Quang Linh Lam
email: quang-linh.lam@g2elab.grenoble-inp.fr

Antoneta Iuliana Bratcu
email: antoneta.bratcu@gipsa-lab.grenoble-inp.fr

Delphine Riu
email: delphine.riu@g2elab.grenoble-inp.fr

Frequency robust control in stand-alone microgrids with PV sources : design and sensitivity analysis

Keywords: Microgrids, frequency control, robust control, multivariable control, uncertainty, sensitivity analysis

In this paper, a robust H∞ control strategy for frequency regulation is proposed in isolated microgrids (MGs) composed of diesel engine generators, photovoltaic (PV) sources, and storage units. First, the linear matrix inequalities (LMI) method is adopted to design a multi-variable H∞ controller which ensures given specifications. In a second step, uncertainties in the storage device state of charge (SoC) are considered and a sensitivity analysis is carried out in order to determine the maximum variation range of SoC for which the dynamic performances are respected. The controller's robustness and performance in the presence of various load disturbances, PV output power variations, and the SoC uncertainty are validated through a series of nonlinear time-domain simulations performed with MATLAB R /Simulink R .

INTRODUCTION

MGs emerge as a major, cost-effective solution for the integration of Distributed Energy Resources (DERs) into power systems, as they offer enhanced global reliability, higher energy efficiency, and various environmental and economic advantages [START_REF] Hatziargyriou | Microgrids[END_REF]. MG stability and control issues are however among the main challenges, especially during islanded mode due to low inertia, uncertainties, and intermittent nature of DERs [START_REF] Olivares | Trends in microgrid control[END_REF]. Maintaining frequency and voltage deviations within their predefined range is critical in autonomous operation mode. High-speed storage systems -e.g., lithium-ion batteries, flywheels or supercapacitors -have thus become necessary, leading to new grid configurations, for which more complex robust control structures are needed for dealing with multiple constraints such as unexpected disturbances and model uncertainties.

H ∞ control theory applied to frequency regulation has attracted the attention of many researchers for several reasons. First, many control objectives such as disturbance attenuation, robust stabilization of uncertain systems or shaping of the open-loop response can be handled by H ∞ -and µ-synthesis techniques [START_REF] Chilali | H∞ design with pole placement constraints : an LMI approach[END_REF]. Then, the solution found is optimal with respect to a defined criterion, which means that if no solution to the control objectives is found, then no solution exists. Finally, sensitivity and robustness analysis to model uncertainties can possibly be associated with H ∞ and µ-synthesis controls.

Some studies on robust control techniques for various MGs have already been conducted. In [START_REF] Goya | Frequency control in isolated island by using parallel operated battery systems applying H∞ control theory based on droop characteristics[END_REF], H ∞ -based droop frequency control is given. System trade-offs are optimized ; however, robustness to model uncertainties is not considered. H ∞ control with loop shaping and H ∞ -based droop control for improving frequency deviation are presented in [START_REF] Singh | Robust H∞ load frequency control in hybrid distributed generation system[END_REF], but only controller sensitivity to parameter uncertainties is analyzed. Parameter uncertainties are considered in designing H ∞ and µ-synthesis controls for the voltage direct single-loop in [START_REF] Kahrobaeian | Direct single-loop µ-synthesis voltage control for suppression of multiple resonances in microgrids with power-factor correction capacitors[END_REF]. Conventional H ∞ -based and µ minimization approaches for the direct single-loop current control scheme are also adopted in [START_REF] Kahrobaeian | Robust single-loop direct current control of LCL-filtered converter-based DG units in grid-connected and autonomous microgrid modes[END_REF]. Multi-variable H ∞ control is utilized in [START_REF] Hossain | Robust control for power sharing in microgrids with low-inertia wind and PV generators[END_REF] to design robust controllers to system nonlinearities and load interactions. [START_REF] Li | Modelling robustness for a flexible gridtied photovoltaic generation system[END_REF] introduces a µ-synthesis robust control method for a gridtied PV generation system taking parameter variations into account. In [START_REF] Bevrani | Robust frequency control in an islanded microgrid : H∞ and µ-synthesis approaches[END_REF], H ∞ and µ-synthesis robust controls under parametric uncertainties for improving secondary frequency control performance are applied.

Our work focuses on the frequency stability problem of MGs with high penetration of renewable energy. Frequency deviation can be considerably reduced by using relatively small storage units, provided that saturation conditions are avoided, by dynamically coordinating storage with other generation sources [START_REF] Han | Robust control for microgrid frequency deviation reduction with attached storage system[END_REF]. This paper proposes a multi-variable H ∞ robust controller for primary frequency regulation that copes with load transients, PV output power variations, and model uncertainties. A sensitivity analysis is then conducted according to system parameters.

The remainder of this paper is structured as follows. Section 2 introduces the studied MG setup. Section 3 is devoted to the proposed control methodology. Dynamic specifications are presented in Section 4. Section 5 provides system modeling for H ∞ control. The detailed process of designing the H ∞ controller, along with some nonlinear time-domain simulations with respect to the rated operating point, are described in Section 6. Section 7 details the sensitivity analysis of robust performance of the H ∞ controller when the supercapacitor voltage v sc -which is an image of its SoC sc -is considered a time-variant parameter ; the controller robustness is then demonstrated through a series of nonlinear time-domain simulations. Some concluding remarks and future work are drawn in Section 8.

MICROGRID STRUCTURE

This paper focuses on an islanded MG constituted by a diesel engine generator, a PV source and a supercapacitor-based energy storage unit. The studied MG setup is illustrated in Fig.

1 [START_REF] Lam | Systematic multi-variable H∞ control design for primary frequency regulation in stand-alone microgrids with high penetration of renewable energy sources[END_REF], where power sources are connected in parallel to a point of common coupling (PCC) and feed a common load. The PV panel bank is connected to the PCC through a chopper -inverter without reversibility allowing only unidirectional power flow, whereas the supercapacitor-based storage device is connected

Control design 3

Control objectives

No

Sensitivity analysis

Sensitivity analysis 4

Robustness specifications satisfied ?

