Abstract
The relationship between structural controllability and observability of complex systems is studied. Algebraic and graph theoretic tools are combined to prove the extent of some controller/observer duality results. Two types of control design problems are addressed and some fundamental theoretical results are provided. In addition new algorithms are presented to compute optimal solutions for monitoring large scale real networks.
1. Introduction
The controllability and observability analysis of dynamical systems has been an active area of research in control theory since the pioneer work of Kalman for the linear time invariant (LTI) case [1]. Since then, progress has been carried out in several directions such as the controllability/observability of a class of nonlinear systems [2–5], some types of fuzzy systems [6, 7], and the structural controllability/observability of LTI systems [8–10], aimed at robust system monitoring.
The structural controllability analysis of LTI systems was initially stated by [8]. Such analysis is intended to model those system properties which only rely on the existence or not of dependencies among inputs, outputs, and state variables; the existence of a dependency is reflected in the model by some nonzero system parameter (which multiplies the corresponding coupling term) but does not depend on the specific value of such parameter. In [8] both linear algebraic and graph characterizations of structural controllability are presented, the second one by means of analyzing the associated directed graph which precisely represents the dependencies among state variables and input signals.
This correspondence between some properties of system dynamics and the structure of the associated directed network has been analyzed in the context of large scale and distributed control systems [11, 12]. Conversely, the same correspondence has led to the study of complex networks from a control theoretic perspective [13]; there, the analysis of a graph has been identified with the structural controllability of an associated LTI system, where the controllability concept can be accordingly interpreted depending on the nature and meaning of the network under study. In this structural LTI system framework, some specific problems concerning the minimum number of required inputs (which corresponds to the number of required controllers or actuators) to guarantee controllability have attracted the attention of several researchers (see [10–16], where some computational solutions have been provided).
The present paper deepens on the relationship between network analysis and the controllability as well as the observability properties of associated dynamical systems. First, the analysis and design of systems regarding their structural properties are formalized. Then, the potential duality between controllability and observability is analyzed in the framework of some design problems, providing new theoretical results which relate both concepts. Finally, properties of maximum matchings (MMs) and strongly connected components (SCCs) are demonstrated, which lead to new computational tools for analyzing complex networks [17–19].
The paper is organized as follows. Section 2 presents the main results on structural controllability of LTI systems. Two problems concerning the optimal design of the control matrix are addressed in Section 3; there, algebraic and graph theoretic tools are combined, and the corresponding computational algorithms are presented. Section 4 considers the observability problem and theoretically demonstrates several duality results which are confirmed via computational simulations. Some fundamental properties of maximum matchings and strongly connected components of the network are demonstrated in Section 5. The algorithms for computing several controllability and observability related properties in complex networks are presented in Section 6. Finally, concluding remarks are summarized in Section 8.
2. Structural Controllability of LTI Systems
This section presents several controllability results for LTI systems of the form where and are given a priori. This is the case for many engineering problems, where physical restrictions define both the relationship between state variables (matrix ) and the possible location of system actuators (matrix ).
First, the classical controllability problem is stated and the need to undertake a structural analysis perspective is motivated. Secondly, some useful results on the structural analysis of matrices are demonstrated; finally, the structural controllability problem is analyzed.
2.1. Classical Controllability
Roughly speaking, system (1) is controllable (in the classical sense) when it is possible to lead the system state variable from any initial point to any arbitrary point in a finite time period.
Classical control theory states that system (1) is controllable if and only if the corresponding controllability matrix satisfies (see [1]). Hence, the classical controllability problem can be formulated as a linear algebra rank condition; this implies that, in some practical cases, the problem may be ill-conditioned and too sensitive to potential parameter variations. Hence, the need of performing robust analyses not affected by modelling errors and/or uncertainties motivates the study of structural properties.
2.2. Structural Properties
In practice, the elements of matrices and may not be precisely known. This leads to the definition of structural properties as those which do not change with variations in the nonzero values of the elements of matrices and . Structural analysis considers two types of entries in the matrices, zero and nonzero entries, and addresses those properties which are preserved no matter what the exact value of the nonzero entries is, except for a set of their values with zero Lebesgue measure in the parameter space; see [8]; such properties are called generic [9]. Hence, the nonzero entries may be represented by a -value (defining then a binary matrix) or, alternatively, the -symbol. This will allow for a straightforward graphical representation of the system as shown in Section 2.3.
