Qianyue Luo

Anh-Tu Nguyen
email: nguyen.trananhtu@gmail.com.

James Fleming
email: j.fleming@lboro.ac.uk

Hui Zhang
email: huizhang285@gmail.com.

- T Nguyen

Unknown Input Observer Based Approach for Distributed Tube-Based Model Predictive Control of Heterogeneous Vehicle Platoons

Keywords: platoon, integrated controller, tube-based model predictive control (TMPC), proportional multiple-integral (PMI) observer

This paper addresses the control problem of heterogeneous vehicle platoons subject to disturbances and modeling errors. The objective is to guarantee spatialgeometry constraints of vehicles in a platoon. We deal with the case where a predecessor-leader following (PLF) communication topology is used and heterogeneous vehicle dynamics is subject to disturbances. To estimate the lumped disturbance, the technique of unknown input proportional multiple-integral (PMI) observer is employed such that both the state and the disturbance are simultaneously estimated. Moreover, tube-based model predictive control (TMPC) is used and the corresponding control law is composed of a feed-forward term, a feedback term, and a disturbance compensation term. The gains in the integrated control strategy are optimized by utilizing the particle swarm optimization (PSO) algorithm with an H∞ performance index of an augmented error system. It is proved that the deviations between the actual system and the nominal system are bounded in a robustly positively invariant (RPI) set, that is, the main objective is guaranteed. With the proposed control strategy, simulations and comparisons are carried out. We can see that the control performance of the proposed strategy is significantly improved while the computational time is reduced compared with existing methods.

I. INTRODUCTION

Intelligent transportation and multi-vehicle coordination are becoming a promising area both from industry and academic research. Among the applications, vehicle platooning has great advantages in terms of efficiency, safety, energy consumption, and environmental protection [START_REF] Hedrick | Control issues in automated highway systems[END_REF]- [START_REF] Al Alam | An experimental study on the fuel reduction potential of heavy duty vehicle platooning[END_REF]. The technique of vehicle platooning is based on the vehicle-to-vehicle (V2V) communication technology. The main function is to make multiple vehicles driving on the same lane with a close range and same velocity [START_REF] Zhou | Range policy of adaptive cruise control vehicles for improved flow stability and string stability[END_REF], [START_REF] Naus | String-stable CACC design and experimental validation: A frequencydomain approach[END_REF].

Generally, vehicle platoon control is more challenging than the traditional vehicle dynamics control. The challenges and differences arise from the following aspects.

1) The models for controller design are different. In order to reduce the computational burden, a simplified model is used for platoon controller design. In the literature, there are second-order models with speed and displacement as two system states [START_REF] Lin | Optimal control of vehicular formations with nearest neighbor interactions[END_REF], [START_REF] Hu | Cyber-physical control for energy-saving vehicle following with connectivity[END_REF], third-order models with one more acceleration state [START_REF] Xiao | Practical string stability of platoon of adaptive cruise control vehicles[END_REF]- [START_REF] Zheng | Stability and scalability of homogeneous vehicular platoon: Study on the influence of information flow topologies[END_REF], and three-state models with lateral dynamics [START_REF] Goli | Evaluation of lateral trajectories with different controllers for multi-vehicle merging in platoon[END_REF], [START_REF] Guo | Adaptive fuzzy sliding mode control for coordinated longitudinal and lateral motions of multiple autonomous vehicles in a platoon[END_REF]. 2) Information from other vehicles is necessary for platoon control in which each vehicle communicates with other vehicles according to the communication topology. The selection and the design of communication topology have a crucial impact on platoon stability [START_REF] Guo | Autonomous platoon control allowing rangelimited sensors[END_REF], [START_REF] Zheng | Distributed model predictive control for heterogeneous vehicle platoons under unidirectional topologies[END_REF].

3) The design objectives are different. The objective of platoon control is not only to ensure the stability of individual vehicle but also to ensure the platoon stability. The overall position error of the platoon should not be amplified as the number of vehicles increased, which is also known as string stability [START_REF] Dunbar | Distributed receding horizon control of vehicle platoons: Stability and string stability[END_REF]- [START_REF] Besselink | String stability and a delay-based spacing policy for vehicle platoons subject to disturbances[END_REF]. Due to the energy-saving characteristics of vehicle platoons, the energy consumption may be considered as an additional control objective [START_REF] Lin | Optimal control of vehicular formations with nearest neighbor interactions[END_REF]. 4) More constraints should be considered for platoon control. Besides original constraints of individual vehicle control, the constraints of the platoon control should include the position, speed and acceleration constraints between the adjacent vehicles.

In addition to different platoon models, different control strategies have been applied for platoon control. The authors in [START_REF] Naus | String-stable CACC design and experimental validation: A frequencydomain approach[END_REF] adopted a frequency-domain design method for a cooperative adaptive cruise control. Sliding mode control strategies were proposed in [START_REF] Gao | Distributed adaptive sliding mode control of vehicular platoon with uncertain interaction topology[END_REF], [START_REF] Guo | Distributed adaptive integratedsliding-mode controller synthesis for string stability of vehicle platoons[END_REF]. An H ∞ control method was employed in [START_REF] Guo | Hierarchical platoon control with heterogeneous information feedback[END_REF], [START_REF]Autonomous platoon control allowing range-limited sensors[END_REF]. Moreover, distributed model predictive control (DMPC) has been applied in various contexts [START_REF] Zheng | Distributed model predictive control for heterogeneous vehicle platoons under unidirectional topologies[END_REF], [START_REF] Liu | Distributed model predictive control for cooperative and flexible vehicle platooning[END_REF], [START_REF] Min | Constrained optimization and distributed model predictive control-based merging strategies for adjacent connected autonomous vehicle platoons[END_REF].

From the application perspective, the platoon control is a multi-objective optimization problem under multiple physical constraints and external disturbances. Though there are considerable existing control strategies for this control problem, few results are obtained by considering simultaneously the physical constraints and external disturbances.

Model predictive control (MPC) has been known as powerful tool to deal with system constraints. However, when the system is subject to persistent disturbances, the constraints may not be satisfied if the disturbances are not explicitly considered in MPC design. When the system is subject to persistent disturbances, Mayne and Langson proposed tube-based model predictive control (TMPC) to solve the optimization problem [START_REF] Mayne | Robust model predictive control of constrained linear systems with bounded disturbances[END_REF]- [START_REF] Mayne | Tube-based robust nonlinear model predictive control[END_REF]. Then, it was applied to applications such as vehicle control in [START_REF] Kayacan | Robust tubebased decentralized nonlinear model predictive control of an autonomous tractor-trailer system[END_REF], [START_REF] Gao | A tube-based robust nonlinear predictive control approach to semiautonomous vehicles[END_REF] by adding feedback terms to make the error caused by disturbance converging to an invariant set [START_REF] Kerrigan | Robust constraint satisfaction: Invariant sets and predictive control[END_REF]- [START_REF] Rakovic | Invariant approximations of the minimal robust positively invariant set[END_REF]. In this work, we employ the tube-based idea to design a robust controller for heterogeneous vehicle platoons subject to both modeling uncertainties and external disturbances. The main contributions of this paper are three-fold: 1) A proportional multiple-integral (PMI) observer is designed to estimate both the system state and the lumped disturbance for DMPC purposes such that the number of physical sensors can be reduced; 2) A compensation term is integrated in the control law to compensate the effects of disturbances such that the conservativeness of the TMPC algorithm is reduced and the robustness performance is improved; 3) H ∞ control performance and particle swarm optimization (PSO) algorithm are adopted to optimally tune both the observer and controller gains, allowing to minimizing the disturbance impacts.

Notation. R and N stand for the set of real numbers and positive integers, respectively. For p ∈ N and a vector x ∈ R n , the p-norm of x is given as

x p = ( |x i | p ) 1/p , while x ∞ = max |x i |; for a vector x ∈ R n and a positive semi- definite matrix Q ∈ R n×n , x Q = x T Qx 1/2
P Q = {x|x + y ∈ P, y ∈ Q}.
x denotes the nominal value of x, and x denotes the observed value of x. The rest of notations will be provided in the paper.

II. MODELING AND PRELIMINARY

Vehicle platoons are composed of one leading vehicle (numbered with 0) and N following vehicles (numbered from 1 to N ) in the same lane, see Fig. 1. The vehicles in a platoon can be heterogeneous or homogeneous. Hereafter, we discuss the three main elements of a model-based control scheme for vehicle platooning: longitudinal dynamics of each vehicle, communication connection topology between vehicles, and goals of vehicle platooning.

A. Vehicle Longitudinal Dynamics

We consider the longitudinal control of vehicles subject to multiple resistances [START_REF] Zheng | Distributed model predictive control for heterogeneous vehicle platoons under unidirectional topologies[END_REF], including wind resistance, rolling resistance, and slope resistance. The longitudinal dynamics of ith vehicle is described as

p i (k + 1) = p i (k) + v i (k)∆t v i (k + 1) = v i (k) + η i m i r i T i (k) -a f i (v i (k)) ∆t T i (k + 1) = T i (k) + (u i (k) -T i (k)) ∆t τ i + w i (k) (1)
where the vehicle p i (k) is the vehicle position, v i (k) is the vehicle velocity, T i (k) is the integrated driving/braking torque, the control input u i (k) represents the desired driving/braking torque, ∆t is the discrete time interval, η i is the transmission efficiency, m i is the vehicle mass, r i is the tire radius, τ i is the inertial lag of longitudinal dynamics, C Ai is the coefficient of aerodynamic drag, g is the gravity constant, f i is the coefficient of rolling resistance, and θ i is the slope angle. The disturbance w i (k) represents the lumped disturbance that contains external disturbances and the modeling errors [START_REF] Sira-Ramirez | On the robust control of buck-converter dc-motor combinations[END_REF], [START_REF] Barmish | On ultimate boundedness control of uncertain systems in the absence of matching condition[END_REF]. Assume that the disturbance is amplitude-bounded, i.e., w i (k) ∈ W i , where W i is a compact interval containing the origin in its interior.

The nonlinearity a f i (v i (k)) is given by

a f i (v i (k)) = C Ai v 2 i (k) + gf i cos(θ i ) + g sin(θ i ). We denote the state vector x i (k) = p i (k), v i (k) T i (k)
, and the system output y i (k) = p i (k). Then, model (1) can be reformulated in the state-space form

x i (k + 1) = A i x i (k) + g i (x i (k)) + B i u i (k) + G i + Hw i (k) y i (k) = Cx i (k) (2)
where

A i =   1 ∆t 0 0 1 ηi∆t miri 0 0 1 -∆t τi   , B i =   0 0 ∆t τi   , G i =   0 -gf i cos(θ) -g sin(θ) 0   , H =   0 0 1   , g i (x i (k)) =   0 -C Ai v 2 i (k) 0   , C =   1 0 0   .
The system is subject to both state and control constraints as

u i ∈ U i , x i ∈ X i ,
where

U i ∈ R 1 is a compact set, X i ∈ R 3×1 is a closed set,
and each set contains the origin in its interior. Without taking into account the bounded disturbance w i (k), the nominal longitudinal dynamics of ith vehicle associated with system (2) is given by

xi (k + 1) = A i xi (k) + g i (x i (k)) + B i ūi (k) + G i ȳi (k) = C xi (k). (3)
Note that for any time-varying variable s(k) of the vehicle system (2), s(k) denotes the corresponding variable of its nominal model (3).

