Leon Miller
email: leonmonroemiller@yahoo.com

Keywords: Faith, Integrity, and Integral Being Arête, Coadunation, Entelechy, Preordain, Telos, Reintegration, Authentic Being

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Introduction

The term integrity applies to an individual's character and to a person whose character is considered admirable because the person is honest, trustworthy and reliable (Graham 2001, 234-235). A person of integrity has a highly developed sense of self and is always true to that self thus is a person who is considered to be a matured character. Such a person is able to consistently act in accordance with chosen values thus always acts in a way that enables the realization of those values. However, it is clear that the concept integrity also has implications connected with what it means to be a well-integrated person. In this sense the term has to do with a character that is able to experience and maintain well-being. Thus, integrity also has to do with what it means to be well-integrated within the fabric of being (i.e. integrity denotes a type of person who understands how to interact with the aspects of existence that are necessary for flourishing in a way that increases beneficial outcomes). If the scope of integrity is broadened it still involves an individual's self-cultivation but it is also indicative of an individual who has achieved integral being. Achieving integral being is evident as the ability to experience complementary interactions with the other aspects of one's social and natural environment. In this respect, integrity-as a trait of the ethical character-has implications that are relational.

The concept integrity is promoted in connection with helping the individual achieve a desired state of being (Loewy 2002, 56-57). The desired state is similar to what Aristotle described as the ability to experience one's highest good. It is also clear that the world's knowledge and wisdom traditions have proposed integrity as a way of enhancing the individual's life experience. The world's wisdom traditions have always indicated that integrity plays a part in providing a person with the sense of a complementary (or harmonic) connection with existence. In addition, in both the East and the West, promoting integrity is intended to help individuals live with less threats to their well-being and in a way that is less harmful to other aspects of existence. It is clear that the most recent views of science and philosophy are indicative of an attempt to generate knowledge that is able to create similar character results and a similar life experience. Thus, knowledge traditions encourage integrity because it helps the individual to realize values that have long been cherished by both Eastern and Western traditions (religious, scientific, and philosophical). This article argues in support of the connection between integrity and the individual character however, in addition, it claims that integrity can be described in relational terms and in a way consistent with contemporary Social Psychological views on self-formation and views regarding the ontology of social existence. The article emphasizes that there is a connection between being a person of integrity, developing good character, and enhanced human interactions, thus an enhanced human experience. This, of course, means that there is a relationship between an individual with a virtuous character and a person who has an enhanced ability to obtain beneficial outcomes in interactions. However, the article stresses that enhanced human interactions occur as a result of the ability to create beneficial interactions with other aspects of one's social and natural environments thus is the outcome of being well-integrated within one's social and natural environments (Horkheimer & Adorno 2002, 1-3). This article describes the ethical aspect of integrity by arguing that it enhances interactions in ways that increases the individual's experience of the things that he or she values highly. However, in addition, the article explains how integrityfrom the perspectives of both Eastern and Western traditionshelps individuals to realize what is considered valuable for enhancing and elevating the human experience. The second section begins with an inquiry into integrity. This section focuses on the connection between integrity and one's character. Integrity is explained in a way that is in accordance with Aristotle's Virtue Ethics (i.e. meaning explained as the ability to achieve one's highest good or most desired outcome). Thus, the second section clarifies the connection between the highest good worth achieving and a self that is fully integrated. Such a well-integrated character is able to flourish, to experience that which is the most desired outcome of or ultimate aim of one's actions, plus to experience happiness and well-being.

Section three argues that integrity provides a regenerative power that enables a person to enjoy a sustained sense of well-being. Being a person of integrity also means avoiding acting in any way that diminishes one's well-being. This amounts to being true to one's natural impulse as ordained by one's neurobiological make-up and acting in accordance with the principles that have been long cherished and promoted by the world's knowledge and wisdom traditions. In other words to be true to human nature the person of integrity continuously acts in a way that is consistent with the biological principles that underlie the ontological basis of the natural order (i.e. principles ordained by the elemental life creating forces of nature). In this sense a person of integrity acts in a way that is true to (thus to realize) values that have been proposed by the world's perennial philosophical traditions. The argument is supported by proposing an expanded conceptualization of the term integrity however one that is consistent with Aristotle's description of the benefits of a virtuous character.

Section four argues that an individual achieves integrity when he or she is integrated with existence in a way that results in the experience of well-being and flourishingin spite of challenges imposed by the nature of human interactions. This argument is supported by the claim that the very biological forces that gave rise to the human organism have also preordained an inherent neural value preference for integral being. In this respect clarity on the concept of integrity has to do with clarity on the relationship between the biological nature of existence and humanity's biological nature. In the fourth section human integrity is referred to as tantamount to Authentic Being. Thus, section four of the article also emphasizes the connection between integrity and authenticity, humanity's neural value preference, and cognition (i.e. the role that integrity plays in directing perception toward the life enhancing, elevating, and enriching opportunities in the environment). Section five summarizes the main points of the article and draws conclusions.

The Ethical Significance of Integrity

The development of a virtuous character is significant because it helps individuals in their effort to creatively navigate a way through life, taking advantage of the beneficial opportunities that reality offers while, at the same time, avoiding that which is harmful. That is to say that the person of integritywho is ethical in the sense of having a virtuous characteris inclined to be attracted to things that enhance human well-being and human interactions but avoids what would diminish one's well-being. Such a character is able to shape the necessary encounters of life (social and environmental encounters) into life enhancing outcomes. This involves learning as much as possible about the nature of things so as to understand how to interact in appropriate accord with the nature of things. In this respect, a virtuous character could be described as a person who understands and pursues that which results in his or her highest good (highest good includes the ability to maximize beneficial exchange in one's interactions).

Most ethicists point out the importance of demonstrating such character traits (usually placing the emphasis on social interactions). In this respect "The point of ethics is to codify how interpersonal and social relationships must reasonably be ordered by people who cannot help caring about the final ends that are most fundamental in the lives of all fully ethical beings" (Frankfort 2006, 198). Thus, if the goal of ethics is excellence in character and action we can think of an ethical person as one who possesses character traits that are magnanimous. If the central ethical question is, "How should one live?" The answer is not only a matter of providing an account of what ought or should not be done. A complete answer will also inform an individual as to what kind of person one ought to be. If the major question ethics is concerned with is what is good then the answer is good is that which helps one to gain insight into what is necessary for achieving the ultimate good (Halfon 1989, 37). "An adequate ethical theory must pertain to one's ethical character in interactions and relationships. A person who lacks this disposition, from the perspective of what it means to be [ethical as a] human being, may be properly referred to as deficient in the qualities of Virtue Ethics" (Sandler 2005, 2-5). This is the case becauseas philosophers of Virtue Ethics arguenot only are social interactions essential to human flourishing but effectively interacting with the greater natural order is also essential for enhancing the human experience (Hirsh et al. 2012, 312). Thus, increasingly Virtue Ethicists include the importance of being well-integrated within the overall natural fabric of existence as a significant aspect of what it means to display a virtuous character. In this sense a person fails to be a good human being if he or she disrupts that which is necessary for good interactions (either socially or in terms of the natural order). From this point of view, the highest good is not only thought of in anthropocentric terms but thought of in terms of the interdependent relationship that humanity has with other biological organisms and non-organic entities. The fact that humans exist in a complex web of interactions means that having a good understanding of the highest good includes consideration of what is involved in being well-integrated within this complex web. To live well one must acquire from nature those things that are needed for nourishment and growth. If the quality of those things is in any way threatened or diminished the quality of the human experience is threatened and/or diminished. Thus, prudence also demands that one does not harm the things that human existence depends on for survival (Sandler 2005, 5).

Defined in these terms an explanation of what it means to be ethical would be expanded to include avoiding those things that could damage one's character, might damage one's own physical well-being, and could damage a relationship with those things needed for flourishing. Rosalind Hursthouse supports the prospect of an expanded explanation of what it means to be ethical by pointing out that ethics and human values are all related to what contributes to improving interpersonal and social interactions thus what helps individuals to be betterintegrated. She also stresses that as the outcome of a certain level of moral and character development one learns to appreciate the common bond that humanity has with all living things (Hursthouse 2007, 163-164). Such a life is worth living because of the happiness it provides, it lends to the flourishing of the individual, it contributes to generating prosperity for society, and results in a life of tranquility.

Aristotle believed that all things (especially all organic things) exist with the ontological necessity of maintaining integrity without which they will begin to deterioratethis includes individuals but is also true for institutional systems, cultures, and societies. Aristotle's naturalism prompted him to claim that prudence (i.e. a type of knowledge demonstrated by an ethical character) is intended to increase an individual's understanding of the teleological significance of the human connection with natural processes plus enhance humanity's participation in natural processes (Falcon 2005, 16). On the basis of this claim one could argue that realizing one's full potential (entelechy) for happiness, flourishing, and well-being is based on understanding the teleological significance of human interactions (e.g. intersubjective and with nature). For, as contemporary social psychologists point out, the research regarding factors that promote or impede the realization of one's full potential indicates that Holistic well-being is based on acting in accordance with humanity's neurological predisposition or value preference for being well-integrated within one's eco-niche (Scheffler 1985, 10-34 & 49).

Because concern for the self is tied to a concern for one's ability to have beneficial interactions with other aspects of existencewhich is necessary for nourishment and growth thus for survivalgood or that which is beneficial for human beings is considered to be that which enables a person to achieve his or her telos. According to Aristotle achieving telos is connected with understanding how to have an appropriate relationship with the natural order (2002,24). Bad (or that which harms) are those things which interfere with achieving telos (or bad can also be considered as that which diminishes the quality of those things on which the human biological organism is dependent for well-being and flourishing). In other words, telos is connected with nature's intentionality (Searle 2002, 79-88). However, experiencing the potential value that nature-human interactions have for enriching human existence is dependent on learning to shape intentionality into beneficial and life enhancing outcomes.

