The Challenge of Deep Models , Inference Structures , and Abstract Tasks ? Werner Horn Department of Medical Cybernetics and Artificial Intelligence , University of Vienna , Freyung 6 , A-1010 Vienna , and Austrian Research Institute for Artificial Intelligence , Vienna . email : werner@ai.univie.ac.at August 31 , 1990 Abstract The paper discusses methodological achievements which have been incorporated into second generation expert systems . The key ideas are -LRB- 1 -RRB- to incorporate more principled knowledge about the domain into a knowledge based system and to reason from these -LRB- first -RRB- principles , -LRB- 2 -RRB- to define the conceptual model explicitly , and -LRB- 3 -RRB- to do some abstraction . Abstraction is done on three levels : the factual knowledge level , the level of inference steps , and the task level . The challenges of these methods are discussed from the viewpoint of medical and technical applications . ? OEFAI-TR-90-17 , appeared in Applied Artificial Intelligence , 5 -LRB- 1 -RRB- 47-56 , 1991 Introduction The great rumor about expert systems started in the late 1970s . It was provoked by systems of great promise for medical diagnostic support and for interpretation , diagnosis and configuration in natural sciences areas and in technical domains . It resulted in a widespread application of expert system technology . The goal was to represent the skills of experienced domain experts in a knowledge based system -LRB- KBS -RRB- and make this knowledge available to less experienced users when solving daily problems . And the goal is still the same today . Capturing the experience of experts focussed on his/her ability to find a quick solution to the problem . This resulted in building systems in a straightforward manner by representing associations between symptoms often seen and a diagnosis -LRB- e.g. for the diagnostic problem -RRB- . Despite the success in several areas of application , a list of problems showed up : ffl While working well in the central area of expertise , the systems reach horrible conclusions on their knowledge boundaries . There is no graceful degradation of performance when coming to these boundaries , as we would expect from an expert . More seriously , the systems are not able to recognize their own boundaries . ffl Besides simple backward tracing of rule invocations there is no chance to provide meaningful and user-tailored explanations . This would require to explain the underlying principles how the problem at hand is solved . ffl Knowledge acquisition is the most pressing problem . In what is commonly called the Feigenbaum bottleneck : there is one knowledge engineer -LRB- the bottleneck -RRB- who filters all the expert 's knowledge . The usual situation can be envisioned by having an expert telling the knowledge engineer some pieces of knowledge which come to his mind during a rather unstructured interview . The knowledge engineer has only a vague idea how these pieces of knowledge fit into the overall picture of the domain represented . As a result we are facing the problems of bad formalization , incorrect representation of knowledge , inconsistencies , and last but not least the open question of completeness . A common cause of these problems is the lack of any understanding what the expert is really doing . The expert systems are representing surface knowledge only . The surface knowledge of the expert is transformed directly into a rule-based or frame-based form of representation . As a result , these systems do not have any idea about the underlying causes of the problem , the inherent structural relations and the tasks the expert is performing when solving the problem . They stay on the surface level . Toward abstract knowledge structures , problem solv - ing methods , task specifications In the early 1980s several people stressed the importance of going away from the direct implementation of surface knowledge . Donald Michie pointed towards high-road programs incorporating structural information in the form of causal representations -LSB- Michie 1981 -RSB- . Peter Hart identified several levels of representation , introducing the term deep knowledge . He put emphasis on the combination of surface and deep level systems in order to get fast performance and a good understanding of what is going on : \ In the near term , success of AI as viewed from the external world is most likely to come from the exploitation of surface systems , but long-term successes will come from continuing research on deep systems and from understanding the relation between deep and surface systems '' -LSB- Hart 1982 , p. 15 -RSB- . Allan Newell got one step further by introducing the knowledge level : \ Knowledge is to be characterized entirely functionally , in terms of what it does , not structurally , in terms of physical objects with particular properties and relations . This still leaves open the requirement for a physical structure for knowledge that can fill the functional role . In fact , that key role is never filled directly . Instead , it is filled only indirectly and approximately by symbol systems at the next lower level '' -LSB- Newell 1982 , p. 105 -RSB- . After nearly ten years of research we are able to see three methodological achievements : 1 . An explicit representation of the application domain by incorporation of structural and functional knowledge into knowledge based systems . These systems are often called second generation expert systems or deep expert systems -LSB- Steels 1985 -RSB- . 