H∞ controller found

Yes

No Yes to the PCC and controlled by means of a reversible chopperinverter, which allows charging or discharging.

Performance specifications

Dynamic specifications 1

PROPOSED CONTROL METHODOLOGY

In order to reduce the time spent for control design, a generalized methodology taking the controller robustness into account is used. Then, it allows minimizing the number of back and forth steps in the design process. This methodology is resumed in Fig. 2 and can be divided into four steps which are detailed below :

-First, performances and robustness levels are specified by the designer according to standards ; -Afterwards, dynamical equations and equivalent averaged models of power-electronic converters are used to describe the behavior of an electrical system. By linearizing these equations for a given equilibrium state a linear statespace model can be obtained ; -In order to respect performances, a control architecture is then designed for nominal values using H ∞ control. Control objectives are introduced using weighting functions over particular closed-loop transfer functions [START_REF] Skogestad | Multivariable Feedback Control : Analysis and Design[END_REF].

An extended model including these weighting functions is then obtained. An optimal controller can now be computed using control tools in the MATLAB R software environment. A multi-variable H ∞ controller is obtained until all desired control objectives are satisfied. For control validation, closed-loop time-domain simulations using the averaged models are convenient ; -Finally, a sensitivity analysis is carried out to test the vali- dation for desired H ∞ controller robustness. If this is not satisfied, the control objectives must be modified and the step 3 is re-conducted.

DYNAMIC SPECIFICATIONS

In the literature, coordinated strategy of the energy storage system and the diesel engine generator in primary frequency control has been proposed in order to improve dynamic performances (lower overshoot, faster response time, smaller steadystate error) of the frequency f grid in case of load power variations. Control design is performed for given frequency performance requirements shown in Fig. 3(a) [START_REF] Lam | Systematic multi-variable H∞ control design for primary frequency regulation in stand-alone microgrids with high penetration of renewable energy sources[END_REF], [START_REF] Lam | Multi-variable H∞ robust control applied to primary frequency regulation in microgrids with large integration of photovoltaic energy source[END_REF]. Another reasonable performance specifications on the DC-bus voltage v dc are given in Fig. 3(b). Robustness specifications will be introduced later.

SYSTEM MODELING FOR H ∞ CONTROL

A cascaded two-level control structure (Fig. 4) -where classical PI-based current tracking controllers are placed on the low control level and receive references from an H ∞ -control-based upper level -is developed to ensure the previous dynamic specifications. The current control level is detailed in [START_REF] Lam | Systematic multi-variable H∞ control design for primary frequency regulation in stand-alone microgrids with high penetration of renewable energy sources[END_REF]. System modeling for H ∞ control loop is given in this Section.

A per-unitized linear averaged model of the storage system together with its power-electronic converters, as well as a linear state-space representation of the considered MG around a steady-state equilibrium point, for H ∞ control loop design, are presented. In this paper, the supercapacitor voltage v sc is regarded as a time-invariant parameter (i.e., v sc = v sce or ∆v sc = 0) in H ∞ controller synthesis. Note that the supercapacitor voltage depends in reality on the supercapacitor current ; a sensitivity analysis considering variation of this parameter is presented in Section 7. Hence, the resulting small-signal state-space model

v R i  est grid f  1 diesel s  1 1 diesel T s  ref diesel P  diesel P  s P  load P  1 2 load Hs D  grid f  est grid f  est grid f  e grid f est grid f sc v s i dc v rd i rq i c  d  q  c  d  q  e c  e d  e q  c   d   q   mes s i mes rd i mes rq i e s i e rd i e rq i mes s i  mes rd i  mes rq i  ref s i  ref rd i  ref rq i  ref s i  ref rd i  0 ref rq i   0 ref dc v   0 ref grid f   mes dc v  mes dc v  e dc v est grid f  mes dc v b i b i b i b f 1 b v 1 b v 1 b i PV P  Fig. 4. Block diagram of the proposed global control structure.
is given as follows

∆ ẋ = A∆x + B 1 ∆u + B 2 ∆w ∆y = C∆x + D 1 ∆u + D 2 ∆w , (1)
where the underline denotes per-unitized values. The state vector ∆x = ∆v dc ∆P diesel ∆f grid T consists of the DCbus voltage variation, the diesel engine generator power variation, and the frequency variation.

∆u = ∆i ref s ∆i ref rd T
is the control input vector composed of the storage device current reference variation and the inverter output current reference variation in d-axis. ∆w = ∆P load -∆P P V is the load and PV output power variation, which represents the disturbance input. ∆y = ∆v dc ∆f grid T is the measured output vector. 1. Note that the v sce value chosen for H ∞ control design corresponds here to the range middle between the minimum and maximum supercapacitor voltages. In this case, v sce = 585 V, where 390 V (SoC sc = 25%) is taken as the minimum and 780 V (SoC sc = 100%) is the maximum values of v sc .

Matrices in (1) are given by

A =    -ω b R dc C dc 0 0 0 -1 T diesel - 1 T diesel s diesel 0 1 2H - D load 2H    , B 1 =    ω b C dc α ce -ω b C dc β de 0 0 1 2H v sce -2R sc i se 0    , B 2 =   0 0 -1 2H   , C = 1 0 0 0 0 1 , D 1 = 0 0 0 0 , D 2 = 0 0 ,
6. H ∞ CONTROL DESIGN

Control design

The design of a multi-variable H ∞ controller for the system (1) is cast into the formalism in Fig. 5, where weighting functions W perf (s) are chosen to guarantee the performance objectives. The practical implementation constraints imposed to the control inputs are dealt with by choosing weighting functions W u (s).