In this respect, according to Rosalind Hursthouse, experiencing eudaimonia requires developing the type of character that [given humanity's biological and neurobiological nature] inclines the individual "To flourishto do well and fare well-in this world in which we inevitably find ourselvesnot just in the particular culture or society we happen to find ourselves in" (Hursthouse 2002, 65). Hursthouse goes on to say that our philosophy of ethics needs to be expanded to include a particular new trait that would specify the importance that interactions have to achieving telos. Humanity's telos is preordained by the primordial biological forces that shape patterns of natural interchange into complex structures of beneficial exchange (i.e. the very basis of life for the human organism and for human society is the ability to form structures of cooperative interchange). The fact that human behavior is directed at (or is about) veridical interaction with the natural order means that human interactions are prompted by an ontological impulse that predisposes individuals to attempt being appropriately integrated with the natural order. Thus, the natural biological forces that generate humanity's biological nature demand nature-human integration and perpetual reintegration.

Individuals are impelled by the impulses of their biological nature to interact with the environment. The necessity of interaction can give the impression that humans are bound by biological necessity. However, the person of integrity experiences complementary interactions with the natural and social environment thus he or she is able to act with more freedom and less impediments in the effort to gain satisfaction (Dewey 1922, 10). In support of this claim the world wisdom traditions (Eastern and Western) have asserted that it is possible to have a wellintegrated connection with one's natural and social environments. From the perspective of the world's wisdom traditions interactions are not only the source of self-formation, nourishment, and growth but also the basis of a heightened sense of well-being and harmony. David Loy points out that Buddhism, for example, stresses that all living things are necessarily intermixed with thus interpenetrate other aspects of existence (2003,5). Loy, in fact, believes that the Buddhist concept of interdependent origination (pratitya samutpada or interpenetration) is not only a fundamental concept of Buddhist beliefs but the basis of Buddhist views on ethics and Social Psychology (2003. 8, 17, & 22). Buddhism is an example of how Eastern and Western perennial philosophies have advocated a conceptual prescription for maintaining a perceptual focus on what is essential for human flourishing. Such a perspective is believed to be the basis of reliable knowledge (faith) plus the key to avoiding what is harmful for human well-being.

G. E. M. Anscombe argued that living well is firstly a matter of having an informed conceptual understanding of why improving human interactions is as important to ethics as utility and duty. Anscombe emphasized the relational aspect of ethics by claiming that it is a "brute fact" that the quality of human existence necessarily depends on the quality of human interactions. She attempted to make this point clear by pointing out the difference between 'is' and 'needs' in terms of an organism's connection with its environment. A brute fact is that an organism needs to be well-integrated with the environment in order to flourish. In other words, to say that a biological organism depends on its environment is to say that it won't flourish unless it is wellintegrated within that environment (Anscombe 1958, 6). What Anscombe emphasized is that the need (the natural urge humans have to engage the environment in an effort to obtain nourishment and sustenance) coupled with the desire is what creates the normative ought (i.e. to enhance survival an individual ought to have a good connection with the environment).

The Western religious traditions describe humanity's initial naturally ordained condition as being well-integrated within the natural order. The belief is that humanity's initial immersion in nature was characteristic of the way humanity's biological nature is constituted thus was a fundamental aspect of how humans initially perceived existence. In this sense it is important to remember that the environment and the perceived experience of the environment are linked together in a way that creates co-dependency (coition or mutuality) between our bodies and things (Merleau-Ponty 2005, 373). Thus, the earliest conceptualization of the nature-human experience reflected a prescription for perceiving and experiencing "at-one-ment" with the natural order. The three Western mono theistic religious traditions clearly assert (concurring with Yoga/Vedanta, Hinduism, Chinese Taoism, and Japanese Shinto beliefs) that this is a preferable perspective from which to perceive and experience existence. They also state that the subsequent alteration of this preferable perspective is unfortunate and should be rectified. Anscombe explained that Western Civilization's foundational conceptualization of justice (although seemingly remote to us) was that justice should be commensurate with ideas of interrelatedness which can also be described as "The balance or ´harmony which keeps things going" (which according to Anscombe modern humanity seems to have lost sense of) (1958,12).

Integrity and Realizing a Desired State of Being

The term integrity applies to a person who has an admirable moral character and whose strength of character is reflected in his or her moral commitments (Halfon 1989, 37 & 55-57). However, the description of the virtuous character, according to Aristotle, includes an individual who experiences a sustained sense of well-being or, put more simply, a person with a virtuous character is able to live well in spite of the challenges he or she is confronted with from the social and natural environments. Based on extending the understanding of ethics to include integrity the term also has to do with a regenerative power that extends longevity thus provides an ability to sustain well-being. In this sense integrity means an ability to hold a biological organism together or to keep its system intact. It is a type of virtue of character that also promotes the integrity of those systems related to, involved with, or connected to it (Gresser 1996, ix).

Integrity is defined as an ability to hold all the parts together so that they are integrated in a way that promotes synergetic interactions. Integrity in this sense means that all the parts are working well in relationship to each other and together they form a well-functioning and well-integrated whole system (Carter 1996, 7). When the term is applied in this way to human actions and interactions it refers to the social value connected with the term integration. Integrity is related to the concept integration in the sense that integrity promotes social interactions that increase outcomes that are beneficial for all stakeholders and a person of integrity facilitates the ability of others to be well-integrated within society (Calhoun 2015, 149-153). Without such successful endeavors to promote integrity a society becomes fragmented, works against itself, and the very foundations of its structure are threatened as a result of its lack of integrity. Aristotle lays the foundation for such a definition of integrity with his political philosophy which states that ethics and "Sociology are elegantly combined to offer a delicate balance between principles and structures, ideals, and reality plus, cultural values and cultural institutions" (McCarthy 2009, 20).

Aristotle grounds his understanding of the social system that best fits human nature on his understanding of physis (i.e. the laws of nature or the principles underlying the natural order). He viewed the desire of an honorable character to live within a friendly and thriving community as prompted by natural impulses. He thought of society as operating best when in accord with natural principles and in this regard he is credited with introducing the concept of natural law (i.e. what is also popularly known as the precursor to Human Rights). Thus, Aristotle thought of a well-integrated society as one that adheres to principles constituted on the basis of natural law as the fundamental basis of social formation. In this respect social integration reflects fundamental natural principles shaped into cultural normative principles that are best suited for promoting beneficial interactions between the members of the society and with the natural order.

The concept integrity also has a meaning that is connected with lacking nothing or complete within itself. Aristotle believed that the reward of a virtuous character is the ability to sustain the experience of happinesswhich is also within itself a state ultimately desired. Aristotle regarded happiness as the experience one enjoys upon achieving self-sufficiency. Thus, Aristotle suggested that becoming a person of integrity is accompanied by realizing how to become self-sufficient. Self-sufficiency can be defined as achieving the level of harmonious interdependence that enables a person to have beneficial, reciprocal, and complementary interchange with the other aspects of one's social and natural environmentthus the person experiences what is sufficient for happiness and flourishing. "Self-sufficiency we define as that which makes life desirable and lacking in nothing; and such we think happiness to be; and further we think it most desirable of all things, without being counted as one good thing among others. Happiness, then, is something final and self-sufficient, and is the end of action" (Aristotle 1996, 12).

Although Aristotle regarded economics as a means of maximizing benefits (i.e. utility) he thought of utility more in value terms rather than in material terms (see Ekelund & Héber 2014, 15). When Aristotle talks about The Good Life he is speaking about experiencing what is sufficient for well-being, happiness, and flourishing. Although referring to self-sufficiency he was thinking in relational terms. In this sense it must be kept in mind that his notion of the good life is what can be achieved within community. Thus, there is tremendous social economic value related to the self-sufficient aspect of the virtuous character. He saw each of the individuals in this arrangement as "A part in relationship to the whole" (Aristotle 1996, 14). In other wordssimilar to what Confucius conceived -Aristotle thought that virtue would be manifest in the individual character but such an individual character would lay the foundation for harmonious household relationships, economic prosperity, and as a result good social relations and a thriving polis.

Aristotle described the phenomenal objects of nature as a composite of the basic constituents of nature. He believed that the ontological nature of existence prompts interactions as part of the measures that perpetually sustain the life process. That is to say that each particular thing exists within a unique eco-niche in which it participates in a larger eco-system (or a larger sphere of essential interactions). As a composite humanity's biological system can also can be considered to be a manifestation of (i.e. or a reflection of) the basic natural constituents that contribute to its composite form. It is in this respect that the ontological nature of existence has preordained humans to exist with a neurobiological predisposition or value preference for perpetually seeking integration or reintegration with elements that shape human existence (see Edelman 2006, 24-25). However beneficial or satisfactory outcomes can only be achieved as a result of appropriately participating in the process of integration (see the definition of integration and reintegration in the explanation of the term on pages 6-7).

Adherence to natural principles has the benefit of providing the individual with a wellintegrated sense of self where "All parts of one's nature are brought into harmony with principle" (Aristotle 1996, 24). Aristotle described the goal of a virtuous character as the experience of arête and eudaimonia (i.e. self-cultivation that results in happiness). Ethicists often describe such a character as a person of integrity because this trait helps one to avoid corrupting influences (in terms of avoiding those things which would diminish well-being). In this respect Aristotle also thought of ethics as related to an ability to maintain one's health. Aristotle certainly thought of happiness, well-being, and flourishing as connected with wellness. If we analyze Aristotle's prescription for experiencing happiness it seems that a virtuous character is reflected in or influences all aspects of a person's life. In other words, Aristotle proposed that there is a connection between ethics, human integrity, and achieving a certain desirable state of being. Aristotle thought that the desirable state that is worth achieving is one in which the person enjoys health as a significant component of the good life. This means that the factors contributing to a virtuous character also contribute to the soundness of a person's overall being. Aristotle made reference to the importance of Holistic well-being by stating that good health is one of the three types of goods. However, he also pointed out that balance (i.e. The Golden Mean) is lost by intemperance which is the outcome of too much or too little, which creates imbalance, and will disturb a person's well-being (Aristotle 1996, 229).

Acts of integrity enable individuals to be true to what was ordained by the biological forces that shape humanity's biological nature. Integrity provides the ability to resist acting in any way that would diminish or alter human nature, it triggers a regenerative power that creates healthy growth, it prompts a resistance to acts that would in any way diminish being well-integrated within one's social and natural environments, and integrity promotes recognition of the essential necessity of continuous integration or reintegration with the basic elements of existence. The good life results from being perpetually reintegrated with life's elemental forces in a way that maintains internal and external integrity or, in other words wholeness, growth, and regeneration is the outcome of being well-integrated within one's eco-niche.

