Member, IEEE Micky Rakotondrabe

Performances inclusion for stable interval systems

This paper presents the performances inclusion on time and frequency domains of SISO stable interval systems. We demonstrate that an interval transfer function included in another interval transfer function will have its performances also included in those of the second one. While the results may be intuitive, the paper provides an analytical demonstration by using interval arithmetic and related tools. These results are of great interest for robust performances analysis and for controller design in parametric uncertain systems.

I. Introduction

The first analysis of interval systems goes back to the works of Kharitonov [START_REF] Kharitonov | Asymptotic stability of an equilibrium of a family of systems of linear differential equations[END_REF] [START_REF] Kharitonov | The Routh-Hurwitz problem for families of polynomials and quasipolynomials[END_REF]. It states that the robust stability of a real (resp. complex) interval polynomial can be deduced from the Hurwitz stability of four (resp. eight) vertex polynomials. Since this primary work, extensions of the robust stability analysis have been emerging. For instance, Barlett et al. [START_REF] Bartlett | Root locations of an entire polytope of polynomials: it suffices to check the edges[END_REF] extend the Kharitonov's theorem to the edge theorem to state the stability of polytope of polynomials. Further, Wang et al. [START_REF] Wang | Edge theorem for multivariable systems[END_REF] present the multivariable edge theorem. In [START_REF] Jaulin | Applied interval analysis[END_REF], Jaulin et al. present the δ-stability condition while in [START_REF] Dahleh | On the robust Popov criterion for interval Lur's systems[END_REF] Dahleh et al. adapt the Popov criterion, both for interval systems. Finally, the extension of the Kharitonov theorem for nonlinear systems is presented by Chapellat et al. [START_REF] Chapellat | On robust stability of interval control systems[END_REF]. Many works also report some elementary tools that may be useful to complete the robust stability: the algorithm to solve a set inversion problem in interval functions [START_REF] Walter | Guaranteed characterization of stability domains via set inversion[END_REF], the H ∞ norm [START_REF] Wang | H∞ performance of interval systems[END_REF] and the envelop of the Nyquist plots [START_REF] Hollot | On the Nyquist envelope of an interval plant family[END_REF] all for interval systems.

More than the stability, a rigorous aim of the robust control design is to maintain the performances in presence of model uncertainty. Hence, the robust perfomances analysis for interval systems can be seen as the natural continuation of the robust stability analysis. The first works considering the robust performances for these systems date back to 1992 when Dahleh et al [START_REF] Dahleh | Robust stability and performance of interval plants[END_REF] synthesize a controller using thirty two point systems. Point systems mean systems whose parameters are point, not intervals. Much later, Okuyama and Takemori [START_REF] Okuyama | Robust performance evaluation for interval systems based on characteristic roots area[END_REF] provide a sufficient condition such that the roots of the characteristic polynomial are contained in a given circular area that can be linked to specified performances. In [START_REF] Bondia | A geometric approach to robust performance of parametric uncertain systems[END_REF],

FEMTO-st Institute, UMR CNRS-6174 / UFC / ENSMM / UTBM Automatic Control and Micro-Mechatronic Systems department (AS2M department) 25000 Besançon -France mrakoton@femto-st.fr

Bondia and Pico employ a geometric approach to give a condition for the robustness of a closed-loop transfer to satisfy the specified time domain performances. The performances analysis in the latter approach were limited to first and second order reference models. More recently [START_REF] Khadraoui | Robust control for a class of interval model: application to the force control of piezoelectric cantilevers[END_REF], robust temporal performances for n th -order transfer functions were addressed. However, the considered numerator was limited to 0-degree. In this paper, we study the robust performances of generalized interval transfer functions. The study is performed both for the time and the frequency domains by using interval arithmetic and related algebraic tools. We especially prove that two interval systems with inclusion relation also have inclusion relation on their (temporal and frequential) performances. For as much as we treat the performances, we assume that the analyzed interval systems are stable. The results are more general than the previous works [START_REF] Okuyama | Robust performance evaluation for interval systems based on characteristic roots area[END_REF][13] [START_REF] Khadraoui | Robust control for a class of interval model: application to the force control of piezoelectric cantilevers[END_REF] since generalized structure of interval systems are considered and both time and frequency domains are treated. These results are useful for an a posteriori robustness analysis, or for the design of robust controllers when having systems with parametric uncertainties. The paper is organized as follows. In section-II, we remind the real and complex interval arithmetic. We also introduce the notations used for interval systems. In section-III, we present the first fundamental result of the paper, which concerns the performances inclusion in the frequency domain. In section-IV, the second fundamental result is presented: the performances inclusion in the time domain. Finally, we present an illustrative example in section-V.

II. Preliminary A. Real interval arithmetic and functions

The real interval arithmetic and functions summarized here are based on [START_REF] Jaulin | Applied interval analysis[END_REF][15] [START_REF] Moore | Interval analysis[END_REF].

A real (closed) interval [x] is a connected, closed subset of R. It is characterized by a lower bound x -and an upper bound x + such as for all x ∈ [x], we have x -≤ x ≤ x + . We say that interval [x] = [x -, x + ] is degenerate iff x -= x + . By convention, a degenerate interval [a, a] is identified by the real (point) number a. In the sequel, the designation point is alternately used to signify degenerate interval. The set of real intervals is denoted by IR.

Given two intervals

[x] = [x -, x + ] and [y] = [y -, y + ], so: [x] + [y] = x -+ y -, x + + y + (1) [x] -[y] = x --y + , x + -y - (2) If [y] = [x], we have: [x] -[x] = [x --x + , x + -x -] = 0, except if x -= x + .
The multiplication and subdivision are:

[x] . [y] = min x -y -, x -y + , x + y -, x + y + , max x -y -, x -y + , x + y -, x + y + (3) and [x] / [y] = [x] . [1/y + , 1/y -] , 0 / ∈ [y] (4) An interval [x] is said to be included in an interval [y], i.e. [x] ⊂ [y], iff [x] ∩ [y] = [x]. We have [x] > [y] if x -> y + . The real interval [x] is positive if x -> 0.
The interval numbers [0, 0] = 0 and [1, 1] = 1 serve as additive and multiplicative identities, respectively. Interval arithmetic is associative and commutative with respect to addition and multiplication. The distributive law for interval does not hold in general. However, the following relation, called subdistributivity, holds:

[x] ([y] + [z]) ⊆ [x] [y] + [x] [z]. Furthermore, if [x]+[y] = [x]+[z],
[x] ⊆ [y] ⇔ a [x] ⊆ a [y] ( 5
)
where a is a real point number.