2 . A thorough description of the inference steps performed during problem solving . An abstract view of these inference steps gives us inference structures -LSB- Clancey 1985 -RSB- and problem solving methods -LSB- McDermott 1988 -RSB- . 3 . A specification of the tasks performed during problem solving . Whereas it is quite clear that the tasks differ from application to application , researchers try to find abstract task descriptions -LSB- Chandrasekaran 1986 , 1987 , 1988 -RSB- . These three achievements are not to be seen as separating items , but rather as methods for knowledge-based systems operating on different levels . The integration of these levels can be seen in the KADS system -LSB- Hayward et al. 1987 , Schreiber et al. 1988 -RSB- . KADS is a system distinguishing knowledge at four different levels : -LRB- 1 -RRB- domain knowledge , describing concepts and elements of the domain and their interrelations ; -LRB- 2 -RRB- the inference structure , describing inferences which can be made . The inference structure relates meta-classes . Meta-classes are an abstracted view of the domain knowledge by describing the role of domain concepts ; -LRB- 3 -RRB- the task knowledge specifies how the goals of the application can be achieved using the available inference steps ; and -LRB- 4 -RRB- the strategic level , which should be able to control the execution of tasks by following specific strategies depending on the situation of the current case or problem . This strategic level is rather small when looking at existing KBS , but it reminds us of the possibility to follow different reasoning or problem-solving strategies . This will be definitely of importance in broad domains for future KBS .1 Leaving the strategic level aside for a moment , we get the componential framework of the layers : domain model , problem-solving methods , and task specification -LRB- see also -LSB- Steels 1990 -RSB- -RRB- . All three contribute essentially to the depth of a knowledge-based system by making explicit what are the knowledge structures of the domain , how do reasoning methods operate on these structures and what are the tasks performed during the problemsolving process .2 Key Ideas Several key ideas which form the basis of these methodological developments are summarized in the following . References are with respect to applications in medical and in technical domains . This should enable us to contrast the biological field versus the technical field in the discussion . Incorporating More Principled Knowledge About Domains The first key idea is to incorporate more principled knowledge about the domain into the KBS . This can be done both by building a detailed model of the domain , and by following the principled ways of reasoning within the domain . Representing more principled knowledge about the domain means -LRB- 1 -RRB- to bring a lot of structure into the model , often using the special form of taxonomies ; and -LRB- 2 -RRB- to relate this structures . This relation can be done by linking the structures due to known causality . In this case we enter the area of causal modeling . The inference process follows the pathways of the causal net during problem solving . The causal net approach has its foundations in early medical -LRB- AIM - -RRB- systems -LRB- CASNET -LSB- Weiss et al. 1978 -RSB- , CADUCEUS -LSB- Pople 1982 -RSB- , ABEL -LSB- Patil et al. 1982 -RSB- -RRB- . It supports the trace of known pathophysiological relations . In addition , it is a convenient way to model the basic constituents of the medical domain : anatomy -LSB- Horn 1989 -RSB- , physiology , pathophysiology , etiology and nosology -LSB- Senyk et al. 1989 -RSB- , thus providing a basis for AIM-systems with the intent to cover broad areas of medical expertise . 1The ` Oxford System of Medicine ' - a decision support system designed for the use by General Practitioners - operates on the strategic level . It is able to modify medical strategies . Generic decision-making schemata are tailored to suit the needs of the tasks to be performed during a consultation -LSB- Glowinski et al. 1989 -RSB- . 2A thorough discussion of factors being related to the depth of expert systems can be found in -LSB- Bylander 1987 , Chandrasekaran et al. 1987 , Klein and Finin 1987 , Keravnou and Washbrook 1989 , Steels 1988 , Washbrook and Keravnou 1990 -RSB- . A second form of relation is the functional relation . It describes the function or behavior of components by giving the function of its subcomponents . More and more detailed description levels can be reached using functional decomposition . This is of interest both for medical diagnosis and technical diagnosis -LSB- Steels 1989 , Sticklen and Chandrasekaran 1989 , Sticklen et al. 1989 -RSB- . A third way of bringing more principled knowledge about the domain into the KBS is by describing the physiological structure of the system . This structure is used to simulate the behavior of the system . The simulation can be done either by using a model describing the physiological mechanisms in the form of qualitative relations -LRB- e.g. electrophysiology of the heart -LSB- Bratko et al. 1989 , Hunter et al. . 