Philosopher of mind John Searle argues thatin the case of the human experiencematter is constituted in a way that value, intent and meaning emerge. He adds that the interesting aspect of thisas far as humans are concernedis the fact that creation's primordial forces motivate human value preferences by building in predisposed intentionality. Searle points out that one of the most amazing features of nature is the fact that organisms -interlaced with other elements of nature in a particular wayare predisposed by nature to continuously seek reintegration with fundamental elements of the environment in their attempt to obtain nourishment, to maintain health, and to grow (Searle 1983, 46). Searle calls the urge to experience regeneration, health, and growth by means of reintegration "one of the most amazing features of nature."

Searle agrees that the human organism is interlaced with elements of nature in a particular way that shapes experience, feelings, and consciousness (1983,37). The human urge to participate in the natural process of reintegration with the environment also prompts a cognitive understanding of how essential managing the nature-human interchange is to maintaining a sense of self. Thus, intentionality is a special feature of the human effort to incorporate various natural elements into an integrated system that will contribute to the human endeavor to shape his or her life into an enjoyable well-integrated experience (Searle 1983, 46). Searle states that it is this "special feature" that highlights the significance of human interactions with the environment by giving them intention, value, and meaning.

Searle acknowledges that the fact that matter can be organized in a way that value, intent and meaning emerge from the particular interaction of mineral and chemical elements of nature is astonishing. The interesting aspect of this mystery (as it relates to the human experience) is the fact that creation's primordial biological forces have built-in a value preference that predisposes humanity with intentionality. This value predisposition motivates the intention to interact in ways that create beneficial, enjoyable, and life-sustaining outcomes. Searle concludes that at present science can state with conviction that organic organisms do reflect intentionality. In the human experience the intention is evident in the value preference for sustaining a wellintegrated experience within a particular environmental context. Thus, he suggests that humans exist with the ontological necessity of attempting to sustain beneficial interactions with as many other aspects of the environment as possible.

Modern research in complementarity conducted by physicists, neurologist, biologists, psychologists, and sociologists confirm that perennial philosophy provides a viable explanation of the link between the biological aspects of the human make-up, how individuals self-organize, the structuring of culture, and the human relationship with the environment (Berntson & Cacioppo 2008, 36-37). That is to say that the whole body (including the human cognitive abilities) is a biological self-organizing system that interacts with the environment in an attempt to shape out of environmental possibilities interactions that best complement humanity's biological organism. The challengeas it has been described by the world's faith traditionsis to maintain a focus on experiencing the human perceptual value preference without being distracted or disturbed by other possibilities that are not preferred (i.e. without being disturbed by the possibility of viewing nature from a diabolical perspective). Integrity enhances the ability to maintain a perceptual focus on the life enriching opportunities in the environment. In other words, because integrity increases the extent to which one perceives complementarity it reduces the disturbance of nature's ambiguity by increasing the probability of anticipation becoming experience.

Nobel Prize-winning neuroscientist Gerald Edelman explains that complementarity is a manifestation of the fundamental connection between nature's biological principles and the value preferences naturally triggered as human cognitive skills developed. The human neurological system (an extension of the nature's biological system) prompts a neurological value preference that reinforces the types of behavior that promise an undisturbed enjoyment of nature's beneficial opportunities (Edelman 1992, 102). The struggle for survival in an unpredictable environment predisposes such neural value preferences. The preference is in favor of the values, proven by natural selection, that contribute to shaping fortuitous encounters into an experience of life sustaining outcomes and life enriching experiences. Ineffectiveness in this regard can bring threat or harm. The threat is maximized to the extent that individuals fail in their attempt to be well-integrated with the other elements in the environment.

Edelman's explanation of neural value preference clarifies why integrity plays such an important role in increasing the beneficial interchange between two interdependent organic systems and why being well-integrated is the preferable way to experience the necessary interaction between nature and humanity. Humans exist with an anxiety resulting from increased awareness of the interplay between how nature is affected by human perceptions and how nature itself shapes perceptions. Because of the "feedback" or "looping" relationship humans have with nature humans are continuously both a product of nature and producing feedback that determines the human perception of nature. In other words humans are ontologically bound to an inescapable naturalness which must be managed appropriately in order to experience the enjoyment of life. Thus, it is because of the inescapable naturalness that the human neurological value preference inclines cognition toward realizing how to be better-integrated with the environment.

Values are what determine a person's conscious reaction to and experience of the external world. As human cognitive skills developed the human value preference favored relating to the various aspects of one's environment in ways that shape interactions into a feeling of satisfaction. However, the desire for satisfaction is coupled with an appreciable aesthetic quality which also prompts a human preference for a sustained and uninterrupted experience of life enriching and life elevating experiences. Sustained enjoyment of life enriching and elevating experiences is what Aristotle described as the good life (Aristotle 1999, 173-177). Studies in perceptual psychology and neurobiology reveal that it is possible to perceive reality in a way that increases the experience of the life enriching and life elevating possibilities that nature affords (Gibson1986, 127; also see Miller 2011, 129-144). That is to say, in terms of an anthropological and social psychological understanding of the role of conceptualization in the human experience, that cognition serves the purpose of directing perception toward possibilities for experiencing internal and external complementarity (Portugali 1996, 39-40). The urge for complementarity is based on the desire to experience nature's perpetual demand for integration and reintegration as not only beneficial but life enriching. Contemporary research in science, philosophy, anthropology, and religion substantiate the claim that the integrity of complex organisms is enhanced by fulfilling the inherent urge for beneficial interchange with nature and in the human case this helps individuals experience a fuller sense of the self. Cooperative interchange can be thought of as a natural biological value preference that establishes the inclination for social cooperation and culture. This means that the continuous commingling between the human organism and various aspects of one's social and natural environments can be thought of as "a tie that binds" all of existence into integral relatedness. Thus, humanity's inherent neurobiological value preference has been conceptualized by the world's wisdom traditions as the conviction that harmonious interaction with the environment is consistent with what was ontologically ordained to be best for human well-being and for the human experience in nature (Miller 2012, 6-15).

The perennial philosophies of the East and West claim that viewing and experiencing existence from the perspective of humanity's inherent value preference is personally beneficial, intellectually beneficial (heightens cognitive abilities), and is socially beneficial. The world's wisdom tradition claim that such a realization is more likely to result in the experience of what might be called "elevated being." Perennial philosophy adds that the fundamental biological principles of creation have encoded the organic elements of nature with intention that can be decoded by humans into an understanding of the teleological significance of existence. From the perspective of the world's wisdom traditions the teleological significance of existence is realizing how to enjoy the sense of "elevated being." This article argues that complex organisms sense this encoded ontological intention as a natural urge to be even better-integrated with one's social and natural environment plus to experience enhanced beneficial interactions. In this respect prudence demands considering that the world's wisdom traditions are right to admonish reconciling the breach between humans and nature in order to enjoy a more elevated human experience.

The Human Neural Value Preference for Integral Being

Aristotle explained his perspective on the highest good achievable by human action in a way that established a conceptual connection between eudaimonia and views on Social Psychology. In Nicomachean Ethics (section ten the first phrase) he stated that a virtuous character results from affirming what is best in an individual (Aristotle 1999, 173). Renowned psychologist Carl Rogers, who emphasized the significance of the relationship between affirmation and authenticity, stated the matter in similar terms, "The most important question for the individual is am I living [in] a way that truly expresses me" (Rogers 1961, 119). This indicates the similarity between Aristotle's views and those of contemporary Social Psychology in that both propose that being a person of integrity is tantamount to achieving authenticity.

Authenticity is related to integrity in the sense that both have to do with the extent to which one is able to shape his or her life pursuits, identity, and actions into an accurate expression one's intrinsic valuesgiven the necessities, possibilities, and constraints of reality (Ferrara 1998, 28). Because acts that are authentic are the basis of experiencing one's chosen values authenticity is also related to acting to express and maintain one's integrity. That is to say that, according to Aristotle, a person of integrity-when making judgements on how to satisfy the demands of human natureis committed to acting authentically thus indicates how the demands of reality can be met in a way that maintains integrity. Authenticity is acting in a way that is faithful to one's intrinsic value commitment thus are acts that are consistent with one's ultimate values. The consistency confirms the result in that acts of integrity authenticate the person's connection with his or her ultimate values. In Aristotle's own words, "By choosing the right means to achieve the End causes the End to be realized" (Aristotle 1935, 305; also see MacIntyre 2007, 149). In this sense a person with a virtuous character chooses to act in a way that is authentic in an effort to be true to oneself and keep the integrity if his or her being intact.

Berkeley University Professor of Philosophy Charles Guignon states that authenticity is becoming the person you are which is the ultimate task of life (2004,2). Guignon goes on to stress that he believesin accordance with Plato's writingsthat when Socrates referred to "know thyself" he was thinking of self-knowledge but in terms of realizing the connection between the essence of one's own being and The Essence of Being. Hinduism refers to this interconnection as the relationship between the individual self and the universal principle-or the laws of nature which are superimposed on every individual. In this respect self-knowledge is accompanied by an awareness of the complementary connection between the self and the natural forces that shape human existence. That is to say that Authentic Being results from opening one's normal level of consciousness to the essence of one's beingwhich is the basis of self-knowledge (see Sartre 1978, 73-85, 92-95, & 617 for an explanation of discovering the true essence of one's being, transcending the ego, the role of valuation in being true to oneself, ethics and morality, and his notion of purity as essential aspects of Authentic Being).

Integritya character trait connected with being true to values-helps a person to realize and experience the things that he or she values. Thus, integrity promotes the experience of happiness by enhancing an individual's ability to satisfy both the value preferences that are shaped by ethical rationality and those that are biologically motivated. This means not only experiencing the instrumental things a person values but it also means experiencing the fullness of one's being. In this respect integrity is related to reliable knowledge (i.e. what Aristotle referred to as epistêmê) because it helps a person to understand how to uncover or realize the authentic nature of his or her being. Acts of integrity are acts that are authentic in that they affirm the person's identity and value commitments (Sartre 1978, 58-59).

Albert North Whitehead asserted that "The whole is co-mingled with the various particulars.

