CHAPTER THREE

THE SEARCH FOR A "NEW" RHETORIC OF THE LEFT: A LOOK AT STRATEGIES OF THE DEMOCRATS AND LABOUR

LORI MAGUIRE

A great deal of attention has been devoted to the electoral problems experienced by the left of centre parties in both Great Britain and the United States in recent times. This provoked a search by a number of prominent figures on both sides of the Atlantic, as to the reasons for these difficulties and to develop a new rhetoric that would appeal more to the voters. The academic Alan Finlayson has commented perceptively on the importance of words in politics:

At the very least we have to acknowledge that politics under democratic constitutions is about some people trying to persuade the rest of us of their virtues or the virtues of their political position. To do so, they employ rhetoric intended to illustrate the ways in which their political programme will be good for us by, for example, associating it with positive ends and characteristics. Anyone who has had any involvement in a political organisation or campaign knows that a central aspect of such activity is the strategic one of trying to find ways in which to connect with the wider public through various images, modes of speech and so forth. [START_REF] Finlayson | Making Sense of New Labour[END_REF] In its essence, political communication is about finding an appealing message and effectively conveying it to the public through the mass media. We will be examining here how leading figures in the two major leftwing parties of the U.S. and the U.K. analysed their repeated defeats of the 1980s (and, in the case of Labour, early 1990s) drawing from this certain conclusions in order to construct (they hoped) a more effective rhetoric and image.

Although a number of dissimilarities exist between the two parties, both the Democrats and Labour began their quest at the same time: Labour"s after its election defeat of 1983 and the Democrats after Reagan"s landslide victory in 1984. In both cases they achieved some success afterwards with Bill Clinton"s election in 1992 and Labour"s triumph in 1997. In neither case have they completely replaced older terminology and, indeed, there has been some return to it recently. We will begin by briefly examining the traditions of the two parties, and then analyse the reasons why many on the Left believed that this "modernisation", as they termed it, was necessary. After this, we will examine the rhetoric of the "new" Left, also known as "triangulation" or the "Third Way" before taking a short look at recent trends.

The Democratic Party of the United States and Britain"s Labour Party differ in certain fundamental ways. To begin with, the Democratic Party is much older and traces its ancestry back to the early days of the Republic. After a long and varied history it only firmly became the party of the Left with Franklin Roosevelt and the New Deal. Class-based rhetoric had not been absent from American politics before Roosevelt but he linked it to phenomenal political success. To some extent, Roosevelt"s Depression era speeches qualify as rhetoric of war for he frequently used military vocabulary, although not directly linked to questions of class. For example, in his first inaugural address, he talked about "the great army of our people dedicated to a disciplined attack upon our common problems". [START_REF]The Essential Franklin Delano Roosevelt[END_REF] He frequently made attacks on the wealthy but, once again, avoided overt class language, talking of the "rulers of the exchange of mankind"s goods" or the "unscrupulous moneychangers". [START_REF] Roosevelt | First Inaugural Address[END_REF] Perhaps the best summary of Roosevelt"s ideas is contained in his speech to the Democratic National Convention in July 1932 in which he announced the New Deal. Roosevelt said: "My program… is based upon this simple moral principle: the welfare and the soundness of the nation depend first upon what the great mass of the people wish and need; and second, whether or not they are getting it." 4 Instead of "capitalists" or even "the rich" or "the workers", Roosevelt preferred to attack particular types of the wealthy and to champion the people. Instead of ideology, pragmatism.

Roosevelt turned the Democratic Party into an umbrella organization linking the have-nots to the middle class and performed -at a cost -the extraordinary achievement of gaining the support both of southern blacks and low-income southern whites. A 1936 Gallup poll showed that, while 42 percent of wealthy voters supported him, the percentage rose to 60 percent among middle-income voters and 76 percent among those in lower income groups. Added to that, 80 percent of trade unionists, 81 percent of unskilled workers and 84 percent of those on relief went for the Democratic Party. [START_REF] Witcover | Party of the People: A History of the Democrats[END_REF] All political parties are coalitions but that of the New Deal, although overwhelmingly successful in its time, was so broad and included so many conflicting interests that it was extraordinarily fragile. This meant that it could not be too radical and that it could splinter fairly easily. Its fragmentation began in earnest in the 1960s as civil rights, the Vietnam War and growing prosperity, began to push its members apart.

In contrast, the Labour Party officially began life only in 1900 and was from the start a consciously class-based party. It was formed from a variety of left-wing organizations including co-operative groups, socialist groups, and trade unions, among others, and began life as the Labour Representation Committee, changing its name in 1906 to the Labour Party. Class-based rhetoric was thus a major element in the party"s presentation of itself from the start, one of its first manifestos declaring that:

The House of Commons is supposed to be the People"s House, and yet the people are not there. Landlords, employers, lawyers, brewers, and financiers are there in force. Why not Labour? [START_REF]Manifesto of the Labour Representation Committee[END_REF] While this rhetoric is stronger than Roosevelt"s, it does not differ that much fundamentally. Note how the word "people" is used rather than a specific class term and how particular groups of the wealthy are denounced rather than the entire class. Neither here nor in Roosevelt"s speeches is wealth itself condemned but the abuse of wealth and power by specific groups. There is an undoubted rejection of Marxist rhetoric here and a belief in the basic soundness of the economic and social system which need only reform. The Labour party"s leaders were always aware of the need to attract middle class voters if they wished to have a majority, and so official party rhetoric was usually restrained. In an in-depth analysis of the 1945 Labour victory, American political scientist, William Newman, came to the conclusion that Labour was "not a class party". He observed that; "If the Labour Party depends mainly on the industrial sections of the country for seats it nevertheless succeeds or fails proportionately in all the economic areas of the country." In fact, he found this to be true of all political parties and insisted that Labour is "a genuinely national party". [START_REF] Newman | Patterns of Growth in the British Labour Vote[END_REF] It must be emphasized, though, that historically the rhetoric of the Left has been more moderate in the U.S. than in Britain for the Labour Party openly embraced the word "socialism"-at least until the late 1980s. This term was never used in mainstream American politics except by Republicans seeking to discredit their opponents. The Labour Party is also a member of the Socialist International which is not the case of the Democratic Party.

The 1970s and 1980s were a critical time for the Left in both nations. Both the Democrats and Labour suffered a similar crisis at approximately the same time. In America this reached a high point with the disastrous candidacy of George McGovern for president in 1972. The voters decisively rejected his program, widely seen as advocating too generous social policies while being far too weak in foreign policy (although it must be said that there was no such rejection of the party at the congressional level). [START_REF] See | s analysis of the problem in The Two Majorities and the Puzzle of Modern American Politics[END_REF] In Britain the crisis reached its zenith later, during Michael Foot"s tenure as Labour leader from 1979 to 1983. Under Foot, who had a few years earlier talked of "the red flame of socialist courage", the party moved to the left and also paid a heavy price both within itself and at the ballot box, as we shall see.

Similar factors have been put forward to explain this decline in popularity, most notably the economic problems of this period which provoked a strong critique of Keynesianism on the Right; a reaction to the social and cultural upheavals of the 1960s; the growth of the middle class and of a "property-owning democracy" and changing demographics due to increasingly suburban population concentrations. Clear evidence exists for the growth of swing voters in both countries. But some have diagnosed the cause as being simply the fracturing of the coalition that sustained each party around certain common policies that had now been put into effect. Everybody wanted "progress" but there was a problem agreeing on what direction the "progress" should take. As early as 1955, the Labour politician Richard Crossman said his party was being "ideologically disintegrated". Because of the success of "Keynesian welfare capitalism", there was no longer much of a demand for socialism: All this happens to the Labour Party because people in Britain are more prosperous and more contented and because peace is breaking out all over the world. We suddenly feel that our mission to save people from cataclysm and disaster has come unstuck. We are missionaries without a mission, or missionaries more and more dubious about the mission. [START_REF] Crossman | The Backbench Diaries of Richard Crossman[END_REF] One of the problems the Left has encountered in both nations has been this need to find a "mission" that would motivate its rank and file and get the support of the electorate. But over the next few years this proved extremely difficult and, as loss piled on loss, the drive to win elections overwhelmed any missionary zeal. [START_REF] Walzer | Politics after the New Labour Decade[END_REF] In the case of the United States, the 1970s saw a reorganisation of the electoral map. In part this was because of population shifts from the traditionally Democratic regions of the Midwest and Northeast to the more Republican leaning Sunbelt. But a number of other Democratic areas became, often in a relatively short period of time, Republican dominated. The classic example is the South which shifted its allegiance, to a large extent, because of racial questions. Pres. Lyndon Johnson was fully aware that this would happen when he signed the Civil Rights Act and the Voting Rights Act. He had been warned that this legislation "will not only cost you the South, it will cost you the election." [START_REF] Branch | Pillar of Fire: America in the King Years, 1963-1965[END_REF] Added to this, Johnson attempted to have both guns and butter: funding both the Vietnam War and the Great Society, which led to higher taxes and contributed to growing inflation and other economic problems. And this, in turn, helped the Republicans to pin the label "tax and spend" on their opponents.

Furthermore, Kevin Phillips, a conservative political analyst, published The Emerging Republican Majority in 1969. He argued that, if the Republicans stressed support for traditional cultural and social values and opposition to a big government agenda-now linked in many people"s minds to racial questions-then they could form a new majority to govern the country. Also in 1969, Newsweek did an examination of "Troubled America" which published the results of a scientific survey showing that almost 80 percent of middle-class Americans thought that, on average, members of racial minorities on welfare had chosen not to work. [START_REF]The Troubled America: A Special Report on the White Majority[END_REF] These and other observations had a profound effect on the Republican Party. In that same year Richard Nixon made a determined appeal to the, as he called it, "Silent Majority"-those who did not oppose the Vietnam War, participate in protest rallies or belong to the counterculture. To attract this "silent majority", Nixon and others attacked court-ordered busing and called for stronger law enforcement procedures.