1989 -RSB- -RRB- or by using a qualitative , constraint model like the QSIM approach -LSB- Kuipers 1986 -RSB- , which is mainly based on differential equations . Qualitative simulation is well suited to model flows -LRB- of liquids , pulses , information , etc. -RRB- both in the medical domain -LSB- Ironi et al. 1989 -RSB- and the technical domain -LSB- Haag 1988 -RSB- . The hard problem is how to make use of the simulation model for a diagnostic system . One possibility is to assume each component to be faulty which does not produce the expected behavior . The second is to explicitly define faults by introducing fault models . Model-based diagnosis currently concentrates on finding single or multiple device faults at hardware troubleshooting -LSB- Davis 1984 , de Kleer and Williams 1987 , Reiter 1987 , de Kleer et al. 1990 -RSB- . A detailed description of the structure and behavior of devices enables to use more principled methods of reasoning . That 's what -LSB- Davis 1983 -RSB- called reasoning from first principles . This works fine for technical domains as the correct or faulty behavior of devices is well defined . In the medical field ` correct ' behavior is often impossible to define exactly . The first principles approach is therefore limited to areas with well described physiological behavior , like renal physiology -LSB- Kunz 1983 -RSB- . Defining the Conceptual Model Explicitly The second key idea is the explicit definition of a conceptual model , which is clearly separated from the implementation . The conceptual model specifies the KBS at the knowledge level . The KADS approach provides interpretation models for creating a high-level functional specification of the problem solving process in the form of a conceptual model . The interpretation models are defined for many different inference methods . Similarly , Chandrasekaran 's generic tasks are conceptual entities , which define a specific function at the knowledge level . The conceptual model is an abstraction of the problem solving behavior of the domain experts . It is completely independant of the implementation of the KBS . In addition to the conceptual model , the KADS approach defines a design model , which is on the same level of abstraction as the conceptual model . It takes into account constraints from performance requirements , appearance of the system -LRB- e.g. user interfaces -RRB- , and integration requirements with existing hardware and software systems . The design model is a specification of the KBS at a level , which -LSB- Steels 1990 -RSB- calls the knowledge-use level . It should support direct implementation of the AI system . Abstraction Abstraction has a central role in designing KBS . Corresponding to the levels of a KBS previously mentioned , there are different forms of abstraction : ffl There is abstraction of factual knowledge . Knowledge is represented at different levels of abstraction supporting the inference steps to be performed at different levels of detail . An early example is the ABEL system -LSB- Patil 1981 , Patil et al. 1982 -RSB- . The importance of abstraction steps for diagnosis was also recognized early in the CA - DUCEUS system for internal medicine -LSB- Pople 1982 -RSB- by introducing planning links associating manifestations with abstract involvement structures . The use of abstractions during diagnosis helps to focus the reasoning process . The resulting improvement of efficiency when applying abstractions on model based diagnosis has been shown by -LSB- Mozetic 1990 -RSB- . But abstraction is not only of high importance to diagnosis . Planning has to look for ways to reuse solutions of one problem for a broad range of new problems . This can be achieved by building abstract plans -LSB- Tenenberg 1986 -RSB- . ffl From the viewpoint of reasoning steps , Pople 's planning links form the basis of the inference step ` match abstract solution ' . That is the second step in the three step procedure : data abstraction -LCB- match abstract solution -LCB- refinement of solution . This procedure was found by -LSB- Clancey 1985 -RSB- to be a fundamental abstract inference method when building the NEOMYCIN system . It is called ` heuristic classification ' . In the meantime several other abstract inference methods and problem solving classes have been identified -LRB- e.g. cover-and differentiate , propose-and-revise , hierarchical design -RRB- . This is of high importance to knowledge acquisition . Using a repertoire of standard problem solving methods it is possible to instantiate the templates of these methods -LSB- Breuker et al. 1987 , Marcus 1988 -RSB- , if the way an expert solves a problem matches one of the abstract problem solving methods . The templates guide the knowledge acquisition process by telling what to look for . ffl Comparable to the idea of abstract inference methods is the idea of abstract task structures : The generic task approach -LSB- Chandrasekaran 1986 , 1987 , 1988 -RSB- has its focus on tasks like diagnosis , classification , or design . They are generic in the sense , that they will be instantiated to real tasks when confronted with a specific application . A generic task defines its function -LCB- the kind of problem it solves -LCB- , the knowledge structure and organization , and the control strategy to accomplish the function of the task . The focus is here on improvement of the knowledge acquisition process , too . Tools are provided for instantiating generic tasks : CSRL -LSB- Bylander et al. 1983 -RSB- , IDABLE -LSB- Sticklen 1983 -RSB- , DSPL -LSB- Brown and Chandrasekaran 1988 -RSB- . If the application problem to solve is associated correctly with a generic task or a combination of tasks , the corresponding tool -LRB- s -RRB- will guide the knowledge acquisition process . What can