The linear time-invariant (LTI) weighting functions, used for a S/KS mixed-sensitivity optimization process to reject distur-Small-signal state-space model G(s)

_ + P _ + H ∞ Controller K(s) K ∆w ∆u ∆z ∆v ref s i  ref rd i  dc v  grid f  0 ref grid f   0 ref dc v   1 e  2 e  3 e  4 e  1 ( ) u W s  2 ( ) u W s  1 ( ) perf W s 2 ( ) perf W s load PV P P   
Fig. 5. Control configuration in the so-called P -K form, where P denotes the plant together with the weighting functions and K denotes the H∞ controller. bance due to the load power variation, are written as follows

       1 W perf (s) = s+ω b Aε s/Ms+ω b , 1 Wu 2 (s) = Au 2 s+ω bc 2 s+ω bc 2 /Mu 2 1 Wu 1 (s) = Au 1 Mu 1 ω bc 11 s+1 Au 1 ω bc 12 s+1 Au 1 ω bc 11 s+1 Mu 1 ω bc 12 s+1 , ω bc 11 Au 1 < ω bc 12 Mu 1 . (2)
The DC-bus voltage variation ∆v dc and the frequency variation ∆f grid are bounded by first-order weighting functions W perf (s). The function 1/W perf (s) can be representative of time-domain response specifications, where the high-frequency gain M s has an influence on the system overshoot, the cutoff frequency ω b tunes the desired response time and the lowfrequency gain A ε allows limiting the steady-state error. That makes equivalences between desired time-domain performances and frequency-domain specifications (Fig. 6). The active power injection or absorption of the storage system is controlled via i s . Thus, the storage device current reference variation ∆i ref s is bounded by a band-pass weighting function W u1 (s). The DCbus voltage is regulated via i rd . The inverter output current reference variation in d-axis ∆i ref rd is bounded by the first-order weighting function W u2 (s). Frequency-domain specifications on these current reference variations are presented in Fig. 7. The parameters of the weighting functions are given in Table 2 and Table 3, where 1 T 0 rdq is the ∆i rd inner closed-loop bandwidth.

The LMI algorithm under the MATLAB R software environment yields a solution after one iteration for a multi-variable

  1 1 perf W j  1 s M 1 A  1 b  1 b  1 s M 1 A    1 S j   2 S j   2 1 perf W j  2 s M 2 A  2 b  2 A  2 b  2 s M
Performance specifications on the DC-bus voltage variation Performance template 1/Wperf (s)

Performance specifications on the frequency variation Performance template 1/Wperf (s) full-order H ∞ controller which has 8 states with a conditioning value of γ = 1301.27 and an H ∞ norm = 3.48. For the studied system shown in Fig. 5, sensitivity functions S 1 (s), S 2 (s) and complementary sensitivity functions KS 1 (s), KS 2 (s) are defined by

   S 1 (s) = ∆v dc (s) ∆P load (s) , S 2 (s) = ∆f grid (s) ∆P load (s) KS 1 (s) = ∆i ref s (s) ∆P load (s) , KS 2 (s) = ∆i ref rd (s) ∆P load (s) . (3)
The singular value plots of the MIMO sensitivity functions S(s), complementary sensitivity functions KS(s), and corresponding templates are shown in Fig. 8. Results show how the shaping of H ∞ performance is straightforward for the fullorder H ∞ controller using the template weighting functions. One can see that control performance weights imposed are wellrespected since the sensitivity functions are placed below the corresponding templates regardless of the frequency value.

Time-domain simulation results

In order to validate the effectiveness of the proposed control approach, MATLAB R /Simulink R time-domain simulations are performed on the nonlinear averaged model. The impact of PV output active power variations on the microgrid frequency is analyzed with or without secondary control (K i = 0 or K i = 0), where K i denotes the inverse of the response time  

1 u W s    2 u W s    1 S j   2 S j   2 KS j   1

KS j

Fig. 8. Singular value plots for the sensitivity functions S 1 (s), S 2 (s) ; complementary sensitivity functions KS 1 (s), KS 2 (s) ; and corresponding templates.

of secondary control of the diesel engine generator. PV output active power variations modeled with different ramp signals are applied as disturbances. The PV output active power decreases by 60 kW from P P Ve = 200 kW at t = 10 s to P P V = 140 kW with different time durations ∆t such as 0 s (step signal), 5 s, 10 s, ..., 60 s as shown in Fig. 9(a). The supercapacitor voltage v sc , initially considered a time-invariant parameter in H ∞ controller synthesis, is regarded now as a time-variant one, i.e., ∆v sc = 0 (its dynamic equation, i.e., ∆ vsc = f ∆v sc , ∆i s being taken into account in the simulation model). From Fig. 9(b) and Fig. 10(a) it can be observed that the MG frequency desired time-domain performances are successfully achieved with respect to choice of the weighting function W perf2 (s). Compared with the case in which only the diesel engine generator participates in primary frequency control, storage device participation has considerably improved the dynamic performances (i.e., lower overshoot, faster response time, smaller steady-state error) of the frequency. The system is also always stable. Secondary control participation of the diesel engine generator brings the frequency back to its nominal value of 50 Hz as shown in Fig. 10(a). Fig. 9(c) and Fig. 10(b) show that the DC-bus voltage desired time-domain performances corresponding to tuning of the weighting function W perf1 (s) are satisfied. Hence, the synthesized H ∞ controller guarantees the desired performance specifications.

SENSITIVITY ANALYSIS

In this Section, the sensitivity analysis of robust performance of the H ∞ controller is considered to answer the question whether the closed-loop system remains robust or not (from a performance point of view) to a given parametric uncertainty level in the steady-state value of the supercapacitor voltage v sce around its design value, 585 V.