Later, Ronald Reagan developed similar themes, using language that, while not overtly racist, contained terms that summoned up, in some people"s minds, a negative image of blacks and other minorities. His attack on "welfare queens" (a story that was largely fictitious) and affirmative action programs and his wish to reduce spending on social programs while increasing crime fighting capabilities, appealed to many whites who considered that African Americans were disproportionately taking advantage of the welfare state-while, at the same time, being the main perpetrators of crime. Furthermore, the Republican Right showed a remarkable gift for caricaturing liberalism, taunting the Democrats as having only one policy: "tax and spend". In this way, it has been argued, the Republicans split the traditional class-based Democratic coalition. [START_REF] Mason | Richard Nixon and the Quest for a New Majority[END_REF] Recent work has tended to criticise this theory as too simplistic and to stress the role of class. For one thing, it does not explain the situation of states like Kansas, which has a radical past and possesses a negligible black population, but has voted Republican in every presidential election since 1948, with the exception of 1964, often by large majorities. Nor does it explain that the South was not so solidly Democratic before 1965, with at least four states lacking a Democratic majority in every election from 1948. Matthew Lassiter, in his book, The Silent Majority: Suburban Politics in the Sunbelt South, argues against a purely race-based interpretation and places greater emphasis on class. He shows how a technically race-neutral language developed to further middle-class aims:

The color-blind and class-driven discourse popularized in the Sunbelt South helped create a suburban blueprint that ultimately resonated from the conservative subdivisions of Southern California to the liberal townships of New England: a bipartisan political language of private property values, individual taxpayer rights, children"s educational privileges, family residential security and white racial innocence. 14 Lassiter links many of the changes in southern voting patterns to more general changes in the US as a whole, which he sees, in turn, as related to the expansion of suburban America. This, of course, is also connected to the growth of the middle class. The late 1940s saw the beginning of a massive demographic shift from rural and urban America towards the suburbs. It also saw an enormous growth in universities and in the number of students. Both of these changes came about, to a large extent, because of the G.I. Bill which provided assistance to veterans. The prosperity of the 1950s continued the evolution towards a largely middle class America with a relatively small group living in poverty-and a large percentage of this group consisting of blacks and immigrants. 15 The Democratic Party shows a complex interrelationship between class, race and geography that makes it difficult for it to find a presidential candidate who pleases voters in all these divisions. 16 In any case, by the mid-1980s, the Democratic Party was demoralized by its failure to win presidential elections (its decline at the Congressional and state levels was considerably less drastic), having lost every election since 1968, except for that of 1976. The 1972 presidential campaign had been the nadir of Democratic prospects with the resounding defeat of George McGovern and his liberal platform. Most Democratic commentators saw this as a decisive rejection of the leftwing, although, as Bruce Miroff has shown, McGovern"s impact on the party, although rarely talked about, has been surprisingly strong. A large number of the party"s leaders since then were involved in one way or another in that campaign. 17 Jimmy Carter, the next Democratic presidential candidate turned away from this legacy, positioning himself in the centre. Add to this the fact that many of its leaders were also aware of the demographic changes we have previously outlined and interpreted the triumphant popularity of Reagan as evidence that America had moved to the right. The old, traditional, classbased rhetoric of the New Deal had, for many, become associated with minority groups and a number of people judged it, therefore, to be counter-productive. Certainly Michael Dukakis, Democratic presidential candidate in 1988, tried to distance himself from these themes, basing his campaign around the so-called "Massachusetts Miracle"-the reinvigoration of the Massachusetts economy during his time as governor. The Republicans, however, worked to paint Dukakis in highly negative colours as a "tax and spend" liberal, weak on crime. 18 The failure of Dukakis to hit back effectively and overcome the image given to him by Republican loyalists convinced even more Democrats that they had to change the way the party was seen and to establish a quick and effective response to such attacks in the future.

Already in 1985, several, mainly southern, Democrats had formed the Democratic Leadership Council (DLC). 19 The DLC argued that the Republicans were winning because they had successfully portrayed the Democrats as a party dominated by narrow interest groups like radical feminists, gay rights or ethnic minorities and thus out of touch with the mainstream of America. This, in turn, led to a philosophy of "tax and spend" which benefitted these minority groups to the disadvantage of the majority. DLC leaders argued that the Democrats had to change the public philosophy of their party. The Democratic Party had to find a way to reconnect with the majority of the nation. As Sam Nunn, one of its founders, put it: "We are going to try to move the party-both in substance and perception-back into the mainstream of American political life." 20 Part of this would mean downplaying the federal government as an engine for reform and emphasizing instead the private sector, local and state governments. Although they certainly do not agree with Reagan"s statement that the federal government "is the problem", the DLC prefers the use of market mechanisms rather than federal bureaucracies to implement policy. In 1990, the DLC published a statement of its basic beliefs:

We believe the Democratic Party's fundamental mission is to expand opportunity, not government… We believe that economic growth is the prerequisite to expanding opportunity for everyone. The free market, regulated in the public interest, is the best engine of general prosperity…We believe the purpose of social welfare is to bring the poor into the nation's economic mainstream, not to maintain them in dependence. 21 Bill Clinton, one of its founders, described the DLC"s goals as "forging a winning message for the Democrats based on fiscal responsibility, creative new ideas on social policy, and a commitment to a strong national defense". 22 It also sent out a strong message on crime.

The problem now, as the DLC saw it, was to convince the electorate that they had a different and original message. Their solution was a vocabulary of change, with repeated use of words like "change" and "modern" while at the same time implying that their programme would be a return to an idealized past, linked to the Democratic Party, that contained the true values of the nation:

In keeping with our party's grand tradition, we share Jefferson's belief in individual liberty and responsibility. We endorse Jackson's credo of equal opportunity for all, and special privileges for none. We embrace Roosevelt's thirst for innovation, and Truman's sense in the uncommon sense of common men and women. We carry on Kennedy's summons to civic duty and public service, Johnson's passion for social justice, and Carter's commitment to human rights. 23 At the same time, the word "class" appeared repeatedly but only in connection with the word "middle". Terms traditionally associated with conservatism were also appropriated, such as "choice", "opportunity" and "responsibility".

In 1990, Bill Clinton, then governor of Arkansas, became chair of the DLC and, in an important speech in May 1991, outlined a number of his ideas. After examining the problems facing the U.S., in particular in relation to education, salary levels, health care and job insecurity, Clinton said:

You may say, well if all these things are out there, why in the world haven"t the Democrats been able to take advantage of these conditions? I"ll tell you why: because too many of the people that used to vote for us, the very burdened middle class we are talking about, have not trusted us in national elections to defend our national interests abroad, to put their values into our social policy at home, or to take their tax money and spend it with discipline. We have got to turn these perceptions around, or we cannot continue a national party. 24 In effect, he accepts the Republican, and notably Reaganite, criticisms of the party as valid and, in so doing, gives them greater force. Democratic programs have been responsible for Democratic losses because they have lost touch with the majority of Americans by focussing too much on the have-nots and not enough on the middle class, described here as "burdened"-an obvious acceptance of right-wing critiques of past Democratic policy on taxation.

He was even more clear in the announcement of his candidacy for the presidency in October 1991:

All of you, in different ways, have brought me here today, to step beyond a life and a job I love, to make a commitment to a larger cause: preserving the American Dream… restoring the hope of the forgotten middle class…reclaiming the future of our children. I refuse to be part of a generation that celebrates the death of Communism abroad with the loss of the American Dream at home. 25 Here we can see the link with an idealized heritage through the use of "American Dream". Words like "restore" and "reclaim" illustrate this decline from past glories. The future of the nation is in jeopardy: America has won the Cold War only to be at risk of losing the peace. Note also the term "forgotten middle class" that obviously echoes Nixon"s famous reference to the "silent majority". They have lost hope although why is not entirely clear. Later in the speech, he called for a "new covenant to rebuild America" which clothes the social contract idea in a typically American religious dress. He also argued that "government"s responsibility is to create more opportunity. The people"s responsibility is to make the most of it." Clinton developed this theme further, using his own life as an example. After talking about the poverty of his mother and grandparents, he said: But we didn"t blame other people. We took responsibility for ourselves and for each other because we knew we could do better. I was raised to believe in the American Dream, in family values, in individual responsibility, and in the obligation of government to help people who were doing the best they could.

Most of the terms used here are traditionally associated with certain strands of rightwing thought, notably "responsibility" (repeated twice), the "individual" and, a more recent addition, "family values". He does, of course, talk about "the obligation of government" but this is clearly limited only to those "who were doing the best they could". We can see here the impact of Reaganism with its implications that many of the people on welfare were simply lazy and taking advantage of the system. The use of such rhetoric may have been a deliberate tactic by Clinton and the DLC: as Republicans moved to the right under Reagan and even further to the right from 1994, the use of terms associated with the right may have been an attempt to capture the centre.

In a number of speeches throughout 1990 and 1991, Clinton insisted on the need for a "modern, mainstream agenda" and used key terms like "expansion of opportunity", "choice", "responsibility and empowerment of poor people" and "reinventing government". Like Dukakis before, he developed the theme of economic management and competence. He repeatedly emphasized that the Republicans had run up a huge deficit and that only the Democrats could be trusted to provide capable economic supervision. The sign "It"s the economy, stupid" became a symbol of the campaign. Over and over again Clinton targeted the middle class. He did not ignore the existence of poverty nor did he deny the need to offer government assistance but placed particular emphasis on limiting its duration and concentrating on the most vulnerable elements of society, notably children. These terms and themes would also appear in Britain and would occupy a large place in Tony Blair"s rhetoric.