we expect ? Let us review the methodological achievements under the aspects of the limitations of first-generation expert systems : ffl Robustness vs. performance : The use of structural domain knowledge and the use of principled reasoning methods based on this fine-grained domain knowledge is the way to achieve robustness when solving uncommon problems . But deep models are very slow performing knowledge structures when applying principled ways of reasoning during problem solving . Their challenge is the ability to serve as a ground for automatic construction of surface models by use of machine learning techniques analyzing cases often seen . In combination we get a fast performing problem solver when confronted with common cases , which can fall back to its robust reasoning methods on complicated situations . ffl Explanability : The knowledge about the tasks and the inference steps the KBS is performing allows for explanations what is really going on . But not much focus has been put to explanation issues by the research community . In addition , explanations are expected to be user-tailored in the sense that they should take into account the level of experience of the user , the facts he/she already knows , and his/her intentions . This results in the need of an explicit user model . User modeling is a current area of AI research , but it will take some time until user models will guide the explanation -LRB- and reasoning -RRB- methods of expert systems in practice . ffl Knowledge acquisition support : The support tools using the specification of domain models , problem solving classes and abstract task descriptions give us a chance to build KBS in a more consistent and complete way by having a specification of the whole problem solving process . They provide a basis to keep the system expandable and maintainable during the whole lifecycle . But -LRB- unfortunately -RRB- the knowledge engineer is needed more than before . At moment direct knowledge transfer from the expert to the KBS is possible only for fixed domains with specific forms of representation -LRB- e.g. , OPAL -LSB- Musen 1989 -RSB- -RRB- . Model-based knowledge acquisition -LSB- Shadbolt and Wielinga 1990 -RSB- will hopefully guide the complex knowledge acquisition process in the form of an active and directive system in the future . Of essential importance is the functional definition of the problem solver at a conceptual level . The specification of KBS at the knowledge level as instantiation of abstract knowledge structures may help us -LRB- 1 -RRB- to create understandable , expandable , and portable systems . We are not captured by the implementation details which often override the conceptual structures of the application . -LRB- 2 -RRB- It may give us new insights to domain models , problem solving classes and task structures , thus , enhancing our repertoire of methods . Comparing the biological versus the technical domain we recognize the influence of medical applications not only to first-generation expert systems , but also to model-based architectures . The comparison of these two domains raises the question : Is there a unifying perspective ? I.e. , are there completely domain-independent architectures ? What speaks against is the fact that we have very different domains -LRB- medical vs. technical -RRB- : analog systems with tolerances and correcting feed-back loops vs. digital systems with 0-1 behavior and errors effecting the whole system ; partial models resulting from an incomplete understanding of biological processes vs. complete technical models which are easy to describe formally ; external -LRB- environmental -RRB- causes of errors vs. internal causes of errors -LRB- material problems -RRB- ; many different knowledge structures vs. complex reasoning processes operating on few knowledge types . This may be the reason why medical applications tend to focus on the extensive task of representing domain knowledge , whereas applications in the technical domains concentrate on finding efficient inference methods . What speaks in favor of the unifying perspective is the fact that many medical applications formed the basis for more abstract methodologies , which have been used lateron in the technical domain : MYCIN ! EMYCIN , NEOMYCIN ! the inference method ` heuristic classification ' , MDX/PATREC ! the generic task architecture . What seems to be identifyable are the same problem-solving methods and abstract domain concepts throughout very different domains . That makes abstract modeling a worthwhile thing to do . But at moment , we have only a very restricted view : -LRB- 1 -RRB- most systems have dealt with the problem of diagnosis , -LRB- 2 -RRB- the domains of application have been very restricted , and -LRB- 3 -RRB- we are at the early beginning of the usage of structure-based knowledge acquisition tools . In conclusion , second-generation architectures are still in the research labs . They will have a hard time to find the way to daily practice . 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O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O

deep models
generic tasks
inference structures
second generation expert systems


