Hussam Atoui

Olivier Sename

Vicente Milanes

John J Martinez

Advanced LPV-YK Control Design with Experimental Validation on Autonomous Vehicles ⋆

Keywords: Linear parameter-varying system, Youla parameterization, Gain-scheduling, Switching control

This paper presents and experimentally validates two new methodologies to perform a smooth switching between a set of Linear Parameter-Varying (LPV) controllers that have been designed separately for different objectives. For a given partition of the scheduling parameter region, a set of LPV controllers (polytopic and grid-based) are designed separately on each subregion based on Youla-Kucera (YK) parameterization. Such kind of parameterization is beneficial to switch or interpolate between multiple controllers without adding any constraints to the design of the local controllers and the switching signals.

The closed-loop system is proved to guarantee stability and performance under arbitrary switching in terms of a set of Linear Matrix Inequalities (LMIs). An application to autonomous vehicle lateral control is chosen for its challenging objectives. Experimental validation and comparison are depicted to illustrate the performance improvement compared to previous LPV control approaches.

Introduction

Linear Parameter-Varying (LPV) control techniques are widely used in different applications as in aerospace [START_REF] Jiang | Gain-scheduled control for morphing aircraft via switching polytopic linear parameter-varying systems[END_REF], [START_REF] He | Simultaneous design of smooth switching state-feedback lpv control[END_REF], engine control [START_REF] Qu | Engine egr valve modeling and switched lpv control considering nonlinear dry friction[END_REF], or autonomous vehicles [START_REF] Atoui | Design And Experimental Validation Of A Lateral LPV Control Of Autonomous Vehicles[END_REF], [START_REF] Atoui | Parameter Varying Approach For A Combined (Kinematic + Dynamic) Model Of Autonomous Vehicles[END_REF]. [START_REF] Hoffmann | A survey of linear parametervarying control applications validated by experiments or high-fidelity simulations[END_REF] presents . The synthesis of an LPV control can be formulated as a Linear Matrix Inequality (LMI) optimization problem using a single Lyapunov function, either quadratic [START_REF] Packard | Gain scheduling via linear fractional transformations[END_REF] or parameter-dependent [START_REF] Wu | Induced l2-norm control for lpv systems with bounded parameter variation rates[END_REF]. The main applied LPV approaches are: 1) Polytopic [START_REF] Apkarian | Self-scheduled h control of linear parameter-varying systems: a design example[END_REF]; 2) Grid-based [START_REF] Wu | Control of linear parameter varying systems[END_REF]; and 3) Linear Fractional Transformation (LFT) [START_REF] Apkarian | A convex characterization of gain-scheduled H∞ controllers[END_REF]. These LPV approaches have been compared and experimentally validated on a real automated vehicle in our previous work [START_REF] Atoui | LPV-Based Autonomous Vehicle Lateral Controllers: A Comparative Analysis[END_REF]. On the other hand, a single Lyapunov function for complex models with large parameter variations may not exist, or if it exists, it could be conservative.

⋆ Institute of Engineering Univ. Grenoble Alpes

Email addresses: hussam.atoui@renault.com (Hussam Atoui), olivier.sename@grenoble-inp.fr (Olivier Sename), vicente.milanes@renault.com (Vicente Milanes), john-jairo.martinez-molina@grenoble-inp.fr (John J. Martinez).

The current paper aims to solve some of the main limitations of the polytopic and grid-based approaches, detailed later, based on Youla-Kucera (YK) interpolating/switching techniques. The interest behind YK concept is to parameterize a set of linear stabilizing controllers K(Q) where each one is parameterized by its corresponding YK parameter Q [START_REF] Niemann | Dual youla parameterisation[END_REF]. This kind of configuration is structured by mapping a set of linear stabilizing controllers onto a Q-based controller. The following two sections discuss the limitations of both polytopic and grid-based approaches, and some solutions that are proposed in the literature.

Grid-based Approach

The grid-based approach uses a single parameterdependent Lyapunov function to solve a set of parameterdependent LMIs (pLMIs). Since a single parameterdependent Lyapunov function could not be efficient for complex designs and large parameter regions, the first solution has been proposed in [START_REF] Lu | Switching lpv control designs using multiple parameter-dependent lyapunov functions[END_REF]. The objective is to design multiple LPV controllers based on multiple parameter-varying Lyapunov functions, each suitable for a specific parameter subregion, and switch between them to achieve better performance. The switching stability has been studied for hysteresis switching and switching with average dwell-time strategies. This methodology enhances the use of the switched LPV techniques in several applications, see for instance [START_REF] Lu | Switching lpv control of an f-16 aircraft via controller state reset[END_REF], [START_REF] Lescher | Switching lpv controllers for a variable speed pitch regulated wind turbine[END_REF], [START_REF] Postma | Air-fuel ratio control of spark ignition engines using a switching lpv controller[END_REF] , [START_REF] Colmegna | Switched lpv glucose control in type 1 diabetes[END_REF].

On the other hand, it has been stated in [START_REF] Lu | Switching lpv control designs using multiple parameter-dependent lyapunov functions[END_REF] that the switched LPV controllers may not provide a smooth transient response during switching, where aggressive performance is obtained at switching instants. Such a case may lead to mechanical damage, decrease material lifetime, or signal saturation which is out of real application objectives. Following this work, several research studies have been involved in solving the switching smoothness. For instance, a bumpless transfer of switching controllers is proposed in [START_REF] Dimon | Bumpless transfer between observer-based gain scheduled controllers[END_REF] followed by some developments in [START_REF] Kinnaert | State resetting for bumpless switching in supervisory control[END_REF], [START_REF] Chen | The smooth switching control for tora system via lmis[END_REF], and [START_REF] Safaei | On controller initialization in multivariable switching systems[END_REF].

Finally, a smooth switching LPV controller has been proposed first in [START_REF] Hanifzadegan | Smooth switching lpv controller design for lpv systems[END_REF]. It is designed in considering adjustable interpolation functions and a higher order differential control signal. An iterative descent algorithm is applied to optimize three decision variables (the parameter-dependent Lyapunov functions, the local controllers, and the interpolation functions). It also augments the problem to two dimensional parameter regions. This concept is developed in the recent works [START_REF] He | Simultaneous design of smooth switching state-feedback lpv control[END_REF] and [START_REF] He | Smooth switching lpv dynamic output-feedback control[END_REF], however, it increases the complexity and the design constraints of the local controllers to achieve their objectives.

Polytopic Approach

The polytopic approach is the most popular among the LPV control approaches. In many fields, especially aerospace, engineers are interested in applying the gain-scheduling method based on optimized designs at different operating points [START_REF] Bianchi | Interpolation for gain-scheduled control with guarantees[END_REF]. On the other hand, it is known to be the most conservative one [START_REF] Hoffmann | A survey of linear parametervarying control applications validated by experiments or high-fidelity simulations[END_REF]. One of the main causes is that the polytopic LPV synthesis requires a constant Lyapunov function to ensure quadratic stability, which increases the problem conservatism.

Moreover, the overbounding of the parameter set is considered as a main cause of conservatism. The operating region of the underlying LPV model is defined by a convex polytope containing the parameter trajectories. This convex parameter region may include vertices that are not attained by the real plant, resulting in conservatism. The reason is that the construction of the polytope is based on the assumption that all parameters vary independently, whereas they could be related to each other by inherent couplings. For example, the known bicycle model describing the lateral dynamics of an autonomous vehicle is parameterized by the scheduling parameters "v x " and "1/v x " [START_REF] Atoui | LPV-Based Autonomous Vehicle Lateral Controllers: A Comparative Analysis[END_REF] (being v x the longitudinal speed). Such a situation might cause unstable models at the polytopic vertices. In addition, the parameters could be physically correlated with each other, such that some combinations of extreme values of the parameters do not occur in real operation. For example, an LPV model describing the vertical flight dynamics of an aeroplane might be parameterized by the external scheduling signals (i.e., parameters) "speed" and "altitude." But usually, the maximum speed is not reached for minimum altitude and vice versa [START_REF] Kwiatkowski | Pca-based parameter set mappings for lpv models with fewer parameters and less overbounding[END_REF].

Several solutions have been investigated in the literature to find a reduced convex parameter region. Several solutions has been investigated in the recent survey [START_REF] Li | Polytopic lpv approaches for intelligent automotive systems: State of the art and future challenges[END_REF]. [START_REF] Azuma | A new lmi approach to analysis of linear systems depending on scheduling parameter in polynomial forms[END_REF] suggests to construct convex polyhedrons along given parameter trajectories, and solve the control design problem using affine parameter-dependent LMIs. Unfortunately, these methods often result in a huge number of vertices or nonconvex parameter sets and thus in increased computational burden. Scheduling Dimension Reduction (SDR) approach is proposed in [START_REF] Kwiatkowski | Pca-based parameter set mappings for lpv models with fewer parameters and less overbounding[END_REF] which reduces the parameter set based on experimental data, and yields the benefit of tailoring a control design to specific trajectories. In [START_REF] Koelewijn | Scheduling dimension reduction of lpv models -a deep neural network approach[END_REF], a Deep Neural Network (DNN) approach is used to develop the SDR methods and achieve higher model accuracy under scheduling dimension reduction. In [START_REF] Zhao | Optimal switching surface design for state-feedback switching lpv control[END_REF] and [START_REF] Zhao | Switching lpv controller design under uncertain scheduling parameters[END_REF], authors develop an LPV switching controller and prove its stability when switching among the overlapped subsets of a polytopic parameter region.