Once he became president, his rhetoric reflected a certain conflict between his DLC motifs and more traditional Democratic themes like communitarianism and the need for government intervention. In his first inaugural address, predictably, he stressed the ideas of renewal and change:

Today we celebrate the mystery of American renewal. This ceremony is held in the depth of winter, but by the words we speak and the faces we show the world, we force the spring, a spring reborn in the world"s oldest democracy that brings forth the vision and courage to reinvent America. When our Founders" boldly declared America"s independence to the world and our purposes to the Almighty, they knew that America, to endure, would have to change; not change for change"s sake but change to preserve America"s ideals: life, liberty and the pursuit of happiness. Though we marched to the music of our time, our mission is timeless. 26 The themes here are traditional in American inaugural addresses, notably, the reference to the Founding Fathers and American democratic traditions, the invocation of God and, of course, the imagery of rebirth. Notice Clinton"s acceptance of the idea that America has a special mission although he does not tell us what that mission is.

His first State of the Union address showed more clearly the tensions between the Roosevelt tradition of the Democratic Party and the DLC"s ideas. Following logically from his campaign, he placed the greatest emphasis on economic policy. While he keeps to his earlier discourse on "opportunity" and "responsibility", he makes sure to add a dose of "community" into the mixture: I believe we will find our new direction in the basic values that brought us here: opportunity, individual responsibility, community, work, family and faith. We need to break the old habits of both political parties in Washington. We must say that there can be no more something for nothing, and we are all in this together. 27 Clinton here asserts that his political ideas, presumably those of the DLC, are genuinely new and different from those of traditional Democratic ones. In fact, he dismisses both parties equally-something which probably irritated members of Congress since many of them symbolized those "old habits". His last sentence here keenly balances the DLC and the Roosevelt tradition: there should be no "Santa Claus state" giving things away to the unworthy but, on the other hand, we all have a responsibility for each other. The role of the state in helping its citizens is thus not denied but it is limited. Later in the speech he explicitly links it to the behaviour of those citizens: "I want to talk about what government can do, because I believe our government must do more for the hard-working people who pay its way." In other words, the taxpayer needs to see a good return on his or her investment. The implication seems to be that if you do not work then you do not deserve assistance from the government. This, of course, reflects Clinton"s repeated emphasis on reforming and limiting welfare (later in the speech he says that "after two years, they [those on welfare] must get back to work-in private business if possible; in public service, if necessary." But note that he does not reject a role for government in putting people back to work.

Of course, Clinton won the presidency without receiving a majority of the vote or increasing Democratic seats in Congress which placed him in a relatively weak position. Certainly the need to work with a Democratic Congress dominated by liberals and his desire to ensure the unity of the party, pushed him to the Left. Many of Clinton"s first actions as president seemed closer to his days campaigning for McGovern than to this DLC rhetoric. His first two years as president disappointed many in the DLC while not satisfying the liberal wing of the party. Public dissatisfaction with several of Clinton"s initiatives-notably the failure of his health care plan and the controversy over his attempt to allow openly gay people to serve in the US Army-contributed to a Republican victory at the congressional elections in 1994. After this slap from the voters, Clinton moved even further to the right and his rhetoric shows this. In his first State of the Union address after the election, he said: "I think we all agree that we have to change the way the Government works. Let's make it smaller, less costly, and smarter; leaner, not meaner." 28 A Republican majority in Congress meant that there was little hope of new benefits and, indeed, Clinton finally agreed to sign a bill limiting the length of time one could remain on welfare. At the same time, he came under vicious attack over his personal life, leading to an impeachment hearing in Congress. In such an atmosphere the legislative achievements of his last years in office were minimal.

Let us then turn to examine the situation of Great Britain. In many ways the demographic evolution of the country resembles that of the United States with the growth of the middle class, the increasing advance of suburbanization and the development of what Margaret Thatcher called "popular capitalism". The first person to use the term "property owning democracy" was also British, the economist J.E. Meade in 1964. Certainly the social and cultural upheavals of the 1960s, although obviously important, were less traumatic than in the United States as there was not the same drama of the Vietnam War and the civil rights movement. Great Britain also has not undergone as radical a change in its electoral map but this is, in part because the Labour Party had not had a notable success in winning with large majorities before Blair. On the other hand, although not as significant as in America, the racial question existed and was growing in magnitude as the Notting Hill riots of 1958, the impact of Enoch Powell and the Brixton and Toxteth riots of 1981 show.

It is well known that, after Labour"s defeat of 1979, when Margaret Thatcher came to power, the party moved to the left, choosing Michael Foot as leader and adopting certain positions viewed as too leftwing by moderate members. [START_REF]Notably unilateral disarmament, withdrawal from Europe and greater nationalization[END_REF] This provoked a rebellion by influential centrist members who created a new party, the Social Democrats (SDP). After a further defeat in 1983, in which Labour and the SDP each got about onequarter of the vote, the left-wing found itself largely discredited but their power within the party remained strong. The right-wing then began a slow campaign to win back control of the party. After Foot"s resignation, Neil Kinnock became leader and he considered his primary objective to be returning the party to the centre. He believed in the need for a radical transformation of the party"s policy, organisation and mentality, without which it would never come back to power. But divisions within the party forced him to proceed cautiously-especially during the first years of his leadership which were complicated by the miners" strike and the activities of Militant Tendency, a Trotskyite group in Liverpool. But the defeat of the miners and of Militant Tendency reinforced Kinnock and weakened the extreme left. He was also helped by the break-up of the Soviet Union which dramatically revealed the weaknesses of the Soviet economy and consecrated the triumph of capitalism and the market. Furthermore, the failure of socialist experiments by the French government from 1981-84 also discredited the economic ideas of the extreme left. By the end of the 1980s, most members of the Labour party were ready to accept the market economy. In 1985, after the defeat of the miners" strike and coincidentally in the same year as the founding of the DLC, Kinnock attacked. 30 In a speech on the future of socialism, he started the process of redefining the policy of the party: certain of his ideas were entirely retaken later by Tony Blair. Kinnock stressed certain values like community and democracy rather than doctrinaire socialism. He even attacked the idea of nationalization and called for a "servant state". In 1986 he published Making Our Way, he even began to talk positively about capitalism and the market, although he saw them as needing reform and management. [START_REF] Kinnock | For more on these changes see Tudor Jones, Remaking the Labour Party from Gaitskell to Blair[END_REF] Kinnock instituted a well publicized policy review which shed Labour"s more extreme positions. [START_REF]For more on its significance and impact see Gerald Taylor, Labour's Renewal ? The Policy Review and Beyond[END_REF] By 1992, Labour had moderate policies and had done its best to attract media attention to them. The result, however, was a fourth election defeat in that year. It seemed that even a movement to the centre could not help Labour win and some began to despair that the party might never form a majority government again. [START_REF] See | To Build a New Jerusalem : The British Labour Party from Keir Hardie to Tony Blair[END_REF] In such a situation it is not surprising to see that Labour chose a more tradition-oriented leader in John Smith who slowed Kinnock"s policies of change. Smith, however, died unexpectedly in 1994 and the young and reform-minded Tony Blair was chosen to replace him.

In an influential article published in 1987, Anthony Heath and S. K. McDonald analysed the demographics of that year"s election and concluded that social change since 1964 had reduced Labour"s vote by five percent. [START_REF] Heath | Social change and the future of the Left[END_REF] At the same time the Conservatives" position had improved by four points. They also argued that the SDP/Liberal Alliance had benefited from the demographic situation, attracting much support from the middle class. Blair and many other Labour figures wholeheartedly accepted this analysis. Labour politician Giles Radice argued that "Labour cannot afford a class approach." 35 Coming to much the same conclusion as the DLC, Radice insisted that:

[upwardly mobile families] do not believe that [Labour] understands, respects or rewards those who want to "get on". Far from encouraging talent and promoting opportunity, Labour is seen as the party that is most likely to "take things away". From the perspective of the aspirant voters, voting Labour is simply not "in their interests". 36 Blair saw Labour"s four election defeats as the voters" punishment for the party"s failure to adapt to social and economic changes in British society. As we have already seen, Blair picked up themes already presented by Clinton in the US and Kinnock in the UK. As with Clinton, we can see the paradoxical assertion that something "new" was also a return to earlier ideas: "Labour has returned to its values and is now seeking the clearest and most effective ways of putting them into practice". 37 In some strange way, "renewal" had become a synonym for "modernisation".

Certainly, "new" quickly became a key word for Blair. As Bill Clinton had talked of "New Democrats" so Blair began to speak of "New Labour" and "modernisation" became one of his mantras. But Blair also realized that he would need a significant gesture to signal major change and so he decided to repeal clause 4 of the Labour Party constitution which committed the party "to secure for the producers by hand or by brain the full fruits of their industry, and the most equitable distribution thereof that may be possible, upon the basis of the common ownership of the means of production of each industry or service" in other words to socialism. Blair succeeded in this and proceeded to develop a rhetoric of the centre for his party.