SoC sc estimation is very important in determining the best operating strategy of the energy storage system. SoC sc is approximated as the ratio between stored energy W * sc (Fig. 11) and maximum storable energy W * scmax as follows (with the internal resistances R sc and R scp being neglected)

SoC sc = W * sc W * scmax = 1 2 C sc v 2 sc 1 2 C sc v 2 scmax = v sc v scmax 2 , (4)
where v scmax = 780 V is the maximum possible supercapacitor voltage. In practice, SoC sc is maintained within the admissible range of [25, 100] % to ensure reliable, efficient, and safe operation of the supercapacitor and prolong its lifespan. The limited variation range, corroborated with (4), indicates that the v sc value must be controlled between 390 V and 780 V.

KS j

From -69% to + 69%  

1 perf H norm W s    2 perf H norm W s    1 u H norm W s    2 u H norm W s  From -69% to + 69%
From -69% to + 69%

From -69% to + 69% The sampled uncertain sensitivity functions S(s), complementary sensitivity functions KS(s) obtained with 33.3% uncertainty in v sce around its design value, 585 V (i.e., SoC sce ∈ [25, 100] % or v sce ∈ [390, 780] V) and corresponding templates of the closed-loop system are presented in Fig. 12. It can be observed that, irrespective of the parametric uncertainty in v sce , these functions are well-located below the corresponding weights, which means that the imposed overshoot performance specifications are guaranteed. MATLAB R /Simulink R time-domain simulations using the nonlinear averaged model are carried out with a PV output active power step disturbance of -60 kW at t = 10 s. The supercapacitor voltage v sc , in the simulation model, appears as a time-variant parameter (i.e., ∆v sc = 0). The initial value of SoC sc is varied between 25% and 100% in simulation. As predicted by analyzing the sampled uncertain sensitivity functions S(s), complementary sensitivity functions KS(s), and corresponding templates in Fig. 12, as well as seen in Fig. 13

CONCLUSION AND FUTURE WORK

A complete time/frequency-domain analysis, including a sensitivity analysis, has been performed for frequency robust control in stand-alone MGs with PV sources. Simulation results show the effectiveness of the proposed control approach. This points out that the synthesized H ∞ controller remains robust in performance to the large uncertainty in the initial value of SoC sc (in particular SoC sce ∈ [25, 100] %), which is quite good for practical operation of the supercapacitor-based energy storage system as long as the corresponding variations of the supercapacitor current are considered admissible.

Future work can concern design of a robust control strategy for PCC voltage regulation. Experimental tests for control validation on a real-time test bench are also envisaged.

Fig. 1 .

 1 Fig. 1. Studied MG setup [12].

Fig. 2 .

 2 Fig. 2. Proposed control methodology.

Fig. 3 .

 3 Fig. 3. (a) MG frequency time-domain performance specifications consequent to a load step of + 5% of the rated load power (+ 60 kW) [12], [14]. (b) DC-bus voltage time-domain performance specifications consequent to a load step.

Fig. 6 .Fig. 7 .

 67 Fig. 6. Translation of temporal norms into frequency-domain specifications.

  Fig. 9. (a) Time-domain responses of PV output active power (b) MG frequency time-domain responses f grid and (c) DC-bus voltage time-domain responses v dc under PV output active power disturbances of -60 kW with respect to the rated operating point.

  response of the microgrid frequency Without energy storage device, without secondary control (K i = 0) H-infinity control, without secondary control (K i = 0) H-infinity control, with secondary control (K i = 2) H-infinity control, with secondary control (K i = 5) H-infinity control, with secondary control (K i = 10) Output active power step of the PV source ofresponse of the DC-bus voltage Without secondary control (K i = 0) With secondary control (K i = 2) With secondary control (K i = 5) With secondary control (K i = 10) step of the PV source of -60 kW Zo om in

Fig. 10 .Fig. 11 .

 1011 Fig. 10. (a) MG frequency time-domain responses f grid and (b) DC-bus voltage time-domain responses v dc under a PV output active power step disturbance of -60 kW with respect to the rated operating point.

Fig. 13 .

 13 Fig. 13. (a) MG frequency time-domain responses f grid and (b) DC-bus voltage time-domain responses v dc under a PV output active power step disturbance of -60 kW taking into account the uncertainty in vsc e (or SoCsc e ).

  (a) and Fig. 13(b), the closed-loop control overshoot performances of the frequency f grid and the DC-bus voltage v dc are preserved regardless of the initial value of SoC sc . Hence, the synthesized H ∞ controller is robust in performance to SoC sce ∈ [25, 100] % (or v sce ∈ [390, 780] V). The time-domain responses of the energy storage device current i s and the inverter output current in d-axis i rd taking into account the uncertainty in v sce are given in Fig. 14(a) and Fig. 14(b), respectively.

Fig. 14 .

 14 Fig. 14. (a) Time-domain responses of the energy storage device current is and (b) Time-domain responses of the inverter output current in d-axis i rd under a PV output active power step disturbance of -60 kW taking into account the uncertainty in vsc e (or SoCsc e ).

Table 1 .

 1 Steady-state real-unit values of the linearized system

	Variable	vsc e	v dce	v rde	vrq e	is e
	Value	585 V	1000 V	400 V	0 V	-2.5 A
	Variable	i rde	irq e	αc e	β de	βq e
	Value	-18.7 A	0 A	0.585	0.4	-0.0027
	Variable	P diesele	P P Ve	P loade	f gride	ω gride
	Value	1.0015 MW	0.2 MW	1.2 MW	50 Hz	100π rad/s
	values are given in Table

where s diesel is the droop value, T diesel denotes the time constant of the diesel engine generator. α c is the average value of the chopper duty ratio, β d is the average value of the inverter switching function in d-axis. H is the MG equivalent inertia constant. D load is the load damping constant. Subscript e indicates the steady-state equilibrium point. Steady-state real-unit

Table 2 .