In [START_REF] Rasmussen | Stable controller interpolation and controller switching for lpv systems[END_REF], a YK configuration is proposed to improve the performance of a polytopic LPV control. On the other hand, [START_REF] Bianchi | Interpolation for gain-scheduled control with guarantees[END_REF] proposes a YK-based gain-scheduled controller by interpolating LTI controllers designed separately at the different vertices of a polytopic parameter region. The interpolation is performed as a function of the varying parameters of the LPV model. Closed-loop quadratic stability and performance are guaranteed at intermediate interpolation points of the convex domain. In [START_REF] Blanchini | Stable lpv realization of parametric transfer functions and its application to gain-scheduling control design[END_REF], a fixed pole-assignment application is introduced using an LPV YK-based method to preserve the closed-loop poles at the same location by interpolating between different controllers. In addition, in our previous work [START_REF] Atoui | Interpolation of multi-lpv control systems based on youla-kucera parameterization[END_REF], an LPV-YK control scheme has been proposed to interpolate between two LPV controllers, each one designed over a full polytopic region, to achieve multiple control performances. On the other hand, all the previous YK-based solutions are still conservative for systems having overbounding polytopic parameter region.

Motivation and Contribution

This paper proposes two advanced LPV-YK control approaches to improve the performance of the switched LPV controllers, with lower conservatism and design complexities. Among all the previous works which have shown successful and smooth LPV switching controllers as in [START_REF] Lu | Switching lpv control designs using multiple parameter-dependent lyapunov functions[END_REF], [START_REF] Chen | The smooth switching control for tora system via lmis[END_REF], and [START_REF] Hanifzadegan | Smooth switching lpv controller design for lpv systems[END_REF], all of them require the re-design of the local LPV controllers using proposed LMIs, de-pending on the switching signals (e.g. hysteresis switching, switching with average dwell-time, etc.). It is worth mentioning that the re-design of all the local LPV controllers together may cause conservatism when increasing the number of subregions or parameter dimensions. The current paper proposes the YK parameterization to:

( The paper is organized as follows: Section 2 designs the new grid-based LPV-YK concept with its stability proof. The partitioned polytopic-based LPV-YK control design with its stability proof is discussed in Section 3. Section 4 implements and experimentally validates both approaches on a real robotized RENAULT ZOE car. Finally, some concluding remarks are given in Section 5.

Notations in this paper are as follows. I[a, b] denotes the integer set from a to b. R stands for the set of real numbers. R m×n is the set of real m × n matrices. The transpose of a real matrix M is denoted by M T . I and 0 denote an identity matrix and a zero matrix, respectively, of appropriate dimensions. diag(X 1 , X 2 , ..., X N ) denotes a matrix with matrices X 1 , X 2 , ..., and X N as diagonal blocks. In the whole paper, the subscript i of a system/matrix/variable of an LPV plant (e.g. G(ρ)) denotes the LPV system/matrix/variable of the plant when the varying-parameter ρ ∈ P i , where P i is a subset of the full parameter region P. In addition, the second subscript j γ ∞ corresponds to the known γ-performance design, whereas γ is a switching signal.

2 Grid-based LPV-YK control design

The following section aims to design a grid-based LPV-YK controller that switches between multiple LPV controllers, each suitable for a specific parameter region. The closed-loop system is proved to remain stable and its performance is optimized over the whole gridded parameter subregions. This approach improves the smooth LPV-switched control domain which has been already studied in the pioneering works [START_REF] Lu | Switching lpv control designs using multiple parameter-dependent lyapunov functions[END_REF], [START_REF] Hanifzadegan | Smooth switching lpv controller design for lpv systems[END_REF], [START_REF] He | Simultaneous design of smooth switching state-feedback lpv control[END_REF], and [START_REF] He | Smooth switching lpv dynamic output-feedback control[END_REF].

LPV Plant and Controllers Description

Consider a Multi-Input-Multi-Output (MIMO) LPV system G(ρ) with m inputs and p outputs:

G(ρ)        ẋ(t) = A(ρ(t))x(t) + B 1 (ρ(t))w(t) + B 2 (ρ(t))u(t) z(t) = C 1 (ρ(t))x(t) + D 11 (ρ(t))w(t) + D 12 (ρ(t))u(t) y(t) = C 2 (ρ(t))x(t) + D 21 (ρ(t))w(t) + D 22 (ρ(t))u(t) (1) where x(t) ∈ R nx , y(t) ∈ R p , u(t) ∈ R m , z(t) ∈ R nz
are the state, output, input, controlled output vectors respectively. w(t) = r n d T ∈ R nw contains the exogenous inputs of the tracking reference r, noise n and input disturbance d. All the state-space data are continuous functions of the parameter vector ρ. Assume that ρ is in a compact set P ⊂ R s with its parameter variation rate bounded by ν k ≤ ρk ≤ ν k for k = 1, 2, ..., s. Moreover, let us assume the following:

• (A(ρ), B 2 (ρ), C 2 (ρ)) triple is parameter-dependent stabilizable and detectable ∀ρ ∈ P. • B T 2 (ρ) D T 12 (ρ) and C 2 (ρ) D 21 (ρ) have full row ranks ∀ρ ∈ P. • D 22 (ρ) = 0.
Suppose that the parameter set P is covered by a finite number of closed subsets {P i } i∈Z N , where the index set Z N = {1, 2, . . . , N }, and P = P i . At the boundaries between each adjacent subsets, there exist at least a single intersecting boundary or an intersecting surface.

Now, assume that (A.3.1).

There exists an LPV output-feedback controller K (0) (ρ) which exponentially stabilizes G(ρ) at the full parameter region P 0 := P. (following the approach in [START_REF] Wu | Induced l2-norm control for lpv systems with bounded parameter variation rates[END_REF]), (A.3.2). Over each parameter subset {P i } i∈Z N , there exists an LPV controller K i (ρ) pre-designed separately and exponentially stabilize G(ρ) over {P i } i∈Z N . Each K i (ρ) is designed to achieve a suitable performance in its corresponding parameter region {P i } i∈Z N The defined LPV controllers K i (ρ) (i ≥ 0) are described over P i as

K i (ρ) :   A k,i (ρ, ρ) B k,i (ρ) C k,i (ρ) D k,i (ρ)   , i ∈ {0, Z N } (2)
where

A k,i (ρ, ρ) ∈ R n k,i ×n k,i , B k,i (ρ) ∈ R n k,i ×m k , C k,i (ρ) ∈ R p k ×n k,i and D k,i (ρ) ∈ R p k ×m k .
The closedloop system performance is achieved in each parameter subregion and meanwhile maintains stability under any switching behavior via YK parameterization concept.

The switching occurs when the parameter trajectory hits one of the subsets boundaries. In our proposed approach, the switching signal could be any continuous/discontinuous switching signal γ. The LPV closedloop system for each pre-defined controller K i (ρ) over P i can be described by

CL i (ρ) :   A cl,i (ρ, ρ) B cl,i (ρ) C cl,i (ρ) D cl,i (ρ)   , i ∈ {0, Z N } (3) where x T cl,i = x T x T k,i ∈ R nx+n k,i (i ≥ 0)
. Notice that, for any i ∈ {0, Z N }, the closed-loop system (3) is required to satisfy the bounded real lemma over P i with a performance level γ ∞,i , i.e. ∥z∥ 2 < γ ∞,i ∥w∥ 2 and a symmetric, positive definite matrix functions X cl,i (ρ), each X cl,i (ρ) is smooth over its corresponding parameter subset P i , such that

        A T cl,i (ρ)X cl,i (ρ) + X cl,i (ρ)A cl,i (ρ) + s k=1 ±{ν k , ν k } ∂Xcl,i ∂ρk X cl,i (ρ)B cl,i (ρ) C T cl,i (ρ) B T cl,i (ρ)X cl,i (ρ) -γ ∞,i I nw D T cl,i (ρ) C cl,i (ρ) D cl,i (ρ) -γ ∞,i I nz         < 0 (4)

Problem Definition

The objective of this work is to obtain exponential stability of the closed-loop system based on YK parameterisation. Moreover, a smooth interpolation scheme K(ρ, γ)

is formulated between multiple pre-designed LPV controllers K i (ρ) (i ∈ Z N ), using the switching vector signal γ = [γ 1 , ..., γ i , ..., γ N ] , where each K i (ρ) is designed to be suitably used for a certain parameter subregion P i . This could be achieved by two steps:

(1) Parameterize each LPV controller K i (ρ) (i ∈ Z N ) with respect to the nominal LPV controller K 0 (ρ), by an LPV-YK parameter Q i (ρ). ( 2) At each boundary of two adjacent subsets P i and P i+1 , the interpolating signals γ i and γ i+1 are adjusted in a way to switch from K i (ρ) to K i+1 (ρ) or vise-versa. As a result, the overall parameterized LPV-YK controller K(ρ, γ) stabilizes G(ρ) ∀ρ ∈ P and for every continuous/discontinuous interpolating signals

γ i , i ∈ Z N .
Figure 1 shows the partitioned parameter region P with intersecting boundaries. The orange solid line represents the chosen nominal LPV controller K 0 (ρ), as defined by assumption (A.3.1). The blue solid lines represent the local LPV controllers K i (ρ) (i ∈ Z N ) as defined in assumption (A.3.2). The overall switching controller is performed using the interpolating signal γ = [γ 1 , ..., γ N ] (γ i ∈ [0, 1] ∀i), and is represented by the LPV-YK controller K(ρ, γ).

