Alejandro Hernan Gonzalez

Antonio Ferramosca
email: ferramosca@santafe-conicet.gov.ar

Germán A A Bustos
email: gbustos@santafe-conicet.gov.ar

Jacinto Luis

L Marchetti

Mirko Fiacchini
email: mirko.fiacchini@gipsa-lab.fr

Darci Odloak
email: odloak@usp.br

A H González

J L Marchetti

Model predictive control suitable for closed-loop re-identification

Keywords: Model predictive control, closed-loop identification, target set control, persistent excitation

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Introduction

Model predictive control (MPC) is typically implemented as a lower stage of a hierarchical control structure. The upper level stages are devoted to compute, by means of a stationary optimization, the targets that the dynamic control stage (MPC) should reach to economically optimize the operation of the process. Since both, the dynamic and stationary optimizations are model based optimizations, a periodic updating of the model parameters are desired to reach meaningful optimums. In this context, a re-identification procedure should be developed in a closed-loop fashion, since the process cannot be stopped each time an update is needed. As it is known, the main problem of a closed-loop identification is that the dynamic control objectives are incompatible with the identification objectives ( [START_REF] Heirung | Towards dual MPC[END_REF]). In fact, to perform a suitable identification, a persistent excitation of the system modes is needed, while the controller takes this excitation as disturbance, and tries to reject this disturbance to stabilize the system.

From a general point of view, the closed-loop identification methods fall into the following main groups ( [START_REF] Soderstrom | System identification[END_REF]). The direct approach ignores the feedback law and identifies the open-loop system using measurements of the input and the output. The indirect approach identifies the closed-loop transfer function and determines the open-loop parameters subtracting the controller dynamic. To do that, the controller dynamics must be linear and known. The joint inputoutput approach takes the input and output jointly, as the output of a system produced by some extra input or set-point signal. Since the last two methods need the exact knowledge of a linear controller, they are not directly applicable for closed-loops under constrained MPC controllers.

Several strategies were developed to perform closed-loop re-identification under MPC controllers: [START_REF] Genceli | New approach to constrained predictive control with simultaneous model identification[END_REF] proposed a controller named Model Predictive Control and Identification (MPCI) where a persistent excitation condition is added by means of an additional constraint in the optimization problem. This strategy, which was explored later in [START_REF] Ballin | Controlador preditivo multivariavel com restricoes de excitacao para identificacao de processos em malha fechada[END_REF], turns the MPC optimization problem nonconvex, and so, most of the well-known properties of the MPC formulation cannot be established. [START_REF] Sotomayor | Closed-loop medel reidentification of processes under MPC whit zone control[END_REF] proposed a strategy that manipulates the steady state target optimization (in the hierarchical MPC control structure) in order to excite the system. In the context of data driven MPC formulations (i.e., MPC that are designed to perform predictions directly from collected data), the subspace identification method is exclusively used ( [START_REF] Overschee | Subspace Identification for linear systems: theory, implementation, applications[END_REF]). In [START_REF] Kadali | A data driven subspace approach to predictive controller design[END_REF], [START_REF] Wahab | Datadriven adaptive model-based predictive control with application in wastewater systems[END_REF] and [START_REF] Mardi | Data-driven Subspace-based Model Predictive Control[END_REF] several approaches are presented, where a closed-loop re-identification is needed to update the data for predictions. Though preliminary studies were made according to the trade-off between stability and excitation, no definitive results were presented.

In general, none of the reports cited in this section have shown results regarding the system stability of the MPC while the system is being re-identified. In this work, based on the concept of stability of an invariant set (as a generalization of stability of a point), a MPC controller with a extended domain of attraction is proposed, which assures stability at the same time that a persistent excitation can be generated to perform a closed-loop re-identification. Some preliminary results regarding the complete strategy presented in this work, were recently presented in [START_REF] González | Model predictive control suitable for closed-loop reidentification[END_REF].

Notation: Matrices I n ∈ R n×n and 0 n,m ∈ R n×m denote the identity matrix and the null matrix, respectively. A C-set is a convex and compact set that contains the origin. A proper C-set is a C-set that contains the origin as an interior point. Consider two sets U ⊆ R n and V ⊆ R n , containing the origin, and a real number λ.

The Minkowski sum U ⊕ V ⊆ R n is defined by U ⊕ V = {(u + v) : u ∈ U, v ∈ V}; the set (U\V) ⊆ R n is defined as U\V = {u : u ∈ U ∧ u /
∈ V}; and the set λU = {λu : u ∈ U} is a scaled set of U. |v| V is the distance from v to V. The boundary of a set U, is defined as ∂U.

Given a continuous function Ψ : R n → R , and γ ≥ 0, the level set

N [Ψ, γ] is defined by N [Ψ, γ] = {x : Ψ (x) ≤ γ}. I m:n denotes the nonnegative integers from m to n. Given x ∈ R n and y ∈ R n , x -y 2 M = (x -y) T M (x -y), with M ∈ R n×n .

Problem statement and preliminaries

Consider a system described by a linear time-invariant discrete-time model

x + = Ax + Bu, y = Cx
where x ∈ IR n is the system state, x + is the successor state, u ∈ IR m is the current control, and y ∈ IR p is the system output. The state, the control input and the output at discrete-time instant k are denoted as x(k), u(k) and y(k), respectively. The system is subject to hard constraints on state and input, (

x(k), u(k)) ∈ Z ∆ = (X × U) ⊂ R n+m for all k ≥ 0, where X ⊂ R n and U ⊂ R m .
Furthermore, the following assumption holds: Assumption 1. Matrix A has all its eigenvalues strictly inside the unit circle, the pair (A, B) is controllable and the state (corresponding to the true plant) is measured at each discrete-time instant. Furthermore, the set X is convex and closed, the set U is convex and compact and both contain the origin in their interior. For simplicity, AX ⊆ λX , with λ ∈ [0, 1).

Previous to the controller formulation, some necessary definitions helpful to generalize the concepts of equilibrium and invariance are introduced. To simplify the notation, we denote system x + = Ax + Bu, (x, u) ∈ Z as Nonautonomous system (N sys ) and system x + = Ax + Bκ(x), (x, κ(x)) ∈ Z, where κ(x) is a state feedback, as Controlled system (C sys ). Accordingly, for a given sequence of control inputs, u = {u(0), • • • , u(j -1)} and a given initial state x(0) = x, the solution of N sys will be denoted as:

x(j) = φ(j; x, u) = A j x(0) + j-1 i=0 A j-i-1 Bu(i), j ∈ I ≥1 .
Similarly, for a given initial state x(0) = x, the solution of C sys will be denoted as:

x(j) = φ κ (j; x) = A j x(0) + j-1 i=0 A j-i-1 Bκx(i), j ∈ I ≥1 . Definition 1. (Control Equilibrium Set -CES) A set Ω ⊆ X is a control equilibrium set for N sys , if for every point x ∈ Ω the condition x + = x holds for some u ∈ U.
The maximal CES, X ss , is given by X ss = (GBU)∩X , where G = (I n -A) -1 . In case of controlled systems, C sys , we simply say that a Control Equilibrium set Ω is an Equilibrium Set -(ES), with u = κ(x). The proper generalization of the concept of equilibrium point is not the concept of equilibrium set, as a mere aggregation of steady-state points, but the concept of invariant set (associated to an equilibrium set), in the sense that both, the equilibrium point and the invariant set are geometric entities such that, if the system reaches them, it remains in them indefinitely ( [START_REF] Blanchini | Set-Theoretic Method in Control[END_REF], [START_REF] Kerrigan | Robust Constraint Satisfaction: Invariant Sets and Predictive Control[END_REF], [START_REF] Kolmanovsky | Theory and conmputation of disturbance invariant sets for discrete-time linear systems[END_REF]):

Definition 2. (λ-Control Invariant Set -λ-CIS) A proper C-set Ω ⊆ X is λ-control invariant, with λ ∈ [0, 1], for N sys , if x ∈ Ω implies x + ∈ λΩ, for some u ∈ U.
Again, in case of controlled systems, C sys , a λ-Control Invariant set is simply a λ-Invariant Set -(λ-IS), with u = κ(x). Furthermore, if λ = 1, the sets are simply Invariant sets, and if λ ∈ [0, 1), the sets are known as Contractive sets. The concept of invariant set, as a generalization of an equilibrium point, makes possible the generalization of the concept of attractivity of an equilibrium point. Then, we can define the attractivity of an IS set as follows ( [START_REF] Rawlings | Model Predictive Control: Theory and Design[END_REF]): Definition 3. (Local attractivity of an IS set) The IS set Ω ⊂ X is locally attractive for C sys if for each x in a vicinity of Ω (that we call the domain of attraction), it follows that |φ κ (j; x)| Ω → 0, φ κ (j; x) ∈ X , κ(φ κ (j; x)) ∈ U as j → ∞.