2.2.1. Algebraic Properties: Generic Rank
We introduce here the concept of a matrix generic rank (denoted by ), which happens to play an important role in characterizing its structural properties. As mentioned earlier, the generic rank of a matrix , say , is the rank of such matrix for all values of its nonzero entries except those that lie in a set of zero measures. We now define some basic concepts aimed to characterize the generic rank of an matrix (with unless stated otherwise).
Definition 1 ([see 16]). An matrix is of form for some , , if for some in the range , contains a zero submatrix of order .
Remark 2 ([see 16]). If has form , then clearly has form for .
The following lemma will be employed in the proof of Theorem 5.
Lemma 3. Given a matrix , let be a matrix structurally equivalent to except for a fixed zero of which has been replaced by an arbitrary nonzero entry in . Then, if is not of form , then is not of form .
Proof. From Definition 1, we have that, given , in the range , does not contain a zero submatrix of order . Hence, based on the way has been constructed from , matrix does not contain a zero submatrix of order either. This means that is not of form .
We can now state the following theorem which provides an alternative way to define the generic rank of a matrix.
Theorem 4 (see [9], Theorem 2.2). For any matrix , it is ,(i)for if and only if is not form ,(ii)for if and only if is of form but not of form .
We end up with the following generic result, which will be useful for structural controllability analysis.
Theorem 5. Given a matrix , let be a matrix structurally equivalent to except for a fixed zero of which has been replaced by an arbitrary entry in . Then .
Proof. Let us consider the case . Then is not of form ; considering Lemma 3, this implies that is not of form which is equivalent to . Let us now consider the case . Then is of form , but not of form . Hence is not of form , which implies that .
(Note that, knowing that is not of form , then if is of form , then ; and if is not of form , then .)
2.3. The Graph Perspective
The matrix binary form suggests a straightforward alternative representation of the system as a graph , where state variables appear as the nodes (or vertices belonging to set ) and the elements of are represented by the existence of a link or edge (nonzero entries correspond to an existing link belonging to set ). Concerning matrix , nonzero entries are reflected as links from an external input to the corresponding node (see Figure 1).
Several system structural properties can be analyzed by referring to its associated graph; in the following, structural controllability is addressed and we emphasize its alternative analysis via a graph theoretic approach.
2.4. Structural Controllability Conditions
In [8] systems of the form are analyzed, where column represents the scalar input influence on the state variables. Structural controllability is analyzed via both matrix and graph theory perspectives. The system (network) is proved to be structurally controllable if and only if all nodes are accessible from the input and the network presents no dilation, which is equivalent to say that the graph is spanned by an input cactus [8, 10].
Structural controllability for multi-input systems defined by a given pair was first addressed in [9] by analyzing two properties of matrix : the first one is related to accessibility and the second one (which is ) relates to the absence of dilations. Fortunately, the problem can be reduced to solely computing the generic rank of the associated extended controllability matrix.
Again, from a graph theory perspective, the system (network) is structurally controllable if and only if there exists a vertex disjoint union of input cacti [10] that covers all the state vertices (see, for instance, [20]).
2.4.1. The Use of Maximum Matchings
In [21] the equivalence between computing the generic rank of a matrix and computing a maximum matching (MM) in over the associated bipartite graph (see [15] for details) is indicated (see Figure 2). A matching is any subset of so that all nodes in have neither more than one incoming edge nor more than one outgoing edge belonging to the matching. A matching is maximum if there are no other larger matchings (i.e., a matching containing a larger number of edges); note that maximum matchings (MMs) need not be unique. A matching is perfect if all nodes of the network have an incoming edge belonging to the matching (i.e., the number of links belonging to the matching equals the number of nodes in the network). Maximum matchings (MMs) will be considered in detail in the following sections, where it will be shown that the equivalence between generic rank evaluation and the determination of a MM is in accordance with the fact that a MM provides a subgraph which guarantees the absence of dilations.
In the next section, some control design problems (on the matrix ) are presented, where both the algebraic and the graph theoretic perspectives can still be employed to address them. Again, the computation of MMs will prove to be an efficient step towards their solution.