Main Results

Based on the statements on LPV concepts and YK parameterization, the overall interpolation scheme which is referred to K(ρ, γ) (see Fig. 4) is designed based on a Parametric Linear Matrix Inequality (PLMI) optimization problem, where its state-space matrices are represented as

K(ρ, γ) :   Ãk (ρ, γ) Bk (ρ, γ) Ck (ρ, γ) Dk (ρ, γ)   (5)
being γ(ρ) the vector of the parameter-dependent switching signals γ i (ρ) (i ∈ Z N ) that are chosen here as follows:

• if γ i (ρ) = 0 ∀i, K(ρ, γ) ≡ K 0 (ρ). • For any ρ ∈ P m , γ m (ρ) = 1 and γ i (ρ) = 0 ∀i ̸ = m, which implies that K(ρ, γ) is equivalent to F l (J(ρ), Q m (ρ)) that recovers K m (ρ) (
(ρ) ∈ R nx×nx , X k,0 (ρ) ∈ R n k,0 ×n k,0
, and matrices V (ρ) and W (ρ) such that for any ρ ∈ P:

A(ρ)X g (ρ) + X g (ρ)A T (ρ) + s j=1 ±{ν j , ν j } ∂X g ∂ρ j + B 2 (ρ)V (ρ) + V T (ρ)B T 2 (ρ) < 0 (6) A k,0 (ρ)X k,0 (ρ) + X k,0 (ρ)A T k,0 (ρ) + s j=1 ±{ν j , ν j } ∂X k,0 ∂ρ j + B k,0 (ρ)W (ρ) + W T (ρ)B T k,0 (ρ) < 0 (7)
And ∀ρ ∈ P, the state-space matrices of K(ρ, γ) in ( 5) are

Ãk(ρ, γ) =     A(ρ) + B2(ρ)Fg(ρ) -B2(ρ)Dq(ρ, γ)C2(ρ) -B2(ρ)Dq(ρ, γ)Fk,0(ρ) B2(ρ)Cq(ρ, γ) -Bk,0(ρ)C2(ρ) Ak,0(ρ) 0 -Bq(ρ)C2(ρ) -Bq(ρ)Fk,0(ρ) Aq(ρ)     Bk(ρ, γ) = B2(ρ)Dq(ρ, γ) Bk,0(ρ) Bq(ρ) T Ck(ρ, γ) = Fg(ρ) -(Dk,0(ρ) + Dq(ρ, γ))C2(ρ) Ck,0(ρ) -Dq(ρ, γ)Fk,0(ρ) Cq(ρ, γ) Dk(ρ, γ) = Dk,0(ρ) + Dq(ρ, γ) (8) where
A q (ρ) = diag(A q,1 , . . . , A q,i , . . . , A q,N ),

B q (ρ) = B q,1 . . . B q,i . . . B q,N T , C q (ρ, γ) = γ 1 C q,1 . . . γ i C q,i . . . γ N C q,N , D q (ρ, γ) = N i=1 γ i D q,i , (9)
being A q,i , B q,i , C q,i , and D q,i the state-space matrices of

Q i (ρ) ∀i ≥ 1 represented in (32), F g (ρ) = V (ρ)X -1 g (ρ), and F k,0 (ρ) = W k,0 (ρ)X -1 k,0 (ρ). Notice that X -1 g (ρ) and X -1
k,0 (ρ) are chosen to be smooth over the whole parameter region P. According to YK concept, the parameterized controller can be formulated as a Linear Fractional Transformation (LFT) system [START_REF] Tay | High performance control[END_REF], i.e. K(ρ, γ) = F l (J(ρ), Q(ρ, γ)) (see Fig. 4), where J(ρ) is defined as:

PROOF.

J(ρ) =        A(ρ) + B 2 (ρ)F g (ρ) 0 0 B 2 (ρ) -B k,0 (ρ)C 2 (ρ) A k,0 (ρ, ρ) B k,0 (ρ) 0 F g (ρ) -D k,0 (ρ)C 2 (ρ) C k,0 (ρ) D k,0 (ρ) I -C 2 (ρ) -F k,0 (ρ) I 0        (10)
and

Q(ρ, γ) = N i=1 γ i (ρ)Q i (ρ) (see Q i (ρ) in (32)). Specif- ically, a defined LPV controller K m (ρ) (m ∈ [1, N ]) can be formulated as F l (J(ρ), Q m (ρ))), i.e. K m (ρ) ≡ F l (J(ρ), Q(ρ, γ)) for γ m = 1 and γ i = 0 ∀i ̸ = m
The following proof is achieved by two steps: 1) Prove that the LPV-YK parameter Q(ρ, γ) is exponentially stable ∀ρ ∈ P and ∀γ; and 2) Prove the closed-loop exponential stability, ∀ρ ∈ P, ∀γ, with γ ∞ -performance level, where γ ∞ = max{γ ∞,i } i∈Z N .

Step 1:

Considering that (A.3.2) and ( 7) are satisfied,the triangular elements of A q,i (ρ) are shown to be exponentially stable ∀ρ ∈ {P i } i≥1 . Following Lemma 2.1, A q,i (ρ), consequently Q i (ρ), is exponentially stable over

P i ∀i ≥ 1. As a result, Q(ρ, γ) = N i=1 γ i Q i (ρ) is exponentially sta- ble over P = P i for every bounded signals γ i ∈ [0, 1].
Step 2:

The LPV-YK closed-loop system CL(ρ, γ) is derived from the LFT interconnection between G(ρ) (1) and K(ρ, γ) (8) (see Fig. 2). The closed-loop state matrix

Qi(ρ) =        A(ρ) + B2(ρ)D k,i (ρ)C2(ρ) B2(ρ)C k,i (ρ) B2(ρ)[D k,i (ρ) -D k,0 (ρ)]F k -B2(ρ)C k,0 (ρ) B k,i (ρ)C2(ρ) A k,i (ρ, ρ) B k,i (ρ)F k,0 0 0 A k,0 (ρ) + B k,0 (ρ)F k,0 (ρ) B2(ρ)[D k,i (ρ) -D k,0 (ρ)] B k,i (ρ) B k,0 (ρ) D (1) k,i C2 -Fg,i C k,i (ρ) [D k,i (ρ) -D k,0 (ρ)]F k,0 (ρ) -C k,0 (ρ) D k,i (ρ) -D k,0 (ρ)        (11)
A cl (ρ, γ) is exponentially stable if ∀ρ ∈ P, ρ ∈ [ν, ν], there exists a symmetric, positive definite matrix function X cl (ρ) such that ∀γ:

A T cl (ρ, γ)X cl (ρ) + X cl (ρ)A cl (ρ, γ) + s k=1 ρk ∂X cl ∂ρ k < 0 (12) Now , let T =        I 0 0 0 0 0 0 I I -I 0 0 0 0 I 0       
be a state transformation matrix which is applied to CL(ρ, γ) without changing its input-output nature, with

T -1 =        I 0 0 0 I 0 -I 0 0 0 0 I 0 I 0 0        .
Due to the block-triangular form of Ācl (ρ, γ) (13), ( 12) is satisfied if the following equations hold ∀ρ ∈ P (check Lemma 2.1):

Y g (ρ)(A(ρ) + B 2 (ρ)F g (ρ)) T + (A(ρ) + B 2 (ρ)F g (ρ))Y g (ρ) + s k=1 ρk ∂Y g ∂ρ k < 0 (14a) Y q (ρ)A T q (ρ) + A q (ρ)Y q (ρ) + s k=1 ρk ∂Y q ∂ρ k < 0 (14b) A T cl,0 (ρ)Y cl,0 (ρ) + Y cl,0 (ρ)A cl,0 (ρ) + s k=1 ρk ∂Y cl,0 ∂ρ k < 0 (14c)
where

Y g (ρ) ∈ R nx×nx , Y q (ρ) ∈ R nq×nq and Y cl,0 (ρ) ∈ R (nx+n k,0 )×(nx+n k,0 ) are symmetric, positive definite matrix functions with X cl (ρ) = T T diag(Y g , Y q , Y cl,0 ) T .
• Inequality (14a) can be deduced from ( 6) by choosing Y g (ρ) = X g (ρ) and considering that F g (ρ) = V (ρ)X -1 g (ρ) in [START_REF] Atoui | Interpolation of multi-lpv control systems based on youla-kucera parameterization[END_REF].

• (14b) is satisfied since Q(ρ, γ) has been proved in

Step 1 to be exponentially stable over P. • (14c) is fulfilled given that K (0) (ρ) exponentially stabilizes G(ρ) according to (A.3.1). Now, assume a sequence of finite switching time over the interval [0, T ] is t 0 , t 1 , . . . , t N with t 0 = 0, knowing that the closed-loop Lyapunov function is

V (x cl ) = x T cl X cl (ρ)x cl .
From the YK basic concept [START_REF] Tay | High performance control[END_REF], a parameterized controller K = F l (J, Q) recovers the performance of its actual controller K. Thus, the closedloop performance of F l (G, K) is equivalent to the performance of F l (G, K). In the current work, it can be deduced that the closed-loop performance of CL(ρ, γ) is equivalent to that of CL i (ρ) (3) within each parameter subset P i . Notice that according to (4), the following inequality describes the performance of

CL i (ρ) ∀ρ ∈ {P i } i∈Z N d dt (x T cl,i X cl,i (ρ)x cl,i ) + 1 γ ∞,i z T z -γ ∞,i w T w < 0 (15)
On the other hand, it is worth mentioning that X cl (ρ) is independent of the switching signal γ, so, for any switch-

ing time t k , V (x cl (t k )) = V (x cl (t - k ))
, and thus

V (x cl (t k )) ≤ V (x cl (t - k )) (16)
Given the initial condition x cl (0) = 0, from ( 15)-( 16), it can be shown that the inequality

V (x cl )+ 1 γ ∞ z T z -γ ∞ w T w < 0, γ ∞ = max{γ ∞,i } i∈Z N (17)
holds within each parameter subset. Integrate on both sides, we get This section presents an LPV-YK control scheme which: 1) formulates a YK-based gain-scheduling between LTI controllers that have been already designed separately at the polytopic vertices of each parameter subset; and 2) switches between the formulated gain-scheduling controllers over the partitioned polytopic subregions. The closed-loop system is proved to guarantee the quadratic stability for any continuous/discontinuous switching signals in terms of a set of Linear Matrix Inequalities (LMIs).