 2 Weighting function parameters W perf (s) W perf 1 (s) for ∆v dc W perf 2 (s) for ∆f grid

	Ms	0.02v dce 0.05P loade	= 1.667	(50.12-50)/f b 0.05P loade	= 0.08
	Aε	0.01v dce 0.05P loade	= 0.833	(50.06-50)/f b 0.05P loade	= 0.04
	ω b	1 40T 0

rdq = 2.61 rad/s 1 20T 0 rdq = 5.22 rad/s (response time tr 1 ≈ 1.2 s) (response time tr 2 ≈ 0.6 s)

Table 3 .

 3 Weighting function parameters Wu (s)

		Wu 1 (s) for ∆i ref s	Wu 2 (s) for ∆i ref rd
	Mu	is max -is e 0.05P loade	= 9.62	i rdmax -i rde 0.05P loade	= 8.46
	Au	0.1Mu = 0.962	0.1Mu = 0.846
		ω bc 11 Au 1	= ωp = 3.04 rad/s	1/(100Ts i ) = 40 rad/s,
	ω bc	(ω bc 11 = 2.93 rad/s) ω bc 12 Mu 1 = 60 rad/s	where Ts i = 1/fs i , fs i is the inverter switching
		(ω bc 12 = 577.2 rad/s)	frequency


M Q Nguyen

O Sename

L Dugard

Comparison of observer approaches for actuator fault estimation in semi-active suspension systems

Keywords: Fault estimation, semi-active damper fault, adaptive observer, fast fault adaptive estimation, LPV observer

In this paper, the actuator fault estimation problem of semi-active suspension systems is considered. For instance, an oil leakage in the damper could cause a reduction of the damping force. The fault estimation requires a modeling of the damper fault (both multiplicative and additive fault models can be used). Three observer-based approaches are compared for fault estimation: an observer using fast adaptive fault estimation (FAFE) approach (used for estimation of additive faults), a parametric adaptive observer (AO) and a switched LPV observer (LPVO) (both intended to estimate mulplicative faults); Since the damper fault estimation is strongly affected by the unknown road disturbances, an H ∞ performance objective is used to reduce the effect of disturbances on the estimation error for performance assessment. Some simulations are performed on a quarter car model to validate these methodologies and a comparison is then given to shows the interest of each method.

I. INTRODUCTION

Fault Detection and Isolation (FDI) has received significant interest in both research and application domains since the last two decades. For complex systems highly equipped with sensors and actuators such as automotive applications, FDI has become one essential issue for reliability and safety. The design of a FDI procedure includes fault detection, isolation and estimation. Among them, fault estimation is a key step in designing a fault tolerant control. Many different approaches to estimate a fault which can be either actuator or sensor malfunction. Let us mention some classical methods, based on the parity space theory [START_REF] Gertler | Fault detection and isolation using parity relations[END_REF] to generate the residues and approximate the fault, or the bank of observers approach [START_REF] Varrier | Détection de situations critiques et commande robuste tolérante aux défauts pour l'automobile[END_REF], or the sliding mode observers [START_REF] Edwards | Sliding mode observers for fault detection and isolation[END_REF]. Recently, a new approach in [START_REF] Shi | Active fault tolerant control of lpv descriptor systems based on extended state observers[END_REF] considered the fault element as a state of the augmented system and designed an extended observer to estimate, at the same time, the state and the fault of system. However, it is limited to constant faults ( ḟ (t) = 0). Then, [START_REF] Zhang | Adaptive observer-based fast fault estimation[END_REF] presented a method allowing to evaluate the timevarying fault by using a fast adaptive fault estimation (FAFE) methodology based on an adaptive observer. However, the authors solved the problem with a regular LTI system without considering the disturbances. Recently, [START_REF] Rodrigues | Actuator fault estimation based adaptive polytopic observer for a class of lpv descriptor systems[END_REF] proposed an adaptive polytopic unknow input observer for time-varying fault estimation, for a class of descriptor LPV systems. Then, several works have been done for fault estimation and fault 1 Univ.

Grenoble Alpes, GIPSA-lab, F-38000 Grenoble, France CNRS, GIPSA-lab, F-38000 Grenoble, France.

{manh-quan.nguyen, olivier.sename, luc.dugard}@gipsa-lab.grenoble-inp.fr tolerant control for LPV systems, such as LPV observer design for state and fault estimation in [START_REF] Abdullah | Sensor-fault-tolerant control for a class of linear parameter varying systems with practical examples[END_REF], [START_REF] Patton | An lpv pole-placement approach to friction compensation as an ftc problem[END_REF], interval observer for LPV systems in [START_REF] Montes De Oca | Robust fault detection based on adaptive threshold generation using interval lpv observers[END_REF], LPV sliding mode observer with fault compensation in [START_REF] Hamayun | An lpv fault tolerant control scheme using integral sliding modes[END_REF], virtual sensor approaches in [START_REF] Montes De Oca | Fault estimation and virtual sensor ftc approach for lpv systems[END_REF].

In the automotive field, the demand is now concerned by safer and more comfortable vehicles. Among all sub-systems impacting the vertical vehicle dynamics, the supension systems play a key role since they ensure the link between the wheels and the chassis (see [START_REF] Fischer | Mechatronic semi-active and active vehicle suspensions[END_REF]). Suspension systems are efficient actuators to considerably improve not only passengers comfort but also car road holding [START_REF] Poussot-Vassal | Survey and performance evaluation on some automotive semi-active suspension control methods: A comparative study on a single-corner model[END_REF]. Therefore, the development of intelligent suspension systems has gained a lot of attention during last years. In particular, semi-active suspensions are today more and more used because of their efficiency, while being less expensive and comsumming less energy than pure active suspensions. These systems consist of a shock absorber which is able to online change its damping characteristics to enhance the vehicle dynamics.

