Mohammed-Tahar Laraba
email: mohammed.laraba@lss.supelec.fr.

Sorin Olaru
email: sorin.olaru@lss.supelec.fr.

Silviu-Iulian Niculescu
email: silviu.niculescu@lss.supelec.fr.

Analysis of PWA control of discrete-time linear dynamics in the presence of variable time-delay

This paper focuses on the robustness problem for a specific class of dynamical systems, namely the piecewise affine (PWA) systems, defined over a bounded region of the state-space X . We will be interested in PWA systems emerging from linear dynamical systems controlled via feedback channels in the presence of varying transmission delays by a PWA controller defined over a polyhedral partition of the state-space. We exploit the fact that the variable delays are inducing some particular model uncertainty. Our objective is to characterize the delay invariance margins: the collection of all possible values of the time-varying delays for which the positive invariance of X is guaranteed with respect to the closed-loop dynamics. These developments can be useful for the analysis of different design methodologies and in particular for predictive control approaches. The proposed delay margins describes the admissible transmission delays for an MPC implementation. From a different perspective, it further provides the fragility margins of an MPC implementation via the on-line optimization and subject to variable computational time.

I. INTRODUCTION

Time-delay dynamics form an important modeling class for networked control systems (NCSs) as well as for many other physical processes where propagation and transport phenomena, heredity and competition in population occur. The presence of communication networks in the closedloop induces varying transmission delays [START_REF] Heemels | Comparison of overapproximation methods for stability analysis of networked control systems[END_REF]. These delays are known to degrade the control performance and can induce instability as documented in the rich control literature dedicated to these subjects [START_REF] Michiels | Stability and Stabilization of Time-Delay Systems (Advances in Design & Control)[END_REF], [START_REF] Gu | Stability of time-delay systems[END_REF].

Roughly speaking, Model Predictive Control (MPC) is a popular constrained control technique based on the resolution of an optimization problem over a receding horizon [START_REF] Maciejowski | Predictive Control with Constraints[END_REF], [START_REF] Johansen | Explicit Nonlinear Model Predictive Control[END_REF]. It constructs at each sampling instant an optimal sequence with respect to a performance index. Unfortunately, using MPC in the presence of time-varying delays leads to complex optimization problems, which are difficult to handle. Linear MPC with constraints is known to result in PWA closed-loop dynamics [START_REF] Bemporad | The explicit linear quadratic regulator for constrained systems[END_REF], [START_REF] Johansen | Approximate explicit constrained linear model predictive control via orthogonal search tree[END_REF]. Recent work dealt with stability of PWA systems using Piecewise Quadratic Lyapunov functions [START_REF] Johansson | Computation of piecewise quadratic lyapunov functions for hybrid systems[END_REF]. The stability problem is usually formulated in terms of linear matrix inequalities (LMIs). Due to the conservatism of such approaches, alternative relaxations can be found in [START_REF] Hovd | Relaxing PWQ lyapunov stability criteria for PWA systems[END_REF], [START_REF] Iervolino | Cone-copositive piecewise quadratic lyapunov functions for conewise linear systems[END_REF].

In this paper, we will focus on the robustness analysis of discrete-time linear dynamics in closed-loop with a PWA control law in the presence of time-varying delays affecting the communication on the feedback channel or induced by the control computation itself. The PWA feedback is generic but it can be obtained, for example, by using a simple explicit MPC design constructed upon the nominal delay-free model. A formal definition of delay margins based on positive invariance is one of the main objectives in this endeavor. The presence of variable input delay induces an uncertainty of exponential type in the closed-loop parameters. All possible delay variations can be covered by embedding the uncertainty within a polytopic model when the maximal delay is known [START_REF] Heemels | Comparison of overapproximation methods for stability analysis of networked control systems[END_REF], [START_REF] Cloosterman | Robust stability of networked control systems with time-varying network-induced delays[END_REF], [START_REF] Gielen | On polytopic inclusions as a modeling framework for systems with time-varying delays[END_REF]. However, the inverse problem of finding the maximal range of delay when a nonlinear control law structure is specified over a bounded region of the original state-space represents an open problem, to the best of the authors' knowledge, and will receive in the sequel a complete characterization.

The main contribution of the paper consists in proposing a constructive method to find the delay margins based on positive invariance of the nominal closed-loop dynamics in the piecewise affine formulation. The procedure describes, by means of set projections, all possible delay values for which the positive invariance (or alternatively D-invariance) of the state trajectories is guaranteed. The paper makes use of some preliminary results in [START_REF] Nguyen | Explicit robustness and fragility margins for linear discrete systems with piecewise affine control law[END_REF] and extends the robustness analysis to PWA systems with variable input delays.

The paper is structured as follows. The class of considered linear continuous time-invariant systems affected by variable delay is introduced in section II. In the same section, the exact discretization of the considered dynamics and the uncertain PWA system obtained in closed-loop are discussed. Section III is devoted to the construction of delay margins in order to ensure invariance in both the original and the extended state-space representations. Lastly, an illustrative example is shown in section IV and concluding remarks are drawn in section V.

Notations: We denote by R, R + , N, Z and Z + the field of real numbers, the field of non-negative real numbers, the set of non-negative integers, the set of integer numbers and the set of strictly positive integer numbers, respectively. For every interval Π of R we define Z Π := Z∩Π. Given m ∈ Z + , by 1 m , we denote the vector of dimension m with all the entries equal to 1 and, by I m ∈ R m×m , the m × m identity matrix. Conv denotes the convex hull operation.

Given two sets X , Y ⊂ R m , X ⊕ Y and X × Y denote the Minkowski sum and the Cartesian product of these two sets, respectively, defined as follows:

X ⊕ Y := {z| ∃(x, y) ∈ (X , Y) such that z = x + y} . X × Y := {(x, y)| x ∈ X and y ∈ Y} .
The unit simplex in R m is defined as:

S m := x ∈ R m + | 1 T m x = 1 .
For a given set X ⊂ Y × Z, int(X ) denotes the interior of X , the projection of X onto Y is defined as:

Proj Y X = {y ∈ Y| ∃z ∈ Z such that (y, z) ∈ X } .
The notions of state-space partition, PWA functions and positive invariance are the classical ones as defined for example in [START_REF] Nguyen | Explicit robustness and fragility margins for linear discrete systems with piecewise affine control law[END_REF], [START_REF] Blanchini | Set invariance in control[END_REF]. Whenever an exponent is associated with a matrix, it will be interpreted as a matrix raised to a power or just an index depending on the context.

II. DYNAMICAL MODEL OF A LINEAR PLANT WITH DIGITAL CONTROL IN THE PRESENCE OF INTERSAMPLE DELAY

A. Dynamical System Consider a nominal linear continuous time-invariant (LTI) system and a sequence of delays (τ k ) affecting the input as follows:

ẋ(t) = A c x(t) + B c u(t) u(t) = u k , ∀t ∈ [t k + τ k , t k+1 + τ k+1 ) . (1)
where For the sake of simplicity of the presentation, we discuss in the sequel only the case when the delay variation is intersample, i.e. smaller than or equal to the sampling period. We will recall next the modeling in discrete-time following the approach in the studies [START_REF] Heemels | Comparison of overapproximation methods for stability analysis of networked control systems[END_REF], [START_REF] Cloosterman | Robust stability of networked control systems with time-varying network-induced delays[END_REF], [START_REF] Gielen | On polytopic inclusions as a modeling framework for systems with time-varying delays[END_REF]. Consider the exact discretization of (1) by exploiting the fact that the control action is piecewise constant, i.e.

A c ∈ R n×n , B c ∈ R n×m , x(t) ∈ R n the
u(t) = u k , ∀t ∈ [t k + τ k , t k+1 + τ k+1 ): x k+1 = e AcTs x k + τ k 0 e Ac(Ts-θ) dθB c u k-1 + Ts τ k e Ac(Ts-θ) dθB c u k (2)
and let k = T s -τ k , and:

A = e AcTs , B = Ts 0 e Ac(Ts-θ) dθB c (3)
∆ ( k ) = Ts Ts-k e Ac(Ts-θ) dθB c = 0 -k e -Acσ dσB c . (4)
Then, the discrete-time model which takes into account the effect of the continuous time-delay variation will become:

x k+1 = Ax k + ∆( k )u k + (B -∆( k ))u k-1 . (5)
In the general case, the variable time delay implies a variable "limit" for the integration for k in (4). One can see that there is no exact link between the samples available for the discrete model and the delay in continuous time, thus leading practically to some parameter-varying dynamical model. When discretizing, we can deal with the variable input delay as an appropriate uncertainty function. All possible delay variations can be covered by confining the induced model uncertainty within a polytopic description. Therefore, a polytopic (simplicial) over-approximation of the uncertainty coming from the variable delay can be constructed (See, e.g. [START_REF] Heemels | Comparison of overapproximation methods for stability analysis of networked control systems[END_REF], [START_REF] Gielen | On polytopic inclusions as a modeling framework for systems with time-varying delays[END_REF], [START_REF] Lombardi | A predictive control scheme for systems with variable time-delay[END_REF]) to obtain finally a polytopic model. It is interesting to note that by setting k = 0 and k = T s , we obtain two extreme realizations of the discrete-time model [START_REF] Johansen | Explicit Nonlinear Model Predictive Control[END_REF].

B. The PWA closed-loop dynamics

The starting point for the present work will be the nominal dynamics corresponding to k = T s (no delay is induced by the network). An explicit PWA control law is designed with respect to this nominal dynamics.

u pwa : X -→ R m u pwa (x) = F i x + g i , ∀i ∈ I N s.t x ∈ X i (6)
The piecewise affine control law obtained will be in turn used in practice for the control of the linear parameter varying dynamics subject to variable delay [START_REF] Johansen | Explicit Nonlinear Model Predictive Control[END_REF]. The closedloop dynamics resulting from applying:

u k pwa = F i x k + g i , for x k ∈ X i u k-1 pwa = F j x k-1 + g j , for x k-1 ∈ X j (7)
will be:

x k+1 = Ax k + (B -∆( k )) [F j x k-1 + g j ] + ∆( k ) [F i x k + g i ] = (A + ∆( k )F i )x k + (B -∆( k ))F j x k-1 + Bg j + ∆( )(g i -g j ), ∀(i, j) ∈ I 2 N such that x k ∈ X i , x k-1 ∈ X j . (8)
It is clear that an extended state-space representation can be constructed for the delay-difference equation ( 8) by introducing some new augmented state vector, i.e.