V (x cl (T )) -V (x cl (0)) + 1 γ ∞ ∥z∥ 2 2 -γ ∞ ∥w∥ 2 2 < 0 Since V (x cl (T )) ≥ 0 and V (x cl (0)) = 0, ∥z∥ 2 < γ ∞ ∥w∥ 2 is achieved. Therefore, the closed-loop is exponentially stable with an achieved performance ∥z∥ 2 < γ ∞ ∥w∥ 2 , where γ ∞ = max{γ ∞,i } i∈Z N . Ācl(ρ, γ) = T Acl(ρ, γ)T -1 =        A(ρ) + B2(ρ)

LPV Plant and Controllers Description

The following approach requires two model assumptions: 1) the system must be strictly proper (D 22 (ρ) = 0); and 2) the input and output matrices B 2 , C 2 , D 12 and D 21 must be parameter-independent [START_REF] Angelis | System analysis, modelling and control with polytopic linear models[END_REF]. From now on, we assume, without loss of generality, that the LPV system is given as:

G(ρ)        ẋ(t) = A(ρ)x(t) + B 1 (ρ)w(t) + B 2 u(t) z(t) = C 1 (ρ)x(t) + D 11 (ρ)w(t) + D 12 u(t) y(t) = C 2 x(t) + D 21 w(t) (18
) Notice that the second assumption doesn't impose any serious constraints since, if needed, it can be fulfilled by filtering the input u and output y (details are given in [START_REF] Apkarian | Self-scheduled h control of linear parameter-varying systems: a design example[END_REF]).

Define P 0 a convex polytope that contains all the parameter trajectories ρ ∈ R np . Let P i , i ∈ Z N = {1, . . . , N }, be the convex polytopic subsets, that could intersect by a boundary or surface, defined in P 0 along the parameter trajectories, i.e. ρ ∈ P = i≥1 P i ⊂ P 0 . Each P i (i ≥ 0) is defined as:

P i := C O {w i1 , ..., w i2 np } ( 19
)
where w ij represent the polytopic vertices of P i ∀j ∈ I[1, 2 np ]. ρ is then scheduled as:

∀ρ ∈ P i , ρ = 2 np j=1 α ij (ρ)w ij , (20)
where ∀i

2 np j=1 α ij (ρ) = 1, α ij (ρ) ≥ 0 ∀i, j, being α ij (ρ)
the scheduling coefficients in the convex region P i .

Fig. 3 shows an example of a convex parameter region defined by two varying parameters ρ 1 ∈ [ρ 1 , ρ 1 ], and

ρ 2 ∈ [ρ 2 , ρ 2 ]
, with ρ 2 = 1/ρ 1 . The convex region P 0 is represented in solid orange, and the convex subregions P i (i ≥ 1) are bounded within dashed green polygons.

The LPV representation of G(ρ) is defined over P 0 as a convex combination of the state-space realizations of the local LTI systems G 0j at the vertices w 0j :

    A(ρ) B 1 (ρ) B 2 C 1 (ρ) D 11 (ρ) D 12 C 2 D 21 0     = 2 np j=1 α 0 j(ρ)     A 0j B 1,0j B 2 C 1,0j D 11,0j D 12 C 2 D 21 0     (21)
Notice that G(ρ) could be defined equivalently in terms of the LTI plants G ij (at w ij ) as, for ρ ∈ P i :

    A(ρ) B 1 (ρ) B 2 C 1 (ρ) D 11 (ρ) D 12 C 2 D 21 0     = 2 np j=1 α ij (ρ)     A ij B 1,ij B 2 C 1,ij D 11,ij D 12 C 2 D 21 0     (22)
Now, assume that (A.4.1). There exists an LPV output-feedback controller K 0 (ρ) which quadratically stabilizes G(ρ) (using the standard polytopic approach in [START_REF] Apkarian | Self-scheduled h control of linear parameter-varying systems: a design example[END_REF]) over the full parameter region P 0 , defined as:

K 0 (ρ) :   A k,0 (ρ) B k,0 (ρ) C k,0 (ρ) D k,0 (ρ)   (23) being,   A k,0 (ρ) B k,0 (ρ) C k,0 (ρ) D k,0 (ρ)   = 2 np j=1 α 0j (ρ)   A k,0j B k,0j C k,0j D k,0j   (24)
where

A k,0 (ρ) ∈ R n k,0 ×n k,0 , B k,0 (ρ) ∈ R n k,0 ×m k , C k,0 (ρ) ∈ R p k ×n k,0 and D k,0 (ρ) ∈ R p k ×m k . (A.4.2). At each vertex w ij (j ∈ I[1, 2 np ], i ≥ 1), a lo- cal LTI controller K ij is designed separately to stabilize the local plant G ij .
It is worth mentioning that, at each intersecting boundary, a unique LTI controller should be designed at the intersecting extremums of the adjacent subsets respectively.

Regarding the example in Fig. 3, the local LTI controllers at the intersecting boundaries w i3 and w (i+1)2 , ∀i ∈ I[1, N -1], are designed similarly. Consequently, the switching between two successive subsets (P i and P i+1 ) undergoes using the same LTI controller (K i3 ≡ K (i+1)2 ). In such a case, the state and control input energies,just before and just after switching, are not affected.

Fig. 3 represents an example of a convex parameter set partitioning. The dashed black curve represents the actual operating conditions of the parameters (ρ 2 = 1/ρ 1 ). The orange solid polygon represents the nominal LPV controller K 0 (ρ) designed using standard polytopic approach over the convex region P 0 . The blue points represent the local LTI controllers

K ij (j ∈ I[1, 4], i ∈ I[1, N ])
designed separately to stabilize G ij at w ij using any LTI control approaches. The green dashed polygons are the subsets chosen along the parameter trajectory. Over each subset P i , a gain-scheduled controller Ki (ρ) is designed based on the YK interpolation of the local LTI controllers K ij . The overall switching scheme between the Ki (ρ) is represented by the LPV-YK controller Kσ (ρ).

Problem Definition

The objective of this work is to:

(1) Design multiple YK-based gain-scheduled controllers Ki (ρ), i ∈ I[1, N ]. Each one is designed by interpolating its corresponding LTI controllers K ij (j ∈ I[1, 2 np ]) based on YK concept (refer to Fig. 3). ( 2) Create an overall switched LPV-YK controller Kσ (ρ) to switch between Ki (ρ), such that Kσ (ρ) quadratically stabilizes G(ρ) ∀ρ ∈ P and for every continuous/discontinuous switching signal σ(t).

The switched closed-loop system is represented as

CL σ (ρ) :   Ãcl,σ (ρ) Bcl,σ (ρ) Ccl,σ (ρ) Dcl,σ (ρ)   (25)

Main Results

Based on the statements of LPV concepts and YK parameterization, the LPV-YK controllers Ki (ρ) are designed based on a Linear Matrix Inequality (LMI) optimization problem, where they are defined as Ki (ρ) :

  Ãk,i (ρ) Bk,i (ρ) Ck,i (ρ) Dk,i (ρ)   ( 26
)
Lemma 3.1. Consider a set of matrices A i corresponding to each vertex of a convex hull

J = C O {w 1 , ..., w 2 np },
The following statements are equivalent:

(i) A i is Hurwitz ∀i ∈ I[1, 2 np ]
(ii) there exist 2 np transformation matrices Z i such that the LPV matrix

Ā(ρ) = 2 np i=1 α i (ρ) Āi = 2 np i=1 α i (ρ)Z i A i Z -1 i ( 27
)
is quadratically stable ∀ρ ∈ J , where ρ =

2 np i=1 α i (ρ)w i such that 2 np i=1 α i (ρ) = 1, α i (ρ) ≥ 0 ∀i.
Proof details are in [START_REF] Bianchi | Interpolation for gain-scheduled control with guarantees[END_REF].

Theorem 3.1. Consider an LPV plant G(ρ) [START_REF] Jiang | Gain-scheduled control for morphing aircraft via switching polytopic linear parameter-varying systems[END_REF], satisfying assumptions (A.3.1) and (A.3.2). Then, the switched LPV-YK controller Kσ (ρ) (26)-( 30) quadratically stabilizes G(ρ) for any ρ ∈ P and for any continuous/discontinuous switching signals

γ i ∈ [0, 1] (∀i ≥ 1), if there exist symmetric, positive definite, constant ma- trices X g ∈ R nx×nx , X k,ij ∈ R n k,ij ×n k,ij
, and matrices

W j ∈ R m×nx and V ij ∈ R m k ×n k,ij such that: A 0j X g + X g A T 0j + B 2 W j + W T j B T 2 < 0 ∀w 0j (28) A k,0 (w ij )X k,ij +X k,ij A T k,0 (w ij ) + B k,0 (w ij )V ij + V T ij B T k,0 (w ij ) < 0 ∀w ij ( 29
)
with the state-space matrices of Ki (ρ) are Ãk,i (ρ) =

2 np j=1 α ij (ρ)     A ij + B 2 F g,ij -B 2 Dq,ij C 2 -B k,0 (w i,j )C 2 -Bq,ij C 2 -B 2 Dq,ij F k,ij B 2 Cq,ij A k,0 (w i,j ) 0 -Bq,ij F k,ij Āq,ij     Bk,i (ρ) = 2 np j=1 α ij (ρ) B 2 Dq,ij B k,0 (w i,j ) Bq,ij Ck,i (ρ) = 2 np j=1 α ij (ρ) [F g,ij -(D k,ij + D q,ij )C 2 C k,0 (w i,j ) -D q,ij F k,ij Cq,ij Dk,i (ρ) = 2 np j=1 α ij (ρ)[D k,0 (w i,j ) + Dq,ij ] (30) ∀i ∈ I[1, N ],
where

Āq,ij = Z ij A q,ij (Z ij ) -1 , Bq,ij = Z ij B q,ij
, and Cq,ij = C q,ij (Z ij ) -1 ∀i, j, and Z ij are state transformation matrices chosen, to satisfy Lemma 3.1, such that

Āq,i (ρ) = 2 np j=1 α j (ρ)Z ij A q,ij (Z ij ) -1 is quadratically sta- ble ∀i ≥ 1. A q,ij , B q,ij , C q,ij and D q,ij are the state- space matrices of Q i (ρ) (32) at the polytopic vertices w ij . In addition, F g,j = W j X -1 g , F k,ij = V ij (X k,ij ) -1 , and F g,ij = F g (w ij ).
PROOF. As mentioned in the previous proof, each parameterized controller can be formulated as a Linear Fractional Transformation (LFT) system [START_REF] Tay | High performance control[END_REF]. Each LPV-YK controller can be written as Ki (ρ) = F l (J(ρ), Qi (ρ)) ∀ρ ∈ P i (see Fig. 4), where J(ρ) is represented in [START_REF] Bianchi | Interpolation for gain-scheduled control with guarantees[END_REF], and Qi (ρ) is the transformed system of Q i (ρ) [START_REF] Tay | High performance control[END_REF].