B. Communication Structure of Vehicle Platoons

With the development of V2V technology, a rapid local vehicular communication can be realized. Individual vehicles are connected via communication to form a vehicle platoon structure. Several communication topologies are available in the literature [START_REF] Zheng | Distributed model predictive control for heterogeneous vehicle platoons under unidirectional topologies[END_REF], which are illustrated in Fig. 2.

Here, the predecessor-leader following (PLF) communication topology is adopted, which has been widely used and discussed [START_REF] Rdnyi | An adaptive spacing policy guaranteeing string stability in multi-brand ad hoc platoons[END_REF], [START_REF] Li | Nonlinear consensus-based connected vehicle platoon control incorporating car-following interactions and heterogeneous time delays[END_REF]. In a PLF topology, each vehicle can receive the information transmitted from the predecessorvehicle and the leader-vehicle. It has been shown that this communication structure can help the platoon state converge to a stable state more securely by directly using the information from the leader-vehicle [START_REF] Zheng | Stability and scalability of homogeneous vehicular platoon: Study on the influence of information flow topologies[END_REF]. The information includes its planned trajectory (or vehicle states) and the desired intervehicle spacing.

C. Platooning Control Objectives

The primary goal of platooning is to maintain a desired safe spacing between vehicles and to drive each vehicle at the same speed on the same lane with the leading vehicle under the effects of external disturbances and modeling errors. For system (2) with the lumped disturbance w i (k) ∈ W i , this goal can be mathematically represented as

lim k→∞   p i (k) -p 0 (k) + i • d 0 v i (k) -v 0 (k) a i (x i (k))   ∈ σ i , i ∈ N, (4)
where d 0 is the desired space between adjacent vehicles, v 0 (k) is the desired speed and

a i (x i (k)) = ηi miri T i (k)-a f i (v i (k))
is the acceleration of ith vehicle, σ i is an invariant set and its size depends on the system disturbance and controller parameters (see Section III).

To evaluate the control performance, we consider two following variables: i) the spacing d si (k) between the adjacent vehicles, ii) the platoon deviation ∆d pi (k), which are respectively defined as

d si (k) = p i-1 (k) -p i (k), ∆d pi (k) = p 0 (k) -p i (k) -id 0 , for i ∈ N.

III. CONTROL PROBLEM FORMULATION

To achieve the control goal specified in (4) for system (2), we propose an integrated control scheme composed of three components: feedback control, disturbance-compensation control and feedforward control, see Fig. 3. Then, the new integrated control law of ith vehicle is of the form

u i (k) = ūi (k) + K i (x i (k) -xi (k)) -B † i H ŵi (k), (5)
where ūi denotes the feedforward control, K i is the feedback gain to be designed, B † i = (B i B i ) -1 B i is the pseudoinverse of the full-collum rank matrix B i , and xi and ŵi are respectively the estimates of x i and w i . The design procedure of the integrated control law ( 5) is described as follows.

x i Disturbance w i Z i K i x i K i B -1 H y i w i x i u i u i X i ,U i K i , L i x i-1 i-1 x 0 K i e i B -

A. Design of Feedback Control and Disturbance-Compensation Control

This section presents the design of an PMI observer to estimate both the state x i and the disturbance w i for the vehicle system [START_REF] Shladover | Automated vehicle control developments in the PATH program[END_REF]. The proposed PMI observer can achieve a high-precision estimation performance for systems with high order time-varying disturbances. In particular, it can provide an asymptotic estimation performance for polynomial disturbances [START_REF] Chen | Disturbance-observer-based control and related methods: An overview[END_REF]. This not only reduces the number of physical vehicle sensors used for feedback control but also allows to construct a disturbance-compensation action in the integrated control law [START_REF] Al Alam | An experimental study on the fuel reduction potential of heavy duty vehicle platooning[END_REF].

1) Proportional Multiple Integral Observer: For state and disturbance estimation, we make use of the following third-order PMI observer:

xi (k + 1) = A i xi (k) + g i (x i (k)) + B i u i (k) + G i + H ŵi (k) + L P i (y i (k) -ŷi (k)) ŵi (k + 1) = ŵi (k) + ŵ1i (k)∆t + L I1i (y i (k) -ŷi (k)) ŵ1i (k + 1) = ŵ1i (k) + ŵ2i (k)∆t + L I2i (y i (k) -ŷi (k)) ŵ2i (k + 1) = ŵ2i (k) + L I3i (y i (k) -ŷi (k)) ŷi (k) = C xi (k) (6)
where L P i , L I1i , L I2i and L I3i are the observer gains to be determined.

e xi (k + 1) = (A i -L P i C)e xi (k) + He wi (k) + δ ĝi (k) e wi (k + 1) = e wi (k) + e w1i (k)∆t -L I1i Ce xi (k) e w1i (k + 1) = e w1i (k) + e w2i (k)∆t -L I2i Ce xi (k) e w2i (k + 1) = e w2i (k) -L I3i Ce xi (k) + w ei (k) (8)
where

δ ĝi (k) = g i (x i (k)) -g i (x i (k)).
2) Analysis of Error Dynamics: Let e xi (k) = x i (k)xi (k) be the state error between the vehicle system (2) and its nominal model ( 3), whose dynamics is given by

e xi (k + 1) = (A i + B i K i )e xi (k) -B i K i e xi (k) + He wi (k) + δ ḡi (k), (9)
where

δ ḡi (k) = g i (x i (k)) -g i (x i (k)).
Let us define the augmented error state vector as

e i (k) = e xi (k) e xi (k) e wi (k) e w1i (k) e w2i (k) .
It follows from ( 8) and ( 9) that

e i (k + 1) = A ei e i (k) + e gi (k) + B ei w ei (k), (10)
where

A ei =       A i + B i K i -B i K i H 0 0 0 A i -L P i C H 0 0 0 -L I1i C 1 ∆t 0 0 -L I2i C 0 1 ∆t 0 -L I3i C 0 0 1       , B ei = 0 0 0 0 0 0 0 0 1 , e gi (k) = δ ḡi (k) δ ĝi (k) 0 0 0 .
Exploiting the Lipschitz property of the nonlinear function g i (x) over the state set X i , it follows that [START_REF] Gao | A tube-based robust nonlinear predictive control approach to semiautonomous vehicles[END_REF]

e gi ∞ ≤ L i (X i ) max evi∈Vi e vi 2 , (11)
where

L i (X i ) is the Lipschitz constant of g i (•) over X i , e vi = 0 v i -vi 0 0 v i -vi 0 0 0 0 , V i is a subset of X i to bound the e vi .
Expression [START_REF] Ploeg | Controller synthesis for string stability of vehicle platoons[END_REF] allows defining the following box B g (V i ) to bound the nonlinear term e gi (•) in system (10):

B g (V i ) = x ∈ R 9×1 : x ∞ ≤ L(X i ) max evi∈Vi e vi 2 .
Then, the augmented error system in (10) can be rewritten as

e i (k + 1) = A ei e i (k) + wei (k) (12)
where wei = e gi (k) + B ei w ei (k), and thus wei

∈ W ei = B g (V i ) ⊕ B ei W ei .
Assume that Z i is a robustly positively invariant (RPI) set [START_REF] Mayne | Robust model predictive control of constrained linear systems with bounded disturbances[END_REF] with respect to system [START_REF] Zheng | Stability and scalability of homogeneous vehicular platoon: Study on the influence of information flow topologies[END_REF], i.e.,

A ei e i (k) + wei (k) ∈ Z i , ∀e i (k) ∈ Z i , ∀ wei (k) ∈ W ei . (13)
We can infer from (13) that if the involved parameters to be optimized are properly tuned, the feedback term and the disturbance compensation term can make the error system [START_REF] Zheng | Stability and scalability of homogeneous vehicular platoon: Study on the influence of information flow topologies[END_REF] tend to be stable.

B. Distributed MPC Feedforward Control

The feedforward control ūi (k) in ( 5) of ith vehicle is determined using a distributed MPC algorithm with the information transmitted from other controllers. Let x * i (k|t) is the nominal optimal trajectory of ith vehicle at time t. To achieve multi-objective control, several specifications are taken into account in the MPC optimization problem, including the safety distance between ith vehicle and adjacent ones, the stability of the vehicle platoon, the acceleration smoothness, and the fuel economy.

1) Design of DMPC Algorithm: For each vehicle i, the state of the nominal system (3) to be optimized is defined as

ξi (t) = β • xi (t), with β = 1 0 0 0 1 0 .
Here, the same sizes of the prediction horizon N p and the control horizon i (k|t) is used as a trajectory variable to parameterize the local optimal control problem. ξ * i (k|t) denotes the optimal trajectory obtained by solving the optimal problem. ξa i (k|t) is the assumed trajectory transmitted from vehicle i to vehicle i + 1. Then, the cost function of the distributed MPC algorithm can be defined as

N
J i (t) = J i ( ξi (t), ūi (t)) = Np-1 k=0 J i ( ξi (k|t), ūi (k|t)), (14)
where

J i ( ξi (k|t), ūi (k|t)) = ξi (k|t) -ξdes,i (k|t) Qi + ξi (k|t) -ξa i-1 (k|t) + d des Ri + ξi (k|t) -ξa i (k|t) Gi + a i ( ξi (k|t)) Fi , and ξdes,i (k|t) = [ ξa 0 (k|t) -i • d des ] is the desired position for ith vehicle, d des = [ d 0 , 0 ] is the desired spacing, Q i , R i , G i , F i are the positive semi-definite weighting matrices.
The safety constraints for vehicle platoons should be also guaranteed with the following constraints:

|p i (k|t) -p i-1 (k|t) + d 0 | < p b , |v i (k|t) -v i-1 (k|t)| < v b , |a i (k|t) -a i-1 (k|t)| < a b , (15)
for k = 0, 1, . . . , N p . The constraints in [START_REF] Guo | Autonomous platoon control allowing rangelimited sensors[END_REF] can be rewritten in a compact form

h(x i (k|t), u i (k|t)) ∈ H i ,
where the bounded set H i can be easily defined with d 0 , p b , v b and a b . Given an RPI set Z i for system [START_REF] Zheng | Stability and scalability of homogeneous vehicular platoon: Study on the influence of information flow topologies[END_REF], the RPI subset Z xi corresponding to the vehicle state x i can be derived from Z i . Then, the constraints of the nominal system (3) can be computed as

xi (k|t) ∈ X i = X i Z xi , ūi (k|t) ∈ U i = U i K i Z xi W i , h(x i (k|t), ūi (k|t)) ∈ H i = H i H i Z xi . ( 16
)
where

H i = diag(2, 2, 2ηi miri ).
The feedforward control design for ith vehicle can be now formulated as the following optimization problem:

min ūp i J i (t) (17)
such that

xp i (k+1|t) = A i xp i (k|t)+g i (x p i (k|t))+B i ūp i (k|t)+G i ξp i (k|t) = β • xp i (k|t) xp i (k|t) ∈ X i ūp i (k|t) ∈ U i , k = 0, 1, . . . , N c -1 ūp i (k|t) = ūp i (k -1|t), k = N c , . . . , N p -1 h(x p i (k|t), ūp i (k|t)) ∈ H i ξp i (N p |t) = ξdes,i (N p |t) T p i (N p |t) = m i r i η i a f i (v p i (N p |t)). (18)
Algorithm 1. The DMPC algorithm designed for vehicle i is shown as follows.