If

f is a function f : R → R, then its interval counterpart [f ] satisfies: [f ] ([x]) = [{f (x) : x ∈ [x]}] (6)
The interval function

[f ] is called inclusion function because f ([x]) ⊆ [f ] ([x]), for all [x] ∈ IR. An inclusion function [f ] is thin if for any degenerate interval [x] = x , [f ] (x) = f (x) . It is minimal if for any [x] , [f ] ([x]) is the smallest interval that contains f ([x]
). The minimal inclusion function for f is unique and denoted by

[f ] * ([x]
). An easy way to compute an inclusion function for f is to replace in the expression of f all x by [x] and all operations on points by their interval counterpart. Thus, one obtains the natural inclusion function.

B. Complex interval arithmetic: the rectangular form

Three forms exist to represent complex interval numbers: 1) the circular form [START_REF] Henrici | Circular arithmetic and the determination of polynomial zeros[END_REF][18] which uses a complex point number and a radius, 2) the polar form [START_REF] Candau | Complex interval arithmetic using polar form[END_REF] which is an extension of the polar form of complex point number, 3) and the rectangular form [START_REF] Alefeld | Introduction to interval computations[END_REF][20] [START_REF] Miodrag | Complex interval arithmetic and its applications[END_REF] whose the real and imaginary parts are interval. It is obvious that when transforming a Laplace transfer function into frequential transfer function (using s = jω), we obtain the rectangular form. Therefore, in the sequel we give a preliminary of the latter and afterwards we employ it.

A complex interval number [I] is characterized by an ordered pair of interval real number (

[A] , [B]), such as [I] = [A] + [B] j = [a -, a + ] + [b -, b + ] j. The negative of I is -[I] = -[A] -[B] j and its conjuguate is [I] * = [A] -[B] j. Let [I] = [A i ]+[B i ] j = a - i , a + i + b - i , b + i j and [J] = [A j ] + [B j ] j = a - j , a + j + b - j , b + j j two complex interval numbers, so: [I] + [J] = ([A i ] + [A j ]) + ([B i ] + [B j ]) j (7) [I] -[J] = ([A i ] -[A j ]) + ([B i ] -[B j ]) j (8)
[I]

• [J] = ([A i ] [A j ] -[B i ] [B j ]) + ([A i ] [B j ] + [B i ] [A j ]) j (9)
The following definitions are provided for complex interval numbers [START_REF] Boche | Complex interval arithmetic with some applications[END_REF].

[I] + [I] * = 2 • [A i ] + 0j (10)
[I]

• [I] * = [A i ] 2 + [B i ] 2 + 0j ( 11
) [I] / [J] = [A i ] [A j ] + [B i ] [B j ] [A j ] 2 + [B j ] 2 + [B i ] [A j ] -[A i ] [B j ] [A j ] 2 + [B j ] 2 j (12)
The division defined above yields a complex interval that is generally far too pessimist. Other definitions of division were therefore introduced, see [START_REF] Rockne | Complex interval arithmetic[END_REF] [START_REF] Lohner | Complex interval division with maximal accuracy[END_REF]. Because their properties, which are more interesting than the definition in this paper, are equivalent we will use the above definition.

If we have two complex intervals [I] and [J], so [I] ⊆ [J] means [A

i ] ⊆ [A j ] and [B i ] ⊆ [B j ].

C. Functions of (rectangular) complex interval

Let [I], [J], [K] and [L] be complex intervals such that

[I] ⊆ [K] and [J] ⊆ [L]. Lemma 2.2: [I] ⊗ [J] ⊆ [K] ⊗ [L] for ⊗ ∈ {+, -, •, /}. Proof: See [20]. Theorem 2.1 (Containment theorem): Let f (x 1 , x 2 , . . . , x n ) be a rational expression in the point variables x 1 , x 2 , . . . , x n . If [X 1 ] ⊆ [Y 1 ], [X 2 ] ⊆ [Y 2 ], . . ., [X n ] ⊆ [Y n ] are complex interval variables, then all inclusion functions [f ] counterpart of f verify [f ] ([X 1 ] , [X 2 ] , . . . , [X n ]) ⊆ [f ] ([Y 1 ] , [Y 2 ] , . . . , [Y n ]) Proof: See [20].
Remark 2.1: Theorem 2.1 holds when [X i ] and [Y i ] are complex intervals. Therefore, it also holds for real intervals.

Corollary 2.1: Let f (x 1 , x 2 , . . . , x n , t 1 , t 2 , . . . , t m ) be a rational expression of the point variables

x 1 , x 2 , . . . , x n , t 1 , t 2 , . . . , t m . If [X 1 ] ⊆ [Y 1 ], [X 2 ] ⊆ [Y 2 ], . . ., [X n ] ⊆ [Y n ], then [f ] ([X 1 ] , [X 2 ] , . . . , [X n ] , t 1 , t 2 , . . . , t m ) ⊆ [f ] ([Y 1 ] , [Y 2 ] , . . . , [Y n ] , t 1 , t 2 , . . . , t m ) Proof: Rewriting t i (i ∈ {1, . . . , m}) by [T i ] such as [T i ] = [t i , t i ] = t i is a degenerate real interval number and knowing that [T i ] ⊆ [T i ],
we apply the Theorem 2.1 and derive the results in Corollary 2.1.

Notations: for a compact notation, we will use:

[X] = [X 1 ] [X 2 ] . . . [X n ] and [Y ] = [Y 1 ] [Y 2 ] . . . [Y n ] . Therefore, [X] ⊆ [Y ]
means that the inclusion relation holds for each element.