Target invariant set for identification

The objective of this section is to propose a set (in the state space) that is invariant under the excitation procedure necessary to perform a suitable identification and, at the same time, can be used as attractive target set (generalized equilibrium) by an MPC controller. As known, to estimate a model from measured input and output data, each (controllable) mode of the system must be excited. To do that, the excitation input signal must contain enough variability. This property is generally indicated by the notion of persistence of excitation ( [START_REF] Ljung | System identification-theory for the user[END_REF]). The persistent excitation input sequences might be of several forms, going from a Pseudo-Random Binary Signal (PRBS) signal to a Filtered Pseudo Gaussian White Noise Signal. A recent formulation proposed a filtered Gaussian inputs signal specifically designed for MPC ( [START_REF] Larsson | On optimal input design in system identification for model predictive control[END_REF]). Independently of the form, the persistent excitation sequences have two main properties: they are bounded, belonging to a compact set smaller than U, and more subtle, they have a persistent-variability behavior. Regarding the first property, we define: Definition 4. (Excitation input set, EIS). An input proper C-set U t ⊂ U ⊂ R m , with enough size to excite the system will be denoted as excitation input set.

The set U t defines a class of sequences u = {u(0), • • • , u(T id -1)} -denoted by C U t -such that u(i) ∈ U t for i ∈ I 0:T id -1 , where T id is the length of the data necessary to perform a suitable identification. Now, regarding the second property of persistent excitation, we define the class of all feasible sequences that fulfill the persistent-variability condition (see [START_REF] Green | Persistence of excitation in linear systems[END_REF], [START_REF] Genceli | New approach to constrained predictive control with simultaneous model identification[END_REF], for the explicit expression of this condition). This class -that we call the persistent excitation class, C pe -is made by sequences of the form u pe = {u pe (0), • • • , u pe (T id -1)}, u pe (i) ∈ U t , i ∈ I 0:T id -1 . Notice that, following the latter classification, C pe is in C U t , but there is an entire class of input sequences that are in C U t and not in C pe . For instance, a constant sequence of T id elements, ū = {ū, • • • , ū}, with ū ∈ U t , belongs to C U t but does not belong to C pe .

The solution of N sys for a particular sequence u pe in C pe , and an initial state x(0) = x, will be denoted as x(j) = φ(j; x, u pe ), j ∈ I ≥1 (persistently excited system). Notice that once a particular input sequence is selected, then the system evolves autonomously, in the sense that no manipulated controls modify the system evolution. We can now define a set in the state space that is invariant to the effect of any persistent excitation sequence. Definition 5. (λ-Invariant set for Persistent Excitation) A proper C-set X pe ⊆ X is a λ-invariant set for persistent excitation, with λ ∈ [0, 1], for N sys , if x ∈ X pe implies φ(j; x, u pe ) ∈ λX pe , for j ∈ I 1:T id , for all u pe ∈ C pe .

The set X pe is well defined and it is what we need to formulate the MPC for re-identification. However, X pe is not easy to characterize, given that the Persistent Excitation condition is not related to a spatial condition, but to a dynamic requirement. Next, an approximation of X pe that is useful for MPC requirements is proposed.

A MPC target set that is an approximation of X pe

Instead of associate an invariant set, X pe , to the class C pe , we could relax the definition and associate an invariant set to the class C U t , which contains C pe . So, the following definition arises:

Definition 6. (λ-Invariant set for identification, λ-ISI) A proper C-set X t ⊆ X is a λ-invariant set for identification, with λ ∈ [0, 1], for N sys , if x ∈ X t implies φ(j; x, u) ∈ λX t , for j ∈ I 1:T id , for all u ∈ C U t .
A λ-ISI set with λ = 1 is simply an ISI set. Since by definition every input sequence of C pe is in C U t , it follows that X pe ⊆ X t , which means that X t is an outer "invariant" approximation of X pe . Clearly, an ISI set definition is a robust invariant set definition adapted to identification requirements. So, it is relatively easy to compute, and we can define the smallest ISI set, for a given system, which is given by the minimal ISI set ( [START_REF] Kolmanovsky | Theory and conmputation of disturbance invariant sets for discrete-time linear systems[END_REF]):

X t = ∞ i=0 (A i BU t ). Remark 1.
Although the set X t could be easy to compute (for polytopic constraint sets), it could be much bigger than X pe . For instance, since Persistent Excitation signals cannot remain constant at a boundary value of U t , then the boundary regions of the set X t (the points far from the origin) that corresponds to the equilibrium subspace X ss are not in X pe . The methodical reduction of the size of set X t (and so, of the conservatism of the formulation) to obtain accurate approaches of X pe could be done in several ways, using for instance the concepts of probabilistic invariant sets ( [START_REF] Kofman | Probabilistic set invariance and ultimate boundedness[END_REF]) and other deterministic concepts as the Pontryagin difference, etc.

Next, some properties regarding the ISI sets are stated.

Property 1. A set X t is a λ-ISI set for N sys if and only if AX t ⊕ BU t ⊆ λX t .
Property 2. Conisder a λ-ISI set X t , with λ ∈ [0, 1], for N sys , and a CES set, X t ss = GBU t ⊆ X , for the same system. Also consider a λ-IS set, Ω ⊂ X , with λ ∈ [0, 1), for the autonomous system x + = Ax, x ∈ X . Then:

(1) X t is also a λ-invariant set for persistent excitation, X pe (contractivity of X t under persistent excitation).

(2) For each x in a vicinity of X t , |φ(j; x, u pe )| X t → 0, φ(j; x, u pe ) ∈ X , as j → ∞ (attractivity of X t under persistent excitation).

(3) X t ss ⊆ X t and furthermore, if m < n, X t ss ⊂ X t (X t includes stationary and transient states).

(4) Provided that (X t ⊕ Ω) ⊂ X , there exists a real δ ∈ (λ, 1), such that if x ∈ (X t ⊕ Ω), then φ(j; x, u) ∈ δ(X t ⊕ Ω) for all j ∈ I ≥1 , for all u pe ∈ C pe (contractivity of X t ⊕ Ω under persistent excitation).

(5) If δX t ⊂ X , for a constant δ > 1, δX t is also an λ-ISI set (contractivity of δX t under persistent excitation).

The proofs of these properties are omitted for brevity.

Target set X t for model mismatch

It should be noticed that the ISI set X t , which will be a parameter of the proposed MPC optimization cost, depends on the model. Since the excitation scenario is precisely given when we suspect that the current model is no longer accurate, a discussion about the effect of the model mismatch on the computation of X t is needed. Two uncertainty descriptions -that are only particular descriptions -were selected to present robust ISI sets.

Parametric uncertainty

Consider a system given by

x + = A(w)x + B(w)u, y = C(w)x, w ∈ W ⊆ R, (1)
where A(w) and B(w) are affine functions of w, i.e., A(w) = A + w Ā, B(w) = B+w B with w belonging to the proper C-set W ⊂ R. Furthermore, assume that the Nominal model is given by

x + = A(w N = 0)x + B(w N = 0)u = Ax + Bu,
and the unknown Real model, is given by

x + = A(w R )x + B(w R )u, for some w R ∈ W.
In this context, there is a minimal ISI set X t (w) associated to each particular model. Now, consider the following Theorem:

Theorem 1. Consider a λ-ISI set X t (w = 0), λ ∈ [0, 1), for x + = Ax + Bu, (x, u) ∈ Z.
Then, there exists a non-empty set W ⊂ R for which the set X t (w = 0) is an ISI set for x + = A(w)x + B(w)u, (x, u) ∈ Z, for all w ∈ W.