While the semi-active suspensions have big advantages in enhancing vehicle dynamics, the necessity of fault diagnosis for these systems is more and more present. In particular, it has been noted that semi-active dampers are more susceptible to faults than their traditional counterpart. Therein, oil leakage is the most common fault, and its effect is a reduction of the damping force. The estimation of suspension faults is an indispensable step in the Fault Tolerant Control (FTC) to prevent handling and comfort deterioration. [START_REF] Sename | Lpv methods for fault-tolerant vehicle dynamic control[END_REF], [START_REF] Nguyen | An lpv fault tolerant control for semi-active suspension-scheduled by fault estimation[END_REF] proposed a LPV/FTC scheduled by fault estimation which is based on the parity space theory and fast adaptive observer approaches respectively. Recently, [START_REF] Hernandez-Alcantara | Fault estimation methods for semi-active suspension systems[END_REF] presented various methods using an observer-based approach and a parameter identification approach to estimate damper fault.

This paper aims at comparing three methodologies to deal with the actuator fault estimation problem of the semi-active suspension systems in the framework of observer based approaches. Note that such a comparison is proposed here for the first time in the case of suspension system. In the first method, the fault is represented as an additive fault, then a fast fault adaptive estimation approach (FAFE approach) is used for fault estimation. In the latter approaches, the damper fault is modeled in a multiplicative way using a effectiveness coefficient. Then, this coefficient can be estimated by a parametric adaptive observer (AO approach) and a switched LPV observer (LPVO approach). The damper fault estimation is strongly affected by the unknown road disturbances. To deal with this problem, an H ∞ performance objective is used to reduce the effect of disturbances on the estimation error for performance assement. The comparison between three methods is given thereafter.

The paper is organized as follows. Section II presents the vehicle modeling and problem statement. Section III, IV and V present the different methods allowing to estimate the damper fault using FAFE, AO and LPVO approaches respectively. In section VI, some simulation results are given. Finally, some conclusions are drawn in the last section. Notations: P T is the transposed of matrix P, 0 and I denote zeros and identity matrix of appropriate dimensions, the star symbol in a symmetric matrix denotes the transposed block in the symmetric position.

II. PROBLEM STATEMENT

The dynamic behavior of a quarter car model with a semiactive suspension, taken into account in this work, is depicted as in Fig. 1: The dynamical equations are governed by:

m s zs = -k s (z s -z us ) -F sa m us zus = k s (z s -z us ) + F sa -k t (z us -z r ) (1)
where F sa is the semi-active damper force, which can be represented by a linear model:

F sa = cż de f ( 2
)
where c is the damping coefficient that must satisfy the dissipativity constraint c min ≤ c ≤ c max .

Let us assume now that a fault occurs on the semi-active damper e.g an oil leakage which induces the effectiveness loss of the damper. Denoting α the efficiency coefficient of the damper, and F sa (t) stands for the force of the faulty damper, then one has the following relation:

F sa (t) = (1 -α)F sa (t) (3)
In this study, α is considered as a constant parameter. For example α = 0 corresponds to a healthy damper, α = 0.8 corresponds to that the damper losses 80% its efficency. α = 1 represents a completely damper failure. Note that (3) is multiplicative fault modeling, but an additive fault modeling could be also used (see section III).

The problem consists now in estimating the damper fault, i.e the coefficient α which could be estimated by several approaches: a fast adaptive fault estimation (FAFE) approach (where an additive fault modeling is used, see section III), an adaptive observer (section IV) and a switched LPV observer (section V).

III. METHOD 1: FAFE APPROACH

In this section, an addittive modeling for damper fault is considered. In fact, let us decompose the total semiactive damper force as: F sa (t) = c 0 żde f (t) + u(t), where c 0 is damping coefficient for the passive damper, żde f is deflection speed and u(t) is control input. Then, the faulty damper force in (3) can be rewritten as follows:

F sa (t) = (1 -α)F sa (t) = c 0 żde f (t) + u(t) + f (t) (4)
where f (t) is the loss of damper force and given by f (t) = -α(c 0 żde f (t) + u(t)). Note that the effectiveness coefficient α is slowly varing or constant, but the additive fault f = -α(c 0 żde f + u) could be time varying even if α is constant. Now, the state space representation of quarter car model (1) while considering a faulty semi-active damper is given as follows:

ẋ(t) = Ax(t) + B 1 w(t) + B 2 u(t) + E f (t) (5) y(t) = Cx(t)
where x = (z s , żs , z us , żs ) T ∈ R n is the state vector, w = z r is the disturbance input, u ∈ R m is the control input, f stand for damper fault. y = [z de f , żde f , z s ] ∈ R p is the output vector. Such a choice of y allows to satisfy Assumption 1 presented later.

A =     0 1 0 0 -k s m s -c 0 m s k s m s c 0 m s 0 0 0 1 k s m us c 0 m us -k s +k t m us -c 0 m us     , B 1 =     0 0 0 k t m us     , B 2 = 0 -1 m s 0 1 m us T , E = 0 -1 m s 0 1 m us T , C =   1 0 -1 0 0 1 0 -1 1 0 0 0  
The fault estimation problem consists now in estimating f (t), and the estimation of α can be given by: α = f /F sa .

Since f (t) can be fast time varying, so in order to get a good estimation of the damper fault, the Fast Adaptive Fault Estimation (FAFE) approach proposed in [START_REF] Zhang | Adaptive observer-based fast fault estimation[END_REF], [START_REF] Rodrigues | Actuator fault estimation based adaptive polytopic observer for a class of lpv descriptor systems[END_REF] is used. Moreover, since the vehicle dynamic is affected by the road disturbance, in order to improve the estimation quality, some extended results using an H ∞ performance is used to attenuate the disturbance effects on the state and fault estimation error.