Lemma 2.3:

If [X] ⊆ [Y ]
and k an independent variable, then

β k=α [f ] ([X] , k) ⊆ β k=α [f ] ([Y ] , k), for any α ≤ β and α, β ∈ Z. Proof: Corollary 2.1 expresses that [f ] ([X] , k) ⊆ [f ] ([Y ] , k). Since [f ] ([X] , k) ⊆ [f ] ([X] , k) and [f ] ([Y ] , k) ⊆ [f ] ([Y ] , k), we can apply Lemma 2.2 to prove that [f ] ([X] , α) + . . . + [f ] ([X] , β) ⊆ [f ] ([Y ] , α) + . . . + [f ] ([Y ] , β). Theorem 2.2: If [X] ⊆ [Y ]
, t an independent variable and f a Rieman integrable function of t, then:

t f to [f ] ([X] , t) dt ⊆ t f to [f ] ([Y ] , t) dt Proof: Replace k in Lemma 2.3 by t o + k t f -to N such as N = β -α + 1 >
0 and multiply the left and the right terms by the positive point number t f -to N , so the following inclusion still holds according to Lemma 2.2:

t f -to N N k=1 [f ] [X] , t o + k t f -to N ⊆ t f -to N N k=1 [f ] [Y ] , t o + k t f -to N
If the original function f is Riemann integrable along t, therefore the previous Riemann sum leads to the result of Theorem 2.2 when N → ∞. Theorem 2.3:

If [X] ⊆ [Y ], t i (i ∈ {1, . . . , m})
are independent variables and f a Rieman integrable function of t i , then:

.

. . [f ] ([X] , t 1 , t 2 , . . . , t m ) dt 1 dt 2 . . . dt m ⊆ . . . [f ] ([Y ] , t 1 , t 2 , . . . , t m ) dt 1 dt 2 . . .

dt m

Proof:

To prove Theorem 2.3, it suffices to use the Theorem 2.2, by noting that

[f X1 ] = [f ] ([X] , t 1 , t 2 , . . . , t m ) dt 1 ⊆ [f Y 1 ] = [f ] ([Y ] , t 1 , t 2 , . . . , t m ) dt 1 and repeating the demonstration until f Xm ⊆ f Y m , where [f Xi ] = [f Xi-1 ] dt i-1 and [f Y i ] = [f Y i-1 ] dt i-1
(i ∈ {1, . . . , m}). the inclusion argument of [I]. This argument corresponds to any interval containing atan [B] [A] and is real interval.

D. Modulus and argument of (rectangular) complex intervals

Let [I] = [A] + [B] j be a rectangle complex interval.

The minimal inslusion argument denoted by [ϕ]

* ([I]), which is unique, corresponds to the smallest interval that contains atan [B] [A]

Let [I] = [A i ] + [B i ] j and [J] = [A j ] + [B j ] j be two rectangle complex interval numbers . Lemma 2.4: If [I] ⊆ [J], then [ρ] ([I]) ⊆ [ρ] ([J]) Proof: Since [A i ] ⊆ [A j ] and [B i ] ⊆ [B j ], therefore [A i ] 2 +[B i ] 2 ⊆ [A j ] 2 +[B j ]
2 according to Remark 2.1. The root square function being increasing monotonic on R + , the following function property is held for any subset:

ρ ([I]) = [A i ] 2 + [B i ] 2 ⊆ ρ ([J]) = [A j ] 2 + [B j ] 2 .

As the two related inclusion functions verify [ρ] ([I]) ⊆

[ρ] ([J]), we demonstrate Lemma 2.4.

Lemma 2.5:

If [I] ⊆ [J], then [ϕ] ([I]) ⊆ [ϕ] ([J]) Proof: Since [A i ] ⊆ [A j ] and [B i ] ⊆ [B j ], therefore [Bi] [Ai] ⊆ [Bj ]
[Aj ] according to Remark 2.1. The arctangent function being increasing monotonic on R, the following function property is held for any subset: ϕ ([I]) = atan [Bi] [Ai] ⊆ ϕ ([J]) = atan

E. Interval systems

All along the paper, we shall be interested by transfer functions, and the state-space, the differential or the Rosenbrock representations shall not be considered. Furthermore, stable interval transfer functions are considered. The reason is that we study robust performances, which implicitely implies the stability.

VI. Conclusion

We analyzed the performances inclusion of interval systems. We have demonstrated that when interval systems are included each other, there is also an inclusion relation between their performances both in the frequency and in the time domains. The results can be used for stability and performances robustness analysis, or for the design of controller dedicated to parametric uncertain systems.

  the cancellation law for addition holds, and [y] = [z]. The same property holds for multiplication: if [x] [y] = [x] [z] and 0 / ∈ [x], thus [y] = [z]. We have the following property:

Lemma 2 . 1 :

 21 Let a be a real point number. If [I] ⊆ [J], then a [I] ⊆ a [J]. Proof: Since a [A i ] ⊆ a [A j ] and a [B i ] ⊆ a [B j ] (according to Eq. 5), therefore a [I] ⊆ a [J].

2 2 . 2 . 2 :

 2222 and is positive real interval. The minimal inclusion modulus denoted by [ρ]* ([I]), which is unique, corresponds to the smallest interval that contains[A] 2 + [B]Definition We define by [ϕ] ([I]) = atan[B] [A]

  . Once again, the related inclusion functions verify [ϕ] ([I]) ⊆ [ϕ] ([J]), and therefore Lemma 2.5 is proven.

Definition 2 . 3 :

 23 We define by interval system denoted by [G] (s), s being the Laplace variable, a system whose the parameters are intervals:[G] (s) = [bm]s m +...+[b1]s 1 +[b0] [an]s n +...+[a1]s 1 +[a0] = m P l=0 [b l ]s l n P k=0 [a k ]s kThe parameters [a k ] and [b l ] are considered to be constant real interval in order to assume linear time invariant (LTI) systems. The notations [G] (s) shall be used if the intervals [a k ] and [b l ] are known. Instead, the notation [G] ([a k ] , [b l ] , s) is used when they are defined by bounds. The system [G 1 ] represents a closedloop system which includes a plant (whose parameters are uncertain and bounded by intervals) and a designed controller. Therefore, it is possible to analyze a posteriori if the controller inside the closed-loop [G 1 ] ensures the performances. According to Lemma 2.6, we have [G 1 ] (s) ⊆ [G 2 ] (s). So, Theorem 3.3 and Theorem 4.1 predict that the controller will efficiently ensure the (frequential and temporal) performances of the closed-loop [G 1 ]. If we plot the bode diagram of the two systems, we obtain the Fig. 1. As we can see in the figure, we have [ρ] ([G 1 ] (jω)) ⊆ [ρ] ([G 2 ] (jω)) and [ϕ] ([G 1 ] (jω)) ⊆ [ϕ] ([G 2 ] (jω)).