Initialization: At time t < 0, all the follower vehicles receive the new desired speed and spacing from the leader vehicle, and solve their local optimal problem [START_REF] Dunbar | Distributed receding horizon control of vehicle platoons: Stability and string stability[END_REF] with R i = 0, G i = 0, for time t = 0. The initial optimal trajectory of the nominal system (3) at time t = 0 is calculated as

x * i (k + 1|0) = A i xi (k|0) + g i (x i (k|0)) + B i ūi (k|0) + G i ,
for k = 0, 1, . . . , N p -1. Meanwhile, the first set of information has been transmitted to other vehicles as

xa i (k|1) = x * i (k + 1|0), k = 0, 1, . . . , N p -1.
Iteration: At any time t > 0, all vehicles follow the following steps to calculate the feedforward control.

Step 1. Apply ū * i (0|t) for the nominal system (3).

Step 2. Compute the optimal state by using the current nominal state xi (t) and the optimal control input ū * i (0|t) in prediction horizon

x * i (k + 1|t) = A i x * i (k|t) + g i (x * i (k|t)) + B i ū * i (k|t) + G i k = 0, 1, . . . , N p -1 x * i (0|t) = xi (t) ξ * i (k|t) = β • x * i (k|t).
Step 3. Compute the assumed trajectory xa i (k|t + 1) and the control input ūa i (k|t + 1) for next step

xa i (k|t + 1) = x * i (k + 1|t), k = 0, 1, . . . , N p -2 x * 0 (N p |t) + i • d des , k = N p -1 (19)
ūa i (k|t + 1) = ū * i (k + 1|t), k = 0, 1, . . . , N p -2 miri ηi a f i (v * i (N p |t)), k = N p -1 (20) Step 4. Transmit xa i (•|t + 1)
to the follower vehicle i + 1, and receive the assumed trajectory from the connected vehicle (predecessor and leader vehicle).

Step 5. Solve the optimal problem [START_REF] Dunbar | Distributed receding horizon control of vehicle platoons: Stability and string stability[END_REF] with the new information to get the ū * i (•|t + 1). Step 6. At the time t + 1, select the control input u * i (0|t + 1) to ith vehicle and return to Step 1 for the next iteration.

To reduce the computational burden, we adopt the Warm-Start method in [START_REF] Wang | Fast model predictive control using online optimization[END_REF] for the optimization problem, which means that the solution at time t is used as the initial solution at time t + 1 as indicated in [START_REF] Gao | Distributed adaptive sliding mode control of vehicular platoon with uncertain interaction topology[END_REF]. This method can effectively increase the computational efficiency, especially for nominal systems that are not affected by disturbances.

2) Recursive Feasibility Analysis of DMPC: Before proving the recursive feasibility of the optimization problem [START_REF] Dunbar | Distributed receding horizon control of vehicle platoons: Stability and string stability[END_REF], a required condition is shown in the following theorem. Theorem 1. [START_REF] Zheng | Distributed model predictive control for heterogeneous vehicle platoons under unidirectional topologies[END_REF] If the platoon system communication topology contains a spanning tree at the leader like the topologies shown in the Fig. 2, the terminal state in the predictive horizon N p of the optimal problem (17) converges to the desired state in at most N steps, that is

ξp i (N p |t) = ξdes,i (N p |t), t ≥ N.
Hence, an initial feasible assumption needs to be stated, which has been widely discussed in previous studies [START_REF] Zheng | Distributed model predictive control for heterogeneous vehicle platoons under unidirectional topologies[END_REF], [START_REF] Dunbar | Distributed receding horizon control of vehicle platoons: Stability and string stability[END_REF].

Lemma 1. For the nominal system (3) of ith vehicle with a perfect state measurement and without disturbance, the feasibility of the open-loop optimal control problem [START_REF] Dunbar | Distributed receding horizon control of vehicle platoons: Stability and string stability[END_REF] with equation ( 14) subject to constrains [START_REF] Öncü | Cooperative adaptive cruise control: Network-aware analysis of string stability[END_REF] at time t = 0 implies its feasibility for all t > 0.

Proof. It is assumed that the optimal solution ū * i (•|t) is found at time t. The feedforward control ū * i (•|t) is held constant for the nominal system (3) after the control horizon N c . Then, we can write ū * i (k|t) = ūss,i , (k ≥ N c ) for this steady-state value.

Consider the system dynamics, we define the steady-state vss,i and Tss,i as the (in this case, unique) solutions of

vss,i = vss,i + η i m i r i Tss,i -a f (v ss,i ) ∆t
Tss,i = Tss,i + ūss,i -Tss,i ∆t τ i which are Tss,i = ūss,i and vss,i = a -1 f η mr Tss,i . Note that for ūss,i > 0 and hence vss,i > 0, this is an asymptotically stable equilibrium, and the domain of attraction includes all v ∈ R, all T ∈ R. The position pi (k|t) will not converge to an equilibrium, but if we change variables pi (k|t) = pi (k|t)kv ss,i ∆t, we can write

pi (k + 1|t) -(k + 1)v ss,i ∆t = pi (k|t) -kv ss,i ∆t + (v i (k|t) -vss,i ) ∆t
and thus

pi (k + 1|t) = pi (k|t) + (v i (k|t) -vss,i ) ∆t.
This new position variable will also converge to a steadystate pss,i as (v i (k|t)-vss,i ) → 0. We will write h, xi = [p i , vi , Ti ] to denote the constraint function and system state accounting for this change of variables.

As we have shown that the system approaches the equilibrium, for k = N c , . . . , N p , we have the set as

S i =              xi ∈ R 3 pi (k+1) = pi (k) + (v i (k) -vss,i )∆t vi (k+1) = vi (k)+ ηi miri Ti (k)-a f (v i (k)) ∆t Ti (k+1) = Ti (k) + ūss,i -Ti (k) ∆t τi ūi (k) = ūi (k -1) h (x i (k), ūi (k)) ∈ H              .
The set S is an invariant set with respect to the dynamics of xi with the optimal solution (x * i (•|t), ū * i (•|t)) at time t. We will construct an (x i (•|t + 1), ūi (•|t + 1)) satisfying constraints at time t + 1. For simplicity, we first assume that ξdes,i has not changed from time t to t + 1, and then modify our constructed solution to account for possible changes in ξdes,i .

Considering (x i (k|t+1), ūi (k|t+1)), for k = 0, . . . , N p -1, all constraints in the optimisation problem at time t + 1 are trivially satisfied by choosing the control input as [START_REF] Gao | Distributed adaptive sliding mode control of vehicular platoon with uncertain interaction topology[END_REF] because these constraints are identical to constraints from time t. For prediction time k = N p , we choose the control input as

ūi (N p -1|t + 1) = ū * i (N p -2|t + 1) = m i r i η i a f i (v * i (N p |t)) = m i r i η i a f i (v i (N p -1|t + 1)). ( 21
)
With the terminal state x * i (N p |t) constrained by [START_REF] Öncü | Cooperative adaptive cruise control: Network-aware analysis of string stability[END_REF], we have

Ti (N p -1|t + 1) = T * i (N p |t) = m i r i η i a f i (v * i (N p |t)) = m i r i η i a f i (v i (N p -1|t + 1)). ( 22
)
Then, combining equations ( 21) and ( 22), the terminal state xi of the system at time t + 1 can be obtained as pi (N p |t + 1) = pi (N p -1|t + 1) = pss,i vi (N p |t + 1) = vi (N p -1|t + 1) = vss,i Ti (N p |t + 1) = Ti (N p -1|t + 1) = Tss,i

The nominal system state xi can be obtained by choosing pi (k|t) = pi (k|t) + kv ss,i . Because S i is an invariant set, which incorporates the constraint set H, this choice satisfies all constraints (18) on xi (N p |t), ūi (N p |t).

Finally, we must account for possible changes in the communicated preceding vehicle states ξdes . If these states have changed, the preceding vehicle acceleration ā(k) has changed for some k. Defining δā(k) as this change, then let δv(k + 1) = δv(k) + δā(k)∆t and δ p(k + 1) = δ p(k) + δv(k)∆t, for all k. We modify our constructed x(k) as

ā(k) → ā(k) + δā(k) v(k) → v(k) + δv(k) p(k) → p(k) + δ p(k)
and choose ū(k) consistently with this change in ā(k). This leaves the differences pi (k)-pi-1 (k), vi (k)-v i-1 (k), āi (k)āi-1 (k) unchanged since the changes in ξdes is transmitted from the leader to all followers (directly or indirectly). Hence, we satisfy the constraints h (x(k), ū(k)) ∈ H at time t + 1 if they were satisfied at t.

3) Stability Analysis of DMPC:

We now prove the stability of the platoon system by analyzing the decreasing properties of the platoon cost function. To this end, the sum of the optimal cost functions is defined as

J * Σ (t) = N i=1 J * i ( ξ * i (•|t), ξa -i (•|t), ū * i (•|t)) = Np-1 k=0 L( ξ * (k|t), ξa (k|t), ū * (k|t)) (23)
where

L( ξ, ξa , ū) = N i=1 ( ξi -ξdes,i Qi + ξi -ξa i-1 + d des Ri + ξi -ξa i Gi + a i ( ξi ) Fi ).
With the terminal state ξp i (N p |t) = ξdes,i (N p |t), t ≥ N provided by Theorem 1, and the feasible control u p i (•|t) = u a i (•|t) provided by Algorithm 1, the distributed optimal cost function can be bounded as

J * Σ (t + 1) ≤ Np-1 k=0
L( ξa (k|t + 1), ξa (k|t + 1), ūa (k|t + 1))

(24) = Np-2 k=0 L( ξ * (k + 1|t), ξ * (k + 1|t), ū * (k + 1|t)).
The equality holds due to the definitions of xa i (k|t + 1) and ūa i (k|t + 1) in ( 19) and [START_REF] Gao | Distributed adaptive sliding mode control of vehicular platoon with uncertain interaction topology[END_REF], respectively. Subtracting the equation ( 23) from ( 24 where

∆ k = L( ξ * (k|t), ξ * (k|t), ū * (k|t)) -L( ξ * (k|t), ξa (k|t), ū * (k|t)) = N i=1 ( ξ * i (k|t) -ξdes,i (k|t) Qi + ξ * i (k|t) -ξ * i-1 (k|t) + d des Ri + ξ * i (k|t) -ξ * i (k|t) Gi + a i ( ξ * i (k|t)) Fi ) - N i=1 ( ξ * i (k|t) -ξdes,i (k|t) Qi + ξ * i (k|t) -ξa i-1 (k|t) + d des Ri + ξ * i (k|t) -ξa i (k|t) Gi + a i ( ξ * i (k|t)) Fi ). (25)
Using the triangle inequality for vector norms with R 1 = 0, the term ∆ k defined in ( 25) is bounded as

∆ k ≤ N -1 i=1 ( ξ * i (k|t) -ξa i (k|t) Ri+1 -ξ * i (k|t) -ξa i (k|t) Gi ).
Then, the weights of the cost function ( 14) can be designed as follows:

G i ≥ R i+1 , i = 0, 1, . . . , N -1. (26)
It follows from ( 26) that the function J * Σ (•) satisfies

J * Σ (t + 1) -J * Σ (t) ≤ -L( ξ * (0|t), ξa (0|t), ū * (0|t)). (27)
Inequality [START_REF] Mayne | Robustifying model predictive control of constrained linear systems[END_REF] shows that the platoon cost function J * Σ (•) is strictly monotonically decreasing. Thus, the asymptotic stability of the platoon system can be guaranteed.

From the above analysis, the states of the nominal system will asymptotically converge to the desired states as

lim k→∞   pi (k) -p0 (k) + i • d 0 vi (k) -v0 (k) āi (x i (k))   =   0 0 0   .
Then, the states of the actual vehicle system will achieve the objective (4) with a set value of σ i defined as

σ i =   2 0 0 0 2 0 0 0 ηi miri   Z xi .