[That is to say that] what is realized is interwoven into the texture of realization" (Whitehead 1948, 89 & 96). Whitehead argued that there is a connection between the essence of one's inner-most being and a life generating force that he described as the essential aspect of one's inner-most being which provides individuals the ability to experience sensation and cognition for without it a person completely lacks awareness and feeling. For Whitehead selfknowledge is defined as the cognitive realization of the essential interconnectedness between one's unique corporal being and the universal essence that sparks the awareness of one's life experience. In other words, according to Whitehead authenticity, integrity, and awareness result from connecting with the essence of one's inner-most being. Whitehead asserts that as a result of reflecting on "Our concrete experience, we see at once that the element of value; of being valuable, of having value, of being an end in itself, something which is for its own sake, [and must be counted as] the most concrete actual intrinsic reality" (Whitehead 1948, 95). "Being a value that is an end within itself" is a statement that indicates Whitehead's understanding of the connection between ethics, integrity, being true to oneself, and what has ultimate value. In other words, Whitehead described self-knowledge not only as authentic being but also as a type of epiphany that culminates in the realization of one's true self. That is to say that he equates self-realization or the experience of authentic being with awakening to the "Brooding presence of the whole onto its various parts" (Whitehead 1948, 88-89;& 1996, 100).

The mythological, anthropological and scientific theories describing the elementary stages of the human experience assert that the fundamental urge that prompts the endeavor to sustain life enhancing relationships with other elements in existence is driven by the impulse toward self-preservation and growth (James 208, 368 -especially the footnote numbered 348). Because of this, in many world cultures and faith traditions integrity is thought of in terms of an ability to experience a sense of wholeness that transcends the individual ego and self-other dualism. In such terms acts of integrity are acts that flow from one's innermost being or they flow out of the inherent nature of one's being. These acts are expressions of the innermost self or, in other words, acts of integrity are acts that flow out of one's inherent inclination toward being one's authentic self. This is the basis of the claim that integrity promotes authentic being or sincerity. When an individual acts with integrity the person is afforded the realization or the actualization of his or her innermost convictions. It is in this sense that human integrity is a manifestation of faith (or what American Pragmatists call reliable knowledge).

Thus, the experience of integral being can be defined as successfully establishing at-one-ment between oneself and the essence of being. Integral beingtantamount to authentic being (thus the reward of integrity) -is the realization of harmony with The Essence of Being. We understand from Whitehead that existence is in a state of flux (a process of constant reintegration, regeneration, and growth) which give rise to infinite creative possibilities. Because the process of life persists perpetually (life cannot be static thus the fact of life demands constant process, growth and [inter]change) each moment is the full, immediate, and complete expression of life (in terms of essential interaction or perpetual reintegration). According to Whitehead acts of integrity shape the possibilities provided by natural forces into life enriching, sustaining, and elevating outcomes. "Creativity is the ultimate principle by which the many enter into complex unity" (Whitehead 1978, 21). Diasetz Suzuki explained the realization of the connection between the essence of one's own being and The Essence of Being in terms adopted by the Japanese from the "Science of the Tao" (Tao-hsüeh) -which Suzuki considered to be the most valuable product of the ancient Chinese worldview. The Tao-hsüeh came to Japan primarily by means of Confucianism but also accompanied with reflections on Chán Buddhismas practiced in ancient China but developed in a unique way in Japan. This aspect of Japanese culture clarifies the Japanese understanding of the connection between integrity, authentic being, sincerity (without pretense), and the ability to sense what the Japanese refer to as Yugen. The ability to sense Yugen is described as an awareness of one's essential connection with a force that sparks consciousness and sensation or a type of epiphany that is triggered by sincerity. Sincerity is otherwise described as a type of purity which results from avoiding anything that would tarnish one's character (Suzuki 2010, 349; also see Trilling 1972, 2-14). Thus, sincerity (authenticity) is connected with the highest principle of the Japanese worldview and has connotations involving the highest personal quality or character trait. "Translated in terms of ethics, [authenticity] is defined as sincerity (ch'eng), and the ideal life consists in cultivating the principle of sincerity" (Suzuki 2010, 51). Such a person develops an intuitive awareness of "the brooding presence" of Yugen.

Conclusion

Integrity has been promoted as an essential aspect of self-cultivation, social psychology, and political economy because of the role it plays in increasing individual well-being and social flourishing. However, Aristotle's perspective on ethics, social psychology, and socialeconomics (i.e. his political economy) proposes that integrity has an expanded meaning that is relational and inclusive of increasing possibilities for beneficial, life enriching, and life elevating interactions. This article has focused on explicating the connection between Virtue Ethics and developing character traits that enable a person to be well-integrated within one's social and natural environments. The world's traditional belief systems lay the foundation for the experience of integrity by promoting being in harmony with the primordial forces of existence and by associating this with what it means to be in harmony with the self (Kierkegaard 1992, 115). They prescribe achieving integrity because it lends to aligning human values and human actions with what has been intended by the forces shaping the natural order.

Many of the world's faith traditions (e.g. Eastern -Yoga, Taoism, and Shinto for example and Westernthe Judeo-Christian and Islamic traditions) have always asserted that harmonious, cooperative interchange with the environment is essential for realizing humanity's fundamental individual and social values. Today contemporary neurobiologists, social psychologists, and philosophers of mind agree that inherent value predispositions are ordained by the fundamental forces shaping the biological aspects of existence (i.e. inherent human values are ordained by the ontological nature of existence). Eastern philosophy asserts that by realizing the essential connection between humans and the natural order of things a person is more likely to experience what might be called "elevated being." Such a realization benefits health, sparks regeneration, increases human vitality, and heightens mental abilities. Thus, it prompts the realization that in order for human actions to fulfill the natural urge to experience a satisfactory outcome one must strive to act in a way that is consistent with humanity's inherent neurobiological value preference.

From the perspective of Eastern philosophy and Western traditions (Judeo-Christian and Islamic) in the earliest human experience integrity afforded the feeling of a nature-human seamless complementarity (a particular sense of being well-integrated within the garden). The feeling of unified wholeness (referred to as coadunation) is the basis of an organism's feeling of integrity and in the human case helps individuals experience a fuller sense of self and what it means to be human (see Adorno 1973, 105-106; & also see Coleridge 1983, 168) . The world's scientific, mythological and philosophical traditions explain this by asserting that all things exist as a composite of natural elements. Ancient Chinese philosophy (Taoism) teaches that humans-as a composite of natural elementscan achieve a higher level of health and vitality by maintaining essential natural elements in a yin-yang balance. Western science affirms this and adds thatin the human casewhen natural elements are formed into a particular composite self-consciousness emerges and when the human organism is held together in integrity higher consciousness (or increased mental ability) is the outcome along with the enhancement of physical abilities, or in short, a fuller appreciation of and experience of nature's regenerative powers.

Wisdom traditions advocate acts of integrity in order to promote the experience of being harmoniously integrated with necessary elements in such a way as to increase the person's enjoyment of life. This is the key to experiencing the highest good worth pursuing (Aristotle 1999, 172-173). Thus, human integrity plays an essential role in successfully participating in the natural process of shaping the integration of life's elements into complex structures that are more fully well-integrated. We know that ineffectiveness in this regard can threaten health and human well-being. The threat is maximized if humanity fails in an attempt to be properly integrated with the other elements in the environment. All of human knowledge has been contrived with the intention of helping humanity act with such integrity. This article has described the nature of phenomenal existence as a perpetual interchange between natural elements. In the case of human existence, a person's physical being can be a described as a composite of the elements of nature. In addition, health, nourishment, and growth demand a constant interchange between the individual and various aspects of nature in the effort to achieve reintegration. This fact makes it evident that the world is not made up of atomistic entities juxtaposed to each other but the nature of reality is essentially relational. In other words in accordance with Aristotelian teleologyas biological an organism's survival, growth, nourishment, and well-being are dependent on being well-integrated. Because of the fact of interdependence humanity's neurobiological value preference has evolved in a way that urges enhanced interactions. In this respect integrity is important to enhanced interactions because it inclines behavior and perception toward recognizing the life-enhancing, life enriching, and life elevating opportunities in one's social and natural environments. Thus, this article defines integrity as "Learning to view the parts in such a way that we perceive them as Holistically integrated, thus the particular quality of the total increases in aesthetic value" (Peirce 1974, 84). Such a perspective on the nature of existence takes us beyond a sense of dualistic contention into improved relationships. It takes us beyond a subject-object split into cooperative union.

Aristotle believed that there is an aspect of appearance that is Universal (i.e. foundational, regenerative, and perpetualeven though forms or its appearance change). He recognized that being true to oneself or being a person of integrity is a matter of being true to one's nature, being authentic, and being sincere. In other words, Aristotle proposed that humanity's highest good (experiencing well-being, happiness, and prosperity) is based on becoming a person who is Holistically well-integrated. In this respect prudence can be defined as making use of one's rational capacity to perceive and experience complementarity (i.e. coadunation). Thus, by becoming a person of integrity one develops the cognitive ability to shape the disparate phenomena of one's existence into integral being (Williams 1975, 491).

Faith traditions, philosophy, and social psychology have promoted living with integrity in an effort to help individuals increase the possibility of enjoying beneficial interactions with others and with the environment. Such beneficial outcomes for individuals and society is based on integrating the knowledge available from faith traditions, the social sciences, and the natural sciences. In this respect enhancing the future of human existence will be based on increasing the amount of integral knowledge available to help "Humanity enjoy a more complementary view of human interactions. Thus, [making] it is possible for humanity to enjoy an enhanced perspective on human existence and human interactions" (Miller 2011, 131). Western and Eastern wisdom traditions claim that the perceptual perspective-that initially afforded humanity a harmonious view of the relationship between self and nature-was shaped by the primordial forces of existence. In both the Western and Eastern traditions the initial harmonious view was partially due to the human natural ability to actually sense compatibility between nature, the self, and culture (Durkheim 1995, 85-86, 100, 133-135, & 208). There was, as well, recognition that nature's abundance provides resources that provide for shaping cultural development in desirable ways. This gave the earliest humans the sense that in spite of the disturbances there were possibilities that nature-human interactions could be supportive of the neurobiological effort to experience a sense of complementarity with natural forces.