Fig. 1 .

 1 Fig. 1. Bode diagram of [G 1 ] (s) and [G 2 ] (s).

Fig. 2

 2 Fig. 2 pictures the step response of the two systems. Once again as predicted by the theory, the step response of [G 1 ] (s) is bounded by the one of [G 2 ] (s).

Fig. 2 .

 2 Fig. 2. Step response of [G 1 ] (s) and [G 2 ] (s).

Acknowledgment

This work is supported by the Conseil Général du Doubs, France.

unknown and to be sought for.

The notion of inclusion of systems should be defined. Consider two interval systems having the same polynomials degrees m and n:

III. Frequential performances inclusion

Performances of systems, of loop transfers and of closed-loop can be defined in the frequency domain. For instance, in the H ∞ control design techniques, upper bounds of the modulus are used to define the maximal settling times, statical errors and overshoots. In this section, we give the inclusion relation in the frequency domain (modulus and argument) of interval systems. The results are fundamental for the robust analysis and control design.

A. Inclusion theorem for the modulus (magnitude or gain)

First, compute the frequential transfer of an interval system. For that, we consider the interval system [G] (s) as defined in Definition 2.3. The frequential transfer is obtained by using s = jω:

which can be rewritten:

such as the real and imaginary parts of the numerator are:

and

The real part [A den ] and imaginary part [B den ] of the denominator have the same structure than Eq. 16 and Eq. 17 If

Proof: We apply the real interval property as in Eq. 5 to

Applying Remark 2.1 to the previous inclusions, we have:

where [A inum ] and [B inum ] are the real and imaginary parts respectively of the numerator of system i (i ∈ {1, 2}), and [A iden ] and [B iden ] their counterparts in the denominator. They are defined by Eq. 16 and Eq. 17.

As a result, we have

Finally, using Lemma 2.2 we have:

Theorem 3.1: Let [G 1 ] (s) and [G 2 ] (s) two interval systems as defined by Eq. 13.

B. Inclusion theorem for the argument (phase)

Afterwards, let us decompose [G] (s) into the multiplication of many first orders systems. Assuming that it is always possible to find imaginary interval roots for any given interval polynomial, it is possible to find imaginary zeros and poles for [G] (s). The system defined in Definition 2.3 can be therefore rewritten as follows:

where [P k ] and [Z l ] are the poles and zeros respectively. They are complex interval numbers.

Finally, the frequential transfer corresponding to Eq. 18 is:

Lemma 3.2: Consider the two interval systems defined by Eq. [START_REF] Bondia | A geometric approach to robust performance of parametric uncertain systems[END_REF]. if

C. The frequential performances inclusion theorem

Bringing together Theorem 3.1 and Theorem 3.2, we have:

and [G 2 ] (s) two interval systems as defined by Eq. 13.

Theorem 3.3 constitutes the first main result of this paper. It states that for two systems such that their parameters are linked by the inclusion relation, their frequential performances will also be linked by the inlcusion relation. These frequential performances are given in the Bode, Nyquist or Black-Nichols diagram.

IV. Temporal performances inclusion In this section, we shall present the time domain counterpart of the previous frequency domain analysis.

Reconsider the interval system [G] (s) as defined in Definition 2.3. So, its impulse response denoted by [g] (t) is given by:

where if

. As e st is a real point number for any s and t, therefore: [G 1 ] (s)e st ⊆ [G 2 ] (s)e st accordingly to Eq. 5. Using Theorem 2.2, we obtain

However, since

Therefore: [g 1 ] (t) ⊆[g 2 ] (t). Theorem 4.1 constitutes the second main result of this paper. It states that for two systems such that their parameters are linked by the inclusion relation, their time domain performances will also be linked by the inlcusion relation. These time domain performances are often defined and/or given with the impulse and step responses.

V. Illustrative example

The system [G 2 ] may represent a reference model whose the time-rseponse, the overshot and the static error are


Lori Maguire

Justification for War: A Comparative Study of How George W. Bush and Tony Blair Presented the Iraq War to Their Respective Citizens

Now this paper may seem a bit out of place in a publication on electoral strategies. The Iraq War enjoyed wide support in the United States and, to a lesser extent, in the United Kingdom for only a short period of time. However, in spite of all the controversies related to the war, both George W. Bush in 2004 and Tony Blair in 2005 secured re-election. Blair was even able to call early elections. In both cases, they presented the threat of Iraqi dictator, Saddam Hussein, in its worst possible light, using exaggeration and insinuations. They did this in a number of, often highly dubious ways, but one of the most important was through their rhetoric. Obviously, each man employs a staff of speech writers and so muchif not mostof what they say has not been written by them, although they have, presumably approved it in advance. An analysis, however, can still yield interesting insights particularly of a speech given by the head of government, for these addresses represent how the government wants their policy to be viewed. As such, they are centrepieces in the bid for media attention and a great deal of care is lavished on them. They are designed to have an influence on public opinion and, indeed, it is important to evaluate their degree of success in that domain.

Both George W. Bush and Tony Blair had expressed interest in toppling Saddam, by military means if necessary, before 11 September 2001. At first glance, George W. Bush did not seem like an interventionist

on foreign policy where: -he talked of America's mission to be a champion of democracy and freedom around the world, not only to make it a better place but also to ensure the security of all peace-loving people. 1 At the time Meyer thought it was just platitudes but the words returned to haunt him during the Iraq crisis. In his memoirs, Meyer asserts that: -It was almost word-for-word identical to what Bush has repeatedly said since becoming President about America's vocation in the wider world.‖ 2 In early 2000, Condoleezza Rice wrote an article for Foreign Affairs in which she singled out three particularly danger nations: Iran, Iraq and North Korea. These three would later become the famous -Axis of evil‖ in Bush's 2002 State of the Union address. She also stated that: -America's pursuit of the material interest will create conditions that promote freedom, markets and peace… The United States has a special role in the world… American values are universal.‖ 3 Here we come across messianic ideas that will reappear later in Bush's speeches. Bush's closeness, from an early date, to certain neo-conservatives was also revealing about his likely preferences in foreign policy. The term neoconservative originally referred to a group of politicians and political thinkers who tended towards the Left on domestic issues but were strongly anti-communist. Over time it has come to refer to those who hold an aggressively moralist foreign policy and who favour unilateral action by the United States.