IV. INTEGRATED CONTROL DESIGN PROCEDURE

This section presents the design of both unknown input observer and feedback control. The implementation of the proposed distributed MPC control scheme is also discussed.

A. Offline Computation

Hereafter, PSO algorithm is exploited to design both observer and feedback gains. Then, the RPI set of the extended error system [START_REF] Zheng | Stability and scalability of homogeneous vehicular platoon: Study on the influence of information flow topologies[END_REF] and the constraints ( 16) of the vehicle nominal system (3) are determined.

1) PSO-Based Control Feedback Design: The following specifications are taken into account in the design of both observer and control feedback gains:

• the stability of the augmented error system [START_REF] Zheng | Stability and scalability of homogeneous vehicular platoon: Study on the influence of information flow topologies[END_REF] under the effect of bounded disturbance w ei , • the size of the RPI sets Z xi and K i Z xi of system [START_REF] Zheng | Stability and scalability of homogeneous vehicular platoon: Study on the influence of information flow topologies[END_REF]. To minimize the effect of the disturbance, the ∞ performance is taken into account in the design of both the feedback control gain and the observer gain. To this end, we consider the controlled output z i (k) = Ex i (k) associated to system [START_REF] Zheng | Stability and scalability of homogeneous vehicular platoon: Study on the influence of information flow topologies[END_REF], where E is selected as diag(100, 1, 0.01, 100, 1, 0.01, 0, 0, 0). Using the bounded real lemma [START_REF] Souza | On the discrete-time bounded real lemma with application in the characterization of static state feedback H∞ controllers[END_REF], the following theorem provides conditions to guarantee an H ∞ control performance for the closed-loop system [START_REF] Zheng | Stability and scalability of homogeneous vehicular platoon: Study on the influence of information flow topologies[END_REF]. Theorem 2. If there exit a positive definite matrix M i , matrices K i , L P i , L I1i , L I2i , L I3i , of appropriate dimensions, and a positive scalar γ i such that the following optimization problem is achievable:

min Pi={Ki,L P i ,L I1i ,L I2i ,L I3i } γ i ( 28
) such that   A ei M i A ei -M i A ei M i B ei E * B ei M i B ei -γ 2 i I 0 * * -I   < 0.
Then, system ( 12) is stable with an H ∞ performance level less than or equal to γ i .

The optimization problem (28) in Theorem 2 is nonconvex due to the nonlinear couplings between decisions variables. Finding a solution for such a nonconvex problem is NP hard. Here, PSO algorithm is exploited to derive the optimal control feedback gain and the observer gains. Note that the initial gains P i (0) are obtained from linear quadratic regulator (LQR) and pole assignment methods. To reduce the computational burden, the particle position P i and velocity V i are constrained

P i,j (n) ∈ [P min , P max ] , j = 1, 2, . . . , N 1 , V i,j (n) ∈ [V min , V max ] , n = 0, 1, . . . , N 2 ,
where j is the particle number, N 1 is the total number of particles, n is the iterations, N 2 is the total number of iterations. The initial state of all search particles are given by

P i,j (0) = P min + r a (P max -P min ) , V i,j (0) = V min + r a (V max -V min ) ,
where r a is a random number drawn from uniform distribution between 0 and 1. The position P i,j and the velocity V i,j are updated as [START_REF] Eberhart | Comparison between genetic algorithms and particle swarm optimization[END_REF] P i,j (n + 1) = P i,j (n) + V i,j (n + 1),

V i,j (n + 1) = w a V i,j (n) + c 1 r a1 (P pb i,j (n) -P i,j (n)) + c 2 r a2 (P gb i (n) -P i,j (n))
, where w a is the inertia weight, P pb i,j (n) is the best previous position of jth particle at iteration n, P gb i (n) is the global best position in the swarm at iteration n, c 1 and c 2 are the accelerated constant, r a1 and r a2 are random numbers drawn from uniform distribution between 0 and 1. Note that after N 2 iterations, the optimal control parameters can be obtained as

P i = P gb i (N 2 ).

2) Computation of Robustly Positive Invariant Set:

The RPI set can be computed following the method in [START_REF] Kerrigan | Robust constraint satisfaction: Invariant sets and predictive control[END_REF]. To this end, we define the following convex compact set F s,i :

F s,i = s-1 j=0 A j ei W ei , with F 0,i = {0}.
Note that the minimal RPI (mRPI) set, denoted by F ∞,i , is given by

F ∞,i = ∞ j=0 A j ei W ei .
The following lemma is useful to compute the RPI set of system [START_REF] Zheng | Stability and scalability of homogeneous vehicular platoon: Study on the influence of information flow topologies[END_REF].

Lemma 2. [START_REF] Kouramas | Control of linear systems with state and control constraints[END_REF] If 0 ∈ W ei . Then, there exists a finite integer s ∈ N + and a scalar α ∈ [0, 1) satisfying

A s ei W ei ⊆ α W ei . ( 29
)
If condition (29) holds, then there exists a convex compact RPI set

F (α, s, i) = (1 -α) -1 F s,i , (30)
such that 0 ∈ F (α, s, i) and F ∞,i ⊆ F (α, s, i).

As shown in [START_REF] Kerrigan | Robust constraint satisfaction: Invariant sets and predictive control[END_REF], the set F (α, s, i) defined in ( 30) is an outer approximation of the mRPI set F ∞,i . Note that the smaller values of s and α lead to a better approximation of F ∞,i . We denote the smallest values of s and α as

s • (α, i) = min{s ∈ N + |A s ei W ei ⊆ α W ei }, α • (s, i) = min{α ∈ R|A s ei W ei ⊆ α W ei }.
In this paper, selecting α = 0.05, we obtain s i = s • (0.05, i) and α i = α • (s • (0.05, i)). Then, the corresponding RPI set is computed as

Z i = F (α i , s i , i) = F (α • (s • (0.05, i)), s • (0.05, i), i).
The multi-parametric toolbox (MPT) [START_REF] Kvasnica | Multi-parametric toolbox (mpt)[END_REF] can be used for the set computations.

3) Nominal Constraints Computation: After computing the invariant set Z i , the sets of tightened constraints X i , U i and H i can be defined for the nominal vehicle system (3) from [START_REF] Zheng | Distributed model predictive control for heterogeneous vehicle platoons under unidirectional topologies[END_REF]. Then, the designed integrated control law (5) can ensure the states x i and control input u i of ith vehicle to satisfy the constraints [START_REF] Guo | Autonomous platoon control allowing rangelimited sensors[END_REF].

B. Online Control Implementation

Once the offline computations are finished, the online implementation steps are performed at the time t ≥ 0 as follows.

Step 1. Use the output y i (t) and control input u i (t) of the ith vehicle to compute the estimated state xi (t + 1) and disturbance ŵi (t + 1) following (6).

Step 2. Compute the nominal system state xi (t + 1) using the nominal state xi (t) and the control input ūi (t).

Step 3. Compute the feedback control and disturbancecompensation control from the results of Steps 1 and 2.

Step 4. Run Algorithm 1 to obtain the optimal control sequence ū * i (0|t + 1), ū * i (1|t + 1),. . . , ū * i (N p -1|t + 1) for the nominal system.

Step 5. Select the first element ūi (0|t + 1) from the optimal control sequence to use for feedforward control ūi (t+1).

Then, compute the integrated control law u i (t + 1) for ith vehicle by combining the results of Step 3.

u i (t + 1) = ūi (t + 1) + K i (x i (t + 1) -xi (t + 1)) -B † i H ŵi (t + 1). Step 6. At the time t + 1, apply the control input u i (t + 1)
to ith vehicle and return to Step 1 for the next iteration.

V. SIMULATION AND COMPARISON RESULTS

This section provides simulation results obtained with two different scenarios and working conditions for vehicle platoons, composed of one leading vehicle and four following vehicles. The proposed control method is compared with the classical DMPC control scheme [START_REF] Zheng | Distributed model predictive control for heterogeneous vehicle platoons under unidirectional topologies[END_REF] and tube-based robust nonlinear MPC (RNMPC) method [START_REF] Gao | A tube-based robust nonlinear predictive control approach to semiautonomous vehicles[END_REF]. These algorithms are implemented in a distributed way like Algorithm 1. The stabilizing state feedback gain of RNMPC method for the error system e xi = x i -xi between (2) and (3) will be selected as the infinite horizon LQR solution K ∞ LQR,i for system (A i , B i ). Therefore, the RNMPC controller is designed as

u RNMPC,i = ūRNMPC,i + K ∞ LQR,i (x RNMPC,i -xRNMPC,i ),
where ūRNMPC,i is the optimal control input for the nominal system (3), which is computed by solving a constrained finitetime optimal control problem with the cost function defined in [START_REF] Guo | Adaptive fuzzy sliding mode control for coordinated longitudinal and lateral motions of multiple autonomous vehicles in a platoon[END_REF]. The K ∞ LQR,i is also used in the computation of the robust invariant set Z RNMPC,i :

F RNMPC (α, s, i) = (1 -α) -1 F RNMPC,s,i = (1 -α) -1 s-1 j=0 (A i + B i K ∞ LQR,i ) j HW i Z RNMPC,i = F RNMPC (α • (s • (0.05, i)), s • (0.05, i), i).
All controllers are implemented using MATLAB 2019a with Intel i5-8250U and 8G RAM. The environmental settings are described in Table I. The parameters of the heterogeneous vehicles are randomly selected according to the passenger vehicles. Then, the nominal vehicle model is different from the actual vehicle model.

η i 0.95 Np 12 m i (kg) 1500+300r ni Nc 5 r i (m) 0.3+0.1r ni p b (m) 2
τ i 0.1+0.1r ni v b (m/s) 5
f i 0.015+0.005r ni a b (m/s 2 ) 5 θ i ( • ) 5+r ni Q i diag(100,1)
C Ai 0.6+0.2r ni R i diag(50,0.5) c 1 2 G i diag(100,1) c 2 2 F i 0.5 N 1 30 N 2 100 v min (m/s) 0 vmax (m/s) 35
a min (m/s 2 ) -6 amax (m/s 2 ) 6
T min (Nm) 0 Tmax (Nm) 1000 This test allows to compare the performance of different controllers under time-varying and time-invariant external disturbances. The simulation scenario and the working condition are specified in Table II.

P i {[
d 0 (m) 20 20 20
v 0 (m/s) 20 20 20
w i (N) 0 500sin( t-i+1 2.9 ) 375
Fig. 4 shows that the proposed control scheme can compensate effectively the effects caused by the external disturbance. In the first 25 seconds, under the impact of the periodic disturbance, the platoon deviation induced by both control methods are fluctuating periodically. The amplitude of the classical DMPC method is ±0.3 m, the RNMPC method is ±0.13 m, while that of the proposed controller is only ±0.02 m. Under the constant disturbance after 25 seconds, the DMPC and RNMPC algorithm leads to a static error, which is not the case of the proposed approach. Moreover, due to the presence of interference, the DMPC method cannot guarantee that the platoon deviation is not amplified as the vehicle number increased with the PLF communication topology even if the speed of the leading vehicle remains unchanged.

The control situation of vehicle spacing under two different controllers is shown in Fig. 5. It can be seen more intuitively that different static errors exist in each adjacent spacing with traditional DMPC, while the proposed controller has a good control effect for all vehicles. Figs. 6 and present a comparison among three controllers in terms of vehicle velocity and acceleration. We can see that the DMPC introduces important oscillations due to disturbance effects and chattering can be observed in both speed and acceleration results. However, the disturbance effect is significantly minimized with the proposed controller, which induces no visible chattering phenomenon in both speed and acceleration results.