Proof. Consider A(w)X t ⊕ B(w)U t = (A + w Ā)X t ⊕ (B + w B)U t ⊆ (AX t ⊕ BU t ) ⊕ w( ĀX t ⊕ BU t ) ⊆ λX t ⊕ wΘ where Θ ∆ = ĀX t ⊕ BU t .
The first inclusion follows from Minkowski sum properties, while the second one follows from the fact that X t (w = 0) is a λ-ISI set, λ ∈ [0, 1), for the Nominal model. Now, since X t and U t are proper C-sets, Θ is also a proper C-set. Then, a value w > 0 does exists such that wΘ ⊆ (1 -λ)X t . Therefore, we have

A(w)X t ⊕ B(w)U t ⊆ λX t ⊕ wΘ ⊆ λX t ⊕ (1 -λ)X t = X t
Then, the non-empty set W ∆ = [0, w] is such that X t is an ISI set for x + = A(w)x + B(w)u, (x, u) ∈ Z, for all w ∈ W. Furthermore, assuming that Θ is 0-simetric, a similar procedure can be followed for -w. In this way, the (non-empty) set W is given now by

W ∆ = [-w, w].
Next, based on the result in Theorem 1, a practical form to compute the robust ISI set is proposed.

Remark 2. (Practical computation of the robust ISI set) Property 2,(4) provides a way to obtain an ISI set like the one required by the hypotesis of Theorem 1. Consider a λ-IS set Ω λ , λ ∈ [0, 1), for the autonomous system x + = A(w = 0)x = Ax, x ∈ X , and take into account that ξ Ω λ , ξ > 0, is also a (contractive) λ-IS set for the same system. Then, the set

X t = X t (ξ, λ) ∆ = X t (w = 0) ⊕ ξ Ω λ ,
where X t (w = 0) is the minimal ISI set for the Nominal system, is a δ-ISI set, with δ ∈ (λ, 1), for the same Nominal system. Now, for given sets W ∆ = [-w, w] and U t , it is desirable to obtain the smallest robust ISI set X t (ξ, λ), which can be done by solving the following problem:

( ξ, λ) = min ξ,λ {(ξ, λ)|(AX t (ξ, λ) ⊕ BU t ) ⊕ w( ĀX t (ξ, λ) ⊕ BU t ) ⊆ X t (ξ, λ)} (2)
The smallest robust ISI set is then given by X t = X t ( ξ, λ)

∆ = X t (w = 0) ⊕ ξΩλ.
Notice that the model described in (1) is not a mere scaling of matrices A and B, since matrices Ā and B are in general different from the nominal matrices. So, notice that a general set X containing all possible minimal invariant sets for the uncertainty system is not useful for a robust formulation of the MPC, since this set is not necessarily an ISI set for each model of the family.

Additive disturbance uncertainty

In this case we have a model of the form x + = Ax + Bu + Dd, where D is the disturbance matrix, d ∈ D is the disturbance vector that describes the uncertainty and D is a proper C-set. In such a case, by selecting a λ-ISI set X t , for the extended system

x + = Ax + B D u d , u d ∈ U t × D,
we obtain a λ-ISI set for the uncertain system x + = Ax + Bu + Dd, (x, u) ∈ Z, for all d ∈ D. This kind of disturbance can include output noise (i.e., identification noise), measurement noise, and other kinds of disturbances that cannot be described by the parametric uncertainty.

The two robust ISI sets presented in this subsection are ISI sets for uncertain systems. However, it should be noted that an explicit robust formulation of the proposed MPC strategy, which assures the feasible robust convergence, is not in the scope of the present approach, and is delayed for future works.

MPC suitable for re-identification (main result)

The objective of the proposed controller is to avoid the conflicts between excitation and control objectives: when the system is outside a target set, the target set is a control target; when the system is inside, the target set is a safe identification set, that is considered as a generalized equilibrium by the controller.

The key concept to achieve such an MPC controller is to propose an MPC cost that (1) penalizes the distance to the target set, and ( 2) is null at every point of the target set, in the same way a standard MPC cost is null at the desired punctual target. This means that the controller does not make a difference between any two points of the target, or, in other words, the controller leaves the system in open loop when it enters the target set. We will start this class of MPC controllers with a controller for tracking control equilibrium sets (CES), which is known as zone control MPC controllers.

MPC for tracking control equilibrium sets (CES)

Consider the EIS set U t ⊂ U, and the associated CES set X t ss = GBU t ⊆ X . This controller is formulated following the strategy proposed in [START_REF] Ferramosca | MPC for tracking zone regions[END_REF][START_REF] Gonzalez | A stable MPC with zone control[END_REF], usually known as zone control. The cost function is given by

V CES N (x, X t ss ; u, u ss , x ss ) = N -1 j=0 ( x(j) -x ss 2 Q + u(j) -u ss 2 R ) + γ |x ss | X t ss ,
where Q > 0 and R ≥ 0 are penalization matrices, γ > 0 is a real number, |x| X t ss is the distance function (from x to X t ss ) and N is the control horizon. Furthermore, x ss = GBu ss is a free stationary state in X ss = GBU. For any current state x ∈ X , the optimization problem P CES N (x, X t ss ) to be solved is given by:

Problem P CES N (x, X t ss ) min u,uss,xss V CES N (x, X t ss ; u, u ss , x ss ) s.t. x(0) = x, x(j + 1) = Ax(j) + Bu(j), j ∈ I 0:N -1 x(j) ∈ X , u(j) ∈ U, j ∈ I 0:N -1 u(j) = u ss , j ∈ I N -1:∞ x ss = GBu ss .
In this optimization problem, x and X t ss are the parameters, while the sequence u = {u(0), • • • , u(N -1)} and x ss , u ss , are the optimization variables. Notice that x ss , u ss are only forced to represent an equilibrium point, but are free to assume any feasible equilibrium value. The control law, derived from the application of a receding horizon policy, is given by κ N (x, X t ss ) = u 0 (0; x), where u 0 (0; x) is the first element of the solution sequence u 0 (x). Now the following Theorem can be established: Theorem 2. Consider that Assumption 1 holds, and consider a given CES X t ss = GBU t ⊆ X , with an associated input set U t . Then, X t ss is an ES for the closed-loop system x(j) = φ κ N (j; x, X t ss ), x(0) = x, j ∈ I ≥1 . Furthermore, X t ss is locally attractive for the same closed-loop system, with a domain of attraction given by X .

Proof. The proof can be seen in [START_REF] Ferramosca | MPC for tracking zone regions[END_REF][START_REF] Gonzalez | A stable MPC with zone control[END_REF].

Notice that this formulation comes directly from the output tracking problem, if we consider output zones of the form CX t ss (being a particular case the set-point tracking, if set X t ss is a single point). The idea to steer the system to an equilibrium set (with independence of the single value of the equilibrium point), suggests the possibility to extend this concept to invariant sets, which also include transient states.

MPC for tracking invariant sets for identification (ISI)

Now, a generalization of the MPC controller for tracking CES sets is presented. The idea is to track and reach sets -i.e., ISI sets -that not only include stationary states, but also transient states, which is necessary to perform a proper identification. We start with a quite general formulation, that is particularized in the next subsections to different applicable cases. The nominal case will be considered, although the extension to the use of robust ISI sets is direct. Consider an ISI set X t for N sys , and the associated EIS set U t . Also consider the following definition Definition 7. (Generalized distance stage cost function) A generalized distance function d X t (x), from x to the ISI set X t , is a function with the following properties: (1) d X t (x) is convex and continuous for all x ∈ X , (2)

d X t (x) = 0 for all x ∈ X t , (3) d X t (x) > 0 for all x ∈ X \X t , (4) d X t (x)
is a Lyapunov function for the excited system x(j) = φ(j; x, u pe ), x(0) = x, j ∈ I ≥1 , for all x ∈ X \X t .