3. Optimal Design of
There are practical situations in which only matrix is known as a characterization of the system dynamics, and there is no a priori restriction about the structure of matrix . This can be interpreted as if any state variable can be directly accessed by a control signal. Then, the selection of an appropriate matrix can be addressed as a design goal.
Different optimization criteria can be defined for the design of matrix . In the following, we formulate two different problems aimed to minimize the control requirements. Both problems can be formulated either in the classical control context (with a specific matrix) or in the structural analysis framework considered in this paper.
3.1. Minimum Number of Required Inputs
The first problem is concerned with minimizing the number of inputs or actuators, independently of the fact that such actuators may need to be connected as an input to more than one state variable.
Problem 6. Find with a minimum number of columns (inputs or actuators) so that is controllable.
Note that, since a column of may have more than one nonzero entry, the number of inputs may be smaller than the number of states directly accessed by an input (i.e., the number of nonzero rows).
Obviously, the solutions to this problem are not unique; and it is straightforward to prove that, given two different solutions and , the number of state variables directly accessed by each of them may be different.
The design of an optimal has not been an important issue in classical control theory since most of the time such matrix is given a priori (or it is restricted to access only a subset of state variables) in real engineering problems.
When structural controllability is considered, the main result concerning the minimum number of required inputs is stated in the following theorem.
Theorem 7. Let us consider the LTI system and let be the minimum number of inputs ( stands for controllers) to make it structurally controllable. Then
Proof. As stated in [8], the system will be structurally controllable if all its variables are accessible from the inputs and the system presents no dilation. The accessibility condition requires having at least one input to the system, which implies that . The condition of no dilation can be expressed as follows:
where is the number of state variables in the system. Since
the structure of the system, described by , determines the conditions imposed to to make the system controllable.
Given , the problem of finding the minimum number of inputs of the system is thus reduced to finding the minimum number of column vectors forming a matrix that satisfies (5). To comply with the accessibility condition, we may face two different cases: if , we need to have at least one column with some nonzero entry; if , the already nonzero matrix selected to satisfy the no-dilation condition may need to add extra nonfixed values to its column vectors, but either of these operations will not affect the no-dilation condition since it will never reduce as stated in Theorem 5. In other words, the range condition expressed in (5) will determine the minimum number of inputs of the system, regardless of the number of variables/vertices affected by them. This result reduces Problem 6 to the rank analysis of (5).
Therefore, can be chosen to comply with (5) just by constructing as many independent columns as , keeping in mind that if we need to have one column. Hence
3.1.1. Computation of : The Maximum Matching Alternative
A priori, the computation of would rely on calculating the generic rank of matrix . Hence, only the no-dilation property must be taken into account to compute , independently of accessibility issues. This implies that, once a matrix satisfying the rank condition has been selected, we may only further require changing some of its zero terms to one (without altering its generic rank and ) to cope with accessibility.
Alternatively, the network theory perspective provides a way of determining the value of by the calculation of MMs on the network associated bipartite graph (see [13]). Such MM, denoted by , need not be unique. Any MM provides a decomposition of the graph into paths and cycles; it can be proved that is the number of right-unmatched vertices of (note also that ) and such value does not depend on the specific that we may have found. Note that any MM only takes into account the no-dilation property and it does not provide information about node accessibility; equivalently, once a set of control inputs has been connected to the right-unmatched nodes, in order to complete the control configuration, we may require adding some new wires from any input(s) to the nonaccessible nodes, without altering the number of required inputs, (see dashed line in Figure 3).
The computation of different s has been analyzed in [14, 15].
3.2. Minimum Number of Directly Controlled States (or Dedicated Inputs)
The second optimization problem associated with matrix is concerned with the minimum number of states that have to be directly controlled with an input signal.
Problem 8. Find with a minimum number of columns so that each column of has only one nonzero entry (i.e., it represents a dedicated input) and is controllable.
In this case, the number of dedicated inputs is exactly the same as the number of states directly accessed by an input. For example, in Figure 3, two states have to be directly accessed; hence, two dedicated inputs are required.