Both the RNMPC and the proposed controller ensure that the states are within the invariant set of the nominal system. However, RNMPC has a larger boundary, which makes the controller unnecessarily conservative.

For illustrations, Fig. 8 depicts the estimation of external disturbances for the first and the last following vehicles in the platoon. We can observe a high estimation performance can be obtained with the designed unknown input observer. This allows for an effective disturbance-compensation control action to improve the overall control performance of the proposed integrated control scheme. In addition, we also consider the comparison of different disturbance observers in the simulation as shown in Fig. 9, which includes PMI observers of different orders and an extended state observer (ESO). Note that PMI observer with an n-order disturbance model is denoted by PnIO in the figure. Furthermore, the root mean square error (RMSE), defined as

RMSE = 1 m m t=1 [w(t) -ŵ(t)] 2 ,
is adopted as the evaluation index to compare the performance differences of the considered observers. The comparison results are shown in Table III. As can be seen from Fig. 9, PMI observers perform better than ESO when dealing with time-varying disturbances. The results in Table III show that the higher-order observers P3IO and P4IO also have a slightly better performance over P2IO observer. However, the observer P4IO did not perform as well as expected due to a large number of involved parameters, which leads to difficulties in observer gain tuning.

B. Scenario 2: Performance Under Vehicle Acceleration and Deceleration

In this scenario, we set a variable speed of the leading vehicle in the presence of external disturbances. It is required that the following vehicles keep the spacing unchanged under the condition that the leading vehicle status changes, so as to detect the control performance of the two controllers under acceleration/deceleration. The simulation conditions of this test scenario are shown in Table IV.

w i (N) 0, t ∈ [ 0, 1.5i -1.5) 30
500 sin( t-1.5i+1.5

4

), t ∈ [ 1.5i -1.5, 50 ] Fig. 10 shows the comparison of the platoon deviation ∆d p obtained with three controllers. In the first 20 seconds, all results show irregular variations in deviation. Note that the DMPC leads to a large error range of ±0.4 m compared to that of the proposed controller (±0.04 m). Moreover, after 20 seconds, the platoon derivation becomes very small only with the proposed controller. Similar remarks can be done for the spacing comparison in Fig. 11. Therefore, the state change of the leading vehicle greatly affects the control performance, but it is still controllable. In this scenario, the vehicle state has a large change as shown in Figs. 12 and 13. However, under the action of the proposed controller, there is no visible chattering behavior in the states of the following vehicles as in the classical DMPC. Moreover, the vehicle state is constrained within a limited boundary during the whole test scenario. The disturbance estimation during the platoon mission change process is shown in Fig. 14. It can be seen that despite the acceleration and the deceleration conditions, the obtained estimation performance is still satisfactory. Note that for this simulation scenario, we also add a platoon control with different communication topologies. The obtained results are shown in Fig. 15, which also indicate an outperformance of the proposed controller compared to two other ones.

Due to the presence of disturbances, the classical DMPC method must transmit the controller information to the receiving vehicles several times in a short time, which also increases the computational burden. The data solved by DMPC in the proposed scheme are all from the nominal system, which can effectively reduce the communication number of times, the calculation time and the difficulty of solving. Moreover, the proposed controller has a better control effect for vehicle platoon as shown in Figs. 4 and10. The test results of the abovementioned data for both test scenarios are summarized in Table V. The comparative study shows a clear outperformance of the proposed controller compared to the classical DMPC scheme in [START_REF] Zheng | Distributed model predictive control for heterogeneous vehicle platoons under unidirectional topologies[END_REF].

VI. CONCLUSIONS

An integrated controller, including feedforward control, feedback control and disturbance-compensation control, has been proposed. This control law combines TMPC, PMI observer, and DMPC to ensure the stability of the vehicle platoon control, and effectively compensate the influence of lumped disturbances. The proposed control scheme exploits the main advantages, performs their respective functions, and complements each other. The proposed controller provides a faster solution speed than a classical DMPC scheme with unknown input. Moreover, our method leads to more reasonable constraint conditions and solution environment than RNMPC scheme, which allows obtaining a better control performance.

As a direction to extend this work, future research will further minimize the impacts of random communication delay in vehicle platoons. Additionally, more applicable scenarios of the proposed controller will be found in future works.

Fig. 1 .

 1 Fig. 1. A scenario and structure of vehicle platoons.

Fig. 2 .

 2 Fig. 2. Common communication topology for vehicle platoons. (a) predecessor-following (PF), (b) bidirectional following (BF), (c) twopredecessor following (TPF), (d) predecessor-leader following (PLF).

Fig. 3 .

 3 Fig. 3. Structure of the proposed integrated control scheme.

  * (k|t), ξa (k|t), ū * (k|t)) = -L( ξ * (0|t), ξa (0|t), ū * (0|t)) +

Fig. 4 .

 4 Fig.4shows that the proposed control scheme can compensate effectively the effects caused by the external disturbance. In the first 25 seconds, under the impact of the periodic disturbance, the platoon deviation induced by both control methods are fluctuating periodically. The amplitude of the classical DMPC method is ±0.3 m, the RNMPC method is ±0.13 m, while that of the proposed controller is only ±0.02 m. Under the constant disturbance after 25 seconds, the DMPC and RNMPC algorithm leads to a static error, which is not the case of the proposed approach. Moreover, due to the presence of interference, the DMPC method cannot guarantee that the platoon deviation is not amplified as the vehicle number increased with the PLF communication topology even if the speed of the leading vehicle remains unchanged.The control situation of vehicle spacing under two different controllers is shown in Fig.5. It can be seen more intuitively that different static errors exist in each adjacent spacing with traditional DMPC, while the proposed controller has a good control effect for all vehicles. Figs.6 and 7

Fig. 5 .Fig. 6 .Fig. 7 .Fig. 8 .Fig. 9 .

 56789 Fig. 5. Adjacent spacing ds. (a) Classical DMPC, (b) RNMPC in [30], (c) Proposed controller.

Fig. 10 .

 10 Fig. 10. Platoon deviation ∆dp. (a) Classical DMPC, (b) RNMPC in [30], (c) Proposed controller.

Fig. 11 .Fig. 12 .Fig. 13 .Fig. 14 .

 11121314 Fig. 11. Adjacent spacing ds. (a) Classical DMPC, (b) RNMPC in [30], (c) Proposed controller.

Fig. 15 .

 15 Fig. 15. Platoon deviation ∆dp (with PF communication topology). (a) Classical DMPC, (b) RNMPC in [30], (c) Proposed controller.

  835, 427, 0.52],[2.5, 41, 900] ,33700,24800,8300}

A. Scenario 1: Performance Under Disturbance Effects

TABLE IV SIMULATION

 IV CONDITIONS OFSCENARIO 2.

	Time (s)	[ 0, 5)	[ 5, 9)	[ 9, 14)	[ 14, 20)	[ 20, 50 ]
	Parameters
	d 0 (m)	30	30	30	30	30
	v 0 (m/s)	30	30 -3t	18	18 + 2t

TABLE V SUMMARY

 V OF COMPARISON RESULTS.

			Scenario 1			Scenario 2
		Classical	Proposed	Relative	Classical	Proposed	Relative
		DMPC	controller	improvement	DMPC	controller	improvement
	Peak platoon deviation (m)	0.2615	0.0174	93.3%	0.3770	0.0411	89.1%
	Average platoon deviation (m)	0.1317	0.0036	97.2%	0.1244	0.0057	95.4%
	Computation time (s)	34.2	11.4	66.7%	40.9	17.3	57.7%
	Communication number (/s)	20	10	50%	20	14	30%

This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TVT.2021.3064680, IEEE Transactions on Vehicular Technology

This work was supported in part by the National Natural Science Foundation of China under Grant U1864201. Qianyue


Rodrigo F Araújo
email: rfaraujo@uea.edu.br

Pedro H S Coutinho
email: coutinho.p92@gmail.com

Anh-Tu Nguyen
email: nguyen.trananhtu@gmail.com

Reinaldo M Palhares
email: rpalhares@ufmg.br

Delayed Nonquadratic L 2 -Stabilization of Continuous-Time Nonlinear Takagi-Sugeno Fuzzy Models

Keywords: Takagi-Sugeno fuzzy models, nonlinear local models, nonquadratic Lyapunov functions, fuzzy control, robust control

This work deals with the design of fuzzy controllers for stabilization of continuoustime nonlinear systems subject to L 2 disturbances, which are represented by nonlinear Takagi-Sugeno fuzzy models, i.e., Takagi-Sugeno fuzzy models with nonlinear consequents. A nonquadratic Lyapunov function is used to derive sufficient design conditions based on linear matrix inequality constraints as well as to reduce the conservativeness when compared to existing control approaches in the literature. Furthermore, the nonquadratic Lyapunov function is defined in terms of an integral membership function, which leads to a delayed nonquadratic L 2 -stabilization condition. This condition avoids the well-known di culties in dealing with time derivatives of membership functions and/or path-independent conditions, found in most of the nonquadratic control approaches for continuoustime Takagi-Sugeno fuzzy models. Two numerical examples are performed to illustrate the reduction in conservativeness provided by the proposed approach.

Introduction

Takagi-Sugeno (TS) fuzzy models have been recognized as an e↵ective way to represent nonlinear systems with an arbitrary precision by means of convex combination of local models, which in general are linear state-space equations [START_REF] Takagi | Fuzzy identification of systems and its applications to modeling and control[END_REF]. By exploiting the convex structure of TS models, stability analysis and control design conditions can be formulated as convex optimization problems under linear matrix inequality (LMI) constraints [START_REF] Nguyen | Fuzzy control systems: Past, present and future[END_REF].

It is well known that exact TS fuzzy models can be obtained to represent nonlinear systems by employing the sector nonlinearity approach [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach[END_REF]. As a consequence, the number of local models (or fuzzy rules) exponentially grows with the amount of the system nonlinearities related to the number of premise variables. Thus, for nonlinear systems with a high number of nonlinearities, the increased number of fuzzy rules leads to a high computational burden to solve stability or stabilization LMI conditions and di culties for online controller implementation if needed.

An alternative to obtain exact TS models with reduced number of fuzzy rules is to consider TS fuzzy models with nonlinear consequent parts, in short: N-TS fuzzy models. In [START_REF] Meng | Output feedback and stability analysis of positive polynomial fuzzy systems[END_REF][START_REF] Tanaka | A sum-of-squares approach to modeling and control of nonlinear dynamical systems with polynomial fuzzy systems[END_REF], TS fuzzy models with nonlinear consequents have been used. However, the consequents are polynomial and sum-of-squares conditions were derived for the control design of nonlinear systems. On the other hand, in [START_REF] Dong | Control synthesis of continuous-time T-S fuzzy systems with local nonlinear models[END_REF][START_REF] Dong | H 1 and mixed H 2 /H 1 control of discrete-time T-S fuzzy systems with local nonlinear models[END_REF], a class of N-TS fuzzy models was proposed from the introduction of sector-bounded nonlinear functions of states to the local models, and the consequent parts can be viewed as local Lur'e-type systems. Thus, conditions were obtained in terms of LMI constraints. This latter TS fuzzy approach is useful to reduce both the design conservativeness and the computational complexity of the analysis and design conditions compared to the traditional TS fuzzy modeling. In addition, such representation has been used successfully in di↵erent applications, such as fault-tolerant control [START_REF] Bessa | TS fuzzy reconfiguration blocks for fault tolerant control of nonlinear systems[END_REF] and interconnected systems [START_REF] Araújo | Distributed control of networked nonlinear systems via interconnected Takagi-Sugeno fuzzy systems with nonlinear consequent[END_REF].