J(ρ) = 2 np j=1 α i j(ρ)        A ij + B 2 F g,ij 0 0 B 2 -B k,0 (w ij )C 2 A k,0 (w ij ) B k,0 (w ij ) 0 F g,ij -D k,0 (w ij )C 2 C k,0 (w ij ) D k,0 (w ij ) I -C 2 -F k,ij I 0        (31)
The following proof is achieved by two steps: 1) Prove that the LPV-YK parameter Qi (ρ) is quadratically stable ∀ρ ∈ {P i } i∈Z N ; and 2) Prove the closed-loop quadratic stability ∀ρ ∈ {P i } i∈Z N , ∀σ.

Step 1:

Knowing that: 1) K ij stabilizes G ij ∀i ≥ 1, ∀j ∈ [1, 2 np ];
and 2) ( 29) is satisfied, it can be shown that the triangular elements of A q,ij ∀i, j are Hurwitz, and consequently, A q,ij is Hurwitz ∀i, j. According to Lemma 3.1, for any i ≥ 1, there exist transformation matrices Z ij such that the transformed system Qi (ρ) is quadratically stable ∀ρ ∈ P i , choose Z ij = (X q,ij ) 1/2 . As a result, Qi (ρ) is quadratically stable ∀ρ ∈ P i .

Step 2:

The switched closed-loop system CL σ (ρ) is derived from the LFT interconnection between G(ρ) and Kσ (ρ) (see Fig. 4).

The switched closed-loop state matrix A cl,σ (ρ) =

2 np j=1
α σj (ρ)A cl,σj is quadratically stable if:

(1) There exist symmetric, positive definite, constant matrices X cl,i such that ∀i ≥ 1

X cl,i Ãcl,i (ρ) + ÃT cl,i (ρ)X cl,i < 0 ∀ρ ∈ P i (33) (2) ∀t k ∈ [0, T ], V (x cl (t k )) ≤ V (x cl (t - k ))
Taking the same state transformation matrix T as in the previous proof, the transformed close-loop state matrix Ācl,i (ρ) is represented in [START_REF] Wu | Control of linear parameter varying systems[END_REF]. Due its block-triangular form, ( 12) is then satisfied if the following equations hold (check Lemma 2 in [START_REF] Xie | Design of lpv control systems based on youla parameterisation[END_REF]):

2 np j=1 α ij (ρ)(Y g,i (A ij + B 2 F g,ij ) + (A ij + B 2 F g,ij ) T Y g,i ) < 0 (35) 2 np j=1 α ij (ρ)(Y q,i Āq,ij + ĀT q,ij Y q,i ) < 0 (36) Qi(ρ) = 2 np j=1 αij(ρ)        Aij + B2D k,ij C2 B2C k,ij B2[D k,ij -D k,0 (wij)]F k,ij -B2C k,0 (wij) B k,ij C2 A k,ij B k,ij F k,ij 0 0 A k,0 (wiij) + B k,0 (wij)F k,ij B2[D k,ij -D k,0 (wij)] B k,ij B k,0 (wij) D k,ij C2 -Fg,ij C k,ij (D k,ij -D k,0 (wij))F k,ij -C k,0 (wij) D k,ij -D k,0 (wij)        (32)
Ācl,i(ρ) =

2 np j=1 αij(ρ)T Acl,ij(γ)T -1 = 2 np i=1 αij(ρ)        Aij + B2Fg,ij B2 Cq,ij -B2(Fg,ij -(Dk,0(wij) + Dq,ij)C2) B2(Ck,0j(wij) -Dq,ijFk,ij) 0 Āq,ij Bq,ijC2 -Bq,ijFk,ij 0 0 Aij + B2Dk,0(wij)C2 B2Ck,0(wij) 0 0 Bk,0(wij)C2 Ak,0(wij)        ( 34
)
2 np j=1 α ij (ρ)(Y i A ij + A T ij Y i ) < 0 ( 37
)
where

Y g,i ∈ R nx×nx , Y q,i ∈ R nq×nq and Y i ∈ R (nx+n k,0 )×(nx+n k,0 ) are symmetric, positive definite, constant matrices, with X cl,i = T T diag(Y g,i , Y q,i , Y i )
T , and

A ij = A ij + B 2 D k,0 (w ij )C 2 B 2 C k,0 (w ij ) B k,0 (w ij )C 2 A k,0 (w ij ) (38)
• Inequality ( 35) is equivalent to ( 28) by choosing Y g,i = X -1 g and W j = F g,ij X g for every ρ ∈ P i . • Since Qi (ρ)) is proved to be quadratically stable ∀i, inequality (36) is satisfied. • (37) is fulfilled given that K 0 (ρ) quadratically stabilizes G(ρ) over P 0 , and consequently over any P i ⊂ P 0 .

Therefore condition (33) is satisfied.

Assume a sequence of finite switching time over the interval [0, T ] is t 0 , t 1 , . . . , t n with t 0 = 0, knowing that closed-loop Lyapunov function as V (x cl ) = x T cl X cl,σ x cl . From the YK basic concept, the closed-loop is written as F l (G, J, Q). Consider a switching between any two adjacent subsets P i and P i+1 ∀i ≥ 1 at time t k , then, the closed-loop dynamics switches from

F l (G, J i , Q i,3 ) to F l (G, J i , Q i+1,2 ), or vise-versa. According to (A.4.2), K i,3 ≡ K i+1,2 ∀i ≥ 1, consequently Q i,3 ≡ Q i+1,2 . Then, for any switching time t k , F l (G, J, Q i,3 ) = F l (G, J, Q i+1,2 ), and thus V (x cl (t k )) = V (x cl (t - k )).
As a result, the switched closed-loop system CL σ (ρ) is quadratically stable ∀ρ ∈ P, and for any continuous/discontinuous switching signal σ. 4 Application to Autonomous Vehicles

The proposed method is applied to the lateral dynamics of an autonomous vehicle, since it contains a nonaffine scheduling parameter which causes an overbounding convex parameter region when implementing the polytopic approach. Recently, the polytopic, grid-based, and LFT approaches has been experimentally validated for lateral control of an automated vehicle in [START_REF] Atoui | LPV-Based Autonomous Vehicle Lateral Controllers: A Comparative Analysis[END_REF]. The lateral bicycle model is chosen, and taking into consideration the steering actuator dynamics.

Modelling Step

In [START_REF] Atoui | LPV-Based Autonomous Vehicle Lateral Controllers: A Comparative Analysis[END_REF], the lateral bicycle model Σ(ρ) has been formulated for gridded and convex parameter regions. The longitudinal speed v x is the varying parameter that varies in [START_REF] Atoui | Parameter Varying Approach For A Combined (Kinematic + Dynamic) Model Of Autonomous Vehicles[END_REF][START_REF] Ravi | H∞ control of linear time-varying systems: A state-space approach[END_REF] m/s. For the gridded parameter region, the statespace representation of Σ(ρ) is written as:

Σ(ρ) ẋ(t) = A Σ (ρ)x(t) + B Σ u(t) y(t) = C Σ x(t) (39)
being

x(t) = v y w , u(t) = δ, B Σ = C f m C f l f I , C Σ = 0 1 , A Σ (ρ) =   - Cr+C f mρ - C f l f -Crlr mρ -ρ - C f l f -lrCr Iρ - C f l 2 f +l 2 r Cr Iρ   , (40)
where v y and w are the lateral and rotational velocities in the vehicle's frame, respectively. δ is the control input, the steering angle of the front tire. C f and C r represent the stiffness of the front and rear wheel-tires. I, m, l f and l r are the vehicle's inertia, mass and the distance from the center of gravity to the front and rear wheel axes respectively.

On the other hand, to represent Σ(ρ) in a convex parameter region, the parameter-varying state-space matrices (i.e. A Σ (ρ)) should be written in affine parameterdependency as:

A Σ (ρ) =   - Cr+C f m ρ 2 - C f l f -Crlr m ρ 2 -ρ 1 - C f l f -lrCr I ρ 2 - C f l 2 f +l 2 r Cr I ρ 2   , (41)
with ρ 1 = v x and ρ 2 = 1 vx . Notice that, in the coming control designs, (40) is used for the gridded parameter regions and (41) is used for convex parameter regions. Notice that the bicycle model is extended by considering an identified steering actuator dynamics Σ act .

Lateral Control Design

In this work, the LPV and LTI controllers are designed using the H ∞ concept. For control design purpose, two weighting transfer functions W e (s) and W u (s) are designed to present the tracking performance and the actuator limitations respectively. Then, the state-space representation of G(ρ) is obtained from the generalized plant shown in Fig. 5.