Disturbance decoupling by measurement feedback for structured systems : a graph theoretical approach Report 92-53 J.W. van der Woude Technische Universiteit Delft Delft University of Technology Faculteit der Technische Wiskunde en Informatica Faculty of Technical Mathematics and Informatics ISSN 0922-5641 Copyright c 1992 by the Faculty of Technical Mathematics and Informatics , Delft , The Netherlands . No part of this Journal may be reproduced in any form , by print , photoprint , microfilm , or any other means without permission from the Faculty of Technical Mathematics and Informatics , Delft University of Technology , The Netherlands . Copies of these reports may be obtained from the bureau of the Faculty of Technical Mathematics and Informatics , Julianalaan 132 , 2628 BL Delft , phone +3115784568 . A selection of these reports is available in PostScript form at the Faculty 's anonymous ftp-site . They are located in the directory / pub/publications/tech-reports at ftp.twi.tudelft.nl Disturbance decoupling by measurement feedback for structured systems : a graph theoretical approach J.W. van der Woude Abstract Structured systems are linear systems of which each of the coefficients either is fixed to zero or is an independent free parameter . In paper we study the well-known disturbance decoupling problem by measurement feedback for such systems . We derive sufficient conditions for the generic solvability of the problem . Generic solvability here means solvability in almost all cases . The conditions will be stated in terms of a directed graph that can be associated to a structured system . The advantage of this is that the conditions then can be verified by means of well-known and efficient combinatorial algorithms . 1 Introduction In this note we study so-called structured linear systems . These are linear systems in which each of the entries of the system matrices either is fixed to zero or is a free parameter . Hence , a structured system is given by the zero-nonzero structure of its matrices . This zero-nonzero structure can be nicely represented by means of a directed graph . In this note we formulate a version of the disturbance decoupling problem by measurement feedback for structured systems . We develop graph theoretic conditions that are sufficient for the solvability of the problem . The outline of this note is as follows . In section 2 we recall some basic facts on disturbance decoupling for linear systems defined in the usual way . Next in section 3 we introduce structured systems and indicate how a structured system can be represented by means of a directed graph . Also we recall some relevant results concerning the disturbance decoupling problem by state feedback for a structured system . In section 4 we present our main result ; sufficient conditions for the generic solvability of the disturbance decoupling problem by measurement feedback for structured systems . The conditions will be stated in terms of the graph that represents the structured system . We conclude this note with some remarks in Section 5 . Upon preparing this document we became aware of the paper -LSB- 1 -RSB- in which also sufficient conditions in graph theoretic terms are derived for the generic solvability of the problem treated in this note . However , we want to point out that the starting point in -LSB- 1 -RSB- are solvability conditions in frequency domain terms , whereas in this note we start from solvability conditions in geometric -LRB- state space -RRB- terms . Although the conditions obtained in -LSB- 1 -RSB- are different from the conditions derived here it may be interesting to compare them . This will be left a subject for future investigations . 2 Disturbance decoupling In this section we recall some of the basic facts on disturbance decoupling . We start with the well-known disturbance decoupling problem by state feedback . To this end , we consider the following linear system 8 > < > : _ x -LRB- t -RRB- = Ax -LRB- t -RRB- + Bu -LRB- t -RRB- + Qd -LRB- t -RRB- ; y -LRB- t -RRB- = Cx -LRB- t -RRB- ; z -LRB- t -RRB- = Hx -LRB- t -RRB- ; with state x -LRB- t -RRB- 2 Rn , control u -LRB- t -RRB- 2 Rm , disturbance d -LRB- t -RRB- 2 Rq , measurement y -LRB- t -RRB- 2 Rp and output z -LRB- t -RRB- 2 Rr . A , B , Q , C and H are real matrices of suitable dimensions . We frequently denote the above system by ? or ? -LRB- A ; B ; Q ; C ; H -RRB- . The system described by the first and the third equation alone we occasionally denote by ~ ? or ~ ? -LRB- A ; B ; Q ; H -RRB- . For ? -LRB- or ~ ? since the measurements are not relevant -RRB- we can pose the following question -LRB- cf. -LSB- 2 -RSB- -RRB- . Question 2.1 Does there exist a state feedback u -LRB- t -RRB- = Fx -LRB- t -RRB- for system ? such that the transfer matrix of the resulting closed loop system is identically equal to zero , i.e. such that H -LRB- sI ? -LRB- A + BF -RRB- -RRB- ? 1Q = 0 ? We say that the disturbance decoupling problem by state feedback -LRB- abbreviated as DDP -RRB- for system ? is solvable precisely then when question 2.1 can be answered affirmatively . The following characterization of the solvability of the DDP is well-known -LRB- cf. -LSB- 2 -RSB- -RRB- . Theorem 2.2 The DDP is solvable for ? , ImQ ? V ? , where V ? denotes the largest -LRB- A ; B -RRB- - invariant subspace in KerH . For more details on -LRB- A ; B -RRB- - invariant subspaces we refer to -LSB- 2 -RSB- . There it is also shown that the subspace V ? can be computed by means of a finite recursive scheme using the matrices A , B and the subspace Ker H . To express the dependency of V ? on A , B and KerH we occasionally write V ? = V ? -LRB- Ker H ; A ; B -RRB- . In the above it is assumed that the full state can be used for the feedback . If this is not possible and only the measurement -LRB- = partial state -RRB- is available for feedback purposes we can pose the following question -LRB- cf. -LSB- 3 -RSB- -RRB- . Question 2.3 Does there exist a feedback compensator for system ? of the form -LRB- _ w -LRB- t -RRB- = Kw -LRB- t -RRB- + Ly -LRB- t -RRB- ; u -LRB- t -RRB- = Mw -LRB- t -RRB- + Ny -LRB- t -RRB- ; with state w -LRB- t -RRB- 2 Rk and real matrices K ; L ; M and N of suitable dimensions such that the transfer matrix from d to z is identically equal to zero , i.e. such that -LSB- H 0 -RSB- -LRB- sI ? '' A + BNC BM LC K # -RRB- ? 