They sympathize with Woodrow Wilson's idealistic desire to spread American values in the worldespecially those related to democracy -but do not accept Wilson's espousal of international organizations. Many of them have long argued that the United States should imitate Israel and use pre-emptive strikes against potential enemiesamong whom Iraq rated very highly. Dick Cheney, the vice-president, Donald Rumsfeld, the former Secretary of Defense, and Paul Wolfowitz, the former under-secretary of defense, have all been associated 1 Christopher Meyer, D.C. Confidential, (London: Weidenfeld & Nicolson, 2005) with the neo-conservatives. Bob Woodward has stated that even before Bush's inauguration, in early January 2001, Vice-President-elect Cheney had approached the outgoing Secretary of Defense about Iraq and expressed the view that, in Cohen's briefing to the president-elect, -Topic A should be Iraq‖. 4 At the end of the month, the National Security Council asked both the State Department and the military to examine options for Iraq. 5 At the time, however, given Bush's weak mandate in the 2000 election, nothing could be done.

Tony Blair has frequently been presented as an almost passive figure in that decisionone who blindly followed the Americans into war. He has been described as Bush's poodle or his -yes‖ man. The reality, however, is considerably more complicated. In 1999, almost two years before Bush became president, Blair addressed the Chicago Economic Club and unveiled what the Public Broadcasting System (PBS), at least, called the Blair Doctrine.

Among the many subjects he discussed was the international situation and, in particular, on the war then taking place in Kosovo. In this discussion, Blair asserted that: This is a just war, based not on any territorial ambitions but on values. We cannot let the evil of ethnic cleansing stand. We must not rest until it is reversed. We have learned twice before in this century that appeasement does not work. If we let an evil dictator range unchallenged, we will have to spill infinitely more blood and treasure to stop him later. 6 Blair emphasizes here the moral dimension of the question and his text is full of emotionally charged words. In particular, -evil‖ appears twice in this short extract alone. Opposed to the -evil‖ of Milosevic stands the morality of the nations who fight against him: their cause is righteous. They seek nothing for themselves and their actions reflect their -values‖ Having established this Manichean division, Blair goes on to allude to the failure of earlier attempts at -appeasement‖another highly charged word that automatically brings to mind the 1930s and Neville Chamberlain's attempts to placate Hitler in order to avoid war. This analogy is increased by the use of the term -evil dictator‖. Although never specifically stated, Blair has summoned up the earlier conflict between Chamberlain and Churchill over how to deal with Hitlerand thus, the tragedy of the Second World Warto support his own position.

The crux of the speech came towards the end where Blair argued that:

No longer is our existence as states under threat. Now our actions are guided by a more subtle blend of mutual self-interest and moral purpose in defending the values we cherish. In the end values and interest merge. If we can establish and spread the values of liberty, the rule of law, human rights and an open society then that is in our national interests too. The spread of our values makes us safer. As John Kennedy put it: -Freedom is indivisible and when one man is enslaved who is free?‖

The word -values‖ appears four times in this short extract. Blair believesand this clearly reflects Blair's conviction and not just that of his speechwriterthat foreign policy must be based on morality. More than this, the use of the word -spread‖, which appears two times, shows that Blair feels that these values, British and American political ideals, must be disseminated throughout the world. There is thus a messianic element to Blair's philosophy.

Note also his argument that -self-interest‖ and -moral purpose‖ coincide in this case, for this reasoning will recur in the Iraq conflict. It is interesting to observe that along with Milosevic, Blair singles out Saddam Hussein in this speech, describing both dictators as -dangerous and ruthless men‖. The British Prime Minister did not suddenly develop a fixation on Saddam because of Bush but had long felt that something had to be done about him. Obviously, though, Blair could not consider such an operation on his own. Britain was constrained by her relative strength to wait for the United States and to play only a supporting role in any operation.

We, therefore, have a situation in which both the leaders of the United States and Britain wished to force Saddam out of power but neither could do so. Bush had no popular mandate from the 2000 election and Britain was certainly too weak to take action on its own. Furthermore, both were hampered by the lack of support for such a war in public opinion and in the legislative of their countries. Into this situation came the attacks of 11 September 2001 and the climate of fear which they aroused in both countries, but especially in America.

Adding to this apprehension was a series of still unexplained anthrax attacks in the United

States not long afterward. Almost immediately after the 9/11 attacks, Wolfowitz pushed for war against Saddam. 7 Nor was Wolfowitz alone in these thoughts for both Rumsfeld, and Cheney seem to have worked for an invasion of Iraq. Public opinion at first prevented an attack on Iraq. Too much evidence existed implicating Al-Qaeda, then based in Afghanistan, in the 9/11 attacks and so the Bush administration was forced to deal with that threat first. But this did not change the Bush government's determination to eliminate Saddam. The influence of the neo-conservatives can be felt in Bush's pronouncements over the next few days in which he developed what has become known generally as the -Bush Doctrine‖. The first element of this philosophy emerged in the president's address to the nation on the very day of the attacks, when he said: -We will make no distinction between the terrorists who committed these acts and those who harbour them.‖ 8 He developed this further in his address to Congress a few days later, saying: -Every nation in every region, now has a decision to make.

Either you are with us, or you are with the terrorists. From this day forward, any nation that continues to harbour or support terrorism will be regarded by the United States as a hostile regime.‖ 9 There are clear Biblical echoes here which, along with statements in other parts of the text, reinforce the idea of a cosmic conflict between Good and Evil.

Strangely enough, although most people considered Osama bin Laden and Al-Qaeda responsible for the tragedy of September 11 th , they did not completely exonerate Saddam from any involvement. Polls taken in September 2001 show that a significant number of Americans also believed that Saddam had played a role. A September 15 th poll revealed that 41% felt this way. Another poll, taken two days later, confirmed that 27% considered Iraq -second most responsible‖. In a 22 September poll, 68% thought it was very important to remove Saddam Hussein from power in order to prevent future terrorist attacks. By November 2001, 74% of the public supported using American troops to remove Saddam. 10 The public was thus not totally out of touch with opinion in the White House.

Over the next year and a half, the administration followed a strategy of increasing and solidifying these perceptions in the American public. The government played on fears of a future attack by insisting that Iraq still possessed weapons of mass destruction and that Saddam Hussein greatly desired to attack the United States. Repeated alerts by Homeland Security increased public apprehension.