The proposed controller cost function will be given by:

V ISI N (x, X t ; u) = N -1 j=0 (αd X t (x(j)) + βd U t (u(j))) + d ter X t (x(N )). ( 3
)
where α and β are positive real numbers and d ter X t (•) is a terminal cost function defined on X ter , and X ter ⊆ X t is the terminal set where the terminal state x(N ) is forced to belong. As usual in MPC design, we need to select a local control action (defined by some conditions) that will act for predictions inside the terminal set. At this moment we simply define a fixed control action û = û(x) ∈ U t . Then, associated to this local control action, the terminal set and terminal cost must fulfill the following conditions:

X ter is IS for x + = Ax + B û, x ∈ X (4) d ter X t (Ax + B û) -d ter X t (x) ≤ -αd X t (x), ∀x ∈ X ter (5)
d ter X t (x) = 0, ∀x ∈ X t . (6)
Different forms for the local control action, the terminal cost and terminal set will be presented in section 5.3. For any current state in the set of states that can be feasibly steered to X ter in N steps (the N -step controllable set to X ter ), x ∈ X N (X ter ), the optimization problem P ISI N (x, X t ) to be solved is given by:

Problem P ISI N (x, X t ) min u V ISI N (x, X t ; u) s.t. x(0) = x, x(j + 1) = Ax(j) + Bu(j), j ∈ I 0:N -1 x(j) ∈ X , u(j) ∈ U, j ∈ I 0:N -1 x(N ) ∈ X ter .
The main differences between this problem and problem P CES N (x, X t ss ) are: (1) the target set is now an ISI set, which includes transient states (i.e., the states can be moved in a certain region with null control cost and without control actions), and (2) the generalized distance functions are now used to penalize the state and input trajectory in the MPC cost. The controller derived from this formulation assures the convergence of the closed-loop system to the ISI set X t , and once the system is there, it only assures that it will remain indefinitely in it. The next Theorems formalize these properties: Theorem 3. Let Assumption 1 hold, and consider an ISI set X t ⊆ X , with an associated EIS set U t . Then, X t is an IS set for the closed-loop system x(j) = φ κ N (j; x, X t ), x(0) = x, j ∈ I ≥1 .

If we compare now the proposed feasible cost at time k + 1 with the optimal one at time k, we have:

V ISI N (x + , X t ; û) -V ISI N 0 (x, X t )=-αd X t (x 0 (0; x)) -βd U t (u 0 (0; x)) +(αd X t (x 0 (N ; x)) + βd U t (û)) + d ter X t (x) -d ter X t (x 0 (N ; x))
Since û ∈ U t by definition, d U t (û) = 0. Furthermore, since x(N ; x) 0 ∈ X ter and x ∈ X ter , then -by conditions (5), d ter X t (x) -d ter X t (x(N ; x) 0 ) ≤ -αd X t (x(N ; x) 0 ). Therefore,

V ISI N (x + , X t ; û) -V ISI N 0 (x, X t ) = -αd X t (x 0 (0; x)) -βd U t (u 0 (0; x)) +αd X t (x 0 (N ; x)) -αd X t (x 0 (N ; x)) = -αd X t (x 0 (0; x)) -βd U t (u 0 (0; x)).

Now, by optimality of the solution to Problem

P ISI N (x + , X t ), at k + 1, we have V ISI N 0 (x + , X t ) ≤ V ISI N (
x + , X t ; û), and so

V ISI N 0 (x + , X t ) -V ISI N 0 (x, X t ) ≤ -αd X t (x 0 (0; x)) -βd U t (u 0 (0; x)). (7)
Since the generalized distance is a positive definite function, [START_REF] González | Stable MPC for tracking with maximal domain of attraction[END_REF] implies that x 0 (0; x) tends to X t and u 0 (0; x) tends to U t as k → ∞, and so, the system converges to the desired ISI set.

Theorems 3 and 4 suggest that an extra requirement to the input, such as a persistent excitation requirement, could be included in the proposed cost function. In fact, what is shown is that, when the system is in X t , any bounded external input disturbance that does not bring the input outside U t , will not produce any effect on the controller.

Including the exciting mode

Given the properties of the latter formulation, several strategies could be followed to excite the system under control, when the state is in X t , and furthermore, to update the model parameters by means of an on line estimation. To do the first, we could add an exciting constraint, as the one used in [START_REF] Marafioti | Enhanced Model Predictive Control: Dual Control Approach and State Estimation Issues[END_REF], [START_REF] Genceli | New approach to constrained predictive control with simultaneous model identification[END_REF], or even a penalization term into the optimization cost, as the one proposed in [START_REF] Heirung | Towards dual MPC[END_REF]. To update the model parameters, a simple recursive least-square (RLS) algorithm ( [START_REF] Ljung | System identification-theory for the user[END_REF]) is recommended for practical applications.

However, since the focus of the present work is mainly in the separation of the objectives in the MPC formulation, we will simply consider a precomputed sequence u pe in C pe (which is also in C U t ) as desired input trajectory. On the other side, no parameter update methods will be discussed. The proposed persistent excitation MPC cost function is as follows:

V EXC N (x, X t , u pe , k; u) = (1 -ρ(x))V ISI N (x, X t ; u) + ρ(x) u(0) -u pe (k) ,
where ρ(x) = 1 if x ∈ X t , and ρ(x) = 0, otherwise. The proposed strategy is a switching control strategy, in which the switching is operated by function ρ(x). Notice that the state x corresponding to the true plant is assumed to be known, and so, the switching function operates on true state measurements. For any initial state x in X N (X ter ), at a given time step k, the optimization problem P EXC N (x, X t , u pe , k), to be solved at each time instant k, is given by:

Problem P EXC N (x, X t , u pe , k) min u V EXC N (x, X t , u pe , k; u) s.t. x(0) = x, x(j + 1) = Ax(j) + Bu(j), j ∈ I 0:N -1 x(j) ∈ X , u(j) ∈ U, j ∈ I 0:N -1 x(N ) ∈ X ter
Notice that the function ρ(x) is a discontinuous function necessary to cancel the persistent excitation in case that an external disturbance takes the system away from the invariant set X t . The following Theorem formalizes the properties of the proposed MPC controller: Theorem 5. Let Assumption 1 hold, and consider an ISI set X t and a persistent excitation sequence u pe in C pe . Then, for any initial state x ∈ X t , the system controlled by the receding horizon MPC control law κ N (x, X t ) = u 0 (0; x), will be persistently excited inside X t , i.e., x(j) = φ κ N (j; x, X t , u pe , j) = φ(j; x, u pe ), x(0) = x, j ∈ I ≥1 . Furthermore, for any initial state x ∈ X N (X ter )\X t , the closed-loop converges to X t . Proof. i) Let us consider an initial state x ∈ X N (X t )\X t . Then, ρ(x) = 0, and so Problem P EXC N (x, X t , u pe , k) is equivalent to Problem P ISI N (x, X t ). Furthermore, for Theorem 4, the closed-loop system will admissibly converge to X t .

ii) Consider now an initial state x ∈ X t . Then ρ(x) = 1, and the persistent excitation penalization is activated. Furthermore, (1 -ρ(x))V ISI N (x, X t ; u) = 0. This means that the cost of problem P EXC N (x, X t , u pe , k) will be:

V EXC N (x, X t , u pe , k; u) = u(0) -u pe (k) .
Now, since X t is an ISI set and the persistent excitation sequence u pe is in U t , then, the system will remain inside X t . So, the cost can be admissibly canceled if and only if u 0 (0; x) = u pe (k) for each time k. This guarantees the persistent excitation of the open-loop system. Remark 3. Another (practical) alternative to implement the strategy is by defining a second λ-ISI set X tt = δX t , δ > 1. Then, if the state is in X tt \X t , the system keeps in open-loop (the control action are forced to be null). The invariant condition of X tt (property 2.( 5)) assures that once the state enter X tt it will keep in X tt , and furthermore, the attractivity condition of X t (property 2.(2)) assures that the state will reach X t .

Operation of the loop

Based on the discussion in 4.1, 4.2 and 4.3, the MPC controller operation will be presented. We have two Operation modes: (1) Control operation mode: in this mode no re-identification is needed, and the MPC for tracking CES set is implemented (Problem P CES N (x, X t ss )); and (2) Re-identification operation mode: this mode is activated only when there is a suspect that the model is not working properly, and a re-identification is needed. In this mode the persistent excitation MPC is used (Problem P EXC N (x, X t , u pe , k)).

Remark 4. Based on Theorem 5, the Re-identification operation mode has in addition two spatially separated modes. When the system is outside X t , it is positively steered to it by the controller; once it is inside X t , it cannot be steered outside this set by the controller, even in an uncertainty scenario as the one described in section 3.2. If an unknown moderate disturbance enters the system and takes the state outside X t -and provided that the current state x is measured or well estimated, then the controller will automatically switch, by means of function ρ(x), to only control the system to X t . On the other hand, if a strong disturbance scenario arises, a Re-identification experiment has no sense, and it is recommended to pass to the Control operation mode.