1 '' Q # = 0 ? We say that the disturbance decoupling problem by measurement feedback -LRB- abbreviated as DDPM -RRB- for system ? is solvable precisely then when question 2.3 can be answered affirmatively . The following chararcterization of the solvability of the DDPM is well-known -LRB- cf. -LSB- 3 -RSB- -RRB- . Theorem 2.4 The DDPM is solvable for ? is solvable , S ? ? V ? , where V ? is as above and S ? denotes the smallest -LRB- C ; A -RRB- - invariant subspace that contains ImQ . For more details on -LRB- C ; A -RRB- - invariant subspaces we refer to -LSB- 3 -RSB- . Also S ? can be computed by means of a finite recursive scheme using the matrices C , A and the subspace ImQ . To express the dependency of S ? on C , A and ImQ we occasionally write S ? = S ? -LRB- Im Q ; C ; A -RRB- . It is well-known that V ? and S ? are each others dual in the sense that S ? -LRB- Im Q ; C ; A -RRB- = -LRB- V ? -LRB- Ker Q > ; A > ; C > -RRB- -RRB- ? where > denotes transpose and ? orthogonal complement . With this duality and theorems 2.2 and 2.4 the following lemma is immediate . Lemma 2.5 The DDPM is solvable for ? -LRB- A ; B ; Q ; C ; H -RRB- , there exist matrices X and Y such that -LRB- i -RRB- Ker Y ? ImX , -LRB- ii -RRB- the DDP is solvable for ~ ? -LRB- A ; B ; X ; H -RRB- and -LRB- iii -RRB- the DDP is solvable for ~ ? -LRB- A > ; C > ; Y > ; Q > -RRB- . 3 Structured systems and graphs In the remainder of this paper we take the system ? -LRB- A ; B ; Q ; C ; H -RRB- as a starting point . In addition we assume that each of the entries in the matrices A ; B ; Q ; C and H either is fixed to zero or is an independent free parameter . Collecting all the free parameters in a parameter vector > = we say that the DDPM -LRB- or DDP -RRB- for system ? is generically solvable if the problem is solvable for almost all values of > = . Here ` almost all ' is to be understood as ` for all except those in some set with Lebesgue measure zero ' . As mentioned before the zero-nonzero structure of the system can be nicely represented by means of a directed graph G . This graph will consist of a set of vertices V and a set of directed edges E and is therefore also denoted G = -LRB- V ; E -RRB- . The set V equals X -LSB- U -LSB- D -LSB- Y -LSB- Z with X = fx1 ; : : : ; xng the set of state vertices , U = fu1 ; : : : ; umg the set of control vertices , D = fd1 ; : : : ; dqg the set of disturbance vertices , Y = fy1 ; : : : ; ypg the set of measurement vertices and Z = fz1 ; : : : ; zrg the set of output vertices . Hence , V consists of n + m + q + p + r vertices . Denoting -LRB- v ; v0 -RRB- for a directed edge from the vertex v 2 V to the vertex v0 2 V the set of edges E is described by EA -LSB- EB -LSB- EQ -LSB- EC -LSB- EH with EA = f -LRB- xj ; xi -RRB- jAij < > g , EB = f -LRB- uj ; xi -RRB- jBij < > g , EQ = f -LRB- dj ; xi -RRB- jQij < > g , EC = f -LRB- xj ; yi -RRB- jCij < > g , EH = f -LRB- xj ; zi -RRB- jHij < > g . In the latter , for instance Aij < > means that the -LRB- i ; j -RRB- th entry of the matrix A is a free parameter . Examples of graphs are given in section 5 and 6 . Consider the graph G = -LRB- V ; E -RRB- and let W ; W be two nonempty disjoint subsets in the vertex set V . We say that there exists a path from W to W if there is an integer t and vertices w0 ; w1 ; : : : ; wt2 V such that w0 2 W , wt 2 W and -LRB- wi ? 1 ; wi -RRB- 2 E for i = 1 ; 2 ; : : : t . The path is then said to have length t and to consist of the vertices w0 ; w1 ; : : : ; wt . Occasionally we denote the path by the sequence of directed edges it consists of , i.e. by -LRB- w0 ; w1 -RRB- ; -LRB- w1 ; w2 -RRB- ; : : : ; -LRB- wt ? 1 ; wt -RRB- . Two paths from W to W are called disjoint if they consist of disjoint sets of vertices . We say that l paths from W to W are pairwise disjoint if each two of them are disjoint . Let W 00 be an additional subset in V and let a set of l pairwise disjoint paths from W to W be given . We say that the total number of vertices in W 00 on the l paths is if the intersection of W 00 and the set of vertices that the paths consist of has size . For a structured version of ? -LRB- A ; B ; Q ; C ; H -RRB- and the corresponding graph G = -LRB- V ; E -RRB- we have the following results -LRB- cf. -LSB- 4 -RSB- , -LSB- 5 -RSB- -RRB- . Theorem 3.1 The disturbance decoupling problem for the structured system ? -LRB- A ; B ; Q ; C ; H -RRB- is generically solvable if and only if a -RRB- the largest number of pairwise disjoint paths from U to Z equals the largest number of pairwise disjoint paths from U -LSB- D to Z , say t , and b -RRB- the smallest total number of vertices in X -LSB- U on any t pairwise disjoint paths from U to Z equals the smallest total number of vertices in X -LSB- U on any t pairwise disjoint paths from U -LSB- D to Z . As an application of theorem 3.1 let the matrix ~ Q be obtained from Q by replacing one free parameter by a fixed zero . In the graph G this