For discrete-time N-TS fuzzy models, nonquadratic stabilization approaches have been developed such as the design of dynamic output feedback fuzzy controllers [START_REF] Dong | Output feedback fuzzy controller design with local nonlinear feedback laws for discrete-time nonlinear systems[END_REF][START_REF] Klug | Fuzzy dynamic output feedback control through nonlinear Takagi-Sugeno models[END_REF], disturbance attenuation [START_REF] Dong | H 1 and mixed H 2 /H 1 control of discrete-time T-S fuzzy systems with local nonlinear models[END_REF], robust fuzzy model predictive control [START_REF] Teng | Robust fuzzy model predictive control of discrete-time Takagi-Sugeno systems with nonlinear local models[END_REF], multiple-parameterization approach [START_REF] Coutinho | A multipleparameterization approach for local stabilization of constrained Takagi-Sugeno fuzzy systems with nonlinear consequents[END_REF] and constrained output feedback control [START_REF] Nguyen | Constrained output-feedback control for discrete-time fuzzy systems with local nonlinear models subject to state and input constraints[END_REF]. On the other hand, for continuous-time N-TS fuzzy models, quadratic Lyapunov functions have been employed to develop conditions for L 2 -stabilization [START_REF] Dong | Control synthesis of continuous-time T-S fuzzy systems with local nonlinear models[END_REF], distributed control [START_REF] Araújo | Distributed control of networked nonlinear systems via interconnected Takagi-Sugeno fuzzy systems with nonlinear consequent[END_REF], fault-tolerant control [START_REF] Bessa | TS fuzzy reconfiguration blocks for fault tolerant control of nonlinear systems[END_REF], and state estimation [START_REF] Pan | A unified framework for asymptotic observer design of fuzzy systems with unmeasurable premise variables[END_REF].

A nonquadratic stabilization condition has been proposed in [START_REF] Moodi | On stabilization conditions for T-S systems with nonlinear consequent parts[END_REF] with a particular Lyapunov function, which is composed of a quadratic term and a nonquadratic one constructed from the integral of the nonlinearities. As a result, path-independent assumptions [START_REF] Rhee | A new fuzzy Lyapunov function approach for a Takagi-Sugeno fuzzy control system design[END_REF] are required, which may introduce extra complexity and conservativeness to the control design approach. Besides, the work in [START_REF] Moodi | On stabilization conditions for T-S systems with nonlinear consequent parts[END_REF] is concerned with obtaining estimates of the domain of attraction and neglects the L 2 control stabilization since the extent of its conditions for this scenario is not trivial. To the best of the authors' knowledge, there is no any other work besides the one in [START_REF] Dong | Control synthesis of continuous-time T-S fuzzy systems with local nonlinear models[END_REF] (quadratic stabilization) considering the main topic of this submission, namely to deal with the problem of L 2 control stabilization for N-TS fuzzy models Therefore, there exists a lack of nonquadratic stabilization conditions for continuous-time N-TS systems, especially when L 2 -stabilization is concerned. Then, the first motivation for this work is to derive nonquadratic L 2 -stabilization conditions for continuous-time N-TS fuzzy systems.

Nevertheless, the major challenge on nonquadratic stabilization of continuoustime fuzzy systems is dealing with the time derivatives of the membership functions [START_REF] Campos | A comparison of di↵erent upper-bound inequalities for the membership functions derivative[END_REF]. To this end, many works in di↵erent scenarios have been done based on the assumption of bounded time-derivatives [START_REF] Campos | A comparison of di↵erent upper-bound inequalities for the membership functions derivative[END_REF][START_REF] Chen | New relaxed stability and stabilization conditions for continuous-time T-S fuzzy models[END_REF][START_REF] Mozelli | A systematic approach to improve multiple Lyapunov function stability and stabilization conditions for fuzzy systems[END_REF][START_REF] Mozelli | Equivalent techniques, extra comparisons and less conservative control design for Takagi-Sugeno fuzzy systems[END_REF][START_REF] Mozelli | Reducing conservativeness in recent stability conditions of TS fuzzy systems[END_REF] or path-independent conditions [START_REF] Moodi | On stabilization conditions for T-S systems with nonlinear consequent parts[END_REF][START_REF] Mozelli | A systematic approach to improve multiple Lyapunov function stability and stabilization conditions for fuzzy systems[END_REF][START_REF] Rhee | A new fuzzy Lyapunov function approach for a Takagi-Sugeno fuzzy control system design[END_REF]. Unlike traditional articles that deal with the time-delay in state variables [START_REF] Kwon | Stability and stabilization of T-S fuzzy systems with time-varying delays via augmented Lyapunov-Krasovskii functionals[END_REF][START_REF] Oliveira | Improved Takagi-Sugeno fuzzy output tracking control for nonlinear networked control systems[END_REF], more recently for TS fuzzy models, the authors in [START_REF] González | Nonquadratic controller and observer design for continuous TS models: A discrete-inspired solution[END_REF][START_REF] Márquez | A non-quadratic Lyapunov functional for H 1 control of nonlinear systems via TS models[END_REF] proposed a nonquadratic Lyapunov function depending on the integral of membership functions, such that artificially delayed membership functions are introduced in the design conditions. The idea of delayed membership func-tions has been exploited to derive less conservative nonquadratic stabilization conditions mainly for discrete-time TS fuzzy models [START_REF] Coutinho | E cient LMI conditions for enhanced stabilization of discrete-time Takagi-Sugeno models via delayed nonquadratic Lyapunov functions[END_REF][START_REF] Coutinho | Generalized non-monotonic Lyapunov functions for analysis and synthesis of Takagi-Sugeno fuzzy systems[END_REF][START_REF] Xie | Relaxed real-time scheduling stabilization of discrete-time T-S fuzzy systems via an alterable-weights-based ranking 305 switching mechanism[END_REF] and, recently, discrete-time N-TS fuzzy models [START_REF] Coutinho | A multipleparameterization approach for local stabilization of constrained Takagi-Sugeno fuzzy systems with nonlinear consequents[END_REF]. Thus, the second motivation for this work is to employ a nonquadratic Lyapunov function as in [START_REF] González | Nonquadratic controller and observer design for continuous TS models: A discrete-inspired solution[END_REF][START_REF] Márquez | A non-quadratic Lyapunov functional for H 1 control of nonlinear systems via TS models[END_REF] to derive a less conservative delayed nonquadratic L 2 -stabilization condition than the results in the related literature.

In this work is proposed new nonquadratic stabilization conditions for N-TS fuzzy models subject to L 2 disturbances. The conditions are based on a general nonlinear control law and a delayed nonquadratic Lyapunov function similar to [START_REF] González | Nonquadratic controller and observer design for continuous TS models: A discrete-inspired solution[END_REF]. The main contributions of this paper are summarized as follows.

• The new delayed nonquadratic L 2 -stabilization approach avoids dealing with time derivatives of the membership functions, and any specific structures imposed by Lyapunov functions such as path-independent conditions [START_REF] Rhee | A new fuzzy Lyapunov function approach for a Takagi-Sugeno fuzzy control system design[END_REF].

• We consider a general nonlinear control law whose structure is entirely independent of the used nonquadratic Lyapunov function. This feature contributes to derive an LMI-based condition with significantly reduced conservativeness for continuous-time N-TS fuzzy systems compared to related existing control results.

The paper is organized as follows. The class of N-TS fuzzy systems and the problem formulation are described in Section 2. Section 3 presents the proposed stabilization conditions. In Section 4, two numerical examples are given to illustrate the e↵ectiveness of the proposed control approach. Section 5 concludes this paper.

Notation. For a square matrix X, X 0 (respectively X 0) means that X is symmetric positive (respectively negative) definite. diag(X 1 , X 2 ) represents a block-diagonal matrix composed of X 1 and X 2 , and "?" represents a symmetric matrix block. Let I p = {1, . . . , p} ⇢ N, with p 2 N. The convex hull of the set S is denoted by co{S}. The following notation is adopted for convex sum:

⌥ ✓ = r X i=1 ✓ i (⇠(t))⌥ i
for any matrix ⌥ i of appropriate dimension. The norm in the L 2 space of contin-85 uous square-integrable functions is defined as:

k⇣(t)k 2 = q R 1 0 ⇣(t) > ⇣(t)dt < 1.
Arguments are omitted when their meaning is clear.

Preliminaries and Problem Formulation

Takagi-Sugeno Fuzzy Models with Nonlinear Consequents

Consider a continuous-time and input a ne nonlinear system given by

ẋ(t) = f 1 (x(t)) + f 2 (x(t))w(t) + f 3 (x(t))u(t) z(t) = f 4 (x(t)) + f 5 (x(t))w(t) + f 6 (x(t))u(t) (1)
Following the sector nonlinearity approach [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach[END_REF], it is known that the nonlinear system (1) can be represented by the following class of N-TS fuzzy models [START_REF] Dong | Control synthesis of continuous-time T-S fuzzy systems with local nonlinear models[END_REF]:

ẋ(t) = A ✓ x(t) + B 1✓ w(t) + B 2✓ u(t) + G ✓ '(x) z(t) = C ✓ x(t) + D 1✓ w(t) + D 2✓ u(t) + L ✓ '(x) (2)
where x(t) 2 X ⇢ R nx is the state vector, X is a convex polytope containing the origin and defines also the validity domain of model [START_REF] Bessa | TS fuzzy reconfiguration blocks for fault tolerant control of nonlinear systems[END_REF],

u(t) 2 R nu is the input vector, w(t) 2 R nw is the L 2 disturbance, '(x) 2 R n' is the vector of sector-bound nonlinearities, z(t) 2 R nz is the controlled output. The constant matrices (A i , B 1i , B 2i , G i , C i , D 1i , D 2i , L i ) of the ith local model are of appro- priate dimensions. The normalized membership functions ✓ i (⇠(t)), for i 2 I r ,
satisfy the following convex sum property:

r X i=1 ✓ i (⇠(t)) = 1, ✓ i (⇠(t)) 0, i 2 I r ,
where r the number of fuzzy rules, and ⇠(t) 2 R nv stands for the vector of premise variables, which depends on the state vector. The convex region X is given as follows:

X = {x 2 R nx : h > m x  1; h m 2 R nx , m 2 I ne }, (3)
where n e is the number of hyperplanes. The following assumption is also considered for system (2).

Assumption 1. The vector '(x(t)) satisfies the sector-boundedness condition given by

' (i) (x(t)) 2 co{0, E i x(t)}, 8i 2 I n'
, which is equivalent to [START_REF] Dong | Control synthesis of continuous-time T-S fuzzy systems with local nonlinear models[END_REF]:

'(x(t)) > ⇤ 1 ✓ (Ex(t) '(x(t))) 0, (4)
where matrix

E = [E > 1 , . . . , E > n' ] > 2 R n'⇥nx is given, and ⇤ ✓ 2 R n'⇥n' is a positive definite diagonal matrix.
As aforementioned, the classical TS fuzzy modeling may lead to an excessive number of fuzzy rules. Using N-TS fuzzy model (2) can be useful to reduce the number of fuzzy rules, since only some nonlinearities are selected as premise variables, while others belong to the sector satisfying Assumption 1. Hence, an exact N-TS fuzzy model with reduced number of local models is obtained.

Problem Formulation

Consider the following nonlinear control law:

u(t) = K ✓ H 1 ✓ x(t) + F ✓ J 1 ✓ '(x(t)), (5)
where the matrices

K ✓ 2 R nu⇥nx , H ✓ 2 R nx⇥nx , F ✓ 2 R nu⇥n' and J ✓ 2
R n'⇥n' , are to be determined.