According to Washington Post reporter Bob Woodward, Bush had decided by 21 November 2001 that Iraq would be attacked.11 He only began to openly suggest this decision, though, in the famous, or infamous, depending on your point of view, State of the Union address of 29 January 2002. Here he announced that: Our nation will continue to be steadfast and patient and persistent in the pursuit of two great objectives. First, we will shut down terrorist camps, disrupt terrorist plans, and bring terrorists to justice. And, second, we must prevent the terrorists and regimes who seek chemical, biological or nuclear weapons from threatening the United States and the world. 12 A few paragraphs later he listed these regimes: North Korea (which gets one sentence); Iran (one sentence) and Iraq (an entire paragraph). In particular, he proclaimed that Iraq had -something to hide from the civilized world‖. Having identified these countries, he proceeded to expose the dangers that, in his opinion, they posed to humanity:

States like these, and their terrorist allies, constitute an axis of evil, arming to threaten the peace of the world. By seeking weapons of mass destruction, these regimes pose a grave and growing danger. They could provide these arms to terrorists, giving them the means to match their hatred. They could attack our allies or attempt to blackmail the United States. In any of these cases, the price of indifference would be catastrophic.

Observe here the use of highly charged, moral vocabulary. The term axis summons up the spectre, once again, of World War II and equates these regimeshowever strange it may seem given their respective might -with the three enemies of that conflict: Germany, Italy and Japan. We have already seen that -evil‖ is a favourite word of Blair when describing certain regimes and their leaders. The other phrases used in relation to the terrorists emphasize both their moral bankruptcy and the danger they pose: -arms‖ and -arming‖; -weapons of mass destruction‖; hatred; -attack‖ and -blackmail‖. They -threaten the peace of the world‖ and are a -grave and growing danger‖. Finally, to ram the point home, the word -catastrophic‖ appears to illustrate what will happen if something is not done. The whole passage of near Churchillian rhetoric creates a Manichean dichotomy that provides a seemingly unanswerable argument for pre-emptive action.

And so began the now famous saga of the celebrated weapons of mass destruction or WMD as they became known. This became one of the major arguments put forward to justify the invasion of Iraq, continuingin a somewhat different form -even after the Americans failed to find these weapons. Just before the attack began, Bush expressed this idea:

The safety of the American people depends on ending this direct and growing threat. Acting against the danger will also contribute greatly to the long-term safety and stability of our world. The current Iraqi regime has shown the power of tyranny to spread discord and violence in the Middle East. A liberated Iraq can show the power of freedom to transform that vital region, by bringing hope and progress into the lives of millions. America's interests in security, and America's belief in liberty, both lead in the same direction: to a free and peaceful Iraq. 13 Notice the appearance of a modified version of the Domino Theory. Systems of government -spread‖ throughout their region and, indeed -spread‖ is a word that recurs repeatedly in Bush's and Blair's speeches. An undemocratic Iraq acts as a conduit for tyranny throughout the region. But, according to Bush and the neoconservatives, a free Iraq will also cause democracy to grow in the area. The war will thus make America safer in two ways: it will get rid of the weapons of mass destruction and the threat posed by Saddam but it will also calm tensions in the Middle East, a major source of instability in the world.

Bush goes on to argue that the United States also has a duty to rid the world of these terrible threats for a higher reason:

We can't stop short [that is, with Afghanistan]. If we stop nowleaving terror camps intact and terror states uncheckedour sense of security would be false and temporary. History has called America and our allies to action, and it is both our responsibility and our privilege to fight freedom's fight. 14 Once again there is a reference to history, a continuing theme in Bush's speeches. He, or at least his speechwriters, seem highly conscious of history and insist that they and their war are part of the movement of history. Their interpretation is one of progress, as liberty spreads throughout the world.

In other speeches, Bush has been even clearer about the origins of that freedom, stating, for example: -The liberty we prize is not America's gift to the world, it is God's gift to humanity.‖15 But America has a special mission. In the same address, he said:

The American flag stands for more than our power and our interests. Our founders dedicated this country to the cause of human dignity, the rights of every person, and the possibilities of every life. This conviction leads us into the world to help the afflicted, and defend the peace, and confound the designs of evil men.

Notice the use of terms with religious overtones here: terms like -dedicated‖, -conviction‖ and, once again, -evil‖. The divine origins of America's mission are evident. At the same time, Bush rejects all assertions that there is a religious dimension to this conflict. He has condemned Al Qaeda for propagating this idea: Some call this evil Islamic radicalism; others, militant Jihadism; still others, Islamofascism. Whatever it's called, this ideology is very different from the religion of Islam. This form of radicalism exploits Islam to serve a violent, political vision: the establishment, by terrorism and subversion and insurgency, of a totalitarian empire that denies all political and religious freedom. These extremists distort the idea of jihad into a call for terrorist murder against Christians and Jews and Hindusand also against Muslims from other traditions, who they regard as heretics. 16 He denies the essential piety of Al Qaeda: it follows an ideology not a religious belief. The debate thus leaves the realm of religion for that of politics. The comparison with fascism and communism is never overtly mentioned but strongly suggested. Like those messianic and ultimately totalitarian ideologies of the twentieth century, radical Islam seeks complete dominance. They work both within and outside a state to find a way to seize control of it. In another speech, he described the conflict as -the clash of ideology‖a word taken up by Samuel Huntington in his book, The Clash of Civilizations. Bush would see it more as a clash between civilization and barbarism.

Both Bush and Blair had, before the invasion repeatedly stated that Saddam's regime possessed WMD and represented a real threat. Most notably, the Blair government issued two documents in September 2002 and in February 2003 that purported to show this. They alleged, among other things that the Iraqis had developed the capability to launch WMD attacks within 45 minutes. The government used these dossiers to convince MPs to back the, although much of the material was later discredited. Blair himself gave a major speech to Parliament on 18 March 2003 bid for support (which he received). He summed up his ideas on the question at the start: So why does it matter so much? Because the outcome of this issue will now determine more than the fate of the Iraqi regime and more than the future of the Iraqi people who have been brutalised by Saddam for so long, important though these issues are. It will determine the way in which Britain and the world confront the central security threat of the 21 st century, the development of the United Nations, the relationship between Europe and the United States, the relations within the European Union and the way in which the United States engages with the rest of the world. So it could hardly be more important. It will determine the pattern of international politics for the next generation. 17

Here there is no talk of the significance and the benefits of spreading democracy. The WMD, of course, pay a visit and their potential for harm is certainly not understated. However, 16 George W. Bush, Speech to the National Endowment for Democracy; 6 October 2005 17 Tony Blair, 18 March 2003, Parliamentary Debates, 6 th Series, vol. 401, col. 761 Blair's emphasis is on the practical effect this conflict will have on world relations, which he believes will be profound.