Candidates for generalized distance functions and the terminal costs

Provided that the proposed MPC formulations are strongly based on the concept of generalized distance functions, two possible candidates that fulfills Definition 7 will be presented in this section.

5.1. Distance from a point to a set Definition 8. Distance from a point to a set. Given an ISI set X t ⊂ X , the distance from x to X t is defined as

|x| X t ∆ = min x∈X t x -x 2 M , M > 0.
Function |x| X t has the following properties:

Property 3. (1) |x| X t is convex and continuous for all x ∈ X , (2) |x| X t = 0 for all x ∈ X t , (3) |x| X t > 0 for all x ∈ X \X t , (4) N [|x| X t , γ] = X t ⊕ B γ , for all x ∈ X \X t , where B γ ∆ = {x ∈ X : x 2 M ≤ γ}, γ > 0.
Property 4. Let Assumption 1 hold, and suppose that X t ⊆ X is an ISI set for N sys . Furthermore, consider that matrix M > 0, associated to the distance function, is such that

A T M A -M = -Q for some Q > 0.
Then, the distance function |x| X t is a Lyapunov function for the excited system x(j) = φ(j; x, u pe ), x(0) = x, j ∈ I ≥1 , for all x ∈ X \X t .

Proof. First, notice that the set B γ defined in Property 3.( 4), is a λ-IS for the system x + = Ax, x ∈ X , with λ ∈ [0, 1), provided that matrix M > 0 is such that A T M A -M = -Q for some Q > 0. Therefore, from Property 2,(4), and Property 3.( 4), it follows that the level sets N [|x| X t , γ] = X t ⊕ B γ , with γ > 0, are δ-IS sets, δ ∈ (λ, 1), for the excited system x(j) = φ(j; x, u pe ), x(0) = x, j ∈ I ≥1 , for all x ∈ X \X t (we assume for simplicity that (X t ⊕ B γ ) ⊂ X ). Consider now a state x ∈ ∂(X t ⊕ B γ ), for some γ > 0, which implies that |x| X t = γ. Then,

x + = (Ax + Bu) ∈ δ(X t ⊕ B γ ), ∀u ∈ U t . This implies that |x + | X t ≤ δγ, an so, |x + | X t ≤ δ |x| X t , with δ ∈ (λ, 1
), which means that |x| X t is a Lyapunov function for the aforementioned system, for all x ∈ X \X t .

Modified Minkowski functional

The Minkowski functional ( [START_REF] Blanchini | Set-Theoretic Method in Control[END_REF]) is defined as: Definition 9. Given an ISI set X t ⊂ X , the Minkowski functional Ψ X t associated to X t is defined as

Ψ X t (x) = inf{µ ≥ 0 : x ∈ µX t }.
The Minkowski functional has a number of useful properties ( [START_REF] Blanchini | Set-Theoretic Method in Control[END_REF]). It also was already used as a part of MPC costs in works as [START_REF] Rakovic | Minkowski terminal cost function fro mpc[END_REF] and [START_REF] González | Stable MPC for tracking with maximal domain of attraction[END_REF]. However, the Minkowski functional is not null inside the set to which it is associated. To achieve this property, we need to introduce the modified Minkowski functional. Definition 10. Modified Minkowski functional. Given a convex set X t ⊂ X that includes the origin as an interior point, and a λ-IS, Ω, with λ ∈ [0, 1), for the system x + = Ax, x ∈ X , the modified Minkowski functional is defined as

ΨX t (x) ∆ = inf{µ ≥ 0 : x ∈ X t ⊕ µΩ}
Function ΨX t (x) has the following properties: Property 5. (1) ΨX t (x) is convex and continuous for all x ∈ X , (2) ΨX t (x) = 0 for all x ∈ X t , (3) ΨX t (x) > 0 for all x ∈ X \X t , (4) N ΨX t (x), γ = X t ⊕γΩ, for all x ∈ X \X t , γ > 0. Property 6. Let Assumption 1 hold, and suppose that X t ⊆ X is an ISI set for N sys . Then, ΨX t (x) is a Lyapunov function for the excited system x(j) = φ(j; x, u pe ), x(0) = x, j ∈ I ≥1 , for all x ∈ X \X t .

Proof. This proof follows a similar procedure to the one of Theorem 4, taking into account Property 5.(4), of the Modified Minkowski functional, and the λ-IS condition of Ω.

Figure 1 shows a schematic plot of a Modified Minkowski functional and a Distance function -together to the level sets -associated to an arbitrary polytopic set. Notice that inside the set the functions are null.

Several choices for the local control action and the terminal cost and set

The easiest choice for the local control action, û, is û ∆ = 0 ∈ U t (this choice is possible since A is stable). Then, the terminal set and terminal cost are X ter ∆ = X and d ter X t (x) ∆ = δd X t (x(j)) ≥ ∞ j=0 αd X t (x(j)), x(0) = x, respectively. Since X is an IS set for x + = Ax, by definition, then X ter fulfills equality (4). On the other hand, since d X t (x) is a Lyapunov function for the system x(j) = φ(j; x, u pe ), x(0) = x, j ∈ I 1:N -2 , for x ∈ X \X t (Properties 4 and 6), and u(j) = 0 ∈ U t , for j ∈ I N -1:∞ , it is possible to select a (large enough) real number δ > 0 such that d ter X t (x) fulfills inequality (5) for all x ∈ X . Equality ( 6) is fulfilled trivially. The drawback of this choice is that the proposed MPC solution could be far from optimality, since no control action is implemented for predictions beyond the horizon N .

The other choice for the local control action is to simply select any fixed û ∈ U t . Then, two options could be considered for the terminal set and terminal cost. The first option is X ter ∆ = X t and d ter X t (x) ≡ 0 (no terminal cost). In this case, X ter fulfills equality (4) and d ter X t (x) fulfills inequality ( 5) and equality (6) trivially. One drawback of this selection is that the domain of attraction of the proposed MPC is given by X N (X t ), which can be considerably small. A second option is X ter ∆ = X t ⊕ δX ⊆ X and d ter X t (x) ≥ ∞ j=0 αd X t (x(j)), x(0) = x. In this case, the real number δ < 0 should be selected to maximize the volume of X ter . Notice that X ter is an ISI set (by Property 2,(4)), and so fulfill condition (4). Furthermore, a simply procedure to obtain the proposed d ter X t (x), which fulfill conditions ( 5) and ( 6), was presented before.

Remark 5. It should be remarked that no prohibitive computational costs are added with the proposed MPC formulation. If the distance function is used as generalized distance function, the optimization problem could be re-arranged to remain a QP problem. If the modified Minkowski functional is used, an additional linear programing problem should be solved.

Simulation results

Nominal case

In this section some simulations results will be presented, to evaluate the proposed control strategy. To this end, a 3-state stable system of the form of N sys is used, with matrices

A =   0.5 0.2 -0.3 0.4 0.3 0.4 -0.3 0.2 0.5   , B =   0.6 0.85 0.5 -0.67 -0.2 0.4   C = -0.54 0.8 0.2 0.3 -1.1 0.7 ,
The constraints of the system are given by: X

= x ∈ R 3 : x ∞ ≤ 9 and U = u ∈ R 2 : u ∞ ≤ 1.25 .
The EIS set has been selected to be U t = u ∈ R 2 : u ∞ ≤ 0.8 , while the persistent excitation signal was selected to be a (bounded) White Noise Signal. The ISI set, X t , was selected with a volume large enough to give some robustness to the controller. Figure 2 (right) shows the relation between the feasible state space X (in light red) and the ISI set X t (in dark red).

The simulations were designed to show the Re-identification operation mode of the controller. Several initial states in X \X t was selected. As can be seen in Figure 2 (left), every (feasible) state is steered to the target set X t , and once the system is inside this set, the exciting procedure is activated. Furthermore, in Figure 2 (right), the state evolution inside the target set is shown. Notice that the state trajectory is around the stationary target X t ss , which is also plotted (shadow plane).