Let us consider now the state space repesentation [START_REF] Zhang | Adaptive observer-based fast fault estimation[END_REF] with additive fault f . The following assumptions are satisfied:

• Assumption 1: rank (CE) = rank(E), and the invariant zeros of (A,E,C) are on the left half plane. • Assumption 2: The fault f (t) and its time derivative are norm bounded i.e: 0 ≤ f (t) < α 1 and 0 ≤ ḟ (t) < α 2 with 0 ≤ α 1 , α 2 < ∞. The state space repesentation (5) includes the road disturbance w = z r which is an unknown input disturbance in the suspension system. Therefore, while estimating the fault actuator f (t), one needs to take into account the effect of this unknown input. To deal with this problem, an unknown input adaptive fault observer is proposed and given as follows:

ż(t) = Nz(t) + Gu(t) + Ly(t) + T 1 E f (t) x(t) = z(t) + T 2 y(t) (6)
ŷ(t) = C x(t)
where z ∈ R n is the state variables of the observer, x ∈ R n the estimated state variables. ŷ ∈ R p is the estimated output vector and f (t) ∈ R r is the estimation of the damper fault f (t). N, G, L, T 1 , T 2 are the observer matrices to be designed to enable the convergence of the state and fault. Denote e x (t) = x(t)x(t), e y (t) = y(t)ŷ(t), e f (t) = f (t) -f (t) as the state, output and fault estimation error respectively. Since there exists a full-row rank matrix

[T 1 T 2 ][ I C ] = I, i.e T 1 = I -T 2 C, then e x = (I -T 2 C)x -z = T 1 x -z and ėx = T 1 ẋ - ż = T 1 (Ax + B 1 w + B 2 u + E f ) -(Nz + Gu + Ly + T 1 E f )
After some manipulations, one has:

ėx = Ne x + T 1 Ee f + + (T 1 A -NT 1 -LC)x + (T 1 B 2 -G)u + T 1 B 1 w
If the following conditions hold:

T 1 A -NT 1 -LC = 0, T 1 B 2 -G = 0,
then, the estimation error dynamic is given by:

ėx = Ne x + T 1 Ee f + T 1 B 1 w (7) e y = Ce x Since T 1 = I -T 2 C, then by denoting K = NT 2 -L, one has N = T 1 A + KC.
The equation [START_REF] Abdullah | Sensor-fault-tolerant control for a class of linear parameter varying systems with practical examples[END_REF], governing the state and fault estimation errors, is affected by the unknown road disturbance w. If T 1 B 1 = 0, the effect of the road disturbance on the estimated states is cancelled. This condition is equivalent to (I -T 2 C)B 1 = 0, where T 2 has to be determined. This equation is solvable if and only if rank(CB 1 ) = rank(B 1 ). Otherwise, the disturbance effect has to be minimized and the problem is to find N such that N is stable and the effect of w on e y is minimized.

In order to minimize the disturbance effect on the estimation error, we use the Bounded Real Lemma for the H ∞ performance index to design the observer.

Then, the following theorem gives an extension to the results in [START_REF] Zhang | Adaptive observer-based fast fault estimation[END_REF] and aims at minimizing the disturbance effect on the estimation error and ensuring the convergence of the estimation error in [START_REF] Abdullah | Sensor-fault-tolerant control for a class of linear parameter varying systems with practical examples[END_REF]:

Theorem 1: Under the assumptions 1-2, given scalars σ , µ, γ 2 > 0, if there exist symmetric positive definite matrices Q, P 1 , and matrices W,U such that the following conditions hold:

  M 11 M 12 M 13 M 22 M 23 M 33   < 0 (8)
and

E T T T 1 Q = UC (9)
where:

M 11 = (T 1 A) T Q + Q(T 1 A) +C T W T +WC +C T C M 12 = - 1 σ [(T 1 A) T Q +C T W T ]T 1 E M 13 = QT 1 B 1 M 22 = - 2 σ E T T T 1 QT 1 E + 1 σ µ P 1 M 23 = - 1 σ E T T T 1 QT 1 B 1 M 33 = -γ 2 2 I
then, the following fault estimation algorithm:

ḟ (t) = ΓU( ėy (t) + σ e y (t)) (10)
can ensure that e x (t) and e f (t) are uniformly bounded, where Γ ∈ R r×r is a symmetric positive definite learning rate matrix.

Proof: Consider the following Lyapunov function:

V (e x , e f ) = e T x (t)Qe x (t) +

1 σ e T f (t)Γ -1 e f (t) (11)
Then, the L 2 induced gain conditon e y 2 < γ 2 w 2 is satisfied if:

V + e T y e yγ 2 2 w T w < 0

Thanks to the Bounded Real Lemma and following the same procedure as in [START_REF] Zhang | Adaptive observer-based fast fault estimation[END_REF], the proof can be infered easily .

Then, it is easy to show that the estimated fault can be deduced from the expression (10) as follows :

f (t) = ΓU e y (t) + σ t t f e y (τ)dτ ( 13
)
where t f is the time since fault occurs. The fault reconstruction in (13) combines a proportional term with an integral one that allows to improve the fault estimation speed. Therefore, to estimate the fault, one needs to solve the 2 conditions in Theorem 1 which consists of a Linear Matrix Inequality [START_REF] Patton | An lpv pole-placement approach to friction compensation as an ftc problem[END_REF] and an equality matrix constraint [START_REF] Montes De Oca | Robust fault detection based on adaptive threshold generation using interval lpv observers[END_REF] which are transformed into the following optimization problem: Min γ subjects to [START_REF] Patton | An lpv pole-placement approach to friction compensation as an ftc problem[END_REF] and

γI E T Q -UC (E T Q -UC) T γI > 0 (14)
Solving this optimization problem, the observer matrices can be computed by: N = T 1 A + KC, K = Q -1 W and the estimation of the fault actuator is performed.