3.2.1. Computation of : Again the Maximum Matching Alternative
In [10], Problem 8 has been formalized by considering a graph theoretic perspective. In fact, is equal to the minimum number of disjoint state cacti that span the network. As stated there, can be indirectly computed by resorting to the relationship between graph cacti decompositions and the more easily computable maximum matchings. One must remember that a MM provides an alternative decomposition of the graph into paths and cycles. Unfortunately, the accessibility information from right-unmatched nodes to cycles is lost in a MM. Hence further analysis is required, where the relationship between the information provided by the MM and the graph strongly connected components (SCCs) becomes crucial.
In [10] it is shown that the minimum number of dedicated inputs is given by where again is the number of right-unmatched vertices with respect to the found maximum matching , is the number of nontop linked strongly connected components (SCC), and is the so-called maximum assignability index of the network (to be explained below).
Each MM found provides a set of right-unmatched nodes that are assigned an external control input. (As mentioned earlier, although the set of right-unmatched vertices may change from one MM to another, its size does not depend on the specific MM found.) Concerning the cycles provided by the matching, some of them may be accessible from a control input and some others may not. Since this accessibility information is not provided by the matching, further analysis is required, knowing that the nonaccessible cycles can only show up within the nontop linked SCCs, in order to determine .
Let be the set of all SCCs and let be the set of all nontop linked SCCs (). Then each specific defines a partition in (where stands for the disjoint union of sets) so that elements of contain vertices which belong to the set of right-unmatched (ru) vertices provided by ; one can interpret that the elements of are directly assigned an external control by so that their accessibility is guaranteed. In this context, the meaning of as the maximum assignability index of the network is formally stated by On the other hand, the elements of do not contain any of the right-unmatched vertices provided by ; hence, additional dedicated input(s) (equivalent to a wiring from any input(s) in the minimum number of required inputs problem) to at least one node belonging to each one of such elements will be required to complete full node accessibility [13]. If dedicated inputs were to be employed to implement such specific matching and associated wiring, the total number of inputs would be where stands for the number of additional wires required. Hence, Problem 8 can be formulated as finding a MM which minimizes . Since the number of required wires also satisfies , we have that Since is solely determined by the network topology (being independent of the obtained ), the solution of Problem 8 requires the computation of (by solving a maximization problem over all possible s).
4. Observability of LTI Systems and Duality Results
We now consider the LTI system defined by where again , , and are given a priori. This system is said to be observable (in the classical sense) if, for any known input , the state space initial condition can be determined in finite time by measuring only the output vector .
It can be shown that for LTI systems matrix does not affect the observability property, which only depends on the relationship between matrices and . Hence, the observability analysis can be addressed relying on a duality property (see [22] for details).
In the following, we address structural observability and associated design issues which will provide similar results to the controllability analysis performed earlier. In addition, duality issues are considered when referring to both structural controllability and observability properties.
4.1. Observability and Optimal Design of
In the same way as for the controllability analysis, there are practical situations, where no restrictions on matrix exist, so that it can be freely selected. Therefore one can formulate diverse problems concerning the design of optimal matrices satisfying different minimality requirements.
Such matrix design problems can be related to the previously presented design problems for matrix , invoking duality. In the following we demonstrate some results concerning the design of both optimal and matrices.
4.2. Minimum Number of Required Inputs and Outputs
Given the LTI system (12), we state the following result concerning Problem 6 and its dual counterpart.
Theorem 9. Consider system (12), where only matrix is predefined (i.e., matrices and and the corresponding dimensions of and can be freely designed); let be the minimum number of inputs to make it structurally controllable and let be the minimum number of outputs ( will stand for observability) to make it structurally observable. Then
(This result was empirically noted in [14].)
Proof. By invoking the duality between the observability and controllability concepts [22], the observability analysis of the system defined by matrix can be performed by studying the controllability of the system defined by . Since the structural properties are grounded on the classical ones, determining the minimum number of outputs to guarantee structural observability in a system defined by is equivalent to determining the minimum number of inputs to guarantee structural controllability of the system defined by .
The dual system will be structurally controllable if
And again, the minimum number of inputs for that new system would be
Since
we conclude that
This proof relies only on algebraic properties of . An alternative proof can be constructed using graph theoretical results and the duality principle. Based on duality, the observability analysis in a given graph , with adjacency matrix , is equivalent to the controllability analysis in a graph whose adjacency matrix is ; that is, a graph with the same set of nodes and whose links in have the directions of links in flipped. We call such a graph the dual graph of .