Figure 3 shows the input, outputs and cost function time evolutions. Notice that there are two clear modes: first, from time k = 0 to time k = 5, the system is steered to the ISI set, with a decreasing cost function. Then, from time k = 6 on, the cost function remains null, which corresponds to a persistent excitation determined by the (bounded) White Noise Signal u pe . The two time periods have been separated using different colors. Notice also, that the input is on its upper bound at the very beginning of the simulation, because the controller objective is to drive the system to the target. Furthermore, after time k = 5, the input remain inside the EIS set U t , denoted by two dotted-lines.

Model-mismatch case

In this section some simulations results are presented, to evaluate the proposed control strategy in a model uncertainty scenario. To this end, a 2-state which corresponds to w R = -0.20. The constraints of the system are given by X = x ∈ R 2 : x ∞ ≤ 17 and U = {u ∈ R : u ∞ ≤ 1}. The EIS set has been selected to be U t = {u ∈ R : u ∞ ≤ 0.65}, while the persistent excitation signal was selected to be a (bounded) White Noise Signal. The robust ISI set, X t , was selected according to Remark 2 (and Theorem 1). It is given by:

X t = X t ( ξ, λ) = X t (w N ) ⊕ ξ Ωλ = X t (w N ) ⊕ 0.85 Ω 0.6
where X t (w N ) is the minimal ISI set corresponding to the Nominal model, and Ωλ is a λ-invariant set for x + = A(w N )x. Figure 4 shows a family of 8 minimal ISI sets, corresponding to parameters w going from -0.2 to 0.1. Furthermore, the Nominal and the Real minimal ISI sets X t (w N ) and X t (w R ), together with the ISI set X t and ξ Ωλ are plotted. The simulations were designed to show the Re-identification operation mode of the controller. To this end several initial states in X \X t was selected. As can be seen in Figure 5 (left), every (feasible) state is steered to the target set X t , and once the system is inside this set, the exciting procedure is activated. Notice that the system enters X t (w N ) and then leaves it, because this set is not associated to the Real model. In fact, the minimal ISI set for the real system is given by X t (w R ) and it can be seen that once the system enters this set, it does not leaves it anymore. The important point here is that X t is also a (non-minimal) ISI set for the Real system, and the state never leaves X t once it enters the set. These facts are depicted in Figure 5 (right). Figures 6 (left) and 6 (right) show the input, outputs and cost function time evolutions. Figure 6 (left) has a time scale that goes only to k = 20 to clearly show how the MPC control the system. From time k = 0 to time k = 5, the system is steered to the ISI set, with a decreasing cost function. Then, from time k = 5 on, the cost function remains null, which corresponds to the injection of the persistent excitation signal u pe . The two time periods have been separated using different colors. Notice also, that the input is on its upper bound in the first time periods, because the controller tries to drive the system to the target. Remark 6. The latter simulation was tested including in the loop a state observer, since in an identification scenario an observer is usually present. However, since the selected systems are of small dimension, and a well tuned observer was used, the obtained responses were not substantially different to the responses in Figure 6.

Conclusions

In this work, some results regarding a new MPC formulation suitable for closed-loop re-identification was presented. The proposed MPC guarantees persistent excitation when the system is inside a target region and guarantees re-cursive feasibility and closed-loop stability to this target region when the system is outside.

The key concept to pursue these two opposite objectives is the concept of attractivity of a robust invariant set, inside of which the excitation of the system can be made. In this way, the controller does not superpose the control and identification objectives, since inside the target region no control is performed and outside the target region no persistent excitation is injected to the system. To account for these properties novel generalized distance functions was proposed to construct the MPC cost function. Furthermore, to prove convergence of the method these functions have shown to be formal Lyapunov functions.

From a practical point of view, a main advantage of the method is that the identification procedure can be made as in open-loop fashion, since once the system is inside the target region, no control actions affect the system. On the other side, a preliminary drawback of the method is that a new robust invariant set needs to be computed if the target change. Future research clearly includes the study of the relation between the invariant target region and the exciting input set, in order to obtain a less conservative formulation.

Figure 1 :

 1 Figure 1: Descriptive plot of the Modified Minkowski Functional and the Distance Function associated to a given set.

Figure 2 :

 2 Figure 2: State evolution (left) outside and (right) inside X t .

Figure 3 : 1 ,

 31 Figure 3: Input, output and cost evolution.

Figure 4 :

 4 Figure 4: Family of 8 minimal ISI sets corresponding to parameters w going from -0.2 to 0.1 and the Robust ISI set X t

Figure 5 :

 5 Figure 5: State evolution (left) outside and (right) inside X t .

Figure 6 :

 6 Figure 6: Input, output and cost evolution: (left) complete simulation, (right) first 20 time steps

Figure 6 (

 6 Figure 6 (right) goes to k = 210 to clearly show the excitation procedure. Notice that after time k = 5, the input remains inside the EIS set U t , denoted by two dotted-lines. The excitation signal has different mean values along the simulation time, to show the invariance condition of the ISI set in this scenario.

Preprint submitted to System and Control LettersDecember 6, 2013

Proof. Consider a state x ∈ X t . Then, by definition of ISI sets, any input sequence û = {u(0), . . . , u(N -1)}, with u(j) ∈ U t , for j ∈ I 0:N -1 , produces a sequence of states that remain in X t . So, considering the definition of the generalized distance function, the input sequence is a trivial optimal solution of P ISI N (x, X t ), with V ISI N (x, X t ; û) = 0. On the other hand, any input sequence û with u(j) / ∈ U t , for some j ∈ I 0:N -1 , produces a cost V ISI N (x, X t ; û) ≥ 0. This means that necessarily u 0 (0; x) ∈ U t . This proves that the MPC cost V ISI N (x, X t ; u) is null along every trajectory starting in an initial state inside X t , and furthermore, u 0 (0; x) is a control input inside U t . From this fact, it directly follows that X t is an IS set for the MPC closed-loop system. Theorem 4. Let Assumption 1 hold, and consider an ISI set X t ⊆ X , with an associated EIS set U t . Then, X t is locally attractive for the closed-loop system x(j) = φ κ N (j; x, X t ), x(0) = x, j ∈ I ≥1 , with a domain of attraction given by X N (X ter ).

Proof. Consider a state x ∈ X N (X ter )\X t , at a given time k. Consider also the solution defined for this state, u 0 (x) = u 0 (0; x), • • • , u 0 (N -1; x) , and the corresponding state sequence x 0 (x) = x 0 (0; x), • • • , x 0 (N ; x) , with x(N ; x) 0 ∈ X ter . The cost function of Problem P ISI N (x, X t ) corresponding to u 0 (x) is given by

(αd X t (x 0 (j; x)) + βd U t (u 0 (j; x))) + d ter X t (x 0 (N ; x)).

Now, consider the successor state x + = Ax + Bu 0 (0; x), at time k + 1, which is obtained by applying the control law κ N (x, X t ) = u 0 (0; x), and define the following sequence: û = u 0 (1; x), • • • , u 0 (N -1; x), û , where û is the local control action. Since no model mismatch is considered for predictions, the successor states x + is equal to x 0 (1; x). This solution has an associated state sequence x = x 0 (1; x), • • • , x 0 (N ; x), x , where x = Ax 0 (N ; x) + B û. Since x 0 (N ; x) ∈ X ter and X ter is an IS set for the system x + = Ax+B û, x ∈ X , then x ∈ X ter . Therefore, sequence û is a feasible solution for problem P ISI N (x, X t ) at time k + 1. The cost function of Problem P ISI N (x + , X t ), at k + 1, for the sequence û, is given by

where x + = x 0 (1; x). So, this cost can be written as a function of x,

(αd X t (x 0 (j; x)) + βd U t (u 0 (j; x)))

+(αd X t (x 0 (N ; x)) + βd U t (û)) + d ter X t (x)


Florence Petroff

The Scots' Response to the American Revolution: A North British Vision of Empire, Constitution, and Representation

As Linda Colley argued persuasively in Britons: Forging the Nation, 1707-1837 (1992), Britishness was invented in the eighteenth century and has since been superimposed on preexisting identities without either replacing or merging with them. Yet eighteenth-century Scottish elites eagerly engaged in Anglicization while preserving many aspects of their distinctiveness. The contours of the resulting "North British" identity have been defined by Colin Kidd ("North Britishness and the Nature of Eighteenth-Century British Patriotisms," Historical Journal 39.2, 1996). Being British was commonly understood throughout the British Empire as enjoying the "English liberties" enshrined in the constitution and protected by the Hanoverian dynasty. Bernard Bailyn, Jack Greene, Pauline Maier, Eliga Gould, and a number of other historians have pointed out that the Thirteen Colonies initially resisted attempts at centralization by Britain in the name of these liberties. Being British also meant taking part in a shared transatlantic culture, as Ned Landsman has shown in From Colonials to Provincials: American Thought and Culture, 1680-1760 (1997), as well as Brendan McConville in The King's Three Faces: The Rise and Fall of Royal America, 1688-1776 (2012).