The arguments used here resemble those of Bush, notably in the evocation of the modified domino theory (-insecurity spreads like contagion‖)made stronger by the imagery of disease -and in the linking of terrorism and the so-called rogue nations. He is somewhat more sceptical than Bush about links between Iraq and terrorism, calling them -loose‖ but maintains that they are -hardening‖. The menace, Blair argues, cannot be dismissed: -The possibility of the two coming togetherof terrorist groups in possession of weapons of mass destruction or even of a so-called dirty radiological bombis now, in my judgment, a real and present danger to Britain and its national security.‖ Another thing is very noticeable here.

While Bush always presents his assertions as absolute truth, Blair is considerably more hesitant and suggests rather than states. For example, in the 18 March 2003 speech he states:

When the inspectors left in 1998, they left unaccounted for 10,000 litres of anthrax; a far-reaching VX nerve agent programme; up to 6,500 chemical munitions; at least 80 tonnes of mustard gas, and possibly more than 10 times that amount; unquantifiable amounts of sarin, botulinum toxin and a host of other biological poisons; and an entire Scud missile programme. We are asked now seriously to accept that in the last few years-contrary to all history, contrary to all intelligence-Saddam decided unilaterally to destroy those weapons. I say that such a claim is palpably absurd… Iraq continues to deny that it has any weapons of mass destruction, although no serious intelligence service anywhere in the world believes it.

On 7 March, the inspectors published a remarkable document. It is 173 pages long, and details all the unanswered questions about Iraq's weapons of mass destruction. It lists 29 different areas in which the inspectors have been unable to obtain information. 18 In a sense, Blair is saying that lack of evidence is evidencegiven the past history of Saddam. Unlike Bush, he never comes out and says that Saddam definitely possesses WMD but presents the case in such a way that it seems obvious.

After this, Blair begins a long and noteworthy analysis of America and of its relationship with Europe in order to attack rising anti-Americanism. He dismisses claims that America is an aggressive power: -I say to my hon. Friend that America did not attack Al-Qaeda terrorist group; the Al-Qaeda terrorist group attacked America.‖ This fact has had a profound impact on American psychology and, he believes, -should have changed the psychology of the world‖. This leads him into a discussion of the current rift within Europe and between parts of Europe and the United States. This discord hurt the effectiveness of the UN. According to Blair:

At the heart of that division is the concept of a world in which there are rival poles of power, with the US and its allies in one corner and France, Germany, Russia and their allies in the other… I believe such a vision to be misguided and profoundly dangerous for our world. I know why it arises. There is resentment of US predominance. There is fear of US unilateralism. People ask, -Do the US listen to us and our preoccupations?‖ And there is perhaps a lack of full understanding of US preoccupations after 11 September. I know all this. But the way to deal with it is not rivalry, but partnership. Partners are not servants, but neither are they rivals. What Europe should have said last September to the United States is this: with one voice it should have said; -We understand your strategic anxiety over terrorism and weapons of mass destruction and we will help you meet it… However, in return‖ -Europe should have said --we ask two things of you: that the US should indeed choose the UN path and you should recognise the fundamental overriding importance of restarting the middle east peace process, which we will hold you to.‖ 19 Blair criticizes those who want to return to the old idea of a balance of power. He sees no interest in having rival poles of power. The United States and Europe should, instead, be partners and work together for world peace and development. Blair does not, however, see things completely in black and white: he admits there are valid reasons for discomfort about American power. They are much stronger than anyone else and some people do proclaim that the United States should be ready to act alone. America needs to be pushed in the right direction but this must be done in a friendly way and with a full understanding of the traumatism the nation has suffered. In the end, Blair believes that all of Europe should have backed the war against Saddam Hussein but only in return for something. They should have made their support conditional on the United States taking the UN path and agreeing to restart 19 Ibid.,col. 770 the Israeli/Palestinian peace process. Tough love would have yielded greater results than simple hostility.

Of course, the failure to find WMD led both Bush and Blair to change the emphasis of their rhetoric and stress some arguments over others and to introduce new reasons. One of the first themes to appear was that of sacrifice. By the summer of 2003 the word was making appearances in Bush's speeches and they became more frequent as the year went by. In November of that year he gave yet another statement of his belief in the advancement of liberty in the world but this time it had a price:

The progress of liberty is a powerful trend. Yet we also know that liberty, if not defended, can be lost. The success of freedom is not determined by some dialectic of history. By definition, the success of freedom rests upon the choices and the courage of free peoples, and upon their willingness to sacrifice. In the trenches of World War I, through a two-front war in the 1940s, the difficult battles of Korea and Vietnam, and in missions of rescue and liberation on nearly every continent, Americans have amply displayed our willingness to sacrifice for liberty. 20 This, of course, had been called forth by the increasing toll of dead and wounded. In the absence of WMD the conflict risked being viewed as useless, or even harmful to America and the Middle East. With an election looming the following year, Bush felt a strong need to shore up support for his position. Among other things, he argued that the turmoil Iraq was experiencing was a normal stage in the creation of a democracy. He used the history of the United States to illustrate this theme:

The eight years from the end of the Revolutionary War to the election of a constitutional government, were a time of disorder and upheaval. There were uprisings, with mobs attacking courthouses and government buildings. There was a planned military coup that was defused only by the personal intervention of General Washington. In 1783, Congress was chased from this city [Philadelphia] by angry veterans demanding back pay, and they stayed on the run for six months… No nation in history has made the transition to a free society without facing challenges, setbacks and false starts. 21 Bush suggests that the United States experienced all this upheaval and became a stable democracy eventually and therefore that Iraq will do the same thing. As time passed, Bush came under more and more pressure to prove that he had a coherent plan to end the conflict. He began to repeat in speech after speech that he had a -coherent plan‖ or a -coherent strategy‖ or sometimes, a -clear strategy‖. He also outlined a vague, generally three point strategy to win the war and repeatedly used the term -victory‖ in his speeches. To some extent it worked: he did win the 2004 election. Blair also won re-election in spite of the Iraq War and in spite of the fact that he came under intense criticism over the Kelly affair. He has generally avoided the theme of sacrifice and there are few new themes in his speeches.