ξ k = x T k x T k-1
T ∈ R 2n . An equivalent extended state-space model will be obtained:

x k+1 x k ξ k+1 = A + ∆( k )F i (B -∆( k ))F j I n 0 n×n x k x k-1 ξ k + Bg j + ∆( k )(g i -g j ) 0 n×1 ; ∀(i, j) ∈ I 2 N such that x k ∈ X i and x k-1 ∈ X j . (9)
The difference equations ( 8) and ( 9) depend on ∆( k ), considered as a parameter varying matrix, lying in a nonconvex subset of R n×m . One can write equations ( 8) and ( 9) in a more compact form as follows:

x k+1 = φ i ( k )x k + θ j ( k )x k-1 + γ ij ( k ), ∀(i, j) ∈ I 2 N s.t x k ∈ X i and x k-1 ∈ X j . (10)
where:

φ i ( k ) = (A + ∆( k )F i ) θ j ( k ) = (B -∆( k ))F j γ ij ( k ) = Bg j + ∆( k )(g i -g j ), (11)
or alternatively for (9) as:

ξ k+1 = Φ ij ( k )ξ k + Γ ij ( k ) ∀(i, j) ∈ I 2 N such that ξ k ∈ X i × X j . (12)
with:

Φ ij ( k ) = A + ∆( k )F i (B -∆( k ))F j I n 0 n×n Γ ij ( k ) = Bg j + ∆( k )(g i -g j ) 0 n×1 (13)
Moreover, one can define the following parameter-varying PWA mappings:

ψ pwa : X × X × R [0,Ts] -→ R n ψ pwa (x, y, ) = φ i ( )x + θ j ( )y + γ ij ( ) ∀(i, j) ∈ I 2 N such that x ∈ X i and y ∈ X j . (14)
Ψ pwa : {X × X } × R [0,Ts] -→ R 2n Ψ pwa (ξ, ) = Φ ij ( )ξ + Γ ij ( ) ∀(i, j) ∈ I 2 N such that ξ ∈ X i × X j . (15)
It is worth to mention that the uncertain piecewise affine systems [START_REF] Iervolino | Cone-copositive piecewise quadratic lyapunov functions for conewise linear systems[END_REF] and ( 12) are defined over the polyhedral partition of the compact sets X (X × X respectively), X = i∈I N X i inherited from the explicit PWA control law design [START_REF] Bemporad | The explicit linear quadratic regulator for constrained systems[END_REF]. Now, the concept of delay margins with respect to dynamical systems, presented in the extended and the original statespace framework, is introduced in terms of sets as follows:

Definition 2.1: The delay margin with respect to the timevarying dynamical system [START_REF] Iervolino | Cone-copositive piecewise quadratic lyapunov functions for conewise linear systems[END_REF], denoted d m , is given by:

d m = { ∈ [0, T s ]| ∀x ∈ X i , y ∈ X j , ψ pwa (x, y, ) ∈ X } .
Definition 2.2: The delay margin with respect to the timevarying dynamical system [START_REF] Gielen | On polytopic inclusions as a modeling framework for systems with time-varying delays[END_REF], denoted D m , is given by:

D m = { ∈ [0, T s ]| ∀ξ ∈ X × X , Ψ pwa (ξ, ) ∈ X × X } .
To summarize, delay margin problem corresponds to the complete description of all possible delay values ∈ [0, T s ], denoted by d m (or by D m ), for which the positive invariance of the set X (or alternatively X × X ) with respect to the time-varying dynamical systems [START_REF] Iervolino | Cone-copositive piecewise quadratic lyapunov functions for conewise linear systems[END_REF], (alternatively (12)), is guaranteed, ∀ ∈ d m (D m ), respectively.

III. CONSTRUCTION OF THE DELAY MARGIN BASED ON THE POSITIVE INVARIANCE

Assume that both X and the elements X i defining the partition in [START_REF] Bemporad | The explicit linear quadratic regulator for constrained systems[END_REF] are bounded polyhedral sets. Then, the polytopes of interest can be described as the intersection of finite number of half spaces. They can also be described as the convex hull of finite point set (vertices) in R n . Let the vertices of polytopes X and X i be:

V(X ) = {v 1 , v 2 , • • • , v q } (16) V(X i ) = {w i1 , w i2 , • • • , w iqi } , ∀i ∈ I N (17)
The vertex representation of these polytopes is expressed in the following:

X = Conv {v 1 , v 2 , • • • , v q } ( 18
)
X i = Conv {w i1 , w i2 , • • • , w iqi } , ∀i ∈ I N (19)
For each region X i of the partition of X , the set containing its vertices is:

W i = V(X i ), ∀i ∈ I N (20)
Let W be the set of all vertices of X i when i ∈ I N :

W = i∈I N V(X i ) (21)
Using only the non redundant elements of W, one can write:

W = {w 1 , w 2 , • • • , w p } , card {W} = p. ( 22
)
The vertices of the polytope X × X are denoted by:

V(X × X ) = v i v j ∈ R 2n , ∀(i, j) ∈ I 2 q (23)
For each region X i × X j of the partition of X × X , its set of vertices:

W ij = [V(X i × X j )] = W i × W j W ij = {w i1 , w i2 , • • • , w iqi } × w j1 , w j2 , • • • , w jqj (24)
Let W X ×X be the set of all vertices of X i ×X j when (i, j) ∈ I 2 N :

W X ×X = (i,j)∈I 2 N V(X i × X j ) (25)
It is worth to mention that there exists some close link between the elements of the two sets W and W X ×X . One can easily write:

W X ×X = w i w j ∈ R 2n , ∀(i, j) ∈ I 2 p . ( 26
)
Based on the above notations, we define the following matrices obtained by storing as columns the non redundant elements of the different sets of vertices using an arbitrary ordering:

V = [V(X )] ∈ R n×q , V X ×X = [V(X × X )] ∈ R 2n×q 2 V i = [W i ] ∈ R n×qi , V ij = [W ij ] ∈ R 2n×(qi×qj ) W = [W] ∈ R n×p , W X ×X = [W X ×X ] ∈ R 2n×p 2 (27)
The image of the set W i using the affine mapping (6) allows the construction of a matrix:

U i = [u pwa (W i )] ∈ R m×qi (28)
The image of the set W via the affine mapping in (6) allows the construction of the matrix:

U = [u pwa (W)] ∈ R m×p . (29)
Let O k p be the p × p matrix whose all entries are equal to zero, except the k th row, which is equal to 1 T p . A. Delay margins in the extended state-space representation

The uncertainty in ( 4) is represented by the matrix ∆( k ) satisfying ∆( k ) ∈ ∆, k ∈ N, with:

∆ = {∆( k )| k ∈ [0, T s ]} (30)
To characterize the delay margins we aim to use a simplicial over-approximation of the matrices ∆ ∈ R n×m in (30).

Based on such an over-approximation of the matrix set ∆, the system can be embedded within a polytopic model with s + 1 extreme realizations:

∆ ∈ Conv {∆ 0 , ∆ 1 , • • • , ∆ s } , (31)
any element of ∆ can be written as convex combinations of generators (corresponding to extreme realizations), i.e.:

∀ k ∈ R [0,Ts] , ∃α ∈ S s+1 such that ∆ = s i=0 α i ∆ i . ( 32
)
More than that, with respect to the set D m ⊂ R [0,Ts] , we have:

∀ k ∈ D m , ∃α ∈ S s+1 such that ∆ = s i=0 α i ∆ i . (33)
For a given ξ k such that x k ∈ X i and x k-1 ∈ X j , the feedback law is known i.e. F i , F j , G i , and G j defining Φ ij ( k ) and Γ ij ( k ) in ( 12) are known.

Proposition 3.1: For a given (i, j) ∈ I 2 N , the matrix

[Φ ij ( k ) Γ ij ( k )] belongs to a polytopic set Ω: Ω = Conv Φ 1 ij Γ 1 ij , • • • , Φ s+1 ij Γ s+1 ij
and there exists a vector α with non negative scalars {α 0 , • • • , α s } such that α ∈ S s+1 satisfying:

[Φ ij ( k ) Γ ij ( k )] = s l=0 α l Φ l ij Γ l ij , (34)
where:

Φ l ij = A + ∆ l F i (B -∆ l )F j I n 0 n×n , ( 35
)
Γ l ij = Bg j + ∆ l (g i -g j ) 0 n×1 . ( 36
)
The proof has been omitted due to space limitations. Theorem 3.2: Consider the uncertain piecewise affine system (12) defined over the polyhedral partition of X ×X . The delay margin is obtained as:

D α m = P roj Ss+1 R ∩ δ ( 37
)
where R and δ are defined as:

δ = α ∈ S s+1 | ∀ k ∈ R [0,Ts] ; ∆( k ) = s i=0 α i ∆ i (38) R = (α, Γ) ∈ R s+1 + × R q 2 ×p 2 + | 1 T Γ = 1 T , E + l=s l=0 α l ∆ l 0 n×m 0 n×m 0 n×m H = V X ×X Γ . (39)
E ∈ R 2n×p 2 , H ∈ R 2m×p 2
are defined, next, in the proof. Proof: The positive invariance of the set X × X with respect to the time-varying dynamical system ( 12) is represented by a set-wise relation:

∀ k ∈ D m ⊂ R [0,Ts] , and ∀ξ k ∈ X i × X j , (i, j) ∈ I 2 N : Φ ij ( k )ξ k + Γ ij ( k ) ∈ X × X , (40)
which is equivalent to:

∀ k ∈ D m ⊂ R [0,Ts] , and ∀ξ k ∈ X i × X j , (i, j) ∈ I 2 N , ∃α ∈ S s+1 such that s l=0 α l Φ l ij ξ k + Γ l ij ∈ X × X . (41)
By substituting ( 35) and (36) in equation ( 41) we obtain:

s l=0 α l A BF j I n 0 n×n + ∆ l 0 n×m 0 n×m 0 n×m F i -F j 0 m×n 0 m×n ξ k + Bg j 0 n×1 + ∆ l 0 n×m 0 n×m 0 n×m g i -g j 0 m×1 ∈ X × X . (42) A BF j I n 0 n×n ξ k + Bg j 0 n×1 + s l=0 α l ∆ l 0 n×m 0 n×m 0 n×m F i -F j 0 m×n 0 m×n ξ k + g i -g j 0 m×1 ∈ X × X . (43)
By expressing the extended state vector ξ k ∈ X i × X j as a convex combinations of the vertices of X i × X j which is known to be polyehdral set, we obtain:

ξ k = qi×qj z=1 β z w z ij for β ∈ S qi×qj (44)
It follows that equation ( 43) is equivalent with:

A BF j I n 0 n×n qi×qj z=1 β z w z ij + Bg j 0 n×1 + s l=0 α l ∆ l 0 n×m 0 n×m 0 n×m F i -F j 0 m×n 0 m×n qi×qj z=1 β z w z ij + g i -g j 0 m×1 ∈ X × X . (45)
For a given vertex in w z ij , i.e. z ∈ Z [1,qi×qj ] , (i, j) ∈ I 2 N we have:

A BF j I n 0 n×n w z ij + Bg j 0 n×1 + s l=0 α l ∆ l 0 n×m 0 n×m 0 n×m F i -F j 0 m×n 0 m×n w z ij + g i -g j 0 m×1 ∈ X × X . (46)
We describe the inclusion (46) explicitly since it is equivalent with the existence of a vector y z ij ∈ X × X such that:

A BF j I n 0 n×n w z ij + Bg j 0 n×1 + s l=0 α l ∆ l 0 n×m 0 n×m 0 n×m F i -F j 0 m×n 0 m×n w z ij + g i -g j 0 m×1 = y z ij . (47)
where the vector y z ij can be expressed as:

y z ij = V X ×X γ z ij such that γ z ij ∈ S q 2 (48)
replacing equations ( 48) in (47), ∀(i, j) ∈ I 2 N and z ∈ Z [1,qi×qj ] , we obtain:

A BF j I n 0 n×n w z ij + Bg j 0 n×1 + s l=0 α l ∆ l 0 n×m 0 n×m 0 n×m F i -F j 0 m×n 0 m×n w z ij + g i -g j 0 m×1 = V X ×X γ z ij . (49)
or in other words, equation ( 49) holds for all non redundant vertices of X i × X j , ∀(i, j) ∈ I 2 N , which means that it holds for all the columns of the matrix W X ×X defined in (27). Exploiting the piecewise affine mapping (6) of the elements of W X ×X , a matrix formulation of the elements

A BF j I n 0 n×n w z ij + Bg j 0 n×1 , (50)
F i -F j 0 m×n 0 m×n w z ij + g i -g j 0 m×1 (51)
in ( 49) is deduced for w z ij ∈ W X ×X :

   A BF j I n 0 n×n w z ij + Bg j 0 n×1 w z ij ∈ W X ×X = A 0 n×n I n 0 n×n W X ×X + BU 0 n×p I p • • • I p (52)
Same procedure can be applied for (51):

   F i -F j 0 m×n 0 m×n w z ij + g i -g j 0 m×1 w z ij ∈ W X ×X = U O 1 p -U U O 2 p -U • • • U O p p -U 0 m×p 0 m×p • • • 0 m×p (53) and V X ×X γ z ij w z ij ∈ W X ×X = V X ×X Γ (54)
with the restriction that each column of Γ is restricted to S q 2 . This restriction can be expressed as linear constraints on the columns of Γ:

1 T Γ = 1 T , Γ ∈ R q 2 ×p 2 + (55)
Finally, equation (49) leads to the matrix formulation:

A 0 n×n I n 0 n×n W X ×X + BU 0 n×p I p • • • I p E + l=s l=0 α l ∆ l 0 n×m 0 n×m 0 n×m U O 1 p -U • • • U O p p -U 0 m×p • • • 0 m×p H = V X ×X Γ E + l=s l=0 α l ∆ l 0 n×m 0 n×m 0 n×m H = V X ×X Γ. (56)
Since the exponential uncertainty corresponds to the values of α in (38), (∆( k ) does not take all values in the embedding), the delay margin is obtained as (37) in terms of α. the proof is complete noticing that the two sets D α m and D m are isomorphic.

B. Delay margins in the original state-space representation Theorem 3.3: Consider the uncertain piecewise affine systems (10) defined over the polyhedral partition of X . The delay margin d m is obtained as follows:

d α m = P roj Ss+1 T ∩ δ ( 57
)
where δ is defined in (38) and T is defined as:

T = (α, L) ∈ R s+1 + × R q×p 2 + | 1 T L = 1 T , E + l=s l=0 α l ∆ l H = V L (58)
and

E = AW O 1 p + BU • • • AW O p p + BU , H = U O 1 p -U • • • U O p p -U .
The proof has been omitted here due to space limitations.

IV. ILLUSTRATIVE EXAMPLE

Consider the following unstable dynamical system:

   ẋ(t) = 1.1 -0.1 1 0 x(t) + 1 0 u(t -h) y(t) = 1 0 x(t). (59)
with h ∈ [0, 0.1]. A discrete model is obtained, using (2)-( 4) with a sampling period T s = 0.1, and the uncertainty 0 < k ≤ 0.1. The uncertainty matrix ∆( k ) has been embedded within a polytopic model with the following 3 vertices:

∆ ∈ Conv 0 0 , 0.0999 -0.0006 , 0.1057 0.0057 (60)
Then, an explicit MPC has been designed for the nominal model (delay-free, h = 0), with a prediction horizon N = 7, in the presence of input and output constraints:

-5 ≤ u k ≤ 5 -5 ≤ y k ≤ 5 (61)
The partition of the obtained PWA control law as well as the resulting over-approximation of the uncertainty are shown in Figure 1.

The delay margin d α m has been computed using (57). Its projection on the plane (α 0 , α 1 ) is shown in Figure 2. The red set and the curved black line represent the sets T and δ Note that some close link between the delay margins and the number of regions in the partition of the PWA control law exists. The number of regions being directly related to the choice of the prediction horizon, one can notice that the delay margin is larger when the prediction horizon is small. Delay margins for different PWA control laws obtained using different prediction horizons are shown in Table I. V. CONCLUSION

In discrete-time modeling framework, the analysis of linear systems affected by uncertain time-varying input delays passes through the characterization of uncertain delay difference equations depending on a parameter varying matrix. The uncertainty appears in the parametrization of an exponential matrix. Many different techniques exist in the literature aiming at constructing an embedding for the uncertainty, and leading to classical polytopic models.

In the present work, we addressed an inverse problem, offering a measure of the delay margin of positive invariance for a closed-loop PWA system in the original (related to Dinvariance) and the extended state-space representations. The result presented in this paper gives a way to tackle the delay margin problem of a nominal PWA control law which can be seen as a relevant issue from the robustness analysis point of view in both feedback communication channels and variable computation-time for real-time optimization-based control.

Fig. 1 .

 1 Fig. 1. Partition of the PWA control law (left), illustration of the embedding of ∆( k ) by a simplex (right)

Fig. 2 .

 2 Fig. 2. Delay margin d α m projected on the plane (α 0 , α 1 )

  continuous system state. Moreover, assume that the system states are sampled periodically with the period T s ∈ R >0 and we denote by t k = kT s the k th sampling instants. The control input u(t) ∈ R m is known for t ∈ [0, τ 0 ), and the control action generated at time t = t k at the controller side is denoted by u k ∈ R m . The possible delay induced by the network at sample instant t k is denoted by τ

k ∈ [τ , τ ], with a lower bound τ ∈ R [0,τ ] and an upper bound τ ∈ R [τ ,Ts] .

VI. ACKNOWLEDGMENTS

The research leading to these results has benefited from the financial support of the European Union's 7 th Framework Programme under EC-GA No. 607957 TEMPO -Training in Embedded Predictive Control and Optimization.


Rajesh Koduri

Pedro Rodriguez-Ayerbe

Sorin Olaru

Robustness margin for piecewise affine explicit control law

Classical robustness margin i.e., gain margin and phase margin, considers the gain variation and phase variation of the model preserving the stability of the closed loop. In this paper, an attempt to find the same kind of margin for a piecewise affine (PWA) controller is done. This controller usually obtained in explicit model predictive control (MPC) is defined over a convex region of the state space X . Starting from the invariance property of the closed loop obtained with a discrete dynamic model and PWA controller in a convex region of the state space, we calculate two robustness margin preserving this invariance property. The first one will be denoted as gain margin corresponding to the variation of the gain of the model guaranteeing the invariance. The second one, that we call, the robustness margin against first order neglected dynamics correspond to the smallest first order neglected dynamic allowed in the system preserving the invariance property.

I. INTRODUCTION

Explicit model predictive control (MPC) belongs to a class of constrained model based optimization methods whose solution is synthesized by a set of piecewise affine (PWA) dynamics. The idea of the explicit MPC [START_REF] Bemporard | The explicit linear quadratic regulator for constrained systems[END_REF][3] [START_REF] Alessio | A survey on explicit model predictive control[END_REF] is to express the on-line quadratic problem in the control design into a multi-parametric Quadratic Programming(mpQP) problem and to reduce the computation of the optimal control inputs to a simple evaluation of a control law stored in a lookup table. Such control laws can be easily implemented for real-time system with fast dynamics and relative small statespace models. Though explicit control law has favorable advantages over the standard MPC, it also comes up with high computational cost for higher order systems and/or for large prediction horizon.

From the analysis point of view, it is important to take into account the capacity of the control law that can cope with disturbances, neglected dynamics or uncertain parameters. This characteristic is termed in control theory as robustness of the controller. In the context of robustness analysis of explicit controller, very few contributions have been made. Few noticeable works include an analysis procedure proposed in [START_REF] Olaru | Explicit robustness margins for discrete-time linear systems with PWA control[END_REF] and [START_REF] Nguyen | Explicit fragility margins for PWA control laws of discrete-time linear systems[END_REF] handles the robustness/fragility of the positive invariance for the dynamics affected by uncertain parameters. On the other hand, there is a substantial work on the robustification of the explicit controllers. The reference [START_REF] Rodrguez-Ayerbe | On the disturbance model in the robustification of explicit predictive control[END_REF] *The work leading to these results has received funding from the People Programme (Marie Curie Actions) of the European Unions Seventh Framework Programme (FP7/2007-2013) under REA grant agreement no 607957 (TEMPO) 1 The Authors are with L2S, CentraleSupelec, CNRS, LIPS, Paris Saclay University, 91192, Gif-Sur-Yvette, France.

rajesh.koduri@centralesupelec.fr, pedro.rodriguez-ayerbe@supelec.fr, sorin.olaru@centralesupelec.fr shows how to improve the robustness of the controller taking disturbances into account in the design phase. In a different perspective [START_REF] Efstratios | Explicit robust model predictive control[END_REF] presented an robust explicit predictive control synthesis which accounts for uncertainties based on dynamic programming.