My work explores the notions of Britishness that the Scots and the colonists forged in the context of the imperial crisis, from the enactment of the Stamp Act in 1765 to the Treaty of Paris in 1783. It contends that two distinct, even competing, conceptions of Britishness were shaped by the Scots and the Americans through processes of identification with England and differentiation with one another. The American Revolution was undoubtedly the catalyst for the emergence of an American identity. Yet before rejecting Britishness, the American Patriots first built up a hybrid identity combining Britishness with Americanness; until the mid-1770s they claimed that they could remain part of the empire while benefiting from the large amount of autonomy that their colonial assemblies had acquired since the late seventeenth century. They asserted their Britishness through differentiation with the French, as all Britons then did, as well as with the Scots. In 1967 Bernard Bailyn briefly mentioned in The Ideological Origins of the American Revolution that a version of the conspiracy theory that pervaded the colonies involved former Prime Minister Lord Bute, a Scot accused of secretly manipulating George III and the government against America (pp. 147-48). The question of what Scotland embodied for American revolutionaries has not drawn much attention since then, possibly because Scotland was hardly ever mentioned in American revolutionary literature. It was, however, discussed in the colonial press. Newspapers throughout the Thirteen Colonies, particularly in Virginia, disseminated an image of Scotland as inherently Jacobite and supportive of tyranny and Roman Catholicism. They reveal a different political culture from that of the pamphlets and display a discourse based on disinformation, fantasy, and fear that originated in the English Scottophobic Grub Street rhetoric of John Wilkes and Charles Churchill. The belief in a "Scottish plot" against America waged by Bute, Scottish ministers, and the Scottish nation was in turn appropriated by Americans such as Rev. Ezra Stiles, whose diary entry for 23 July 1777 states: "Let us boldly say, for History will say it, that the whole of this War is so far chargeable to the Scotch Councils, & to the Scotch as a Nation (for they have nationally come into it) as that had it not been for them, this Quarrel had never happened. Or at least they have gloried in the Honor of exciting & conduct[ing] these Measures avowedly by their Earl of Bute behind the Curtain." Scotland was not just considered an enemy to the American Patriots' cause. It was also the antithesis of the Britishness that the Americans identified with through binary oppositions such as Hanover/Stuart, liberty/tyranny, and Protestantism/Catholicism.

Conversely, the Scots considered themselves the opposite of the American rebels and predominantly opposed the American Revolution. The perceptions of the American Revolution in Scottish public opinion were explored by Dalphy Fagerstrom in 1954 ("Scottish Opinion and the American Revolution," William and Mary Quarterly, 11.2) and D. B. Swinfen in 1977 ("The American Revolution in the Scottish Press," in Scotland, Europe, and the American Revolution, ed. Owen Dudley Edwards and George Shepperson). My approach is based on a more extensive study of the Scottish pamphlet literature in order to investigate the various reasons why the Scots stood against the Americans. At least sixty-six works on America were published either by a Scot or anonymously in Scotland between 1768 and 1784, with a peak of forty-two between 1776 and 1779. Thirty-six authors produced one, two, sometimes three pamphlets, sermons, or books discussing the conflict with the colonies. A number of them took part in the Scottish Enlightenment, including prominent figures such as Adam Smith and Adam Ferguson and less familiar ones such as William Barron. About half these authors published on a variety of other topics, including religion, history, husbandry, poetry, economic improvement, and literature. They were educated professionals who

worked as teachers, lawyers, or clergymen. With the exception of a few officers of the Crown and the loyalists George Chalmers and John McAlpine, none had ever been to America. Some, like George Johnstone, Alexander Wedderburn, and Lord Mansfield, were pursuing careers as politicians, judges, or civil servants in London. Ministers in the Church of Scotland account for almost half these authors, dominated by Moderate Party clergymen such as Alexander Carlyle, George Campbell, Adam Ferguson, Alexander Gerard, and Thomas Somerville. A majority of these Scottish authors argued that the Americans were wrong in declaring themselves independent. They believed them to be manipulated by a faction of ungrateful American politicians apt to endanger the empire and the wellbeing of all its inhabitants for the sake of their personal ambitions. Some expected Britain to bring the rebels back to their senses by force, others believed that the colonists could be convinced by sound reasoning. The Johnstone brothers, who both sat in Parliament in the 1770s, first appeared as "friends of America" when they advocated the right of the colonists to tax themselves and condemned the Coercive Acts of 1774. George, known as Governor Johnstone, reversed his opinion after the creation of the United States and its alliance with Britain's rival, France. In some of the most thoughtful and articulate pamphlets on the topic, his brother Sir William Pulteney recounted the growing tensions between Britain and its colonies and proposed a peace negotiation in 1779 based on the granting of fiscal and legislative autonomy to the colonies, assuming independence could still be reversed. John Erskine and William Thom in the Popular Party disapproved of the war but did not support the Declaration of Independence. All told, only two Scots openly endorsed American resistance to Britain's policy throughout the Revolution. George Dempster, politician and agricultural improver, expressed his staunch belief that the American Patriots were in the right, even after they seceded from the empire. James Murray, a Presbyterian minister at Newcastle upon Tyne, kept defending the American Patriots in sermons and in An Impartial History of the Present War in America (1778).

The Scots did not actually engage in a direct debate with the American pamphleteers, whom they hardly ever quoted or mentioned. Rather, they tried to refute Richard Price's Observations on the Nature of Civil Liberty (1776) and other British pamphlets. Thus, they addressed the American controversy as Britons speaking to other Britons, in an attempt to convince them that the American rebellion was unconstitutional and that the empire had to be preserved. The American controversy provided Scottish elites with an opportunity to display their British patriotism. They professed their commitment to the constitution and the principle of parliamentary sovereignty as well as their Hanoverian loyalty. The Americans were accused of being far worse rebels than the Scottish Jacobites in 1745 because they threatened not only the dynasty but also the constitution and "English liberties." The natural rights that the Declaration of Independence cited were considered "an inalienable right to talk nonsense," according to the comments inserted in the text of the Declaration in the Scots Magazine of August 1776. The colonists' grievances and claims were seen as a ploy that could not disguise the real motive of Congress: a quest for power. The American republic that pretended to ensure the liberties of its citizens challenged the belief in the perfection of the British constitution which, most Scots believed, remained the only real source of liberty. The Scottish approach to the issue of the American Revolution combined a commitment to Britishness with a desire to promote the interests of the Scottish nation. Sir John Dalrymple and Alexander Carlyle called for the right to establish a Scots militia and to expand recruitment of Highlanders in the army. Robert Alves, Charles Nisbet, and John Stevenson glorified the part played by the Scots, especially the Highlanders, in the patriotic war against the rebels. British patriotism mixed with Scottish patriotism, as in Stevenson's description of the Highlanders in his Letters in Answer to Dr. Price's Two Pamphlets on Civil Liberty (1778): "they fought, they bled, and they conquered as Britons" (p. 149).

As Emma Macleod has shown in British Visions of America, 1775America, -1820America, (2013)), American independence met with dismay and disbelief in Britain. Consequently, most of the Scots who published their reaction to the American Revolution from 1776 onward discussed matters that had been settled before 1776 in the eyes of the American Patriots. To the American claim for "no taxation without representation," the Scots answered that not only could the colonists be rightfully taxed by Parliament but also that they were not entitled to reject its authority since they were represented virtually. The empire was described as a community in which each part had to contribute to the defense of the whole through taxation in order to maintain its dominance. Scottish pamphleteers articulated the vision of a centralized empire, united by a unique fiscal system and transatlantic commerce. Adam Smith even suggested in The Wealth of Nations (1776) that the colonists should be given the right to elect their own representatives in Parliament, an idea that did not generate much enthusiasm in Great Britain, much less in America. Even though they were not in favor of an imperial Parliament, those who discussed the imperial crisis generally assumed that the empire should somehow be reorganized on the model of the Union.