Professor Edward Younkins points out that a person who holds rational values, selects ends and means consonant with the nature of existence, and with the integrity of the essence of his or her own being increases the possibility of achieving his or her desired outcome (1998,2). For this reason, it can be argued that acts of integrity are acts that are life enhancing, elevating, and enriching. Such acts help individuals to be true to what was ordained by the forces of creation thus be true to the essence of their own being. In other words, such acts enable individuals to experience a harmony (or complementarity) between the biological principles comprising human nature and the biological principles that shape the human experience in/of nature. That spark of life that enlivens human being also stimulates awareness that the nature of Being (i.e. a composite of natural elements) is in essence a co-mingling of natural elements. Thus, for a person of integrity internal and external balance and harmony reflect each other.


Anton Korniienko

Xavier Bombois

Håkan Hjalmarsson
email: hakan.hjalmarsson@ee.kth.se

Gérard Scorletti

Gérard Scorletti Hierarchical Robust

Hierarchical Robust Analysis for Identified Systems in Network

is denoted as diag i (X i ) with (block-)diagonal elements X i (i = 1, ..., N ). For a complex number y, we denote yy * by y 2 while σ(A) denotes the maximal singular value of a complex matrix A.

II. PROBLEM STATEMENT

Let us consider a network of N mod single-input single-output (SISO) subsystems S i (i = 1...N mod ) operated in closed loop with a SISO decentralized controller K i (i = 1...N mod ):

S i (θ i ) : y i (t) = G i (s, θ i )u i (t) + v i (t) (1)
u i (t) = K i (s)(r i (t) -y i (t)) (2)
r(t) = A ȳ(t) + B ref (t) (3)
where s, in order to keep the discussion as general as possible and to consider both cases, defines the Laplace variable s in the continuous time domain or the shift variable z in the discrete time domain. The vector θ i ∈ R n θ i represents the parameter vector of the ith system. We will distinguish hereafter between a variable θ i ∈ R n θ i , its unknown true value, θ i,0 ∈ R n θ i , and its estimated value, θi ∈ R n θ i . Let us also define θ = [θ 1 , . . . , θ N ] T ∈ R n θ , θ 0 ∈ R n θ and θ ∈ R n θ : the stacked version of the previous parameter vectors, with n θ = i n θi . The signal u i is the input applied to the system S i and y i is the measured output. This output is made up of a contribution of the input u i and of a disturbance term v i that represents both process and measurement noises and is modeled as a stochastic random process [START_REF] Ljung | System identification[END_REF]. The different true systems are thus described by transfer functions G i (s, θ i,0 ). Moreover, the vector v ∆ = (v 1 , v 2 , ..., v N mod ) T is assumed to have mutually independent components v i .

The subsystems S i (θ i,0 ) in (1) may all represent the same type of subsystems combined into the network in order to achieve some global goals. Due to industrial dispersions, the unknown parameter vectors θ i,0 may, of course, be different for each i, the same applies to the order of the transfer functions G i .

In this technical report, the interconnection form used in formation control or multi-agent systems (see e.g. [START_REF] Fax | Information flow and cooperative control of vehicle formations[END_REF], [START_REF] Korniienko | Performance control for interconnection of identical systems: Application to PLL network design[END_REF]) is under consideration. Each subsystem S i (θ i ) is operated with a decentralized controller K i (s), see [START_REF] Ljung | System identification[END_REF], and the signal r i is a locally available reference signal that will be computed via [START_REF] Doyle | Analysis of feedback systems with structured uncertainties[END_REF]. The matrices A and B in (3) represent the interconnection (flow of information) physically present in the network. Furthermore, r, ȳ are defined in the same way as v above. A possible main global objective of the network could be the tracking performance: each output y i (t) has to approach in a specified time the reference signal: ref i (t) = ref (t). However, the external reference signal ref (t) is generally only available (throughout r i ) at one or a few nodes of the network, which is defined by the matrix B.

As an example, let us consider the network in Fig. 1 (consider δ = 0 for this part) with N mod systems connected in a chain, all of the form (1) and all with a decentralized controller K i , see [START_REF] Ljung | System identification[END_REF]. These local closed loops are represented by a circle and are detailed in Fig. 1. In order to be able to track the external reference ref even though this reference is only available at Node 1, a number of nodes are allowed to exchange information (i.e. their measured outputs) with some other neighboring nodes. The arrows between the nodes in Figure 1 indicate the flow of information. For example, Node 2 sends its output to/receives the outputs from Nodes 1 and 3 while Node 1 receives the output of Node 2 and from the external reference signal and sends its output only to Node 2. The local reference signal r i of Node i will be computed as a linear combination of the received information at Node i. More precisely, to define all outputs y i , A and B in (3) are chosen as [START_REF] Fax | Information flow and cooperative control of vehicle formations[END_REF], [START_REF] Korniienko | Performance control for interconnection of identical systems: Application to PLL network design[END_REF]:

A =      0 1/2 0 • • • 0 1/2 0 1/2 • • • 0 . . . . . . . . . . . . 0 0 0 1/2 0 1/2 0 0 0 1 0      B =     1/2 0 . . . 0     . (4)
The matrix A is called the normalized adjacency matrix in the literature [START_REF] Fax | Information flow and cooperative control of vehicle formations[END_REF] and it can be easily obtained for any interconnection topologies. Using (3), it is possible to define the local tracking error signals e i = r i -y i and it can be proven that such an interconnection allows good tracking if different loops [K i G i ] are designed to make the tracking error e i as small as possible. Our objective is thus to design (or redesign) local controllers K i ensuring this global objective for a given interconnection topology A, B and given subsystem dynamics G i (s, θ i,0 ), see ( 1)-( 3). Let us first define general performance specifications that cover the expressed tracking performance objective but also other additional specifications. To do so, let us introduce performance input w and output z and a (possible dynamic) interconnection matrix M such that r z = M ȳ w

Different components of the matrix M depend on the information flow in the network, i.e. matrices A and B, as well as on the specific performance measure, as will be detailed in Section V. In this article, we focus on the performance specifications expressed in the frequency domain, see [START_REF] Skogestad | Multivariable Feedback Control, Analysis and Design[END_REF]. For this purpose, let us further define the local, independent from the network, transfer function T i and the global transfer function of the network T w→z between local (r i → y i ) and global ( w → z) signals respectively. Based on (1), ( 2) and ( 5) the following expression are obtained :

T i (s, θ i ) = K i (s)G i (s, θ i ) 1 + K i (s)G i (s, θ i ) T w→z (s, θ) = diag i (T i (s, θ 1 )) M
The global performance specification will be deemed satisfactory if: ∀ω, σ (T w→z ( , θ)) < W (ω) (6) where defines jω in the continuous time domain or e jω in the discrete time domain. It is thus necessary to design (or redesign) the local controllers in order to ensure (or improve) the network performance and respect [START_REF] Braatz | Computational complexity of µ calculation[END_REF] with θ = θ 0 . However, since θ 0 is unknown, it will be necessary to identify a model for each of the systems S i (θ i,0 ). We assume that there is an identification procedure leading to a consistent parameter vector estimate θi of each subsystem true parameter vector θ i,0 as well as an estimate of the corresponding covariance matrices P θi . Such an identification procedure exists in open or closed-loop for each module independently, see [START_REF] Ljung | System identification[END_REF], [START_REF] Barenthin | Identification for control of multivariable systems: Controller validation and experiment design via LMIs[END_REF], or when the modules are connected to the network [START_REF] Bombois | Optimal identification experiment design for the interconnection of locally controlled systems[END_REF]. It implies with some probability that the true parameter vector θ i,0 belongs to some uncertainty set U i defined as :

U i = {θ i | (θ i -θi ) T P -1 θi (θ i -θi ) < χ} (7)
with a constant χ given the probability level we would like to ensure and the number of elements in the parameter vector θ 0 . We also assume that there is a design procedure allowing to compute local controllers K i (s) such that the nominal global transfer function T w→z (s, θ), with θ = θ an estimate of θ 0 , respects the frequency dependent bound [START_REF] Braatz | Computational complexity of µ calculation[END_REF]. Such design procedures could be found in [START_REF] Korniienko | Performance control for interconnection of identical systems: Application to PLL network design[END_REF], [START_REF] Scorletti | An LMI approach to decentralized H∞ control[END_REF].

Of course since θ is not necessarily equal to θ 0 this will not necessarily ensure the constraint (6) for the true system. In order to ensure the performance of the true system, in this article we would like to solve the following worst-case robustness analysis problem. Since θ i,0 ∈ U i for all i, it is possible to ensure [START_REF] Braatz | Computational complexity of µ calculation[END_REF] with θ = θ 0 by computing the worst-case gain of T w→z ( , θ), evaluated in terms of maximum singular values, ∀θ i ∈ U i . Similarly to the robustness analysis approaches [START_REF] Doyle | Analysis of feedback systems with structured uncertainties[END_REF], [START_REF] Safonov | Stability margin of diagonaly perturbed multivariable feedback systems[END_REF], [START_REF] Fan | Robustness in the presence of mixed parametric uncertainty and unmodeled dynamics[END_REF], this computation will be performed frequency by frequency assuming an appropriate definition of the frequency gridding vector Ω = {ω 1 , . . . , ω Nω } and that the properties ensured ∀ω j ∈ Ω imply that they are ensured ∀ω ∈ R.

Problem 2.1: Given system (1)-( 3), [START_REF] Skogestad | Multivariable Feedback Control, Analysis and Design[END_REF], given uncertainty sets [START_REF] Fan | Robustness in the presence of mixed parametric uncertainty and unmodeled dynamics[END_REF], compute for each ω j ∈ Ω:

min θi∈Ui(i=1...N mod ) γ(ω j ) subject to σ (T w→z ( j , θ)) < γ(ω j ) (8)
If the minimal solution of the previous problem respects γ(ω j ) ≤ W (ω j ) for all j, then the computed controllers ensure that the true system T w→z ( , θ 0 ) respects the frequency dependent bound in [START_REF] Braatz | Computational complexity of µ calculation[END_REF] and thus the global performance.