The longitudinal speed is assumed to vary within the range ρ ∈ P = [START_REF] Atoui | Parameter Varying Approach For A Combined (Kinematic + Dynamic) Model Of Autonomous Vehicles[END_REF][START_REF] Ravi | H∞ control of linear time-varying systems: A state-space approach[END_REF] m/s, with a variation ρ ≤ 5 m/s 2 . Since the system dynamics changes significantly in this speed range, it could be conservative to design a single LPV controller over the full parameter region. Thus, the parameter region is divided into 5 subsets as: (44)

Grid-based control design

This section designs the proposed grid-based LPV-YK controller accordingly to the next steps:

• The nominal controller K 0 (ρ) is designed using W e,0 and W u,0 over the full parameter region P satisfying (A.3.1). Following (A.3.2), the controllers K i (ρ) are designed separately over P i (i ∈ [START_REF] Angelis | System analysis, modelling and control with polytopic linear models[END_REF][START_REF] Atoui | Parameter Varying Approach For A Combined (Kinematic + Dynamic) Model Of Autonomous Vehicles[END_REF]), with the same weighting functions W e and W e . • Following the conditions of Theorem 2.1, the LPV state-feedback gains F g (ρ) and F k,0 (ρ) are designed using the LMIs ( 6)- [START_REF] Atoui | LPV-Based Autonomous Vehicle Lateral Controllers: A Comparative Analysis[END_REF], where the multiple parameter-dependent Lyapunov functions at each subset are specified as affine and smooth functions of scheduling parameters. That is,

X g (ρ) = X 0 g + X 1 g ρ, X k,0 (ρ) = X 0 k,0 + X 1 k,0 ρ,
where matrices X j g and X j k,0 , j = 0, 1 are the optimization variables to be determined.

• J(ρ) and Q i (ρ) (i ∈ [START_REF] Angelis | System analysis, modelling and control with polytopic linear models[END_REF][START_REF] Atoui | Parameter Varying Approach For A Combined (Kinematic + Dynamic) Model Of Autonomous Vehicles[END_REF]) are obtained from ( 10)- [START_REF] Tay | High performance control[END_REF].

• Q(ρ, γ) = 5 i=1 γ i (ρ)Q i (ρ)
, where γ i (ρ) is switched between {0,1} when ρ(t) touches the switching instants. Then, the LPV-YK controller K(ρ, γ) is ready to be implemented.

Partitioned polytopic-based control design

In this section, G(ρ) is written as a convex combination of the vertices of a triangular polytope P (0) = C O {(ρ 1 , ρ 2 ), (ρ 1 , ρ 2 ), (ρ 1 , ρ 2 )} as shown in Fig. 6. The following section aims to design proposed partitioned polytopic-based LPV-YK control. Five triangular convex subsets (P 1 . . . P 5 ) are chosen along the parameter trajectory as shown in Fig. 6. At the polytopic vertices

w ij ∀i ∈ I[1, 3], ∀j ∈ I[1, 5
], LTI controllers K ij are designed separately using the same weighting functions W e and W u . In addition, a polytopic LPV controller K 0 (ρ) is designed, based on (A.4.1), using W e,0 and W u,0 .

For the partitioned polytopic-based LPV-YK control design, the following steps are done:

• According to the method explained in Section 3.3, the LPV polytopic-based state-feedback controller F g (ρ), and the LTI state-feedback controllers F (0) k,i , ∀i ∈ I [START_REF] Angelis | System analysis, modelling and control with polytopic linear models[END_REF][START_REF] Atoui | Design And Experimental Validation Of A Lateral LPV Control Of Autonomous Vehicles[END_REF], can be designed using an LMI-based state-feedback approach (pole-placement constraints or Linear Quadratic Regulator).

• J(ρ) and Q(ρ, γ) are obtained from ( 31)-( 32) and as mentioned in the proof of Theorem 3.1.

As a result the partitioned polytopic-based LPV-YK control scheme Kσ (ρ) is designed, and σ switches when ρ hits the subsets boundaries.

Experimental Results

The experiments shown here have been carried out on a robotized electric Renault ZOE vehicle shown in Fig. 7. It is prepared for lateral and longitudinal controls by computer-controlled steering and pedal actuators. Vehicle speed and the global coordinates are measurable using GPS and IMU. The vehicle is employed using a dSPACE MicroAutoBox. The test results of the designed controllers are discussed concerning their implementation and the analysis of the obtained performance.

In addition to the grid-based and partitioned polytopicbased LPV-YK controllers, two more controllers are tested for comparison purpose: 1) LPV-switched controller designed using the Theorem presented in [START_REF] Lu | Switching lpv control designs using multiple parameter-dependent lyapunov functions[END_REF]; and 2) A gain-scheduled LPV-YK controller that is designed by interpolating the LTI controllers at the vertices of P 0 , as presented in [START_REF] Chen | The smooth switching control for tora system via lmis[END_REF]. These two controllers are chosen to compare and show the improvement of our proposed approaches. The tests are done in a private test track in Satory (France) as shown in Fig. 8. This track contains bad road conditions and road-inclinations which allows to evaluate the controller robustness. The first part of the test describes the response of the controllers at a straight highway with high speeds. The second part concerns the precision of lateral control at optimal speeds chosen coherently depending on the road curvature. The longitudinal speed is considered as an external parameter of the LPV mode that is shown in Fig. 9. Fig. 10 presents the operating parameter subset that switches according to the speed evolution.

In the next two subsections, the vehicle performance is analyzed and compared between the grid-based LPV-YK controller and the LPV-switched controller [START_REF] Lu | Switching lpv control designs using multiple parameter-dependent lyapunov functions[END_REF], and between the partitioned polytopic-based LPV-YK controller and the gain-scheduled LPV-YK controller. Notice that, in all the tests, a restarting mode of the navigation systems appears at the end of the highway. Consequently, the results analysis doesn't concern the boarded part marked "NAV".

Grid-based LPV-YK control

Fig. 11a shows that both controllers achieve minimized lateral error. However, Fig. 11b and 11c emphasize a 11c show that the switching effect is negligible using the proposed grid-based LPV-YK controller, which is not the case for the switched LPV controller. Moreover, it is worth mentioning that the control input response at high speeds is not noisy. However, it has been shown previously that designing a single LPV grid-based controller over the whole parameter region could cause noisy performance at high speed as discussed in [START_REF] Atoui | LPV-Based Autonomous Vehicle Lateral Controllers: A Comparative Analysis[END_REF]. 12a that both controllers, the proposed partitioned and gain-scheduled LPV-YK, could successfully achieve lateral error minimization. On the other hand, the gain-scheduled LPV-YK controller leads to higher steering oscillations, compared to the partitioned LPV-YK controller, as shown in Figs. 12b and12c. When t ∈ [28.3, 28.5]s, it is observed in Fig. 10 switches hysterically back and forth between the subsets P (3) and P (4) . However, the hysterical switching of the partitioned LPV-YK controller between both subsets didn't affect the performance of the steering input and consequently the yaw rate (see the zoomed part in Figs. 12c and12b). This is expected since, at that instants, the operating controllers at that intersecting boundary are dynamically equivalent, i.e. F l (G, J 33 , Q 33 ) ≡ F l (G, J 42 , Q 42 ).

Conclusion

This work has proposed advanced LPV-YK switching methods to obtain smooth switching between parameter subregions. The switching signal is parameterdependent which can be any continuous/discontinuous signal. The main advantages behind these approaches, compared to the previous LPV-switched controllers are: 1) The pre-designed LPV controllers are parameterized with respect to a nominal LPV controller, instead of requiring the redesign in new constrained LMIs; 2) The design conditions proposed for LPV-YK switching control are more smooth and can be satisfied assuming the LPV stabilizability of the system; 3) The local controllers can be designed using different control approaches, i.e. PID, H ∞ , etc; and finally 4) The proposed LPV-YK control schemes doesn't pose any conditions or limitations on the switching signals.

As a future work, an interest appears to study the choice of Q dynamics, i.e. how to choose the best design of the state-feedback gains F g and F k,0 , and how they affect the closed-loop performance.

Fig. 1 .

 1 Fig. 1. LPV-YK gridded controller

Fig. 2 .

 2 Fig. 2. Grid-based generalized LPV-YK configuration

Fig. 3 .

 3 Fig. 3. Partitioned polytopic regions 3 Partitioned polytopic-based LPV-YK control design

Fig. 4 .

 4 Fig. 4. Partitioned polytopic-based generalized LPV-YK configuration

Fig. 5 .

 5 Fig. 5. Generalized plant G(ρ)

ρ ∈ [ 5 ,

 5 10] ∪[START_REF] Bianchi | Interpolation for gain-scheduled control with guarantees[END_REF][START_REF] Hanifzadegan | Smooth switching lpv controller design for lpv systems[END_REF] ∪[START_REF] Hanifzadegan | Smooth switching lpv controller design for lpv systems[END_REF][START_REF] Koelewijn | Scheduling dimension reduction of lpv models -a deep neural network approach[END_REF] ∪[START_REF] Koelewijn | Scheduling dimension reduction of lpv models -a deep neural network approach[END_REF][START_REF] Niemann | Dual youla parameterisation[END_REF] ∪[START_REF] Niemann | Dual youla parameterisation[END_REF][START_REF] Ravi | H∞ control of linear time-varying systems: A state-space approach[END_REF] (42) Consider the following weighting functions to be used in the control design:W e,0 (s) = s+2 2s+0.002 , W u,0 (s) = s+5 0.01s+5(43)

Fig. 6 .

 6 Fig. 6. Partitioned parameter region

Fig. 7 .Fig. 8 .Fig. 9 .Fig. 10 .

 78910 Fig. 7. Renault ZOE automated vehicle

  Fig. 11. Grid-based LPV-YK control

  Fig. 12. Partitioned polytopic-based LPV-YK control 4.3.2 Partitioned polytopic-based LPV-YK control It is shown in Fig.12athat both controllers, the proposed partitioned and gain-scheduled LPV-YK, could successfully achieve lateral error minimization. On the other hand, the gain-scheduled LPV-YK controller leads to higher steering oscillations, compared to the partitioned LPV-YK controller, as shown in Figs.12b and 12c. When t ∈ [28.3, 28.5]s, it is observed in Fig.10switches hysterically back and forth between the subsets P(3) and P(4) . However, the hysterical switching of the partitioned LPV-YK controller between both subsets didn't affect the performance of the steering input and consequently the yaw rate (see the zoomed part in Figs.12c and 12b). This is expected since,

  ∞ ∥w∥ 2 , where γ ∞ = max{γ ∞,i } i∈Z N , for any continuous/discontinuous bounded switching signals γ i ∈ [0, 1], if there exist symmetric, positive definite, parameter-dependent matrix functions X g

	Theorem 2.1. Consider an LPV plant G(ρ) (1), and
	that the assumptions (A.3.1) and (A.3.2) are satis-
	fied. Let K 0 (ρ) be the nominal LPV controller designed
	over the full parameter region P 0 . Then, the param-eterized LPV-YK controller K(ρ, γ) (5)-(8) exponen-
	tially stabilizes G(ρ), with an achieved performance
	∥z∥ 2 < γ
	refer to Fig.
	4).
	Lemma 2.1. Assume that the matrices A 11 (ρ) and
	A 22 (ρ) are exponentially stable, every continuous and
	bounded block triangular matrix whose diagonal matrices
	consist of A 11 (ρ) and A 22 (ρ) is also exponentially stable.