Every MM of (considered merely as a set of links, neglecting their directions) is also a MM of . Also, is composed by a disjoint union of paths and cycles, so that the number of required inputs is determined by the size of such paths and cycles. Since flipping the directions of links does not change the number and size of those paths and cycles, we have . The sensors would be connected to the right-unmatched vertices determined by in or equivalently to the left-unmatched vertices determined by in .
Note that this result does not imply that the number of required wirings should be the same, since it will depend on the accessibility of the cycles provided by , which can change from to (the directions of links do matter when determining accessibility), as illustrated in the following subsection.
4.3. Minimum Number of Dedicated Outputs
Given the LTI system (12), we now consider the dual counterpart of Problem 8, that is, the required dedicated outputs for guaranteeing observability.
Based on duality it can be shown that the minimum number of dedicated outputs (sensors) is given by where again corresponds to the left-unmatched vertices in provided by , is the size of the set composed by the nonbottom linked SCCs, and is the maximum assignability index of the network (now also referred to as the nonbottom linked SCCs).
A parallel reasoning to the one carried out for controllability can be performed for the observability analysis, where the left-unmatched vertices play the role of the previous right-unmatched ones and the nonbottom linked SCCs play the role of the previous nontop linked ones.
4.4. Dedicated Inputs versus Dedicated Outputs
It is obvious that, in general, need not be equal to ; by the same way, may not be equal to . Accordingly, the number of required wirings may be different from . Therefore we get the following.
Remark 10. Consider system (12) and let be the minimum number of dedicated inputs to make it structurally controllable, and let be the minimum number of dedicated outputs to make it structurally observable. Then may or may not to be equal to .
For instance, if , then , whereas if , then (see Figure 4).
(This result was also empirically discovered in [14].)
The difference of value between and suggests that the relationship between these two quantities can shed some light on a further characterization of the network properties.
5. Properties of Maximum Matchings and Strongly Connected Components
In this section some fundamental results are presented for addressing the practical solution of the two problems presented in Section 3. In order to simplify the exposition, the controllability problem will be considered to illustrate the results. Note that the whole reasoning applies also to the observability analysis, which is performed by merely applying the same reasoning to the dual network.
As mentioned earlier, can be obtained via the computation of a MM. We will see that MMs are also crucial for the computation of together with the properties of the SCCs. In the following some fundamental results concerning the properties of MMs and the network SCCs are presented.
5.1. Properties of Maximum Matchings
We begin by stating some properties which characterize the structure of the set of possible MMs; precisely, the construction of a MM from another one by only performing few changes is addressed, which will lead to characterize similarities between different MMs. In order to make the notation easy, the MM, as subgraph of , will be defined with representing their set of links.
Given a MM so that one of its right-unmatched nodes in has an incoming link in , the following results address the possibility of constructing a new MM whose is obtained by just swapping such node of by another node in .
Lemma 11. Let be a MM and let be the set of right-unmatched nodes of . Let be such that there exists a link (going from some node to ). Then, there exist a node and a MM such that implying that .
Proof. Let us consider the subgraph with . Obviously, must contain some , a second outgoing link from (if not, would become a matching with more links than , leading to a contradiction with the maximality of ). By removing such link we obtain a new subgraph with the same number of links as and satisfying again the no-dilation condition (i.e., a new MM), which satisfies .
Remark 12. Note that if and there exists such that , then Lemma 11 does apply, implying the existence of , where has been swapped in by another single node to form .
Lemma 13. Let be a MM and let be the set of right-matched nodes of . Let be such that there exists with . Then, there exist a node and a MM such that .
Proof. Let us consider , the link right-matching node . Note that since . Let us now consider such that . Note that must contain such a link; otherwise, would be a valid matching, contradicting being maximum. If we construct , then we face two possibilities.(1)If , then would be the matching we are looking for such that .(2)If , then would have two incoming links to . Let be such that (note that since and ). Let also , , such that (note that such link must exist; otherwise, would be a valid matching, leading to a contradiction). We construct , where again we can have two possibilities: if , then we are done with and . Otherwise, we could apply the same reasoning recursively until some node is encountered such that allowing its swapping with .
