block B chooses (nondeterministically) some deterministic sub-block B0 E B. When a signal starts flowing into the input x1, the block B0 tries to choose the initial values for the signals of y1, y2, x2 so that they satisfy the operation of the block B0 (Op(B0)) and the signals of x2 and y2 have the same values. If such initial values do not exist, the block B0 terminates (the output signal bunch is nowhere defined). Otherwise, B0 continues to operate in the similar way until either the signals of y1, y2, x2 cannot be continued, or the input signal (x1) ends.
Definition 11. Let ((Bj )j∈J , ) be a regular block diagram.
A block B is a composition of (Bj )j∈J under the interconnection relation
, if
– In(B) = (Sj∈J {j} × In(Bj ))\range( ),
– Out(B) = (Sj∈J {j} × Out(Bj ))\dom( ),
– Op(B)(i) is the set of all Out(B)-signal bunches o such that there exist
deterministic causal sub-blocks Bj0 E Bj , j ∈ J and an indexed family
(ij , oj )j∈J ∈ Xm(i) such that
(
(j0, x0) implies oj [x] = ij0 [x0].
In this definition, ij and oj denote the input and output signal bunches of the j-th block. The set Xm(i) contains maximally extended (in sense of the relation
J,2) indexed families of signal bunches defined on a subset of the domain of i
(the input signal bunch of B) which satisfy constraints imposed by the
interconnection relation. Any such family gives a possible output of B for the given i by
the condition (
j
It is clear that any two compositions of (Bj )j∈J under are semantically identical.
Let B be a deterministic causal block. Let c ⊆ Sb(In(B), W ) be a non-empty -chain, i∗ be its supremum (in sense of ), and o∗ ∈ Op(B)(i∗). Then o∗ is a supremum of Si∈c Op(B)(i) (in sense of ).
Proof. Follows from Definition 6.
Theorem 3. Let ((Bj )j∈J ,
(
tive.
The proof follows from Lemma 2 and Definition 11.
) be a regular block diagram. Then , then B is strongly nonanticipa3.3
Above we have considered a block as an abstract model of a real component. However, it can also be considered as a specification of requirements for a component. Let Bspec, Bimpl be two strongly nonanticipative blocks. Let us call them a specification block and implementation block respectively.
Bimpl is a refinement of Bspec, if Bimpl is a sub-block of Bspec.
I.e. an implementation should have the the same input and output names as a specification, and for each input, an output of an implementation should be one of the possible outputs of a specification. We generalize this to diagrams.
Let D = ((Bj )j∈J , ) and D0 = ((Bj0 )j∈J0 , 0) be regular block diagrams.
for each j ∈ J , and the relations
D0, if J = J 0, Bj is a refinement of and 0 coincide.
B0 j Theorem 4. (Compositional refinement) Let B be a composition of (Bj )j∈J under and B0 be a composition of (Bj0 )j∈J0 under 0. If D is a refinement of D0, then B is a refinement of B0.
The proof follows from Definition 9, 11, and transitivity of the sub-block relation.
This theorem can be considered as a foundation of a modular approach [29, 30] to system design: 1. Create specifications of the system components ( Bj0 , j ∈ J 0) and connect them ( 0), as if they were real components. 2. Analyze a composition of specifications ( B0) to ensure that any of its implementations (B00 E B0) satisfies requirements to the final system. 3. Create an implementation (Bj ) for each specification ( Bj0 ). 4. Connect implementations (according to 0). Then the composition of implementations (B) is a final system which satisfies design requirements. We consider the steps 1 and 3 domain- and application-specific. The conclusion of the step 4 is addressed in Theorem 4. Step 2 requires some verification method which depends on the nature of requirements.
One of the most basic and common requirements is that the operation of B0 is defined on all input signal bunches which are possible in the context of a specific application of this composition. This trivially holds because of Theorem 3 and our definition of a block. However, B0 may terminate abnormally on some or all input signal bunches of interest (as we have noted, we interpret the situation when o ∈ Op(B˜)(i) and dom(o) ⊂ dom(i) for some block B˜ as abnormal termination of B˜ on i). So the requirement can be reformulated as follows: B0 never terminates abnormally on any input signal bunch from a given set IN (this implies that the same property holds for B). We will call this property as well-definedness of the operation of B0 on IN and study it in the next subsection. 3.4
Let B be a block and IN be some set of In(B)-signal bunches.
Definition 14. The operation of B is well-defined on IN , if dom(i) = dom(o) for each i ∈ IN and o ∈ Op(B)(i).
Let D = ((Bj )j∈J , ) be a regular block diagram and B be a composition of (Bj )j∈J under . Let F be the set of all families of blocks of the form (Bj0 )j∈J , where for each j ∈ J , Bj0 is a deterministic causal sub-block of Bj .
For each In(B)-signal bunch i and a family of blocks F = (Bj0 )j∈J let XF (i)
be the set of all indexed families of pairs of signal bunches u = (ij , oj )j∈J which
satisfy conditions (
Let XmF(i) denote the set of all J,2-maximal elements of XF (i). For any
indexed family of signal bunches u = (ij , oj )j∈J let O(u) denote the set of all
Out(B)-signal bunches o which satisfy conditions (
For any u = (ij , oj )j∈J ∈ XF (i), the domains of ij , oj for all j ∈ J coincide. Denote by cdom(u) this common domain (we assume cdom(u) = T , if J = ∅).
From Definition 11 we have Op(B)(i) = SF ∈F Su∈XmF(i) O(u) for each i. Then from Definition 14 we get the following simple criterion:
F ∈ F , and u ∈ XF (i), if cdom(u) ⊂ dom(i), then u ∈/ XmF(i). i ∈ IN , This criterion means that B is well-defined, if each u ∈ XF (i), the common domain of which does not cover dom(i), is extendable to a larger u0 ∈ XF (i) (in sense of J,2). We will call it a local extensibility criterion, because, basically, to prove well-definedness, we only need to show that the members of the family u can be continued onto a time segment [0, sup cdom(u) + ] for some small > 0 (under constraints imposed by the interconnection relation ). Locality is especially useful when a block diagram contains ”delay” blocks (possibly working as variable delays), because constraints imposed by connections between blocks reduce over small time intervals.
A drawback of this criterion is that it requires checking local extensibility
of signal bunches satisfying the I/O relations (IO(Bj0 )) of arbitrarily chosen
deterministic causal sub-blocks Bj0 E Bj (condition (
For this reason, we seek for a condition of well-definedness in terms of I/O relations of the constituents of the composition (IO(Bj ), j ∈ J ).
Let X(i) denote the set XF (i), where F = (Bj )j∈J . Note that F may not be a member of F .
Theorem 6. The operation of B is well-defined on IN if for each i ∈ IN and u ∈ X(i) there exists u0 ∈ X(i) such that u J,2 u0 and cdom(u0) = dom(i). The proof follows from Theorem 5 and Definition 9.