From the start, Blair rejected ideology, insisting that the Labour Party had been founded to improve the living conditions of the mass of the population and not to promote socialism. In one speech, he argued that the 1945 Labour government was the greatest peace-time government of the 20 th century not because it introduced a socialist program but because it put into effect the British people"s aspirations for a better existence. [START_REF]The Radical Coalition[END_REF] This, indeed, should be the aim of government. According to Blair, during the 1970s and 1980s, many in Labour lost contact with the people"s wishes and became obsessed by ideology and, for this reason, the party suffered four consecutive election defeats. [START_REF]New Labour, New Life for Britain[END_REF] Labour, Blair argued, had attracted voters because it had been a party of change but, over the years, the insistence on ideology had become a refusal of change. He described the Labour left as being "conservatives of the left" while he called for a "radicalism of the centre". [START_REF]for example[END_REF] Rhetorically, Blair has borrowed from the traditions of the Liberal Party, echoing both Gladstone and Lloyd George at times. He made a point to laud the achievements of past Liberal reformers at the Labour celebrations for the 50 th anniversary of the Attlee government:

The ultimate objective is a new political consensus of the left-of-centre, based around the key values of democratic socialism and European social democracy, firm in its principles but capable of responding to changing times, so that these values may be put into practice and secure broad support to govern for long periods of time. To reach that consensus we must value the contribution of Lloyd George, Beveridge and Keynes [all noted Liberals] and not just Attlee, Bevan or Crosland. We should start to explore our own history with fresh understanding and an absence of preconceptions. [START_REF] Also | The Character of New Labour[END_REF] The key terms here are "consensus" and "broad support". Blair presents his philosophy as one of pragmatism: be flexible, adapt to change, and respond to what the voters want. At all costs avoid an ideological straitjacket. To make his point even clearer, Blair also echoed the Conservatives: So, yes, there has been a revolution inside the Labour party. We have rejected the worst of our past, and rediscovered the best. And in rediscovering the best of our past, we have made ourselves fit to face the future, and fit to govern in the future. There is a big idea here. It is about creating a society that is genuinely One Nation in which we seek to realise the potential of all our people. 42 Once again we see the idea that only by keeping to the values of the past can one create a good future. Note his use of the term "One nation", an expression strongly associated with the Conservatives. Blair certainly demonstrates here his sympathy with certain strands of Conservative thought and his fascination with Margaret Thatcher (although one would never associate the term "one nation" with her) is well-known-to such an extent that Anthony Seldon devoted one chapter of his biography of Blair to the subject. [START_REF] Seldon | [END_REF] In one pamphlet, Blair went so far as to commend some aspects of the Thatcher government, saying: Some of its reforms were, in retrospect, necessary acts of modernisation, particularly the exposure of much of the state industrial sector to reform and competition. 44 Notice his praise of "competition" in particular. Of course, he immediately counters this by attacking the Thatcher government which he accuses of "damaging key national services, notably education and health".

Blair, in fact likes to say that his socialism is moral and not economic:

Since the collapse of communism, the ethical basis of socialism is the only one that has stood the test of time. This socialism is based on a moral assertion that individuals are interdependent, that they owe duties to one another as well as to themselves, that the good society backs up the efforts of the individuals within it, and that common humanity demands that everyone be given a platform on which to stand. 45 He is not egalitarian but communitarian for he continually talks about the importance of the community with which the individual evolves. Blair likes to underline the need for cooperation between all members of society in order to assure a well functioning economy-for, like the Democrats, New Labour stresses its economic competence. In an early speech to the TUC, Blair subscribed to the following principle:

Business and employees, your members, aren"t two nations divided.

That"s old style thinking. That"s the thinking of the past. Britain works best when business and unions work together. 46 The citizen, Blair also believes, has rights as well as duties towards other citizens and so, as with Clinton, we find the lexical field of responsibility.

Like the Democrats, Blair picked up on fears of law-breaking, famously saying that he would be "tough on crime; tough on the causes of crime". Blair has always insisted on the need to maintain the welfare state and he certainly has significantly increased spending in certain areas, most notably the health service. But he does not want to be considered as hostile to the rich. In a speech on social justice in 1994, Blair observed:

Because we [Labour] were anti-poverty, we were portrayed as antiwealth. Because we were concerned with lifting up the less successful, we were seen as attacking those who aspired to do better. Because we campaigned for adequate benefits, we were said to be unconcerned about the working poor who were taxed to pay for benefits. Because we wanted to defend the welfare state, people came to assume that we did not think it could be improved. We were seen as interested more in protecting the gains of the past, rather than building on them.

But like Clinton he sees the need for changes and for limiting certain benefits.

Blair won a large victory in 1997 and immediately began work on an important series of reforms, notably in the constitutional domain. In part because of the differing nature of the political systems, Blair enjoyed significantly greater success than Clinton. Realising that an emphasis on newness and moderation became more difficult once in power, Blair sought to present himself and his party in somewhat different terms. His first attempt became known as the famous "Third Way". According to Blair:

The Third Way is not an attempt to split the difference between Right and Left. It is about traditional values in a changed world. [START_REF] Blair | The Third Way: New Politics for the New Century[END_REF] So, once again newness is a return to tradition. But the term "Third Way" was found to confuse voters and largely disappeared from Labour discourse in subsequent years. [START_REF] Fielding | The Labour Party: Continuity and Change[END_REF] Blair made a number of other attempts after this but none were particularly successful. Blair became obsessed with framing political discourse and came to believe in the need for permanent campaigning. As Labour Party adviser, Philip Gould, explained:

You must always seek to gain and keep momentum, or it will pass immediately to your opponent. Gaining momentum means dominating the news agenda, entering the new cycle at the earliest possible time, and repeatedly re-entering it, with stories and initiatives so that subsequent news coverage is set on your terms. [START_REF] Gould | The Unfinished Revolution: How the Modernisers Saved the Labour Party[END_REF] The problem, though, was that the press quickly became sceptical about New Labour"s spin and began to report critically on Blair"s announcements-to the point of even questioning whether Cherie Blair"s pregnancy was a political manoeuvre. This cynicism was then communicated to the public who started to distrust Labour-a feeling which only increased after the invasion of Iraq and revelations of how the government had misrepresented information. It did not, however, stop Labour from being re-elected in 2001 and 2005.

But, in the end, how "new" are New Labour and the New Democrats? And how does their rhetoric relate to their actual policy positions and actions in power? It has to be stated that most elements of the DLC programme are acceptable to the liberals, which may help explain the failure of the Left to make an effective counter attack. The New Democrats supported welfare reform but wanted it to be achieved through greater investment in education and job training. They called for more environmental protection, supported abortion, family leave, hand gun control and national health care. As Jim Hale has pointed out, they differ mainly from liberals in the areas of trade and affirmative action. He has even argued that: "the DLC has fleshed out a liberal-leaning platform couched in soothing centrist rhetoric." 50 Certainly the first two years of the Clinton presidency can be used to argue such a position.

The same argument can be made in relation to New Labour. Blair continually proclaimed that his party had definitely broken from the past, and many agreed with him, seeing New Labour as an extension of Thatcherism. [START_REF] The | Lawson invented the term "neo-Labour", arguing that Blair"s party was simply continuing Thatcherism. See also Stuart Hall "The Great Moving Nowhere Show[END_REF] It did not take long for reassessments to emerge and a number of scholars began to question how different from the Labour past it really was. David Rubinstein, for example, argued that Blair"s objectives were little different from those of the Attlee or Wilson governments. [START_REF] Rubinstein | A New Look at New Labour[END_REF] Steven Fielding even said of the revision of Clause Four: Widely hailed as New Labour"s defining moment, this involved deleting from the party"s constitution words that enjoyed nothing more than a questionable relevance. They were moreover replaced by another set of words that still allowed the possibility of state ownership. Furthermore, while the new clause endorsed the market, that was something party leaders had done decades before. Consequently, if the revision of Clause Four was the ultimate expression of "New" Labour, then "New" Labour was nothing new. 53 Certainly Blair"s governments focussed on a number of issues associated with the Left, such as health and education and sharply increased social spending (especially health care). Blair himself repeatedly said that education as his top priority, although he continued such Thatcher and Major reforms as a national curriculum and key stage tests (which Labour had initially opposed). With regard to social policy, the Blair governments did make some attempt at lowering income inequality which had grown during the Thatcher years. Blair chose to focus on target groups, notably children and pensioners, and did have some success in improving their living standards. [START_REF]For more on this see Kitty Stewart[END_REF] Certainly Blair made no major changes like those of the post-war Labour government and was content with piecemeal improvements to social welfare, sharing much of the philosophy of the New Democrats. However, this does not make New Labour "new"a significant and usually dominant proportion of the party since 1951 had always believed that after Attlee, the emphasis should be on improvements to what already existed than in the creation of new structures. So, as in the case of Clinton"s New Democrats, it can certainly be argued that the emphasis on novelty and modernisation was primarily a rhetorical device.