Remark 1. The control law proposed in [START_REF] Dong | Control synthesis of continuous-time T-S fuzzy systems with local nonlinear models[END_REF] can be obtained from (5) by choosing H ✓ = I and J ✓ = I. Moreover, controller (5) is similar to the one used in [10, Corollary 12] and [START_REF] Coutinho | A multipleparameterization approach for local stabilization of constrained Takagi-Sugeno fuzzy systems with nonlinear consequents[END_REF] for the discrete-time case with the particular choice

J ✓ = ⇤ ✓ .
Hence, the new control law (5) is more general than those in previous related works.

Substituting (5) into (2) leads to the closed-loop system of the form

ẋ(t) = A cl x(t) + G cl '(x(t)) + B 1✓ w(t) z(t) = C cl x(t) + L cl '(x(t)) + D 1✓ w(t) (6)
where

A cl = A ✓ + B 2✓ K ✓ H 1 ✓ , G cl = G ✓ + B 2✓ F ✓ J 1 ✓ C cl = C ✓ + D 2✓ K ✓ H 1 ✓ , L cl = L ✓ + D 2✓ F ✓ J 1 ✓ .
This paper considers the L 2 -stabilization control problem for N-TS fuzzy systems (2), i.e., we aim to design a nonlinear control law [START_REF] Coutinho | A multipleparameterization approach for local stabilization of constrained Takagi-Sugeno fuzzy systems with nonlinear consequents[END_REF] such that the closedloop system has the following properties.

(i) If w(t) = 0, for 8t 0, the origin of the closed-loop system ( 6) is asymp-110 totically stable.

(ii) If w(t) 6 = 0, for 8t 0, the closed-loop system ( 6) is L 2 stable from w(t) to z(t) with L 2 gain less than or equal to under zero initial condition:

kzk 2  kwk 2 , 8t 2 [0, 1). ( 7
)
For control design, we consider the following nonquadratic Lyapunov function:

V (x) = x > r X i=1 ⌫ i (⇠)P i ! 1 x = x > P 1 ⌫ x, (8)
where P i 0, for i 2 I r , and ⌫ i (⇠(t)) is defined as

⌫ i (⇠(t)) = 1 ↵ Z t t ↵ ✓ i (⇠(⌧ ))d⌧ 0. ( 9
)
The given scalar ↵ > 0 is an artificial time delay. Note that the convex sum property holds for the integral function

⌫ i (⇠(t)) since r X i=1 ⌫ i (⇠(t)) = 1 ↵ Z t t ↵ r X i=1 ✓ i (⇠(⌧ )) ! d⌧ = 1. (10)
Furthermore, the time derivative of

⌫ i (⇠(t)) is given by ⌫i (⇠(t)) = 1 ↵ (✓ i (⇠(t)) ✓ i (⇠(t ↵))) , (11)
with the initial function given by

x(t) = (t), t 2 [ ↵, 0], 2 C ([ ↵, 0], R),
where (P) is an arbitrary matrix expression related to P or not. Then, there exist matrices P 0 and Z such that 2

4 (P) ? Z > A P Z Z > C 3 5 0. ( 13
)
3. Delayed Nonquadratic L 2 -Stabilization of N-TS Fuzzy Models

The following theorem provides su cient conditions to design a nonlinear controller (5) for system (2).

Theorem 1. Given an N-TS fuzzy model (2), positive scalars µ, ↵ and a sufficiently small ✏. If there exist positive definite matrices P i , positive definite diagonal matrices ⇤ i , symmetric matrices M kl ij , and matrices K i , F i , H i , J i , of appropriate dimensions, such that

⌥ kl ii M kl ii , i,k,l 2 I r , (14)
⌥ kl ij + ⌥ kl ji M kl ij + M kl ij >
, i,j,k,l 2 I r and i 6 = j, (15) 2 6 6 6 4

M kl 11 • • • M kl 1r . . . . . . . . . M kl r1 • • • M kl rr 3 7 7 7 5 0, k,l 2 I r , (16)
2 4 1 h > m H j ? P k + H j + H > j 3 5 ⌫ 0, j,k 2 I r , m 2 I ne , (17)
where

⌥ kl ij = 2
? ? 0 ✏EH j (1 ✏)J j 0 ⇤ i ? ✏ 1 D > 1i 0 0 B > 1i 0 µ✏ 1 I 3 7 7 7 7 7 7 7 7 7 7 7 7 5
,

21 = H > j C > i + K > j D > 2i , 22 = P l ✏↵ 1 (P i P k ) H j H > j , 31 = J > j L > i + F > j D > 2i , 33 = ⇤ i J j J > j , 42 = H j + ✏(A i H j + B 2i K j ), 43 = G i J j + B 2i F j .
Then, if w(t) = 0, the origin of the closed-loop system (6) is asymptotically 120 stable and the region

L V = {x 2 R nx : V (x)  1}, L V ⇢ X , is included in its region of attraction. Moreover, if w(t) 6 = 0, then system (6) is L 2 -stable from
w(t) to z(t) with an L 2 -gain less than or equal to = p µ.

Proof. By the relaxation result in [START_REF] Xiaodong | New approaches to H 1 controller designs based on fuzzy observers for T-S fuzzy systems via LMI[END_REF], it follows from ( 14), ( 15) and ( 16) that

r X l=1 r X k=1 r X i=1 r X j=1 ⌫ l (⇠(t))✓ k (⇠(t))✓ i (⇠(t))✓ j (⇠(t))⌥ kl ij 0 ( 18
)
where we denote ✓ k (⇠(t)) = ✓ k (⇠(t ↵)). Note that the satisfaction of inequality [START_REF] Mozelli | A systematic approach to improve multiple Lyapunov function stability and stabilization conditions for fuzzy systems[END_REF] guarantees that J ✓ + J > ✓ 0, and H ✓ + H > ✓ 0, for a su ciently small scalar ✏. This, in turn, guarantees the existence of J 1 ✓ and H 1 ✓ , thus the validity of the control expression [START_REF] Coutinho | A multipleparameterization approach for local stabilization of constrained Takagi-Sugeno fuzzy systems with nonlinear consequents[END_REF]. Note also that inequality (18) can be represented similar to (13) as 2 6 6 6 4

(P) ? ?

Z > A P Z Z > C ? ⌃ 0 µ✏ 1 I 3 7 7 7 5 0, (19) 9 with
P = diag(P ⌫ , ⇤ ✓ ), (P) = diag(✏ 1 I, P ⌫ , ⇤ ✓ ), Z = diag(H ✓ , J ✓ ), C = diag(✏↵ 1 (P ✓ P ✓ ), 0), A = 2 4 C > cl I + ✏A > cl ✏E > L > cl ✏G > cl (1 ✏)I 3 5 , P ✓ = r X i=1 ✓ i (⇠(t))P i , ⌃ = h ✏ 1 D > 1✓ B > 1✓ 0 i .
Applying

with

⇥ 31 = P ⌫ + ✏P ⌫ A > cl , ⇥ 33 = P ⌫ ✏↵ 1 (P ✓ P ✓ ), ⇥ 41 = ⇤ ✓ G > cl , ⇥ 42 = (1 ✏)⇤ ✓ , ⇥ 63 = C cl P ⌫ , ⇥ 64 = L cl ⇤ ✓ .
Pre-and postmultiplying inequality [START_REF] Mozelli | Reducing conservativeness in recent stability conditions of TS fuzzy systems[END_REF]

I + ✏A > ? 0 ⇧ 3 7 7 7 5 0, (21)
where

A = 2 4 A cl G cl E I 3 5 , P = diag(P 1 ⌫ , ⇤ 1 ✓ ), = 2 4 B > 1✓ P 1 ⌫ 0 C cl L cl 3 5 , ⇧ = 2 4 µI ? D 1✓ I 3 5 , = diag ✏ 1 P 1 ⌫ + ↵ 1 P 1 ⌫ (P ✓ P ✓ )P 1 ⌫ , ✏ 1 ⇤ 1 ✓ . Note that Ṗ 1 ⌫ = P 1 ⌫ Ṗ⌫ P 1 ⌫
and Ṗ⌫ = ↵ 1 (P ✓ P ✓ ). Then, it follows that

✏ 1 P 1 ⌫ + ↵ 1 P 1 ⌫ (P ✓ P ✓ )P 1 ⌫ = ✏ 1 P 1 ⌫ Ṗ 1 ⌫ . (22)
Hence, inequality ( 21) can be rewritten in the form 2 6 6 6 4

✏P 1 ? ?

I + ✏A > ✏ 1 P + C ? 0 ⇧ 3 7 7 7 5 0, (23)
with C = diag( Ṗ 1 ⌫ , 0). By Schur complement lemma, we can prove that inequality ( 23) is equivalent to

2 4 A > P + PA + C + ✏A > PA ? ⇧ 3 5 0. ( 24
)
Since ✏A > PA 0, then it follows from inequality ( 24) that 2 6 6 6 6 6 6 4

⌅ ? ? ? G > cl P 1 ⌫ + ⇤ 1 ✓ E 2⇤ 1 ✓ ? ? B > 1✓ P 1 ⌫ 0 µI ? C cl L cl D 1✓ I 3 7 7 7 7 7 7 5
0, [START_REF] Rhee | A new fuzzy Lyapunov function approach for a Takagi-Sugeno fuzzy control system design[END_REF] with

⌅ = P 1 ⌫ A cl + A > cl P 1 ⌫ + Ṗ 1 ⌫ .
Applying Schur complement lemma to [START_REF] Rhee | A new fuzzy Lyapunov function approach for a Takagi-Sugeno fuzzy control system design[END_REF], then pre-and postmultiplying the result with

h x > '(x) > w >
i and its transpose, the following inequality can be obtained:

V (x) < µw > w z > z 2'(x) > ⇤ 1 ✓ (Ex '(x)), (26)
where V (x) is the time derivative of the Lyapunov function ( 8) along the trajectory of system [START_REF] Coutinho | E cient LMI conditions for enhanced stabilization of discrete-time Takagi-Sugeno models via delayed nonquadratic Lyapunov functions[END_REF]. From ( 4) and ( 26), it follows that

V (x) + z > z < µw > w. ( 27
)
We distinguish two following cases.

• If w(t) = 0, for 8t > 0, then V (x(t)) < z(t) > z(t) < 0, for 8x(t) 6 = 0.

125 Then, the origin of the closed-loop system ( 6) is asymptotically stable.

• If w(t) 6 = 0, for 8t > 0, then integrating both sides of ( 27) from 0 to 1 and considering that V (x(1)) > 0, it follows that

Z 1 0 z(t) > z(t)dt  µ Z 1 0 w(t) > w(t)dt + V (x(0)). ( 28
)
Under zero initial conditions, it is clear that (28) leads to inequality [START_REF] Coutinho | Generalized non-monotonic Lyapunov functions for analysis and synthesis of Takagi-Sugeno fuzzy systems[END_REF], which means that the closed-loop system ( 6) is L 2 -stable from w(t) to z(t) with an L 2 -gain less than or equal to = p µ.

Finally, it follows from ( 17) that L V ⇢ X . Thus, since V (x) < 0, for w(t) = 0, 8t, then it is included in the region of attraction. Hence, if x(0) 2 L V , the state trajectories will never leave L V . This concludes the proof.