In this paper, we consider the inherent robustness properties of explicit predictive control described as piecewise affine (PWA) control laws for a class of linear discrete time systems. First, we present a numerical method to compute a gain margin set for a discrete-time system stabilized by a continuous PWA dynamics with respect to the invariance property. The desired gain margin set is a polytope which characterize for variations of system gains preserving the invariant characteristics of the controller. Second, we analyze for the dynamic system affected by first order neglected dynamics. The robustness margin of the controller is defined by a set which characterize for a range of parameters preserving the invariance of the controller.

The paper is organized as follows: In section III, the system description for explicit model predictive control is briefly discussed. Then in section IV, we introduce some important notations and tools which are used throughout the paper. In section V, we discuss about the gain margin set for a given PWA control law with two examples. In section IV, we present the robustness margin of the PWA control law affected by the first order neglected dynamics. Finally, in section VII, we sum up the paper with conclusion.

II. BASIC NOTATIONS

This section addresses some basic notations and definitions. A vector is noted

x ∈ R n , x = [ x 1 • • • x n ] T and a matrix A ∈ R n×m , A = [a ij
]. An identity matrix is represented by I n , where the subscript n denotes the dimension of that matrix. The sets R, R + , Z, N and N >0 denote set of real numbers, set of non-negative real numbers, set of integers, set of non-negative integers, set of positive integers, respectively. We denote R n a Euclidean space and x ∈ R n a vector with n elements. A polyhedron is the (convex) intersection of a finite number of open or closed half-spaces and a polytope is a bounded and closed polyhedron. A mapping function f : R n → R m is said to be positively homogeneous of the first degree, if f (αx) = αf (x), ∀ α ∈ R + and ∀ x ∈ R n . 1 is a vector with its elements equal to 1. For a N ∈ N + , I N denotes the set of integers,

I N := { i ∈ N + | i ≤ N }. P roj R c2 S
is a projection mapping of a set S onto a subset, S ⊂ R c1 → R c2 for c1 > c2 and will be considered to operate on the first c2 coordinates of R c1 . Finally, convh denotes the convex hull. III. SYSTEM DESCRIPTION Consider a linear discrete-time system given by,

x(k + 1) = Ax(k) + Bu(k) (1a) y(k) = Cx(k) (1b)
Where, x(k) ∈ R n , u(k) ∈ R m and y(k) ∈ R s denote the state vector, input variables and output variables respectively at time k. The constraints on the system states and input variables are represented by,

X = {x : H x x ≤ h x , H x ∈ R nx×n , h x ∈ R nx } (2a) U = {u : H u u ≤ h u , H u ∈ R mu×m , h x ∈ R mu } (2b)
The state and input constraints are bounded polytopic sets. For explicit MPC, the quadratic cost function of the standard MPC with the terminal and stage costs is transformed into a multi-parametric Quadratic Programming (mp-QP) problem [START_REF] Alessio | A survey on explicit model predictive control[END_REF]. Considering a discrete-time system given by (1) subject to constraints (2), the equivalent mp-QP problem is given as

J(x, z) = min z 1 2 z T Hz + x T F T z + 1 2 x T Y x (3a) s.t Gz ≤ W + Sx. (3b)
The optimal solution for (3) is represented by a finite set of affine functions defined over the polyhedral partition of the set X , where

P N (X )= [X 1 , X 2 • • • X N ]
is called the polyhedral partition of the set X . The polyhedral X i ⊂ R n are called critical regions or components of the partition. The polyhedral regions X i are non overlapping i.e., X i ∩ X j = ∅, for i = j. The non-overlapping property ensures the unique solution for the state x(k) ∈ X i , ∀i ∈ I N . Two neighboring regions X i and X j share some vertices or facets and thus the optimal solution over the set X exhibits the continuity property. Definition 1: A mapping function

f pwa : {R n → R m |A i x + b i , ∀x ∈ X i , i ∈ I N } (4)
defined over the polyhedral partition of the set X is called a piecewise affine function of the polyhedral partition. The explicit control law, solution of (3), is synthesized in terms of the piecewise affine function defined over the polyhedral partition of the set X and it can be described by

u pwa (x(k)) = F i x(k) + g i , ∀x(k) ∈ X i . (5)
Definition 2: A closed and bounded set P ⊂ X is called positively invariant with respect to the system (1) in closed loop with the control law u pwa (x(k)) if for any x(0

) ∈ P, it follows x(k) ∈ P, ∀k ∈ Z [1;∞] .

IV. PRELIMINARIES

This section introduces few matrices which will be used throughout the paper. The vertex representation of the polyhedral sets X and X i are given by

X = convh{w 1 , w 2 • • • w r } (6a) X i = convh{w i1 , w i2 • • • w iri } (6b) A matrix W ∈ R n×r stores the vertices of the polyhedron X ∈ R n . W = [w 1 , w 2 , • • • , w r ] (7)
We introduce a matrix V ∈ R n× N i=1 ri to store the vertices of all the partition in the state space

X i ⊂ X , ∀i ∈ I N . V = [w 11 , w 12 , • • • , w iri , • • • , w N r N ] (8)
Like stated before, neighboring partitions have vertices in common, and therefore, the repeated vertices are removed from the matrix V and therefore the matrix V contains nonidentical column vectors.

V = [w 1 , w 2 , • • • , w p ] ∈ R n×p (9)
With the help of affine mapping f (x(k)) = u pwa (x(k)), a matrix U ∈ R m×p is used to store the control input for each column of matrix V .

U = u pwa (V ) (10)
V. GAIN MARGIN

In the following, we describe the gain margin set for the system (1) stabilized with the help of a state feedback explicit control laws. The construction of the gain margin set is based on vertex representation of the regions forming the partition X .

Definition 3: Consider a discrete time linear system (1) with a continuous PWA control law [START_REF] Nguyen | Explicit fragility margins for PWA control laws of discrete-time linear systems[END_REF], such that the state space set X is positively invariant. The Gain Margin is represented by the set

K ⊂ R m , such that x(k + 1) = Ax(k) + B(I m + diag(δ K ))u pwa (x(k)) ∈ X , ∀x(k) ∈ X and δ K ∈ K, δ K ∈ R m .
The set K ⊂ R m is a set which contains the input channels gain variations δ K such that for any point inside the set K, the invariance characteristics of the set X are preserved.

Theorem 1: Consider a discrete-time linear system (1) with a piecewise affine state feedback control law defined in [START_REF] Nguyen | Explicit fragility margins for PWA control laws of discrete-time linear systems[END_REF]. The gain margin K of the controller is defined by the set,

K = ∩ p q=1 K q (11)
Where K represents the gain margin set and K q the local gain margin for the vertex w q for some q ∈ I p . Proof: Starting from the PWA closed loop dynamics assuring the invariance of the set X given by,

x(k + 1) = Ax(k) + Bu pwa (x(k)) ∈ X (12)
with the addition of δ K in [START_REF] Kvasnica | Stabilizing polynomial approximation of explicit MPC[END_REF], ∀δ K ∈ K we obtain:

x(k + 1) = Ax(k) + B(I m + diag(δ K ))u pwa (x(k)) ∈ X (13
) It is also possible to exploit the structure of the PWA closedloop dynamics, by introducing the parametric variations on the control gains preserving the invariance and boundedness properties of the controller. Suppose that û = u pwa (x(k)) + δ u , where the term δ u ∈ R m describes admissible control input variations,

x(k + 1) = Ax(k) + Bu pwa (x(k)) + Bδ u ∈ X (14)
By relating ( 13) and ( 14) we obtain:

Ax(k) + Bu pwa (x(k)) + Bδ u = Ax(k) + B(I m + diag(δ K ))u pwa (x(k)) (15)
And,

u pwa (x(k)) + δ u = (I m + diag(δ K ))u pwa (x(k)) (16)
By rewriting (16) with analogy to each column vectors of the matrix V , ∀q ∈ I p (vertices in the set X i ), we obtain:

δ uq = diag(δ Kq )u pwa (w q ) (17)
The admissible input δ uq belongs to a set ∆U q ⊂ R m and δ Kq ∈ K q . A matrix M q = diag(u pwa (w q )) ∈ R m×m is uniquely defined based on the value of the control action u pwa (w q ) such that

∆U q ⊇ M q K q (18)
Exploiting the polyhedral structure of the admissible input variations,

∆U q = {u : | Ĥu u ≤ ĥu } (19)
where Ĥu ∈ R du×m and ĥu ∈ R du one can obtain the local set of gain variation for the vertex w q as:

K q = {z ∈ R m |∃u ∈ ∆U, M q z = u} (20)
Finally,

K q = {z ∈ R m | Ĥu M q z ≤ ĥu }. (21)
The collection of sets K q are independent of each vertices of the set in X i and the intersection of these independent set gives the global set K. In order to compute K q , we first need to compute explicitly the sets ∆U q .

Remark 1: Even if no control gain variation is admissible in (13), the null vector 0 m ∈ K is admissible as long as it corresponds to the set invariance of the original PWA control law.

A. Construction of ∆U q set:

In this subsection, a description for the ∆U q set is constructed based on forward mapping of the vertex in the PWA partition.

Theorem 2: Consider a linear discrete-time system (1) stabilized by a piecewise affine control law [START_REF] Nguyen | Explicit fragility margins for PWA control laws of discrete-time linear systems[END_REF]. The set ∆U q of admissible input variations at the vertex w q is obtained by

∆U q = P roj U H q (22)
U ∈ R m denotes the input constraint set. The polyhedral set H q is described by:

H q =    (δ u , λ) ∈ R m × R r , and [A B] x(k) u pwa (x(k)) +Bδ u = W λ    (23)
Proof: Let us recall the equation ( 14) which resumes the positive invariance for the set X ,

[A B] x(k) u pwa (x(k)) + Bδ u ∈ X (24)
From the definition of invariance, (24) can be expressed using the vertices of the set X as convex combination and corresponding variables λ = [λ 1 , λ 2 , • • • , λ r ] and equality constraints 1 T λ = 1 such that

[A B] x(k) u pwa (x(k)) + Bδ u = W λ. (25)
B. Computation of ∆U q set:

Based on the value of the control action u pwa (w q ). Consider the objective of finding the ∆U q set for each column vector of the matrix V independently, ∀q ∈ I p .