Let us now consider how successful the move to the centre was for both the Democrats and Labour. A 2001 poll showed that voters saw Blair"s party as preoccupied by the middle class and their interests. [START_REF] Also | The Rise of New Labour: Party Policies and Voter Choices[END_REF] However a detailed analysis by Heath, Jowell and Curtice established that Labour support had already started to grow under John Smith among voters who saw themselves as centrists-and thus before Blair began his ideological repositioning. 56 On the other hand, they found that:

In contrast, between 1994 and 1997 Tony Blair and New Labour lost ground somewhat on the left, made only modest gains on the centre-left, and had their largest gains on the right of the spectrum. Unlike the 1979-83 and the 1992-4 periods, when gains and losses tended to take an acrossthe-board character, the 1994-97 gains were highly unevenly distributed and we have no hesitation in attributing them to New Labour"s ideological shift rather than to its perceived competence (or to the Conservatives" perceived incompetence). [START_REF] Also | Why Parties Fail to Learn: electoral defeat, selective perception and British party politics[END_REF] Their conclusion then is that New Labour did, indeed, draw some support from the Conservatives. But, as Pippa Norris has pointed out, resolutely occupying the centre has been "both a blessing and a curse". She argues that it has been "the bedrock of his popular success and yet the limit of what he can do with his popularity" for being a centrist does not make one an inspiring visionary or give a candidate a mandate for "radical policy change" which will ensure a place in history. [START_REF] Norris | Elections and Public Opinion[END_REF] Evidence is less clear in relation to the New Democrats. Exit polls show that in 1992 Clinton had higher support than Dukakis among the white middle class but the difference was small-only one or two percent. 59 Congressional Quarterly, in its analysis of the 1992 election, found that social class was not a major factor: Except for race, all of the social factors we have examined-region, union membership, social class, and religion-have declined in importance during the postwar years. The decline in regional differences directly parallels the increase in racial differences.... The Democratic Party"s appeals to blacks may have weakened its hold on white groups that traditionally supported it... But the erosion of democratic support among union members, the working class and Catholics results from other factors as well. During the postwar years, these groups have changed... Differences in income between the working class and the middle class have diminished. 60 It would seem, therefore, that the New Democrats" demographic analysis was correct. The problem, though, is that Congressional Quarterly can find little evidence of their strategy working. The Democrats did, indeed, increase their percentage of the vote among the white working class and other target groups but "the Democrats won with lower absolute levels of support among most of these groups than they had won in previous Democratic victories." 61 Clinton"s election in 1992 has usually been seen more as a criticism of Bush and the result of a three-way race than a positive endorsement of the Democrat.

The 1994 congressional elections were a complete disaster for the Democrats. Clinton analysed the problem as being fundamentally one of communication which the Republicans, thanks to their Contract with America, had been better at. According to Clinton: "From 1994 on... the side without a national message would sustain unnecessary losses." 62 He did admit to a series of mistakes, stating in his memoirs that he should have postponed health care reform when a Republican filibuster became clear and concentrated instead on welfare reform which "would have been popular with alienated middle-class Americans who voted in droves for Republicans." 63 By not doing so he believed this allowed the media to amplify his errors: "my victories were minimized, my losses were magnified, and the overall impression was created that I as just another pro-tax, big-government liberal, not the New Democrat who had won the presidency." 64 So his analysis was clear: the Democrats lost because of his failure to embrace strongly the New Democratic programme and communicate this effectively to the public. Sen. Diane Feinstein agreed: "We Democrats have listened to the 15 percent of the people who had no [health insurance] coverage. Republicans listened to the 85 percent of the people who had coverage." 65 But in spite of his subsequent move to the centre, his signing of the Republican welfare reform bill and a booming economy, in 1996 Clinton failed to get a majority of the popular vote with only 49 percent supporting him. Congressional Quarterly did, however see an "increasing part of the middle class voting Democratic in that election". [START_REF] Abramson | Change and Continuity in the 1992 Elections[END_REF] At the same time, though, the Democrats gained only eight seats in the House and actually lost two in the Senate.

In the period since 2000 profound differences have developed between eh two countries. To begin with, if the impeachment of Clinton had revealed deep polarisation in the U.S., the contested presidency of George W. Bush made the situation worse. The Iraq War played a major role here. Discourse related to this conflict was examined in some detail in the companion volume to this book, Foreign Policy Discourse in the "New World Order" in the United States and the United Kingdom so it will not be considered in any depth here. Its impact, however, in motivating and radicalizing the American Left should not be underestimated. The rejection of the Iraq War, torture, and Guantanamo Bay discredited the Republicans among many voters and encouraged the Democratic victory at the congressional elections in 2006. When added to a growing economic crisis, it led to victory for the Democrats in the presidential election of 2008 as well. The election of the first black president in American history was an immensely significant development but it has led to greater polarisation in political discourse.

Although Britain has not experienced as strong a polarisation, it has other things in common with the United States, notably the Iraq War. Tony Blair resigned as Prime Minister in June 2007 having lost much of his popularity because of his enthusiastic support for the invasion of Iraq. He was replaced by his Chancellor of the Exchequer, Gordon Brown. Brown"s major advantage probably was that he had been the architect of economic policy and, thus, of the prosperity of the time. When the economic crisis came, Brown found himself in a difficult position because he had to justify his own economic decisions and, therefore, could not return to traditional Labour rhetoric. He did try at times to strike a populist tone, notably when asking for a special tax on bonuses, but he was continually hampered by his own past rhetoric and championship of the market. His language has been carefully balanced:

The first choice was this: whether markets left to themselves could sort out the crisis; or whether governments had to act. Our choice was clear: we nationalised Northern Rock and took shares in British banks, and as a result not one British saver has lost a single penny. That was the change we chose. The change that benefits the hard working majority, not the privileged few. And we faced a second big choice-between letting the recession run its course, or stimulating the economy back to growth. And we made our choice: help for small businesses, targeted tax cuts for millions and advancing our investment in roads, rail and education. That was the change we chose-change that benefits the hard working majority and not just a privileged few. [START_REF] Brown | Labour Party Conference[END_REF] Here we see several of Blair"s favourite terms: "choice", "change", "hard working majority", "small businesses" and "tax cuts". But they are offset by traditional Labour themes like nationalisation and by a repeated attack on "a privileged few". It is one of the obvious differences between British and American left-wing rhetoric that, while Obama took the same actions as Brown with regard to the banks, he would never have dared to use the word "nationalisation". Brown, unlike pre-Blair Labour leaders, does not present nationalisation as something good but as the necessary response to a major crisis. The decision was made not in the interests of ideology but in those of "British savers". Let us also examine his use of "privileged few". Looking at earlier New Labour and New Democrat rhetoric one has the impression-although this is never clearly stated-that if a privileged few exists it consists not of the rich but of certain unspecified elements of the non-working poor, which obviously has a link with Reaganism. With the current economic crisis, "the privileged few" seems to be once again, the rich-or at least certain of them.

Often electorates turn to the left in times of economic crisis but given Labour"s own role in the period leading up to the crisis, this has not occurred. In the elections of May 2010, Nick Clegg and the Liberal Democrats clearly hoped to capitalize on disillusionment with both Labour and the Conservatives but obviously failed to do so. The results showed some fear at the idea of the Conservatives returning to power, a loss of confidence in Labour, and an uncertainty about any other possibility. All of this, of course, complicates economic recovery and makes twice as difficult Labour"s search for an effective discourse in the present atmosphere.

The economic crisis has also had an impact on discourse in the United States. Certainly there have been a number of attacks on Wall Street and the nation"s economic aristocracy but Obama"s criticisms have been notably tepid. In a major speech on financial reform, his moderation was evident: I believe in the power of the free market. I believe in a strong financial sector that helps people to raise capital and get loans and invest their savings. That"s part of what has made America what it is. But a free market was never meant to be a free license to take whatever you can get, however you can get it. That"s what happened too often in the years leading up to this crisis. Some-and let me be clear, not all-but some on Wall Street forgot that behind every dollar traded or leveraged there"s a family looking to buy a house, or pay for an education, open a business, save for retirement. What happens on Wall Street has real consequences across the country, across our economy. [START_REF] Obama | Remarks on Wall Street Reform[END_REF] This is a weaker criticism than Roosevelt"s, although not fundamentally different. The basic point is the same: some are abusing the system and average people are suffering because of it. The function of government, especially at the national level, is to prevent this from happening. But while Obama"s rhetoric has been notably mild, his administration has achieved a remarkable amount in Congress: the stimulus bill, student loan reform, credit card reform and, of course, major health care reform which had eluded many past presidents.

All of this leads one to believe that political rhetoric is extremely fluid and, indeed, limited and repetitive. Tony Blair uses terms associated with Margaret Thatcher and other Conservatives. Clinton appropriates rightwing rhetoric while the Republican right re-uses radical leftwing terminology in a totally different context. While there are some ideas and expressions associated with one party more than another, this can change. If something seems to succeed, to appeal to people, then the other party will borrow it-although they tend to adapt it a bit. The essential motivation for any political party is to win elections. If one party fails to do so for a long period of time, they will obviously study the methods and rhetoric of the winning party and try to adjust theirs to make it more attractive to the voters. Both Britain and the United States have tended towards the right since the 1970s for a number of reasons, demographic, and political, among others and, in order to win, the left (at least some of it) has tried to adjust its policies and rhetoric to this situation.

What this brief study has tried to show is that, while there are important differences between the two nations, there is a remarkable similarity in the development of political ideas. The New Democrats and New Labour developed at the same time after suffering a similar period of electoral disappointment. In both cases, their discourse moved to the centre in an attempt to woo voters and they both made a show of breaking from past party orthodoxy and instituting something new. However, it would be wrong to imagine that past rhetoric was significantly more incendiary or oriented towards class warfare or that policy changed dramatically. The speeches and the policies of past Democratic or Labour leaders have not been significantly different from those of New Labour or the New Democrats. Politics is the art of the possible certainly, but it is also the art of repackaging the old and presenting it as the new.

have seen a bizarre reversal of class-consciousness in states like Kansas. The journalist Thomas Frank has argued that the Republican right has appropriated the traditional rhetoric of the left, changing the word "capitalism" into "liberalism" (in its American sense). He asserts that, "in some ways, the backlash vision of life is nothing more than an old-fashioned leftist vision of the world with the economics drained out." Average people are portrayed as helpless pawns in a world controlled by an eastern liberal elite-frequently accused of being unmanly, effete, and lover of French things. See Thomas Frank, What's the Matter with Kansas? (New York: Metropolitan Books, 2004) 129-30. Voting against what is generally perceived to be a voter"s economic interest is not limited to America. Studies have repeatedly shown that about one-third of the working class in the UK always votes Conservative. 15 In 1956 the Census Bureau announced for the first time that a majority of the population was white collar. Byron Shafer, for example, has argued that: "A different structure to the economy meant by definition a different occupational structure for American society and a different occupational structure meant a different class structure as well." The Two Majorities and the Puzzle of Modern American Politics, 11 16 For a more detailed discussion of factions within the Democratic party see Nicol Rae, Southern Democrats (Oxford: Oxford University Press, 1994) 17 See Bruce Miroff, The Liberals' Moment: the McGovern Insurgency and the Identity Crisis of the Democratic Party. (Lawrence: University of Kansas, 2007) 18 He became the victim of a smear campaign initiated by a supposedly independent group with no obvious ties to the Bush campaign. This group made a television commercial showing convicted killer Willie Horton who, during a furlough from a Massachusetts prison, raped a woman and assaulted her husband. Horton, of course, was blackthus playing to racial fears. Since Dukakis had been governor of Massachusetts at this time, the ad implied that he was somehow responsible for this tragedy. 19