Lemma 14. Let be a MM and let be the set of right-matched nodes of . Let be such that there exists with . Then, there exists a set of nodes and a MM such that .
Proof. From the previous Lemma 13, we can construct a MM such that for some . Applying again the same reasoning of Lemma 13, we can construct a new MM such that , where again which guarantees that . The procedure can be applied repeatedly for each to obtain a new such that with which guarantees that . When , the desired result holds.
These lemmas will allow for an efficient search of appropriate MMs.
5.2. Properties of the Elements of
We now address some properties of , the set of nontop linked SCCs, from the point of view of their relationship with the different MMs which can be defined in the network.
For every , let represent the set of its vertices or nodes. For any MM , let be the set of nodes of having an outgoing link in . Then we can define and to be a partition of into two subsets: nodes whose outgoing links in are between nodes of and those whose outgoing links leave , respectively, so that . Note that , the set of right-unmatched nodes of for , satisfies (hence .
If , all nodes of are right-matched for and we define . We can then define the following subset of : We will see that the elements of accept a perfect matching; hence, they may end up being inaccessible from any input in a given MM, requiring an additional dedicated input. Therefore, further analysis of this type of subgraphs is required.
The following theorem analyzes the existence and similarity among different MMs when focused on the elements of .
Theorem 15. If (equivalently, accepts a perfect matching), then (1) (equivalently, ) for all ;(2)given any , it is possible to construct an alternative so that is any arbitrary subset of ( taking the corresponding arbitrary value between and ) and is the same as for links not outgoing from nodes of .(In particular, one can construct such a so that , meaning that ).
Proof. (1) Let us first consider the existing such that . Then every node of must have an input link belonging to , necessarily coming from another node of . Therefore, so that defines a perfect matching in ; note also that , so that all links outgoing from do not belong to (). The same can be said for any other satisfying .
Let now be an alternative MM so that (i.e.,). We will show now that .
Note that . On one hand, if , we would have and could define with the (known existing) perfect matching in so that and which would allow to preserve the same links as in the rest of the network; this would imply leading to a contradiction. On the other hand, if , we would have leading also to a contradiction.
Therefore and .
(2) Let be any MM; note that, from 1, . We begin by constructing such that , in two parts. On one hand, would contain the (known existing) perfect matching in so that . Since , this would allow completing by keeping the same links as in the rest of the network (satisfying ).
We can now construct so that is any arbitrary subset of . Since for each of its nodes, we can remove the (known existing) outgoing link in and restore the corresponding link in . This again would allow completing by keeping the same links as and in the rest of the network.
Finally, note the true equivalence in the theorem statement: for any , we have that if and only if accepts a perfect match. If , we have already seen that defines a perfect matching in . Alternatively, consider that accepts a perfect match. As shown above, given any , either or we can construct such that .
We now formulate a result illustrating the existence of MMs which can make or not, one by one, the elements of to be right-unmatched.
Corollary 16. Let , for some .(1)If there exist and satisfying and , then there may not exist satisfying .(2)The other way around, if there exists satisfying , then there must exist and satisfying and .
Proof. (1) The first part of the corollary is obvious due to the interdependence of the outgoing links in the elements of . For instance, let us consider
whose graphical representation can be found in Figure 5. There exists such that is the subgraph gathering nodes , and gathers . There exist and satisfying and , but no satisfying .
(2) Given , such that , then, by Theorem 15, we can construct a new MM (let us call it ) such that , being the same as for links not outgoing from nodes of (this includes all links involving nodes of , since there cannot be links from nodes of to nodes of ). Therefore .
The same reasoning can be applied to justify the existence of .
We now consider the optimality with respect to : a MM is optimal if for any other . The following result provides information about the existence of optimal solutions in a standard form.
Corollary 17. Let ; then there exists an optimal such that .
Proof. Let us consider being optimal and . By Theorem 15, we can construct a new MM (let us call it ) such that , being the same as for links not outgoing from nodes of . Note that neither nor require a wiring on , and the required wirings in the rest of the network remain unchanged. Hence, invoking (10), we have .