corresponds with the deletion of an edge from the set EQ -LRB- an edge from D to X -RRB- . By this the largest number of pairwise disjoint paths from U -LSB- D to Z does not increase and the smallest total number of vertices in X -LSB- U on t pairwise disjoint paths from U -LSB- D to Z does not decrease . From theorem 3.1 it therefore follows that if the DDP is generically solvable for ? -LRB- A ; B ; Q ; C ; H -RRB- then it is also generically solvable for ? -LRB- A ; B ; ~ Q ; C ; H -RRB- . The same holds if ~ Q is obtained from Q by deleting an entire column . The latter corresponds in G with the deletion of a vertex from D and the edges in EQ starting from this vertex . Denote ei for the ith unit column and gi for a column with a free parameter at the ith position and fixed zeros elsewhere . Repeated application of the previous implies that if ei > Q < > -LRB- so the ith row of Q is not a zero row -RRB- and if the DDP is generically solvable for ? -LRB- A ; B ; Q ; C ; H -RRB- then the DDP is also generically solvable for ? -LRB- A ; B ; gi ; C ; H -RRB- . Motivated by this we define I ? = fi 2 nj the DDP is generically solvable for ? -LRB- A ; B ; gi ; C ; H -RRB- g ; where n = f1 ; 2 ; : : : ; ng . Clearly , I ? only depends on A ; B , H . By linearity it follows that the DDP is generically solvable for ? -LRB- A ; B ; Q ; C ; H -RRB- for any Q that has nonzero rows with indices in I ? . Hence , we have obtained the following sufficient condition . Proposition 3.3 The DDP for ? -LRB- A ; B ; Q ; C ; H -RRB- is generically solvable if ej > Q = for all j 2 nnI ? . To check whether or not i 2 I ? the graph associated to ? -LRB- A ; B ; gi ; C ; H -RRB- can be used . The graph has one disturbance vertex -LRB- jDj = 1 -RRB- and every path starting in that vertex next passes through the state vertex xi . Now assume that the the largest number of pairwise disjoint paths from U to Z equals t and the smallest total number of vertices in X -LSB- U on t of pairwise disjoint paths from U to Z equals ? . From theorem 3.1 it then follows that I ? = fi 2 nj the largest number of pairwise disjoint paths from U -LSB- fxig to Z equals t and the smallest total number of vertices in X -LSB- U on t pairwise disjoint paths from U -LSB- fxig to Z equals ? : g The computation of I ? can be done in an efficient way using algorithms from combinatorics based on the max flow min cut theorem and on the theory of minimal cost flows in transportation networks -LRB- cf. -LSB- 6 -RSB- chapt . 4 , -LSB- 7 -RSB- chapt . 2 -RRB- . Dual to I ? we define J ? = fj 2 nj the DDP is generically solvable for ? -LRB- A > ; C > ; gj ; B > ; Q > -RRB- g with gj as before . Clearly , it follows from the previous that J ? = fj 2 nj the largest number of pairwise disjoint paths from D to Y -LSB- fxjg equals s and the smallest total number of vertices in X -LSB- Y on s pairwise disjoint paths from D to Y -LSB- fxjg equals ? : g Here s denotes the largest number of pairwise disjoint paths from D to Y and ? the the smallest total number of vertices in X -LSB- Y on s of pairwise disjoint paths from D to Y . 4 Main result In this section we present some graph theoretic sufficient conditions for the generic solvability of the DDPM . Let ? -LRB- A ; B ; Q ; C ; H -RRB- and G = -LRB- V ; E -RRB- be as above and let I ? and J ? be as defined in the previous section . Then our main result consists of the following . Theorem 4.1 If I ? -LSB- J ? = n then the DDPM is generically solvable for ? -LRB- A ; B ; Q ; C ; H -RRB- . Proof Assume I ? = fi1 ; i2 ; : : : ; iag and J ? = fj1 ; j2 ; : : : ; jbg . Denote XI ? = -LSB- ei1 ; ei2 ; : : : ; eia -RSB- and YJ ? > = -LSB- ej1 ; ej2 ; : : : ; ejb -RSB- . Then , because I ? -LSB- J ? = n , it follows that Ker YI ? ? ImXJ ? . By the definition of I ? and J ? and using linearity it follows that the DDP is generically solvable for both ~ ? -LRB- A ; B ; XI ? ; H -RRB- and ~ ? -LRB- A > ; C > ; YJ ? > ; Q > -RRB- . This means that for almost all values of the free parameters in A ; B ; Q ; C and H the DDP is solvable for both ~ ? -LRB- A ; B ; XI ? ; H -RRB- and ~ ? -LRB- A > ; C > ; YJ ? > ; Q > -RRB- . Hence , by lemma 2.5 , the DDPM is solvable for almost all vallues of the free parameters . Therefore , the DDPM is generically solvable for ? -LRB- A ; B ; Q ; C ; H -RRB- . In theorem 4.1 we have presented some sufficient conditions for the solvability of the DDPM . It turns out that for all systems that we studied upto now the conditions were necessary too . However , we do not yet have been able to prove or disprove the necessity of the conditions . 5 Worked example In this section we illustrate the results of section 4 . For that purpose we consider a structured system ? -LRB- A ; B ; Q ; C ; H -RRB- taken from -LSB- 1 -RSB- with n = 8 ; m = 2 ; q = 2 ; p = 2 ; r = 2 and with the matrices A = 2 6666666666664 > = 1 > = 2 > = 3 > = 4 > = 5 > = 6 3 7777777777775 ; B = 2 6666666666664 > = 7 > = 8 > = 9 3 7777777777775 ; Q = 2 6666666666664 > = 10 > = 11 3 7777777777775 ; C = '' > = 12 > = 13 > = 14 # ; H = '' > = 15 > = 16 # : The associated graph G can be depicted as below . Recall that X = fx1 ; : : : ; x8g , U = fu1 ; u2g , D = fd1 ; d2g , Y = fy1 ; y2g and Z = fz1 ; z2g . Note that f -LRB- u1 ; x1 -RRB- ; -LRB- x1 ; z1 -RRB- g and f -LRB- u2 ; x2 -RRB- ; -LRB- x2 ; x3 -RRB- ; -LRB- x3 ; z2 -RRB- g form a pair of disjoint paths from U to Z . Since jZj = 2 -LRB- jZj denotes the number of elements of Z -RRB- , the largest number of pairwise