  For more on the DLC see Kenneth Baer , Reinventing Democrats: The Politics of Liberalism from Reagan to Clinton. (Lawrence: University Press of Kansas, 2000) 20 Janet Hook, "Officials Seek Moderation in Party"s Image", CQ Weekly Report, 9 March 1985 21 DLC, New Orleans Declaration, 1 March 1990, http://www.dlc. org/ndol_ci.cfm?kaid=86&subid=194&contentid=878 (accessed 21 June 2010) 22 Bill Clinton, My Life (New York: Knopf, 2004) 319 23 Ibid. /www.presidentialrhetoric.com/historicspeeches/clinton/first_inaugural.html (accessed 29 June 2010)

	24	DLC"s Cleveland Convention, 6 May 1991, http://www.dlc.org/
	ndol_ci.cfm?kaid=86&subid=194&contentid=3166 (accessed 21 June 2010)
	25	Little	Rock,	3	October	1991,	http://www.4president.org
	/speeches/billclinton1992announcement.htm (accessed 29 June 2010)
	26	First	Inaugural	Address,	20	January	1993,
	http:/


Sofiane Khadraoui
email: sofiane.khadraoui@femto-st.fr

Member, IEEE Micky Rakotondrabe

Member, IEEE Philippe Lutz
email: plutz@femto-st.fr

PID-Structured Controller Design for Interval Systems: Application to Piezoelectric Microactuators

This paper addresses the modeling and robust PID controller design for piezoelectric microsystems. Piezoelectric cantilevers, used as microactuators in micromanipulation and microassembly contexts, are particularly concerned. Due to their small sizes, these systems are very sensitive to environment (temperature, vibration, etc.) and to usury during functioning. Their behaviors often change because of the parameters variation. For that, linear modeling with uncertainty has been used to account the uncertainties, then classical H∞ and µ-synthesis approaches were applied. These techniques were efficiency but they were of high order which is not suitable for embedded microsystems. Furthermore, when the number of uncertain parameters increases, the modeling of microsystems became delicate and difficult.

In this paper, we propose to model the uncertain parameters by bounding them with intervals. Afterwards, we propose to design a robust PID controller by using interval arithmetic and related tools in order to ensure the specified performances. In addition to thesimplicity of the uncertainties modeling, the derived controller is of low order. The controller synthesis is formulated as a set-inversion problem. An application to the control of piezoelectric microactuators proves the efficiency of the proposed method.

I. Introduction

The development of high performances miniaturized systems -which is of great interest for various applications -presents technological and scientifical challenges. These miniaturized systems called microsystems are very sensitive to the environmental conditions (temperature, vibrations, etc.) and to the interaction and contact with surrounding systems (objects, other microsystems). The models of their behaviors are therefore subject to change and uncertainties that should be taken into acocunt during the controller design. Among these microsystems, piezoelectric microgrippers are used to manipulate and assembly or characterize artificial micro-objects and biological cells with sizes ranging between 10µm to 1mm. A microgripper is composed of two piezoelectric cantilevered microactuators (piezocantilevers) [START_REF] Haddab | A microgripper using smart piezoelectric actuators[END_REF] [START_REF] Agnus | A smart microrobot on chip: design, identification and modeling[END_REF]. In fact, on the one hand piezoelectric materials are widely used because of their high resolution, large bandwidth and high force density [START_REF] Devasia | A survey of control issues in nanopositioning[END_REF], but on the other hand their high sensitivity makes the developed piezoelectric microactuators lose the accuracy.

To achieve the required performances in micromanipulation and microassembly tasks, linear modeling with ∆matrix uncertainties have been used and classical robust control laws (H 2 , H ∞ and µ-synthesis) were applied for each piezocantilever [START_REF] Sebastian | Achieving Subnanometer Precision in a MEMS-Based Storage Device During Self-Servo Write Process[END_REF]- [START_REF] Iorga | A Review of H∞ Robust Control of Piezoelectric Smart Structures[END_REF]. The efficiency of these advanced methods was proved in several applications (SISO and MIMO microsystems). However their major disadvantage is the derivation of high-order controllers which are time consuming and which limit their embedding possibilities, as required for real packaged microsystems.

A possible alternative to classical robust control laws is the use of interval analysis which is a suitable tool dealing with parametric uncertain models. These parametric uncertain models have known structures with unknown parameters, but their values are assumed constant within given intervals. Their possible values are usually bounded by intervals. The principle of the controller design is therefore based on the combination of the interval arithmetic with a linear control theory. In addition to its principle simplicity to model the uncertain parameters, the main advantage is the derivation of low-order controllers.

The first idea on interval arithmetic has been proposed in 1924 by Burkill and 1931 by Young, then later in 1966 with R.E. Moore's works [START_REF] Moore | Interval Analysis[END_REF]. Since, several applications on interval analysis have been raised. Since then, several applications appeared on the subject. Some of them relates to guaranteed estimation, robust stability and controllers design. The works in [START_REF] Jaulin | Set inversion via interval analysis for nonlinear bounded-error estimation[END_REF] [10] deal with guaranteed parameters estimation based on the SIVIA algorithm (Set Inversion Via Interval Analysis). In [START_REF] Jaulin | Applied Interval Analysis[END_REF] [12] [START_REF] Kharitonov | Asymptotic stability of an equilibrium position of a family of systems of linear differential equations[END_REF], the stability analysis of the closed-loop with a given controller was proposed using the Routh's criteria and/or the Kharitonov's theorem. Concerning the design of controller, Chen and Wang [START_REF] Chen | Robust controller design for interval process systems[END_REF] proposed a systematic computational technique to design of robust stabilizing controller for interval systems basing on the transformation of the robust controller design problem into an equivalently non-linearly constrained optimization problem. In [START_REF] Smaginaa | Using interval arithmetic for robust state feedback design[END_REF] an approach of state feedback control combined with the intervals for the parameters model was proposed to synthesize a controller that ensures the stability. In [START_REF] Bondia | Guaranteed tuning of PID controllers for parametric uncertain systems[END_REF], a P ID controller that ensures robust performances was proposed by using a setinversion problem. The method suffers from the computational complexity, particularly when using high-order interval systems. Chen and Wang [START_REF] Chen | A two-degrees-of-freedom design methodology for interval process systems[END_REF] also proposed a robust method to control interval systems. In their work, two controllers were necessary: a robust controller stabilizing the feedback first, and then a pre-filter must be computed to ensure the wanted performances. Li et al. [START_REF] Li | Interval Model Control of Consumable Double-Electrode Gas Metal Arc Welding Process[END_REF] proposed a control algorithm prediction-based interval model that was efficiently applied to a welding process. In our previous work [START_REF] Khadraoui | Robust control for a class of interval model: application to the force control of piezoelectric cantilevers[END_REF], a robust controller for interval systems with zero-order numerator was proposed. Its main advantage relative to the other existing works is that the order of the system's denominator is not limited and the derived controller has a low-order. This later work also proved that interval analysis and related controller design could be very promising for modeling and control microsystems.

This paper deals with the interval modeling and robust control design for piezoelectric microactuators. While we bound the uncertain parameters with intervals, a PID structure controller is proposed. Contrary to the previous work [START_REF] Khadraoui | Robust control for a class of interval model: application to the force control of piezoelectric cantilevers[END_REF], the proposed approach is extended to general transfer function, i.e. no limitation on the degrees of both numerator and denominator. Despite the limitation to SISO systems and the account of only parametric uncertainties, the proposed approach proposes a low-order cotroller (PID) that is suitable for real-time embedded microsystems.

The paper is organized as follows. In section-II, preliminaries related to interval analysis and systems are provided. Section-III is dedicated to the tuning of the robust P ID controller. In section-IV is concerned with the application of the proposed method to control piezoelectric microactuators. Finally, the experimental results end the paper.

II. Mathematical preliminaries A. Basic Terms and Concepts on intervals

More details on the preliminaries given here can be found in [START_REF] Moore | Interval Analysis[END_REF] or [START_REF] Jaulin | Applied Interval Analysis[END_REF].

A closed interval number denoted by [x] corresponds to a range of real values, it can be represented by the left and right endpoints x -and x + respectively:

[x] = [x -, x + ] = x ∈ R\x -≤ x ≤ x + (1)
An ordinary real number x can be represented by a degenerate interval [x, x] where x -= x + .