What was the impact of Iraq on their electoral strategies? It seems likely that Bush's precipitation in invading Iraq was linked to the 2004 campaign, at least in part. In particular, he insisted on a March 2003 date. Any war had to end before the hot summer months and, so, if Bush had waited, he would have faced an autumn offensiveonly one year before the election. The war would then have had an even greater impact on the campaign. During the initial period of the war from 19 March to the official end of hostilities on 19 May, support stayed high among the US public, averaging 72%. But after that backing began to drop steadily. 22 From June 2003 to June 2004 (with the exception of the capture of Saddam) polls showed that presidential approval ratings continually declinednotably because of the growing toll of US deaths. However, sovereignty was restored to Iraq and preparations for elections there started in June 2004five months before the American presidential election.

Polls showed an immediate recovery of optimism about Iraq among much of the US public.

Christopher Gelpi has noted that -from July to November 2004, there were 300 deaths‖ [of US servicemen] but Bush's approval rating remained unchanged. 23 It is difficult to believe Numerous reasons have been advanced for this change, most notably the desire to support the troops and the absence of media criticism of the government's position. 28 The July 2003 suicide of David Kelly began a difficult period for the government which faced, soon after, the Hutton inquiry into Kelly's death, which reported in January 2004 and then the Butler Report on intelligence which appeared in July of that year. At this point, foreign policy began to feature less and less often in Blair's speeches. Although the war was hugely unpopular, British losses were significantly lower than American ones and most people were satisfied with their living standards and the government's domestic policies. At the same time, the public does not appear to have found either the Conservatives or the Liberal Democrats very convincing. Although the public clearly did not believe Blair's earlier justifications for war, they did not appear to significantly penalize him in the May 2005 election in which Labour won an unprecedented third straight victory.

During the 2004 elections in the United States and those of 2005 in the United Kingdom, the public expressed great unhappiness over the Iraq War in both countries but, in spite of this, Bush and Blair won re-election. Numerous reasons can be advanced for this. In the United States, the return of sovereignty convinced many Americans that their nation's involvement was nearly over. Furthermore, fears of another attack remained strong in America after 9/11 and Bush certainly played on this. He was helped by a tape from Bin Laden which appeared just before the election. In both nations, media coverage tended to reinforce the government's arguments. Once gain in both nations losses were relatively small and there was no draft that would have affected the general population. Still, the 2004 election revealed a massive cleavage, notably geographic, in the United States: urban vs. rural and the coasts vs. the centre and the south. We may also attribute much of Blair's victory to the general contentment of the British public with regard to domestic issues. Interestingly enough, 28 See Lewis for more on this.

  20 George W. Bush, Speech to the National Endowment for Democracy, 6 November 2003 21 George W. Bush, Speech to the Philadelphia World AffairsCouncil, 12 December 2005

Bob Woodward, Plan of Attack (London: Simon & Schuster, 2004), 9

See Douglas Foyle, -Leading the Public to War? The Influence of American Public Opinion on the Bush Administration's Decision to Go to War in Iraq‖ in International Journal of Public Opinion Research, vol. 16, N°3, 269-294

Tony Blair, Speech to the Chicago Economic Club, 22 April 1999, at www.pbs.org/newshour/bb/international/jan-june99/blair_doctrine4-23.html

Woodward, Bush at War (London: Simon & Schuster, 2002)

8 George W.Bush, 11 September 2001. This and other speeches by Bush can be found at the White House website. 9 George W. Bush, Address to a Joint Session ofCongress, 20 September, 2001

See Gallup/CNNN/USA Today poll, 15 Sept 2001 ; Wirtheir Quorum Poll, 17 September 2001 ; Gallup/CNN/USA Today poll ; Gallup poll, November 2001. For more analysis of these polls see Foyle

Woodward, Plan of Attack, 1

George W. Bush, State of the Union Address, 29 January 2002

George W. Bush, 26 February 2003

George W. Bush, State of the Union Address 2002

George W. Bush, State of the Union address, 28 January 2003

Ibid., col. 762

See Richard Eichenberg, -Victory Has Many Friends: U.S. Public Opinion and the Use of Military Force,1981-2005‖ in International Security, vol 30, N° 1 (Summer 2005)

Christopher Gelpi « The Cost of War: How Many casualties will Americans Tolerate? » Foreign Affairs, Jan/Feb 2006, vol. 85, N° 1

that the timing of the restoration of Iraqi sovereignty had no link with the upcoming presidential election in the United States. British public opinion, as reflected in polls and demonstrations, showed great hostility to the war before the actual invasion of Iraq but changed quickly once action began. MORI surveys confirm that in mid-March 63 per cent opposed the war, while 26 per cent favoured it.

Two weeks later, after the invasion had started, the figure was 56 per cent in favour and 38 per cent opposed, although it has to be noted that there were significant differences in the phrasing of the questions between the two surveys. [START_REF] Justin | Television, Public Opinion and the War in Iraq : The Case of Britain[END_REF] Numerous other polls showed that people who had opposed the war beforehand, supported it once hostilities had begun. Polls before the war repeatedly showed that large majorities supported going to war if clear evidence of WMDs was found and if the Security Council voted for it. 25 This may explain Blair's insistence on blaming the French. A survey on 18 March 2003 found that 50 per cent of those surveyed considered Chirac responsible for the Security Council's failure to reach an agreement as opposed to 35 per cent who faulted Bush. [START_REF] Kellner | Iraq -the Public and the War: a Report on yougov Opinion Surveys before, during and after the conflict‖[END_REF] Certainly the pro-war camp, led by Blair, presented their arguments well and had the immense advantage of receiving little criticism. Except for Robin Cook, who attacked the invasion in his resignation speech, major political figures and the media in Britain generally accepted the government's arguments.

On 15 April 2003 The Guardian observed that: Support for the war among British voters has surged to a new record level of 63 per cent, according to results of this week's Guardian/ICM war tracker poll. The sevenpoint rise in support for military action since the fall of Baghdad confirms the war has been accompanied by one of the most drastic shifts in public opinion in recent British political history. 27