Remark 2. One of the major di culties when deriving the design conditions in Theorem 1 is to deal with the term Ṗ⌫ = P r i=1 ⌫i (⇠)P i in [START_REF] Nguyen | Fuzzy control systems: Past, present and future[END_REF]. Exploiting the integral membership functions ⌫ i (⇠), it follows from (11) that Ṗ⌫ = ↵ 1 (P ✓ P ✓ ), which is a convex term. This avoids assuming conservative bounds for the time derivative of the membership functions ✓ i (⇠) or considering path-independent Lyapunov functions to obtain numerically tractable LMI design conditions. Remark 3. Based on Theorem 1, the L 2 -gain upper-bound , with = p µ, can be minimized by solving the following optimization problem:

min i µ subject to (14)-(17), with i = (P i , ⇤ i , K i , F i , H i , J i ). ( 29
)
Remark 4. Note that for a su ciently small scalar 0 < ✏ ⌧ 1, conditions given in [START_REF] Pan | A unified framework for asymptotic observer design of fuzzy systems with unmeasurable premise variables[END_REF] and (25) are equivalent since the only di↵erence between them is the term ✏A > PA . Then, less conservative results are expected from (29) with 0 < ✏ ⌧ 1, which can be determined with a simple gridding algorithm.

Numerical Examples

To illustrate the relaxation provided by the proposed approach, comparisons with the control results in [START_REF] Dong | Control synthesis of continuous-time T-S fuzzy systems with local nonlinear models[END_REF] are performed hereafter. Note that the method [START_REF] Dong | Control synthesis of continuous-time T-S fuzzy systems with local nonlinear models[END_REF] is the only work currently available in the literature addressing L 2 -stabilization for N-TS fuzzy models. Also, the application to a real-world example is employed to discuss the stabilization and the trade-o↵ between the parameter ✏ and the obtained L 2 -gain. All optimization problems were done using Yalmip toolbox and Mosek solver.

Example 1

Consider a nonlinear system whose N-TS fuzzy model ( 2) is described as follows [START_REF] Moodi | On stabilization conditions for T-S systems with nonlinear consequent parts[END_REF]Example 3]:

A 1 = 2 4 2 10 2 0 3 5 , A 2 = 2 4 a 5 1 2 3 5 , B 11 = B 12 = h 1 1 i > , B 21 = B 22 = h b 2 i > , G 1 = G 2 = 2 4 0.1b 0 0 0.1a 3 5 , E = 2 4 1 0 0 1 3 5 ,
where

x 1 , x 2 2 [ 2, 2]
. The controlled output of the system is defined by

z(t) = h x 1 (t) x 2 (t) i >
. For an artificial time-delay ↵ = 0.003 s and ✏ = 10 3 , the feasibility regions obtained by solving the optimization problem (29) in Theorem 1 and Corollary 1 of [START_REF] Dong | Control synthesis of continuous-time T-S fuzzy systems with local nonlinear models[END_REF], for di↵erent values of a and b, are shown in Figure 1. Observe that the feasibility region obtained with the proposed condition is clearly enlarged when compared to the control approach in [START_REF] Dong | Control synthesis of continuous-time T-S fuzzy systems with local nonlinear models[END_REF].

Also, by taking b = 13, the values of the upper-bound of the L 2 -gain for di↵erent values of a are depicted in Figure 2. It can be noticed that the proposed approach provides smaller upper-bounds for all values of a. In particular, as a is increased, this improvement is even more evident, which clearly illustrates the conservativeness reduction in comparison with [8, Corollary 1].

Finally, the smallest value of obtained for all parameters a and b considered in Figure 1 and the related computational complexity are depicted in Table 1.

It is worth mentioning that the constraint [START_REF] Moodi | On stabilization conditions for T-S systems with nonlinear consequent parts[END_REF], with H j = P k = P , is added to the approach in [START_REF] Dong | Control synthesis of continuous-time T-S fuzzy systems with local nonlinear models[END_REF] to account the validity domain X of the N-TS fuzzy model. The numerical complexity is evaluated in terms of number of decision variables N d , and number of LMI rows N l . For Theorem 1, these values are given by

N d = 1 + (n ' + (n u + n x )n x + (n u + n ' )n ' )r + 1 2 n x (n x + 1) + r 2 N ⌃ (rN ⌃ + 1) r, N l = 1 2 r 3 (r + 3)N ⌃ + (1 + n x )r 2 n e ,
with N ⌃ = 2n x + 2n ' + n z + n w .

Table 1: Computational complexity and comparison of the smallest upper-bound of the L 2 gain , considering the variation of parameters in Figure 1.

Design approach (solution of ( 29)) Although the proposed control approach has a slightly higher computational complexity than the design condition in [START_REF] Dong | Control synthesis of continuous-time T-S fuzzy systems with local nonlinear models[END_REF], it provides a significant improvement 165 in terms of both the enlargement of the feasibility region and the reduction of the upper-bound of the L 2 -gain. This clearly illustrates that the proposed condition to design the control law [START_REF] Coutinho | A multipleparameterization approach for local stabilization of constrained Takagi-Sugeno fuzzy systems with nonlinear consequents[END_REF], regarding the nonquadratic Lyapunov function [START_REF] Dong | Control synthesis of continuous-time T-S fuzzy systems with local nonlinear models[END_REF], leads to a conservativeness reduction in comparison to [START_REF] Dong | Control synthesis of continuous-time T-S fuzzy systems with local nonlinear models[END_REF].

N d N l log(N 3 d N l ) [8,

Example 2 170

Consider the two-tank system [30, Example 2] illustrated in Figure 3 with the following incremental model:

2 4 ẋ1 (t) ẋ2 (t) 3 5 = 2 4 a1 A 2g f (x1,x2) a1 A 2g f (x1,x2) a1 A 2g f (x1,x2) a1 A 2g f (x1,x2) 3 5 2 4
x 1 (t)

x 2 (t)

in which x 1 (t) = h 1 (t) h 1e , x 2 (t) = h 2 (t) h 2e with h 1 (t) and h 2 (t) denoting the water level of each tank, respectively, and h 1e and h 2e are the given operating points. v = v(t) v e and v(t) denotes the flow rate of the pump (control input).

w(t) represents an exogenous disturbance which can be considered as an extra flow beyond the control input, A = 100 cm 2 is the horizontal section, k = 4 is a constant, a 1 = 1 cm 2 is the section of the connecting the tanks, a 2 = 0.7 cm 2 is the section of the outlet valve, g = 981 cm/s 2 is the gravitational acceleration constant. The control objective is to keep the water level of the two tanks in the operating point

h h 1e h 2e i > = h 9 6 i > cm. Also, f (x 1 , x 2 ) = p 2g(x 1 x 2 + h 1e h 2e ) + p 2g(h 1e h 2e ), '(x 2 ) = x 2 p 2g(x 2 + h 2e ) + p 2gh 2e . v(t) h 1 (t) h 2 (t)
Tank 1 Tank 2 Assume that

|x 1 |  8, |x 2 |  5. Then, the premise variable ⇠ = 1 f (x1,x2
) 2 [0.0054, 0.0130] and the sector nonlinearity '(x 2 ) 2 co{0.0039x 2 (t), 0.0065x 2 (t)}.

After a loop-transformation, it follows that '(x 2 ) = '(x 2 ) 0.0039x 2 . Note that '(x 2 ) 2 co{0, ⌦x 2 } with ⌦ = 0.0026. We consider v(t) ⌘ u(t). Hence, system (30) can be rewritten in the form 2 4 ẋ1 (t) ẋ2 (t)

3 5 = 2 4 a1 A 2g f (x1,x2) a1 A 2g f (x1,x2) a1 A 2g f (x1,x2) a1 A 2g f (x1,x2) 0.0078ga2 A 3 5 2 4 x 1 (t)
x 2 (t)

Then, using the sector nonlinearity approach [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach[END_REF], the corresponding local statespace matrices of the N-TS fuzzy model (2) for system [START_REF] Xiaodong | New approaches to H 1 controller designs based on fuzzy observers for T-S fuzzy systems via LMI[END_REF] are given by Note that, for ✏  10 2 , the upper bound of the L 2 -gain is lower than the one obtained from the approach in [START_REF] Dong | Control synthesis of continuous-time T-S fuzzy systems with local nonlinear models[END_REF], which is equal to = 1.7205. The best case corresponds to = 1.7102 obtained when ✏ = 3 ⇥ 10 3 . This numerically 180 confirms the statement in Remark 4.

A 1 = 2
Consider the corresponding controller, Fig. 5 depicts the trajectories of the water levels (h 1 (t) and h 2 (t)) of the two tanks when the following disturbance is applied:

w(t) =

Conclusions

The stabilization problem for nonlinear Takagi-Sugeno fuzzy systems subject the proposed control approach, the application to a real system model, and the trade-o↵ between the L 2 -gain and the ✏ parameter introduced in the relaxation of the design constraints. Future directions focus on developing a delayed nonquadratic approach to design observers for the class of nonlinear Takagi-Sugeno fuzzy models [START_REF] Pan | A unified framework for asymptotic observer design of fuzzy systems with unmeasurable premise variables[END_REF].

1 . 2 4

 12 where C ([ ↵, 0], R) is the Banach space of real continuous functions on theinterval [ ↵, 0] with max t2[ ↵,0] | (t)|.The following technical lemma, adapted from [12,Lemma 2], is useful to Lemma Consider matrices of appropriate dimensions A and C. If there exists a matrix P 0 such that

Figure 1 :

 1 Figure 1: Feasibility regions provided by Corollary 1 [8] ('⇥') and Theorem 1 ('⇥' and ' ') -Example 1.

Figure 2 :

 2 Figure 2: Comparison of upper-bound of the L 2 gain ( ) for Corollary 1 [8] ('⇥') and Theorem 1 ('⇥' and ' '), considering b = 13 -Example 1.

Figure 3 :

 3 Figure 3: Two-tank system schematic -Example 2.

3 5 , 8i 2 I 2 .

 32 Let us consider the controlled output as same of the previous example and the artificial time delay ↵ = 0.001 s. Fig.4depicts the relation between the value of the scalar ✏ and the minimum value of the L 2 -gain obtained from the optimization problem[START_REF] Tanaka | A sum-of-squares approach to modeling and control of nonlinear dynamical systems with polynomial fuzzy systems[END_REF]. Note that the improvement achieved by applying Theorem 1 is not a monotonically increasing function of ✏. Moreover, to obtain a better 175 result of the upper-bound of the L 2 -gain , one should solve the optimization problem (29) by iterating over ✏, see also Remark 2 in[START_REF] Shaked | Improved LMI representations for the analysis and the design of continuous-time systems with polytopic type uncertainty[END_REF].

  that the designed controller e↵ectively attenuates the disturbance e↵ect and guarantees the closed-loop stability.

Figure 4 :

 4 Figure 4: Relation between ✏ and the L 2 -gain in Example 2.

Figure 5 :

 5 Figure 5: Response of the water levels h 1 (t) and h 2 (t) in Example 2.

185 to L 2

 2 disturbances has been investigated in this paper. A nonquadratic Lyapunov function with integral membership functions have been considered and su cient delayed design conditions have been obtained. The considered non-quadratic Lyapunov function avoids assuming bounds for the time derivatives of the membership functions or path-independent conditions, which are sources of conservativeness. Moreover, based on appropriate matrix transformations, the control law structure can be entirely independent of the Lyapunov function in a new fashion, contributing to reduce the design conservativeness. Two examples have been provided to illustrate the reduction of conservativeness by

derive delayed nonquadratic stabilization conditions for N-TS fuzzy systems.

Acknowledgement

This work was supported by the Brazilian agencies CNPq (Grant Number: 307933/2018-0) FAPEMIG (Grant Number: PPM-00053-17), and CAPES.