Ax(k) + Bu pwa (x(k)) + Bδ u = W λ (26)
Where λ = [λ 1 , λ 2 , • • • , λ r ] and 1 T λ = 1 with λ i ≥ 0 ∀i ∈ I r . Clearly (26) can be represented by a system of linear equalities with the variable vector Γ, where the vector Γ contains the required variable δ u along with the variables

[λ 1 • • • λ r -1]. Γ = [δ u , λ 1 , λ 2 , • • • , λ r-1 ] ∈ R m+r-1 (27a) λ r = 1 - r-1 i=1 λ i , λ i ≥ 0. (27b)
A linear programming problem with ( 26) and ( 27) as linear equality constraints can be setup,

      B(1, :) . . . B(n, :)    - Ŵ    Γ = w r -(Aw q + Bu pwa (w q )) ( 28
)
where Ŵ is defined by the notion

W = [w 1 • • • w r ] Ŵ = [w 1 -w r , • • • , w r-1 -w r ] ∈ R n×r-1
The system of linear equalities (28) is defined in the form HΓ = h, where H ∈ R n×(m+r-1) and h ∈ R n×1 , and has the solution

Γ = H z t + h z , (29)
where H z = null(H) is nothing but the orthonormal basis for the null space of matrix H and h z is the solution set for the linear programming problem. Subsequently, we establish two polyhedral sets from the matrices H z , h z with a variable t. The first polyhedral set corresponds to the required set ∆U q . Recalling the non-negativity constraints defined in (27), the second polyhedral is defined with the non-negativity property of the elements in the solution vector

Γ(m + 1 : m + r -1) = [λ 1 • • • λ r-1 ] T ≥ 0, H (1) = {t | H (1) z t ≤ h (1) z } (30a) H (2) = {t | -H (2) z t ≤ h (2) z }. (30b)
The matrices appearing in (30) are related to a decomposition

H (1) z = H z (1 : m, :), h (1) z = h z (1 : m), H (2)
z = H z (m + 1 : m + r -1, :) and h

(2) z = h z (m + 1 : m + r -1). And finally the set ∆U q is nothing but a linear transformation of the set H (2) , ∆U q = H (1) z H (2) + h (1) z .

The vertices of ∆U q is computed by applying the transformation to all the vertices of the set H (2) . Corollary 1: The set K q representing the gain matrix set is a polyhedral ∀q ∈ I p .

Proof: The set H q and ∆U q used in the description for K q are polyhedral and by this virtue K q inherits this property.

Remark 2: we note the analogy between the results reported in [START_REF] Nguyen | Explicit fragility margins for PWA control laws of discrete-time linear systems[END_REF] and (22). The work of [START_REF] Nguyen | Explicit fragility margins for PWA control laws of discrete-time linear systems[END_REF] focuses on the computation of fragility margin which is characterized for the parametric variations in the space of F i for each individual regions. However, in the present framework we interpret those findings to compute the variations in the control input for each individual vertices in the set X .

Remark 3: Regarding the computation of the set K, one need to solve p LP problems with a set of linear equations given in (28), where p is the number of non-identical vertices in the set X . Figure 1 shows the state trajectories for δ K = 0.3 which is a value inside the computed gain margin and it is observed that the trajectories are inside the invariant set thus confirming the theoretical result. From figure 2, it is noticeable that the controller is no more invariant for δ K = -0.25.

Multi Input system: Consider a linear discrete system with two inputs and two outputs,

x(k + 1) =   1.2 -1. 0 0 -1.2 0.5 0.2 0.4 0   x(k) +   1.0 0.2 0.5 0 0 0.7   u(k), y(k) = 1 0 0 0 0 1 x(k).
The weight applied on the control inputs and state vectors are

Q =   5 0 0 0 1 0 0 0 1   and R = 0.5 0 0 1 .The input constraint -2 -2 ≤ u(k) ≤ 2 2
and the output constraint

-2 -2 ≤ y(k) ≤ 2 2
. The prediction horizon chosen is 2. The resulting controller has 37 regions. Figure 3a shows the set K for ( 5) assuring the invariance of the set X . However, the invariance property of the controller is fragile to unmeasured input perturbation.

For instance, choosing δ K = diag(-0.01, -0.01) results in the controller loosing the invariance. To prove the positive side of gain margin analysis, a new explicit controller is chosen by adapting the gain with an admissible variation from the set K, shown in figure 3a, and which preserves its definition domain over the set X . The new controller is adapted in a straightforward manner to u µ pwa (x(k)) = (I m + diag(δ K )) * (F i x(k) + g i ) ∀i ∈ I N and δ K = diag(0.2, 0.2). Subsequently, we computed the gain margin set for this controller and present an illustration of its corresponding admissible variation in figure 3b. It is noticeable from the figure that the new controller is more robust to input disturbances and practically we obtained a translation of the point corresponding to the nominal controller without altering the shape of the set K. The new controller also violates the input constraints but the violation can be omitted by saturating the control inputs.

VI. FIRST ORDER NEGLECTED DYNAMICS

In the following, we analyze the robustness of the controller (5) defined over the polyhedral set X affected by a first order neglected dynamics in closed loop. Our approach is to present an admissible set for the dynamic variations of the neglected parameters or variables assuring the invariance of the set X .

A. Problem Formulation

In this section, a problem is formulated for the first order dynamics that perturb the closed loop system as shown in figure 4. The neglected dynamics can be represented by a first order equation and it can be written in the form,

x I (k + 1) = αx I (k) + (1 -α)u * (k) (32a) u(k) = αx I (k) + (1 -α)u * (k) (32b)
The neglected dynamics in closed loop model can be described by,

x(k + 1) = Ax(k) + αBx I (k) + B(1 -α)u * (k) (33)
The augmented model is

x e (k + 1) = A e x e (k) + B e u(k) (34a) y e (k) = C e x e (k) (34b)
and,

x e = x x I , A e = A αB 0 1×n α B e = B(1 -α) 1 -α , C e = [C, 0] (35)
where x e ∈ R ne , A e ∈ R ne×ne and B e ∈ R ne×m . Starting from a piecewise affine control (5) defined over the polyhedral partition of the set X with X i ∈ X ∀i ∈ I N for the linear discrete time system (1) the aim is to investigate the robustness of the control synthesis (5) for the system (34) and (35) affected by the neglected dynamics. The objective being to find the largest set of α assuring the invariance of the set X , first we construct vertices for the augmented state with the help of the vertices in the set X i ∀i ∈ I N . Second, we analyze the robustness of the control synthesis (5) with respect to the invariance characteristics for the set X .

B. Construction of vertices for the augmented state

Before entering into the technical details of the construction, we discuss about the extension of vertices for the augmented state.

Recall the vertices of the set X and X i defined by the column vectors of matrices [START_REF] Efstratios | Explicit robust model predictive control[END_REF] and [START_REF] Hovd | Low Complexity constraint control using contractive sets[END_REF]. The piecewise affine control law u pwa (x(k)) for the column vectors of the matrix V is given by the matrix U [START_REF] Herceg | Multi-Parametric Toolbox 3.0[END_REF]. The construction of vertices for the augmented state x I can be done in two ways.

• Method 1: Initialize the values of the state x I (k) for each vertex corresponding to the column vectors of matrix V with the piecewise affine control law. For instance, w Ip = u pwa (w p ) ∀p ∈ I p . • Method 2: Initialization of the state x I (k) for each vertex corresponding to the column vectors of matrix V can be carried out with the help of minimum and maximum values of the control input. Method 2 produces twice the number of vertices as compared to method 1. For each column vector of matrix V , we obtain two different vertices i.e., the values of augmented state vector x e for each non-identical vertices in X i is [w p , u min ] and [w p , u max ] ∀p ∈ I p . In the following, we investigate the robustness problem for method 1.

C. Controller for extended state system

The controller for the extended model ( 34 where F e i = [F i 0] ∈ R m×ne and g e i = g i ∈ R m×1 . The controller has no impact on the state x I . This is just an extension of ( 5) to accommodate the extended state and therefore the characteristics of the original controller (5) are preserved.

D. An admissible set for the first order neglected dynamics:

We introduce few matrices which will be used in the construction. The extended model (34)-(35) can be written as a convex combination of the extreme realization obtained with α = 0 and α = 1.

For α = 0,

A e 1 = A 0 n×m 0 m×n 0 m×m , B e 1 = B 1 m×m (37)
For α = 1,

A e 2 = A B 0 m×n 1 m×m B e 2 = 0 n×m 0 m×m (38)
Therefore,

A e = (1 -α)A e 1 + αA e 2 , B e = (1 -α)B e 1 + αB e 2 (39)
Since our objective follows the computation of maximum α value assuring the invariance of the set X , we just need the state and input values in order to produce for the state vector x(k + 1) in (33).

A

1 = A 0 n×m ∈ R n×ne A 2 = A B ∈ R n×ne B 1 = B and B 2 = [0 n×m ].
Similar to (9), we introduce a matrix V

(1) e ∈ R ne×p which stores the vertices of the state vector x(k) and the PWA control input associated with it for all the non-identical vertices in X i ∈ R n , i ∈ I N in order to create a matrix that contains the vertices for the state vector x e .

V (1) e = [V, U ] (40)
In a second step, the values of the state vector x e (k + 1) are found by exploiting (40) and (36) for column vectors of the matrix V

(1) e in a closed loop formulation. This leads to the matrix V

(2) e ∈ R ne×p . V (2) e = AV (1) e + BU (41)
Similar to the notation in [START_REF] Herceg | Multi-Parametric Toolbox 3.0[END_REF], a PWA image for the matrix V

(2) e can be found with the control law (36) and stored in a matrix U

(2) e ∈ R m×p . U (2) e = u e pwa [V (2) e ] (42)
Definition 4: Consider a linear discrete time system affected by the first order neglected dynamics (34)-( 35) and stabilized via a PWA state feedback control law (36). A margin for first order neglected dynamics is characterized by a set Ω α ∈ R m which contains the values of parameter α (for each input channel) such that the invariance property of the set X is assured.