A vector (box) of n interval parameters is denoted by:

[x] = [[x 1 ], [x 2 ], ..., [x n ]] (2)
The width of an interval [x] is given by:

w([x]) = x + -x - (3)
The midpoint of [x] is given by:

mid([x]) = x + + x - 2 (4)
The radius of [x] is defined by:

rad([x]) = x + -x - 2 (5)
1) Operations on intervals: The elementary mathematical operations are also extended to intervals, the operation result between two intervals is an interval containing all the operations results of all pairs of numbers in the two intervals. So, if we have two intervals [x] = [x -, x + ] and [y] = [y -, y + ] and a law ⊗ ∈ {+, -, * , /} , we can write:

[x] ⊗ [y] = {x ⊗ y |x ∈ [x], y ∈ [y] } (6)
B. Interval system Definition 2.1: A linear system under parametric uncertainties is often modeled by interval system. A

SISO interval system [G](s, [a], [b]) is a family of systems: [G](s, [a], [b]) =        m j=0 b j s j n i=0 a i s i b j ∈ [b - j , b + j ], a i ∈ [a - i , a + i ]        (7) such as: [b] = [[b 0 ], ..., [b m ]] and [a] = [[a 0 ], ..., [a n ]
] are two boxes (interval vectors) and s the Laplace variable.

The following lemma which is a result for interval functions is due to [START_REF] Moore | Interval Analysis[END_REF].

Lemma 2.1:

(Containment Theorem) Given [F ]([x]) a rational expression in the interval variables [x] = [[x 1 ], ..., [x n ]]. Let [y] = [[y 1 ], ..., [y n ]] be a box of interval variables, if [y] ⊆ [x], i.e. [y 1 ] ⊆ [x 1 ], ..., [y n ] ⊆ [x n ], then [F ]([y]) ⊆ [F ]([x]). Proof: see [8]
The following theorem is a straightforward consequence of Lemma 2.1.

Theorem 2.1: Given two SISO, linear and stable inter-

val transfers [G 1 ](s, [α], [β]) and [G 2 ](s, [γ], [λ]
) defined as in Definition 2.1. The two systems have the same structure (same degree for their numerators, idem for their denominators).

If [α] ⊆ [γ] and [β] ⊆ [λ], then [G 1 ](s, [α], [β]) ⊆ [G 2 ](s, [γ], [λ]). Proof: Noting that s = [s, s] = [s] and [s] ⊆ [s], and applying Lemma 2.1 with [F ]([x]) = [G 2 ](s, [γ], [λ]) and [F ]([y]) = [G 1 ](s, [α], [β]), where [x] = [[s], [γ], [λ]] and [y] = [[s], [α], [β]], we obtain: [y] ⊆ [x] ⇒ [F ]([y]) ⊆ [F ]([x])
which leads for any s to:

[α] ⊆ [γ] [β] ⊆ [λ] ⇒ [G 1 ](s, [α], [β]) ⊆ [G 2 ](s, [γ], [λ])

C. Performances of interval systems

The following theorem [START_REF] Rakotondrabe | Performances inclusion for stable interval systems[END_REF] defines the inclusion of the time/frequency domains performances of interval systems. Proof: see [START_REF] Rakotondrabe | Performances inclusion for stable interval systems[END_REF] III. Computation of the controller This section aims to design P ID controllers ensuring performances for a parametric uncertain system. The interval arithmetic and related tools are used for that.

A. Problem statement

Consider the closed-loop with an interval system [G](s, [a], [b]) as shown on (Fig. 1). The controller must ensure some given performances for the closed-loop whatever the parameters a i and b j ranging in [a i ] and [b j ] respectively. [H cl ](s, [p], [q]) denotes the closed-loop transfer.

+ - c y y c y y PID [G](s,[a],[b]) [H ](s,[p],[q]) U ε cl Fig. 1. Closed-loop transfer H cl .
In the sequel, [G](s, [a], [b]) will be written as:

[G](s, [a], [b]) = m j=0 [b j ]s j n i=0 [a i ]s i (8) with [a] = [[a 0 ], ..., [a n ]], [b] = [[b 0 ], ..., [b m ]] and m ≤ n.

B. Computation of the closed-loop model

Let us define a fixed-order controller P ID with adjustable parameters

[θ] = [[K p ], [K i ], [K d ]] as follows: [C](s, [θ]) = [K p ] + [K d ]s + [K i ] 1 s (9)
The closed-loop model can be computed using the interval model [START_REF] Moore | Interval Analysis[END_REF] and the controller (9) as follows:

[H cl ](s, [a], [b], [θ]) = 1 1 [C](s, [θ])[G](s, [a], [b]) + 1 (10) After replacing [G](s, [a], [b]) and [C](s, [θ]), we get: [H cl ](s) = m+2 j=0 [α j ]s j n i=0 [a i ]s i+1 + m+2 j=0 [α j ]s j (11)
Such as the coefficients of the box [α j ] for j = 0, ..., m+ 2 are dependent and are function of the boxes [b] and [θ].

After developing [START_REF] Jaulin | Applied Interval Analysis[END_REF] and factorizing the last coefficient of the numerator (or the denominator), we obtain:

[H cl ](s, [p], [q]) = 1 + e j=1 [q j ]s j r i=0 [p i ]s i

C. Feasible controller parameters

The objective consists to compute the set Θ of controller parameters [START_REF] Jaulin | Set inversion via interval analysis for nonlinear bounded-error estimation[END_REF] for which the set of all possible closed-loop behaviors ( 12) is included inside the set of all wanted behaviors defined by an interval reference model [H]:

Θ = {θ ∈ [θ] |[H cl ](s, [p], [q]) ⊆ [H](s) } (13)
The condition [H cl ](s, [p], [q]) ⊆ [H](s) can be checked by applying the parameter by parameter inclusion as given in Theorem 2.1. For that, the interval reference model [H] must have the same structure than [H cl ] (12). Therefore, we use as reference model: Based on ( 12), ( 14) and Theorem 2.2, the problem (13) can be reduced to the problem of finding Θ such as:

[H](s, [w], [x]) = 1 + e j=1 [x j ]s j r i=0 [w i ]s i
Θ = θ ∈ [θ] [q j ] ⊆ [x j ], for j = 1, ..., e [p i ] ⊆ [w i ], for i = 0, ..., r (15)
Remark 3.2: The number of unknown parameters in ( 9) is 3 while the number of inclusions is r + e + 1 (see [START_REF] Smaginaa | Using interval arithmetic for robust state feedback design[END_REF]). Since e = m + 2 and r = max(n + 1, m + 2), we can write r + e + 1 > 3. Therefore, there are more inclusions than unknown variables.

The problem of finding the set Θ so that (15) holds, is known as a set-inversion problem which can be solved using set inversion algorithms. An algorithm that can be used to solve such problem is the SIVIA algorithm [START_REF] Jaulin | Set inversion via interval analysis for nonlinear bounded-error estimation[END_REF].

IV. Control of piezocantilevers

This section is focused on the application of the proposed method to control piezoelectric microactuators (piezocantilevers) used in microgrippers. We particularly use unimorph piezocantilevers due to their ease of fabrication relative to multimorph ones. A unimorph piezocantilever is made up of one piezoelectric layer (often Lead Zirconate Titanate: PZT) and one passive layer (Copper). Indeed, the piezoelectric layer is used to actuate or produce energy while the non-piezoelectric layer (passive) is used to add stiffness as well as make the beam more durable. When a voltage U is applied to the piezoelectric layer, the cantilever expands/contracts which causes a global deflection δ (Fig. 2). Besides, a force F applied at the tip of piezocantilever may also cause a charge between the electrodes of the piezoelectric layer. In this situation, energy can be produced from the electrodes.

Piezocantilevers can be modeled by a transfer function with varying parameters. Unfortunately, such models with interval parameters are difficult to obtain. In addition, it is known that small differences in dimensions (somes microns) of similar piezocantilevers due to the imprecision of the microfabrication process, generate non-negligible difference on their model parameters. So, instead of having one piezocantilever with varying parameters during the experiment, we use two (or more) similar piezocantilevers. Thereafter, the derivation of the interval model [G](s, [a], [b]) is based on the two models of the used piezocantilevers. This interval model is then used to design controller that ensures performances not only for the both piezocantilevers but also for a set of piezocantilevers.

B. Modeling and identification

The linear relation that relates the output deflection to the input voltage U applied to piezocantilevers [START_REF] Rakotondrabe | Quadrilateral modelling and robust control of a nonlinear piezoelectric cantilever[END_REF]:

δ = G(s)U ( 16
)
where, for us, the transfer functions G 1 (s) and G 2 (s) that model the two piezocantilevers must be identified.

For the identification, a step voltage U = 20V is applied to each piezocantilever. A second order was chosen for each model because of its sufficiency to account (the first) resonance and its simplicity (low order). Using the output error method and the matlab software, we obtain:

G 1 (s) = 8.0313×10 -8 s 2 +1.808×10 -4 s+1
9.794×10 -8 s 2 +5.24×10 -6 s+1.44 G 2 (s) = 7.042×10 -8 s 2 +1.809×10 -4 s+1 8.802×10 -8 s 2 +5.364×10 -6 s+1.291 [START_REF] Chen | A two-degrees-of-freedom design methodology for interval process systems[END_REF]

C. Derivation of the interval model

Let us rewrite each model G i (s) (i = 1, 2) as follows:

G i (s) = b 2i s 2 + b 1i s + b 0i a 2i s 2 + a 1i s + a 0i (18)
The interval model [G](s, [a], [b]) which represents a family of piezocantilever models is derived using the two point models G i (s). Considering each parameter of G 1 (s) and the corresponding parameter in G 2 (s) as an endpoint of the interval parameter in [G](s, [a], [b]), we have: In order to increase the stability margin of the closedloop system and to ensure that the interval model really contains the models [START_REF] Chen | A two-degrees-of-freedom design methodology for interval process systems[END_REF], we propose to expand the intervals of the model [START_REF] Khadraoui | Robust control for a class of interval model: application to the force control of piezoelectric cantilevers[END_REF]. However, it is noticed that when the interval width of the parameters in the model is too large, it is difficult to find a controller that respects both the stability and performances of the closed-loop. After some trials of controller design, we choose to expand the interval width of each parameter of (19) by 10%. 10% represents the maximal value allowed in this application. Finally, the extended parameters of the interval model which will be used for the controller design are given by:

[G](s, [a], [b]) = [b 2 ]s 2 + [b 1 ]s + [b 0 ] [a 2 ]s 2 + [a 1 ]s + [a 0 ] (19)

D. Definition of the specifications

Consider the following specifications for the closedloop. These specifications correspond to the requirement in micropositioning tasks for microassembly and micromanipulation that use piezoelectric microgrippers.