Remark 18. Let and let be optimal with . Let be such that , being the same as for links not outgoing from nodes of . Then is not optimal since a new wiring is required and .
Nevertheless, there may exist another optimal such that , but should be necessarily different from for links not outgoing from nodes of .
5.3. Compatibility
We are now ready to state the final results which will determine the steps of the algorithms for searching optimal solutions .
Definition 19. Let . We say that is top-assignable if and only if there exists a MM such that .
Note that we only need to consider top-assignable elements of in the search for an optimum .
Definition 20. Let be top-assignable. We say that are compatible if and only if there exists a MM such that .
By Theorem 15 it is equivalent to guarantee that there exists a such that , . Note that all unitary sets of the form with assignable are also compatible; the definition provides new insights when being particularized for pairs (i.e., pairwise compatibility implicitly addressed in Corollary 16).
We say that are incompatible if they are not compatible.
The following lemma proves a fundamental property of compatibility and incompatibility.
Lemma 21. Let so that . Let be two different nonempty subsets of , such that and are subsets of so that all elements of are compatible among them and all elements of are also compatible among them. Let us consider without loss of generality. Then, there exists compatible such that and (equivalently, ).
Proof. From the hypotheses, there must exist the following MMs:(i) such that (since );(ii) such that (since is compatible);(iii) such that (since is compatible).
Let us consider the subgraph , where all nodes of are removed from together with the links outgoing from them. Note that restricted to each defines a perfect submatching on it. Hence, restricted to defines a maximum submatching on it; otherwise, a matching larger than could be constructed on the whole graph by adding to the new larger submatching the subgraphs with their corresponding perfect submatchings.
Let us consider now and , the corresponding submatchings of and on , respectively. By Theorem 15, all these submatchings are maximum in having all size . By construction has right-unmatched nodes (let us call them ) each one being the destination of the corresponding link outgoing ; by the same way has right-unmatched nodes (let us call them ) destination of the links outgoing in . Let ; then for all , we have , , and we can apply Lemma 14 to obtaining (which satisfies ) and such that . Then, we have that, for , and . Completing such submatching with the corresponding submatchings, , , and , we would complete the required MM to end the proof.
The following corollaries prove some relationships when the sets are modified element by element; they also show that pairwise incompatibility, besides being symmetric, is also a transitive property.
Corollary 22. Let be top-assignable, so that are compatible and , are incompatible. Then, there exists such that are compatible.
Proof. Calling and we only need to apply Lemma 21.
Corollary 23. Let be top-assignable, so that is incompatible with both and , respectively. Then and are incompatible.
Proof. Keeping the assumptions, we will consider and to be compatible; then, applying Lemma 21, we know that either or must be compatible, which leads to a contradiction.
Corollary 24. Let be top-assignable and incompatible. If is compatible with , then is also compatible with .
Proof. If we consider and incompatible, the first assumption and the transitivity property would imply and being incompatible, which contradicts the second assumption.
Finally, Lemma 21 allows for a useful characterization of the set of possible optimal matchings.
Theorem 25. Let be top-assignable and compatible. Then such that , for (i.e., ).
Proof. Let us consider optimal, so that , where obviously (otherwise such that would provide a larger set of right-unmatched components contradicting the optimality of ). Then applying Lemma 21, we can construct a new compatible set satisfying and . Since is optimum, then and MM is also optimum, satisfying the theorem statement.