Theorem 3: Consider the extended system (34) subject to first order neglected dynamics stabilized by a piecewise affine control law (36). The admissible set of parameters for the neglected dynamics is given as the projection,

Ω α = P roj α T (43)
where T denotes the polyhedral set:

T = (α, Γ) ∈ R × R r×p | (1 -α)(A 1 V (2) e + B 1 U (2) e ) + α(A 2 V (2) e + B 2 U (2) e ) = W Γ (44)
Proof: If Ω α describes admissible set for neglected dynamics ∀α ∈ Ω α then ((1 -α)A 1 + αA 2 )x e (k) + ((1α)B 1 + αB 2 )u e pwa ∈ X , ∀k ∈ Z [1;K] . We remark that the computation of the admissible set Ω α corresponds to the variations between the nominal model ( 1) and the model affected by the neglected dynamics (34).

(1 -α)(A 1 x e (k + 1) + B 1 (F e i x e (k + 1) + g e i )) + α(A 2 x e (k + 1) + B 2 (F e i x e (k + 1) + g e i )) ∈ X (45)

By replacing x e (k) with the column elements of the matrix V

(2) e and similarly the PWA function u e pwa = F e i x e (k + 1) + g e i with the column vectors of matrix U

e . Equation ( 45) is further modified into:

(1 -α)(A 1 V (2) e (:, q) + B 1 U (2) e (:, q)) + α(A 2 V (2) e (:, q) + B 2 U (2) e (:, q)) ∈ X (46)

For q ∈ I p , by representing the state vector x ∈ X as the convex combination of column vector of the matrix W , one can write r l=1 γ l W (:, l) and r l=l γ l = 1 and introducing this term in (46) obtain (1 -α)(A 1 V (2) e (:, q) + B 1 U (2) e (:, q)) + α(A 2 V (2) e (:, q) + B 2 U (2) e (:, q)) = W γ (47) with a vector γ = [γ 1 , • • • , γ r ] ∈ R r . If (47) holds for one column vector of matrix V

e , then it holds for all the column vectors of the matrix V

Example

For the single input system given in the previous section, the margin for the first order neglected dynamics is represented by the set Ω α = [{0, 0.267}]. The invariance property of the controller is preserved for any input filter with parameters in these values.

VII. CONCLUSION

In this work the gain margin set assuring the invariance of the closed loop obtain for a linear discrete time system controlled by a piecewise affine control law has been computed. For a discrete time system affected by first order neglected dynamics, a robustness margin has been also deduced assuring the invariance property. It is worth to be mentioned that this analysis procedure proposed does not extend to analysis to the convergence (asymptotic stability properties). Future work will include the robustness analysis for PWA control law assuring the contractivity.

1 .

 1 The prediction horizon chosen is 2. The input constraint -5 ≤ u(k) ≤ 5 and the output constraint -5 ≤ y(k) ≤ 5. Solving the mp-QP yields 13 controllers. The gain margin set K in (11) is computed and the value of δ K lies between [-0.2178, 0.3051].

Fig. 1 :

 1 Fig. 1: state trajectories for δ K = 0.3

Fig. 2 :

 2 Fig. 2: state trajectories for δ K = -0.25

Fig. 3 :

 3 Fig. 3: Gain Margin set

Fig. 4 :

 4 Fig. 4: Closed loop system with first order neglected dynamics

  ) can be represented by, u e pwa (x e (k)) = F e i x e (k) + g e i (36)

+ B 2 U

 2 of the piecewise affine defined by the matrix U(2) e . Equation (47) can be written as,(1 -α)(A 1 V (2) e + B 1 U (2) e ) + α(A 2 V (2)e (2) e ) = W Γ (48) with Γ ∈ R r×p and the proof is completed.


Iris Ballesteros-Tolosana

Sorin Olaru

Pedro Rodriguez-Ayerbe

Guillermo Pita-Gil

Renaud Deborne

Comparison of optimization-based strategies for constrained control of Auto-Steering systems

This paper presents a comparison between two approaches addressing vehicle lateral dynamics control. In the framework of Auto-Steering for target tracking application, Model Predictive Control (MPC) and Interpolation Based Control (IBC) are studied on a similar model-based design. Both control strategies are of interest by their ability to treat problem constraints and actuator limits in a systematic manner, as well as their capability to include changes of the behavior when the speed of the vehicle changes, yielding a linear parameter varying system. Speed variation for the MPC controller brings an uncertain model, that will be described by a polytopic class of dynamics. Thanks to the online measurement of the parameter, the system dynamics will be computed at each sample time to solve the optimization problem. Parameterdependent Lyapunov functions and positive set invariance theory are used to ensure stability and feasibility. Interpolation Based control is a different approach, whose principle is to use a control action constructed as an interpolation between two computed extreme values. Each time, two linear programming problems are solved, which makes this approach to be a suitable trade-off between performance and computation cost.

I. INTRODUCTION

In the recent years, Advanced Driving Assistance Systems (ADAS) have gained great importance in the automotive industry. These are complex control systems that are able to compensate or take control over the vehicle dynamics, in order to avoid potential dangerous situations, or spare the driver from tedious quotidian driving situations. Driving conditions are never the same, so it becomes a critical feature to come up with robust control strategies that always keep a correct performance and ensure the safety of the system by constraints handling. This study presents two different generic approaches for the control design of a vehicle Auto-Steering system, where the uncertainty produced by the variation on the speed is considered explicitly.

The exposition is organized as follows. Section II provides an overview of the auto-steering system and the dynamical model used. In the subsequent, the two studied methods are detailed: in Section III Model Predictive Control is considered and continuing with the study, Interpolation Based Control is applied in Section IV. After that, Section V shows simulations that have been carried out in order to test the methods and draw the conclusions in Section VI. 1 Laboratory of Signals and Systems (L2S), Centrale-Supèlec, CNRS, Université de Paris-Saclay, France. {sorin.olaru, pedro.rodriguez}@centralesupelec.fr 2 RENAULT SAS, DEA-SAD5, Development ADAS control laws.

{iris.ballesteros-tolosana, renaud.deborne, guillermo.pita-gil}@renault.com

II. AUTO-STEERING SYSTEM AND CONTROL PROBLEM FORMULATION

A. Auto-steering System

Several auto-steering control systems have been developed in the literature and industry. These active control systems act on the lateral vehicle dynamics control in order to offer safety, like the Lane Keeping Assistance [START_REF] Enache | Assistance préventive à la sortie de voie[END_REF] or Stability Control Systems [START_REF] Doumiati | Integrated vehicle dynamics control via coordination of active front steering and rear braking[END_REF], and comfort, such as Target Tracking or Lane Centering Assistance. All of them cooperate with the longitudinal vehicle dynamics control, that is an independent control system which will be in charge of the vehicle's longitudinal speed, acceleration and distance with the preceding vehicle, if any. These longitudinal control variables are measured or estimated at each sample time and will be used as external parameters in the lateral control systems. This means that in the present work the control structure is divided in two parts, where lateral and longitudinal dynamics of the vehicle are assumed to be decoupled.

The auto-steer system object of study in this application case is comprised in the Traffic Jam Pilot (TJP), that is in charge of the vehicle steering angle at low speed when a target vehicle is detected, that is, the vehicle which runs ahead of our vehicle. The controller goal is to follow this target lateral position and yaw angle. Vehicle dynamics have been studied and targeted in different contexts, [START_REF] Sename | Robust control and linear parameter varying approaches: application to vehicle dynamics[END_REF], [START_REF] Fergani | Study and comparison of non linear and LPV control approaches for vehicle stability control[END_REF], [START_REF] Bokor | Linear parameter varying systems: A geometric theory and applications[END_REF], being common knowledge in both mechanical and control fields. In [START_REF] Carvalho | Automated driving: The role of forecasts and uncertainty-a control perspective[END_REF] the commonly used models for vehicle steering control are outlined. In the present work, a dynamic model based on the interacting lateral forces between the vehicle and the road surface through the tires is used. The tire forces modeling is an extensive topic itself [START_REF] Pacejka | Tire and vehicle dynamics[END_REF]. In this paper, linear and uncoupled tire forces behavior is considered with the objective of control design for normal driving conditions. We are not interested in the control of the vehicle in large slip angles and other such extreme conditions [START_REF] Di Cairano | Vehicle yaw stability control by coordinated active front steering and differential braking in the tire sideslip angles domain[END_REF].

The adopted dynamical model is based on the known bicycle model, as it exhibits a good compromise between complexity and performance. In [START_REF] Rajamani | Vehicle dynamics and control[END_REF] the lateral dynamics of a vehicle bicycle model moving along an infinite surface are given. However, in order to represent the dynamics of our system, it is convenient to locate the model in a different reference, that is, the target vehicle and a virtual perfect following vehicle (Fig. 1). Thus ψ will be related to the relative angle between the perfect following vehicle and the controlled one. Following this last remark we adopt ψ rel for this angle and denote by y CoG the lateral difference between the trajectory of the ego vehicle and the perfect following one. y cam stands for the lateral offset between our vehicle and the target vehicle. Thereby, the tracking objective of the auto-steering control problem can then be expressed as a problem of stabilizing the dynamics of the controlled vehicle with respect to the virtual perfect following one.

After taking this into consideration, the conceived model is expressed as ẋ(t) = Ax(t) + Bu(t), y(t) = Cx(t) with:

A =      -(C f l 2 f +Crl 2 r ) Izv (C f l f -Crlr) Iz -(C f l f -Crlr) Izv 0 1 0 0 0 (Crlr-C f l f ) mv (C f +Cr) m -(C f +Cr) mv 0 0 0 1 0      B =     C f l f Iz 0 C f m 0     C = 1 0 0 0 0 x m 0 1
(1) Where x(t) is the state vector [ ψ, ψ rel , ẏCoG , y CoG ] T ∈ R n , u is the control input ∈ R p which represents the steering wheel angle δ c and y ∈ R m holds by the vector of system outputs, that will be [ ψ, y cam ] T . Complete measurements of the states are considered in the present study. The transition matrix A ∈ R n×n and input matrix B ∈ R n×p depend on several vehicle parameters: C f and C r denote the cornering stiffness of the front and rear wheels respectively. The distances from the front and rear axis to the center of gravity are l f and l r , m stands for the vehicle mass and I z for the total yaw moment of inertia. In addition to this, the observation matrix C ∈ R m×n depends on x m which is the longitudinal distance with respect to the followed target. All of them are considered known and fixed, so they do not introduce any uncertainty to the model.