Problem 2.1 is close to the well-known problem of worst-case robustness analysis (or µ-analysis procedure) from the Robust Control Community [START_REF] Skogestad | Multivariable Feedback Control, Analysis and Design[END_REF]. However the uncertainty sets [START_REF] Fan | Robustness in the presence of mixed parametric uncertainty and unmodeled dynamics[END_REF], representing ellipsoids in parameter space, are not the traditional ones considered in this field. The adaptation of traditional worst-case robust analysis methods to the case of the uncertainty set obtained from the identification can be found in [START_REF] Bombois | Robustness analysis tools for an uncertainty set obtained by prediction error identification[END_REF]- [START_REF] Barenthin | Identification for control of multivariable systems: Controller validation and experiment design via LMIs[END_REF]. However direct application of these results in the case of a large-scale network system, i.e. when N mod is large, is not possible due to the high system complexity implying prohibitive computation time. As was mentioned in the introduction, the main contribution of this technical report is to extend these methods to the network context i.e. to derive tractable robustness performance analysis conditions while keeping computation time reasonable.

III. HIERARCHICAL ANALYSIS APPROACH A. Keys ideas

As was discussed previously, the direct application of the worst-case analysis method will result in a prohibitive computation time for large scale networks. To avoid this, we propose to use the hierarchical robustness analysis approach of [START_REF] Dinh | Embedding of uncertainty propagation: Application to hierarchical performance analysis[END_REF], [START_REF] Dinh | Convex hierarchical analysis for the performances of uncertain large-scale systems[END_REF].

The main idea of the hierarchical approach is to decompose the network into two or more hierarchical levels and to perform the robustness analysis level by level by propagating the analysis results from one level to another. For some network systems such decomposition appears naturally, as for example for the system under consideration in this technical report : (i) local hierarchical level : subsystem dynamics T i (s, θ i ) defined by ( 1) and ( 2) and (ii) global hierarchical level: the global information exchange (3) and [START_REF] Skogestad | Multivariable Feedback Control, Analysis and Design[END_REF]. The robustness analysis at each hierarchical level allows to embed the subsystem dynamics with a possibly complex non-linear dependence on the uncertainty, into a much simpler subsystem description with a convex dependence on the uncertainty. We will call it the embedding procedure in the sequel. Then in the next hierarchical level, the subsystem is replaced by this simple description and the procedure is repeated once again until reaching the last hierarchical level. The last step consists in the worst-case robustness analysis based on the propagated subsystem descriptions in order to evaluate the global network performance i.e. solve the Problem 2.1. The complexity and time computation reduction is ensured thanks to the embedding procedures and by the fact that all embeddings at each hierarchical level are independent and thus can be easily performed in parallel.

In this technical report, a two level hierarchical structure (local and global) is under consideration. Before formalizing this approach separately for the local and global hierarchical levels, let us first define what we mean by subsystem dynamics and subsystem description.

Since the performance measure in this technical report is expressed in the frequency domain, see ( 6), the subsystem dynamics are defined by the structured frequency response set T s i (ω) of the subsystem transfer function at frequency ω:

T s i (ω) = {T i ( , θ i ) | θ i ∈ U i } (9)
The subsystem description in turn is defined by an uncertainty set

T i (x i (ω), y i (ω), z i (ω)) of complex numbers ∆ i (ω) ∈ C that respects a frequency dependent quadratic constraint imposed by x i (ω) ∈ R, y i (ω) ∈ C, z i (ω) ∈ R: T i (x i (ω), y i (ω), z i (ω)) = {∆ i (ω) | ∆ i (ω) 1 * x i (ω) y i (ω) y i (ω) * z i (ω) ∆ i (ω) 1 ≤ 0 (10)
Let us introduce the following definition characterizing the frequency response of a system : Definition 3.1 (Dissipativity): An LTI system H(s) is {x(ω), y(ω), z(ω)} -dissipative at ω for some x(ω) ∈ R, y(ω) ∈ C, z(ω) ∈ R, if its frequency response H( ) respects the following quadratic constraint at ω:

H( ) 1 * x(ω) y(ω) y(ω) * z(ω) H( ) 1 ≤ 0.
If the following additional constraint is imposed on x(ω), then the corresponding quadratic constraint defines a convex set :

x(ω) ≥ 0.

(11) Please note that, in the case of x(ω) > 0, by Definition 3.1 and the Schur complement [START_REF] Boyd | Linear Matrix Inequalities in Systems and Control Theory[END_REF], the following constraint is implied: y 2 (ω) ≥ x(ω)z(ω). When x(ω) = 0, no constraint is imposed on y(ω) and z(ω). In order to reduce the computational complexity, the convexity constraint [START_REF] Barenthin | Identification for control of multivariable systems: Controller validation and experiment design via LMIs[END_REF] will be used in the sequel.

If each subsystem T i (s, θ i ) is {x i (ω), y i (ω), z i (ω)} -dissipative for some frequency dependent x i (ω), y i (ω), z i (ω) and for all θ i ∈ U i and ∀ω, we then obtain the following embedding

T s i (ω) ⊂ T i ((x i (ω), y i (ω), z i (ω))
, ∀ω; and the frequency responses of the uncertain subsystems T 1 (s, θ i ), . . . , T N mod (s, θ i ) generated by varying θ i ∈ U i , can be replaced in the global hierarchical level by the corresponding subsystem description

T i ((x i (ω), y i (ω), z i (ω)).
Of course, since the set T i ((x i (ω), y i (ω), z i (ω)) is in general larger than the set T s i (ω) the result of the corresponding worst-case analysis might be conservative. In order to reduce this conservatism, it is important to choose suitable x i (ω), y i (ω), z i (ω) for each subsystem defining as tight embedding as possible. It is also possible to compute several complementary triplets x k i (ω), y k i (ω), z k i (ω) for k = 1 . . . N d defining therefore N d dissipativity properties for each subsystem. It allows to define for each subsystem a basis of dissipativity properties (a set of subsystem descriptions) and propagate it to the global hierarchical level. Such a suitable choice in the context of the uncertainty set [START_REF] Fan | Robustness in the presence of mixed parametric uncertainty and unmodeled dynamics[END_REF] obtained through an identification procedure is presented in the next subsection while Subsection III-C presents how the embeddings are combined and propagated in a global hierarchical step in order to efficiently solve Problem 2.1. It is clear that the more dissipativity characterizations are used for each subsystem, the more the conservatism is reduced. Of course, the price to pay for this is the increase of computation time. For this reason it is important to find appropriate triples x k i (ω), y k i (ω), z k i (ω) at each hierarchical step.

B. Local Step

In this subsection we present how to efficiently compute different dissipativity triplets x, y, z at a given frequency ω such that an uncertain system T (s, θ i ) is {x(ω), y(ω), z(ω)} -dissipative, ∀θ i ∈ U i with U i defined in [START_REF] Fan | Robustness in the presence of mixed parametric uncertainty and unmodeled dynamics[END_REF].

For this purpose let us define the following factorization of the transfer function T (s, θ i ), suitable for the system identification [START_REF] Bombois | Robustness analysis tools for an uncertainty set obtained by prediction error identification[END_REF]:

T (s, θ i ) = e(s) + Z N (s)θ i 1 + Z D (s)θ i (12)
with θ ∈ R n θ and then present the following Lemma. Lemma 1: Given the uncertain LTI system T (s, θ i ) in [START_REF] Dinh | Embedding of uncertainty propagation: Application to hierarchical performance analysis[END_REF], it is {x(ω), y(ω), z(ω)} -dissipative for all θ i ∈ U i and for given ω, x(ω) ∈ R, y(ω) ∈ C, z(ω) ∈ R respecting [START_REF] Barenthin | Identification for control of multivariable systems: Controller validation and experiment design via LMIs[END_REF], if and only if (i) in the case of x(ω) > 0 :

-α(ω)

λ(ω) λ * (ω) -A 1 (ω) -ξ(ω)B + jX (ω) ≤ 0 (13)
(ii) in the case of x(ω) = 0 :

A * 2 (ω)y(ω) + y * (ω)A 2 (ω) + A 1 (ω)z(ω) -ξ(ω)B + jX (ω) ≤ 0 (14)
with λ(ω) = ZN ( ) + y(ω)

x(ω) ZD( ) e( ) + y(ω)

x(ω)

,

A1(ω) = Z * D ( )ZD( ) Z * D ( ) ZD( ) 1 , A2(ω) = Z * D ( )ZN ( ) Z * D ( )e( ) ZN ( ) e( ) , α(ω) = y 2 (ω) x 2 (ω) -z(ω) x(ω) , B = P -1 θ i -P -1 θ i θi -θT i P -1 θ i θT i P -1 θ i θi -χ and some ξ(ω) ≥ 0 ∈ R, X (ω) = -X T (ω) ∈ R n θ ×n θ .
Proof: For the sake of conciseness, we will drop the frequency argument ω and in the variables. By definition of dissipativity, T (s, θ) is {x, y, z} -dissipative ∀θ ∈ U , is equivalent to :

(i) T (θ i ) + y x * T (θ i ) + y x ≤ y 2 x 2 - z x , for x > 0 (ii) (y * T (θ i )) * + y * T (θ i ) + z ≤ 0, for x = 0, ∀θ i ∈ U i .
Using factorization [START_REF] Dinh | Embedding of uncertainty propagation: Application to hierarchical performance analysis[END_REF], and compact notation θ = [θ T i 1] T , the previous inequalities are equivalent to

(i) θT -A 1 -λ * -1 α λ θ ≤ 0, ∀θ i ∈ U i ( 15
) (ii) θT (A * 2 y + y * A 2 + A 1 z) θ ≤ 0, ∀θ i ∈ U i (16)
while the constraint θ i ∈ U i is equivalent to θT B θ < 0. Consequently, by virtue of the S-procedure [START_REF] Boyd | Linear Matrix Inequalities in Systems and Control Theory[END_REF] and Lemma 2 in [START_REF] Bombois | Least costly identification experiment for control[END_REF], [START_REF] Safonov | Propagation of conic model uncertainty in hierarchical systems[END_REF] or ( 16) holds if and only if there exist ξ ≥ 0 and X = -X T such that

(i) -A 1 -λ * -1 α λ -ξB + jX ≤ 0 ( 17
) (ii) A * 2 y + y * A 2 + A 1 z -ξB + jX ≤ 0 (18)
The last constraint is exactly condition [START_REF] Laib | Phase iqc for the hierarchical performance analysis of uncertain large scale systems[END_REF]. Due to convexity constraint [START_REF] Barenthin | Identification for control of multivariable systems: Controller validation and experiment design via LMIs[END_REF], with non zero x = 0, α > 0, and the application of the Schur complement [START_REF] Boyd | Linear Matrix Inequalities in Systems and Control Theory[END_REF] shows that ( 17) is equivalent to [START_REF] Dinh | Convex hierarchical analysis for the performances of uncertain large-scale systems[END_REF]. This concludes the proof. Please note that the sufficiency of Lemma 1 can be proved using the result of [START_REF] Dinh | Convex hierarchical analysis for the performances of uncertain large-scale systems[END_REF] (see Corollary 2.2). As is shown in the proof of Lemma 1, the result of [START_REF] Dinh | Convex hierarchical analysis for the performances of uncertain large-scale systems[END_REF] is adapted to the case of uncertain vectors that belong to an ellipsoid which recovers sufficient and necessary conditions of {x(ω), y(ω), z(ω)} -dissipativity. This lemma is an extension of the robustness analysis result of [START_REF] Bombois | Optimal identification experiment design for the interconnection of locally controlled systems[END_REF] and will be used to generate different types of embeddings.