disjoint paths from U to Z is equal to 2 . In fact , because jZj = 2 the largest number of pairwise disjoint paths from U -LSB- fxig to Z is equal to 2 for any i 2 n. Furthermore , the smallest total number of vertices in X -LSB- U on a pair of disjoint paths from U to Z equals 5 . The above given pair of paths from U to Z is unique and has 5 vertices in X -LSB- U . Since jDj = 2 there are only pairs of disjoint paths from D to Y . The paths f -LRB- d1 ; x4 -RRB- ; -LRB- x4 ; y1 -RRB- g and f -LRB- d2 ; x7 -RRB- ; -LRB- x7 ; x8 -RRB- ; -LRB- x8 ; y2 -RRB- g form the only one of such pairs . Also there can only be pairs of disjoint paths from D to Y -LSB- fxjg for any j 2 n . The smallest total number of vertices in X -LSB- Y on the unique pair of disjoint paths from D to Y equals 5 . It is easy to see that I ? = f4 ; 5 ; 6 ; 7 ; 8g and J ? = f1 ; 2 ; 3 ; 5 ; 6 ; 8g . For instance , 3 62 I ? . Indeed , the smallest total number of vertices in X -LSB- U on a pair of disjoint paths from U -LSB- fx3g to Z equals at most 3 as follows from the pair f -LRB- u1 ; x1 -RRB- ; -LRB- x1 ; z1 -RRB- g and f -LRB- x3 ; z2 -RRB- g. Furthermore , 3 2 J ? since the smallest total number of vertices in X -LSB- Y on a pair of disjoint paths from D to Y -LSB- fx3g equals 5 . Indeed , there are only two pairs of disjoint paths from D to Y -LSB- fx3g . One of these pairs is given above and the second is formed by the paths f -LRB- d1 ; x4 -RRB- ; -LRB- x4 ; y1 -RRB- g and f -LRB- d2 ; x7 -RRB- ; -LRB- x7 ; x8 -RRB- ; -LRB- x8 ; x2 -RRB- ; -LRB- x2 ; x3 -RRB- g . The total number of vertices in X -LSB- Y contained by this last pair is 6 . - - - - - - - - - - ? ? ? HHHHHHHHHHHj @ @ @ @ @ @I ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? > = d2 s u2 s u1 s d1 s y2 s z2 s z1 s y1 s x7s x8 s x2s x3 s x1 s x6 s x5 s x4 s Since I ? -LSB- J ? = n it follows from theorem 4.1 that the disturbance decoupling problem by measurement feedback is generically solvable . To compute a feedback compensator that solves the DDPM the actual values of the entries in A ; B ; Q ; C and H are required . 6 Remarks and conclusions In this note we developed a sufficient condition for the generic solvability of the DDPM for structured systems . The conditions can be verified by means of efficient algorithms based on the max-flow-min-cut theorem and on results concerning minimal cost flows in transportation networks -LRB- -LSB- 6 -RSB- , -LSB- 7 -RSB- -RRB- . In structured system approach we only -LRB- may -RRB- use the zero-nonzero structure of the system matrices . This means however that the algorithm derived in this note also can serve as an additional tool in the case that one is considering the DDPM for a system defined in the usual way , i.e. a system of which the system parameters are known . Focussing on the conditions in theorem 2.2 and proposition 3.3 , or in theorems 2.4 and 4.1 , we may be tempted to compare v?g and jI?j where v?g denotes the generic dimension of V ? . However , an equality between v?g and jI?j does not hold in general . It easily follows from theorem 2.2 and proposition 3.3 that v?g > = jI?j , but the inequality may be strict as we see in the next example . Therefore , consider a system ? -LRB- A ; B ; Q ; C ; H -RRB- with A = 2 64 > = 1 3 75 ; B = 2 64 > = 2 > = 3 3 75 ; H = -LSB- > = 4 > = 5 -RSB- and Q , C not specified . The relevant part of the corresponding graph can be depicted as follows . - ? ? ??? - - @ @ @@ Rs s s s s u x1 z x2 x3 It easily follows that I ? = fx2g , so jI?j = 1 . Further , it can be computed by means of the recursive scheme in -LSB- 2 -RSB- that V ? is spanned by the vectors -LSB- 0 1 0 -RSB- > and -LSB- > = 5 ? > = 4 -RSB- > for almost all values of > = 1 , > = 2 , > = 3 , > = 4 , and > = 5 . Hence , in general we can only conclude v?g > = jI?j . Similarly , in general only n ? s?g > = jJ?j with s?g denoting the generic dimension of S ? . Using results from -LSB- 8 -RSB- the generic dimensions v?g and s?g can be computed by means of other graph theoretical techniques . However , due to space limitations this can not be treated here . References -LSB- 1 -RSB- C. Commault , J.M. Dion , M. Benahc?ene , `` Output feedback disturbance decoupling graph interpretation for structured systems '' , Internal Report LAG , Laboratoire d'Automatique de Grenoble , 1992 . -LSB- 2 -RSB- W.M. Wonham , Linear Multivariable Control : a geometric approach , 3rd ed. , Springer Verlag , New York , 1985 . -LSB- 3 -RSB- J.M. Schumacher , '' -LRB- C ; A -RRB- - invariant subspaces : some facts and uses '' , Report nr . 110 , Wisk . Sem . VU , 1979 . -LSB- 4 -RSB- J.W. van der Woude , `` On the struture at infinity of a structred system '' , Lin . Alg . Appl. , Vol . 148 , 1991 , pp. 145-169 . -LSB- 5 -RSB- C. Commault , J.M. Dion , A. Perez , `` Disturbance rejection for structured systems '' , IEEE Trans . Aut . Control , Vol . 36 , 1991 , pp . 884-887 . -LSB- 6 -RSB- M.Gondran , M. Minoux , Graphs and Algorithms , Wiley , Chichester , 1984 . -LSB- 7 -RSB- H. Walther , Ten Applications of Graph Theory , Reidel , Dordrecht , 1984 . -LSB- 8 -RSB- C. Commault , J. Descusse , J.M. Dion , J.F. Lafay , `` New decoupling invariants : the essential orders '' , Int . J. Control , Vol . 44 , 1986 , pp. 689-700 .

B I O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O B I O O O O O O B I O O O O O O O O O O O O O O O O O O B I O O B I O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O 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