6. A New Algorithm for Computing
The proposed algorithm for locating an optimal is as follows.(1)Determine .(2)Determine all elements of accepting a perfect matching; for each , we remove the links outgoing and compute a maximum submatching in . If such matching is perfect, then accepts a perfect matching. Let be the elements of accepting a perfect matching. We call the subgraph, where are removed from together with the links outgoing from them.(3)For all elements of , determine the set of top-assignable elements ; for doing so, we apply procedure P1. If a given happens to be assignable, then a maximum submatching is already available which might also give additional information about assignability of other 's as well as compatibility among them.(4)Construct as the maximum set of assignable and compatible elements provided by the previous step (the index being reordered without loss of generality). Note that if some assignable element has been found, will contain at least one element. If no is assignable, then , implying that we are done since all elements of require a dedicated input.(5)For , check compatibility of (applying procedure P2). If is compatible, then .(6).(7)By (8) and keeping in mind , is directly obtained as
The basic procedures are as follows.(P1)Given , this procedure determines if it is top-assignable. We first compute a maximum submatching in , called ; is to be employed as a reference of the attainable MM size (note that , where refers to the whole network, and it is obtained by adding to the perfect submatchings corresponding to each ). Then, for all , we check the existence of a maximum submatching in having one right-unmatched node belonging to the set of destination nodes of links outgoing (i.e., for each destination node, the links entering it are removed and the existence of a MM in such new graph is checked. Note that one may need to check for all the destination nodes associated with ). If such maximum submatching is found, we define to be assignable. If is top-assignable, this procedure will provide at least one maximum submatching associated with one of such destination nodes.(P2)Given and top-assignable, determine if they are compatible. For doing so, we only need to consider the existence of a maximum submatching in which has one right-unmatched node belonging to each set of destination nodes of links outgoing each and one right-unmatched node (different from the previous one) belonging to the set of destination nodes of links outgoing . Note that one may need to check for all the possible pairings of different destination nodes associated with and , respectively. If and are compatible, this procedure will provide at least one maximum submatching associated with a pair of such nodes. The search of such maximum submatching can be (sometimes) simplified by using available maximum submatchings and from P1 and following the procedure proposed in Lemma 13.
6.1. Suboptimal Solutions
Finding the control configuration with the minimum number of dedicated inputs might be computationally expensive for large networks; hence the consideration of a suboptimal solution can be useful. Considering the expression proposed in [10], a suboptimal solution could be proposed, requiring dedicated inputs. This upper bound can be computed by determining the nontop linked SCCs of and performing a MM search on the network. Since the MM search dominates the complexity of the algorithm, computing such suboptimal solution takes time.
Analogously, given the definition of in step (10) of the algorithm proposed on Section 6, the minimum number of required inputs can also be expressed as
Note that a new smaller upper bound to can also be derived from this expression by just computing and . While the latter is already available at step (2) of the algorithm, can be obtained by just performing a MM search on .
This suboptimal solution computes the MMs of subgraphs and that define a partition on . Since the time complexity of a MM search is superlinear, finding a MM for each of the subgraphs is faster than computing a MM of the whole network. This means that this latter upper bound is not only closer to the optimal solution but also less computationally expensive.
7. A Comparative Example
The following example illustrates the behavior of the new proposed algorithm. If we consider the network in Figure 6, the application of a simple MM-based algorithm plus direct wiring keeping track of accessibility (see [15] for details) may provide (depending on the obtained MM) a different number of required dedicated input signals, ranging from four (corresponding to a solution with two unmatched nodes and only two wirings, in the most favourable case) to eight (two unmatched nodes plus six wirings, in the worst case). If we combine the MM-based algorithm with another one which also determines the SCCs for an ordered accessibility track (see [16]), we may obtain (again depending on the obtained MM) solutions ranging from four dedicated inputs (corresponding to an optimal solution) to six (two unmatched nodes plus four wirings, in the worst case).
The new proposed algorithm always leads to an optimal solution by first determining ; then, applying procedure P1, it finds that is not assignable whereas are assignable. Finally, applying procedure P2, it determines that are pairwise compatible but is not compatible. These results lead to selecting (among others) the optimal MM shown in the figure such that . (Note that nodes and are controlled since they constitute a cycle which is accessible from either or .)
8. Concluding Remarks
Structural controllability and observability of complex directed networks have been analyzed by combining algebraic and graph theoretic tools. Two different design problems have been addressed and the extent of some controller/observer duality results has been demonstrated. In addition, some results concerning the structure of optimal solutions and their relationship with respect to MM have also been proved; these results have led to new algorithms to efficiently compute optimal and suboptimal solutions for monitoring large scale real networks.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors want to acknowledge the financial support of Orange (Spain and France), in the framework of Cátedra Orange, at the ETSI Telecomunicación, in the Universidad Politécnica de Madrid (UPM), Spain. This work has been also partially supported by Projects MTM2010-15102 of Ministerio de Ciencia e Innovación and Q10 0930-144 of the UPM, Spain.