PROOF. Please see appendix A.

The following theorem aims to prove that K(ρ, γ) exponentially stabilizes G(ρ) for every ρ ∈ P and for every continuous/discontinuous switching signals γ i .

Acknowledgements

Authors express their gratitude to RENAULT research department for its support in developing experimental tests. Special thanks to Alexendre ARMAND who helped in achieving the experimental results. This paper reflects solely the views of the authors and not necessarily the views of the company they belong to.

A Proof of Lemma 2.1 PROOF. Consider an upper triangular matrix:

According to the assumption, there exists bounded positive definite matrix functions X 1 (ρ) and X 2 (ρ), positive real numbers α 1 and α 2 , satisfying the following inequalities [START_REF] Ravi | H∞ control of linear time-varying systems: A state-space approach[END_REF]:

2) Since the off-diagonal matrix A 12 (ρ) is assumed to be bounded, there exists a positive real number α 3 satisfying

A(ρ) is said to be exponentially stable if there exists a positive definite matrix function X(ρ) such that

Then,

Next, let us find a positive real number λ > 0 which satisfies (A.4). Using Schur complement, inequality (A.4) is equivalent to the following two inequalities:

The first inequality holds for any λ > 0 (from (A.2)). Considering (A.2) and (A.3) in (A.5), then

Therefore, π < 0 for any λ > 0 satisfying α 2 α 2 3 /λ < α 1 . A similar proof could be deduced for the lower triangular matrices.


Jieli Lyu

Wajdi Chaabani

Evgeny Modin

Andrey Chuvilin

Thomas Bizien

Frank Smallenburg
email: frank.smallenburg@universite-paris-saclay.fr

Marianne Impéror-Clerc
email: marianne.imperor@universite-paris-saclay.fr

Doru Constantin
email: constantin@unistra.fr

Cyrille Hamon
email: cyrille.hamon@universite-paris-saclay.fr

Double-lattice packing of pentagonal gold bipyramids in supercrystals with triclinic symmetry

Keywords: Supercrystal, Small angle X-Ray scattering, Evaporation-induced selfassembly, FIB-SEM tomography, Surface enhanced Raman scattering, Gold nanoparticles

Pentagonal packing is a long-standing issue and a rich mathematical topic, brought to the fore by recent progress in nanoparticle design. Gold pentagonal bipyramids combine five-fold symmetry and anisotropy and their section varies along the length. In this work, we obtain colloidal supercrystals of pentagonal gold bipyramids in a compact arrangement that generalizes the optimal packing of regular pentagons in the plane. Multimodal investigations reveal a two-particle unit cell with triclinic symmetry, a lower symmetry than that of the building blocks. Monte Carlo computer simulations show that this lattice achieves the densest possible packing. Going beyond pentagons, further simulations show an odd-even effect of the number of sides on the packing: odd-sided bipyramids are non-centrosymmetric and require the double-lattice arrangement to recover inversion symmetry. The supercrystals display a facetdependent optical response that is promising for sensing, metamaterials applications and for fundamental studies of self-assembly processes.

Introduction

Since Greek antiquity, scientists have been fascinated by the way identical bodies pack together in regular assemblies. Determining their optimal packing, i.e. the one with the highest possible volume fraction is still a challenging task, even for bodies with simple geometrical shapes like convex polyhedra or polygons in the plane. Recently, this quest for optimal packing has been fueled by the design of nanoparticles with increased structural complexity, which typically selfassemble into dense supercrystals. However, the vast majority of these efforts have focused on isometric particles (with a similar size along the three space directions) [1] and elongated shapes have received much less attention. [2] Fivefold symmetry is ubiquitous, from human-made architecture to floral symmetries. [3] In two dimensions, pentagonal packing is a long-standing issue and a rich mathematical topic, started by the early work of Johannes Kepler. [START_REF] Kepler | The harmony of the world[END_REF] Tiling the plane with pentagons can be achieved either with non-regular pentagons in 15 different ways [START_REF] Rao | [END_REF] or in combination with other polygonal shapes, giving rise to the fascinating aperiodic Penrose tilings. [6] The optimal packing of regular pentagons is a periodic lattice with a packing fraction around 0.92 called the 'pentagonal ice ray'. [7] First reported in China as a decorative pattern, [START_REF] Dye | A grammar of Chinese lattice / by Daniel Sheets Dye[END_REF] it consists of two vertical columns of pentagons pointing in opposite direction and stacked along their edges. [7] Since the two orientations of pentagons form two sublattices that are related to each other via an inversion center, this results in a double-lattice packing. [START_REF] Kuperberg | [END_REF] This organization has also been observed in numerical simulations of hard pentagons. [10] Fivefold symmetry is regularly encountered in the shape of gold nanoparticles (NPs), as a result of crystal twinning during particle synthesis. For instance, gold bipyramids consist of two pentagonal pyramids base-stacked together. [11] Pentatwinned gold bipyramids were first obtained as a byproduct in gold nanorod synthesis, [12] but since then high-yield synthesis has been achieved by seed-mediated growth. [13] The self-assembly of pentagonal bipyramids has been studied within monolayers, [14] but their self-assembly into 3D supercrystals has not yet been elucidated. In contrast to many other elongated particles, the cross-section of a pentagonal bipyramid is not constant along the symmetry axis, and hence their packing in 3D space does not reduce to its two-dimensional counterpart. [2] Allowing the section to vary along the length increases the complexity of the problem, as demonstrated by the case of ellipsoids. [15] For ellipsoids, the unit cell of the densest lattice contains a motif of two inequivalent particles having two different orientations. [START_REF] Trovato | Proc. Natl. Acad. Sci. U.S.A[END_REF] Although the stacking of ellipsoid-like particles such as bipyramids, prolate spheroids and bicones has been investigated numerically, [1c, 15-17] there is to date no experimental evidence on the three-dimensional packing of elongated particles with varying cross-section. This may be due to their relative experimental scarcity: notable exceptions are hematite nanoparticles [18] and ellipsoidal microparticles, [19] but crystallization also requires a high size and shape homogeneity of the particle ensemble, [20] which was difficult to achieve until about fifteen years ago. Furthermore, investigating the structure of the supercrystals by electron microscopy in real space is challenging, as the electrons typically interact too strongly with the nanoparticles to penetrate the bulk of supercrystals and reveal their three-dimensional order.

In this work, we obtain long-range supercrystals of pentagonal gold bipyramids with a packing fraction of 0.835. Their structure is a double-lattice packing, where all bipyramids are arranged parallel to each other but have two different in-plane orientations of the pentagonal crosssection. This double lattice generalizes the optimal packing of regular pentagons in the plane, the 'pentagonal ice ray', in the sense that the two orientations of the pentagonal bipyramids are related to each other by an inversion center. Synchrotron-based small-angle X-ray scattering (SAXS) and focused ion beam-scanning electron microscopy (FIB-SEM) tomography analysis reveal a two-particle unit cell with triclinic symmetry. Monte Carlo (MC) computer simulations show that this arrangement corresponds to the densest possible packing. The crystalline order of the bipyramids results in a facet-dependent optical response, which is demonstrated by surface-enhanced Raman scattering (SERS) spectroscopy.

Results and Discussion

Pentatwinned gold bipyramids (AuBPs) were obtained by seed-mediated growth and were uniform in size, as shown by a relative standard deviation of the length and width below 8% (e.g. length: 75.3±3.6 nm, base width: 25.2±1.5 nm). We optimized the conditions to crystallize the AuBPs via the slow evaporation (i.e. 48h) of a sessile droplet, and investigated their nanostructuration. The optical and structural characterization are provided in the Supporting Information (Sections 1 and 2). Results show that not only the detailed shape of the bipyramids matters (anisotropy, truncation of the tips) but also that the surfactant (i.e. hexadecyltrimethylammonium chloride (CTAC)) concentration has to be tuned carefully to obtain supercrystals. Figure 1 shows a combined SAXS and SEM analysis of a sample comprising AuBPs supercrystals. E-G) SEM micrographs of an AuBP supercrystal viewed at different magnification and different orientations. In E), the supercrystal was sliced by FIB to reveal the NPs' organization from the top (F) and from the side (G). The insets in F, G) shows the lattice parameters a, b, c, β, γ of the triclinic unit cell. H) SAXS data with experimental structure factor, S(q). Vertical lines indicate the expected positions of some Bragg peaks of the triclinic lattice. I) Threedimensional reconstruction of a portion of the supercrystal. In SAXS and SEM, 64 mM in Au 0 and 16 mM in Au 0 were used for the preparation of the sample, respectively. A fixed concentration of CTAC of 2.5 mM was used in all cases.

Low-magnification optical microscopy images show numerous supercrystals, a few micrometers in size and with the same rhomboidal shape, scattered randomly on the substrate.

In the 2D SAXS image (Figure 1D), the assemblies exhibited Bragg spots, indicating the presence of supercrystals within the footprint of the X-ray beam (300x500µm²). We also obtained the structure factor S(q) as the ratio between the intensity scattered by the supercrystal and the one for isolated gold bipyramids. As shown in Figure 1H, the Bragg peak positions are in agreement with a triclinic lattice. In order to confirm the lattice symmetry, AuBPs supercrystals were investigated by SEM. Analysis of the surface of the supercrystal revealed that all bipyramids are oriented in the same direction and parallel to the substrate (Figure 1F).