• no overshoot, • settling time 1ms ≤ tr 5% ≤ 30ms,

• static error |ε| ≤ 1%. As the desired system behavior is without overshoot, we can use two first-order systems to create an interval reference model. The used time constants for both systems are 1 3 ms and 10ms, while -0.01 and 0.01 for their statical errors.

E. Controller structure

In this application, we consider a PI (Proportional-Integral)structure because of its low-order (two parameters). It is a particular case of PID controllers where the derivative action K d is set to zero. Though PID structure can be easily used as presented in Section III.

[C](s,

[K p ], [K i ]) = [K p ]s + [K i ] s (21)
such as K p and K i are the parameters to be adjusted and representing the proportional and the integral gains repsectively.

F. The closed-loop and the reference models computation

The general model of the closed-loop is given by [START_REF] Jaulin | Interval constraint propagation with application to bounded-error estimation[END_REF]. In our case, the closed-loop transfer is obtained using the model with interval parameters in [START_REF] Rakotondrabe | Performances inclusion for stable interval systems[END_REF] and the controller (21):

[H cl ](s, [p], [q]) = [q 3 ]s 3 + [q 2 ]s 2 + [q 1 ]s + 1 [p 3 ]s 3 + [p 2 ]s 2 + [p 1 ]s + [p 0 ] (22)
such as:

[q 3 ] = [Kp][b2] [Ki] [q 2 ] = [Kp][b1] [Ki] + [b 2 ] [q 1 ] = [Kp] [Ki] + [b 1 ] [p 3 ] = [a2]+[Kp][b2] [Ki] [p 2 ] = [a1]+[Kp][b1] [Ki] + [b 2 ] [p 1 ] = [a0]+[Kp] [Ki] + [b 1 ] [p 0 ] = 1
The computation of the interval reference model is based on the required specifications and the structure of the closed-loop (22). As we said before, a first order interval model would be considered (see Section IV-D):

[H](s, [K], [τ ]) = [K] [τ ]s + 1 ( 23
)
where the parameters [K] and [τ ] define the static error and settling time respectively and are deduced from specifications as follows:

• [K] = 1 + ε = [0.99, 1.01], • [τ ] = [tr 5% ] 3 = [0.33ms, 10ms].
However, the application of parameter by parameter inclusion [START_REF] Smaginaa | Using interval arithmetic for robust state feedback design[END_REF] requires that the reference model has the same structure than the closed-loop (22): the same degrees for the numerator, also for the denominators. [H cl ](s, [p], [q]) has a degree of 3 for both numerator and denominator. Thus we add some poles and zeros far from the imaginary axis to (23)

[H](s, [K], [τ ]) = [K]( [τ ] 10 s + 1) 3 ([τ ]s + 1).( [τ ] 10 s + 1) 2 (24)
which can also be rewritten as follows:

[H](s, [w], [x]) = [x 3 ]s 3 + [x 2 ]s 2 + [x 1 ]s + 1 [w 3 ]s 3 + [w 2 ]s 2 + [w 1 ]s + [w 0 ] (25)
such as:

[x 3 ] = 0.001[τ ] 3 [x 2 ] = 0.03[τ ] 2 [x 1 ] = 0.3[τ ] [w 3 ] = 0.01[τ ] 3 [K] [w 2 ] = 0.21[τ ] 2 [K] [w 1 ] = 1.2[τ ] [K] [w 0 ] = 1 [K]

G. Acheivement of robust performances

The controller defined in (21) ensures the required specifications in Section IV-D if its parameters [K p ] and [K i ] meet the following inclusions:

[Kp][b2] [Ki] ⊆ 0.001[τ ] 3 [Kp][b1] [Ki] + [b 2 ] ⊆ 0.03[τ ] 2 [Kp] [Ki] + [b 1 ] ⊆ 0.3[τ ] [a2]+[Kp][b2] [Ki] ⊆ 0.01[τ ] 3 [K] [a1]+[Kp][b1] [Ki] + [b 2 ] ⊆ 0.21[τ ] 2 [K] [a0]+[Kp] [Ki] + [b 1 ] ⊆ 1.2[τ ] [K] 1 ⊆ 1 [K] (26)

H. Derivation of the controller

Let us now compute the controller parameters. The problem given in (26) is known as a set-inversion problem which can be solved using SIVIA algorithm [START_REF] Jaulin | Set inversion via interval analysis for nonlinear bounded-error estimation[END_REF] if an initial box is provided. We denote S c the set parameters of the controller that satisfy these conditions. After the application of the SIVIA algorithm implemented in the Matlab-Software, with an initial box

[K p0 ] × [K i0 ] = [0.1, 0.6] × [0.

I. Experimental results

For the implementation of the controller, (point) parameters K p and K i must be taken from the set solution S c . In this application, we test two point controllers:

C 1 (s) = 0.1s+200 s C 2 (s) = 0.2s+400 s (27)
These two controllers are implemented for the two piezocantilevers. Fig. 5 shows the experimental results when a step reference of 20µm is applied. In this figure, we also have the temporal envelope of the reference model [H](s, [K], [τ ]). We mean by the envelope of [H](s, [K], [τ ]) the step responses of two tranfer functions H 1 (s) and H 2 (s) such as: 1) H 1 (s) has the minimal constant time τ = 0.33ms and the maximal static gain K = 1.01, 2) and H 2 (s) has the maximal constant time τ = 10ms and the minimal static gain K = 0.99. As shown in Fig. 5, the controllers have played their roles and ensure the specifications. Indeed, experimental settling times are tr 1 = 17.7ms and tr 2 = 7ms with C 1 (s) and C 2 (s) respectively, and the static errors are neglected and belong to the required interval |ε| ≤ 1%. Step response of the interval reference model and experimental results on the two piezocantilevers using C 1 (s) and C 2 (s).

V. Conclusion

The main contribution of this paper was the interval modeling and robust PID controller design for piezoelectric microsystems. These microsystems are known to be sensitive to usury functioning and to environmental disturbances making their models uncertain during micromanipulation or microassembly tasks. We therefore introduced interval techniques to model the uncertain parameters and to compute robust control law. The main advantage to use intervals is the ease and natural way to bound the parametric uncertainties. The proposed controller design is advantageous for deriving low-order robust controllers which are necessary to develop real packaged microsystems. Finally, the experimental results demonstrated the efficiency of the proposed method.

Theorem 2 . 2 :

 22 Let two SISO, linear and stable interval transfers [G 1 ](s, [α], [β]) and [G 2 ](s, [γ], [λ]) with the same structure. If [α] ⊆ [γ] and [β] ⊆ [λ], then the time and the frequency domains responses of [G 1 ](s, [α], [β]) are bounded by those of [G 2 ](s, [γ], [λ]). These responses define the time and frequency performances respectively.

( 12 )

 12 where e = m + 2 and r = max(n + 1, m + 2) and where the boxes of interval parameters [q] and [p] are function of the boxes [a], [b] and of the controller parameters [θ].

( 14 )

 14 Remark 3.1: [w] and [x] are two interval boxes chosen by the user from the specifications.

Fig. 2 .Fig. 3 .

 23 Fig. 2. Piezocantilever principle.

  such as: [b 2 ] = [min(b 21 , b 22 ), max(b 21 , b 22 )] [b 1 ] = [min(b 11 , b 12 ), max(b 11 , b 12 )] [b 0 ] = [min(b 01 , b 02 ), max(b 01 , b 02 )] [a 2 ] = [min(a 21 , a 22 ), max(a 21 , a 22 )] [a 1 ] = [min(a 11 , a 12 ), max(a 11 , a 12 )] [a 0 ] = [min(a 01 , a 02 ), max(a 01 , a 02 )] After the numerical application, we obtain: [b 2 ] = [7.042, 8.0313] × 10 -8 [b 1 ] = [1.808, 1.809] × 10 -4 [b 0 ] = 1 [a 2 ] = [8.802, 9.794] × 10 -8 [a 1 ] = [5.24, 5.364] × 10 -6 [a 0 ] = [1.291, 1.44]

[b 2

 2 ] = [6.992, 8.08] × 10 -8 [b 1 ] = [1.807, 1.809] × 10 -4 [a 2 ] = [8.753, 9.844] × 10 -8 [a 1 ] = [5.234, 5.37] × 10 -6 [a 0 ] = [1.283, 1.448]

4 . 1 :

 41 [START_REF] Haddab | A microgripper using smart piezoelectric actuators[END_REF], 500], we obtain the subpaving given in Fig.4. The dark colored subpaving (S c ) corresponds to the set parameters [K p ] and [K i ] that define a family of controllers ensuring performances for the interval model.Remark Any choice of the parameters [K p ] and [K i ] in the dark colored subpaving S c (see Fig.4) satisfies the inclusions (26) and consequently ensures specifications in Section IV-D.Remark 4.2: If the set-inversion problem is not feasible, i.e. S c = ∅, the initial box of the parameters must be changed and/or one must modify the specifications.

Fig. 4 .

 4 Fig. 4. Set parameters [Kp] × [K i ] ensuring performances.

  Fig. 5. Step response of the interval reference model and experimental results on the two piezocantilevers using C 1 (s) and C 2 (s).