The concept of representation exemplifies the discrepancy between Scottish and American visions of empire and constitution. The colonists believed they were not represented in the Westminster Parliament because their understanding of representation was based on their political experience within the colonies. Their assemblies were regarded by the American Whigs as the only bodies that represented them. These political bodies had grown into the main institutions in colonial America by enlarging their prerogatives at the expense of governors and councils. Not only were they geographically close, whereas the Westminster Parliament was more than three thousand miles away, but they defended the interests of their constituents. Bernard Bailyn has argued that the American concept of representation drew on the English medieval notion of representation seen as a form of attorneyship, in which the electors delegated power to the elected. The representatives in the colonial assemblies were accountable to their voters, whose interests they defended, sometimes even receiving instructions from them (Ideological Origins of the American Revolution, p. 164). It was a general rule that candidates had to live in the county, parish, or precinct where they were running for an election. In Virginia, they had at least to possess land in the county as it was the case for George Washington in 1758 when he was elected to the Virginia House of Burgesses in Frederick County even though he lived in the neighboring county of Fairfax (see Richard R. Beeman, The Varieties of Political Experience in Eighteenth-Century America, 2004, p. 39). An average of 75 percent of the white adult male population having access to vote, the colonial assemblies had a direct connection with a large part of the community, even though many voters did not care to exert their right. Representation was also conceived on the local scale in eighteenth-century America because the colonists identified first and foremost with their colony. From the failure of Benjamin Franklin's Albany Plan in 1754, the building of a collective American identity and the unification of all colonies was a long, arduous process. It is the conflict with Britain that actually stimulated the creation of the Stamp Act Congress in New York in 1765 and the Continental Congress from 1774. In the American vision of empire, sovereignty was shared, and each colony acted as a state within a confederation under the sole authority of the Crown. There was the historical precedent of Scotland and England between 1603 and 1707, Benjamin Franklin argued in 1768 in a letter to his son William and in the press (The Papers of Benjamin Franklin, vol. 15, ed. William B. Willcox, 1972, pp. 74-78; A Briton, "Arguments Pro and Con: I," London Chronicle, 18-20 October 1768; Boston Evening-Post, 11 April 1774, p. 1). Why should America be ruled by the Westminster Parliament, asked John Adams, when Scotland was not at the time of the union of the crowns: "So the Scots held their lands of him who was then king of England, his heirs and successors, and were bound to allegiance to him, his heirs and successors, but it did not follow from thence that the Scots were subject to the English parliament" (Novanglus, "XII. To the Inhabitants of the Colony of Massachusetts-Bay, 17 April 1775," Novanglus, and Massachusettensis, or, Political Essays: Published in the Years 1774 and 1775, on the Principal Points of Controversy, between Great Britain and Her Colonies, 1819, pp. 133-39).

Contrary to the Americans, most eighteenth-century Scots regarded the incorporating union as an improvement compared to the time when they had their own Parliament. They conceptualized a notion of representation on the scale of Britain which was not based on the electoral process. In Britain the voting population was much more limited than in America, even more so in Scotland where there were fewer than three thousand electors for a population of one million people. The unequal distribution of the franchise between men and between burghs and counties was not considered an issue, for it was understood that MPs were not accountable to their electors since, once elected, they became "representatives of the state at large; not only their own constituents, but every individual member of the community" (Free Thoughts on the American Contest, 1776, p. 22). They defended the interests of the whole nation when voting on laws and taxes that everyone was subjected to, including themselves. Representatives were in any case increasingly less connected to the constituency that elected them in eighteenth-century Britain since many did not live there; some 60 Scots even sat for an English or a Welsh constituency between 1760 and 1790, 130 between 1790 and 1820. The Scots thus felt as much under the protection of Parliament against oppression and unfair taxation as any part of England or Wales, even though they elected only 45 MPs out of 558 in the House of Commons. They were committed to centralization, believing the empire needed a single head to keep all its parts together, and their attitude toward authority was the opposite of American distrust of central power (see Jack P. Greene, Peripheries and Center: Constitutional Development in the Extended Polities of the British Empire and the United States, 1986; Ned C. Landsman, "British Union and American Revolution: Imperial Authority and the Multinational State," in The American Revolution Reborn, ed. Patrick Spero and Michael Zuckerman, 2016; Paul Tonks, "Rethinking the Eighteenth Century Province and Periphery: A Historiographical Reflection on Scotland, Scottish Thought, and the Government of the British Atlantic Empire," Korean Journal of British Studies, no. 22, Dec. 2009).

Scottish elites believed their interests were better protected in the Westminster Parliament, which provided a stronger counterweight to the Crown's prerogative than they had enjoyed in their pre-Union unicameral Parliament or in local institutions such as town councils. They adopted the notion that virtual representation included the colonists, as George Grenville's government maintained. James Macpherson, who was pensioned by the government, argued in The Rights of Great Britain Asserted against the Claims of America: Being an Answer To the Declaration of the General Congress (1776) that, like the Americans, most British subjects did not vote, and he concluded: "the truth is, Representation never accompanied Taxation in any State" (p. 4). In his Letters in Answer to Dr. Price's Two Pamphlets on Civil Liberty, John Stevenson added that the members of the House of Commons "represent actually and virtually, not only all the inhabitants of this island, but also every individual throughout the colonies" (p. 24).

Liberty was not seen as contingent upon the franchise by most Scottish pamphleteers, who contended that Britons were free because they had a Parliament that checked the Crown's power, no matter how it was elected.

Harry T. Dickinson observed in Liberty and Property: Political Ideology in Eighteenth-Century Britain (1977) that a movement for electoral reform swept through England from the 1770s, partly under the influence of American revolutionary ideas (pp. 217-18). Since they considered the British system of representation as nearly perfect, the Scots who wrote about the American controversy did not support that movement, with the exception of James Burgh, a member of the radical society of London. Burgh's 1500-page work Political Disquisitions (1774-75) called for reform that would correct the many flaws and inequalities in the electoral system and reduce the influence of the aristocracy, for example by putting an end to government intrusion in elections and establishing a more equitable electoral franchise at all levels of society. He believed that most people who paid indirect taxes on malt, soap, and other necessities were not represented in Parliament since they could not vote. These radical views were hardly supported in Scotland. The burgh reform movement had connections with the Society for Constitutional Information and Chistopher Wyvill's Yorkshire Association, but it followed a different path. Its appeal for an enlargement of the ruling elite in Scottish burghs stemmed not from a hope to reform the constitution but from a wish to modernize local institutions, which were seen as archaic compared to those in England.

The Scottish reception of the American Revolution was generally unsympathetic, as the Americans correctly perceived, but it was not rooted in Jacobitism, as the Americans believed. On the contrary, the Scots rejected Whig American thought because it questioned the British conception of liberty and empire that they endorsed. The American notions of natural rights, direct representation, and shared sovereignty within the empire induced the Scots to assert their commitment to the principle of parliamentary sovereignty over the empire, the belief in the perfection of the constitution, and loyalty to the Hanoverian dynasty, which were the fundamentals of the emerging British identity. The imperial crisis arguably weakened the empire but was also a catalyst for Britishness, which was built not merely in opposition to a French "Other" but also in opposition to an American one.

This article is based on Florence Petroff's doctoral thesis, "Le Miroir atlantique. L'Ecosse et l'Amérique dans la crise impériale: regards croisés et identités hybrides au sein du monde britannique (1765-1783)"/"Scotland and America in the Imperial Crisis: Mutual Perceptions and Hybrid Identities within the Anglo-American World (1765-1783)," for which she was awarded a PhD in history from the University of Paris 8 in 2020. In September 2021 Florence will take up a faculty position in the History Department at the University of La Rochelle. She is planning to publish her thesis in French and a few articles in English. Her research interests focus on the circulation of political thought, opinions, and representations within the British Atlantic world. She would welcome any comments or be happy to engage in a discussion of those topics (florence.petroff@gmail.com).