Finally, one of the main parameters appearing in (1) is the speed of the vehicle. In practice, the varying parameter will be defined as the inverse of the longitudinal speed of the vehicle, ν = 1/v. The speed range in which the auto-steer system will be active, [1-40 km/h], is known, so the parameter is bounded, ν ∈ V. Its time-varying (and measured) characteristic yields a linear parameter varying (LPV) model. In the following, discrete-time formulation is used. For doing so, the forward Euler discretization method based on a truncated Taylor series expansion with T s = 0.01s has been applied to the continuous-time system [START_REF] Enache | Assistance préventive à la sortie de voie[END_REF].

There are several uncertainty sources when modeling a system, and consequently various ways to describe them when considered in the control problem formulation [START_REF] Kothare | Robust constrained model predictive control using linear matrix inequalities[END_REF], [START_REF] Bemporad | Robust model predictive control: A survey[END_REF]. In this paper, a linear parameter-varying formulation will be considered which can in turn be embedded in a polytopic uncertainty approach, as any operating speed can be obtained as a convex combination of the maximal and minimal ones due to the monotonicity of the curve. For prediction purposes, the system dynamics will be described by the parameter varying model

x k+1 = A(ν k )x k + Bu k y k = Cx k (2)
With A(ν k ), B, C described by the discrete time formulation of (1), and

A(ν k ) = nv i=1 α i A(ν i ) with nv i=1 α i = 1, α i ≥ 1 and n v = 2.

C. Constrained control objectives

Industrial applications involve constraints that need to be taken into account at the design stage in order to fulfill security conditions or actuator limitations. The control design methods considered next are appealing for this constraint handling capability, which allows the system to guarantee a correct performance and to work closer to the limits without running any risks. In practice the dynamical system has limitations with respect to the states, outputs and in the control action, that is

x ∈ X, y ∈ Y, u ∈ U (3)
where X ⊂ R n , Y ⊂ R m and U ⊂ R p are the bounded sets of admissible states, outputs and inputs, respectively, all of them containing the origin in their interior..

III. CONSTRAINED MODEL PREDICTIVE CONTROL

A. Problem formulation

A conventional Model Predictive Control (MPC) strategy is used [START_REF] Rawlings | Model predictive control[END_REF]. Each time, a finite horizon Optimal Control Problem (OCP) is formulated, which in this case consists of solving a quadratic programming (QP) to minimize the control input rate along the prediction horizon subject to linear equality and inequality constraints.

min ∆U J(x k , ∆U ) = xN 2 P + N -1 k=1 xk 2 Q + ∆u k 2 R s.t. xk+1 = Ā(ν k )x k + B∆u k y k = C xk x ∈ X, y ∈ Y, ∆u ∈ ∆U xN ∈ Ω (4)
where ∆U = [∆u T k , ∆u T k+1 , . . . , ∆u T k+N -1 ] T , using an augmented state formulation, with

xT k = [x T k u T k-1 ] T and considering the known velocity form, ∆u k = u k -u k-1 ,
which has additional interesting properties from the offset free tracking point of view [START_REF] Pannocchia | The velocity algorithm LQR: a survey[END_REF]. Q 0 and R 0 represent respectively the positive definite state and input weighting matrices along the prediction horizon N [START_REF] Chiriac | A model predictive control approach for the pantograph-catenary positioning system[END_REF]. Moreover, a quadratic terminal cost and terminal set Ω associated to an stabilizing gain for the unconstrained system are considered to ensure system's recursive stability and admissibility with respect to constraints and will be further studied in (III-B). Finally, in order to formulate this finite-time OCP, only the current speed is known (II-B), so the system matrices in (2) are computed at each sample time and are considered constant along the prediction horizon based on the measured ν k . This constant parameter profile along the prediction horizon is feasible with respect to the available information and preferable to a robust MPC design which is conservative from the performance point of view.

B. Feasibility, cost function and stability

A classical construction for the guarantee of stability and recursive feasibility in predictive control is the use of the so-called terminal set -terminal cost in the design [START_REF] Mayne | Constrained model predictive control: Stability and optimality[END_REF]. These constraints are intended to transform the cost function into a pseudo-infinite cost-to-go and thus play the role of a Lyapunov function for the closed loop system.

1) Terminal cost and stabilizing gain for unconstrained system: Taking into account the parameter-varying nature of the dynamics, the terminal cost will be based on the construction of a parameter-dependent Lyapunov function. In [START_REF] Daafouz | Parameter dependent Lyapunov functions for discrete time systems with time varying parametric uncertainties[END_REF] necessary and sufficient conditions for the computation of such functions are derived. Then, in [START_REF] Cairano | Model adjustable predictive control with stability guarantees[END_REF] this formulation is modified in order to achieve a finite-time receding-horizon formulation, by means of enforcing a cost decrease leading to a parameter-varying terminal cost and a stabilizing control law for the unconstrained system. The LMI designed to obtain a parameter-dependent terminal cost for ν ∈ V is recalled here for completeness. We point the interested readers to [START_REF] Cairano | Model adjustable predictive control with stability guarantees[END_REF]- [START_REF] Mao | Robust stabilization of uncertain time-varying discrete systems and comments on an improved approach for constrained robust model predictive control[END_REF] for further discussions on the subject and proof for Lemma 1.

Lemma 1: Let the symmetric positive definite matrices S i , S l and G i , Y i , Y l of appropriate dimensions, be such that

    G i + G i T -S i (A i G i + BY i ) T Y i T G i T (A i G i + BY i ) S l 0 0 Y i 0 R -1 0 G i 0 0 Q -1     > 0 (5
) for all i = 1, ..., n v and l = 1, ..., n v . Then G i is full rank and the parameter-dependent Lyapunov function is given by

P i = S i -1
, and the parameter-dependent stabilizing gain by

K i = Y i G i -1 which yields the stabilizing parameter dependent control law u k = nv i=1 K i (ν)x k .
The number of degrees of freedom introduced in the problem through the matrices G i , S i , S l can be adapted to obtain a common Lyapunov function for all the parameter values, denoted P , by imposing S i = S l = S and G i = G or a parameter-dependent one P (ν), that will change depending on the current measured value of the speed.

2) Invariant sets: Set invariance theory is largely used in control applications [START_REF] Blanchini | Survey paper: Set invariance in control[END_REF], in order to assess critical features such as feasibility, stability or robustness.

Maximal output admissible set (MOAS) [START_REF] Gilbert | Linear systems with state and control constraints: The theory and application of maximal output admissible sets[END_REF]: considering a constrained stable linear dynamical system x k+1 = A cl x k , y k = Cx k , a set Ω is output admissible if x 0 ∈ Ω implies that x k ∈ Ω for all future times, k > 0 and no constraints violation takes place. This is equivalent to

Ω = {x ∈ R n : CA k cl x ∈ Y }.
We are interested in transition matrices A cl , corresponding to the closed loop form, that is, Ā(ν) + BK(ν) where the stabilization is done with the parameter-dependent gain K(ν) (Lemma.1). In [START_REF] Gilbert | Linear systems with state and control constraints: The theory and application of maximal output admissible sets[END_REF] and [START_REF] Borrelli | Predictive Control for linear and hybrid systems[END_REF], the properties of this kind of robust invariant sets are characterized and recursive algorithms are introduced in order to compute them.

Two MOAS have been computed,

Ω P = {x ∈ R n : H Ω P x ≤ b Ω P } and Ω P (ν) {x ∈ R n : H Ω P (ν) x ≤ b Ω P (ν) }.
Both are tested in the present study as terminal sets for the optimization problem (4). For Ω P , the parameter-dependent stabilizing gain is obtained with the common Lyapunov function P . Ω P (ν) is computed using a parameter-varying control based on the parameter-varying Lyapunov function P (ν). As expected, Ω P proves to be quite restrictive (Ω P ⊂ Ω P (ν) ). This is further analyzed in Section (V).

N-step controlled invariant set (CIS): given a positive invariant set Ω ⊂ X with respect to a pre-defined control law, a set C N is N-step controlled invariant if any x 0 ∈ C N can be driven to Ω in N-steps while satisfying the constraints by applying admissible control inputs, u k ∈ U . This set can be computed for the constrained LPV system (2), [START_REF] Sename | Robust control and linear parameter varying approaches: application to vehicle dynamics[END_REF], by means of recursive pre-image set construction with respect to Ω and for each extreme realization of the polytopic uncertainty description [START_REF] Blanchini | Set-theoretic methods in control[END_REF], [START_REF] Takács | Reachability analysis and control synthesis for uncertain linear systems in MPT, author=Kvasnica[END_REF]. Such a N-step controlled set is denoted here by

C N = {x ∈ R n : H C x ≤ g C }.
If the procedure is initiated with a robust control invariant set Ω then C N is robust control invariant. Further details on such constructions and the available tools can be found in [START_REF] Borrelli | Predictive Control for linear and hybrid systems[END_REF], [START_REF] Blanchini | Set-theoretic methods in control[END_REF].

In the case of lateral vehicle dynamics, the CIS characterization leads to a complex set in R 5 , and the computational load becomes excessive. Due to this limitations, the 3-steps controlled invariant set, C 3 will be used in the present study. In (Fig. 2, the MOAS Ω P (ν) and the CIS C 3 for the LPV system (2) is shown via cutting with the state constraints polyhedron X (3). It can be seen that the limits for ψ rel are relatively small compared to the admissible values at the cutting values, so the initial conditions for this state will highly condition the feasibility of the control problem. The limits for the rest of the states are close to the state constraints and are not showed here due to limited space.

3) Recursive feasibility: As mentioned above, the parameter ν(k) is known at each moment in time but no other information is available for the prediction, so the MPC problem is solved for the system related to the current speed, and considered constant along the whole prediction horizon. This is an issue with respect to the recursive feasibility, which has to consider the worst case parameter variation. Indeed, the tail of the optimal sequence would continue being feasible only if the speed is constant. In order to ensure recursive feasibility, one solution would be to take into account all the possible combinations of parameters realization along the prediction horizon. Nevertheless, this option would lead to a high computational load, as the number of possible prediction model combinations would increase exponentially with the prediction horizon length.