Further investigation of the internal structure was carried on by FIB-SEM tomography, allowing the 3D rendering of a portion of the supercrystal (Video S1). [21] 2D analysis of the images allowed identifying the lattice vectors of the triclinic unit cell (Figure 1F-G). 3D analysis of the supercrystal revealed that all layers were identical and were stacked in an ABA fashion, with a relative offset along the 2c+a triclinic lattice direction. From the SEM image analysis, we confirmed that the supercrystal belongs to the triclinic symmetry, in agreement with the SAXS indexing of the Bragg peaks. Additional SEM images of the sample (in 2D and 3D) are provided in the Supporting Information (Section 4). Next, we performed MC computer simulations in order to investigate the effect of the detailed shape of the bipyramids on the lattice symmetry. In particular, we use a crystal structure prediction technique based on MC simulations of small numbers of particles in a simulation box with a variable shape. [22] Here, we make the assumption that the interactions between the bipyramids are dominated by the hard cores, and neglect other possible interactions due to e.g. charges, Van der Waals attractions, or ligands. Note that since the bipyramids used in this work are large compared to the size of the CTAC ligands, we do not expect the ligands to modify the nature of the interaction. Hence, our model consists of perfectly hard truncated bipyramids, with varying number of vertices on the polygonal base: n= 3, 4, 5, 6, 7, 10, 12 and 20. For each particle shape, we simulate on the order of 300 compression runs, in which a small variable unit cell containing between 2 and 6 particles is compressed slowly under gradually increasing pressure. After compression, we examine the final structures and calculate their packing fractions. In practice, structures that are commonly found using this method tend to be good candidates for entropically favored phases in the system, and the densest-packed phase is expected to be the stable phase at sufficiently high packing fractions. [22][23] Note that the anisotropy and the truncation of the object are closed to the ones obtained experimentally and, if these parameters were varied, the simulation outcome would change, especially if the anisotropy decreased significantly. We show the results of the simulations in histograms of the number of observations of predicted structures as a function of packing fraction, together with two snapshots of the corresponding densest structure (Figure 2A-B). Interestingly, five recurrent structures were found for n=5, but only one for both n=4 and n=6 and at most three for the other shapes. For n=4 and n=6, the packing fraction of the predicted structure was about 0.94 and 0.87 respectively. For n=5, the packing fraction of the recurrent structures varied between 0.69 and 0.83 (Figure 2A). In Figure 2C, we plotted the packing fraction of the densest recurrent structure as a function of n. For n<10, results show an odd-even effect of the number of sides on the packing density: bipyramids with an odd number of vertices yield less dense structures. We attribute this to the non-centrosymmetry of the odd shapes, imposing lattices with two NPs in the unit cell in a double-lattice configuration, [START_REF] Kuperberg | [END_REF] whereas those of centrosymmetric nanoparticles (i.e. with even n) have only one NP in the unit cell (Figure 2B).

Pentagons do not tile the plane, inducing the emergence of more open lattice polymorphs compared to the other shapes. For n<10, the packing fraction of the densest structure evolves significantly with n (i.e. between 0.83 and 0.92) whereas for n≥10, it converges towards 0.84.

It is noteworthy that, although these NPs are faceted, for large n the problem converges to the closest packing of perfectly rounded truncated bicones. In this limit (n→∞), the densest phase has a packing fraction of approximately 0.84, but the second densest (and quite frequently occurring) structure, with a packing fraction of 0.77, corresponds to an ellipsoid packing (the densest to date), in which the particles are stacked in alternating layers with differing orientations (Figure S4.3). [START_REF] Trovato | Proc. Natl. Acad. Sci. U.S.A[END_REF] Our results point towards an even denser packing than previously reported for ellipsoids due to the noncircular cross section of the objects. Overall, two regimes for the structuration of faceted bipyramids were identified, depending on n: a "faceted" regime for n<10 and a "smooth" regime for n≥10. Among these different shapes, bipyramids with n=5 exhibit a particularly rich variety of observed structures and low maximum packing, confirming the specificity of the pentagonal cross section. We next compared the simulation results with the experimental analysis in real space (SEM) and in reciprocal space (SAXS). Among the five predicted structures for pentagonal bipyramids, the densest one matched the lattice determined experimentally. In agreement with the simulations, we find a double-lattice arrangement, where all pentagonal bipyramids are parallel to each other but have two different orientations, colored in blue and red in Figures 2 and3. The periodic arrangement is a triclinic lattice with two particles in the unit cell with each type of orientation (Figure 3A-B). The two particle orientations are related by an inversion center, so their pentagonal cross-sections are turned by an angle of = 36° (Figure 3B). The whole structure can be decomposed into two identical lattices, each one containing all particles with the same orientation (blue or red). The space-group is P1 , with the inversion center located in-between the centers of the two particles in the unit cell, in agreement with the SAXS model.

Note that the long axis of the bipyramids is along a direction close to a+2c. The structure can also be decomposed in alternate layers with the two orientations (ABA stack), as can be seen from the projection of the structure onto the (a,b) plane displaying pentagons with the two orientations (Figure 3D). We conclude that the supercrystals are oriented along the c and c-a vectors of the triclinic lattice, by superimposing the model lattice composed of a few NPs on a SEM image of one supercrystal (Figure 3E). The agreement between the model and the experiment is very good, as shown by the identical orientation of the particles at the nanoscale and a similar angle of 30° between the micrometer-sized rhomboidal tips and the angle defined by the c and c-a vectors of the unit cell. The lattice parameters of the triclinic unit cell were found to be a = 41.35 nm; b = 37.24 nm; c = 47.74 nm; α = 105°; β=109°; γ=90° (Figure S2.9).

The experimental packing fraction was determined as 0.835, in excellent agreement with the one determined from the MC computer simulations (0.83). The triclinic pinacoidal lattice (P1 ) defines a low-symmetry architecture ,with only the center inversion as symmetry operation. [START_REF] Sands | Introduction to Crystallography[END_REF] To the best of our knowledge, this is the first experimental realization of the shape-directed self-assembly of a triclinic supercrystal. [START_REF] Dshemuchadse | [END_REF] Finally, the optical properties of the supercrystals were investigated by SERS, allowing the detection of the spectral fingerprint of a probe molecule in contact with the plasmonic surface. [26] As an analyte, we used Crystal Violet (CV), which is resonant with the laser frequency, resulting in surface-enhanced resonance Raman scattering (SERRS). This analysis was performed on a faceted supercrystal (Figure 4A-B). The SERRS signal intensity was uniform on each facet due to the periodic arrangement of the nanoparticles (Figure 4C).

Moreover, the average signal is found to be facet-dependent, with the (12 1) facets displaying the highest SERRS intensity (Figure 4D). This phenomenon can be ascribed to the varying local arrangement of the NPs on the different facets, directly affecting electromagnetic hotspots formation at the interparticles gaps. [26] The orientation of the longitudinal axis of the bipyramids with respect to the surface plays a key role: When this axis is parallel to the surface facet (i.e.

the (010) one), the Raman enhancement is small (201 CPS) whereas a much higher enhancement (560 CPS) is observed when the bipyramids point outwards (i.e. the (12 1) facet) (Figure 4). [27] In contrast, mapping the SERRS intensity in a similar way on gold nanosphere supercrystals resulted in a uniform response (Figure S6.4). Here, we show facet-dependent SERS properties that originate from the anisotropy of the AuBPs, opening an avenue for tailored SERS performance using supercrystals with "spiky" and "smooth" facets, displaying respectively high and low SERS efficiency.

Conclusion

This work shows that fivefold symmetry and anisotropy can be combined to crystallize complex nano-objects into triclinic pinacoidal supercrystals. Pentagonal gold bipyramids pack into crystals with two particles per unit cell, corresponding to the densest packing predicted by Monte Carlo (MC) computer simulations. Further simulations show how the cross section influences the arrangement (symmetry and packing fraction) of the particles, beyond pentagonal bipyramids. Objects with an odd-sided base are non-centrosymmetric but the two-particle unit cell retrieves inversion symmetry, explaining the double-lattice arrangement. Surprisingly, their packing fraction is lower than that of even-sided particles. The double-lattice packing generalizes the optimal packing of regular pentagons in the plane, the 'pentagonal ice ray' in a non-trivial manner (as can be seen in Figure 3D), since the section of the bipyramids is not constant along their long axis. This gold bipyramid supercrystal has a facet-dependent optical response, with spiky and smooth facets displaying respectively high and low SERS efficiency.

Its low-symmetry triclinic structure is also useful for metamaterials applications and for fundamental studies of self-assembly processes.

Figure 1 :

 1 Figure 1: Multiscale characterization of pentagonal gold bipyramid supercrystals

Figure 2 :

 2 Figure 2: Monte-Carlo computer simulation of the packing of truncated bipyramids with

Figure 3 :

 3 Figure 3: Comparison between the model and the experiment. A-B) P1 unit cell predicted

Figure 4 :

 4 Figure 4: Facet-dependent surface enhanced resonance Raman scattering on a gold

Acknowledgments

The CNRS is acknowledged for funding and support. The authors thank Claire Goldmann for continuous experimental support and Stéphan Rouzière for the use of the MORPHEUS platform. J. L. acknowledges financial support by the China Scholarship Council (CSC) for her PhD. This work has benefited from an Investissements d'Avenir grant from Labex PALM (ANR-10-LABX-0039-PALM). We acknowledge SOLEIL for the provision of synchrotron radiation facilities (experiment 20201118, SWING beamline). The present work has benefited from the electronic microscopy facility of Imagerie-Gif, (http://www.i2bc.parissaclay.fr), member of IBiSA (http://www.ibisa.net), supported by "France-BioImaging" (ANR10-INBS-04-01), and the Labex "Saclay Plant Science" (ANR-11-IDEX-0003-02).

Supporting Information

Supporting Information is available from the Wiley Online Library or from the author.