We will now consider two types of embedding: the disc and the band embedding, and formulate a convex optimization problem to compute them. Please note that thanks to Lemma 1, it is possible to study other types of embedding, as for example cone embedding [START_REF] Laib | Phase iqc for the hierarchical performance analysis of uncertain large scale systems[END_REF], half planes etc.

1) Disc Embedding: Given system T (s, θ) in ( 12), its frequency response set {T ( , θ i )

| θ i ∈ U i } is embedded in a disc set at ω if |T ( , θ i ) -c(ω)| ≤ ρ(ω), ∀θ i ∈ U i (19)
where c(ω) ∈ C is the center of the disc and ρ(ω) ∈ R is its radius, see [START_REF] Dinh | Convex hierarchical analysis for the performances of uncertain large-scale systems[END_REF]. The size measure of this embedding is the radius of the disc, and the problem of the computation of the tightest embedding can be formulated as follows assuming appropriate gridding Ω. Problem 3.1: Given system (12) and its uncertainty set [START_REF] Fan | Robustness in the presence of mixed parametric uncertainty and unmodeled dynamics[END_REF], compute for each ω j ∈ Ω: min

ρ(ωj ),c (ωj )
ρ(ω j ) subject to [START_REF] Scorletti | An LMI approach to decentralized H∞ control[END_REF] with ω = ω j

This problem is efficiently solved by the following theorem. Theorem 3.1 (Disc embedding): Given system (12) and its uncertainty set [START_REF] Fan | Robustness in the presence of mixed parametric uncertainty and unmodeled dynamics[END_REF], Problem 3.1 is solved by the following convex optimization problem: min

ρ 2 (ωj ),c (ωj )
ρ 2 (ω j ) s.t. ( 13) is holds with ω = ω j and ( 20)

x(ω j ) = 1, y(ω j ) = -c(ω j ), z(ω j ) = c 2 (ω j ) -ρ 2 (ω j ). Proof:
The roof is straightforward after replacing the value of x, y, z and applying Lemma 1. Please note that in this case α(ω j ) = ρ 2 (ω j ) > 0 and λ(ω j ) = Z N ( j ) -c(ω j )Z D ( j ) e( j ) -c(ω j ) , implying affine dependence on the decision variables. As a consequence, the optimization ( 20) is an LMI optimization and can be solved efficiently.

2) Band Embedding: Given system T (s, θ i ) in ( 12), its frequency response set {T ( , θ i )

| θ i ∈ U i } is embedded in a band set at ω if ∀θ i ∈ U i 2a 2 (ω) ≤ T * ( , θ i )n(ω) + n * (ω)T ( , θ i ) ≤ 2a 1 (ω), (21)
where n(ω) ∈ C is the complex number which defines the vector -→ n = [Re(n), Im(n)] T giving the band orientation in complex plain (it is perpendicular to both band hyperplanes) and a 1 (ω), a 2 (ω) ∈ R are the signed distances of the two band hyperplanes to the origin multiplied by |n|, see [START_REF] Dinh | Convex hierarchical analysis for the performances of uncertain large-scale systems[END_REF]. The size measure of this embedding is the band width d(ω) = a 1 (ω) -a 2 (ω) (see [START_REF] Dinh | Convex hierarchical analysis for the performances of uncertain large-scale systems[END_REF] and Fig. 4 for illustration), and the problem of computation of the tightest embedding can be formulated as follows assuming again appropriate gridding Ω. Problem 3.2: Given system (12) and its uncertainty sets [START_REF] Fan | Robustness in the presence of mixed parametric uncertainty and unmodeled dynamics[END_REF], compute for each ω j ∈ Ω: min n(ωj ),a1(ωj ),a2(ωj )

a 1 (ω j ) -a 2 (ω j ) subj. to (21) with ω = ω j
This problem is efficiently solved by the following Theorem. Theorem 3.2 (Band embedding): Given system (12) and its uncertainty sets [START_REF] Fan | Robustness in the presence of mixed parametric uncertainty and unmodeled dynamics[END_REF], Problem 3.2 is solved by the following convex optimization problem: min a1(ωj ),a2(ωj ),n(ωj )

a 1 (ω j ) -a 2 (ω j ) (22)
s.t. ( 14) holds with ω = ω j and

x 1 (ω j ) = 0, y 1 (ω j ) = n(ω j ), z 1 (ω j ) = -2a 1 (ω j )
and ( 14) holds with ω = ω j and

x 2 (ω j ) = 0, y 2 (ω j ) = -n(ω j ), z 2 (ω j ) = 2a 2 (ω j ). Proof: The proof is straightforward after replacing the value of x 1 , y 1 , z 1 , x 2 , y 2 , z 2 and applying Lemma 1 twice. Please note that, in this case as well, the dependence on the decision variables is affine. As a consequence, the optimization ( 22) is an LMI optimization and can be solved efficiently.

C. Global Step

In this subsection, we assume that for all ω j ∈ Ω and for each subsystem T i (s, θ i ), several embeddings are found in the local step. We thus obtain N d dissipativity triplets x k i (ω j ), y k i (ω j ), z k i (ω j ) for k = 1, . . . , N d , for each subsystem j = 1, . . . , N mod and for all ω j ∈ Ω. The next theorem allows to compute an upper bound γ U B (ω j ) on the maximum amplification γ(ω j ) of Problem 2.1.

Theorem 3.3: Given system (1)-( 3),( 5), a frequency ω j and given [START_REF] Bombois | Robustness analysis tools for an uncertainty set obtained by prediction error identification[END_REF] and ( 10))

x k i (ω j ), y k i (ω j ), z k i (ω j ) such that T s i (ω j ) ⊂ T i ((x k i (ω j ), y k i (ω j ), z k i (ω j )) for k = 1, . . . , N d , i = 1, . . . , N mod (see
The upper bound γ U B (ω j ) on the maximum amplification γ(ω j ) of Problem 2.1 is the solution of the following LMI optimization problem:

min γ2 (ω j ),T k ω ,(k=1...N d ) γ2 (ωj) s.t. M ( ) I * N (γ 2 (ωj)) M ( ) I > 0, with (23)
N (γ 2 (ωj))

∆ =     T k ω Z k d 0 0 -I T k ω Y k d 0 0 0 * T k ω Y k d 0 0 0 T k ω X k d 0 0 γ2 (ωj)I    
with strictly definite positive diagonal matrices T k ω ∈ R N mod ×N mod , and k = 1 . . . N d .

X k d = diag i (x k i (ωj)), Y k d = diag i (y k i (ωj)), Z k d = diag i (z k i (ωj))
Proof: This theorem can be straightforwardly deduced from the separation of graph theorem [START_REF] Safonov | Stability and Robustness of Multivariable Feedback Systems[END_REF] and from the results in [START_REF] Dinh | Convex hierarchical analysis for the performances of uncertain large-scale systems[END_REF]. It follows from the fact that the constraint ( 23) is a sufficient condition for σ (T (ω j ) M(ω j )) < γ 2 U B (ω j ), with

T (ω j ) = diag i (∆ i (ω j )) , ∀∆ i (ω j ) ∈ k T i (x k (ω j ), y k (ω j ), z k (ω j )) to hold.

IV. COMPUTATIONAL COMPLEXITY

As already mentioned earlier, Problem 2.1 can be solved directly by the the method proposed in [START_REF] Scorletti | Improved efficient analysis for systems with uncertain parameters[END_REF], [START_REF] Barenthin | Identification for control of multivariable systems: Controller validation and experiment design via LMIs[END_REF]. Let us call this approach the direct worst case analysis approach. In fact, in the case of a SISO global transfer function T w→z (s, θ), Lemma 1 together with Theorem 3.1 can be seen as a generalization of the result [START_REF] Scorletti | Improved efficient analysis for systems with uncertain parameters[END_REF], [START_REF] Barenthin | Identification for control of multivariable systems: Controller validation and experiment design via LMIs[END_REF]. Indeed, considering factorization of T w→z (s, θ) similar to [START_REF] Dinh | Embedding of uncertainty propagation: Application to hierarchical performance analysis[END_REF], defining overall parameter vector θ and its uncertainty similar to (7), Problem 2.1 is equivalent to Problem 3.1 with center c(ω j ) = 0 and γ(ω j ) = ρ(ω j ), ∀j. It can thus be efficiently solved by convex optimization in Theorem 3.1. This result can be generalized to the case of Multi-Input, Multi-Output (MIMO) global transfer function T w→z (s, θ). However the necessity part of the result will be lost so let us focus here on the SISO case only.

In this technical report, the hierarchical worst case analysis approach is proposed. The main interest of the hierarchical approach is the computational time reduction in comparison to the direct one and in the case of large scale network N mod 1. To evaluate this reduction in both cases, independently of the computational facilities, let us assume that the computational time is equal to the algorithm complexity and let us investigate its evolution as a function of the subsystem number N mod . For interior point methods for LMI optimization, it is a cubic function of the decision variable number n [START_REF] Nesterov | Interior-point polynomial methods in convex programming[END_REF]:

t = O(n m ) ∆ = α m n m + . . . + α 1 n + α 0
with some non-negative α i and m = 3.