In the present study we adopt a slightly different approach but that will not increase the computational burden. An additional constraint is considered in the nominal MPC problem (4): the one step-ahead predicted state belongs to the robust controlled invariant set (III-B.2) x k+1 ∈ C N . This ensures that independently of the parameter evaluation on the next time step, the problem remains feasible. More than that, the fact that the predicted nominal trajectory enters in the terminal set, ensures the optimality on a pseudo-infinite horizon for the time-invariant prediction model.

IV. INTERPOLATION BASED CONTROL

This section is introduces the main idea of a relatively novel constrained control approach [START_REF] Nguyen | Implicit improved vertex control for uncertain, time-varying linear discretetime systems with state and control constraints[END_REF], whose online computational load is much lower than in the previous method. The main idea is to apply a control action that is interpolated between the stabilizing unconstrained controller and the control in the controlled invariant set.

A. Problem formulation

The IBC technique is based on the fact that any state inside the CIS can be expressed as the linear combination of two states:

x k = η k x c k + (1 -η k )x o k , where x c k ∈ C N , x o k ∈ Ω and 0 ≤ η ≤ 1.
In a similar way, the following control law is considered:

u k = η k u c k + (1 -η k )u o k
where u c k is obtained by solving an (optimization-based) control problem which exploits the controlled-invariance and u o k will be obtained using a parameter-varying stabilizing feedback gain K(ν k ).

In order to obtain the input signal u k , the first step is to compute the coefficient η k . In order to do so, a nonlinear optimization problem is build, whose objective will be to minimize this coefficient in order to be as close as possible to the unconstrained local controller. 6) in a LP, the following change of variables can be done [START_REF] Nguyen | Control synthesis for switched systems with control and state constraints[END_REF]:

η * k = min xc k ,xo k ,η k {η k } s.t.        H C x c k ≤ b C H Ω x o k ≤ b Ω x k = η k x c k + (1 -η k )x o k 0 ≤ η k ≤ 1 (6) To transform (
r c k = η k x c k , r o k = (1 -η k )x o k . Hence, η * k = min rc k ,η k {η k } s.t.    H C r c k ≤ η k b C H Ω (x k -r c k ) ≤ (1 -η k )b Ω 0 ≤ η k ≤ 1 (7)
Once the optimal interpolation factor η * k is known, the states x c k and x o k are also obtained as a by-product. The only missing element is the admissible input signal corresponding to x c k , that is, an input signal u c k that will keep a state that belongs to the border of C N inside it. A simple one-step optimization problem constructs this control action by optimizing the contraction factor γ k :

γ * k = min uc k ,γ k {γ k } s.t.    H C ( Āi x c k + Bu c k ) ≤ γ k b C H U u c k ≤ b U 0 ≤ γ k (8)
with i = 1, 2, ...n v . As it can be seen, the IBC method consists of a pair of linear programming problems with n+1 arguments, which are considerably simpler than the N -step MPC optimization (QP).

B. Stability and feasibility

The interpolation based control scheme ensures asymptotic stability for all initial states inside the controlled invariant set, C N , with the interpolation factor η k playing the role of a Lyapunov function on C N \ Ω. By exploiting the controlledinvariance properties of C N it can be shown that the problem (8) is always feasible, [START_REF] Nguyen | Control synthesis for switched systems with control and state constraints[END_REF].

From the construction of the invariant set in (Fig. 2), the strict inclusion of the CIS, C N , in the state constraints was obvious and it could happen that the states to control lay out of its boundaries, which would bring up an unfeasible problem when solving [START_REF] Pacejka | Tire and vehicle dynamics[END_REF] in X \ C N . A simple way to avoid the unfeasibility for the states out of C N , is to soften the constraints in order to contain the current state. The minimum scaled polyhedron C N that contains the current system state x k is computed form the original C N solving the following LP:

β * k = min{β} s.t. H C x k ≤ βb C (9)
Once β * k is known, the scaled set for the current iteration is defined

C N = {x ∈ R n : H C x ≤ b C } with H C = H C and b C = β * k b C .
This artifact ensures the feasibility in (8) as long as the state constraints are feasible. However the stability will be guaranteed only for β * < 1.

V. SIMULATION STUDIES AND ANALYSIS

In this section the two controllers derived in Sections III and IV have been tested on a extensive range of initial conditions. In order to present a relevant comparison for a usual configuration, a scenario with a trapezoidal profile of varying speed between [1-40 km/h] has been defined. The simulated vehicle will be running in a straight road while the ego vehicle is initialized at a perturbed initial state.

A. Basic analysis of the MPC feasibility

The safe behavior of the controller is a critical feature when driving an application in which human life is involved. We want to ensure robust feasibility of the problem from the design stage. For doing so, we have to be able to steer the system state to the terminal set Ω in N steps ahead in time, fulfilling all the system input and output constraints no matter the (bounded) speed of the vehicle. This feature is influenced by three arguments of the MPC design: first, the initial conditions x 0 . Second, the length of the prediction horizon N and last, the size of the MOAS, Ω. If we start at initial conditions that are far away from the MOAS, we will need to increase the prediction horizon, in order to get a feasible problem with x N ∈ Ω. At the same time, the size of the MOAS will make this task easier to achieve at the price of an increase complexity of the respective polyhedral set. This first test analyzes the influence of the prediction horizon length and the terminal set constraint size. For doing that, we have run a simulation starting from the possible maximum values of the initial conditions x 0 = [0.05, 0.05, 0.2, 0.3, 0.15] T , checking which would be the necessary prediction horizon length in order to derive a feasible optimization problem when the terminal set is Ω P or Ω P (ν) . With the first MPC choice, the MPC has not been able to solve the problem, even considering a prediction horizon up to a prediction window of N = 500 steps, which would anyways yield a problem without a practical meaning, due to the huge dimension of the optimization problem. On the other hand, the MPC controller with Ω P (ν) provides a solution when N ≥ 173. This is mainly due to the difference on the size of the terminal sets Ω. In addition to this, it has been showed in Fig. 2 that the size of Ω P (ν) in the ψ rel dimension is very restrictive. That means that the domain of attraction for this state is relatively low, constraining the initial conditions for this angle at the 10% of its maximal value, 0.5 [rad]. This means that this state will be the most sensitive from the prediction horizon point of view.

B. Analysis of the IBC feasibility

In this case the control technique is not affected by a prediction horizon but by the size and complexity of the positive invariant sets used in the formulation. This controller ensures feasibility to those points that belong to the original CIS, C N or the scaled one C N , (IV-B). In order to check the closed loop behavior, we have performed several simulations for a grid of initial conditions, varying from [0, 0, 0, 0, 0] T to the maximum values inside a R 5 box. Four different IBC controllers have been tested. In Table I it can be seen that the use of a bigger MOAS, Ω P (ν) and the scaling technique for C N provides an improved feasibility.

C. Numerical simulation

In order to allow a detailed analysis, a simulation case is presented together with the time-signals. The initial conditions have been fixed to x 0 = [0.025, 0.02, 0.15, 0.25, 0] T , which represent a common situation in which the system would be activated. For the MPC, the prediction horizon has been fixed according to the time to collision, that is often fixed at 2[s]. This corresponds to N = 200 (II-B). Finally, just the acceleration phase is shown in the simulation. The first MPC (using P for the terminal cost formulation and Ω P for the terminal set constraint) has a domain of attraction which makes the MPC policy unfeasible for this case study, being unable to solve the problem for such initial conditions, so it has been discarded from the test. At the same time, the IBC controller that use a fixed C N is not applicable: the initial point is out of the C N , so the strategies that use a fixed C N are not applicable. In Figures 3,4 and 5, the time-trajectory for the MPC controller using P (ν) for the terminal cost formulation and Ω P (ν) for the terminal set constraint and the IBC using the scaling technique are shown. In Figure 6, the IBC interpolation factor is shown. It is interesting to note that η k = 0 after 2, 5 [s], implying that from that time instant, the state of the closed-loop system is in the invariant set Ω. Its monotonic decrease and the positivity confirms the Lyapunov interpretation given in IV-B.

VI. CONCLUDING REMARKS AND FUTURE WORK

Two approaches for the design of auto-steering target tracking control have been considered, when the system dynamics are described by a parameter-varying model where the parameter, vehicle's speed, is bounded and measured. On one hand the MPC has been tested, starting from a recognized technique with a mature theoretical background. The online resolution of an optimization problem in the available embedded control units for the ADAS becomes nowadays feasible due to the computational power available.

As an alternative optimization-based control, the IBC proposes a computational complexity on a par with the one step horizon MPC controller, while keeping the constraints handling feature. This makes this controller an interesting candidate for applications where the computation power available is not high. The main problem of this kind of controller is the complexity of the positive controlled invariant set, C N . Although the computation of this set is made offline and does not affect the online solver, this set increases tremendously its complexity each step we take backwards when computing it the backward reachability construction (III-B.2) thus, its size is relatively small comparing to the admissible state space set X. This drawback has been improved by a scaling technique that increases the domain of attraction of the interpolation based controller. However, this type of controller would not be advisable for a system with such complex positive invariant sets.

From the application point of view, the system studied in the present work has been simplified in order to provide an scenario where the attention is focused in the variation of the speed. There are more features which will be included in future controllers. These include actuator dynamics and additive disturbances, like wind forces acting on the vehicle or the curvature of the road.

Fig. 1 :

 1 Fig. 1: Auto-steering system for target tracking

Fig. 2 :

 2 Fig. 2: Invariant sets: cut [x 2 , x 3 , x 4 , 0, 0]

Fig. 3 :

 3 Fig. 3: States trajectories

Fig. 6 :

 6 Fig. 4: Outputs trajectory

TABLE I :

 I Interpolation based control tests

	MOAS	C N	N • feasible x 0	N • unfeasible x 0
	Ω P	Fixed Scaled	50.76% 57.36%	49.24% 42.63%
	Ω P (ν)	Fixed Scaled	80.24% 85.23%	19.76% 14.77%

ACKNOWLEDGMENT

The work leading to these results has received funding from the People Programme (Marie Curie Actions) of the European Unions Seventh Framework Programme (FP7/2007-2013) under REA grant agreement no 607957 (TEMPO).