Supposing that each subsystem is SISO with the same size ∀i, n θi = nθ of parameter vector θ i,0 , the amount of decision variables of the direct worst case analysis approach is equal to

2 + n θ (n θ -1) 2 = 2 + N mod nθ (N mod nθ -1) 2
which gives polynomial dependence of the order 6 for the direct approach time computation t direct with respect to the number of subsystems N mod :

t direct = O(N 6 mod ).
For the hierarchical approach, with two hierarchical levels, we obtain N mod n d local embeddings with 3 + nθ (n θ -1)

2

decision variables each and one global analysis with N mod n d decision variables. It gives the overall computation time :

t hierarch. = N mod n d O(n 6 θ ) + O(N 3 mod ) ≈ O(N 3
mod ) and in the case of parallel computation of local embeddings :

t hierarch. parallel = n d O(n 6 θ ) + O(N 3 mod ) ≈ O(N 3 mod ).
Therefore, if the parallel computation of the local subsystem embeddings is allowed by available computational facilities, the time computation reduction is even better. As a consequence, the hierarchical approach for N mod 1 is much more efficient from computational point of view while, as illustrated in the next section, keeping reasonable conservatism level with an appropriate choice of embeddings.

V. NUMERICAL EXAMPLE

Let us now consider an illustration example of an Automated Highway System (AHS): a platoon of autonomous cars following external reference signals as in [START_REF] Seiler | Disturbance propagation in vehicle strings[END_REF]. Each car's simplified model dynamics is described by [START_REF] Cao | An overview of recent progress in the study of distributed multi-agent coordination[END_REF], with G i (s, θ) = ki s 2 (τis+1) and true parameter vector θ i,0 = [τ i , k i ] T where τ i , k i were randomly chosen around 0.105 and 0.95 respectively with uniform ±10% distribution. Each system is controlled by the same initial decentralized controller K init (s) = 2s+1 0.05s+1 taken from [START_REF] Seiler | Disturbance propagation in vehicle strings[END_REF], see [START_REF] Ljung | System identification[END_REF]. There are N mod = 5 cars in the network which are allowed to exchange information according to bidirectional chain topology, see [START_REF] Seiler | Disturbance propagation in vehicle strings[END_REF], as depicted in Fig. 1 and defined by [START_REF] Ljung | System identification[END_REF].

The main objective of the network is that each car follows a ramp reference signal ref (t), available only for the first car, shifted by a constant value δ i = iδ, ∀i, while keeping string instability (oscillation propagation through the network) limited [START_REF] Seiler | Disturbance propagation in vehicle strings[END_REF]. It can be shown that this tracking performance specification is equivalent to the ability of each car to track the same ramp signal ref (t) ensuring that all local tracking errors e i = r i -y i go to zero in steady-state. It is sufficient to locally apply a constant shift -δ = 0 at the input of each subsystem, as depicted in Fig. 1, and to perform a suitable change of variable y i → y i -δ.

As a consequence, let us define performance input w(t) = ref (t) and performance output z(t) = r(t) -ȳ(t). It thus determines the interconnection topology [START_REF] Skogestad | Multivariable Feedback Control, Analysis and Design[END_REF] with

M = A B A -I B
. If the maximum singular value of T w→z (s, θ o ) has a slope of +40 dB/dec at low frequency range, then the tracking performance is ensured, see [START_REF] Skogestad | Multivariable Feedback Control, Analysis and Design[END_REF]. Moreover, a lower gain ensures a better tracking speed and the resonance peak limitation reduces the effects of string instability [START_REF] Seiler | Disturbance propagation in vehicle strings[END_REF]. The maximal singular value of the true system T w→z (s, θ o ) with initial controller K init is represented by orange dash-dotted line in Fig. 2. In order to improve the tracking performance of the network and to reduce the oscillation effects provoked by the string instability, let us impose the frequency constraint [START_REF] Braatz | Computational complexity of µ calculation[END_REF] with W (ω) represented in Fig. 2 by the red dashed line. To satisfy this constraint, first an identification procedure is performed leading to a consistent parameter vector estimate θi of each subsystem true parameter vector θ i,0 as well as an estimation of the corresponding covariance matrices P θi ensuring [START_REF] Fan | Robustness in the presence of mixed parametric uncertainty and unmodeled dynamics[END_REF]. Due to the presence of a double integrator in the car transfer function model, this identification experiment has to be performed in closed loop with a stabilizing controller either independently for each module (see [START_REF] Ljung | System identification[END_REF], [START_REF] Barenthin | Identification for control of multivariable systems: Controller validation and experiment design via LMIs[END_REF]) or in the network (see [START_REF] Bombois | Optimal identification experiment design for the interconnection of locally controlled systems[END_REF]). The results of the latter method are presented in Fig. 3 where the controllers were chosen as K i (s) = K init (s), ∀i. Different discrete-time white noise excitation signals of length N id = 1000, sampling time T s = 0.01 sec and variance 10 are added via a zero order hold to the references r i of each closed-loop systems T i (s, θ i,0 ). The measured discrete signal y i is also perturbed by generated mutually independent white noise discrete signals v i with variance of 4 each modeling the measurement noise effects. A standard, prediction-error identification criterion is used, see [START_REF] Ljung | System identification[END_REF]. Notice that in this example the continuous transfer function parameters k i and τ i and the corresponding covariance matrices could be directly identified since the car transfer function model is rather simple. An adapted to the subsystem dynamics optimization, taking into account zero-order hold effects, had to be applied in order to identify directly continuous transfer function parameters k i and τ i and corresponding covariance matrices.

A new improved decentralized controller is designed based on the H ∞ framework [START_REF] Korniienko | Performance control for interconnection of identical systems: Application to PLL network design[END_REF] Our problem is now to efficiently test if the constraint is satisfied by the true system by solving Problem 2.1 for properly chosen Ω. To do so, the proposed hierarchical approach is used. The results of the local step embeddings for the first system and at 0.15 Hz are presented in Fig. 4 where the borders of the minimum radius disc embedding (green full circle) and of the tightest band (red full lines) are presented. For the sake of illustration reason, we show the borders of the structured uncertainty set T s 1 (red dots), the estimated T 1 ( θ1 ) (blue cross) and the true T 1 (θ 1,0 ) (black round) value of the corresponding frequency responses evaluated at ω = 0.15 Hz. Notice that disk center c(ω) = T 1 (jω, θ 1,0 ). The results are found by solving the LMI optimization problems [START_REF] Boyd | Linear Matrix Inequalities in Systems and Control Theory[END_REF] and [START_REF] Safonov | Stability and Robustness of Multivariable Feedback Systems[END_REF]. Similar results are obtained for other subsystems and other frequencies from Ω. The global step analysis results are presented in Fig. 5 for two cases : computed γ U B based on the propagation of (i) disc embedding only (blue rounds) and of (ii) disc and band embeddings (red dots). Fig. 5 also presents some Monte-Carlo samples i.e. the maximal singular value of T w→z (s, θ) for randomly chosen θ i ∈ U i . As we can see, the worst-case bounds are respected. Surprisingly even though the disc embedding set is much bigger than the intersection of disc and band sets (see Fig. 4), the overall upper bound γ U B is not improved a lot, see Table I. It is due to the fact that, in this application, the phase uncertainty information, mostly captured by the band embedding, is much less important than the gain uncertainty information, mostly captured by the disc embedding. The corresponding computation times are also given Table I for both serial and parallel computation of local embeddings. Finally, maximal singular values of the true system T w→z (s, θ o ) with the new controller are represented by the blue solid line in Fig. 2.

VI. CONCLUSIONS

In this technical report we proposed robustness analysis method adapted to the uncertainty sets constructed by identification in a network context. The type of network in this system is usual in the literature of multi-agent systems and the size of the network plays a crucial role in the robustness analysis complexity. In order to manage the trade-off between the computation time and the precision of the obtained result, the hierarchical robustness analysis approach was proposed and illustrated in the case of SISO subsystems. Future extension is the MIMO subsystem case with an appropriate choice of hierarchical structure (with possibly more than two hierarchical levels) in order to even better address the mentioned trade-off. This technical report is the first step needed to built identification experiment design for control in network context.

Fig. 1 .

 1 Fig. 1. Example of a network

Fig. 2 .

 2 Fig.2. Maximal singular value of the true system T w→z (s, θo) for initial controller (orange dash-dotted line), improved controller (blue solid line) and imposed frequency constraint W (ω) (red dashed line).

Fig. 4 .

 4 Fig. 4. Local step embedding results. Borders of structured uncertainty set T s 1 (red dots), of the minimum radius disc embedding (green full circle), of the tightest band (red full lines), circle center (green dot), estimated frequency response (blue cross) and true frequency response (black round).

Fig. 5 .

 5 Fig.5. Global step analysis results. Upper bounds computed by (i) propagation of disc embedding only (blue rounds), by (ii) propagation of disc and band embeddings (red dots), Monte-Carlo samples of maximal singular value of T w→z (s, θ), for some θ i ∈ U i .

  Fig.3. Identification results. True parameter vectors θ i,0 (green dots), its estimated values θi (red crosses), and corresponding ellipsoid set borders (full lines) for χ chosen to ensure 95% probability.

	1.02
	1
	0.98
	0.96
	0.94
	0.92
	0.9
	0.88
	0.86
	0.08	0.085	0.09	0.095	0.1	0.105	0.11	0.115	0.12	0.125

,

[START_REF] Scorletti | An LMI approach to decentralized H∞ control[END_REF]

:

K(s) =

12111(s + 10)(s 2 + 0.9s + 0.4) s(s 2 + 111.6s + 6230) .

It ensures that the nominal global transfer function T w→z (s, θ), with θ = θ, respects the frequency dependent bound in

[START_REF] Braatz | Computational complexity of µ calculation[END_REF]

, see Fig.

2

.

TABLE I

 I

	HIERARCHICAL WORST-CASE ANALYSIS RESULTS
		disc only	disc + band difference
	γ U B @ 0.13 Hz	-11.83 dB	-12.04 dB	1.8%
	γ U B @ 0.15 Hz	-12.64 dB	-12.92 dB	2.2%
	γ U B @ 0.17 Hz	-14.28 dB	-14.44 dB	1.1%
	Overall Time	15.33 sec	19.04 sec	-24.2%
	Overall Time (Parallel)	11.85 sec	14.43 sec	-21.8%

ACKNOWLEDGMENT

This work originally was supported by a grant from the Région Rhône-Alpes.