IV. METHOD 2: ADAPTIVE OBSERVER (AO) APPROACH

This part shows the procedure to estimate a multiplicative damper fault. As section III, the total damper force is still written as: F sa (t) = c 0 żde f (t) + u(t), where c 0 is damping coefficient for the passive damper, u(t) is control input. From (3), one has:

F sa (t) = (1 -α)F sa (t) = (1 -α)(c 0 żde f (t) + u(t)) (15)
Then the state space representation of the quarter model (1) under the damper fault is given by:

ẋ(t) = Ax(t) + B 1 w(t) + B 2 u(t) + Ψ(y, u,t)α (16) y(t) = Cx(t)
where x = (z s , żs , z us , żs ) T ∈ R n is the state vector, w = z r is the disturbance input, u ∈ R m is the control input, α stand for damper fault. y = [z de f , żde f , z s ] ∈ R p is the output vector,.

A =     0 1 0 0 -k s m s -c 0 m s k s m s c 0 m s 0 0 0 1 k s m us c 0 m us -k s +k t m us -c 0 m us     , B 1 =     0 0 0 k t m us     , B 2 = 0 -1 m s 0 1 m us T , C =   1 0 -1 0 0 1 0 -1 1 0 0 0   Ψ = -0 -1 m s 0 1 m us T
(c 0 żde f + u). Since the multiplicative fault α is slowly varying or constant, the fault estimation can be considered as a problem of This problem can be solved based on an adaptive observer for joint state-parameter estimation [START_REF] Zhang | Adaptive observer for multiple-input-multiple-output (mimo) linear time-varying systems[END_REF]. However, the effect of the disturbances is not taken into account explicitly. Moreover, regarding to vertical vehicle dynamics, the system is always affected by the unknown road disturbances. Therefore, the developements in [START_REF] Zhang | Adaptive observer for multiple-input-multiple-output (mimo) linear time-varying systems[END_REF] are adapted to our problem and applied for the automotive applications. An H ∞ performance is here proposed to attenuate the disturbance effects on the state and fault estimation.

Let us consider now system [START_REF] Hernandez-Alcantara | Fault estimation methods for semi-active suspension systems[END_REF] with multiplicative fault α. It is assumed that the matrix pair (A,C) in ( 16) is observable. Following [START_REF] Zhang | Adaptive observer for multiple-input-multiple-output (mimo) linear time-varying systems[END_REF], an adaptive observer of the following structure can be proposed to estimate both states and fault α:

   Υ(t) = (A -KC)ϒ(t) + Ψ(y, u,t) ẋ = A x + B 2 u + Ψ(y, u,t) α + K(y -C x) + ϒ(t) α α = Γϒ C Σ(y -C x) ( 17
)
where K is the adaptive observer gain to be determined. Γ, Σ are some bounded symmetric positive-definite matrices to be tuned. Since the quarter car model is a LTI system, K, Γ, Σ are considered constant. ϒ(t) satisfies the following assumption:

Assumption 1: ϒ ∈ R n × R p is a matrix of signals generated by the ordinary differential equation (ODE) system

Υ(t) = [A -KC]ϒ(t) + Ψ(t) (18)
Morever, to apply the adaptive observer approach, an important required condition is that Ψ(y, u,t) is persistently exciting. The following lemma is used to get the convergence of the adaptive observer. Lemma 1: [START_REF] Zhang | Adaptive observer for multiple-input-multiple-output (mimo) linear time-varying systems[END_REF] Assume that Ψ(t) is persistently exciting, so that there exist positive constants α, β , T and some bounded symmetric positive definitve matrix Σ ∈ R m × R m such that, for all t, the following inequalities hold:

αI ≤ t+T t ϒ T (τ)C T (τ)Σ(τ)C(τ)ϒ(τ)dτ ≤ β I (19)
then the system: ż(t) = -Γϒ C ΣCϒz(t) is exponentially stable Let us denote e x (t) = x(t)x(t), e α (t) = α(t) -α(t) the state and fault estimation errors respectively.

Since α is constant or slowly varying, then α 0, then (see [START_REF] Zhang | Adaptive observer for multiple-input-multiple-output (mimo) linear time-varying systems[END_REF]):

e α = α -α = - α ėx = ẋ -ẋ = (A -KC)e x + Ψ(y, u)e α + B 1 w -ϒ α (20)
Let us now define η as a combination of state and fault estimation errors, i.e:

η = e x -ϒe α , e x = η + ϒe α = η -ϒ α then: η = (A -KC)η + B 1 w (21) ėα = -Γϒ C ΣCη -Γϒ C ΣCϒe α (22)
As seen in the above equations, the estimation errors e x , e α are affected by the unknown road disturbance w(t). Then, the disturbance effect has to be minimized. It is worth noting that from the Lemma 1, the homogenous term of (22), i.e ėα = -Γϒ C ΣCϒe α is exponentially stable. Therefore, if η(t) -→ 0, then e α -→ 0, and e x = η + ϒe α -→ 0 provided that ϒ is bounded. However, due to the presence of the road disturbance w, in this work, we aim at minimizing the effect of w on the estimation errors e x , e α . To this aim, the effect of w on η in ( 21) is minimized as follows:

Proposition 1: If there exists a matrix K ∈ R n×m so that the system:

η(t) = [A -KC]η(t) + B 1 w(t) (23)
satisfies the following conditions:

• when w(t) ≡ 0, the system is exponentially stable • when w(t) = 0, the following L 2 -induced gain performance criterion is satisfied :

min γ s.t η 2 < γ w 2 (24)
where . 2 stands for L 2 norm. Or equivalenty, if there exist a matrix Y such that

  A P + PA -C Y -YC PB 1 I -γ 2 I 0 -I   < 0 (25)
then state and fault estimation can be performed using the adaptive observer [START_REF] Zhang | Adaptive observer for multiple-input-multiple-output (mimo) linear time-varying systems[END_REF], and the gain K is given by K = P -1 Y . Proof: Let us consider a candidate Lyapunov function V (η) = η T Pη for the system (23). In order to satisfy the condition in (24), then the following inequality should be ensured:

V + η T ηγ 2 w T w < 0 (26)

By applying The Bounded Real Lemma, it leads to:

  (A -KC) P + P(A -KC) PB 1 I -γ 2 I 0 -I   < 0 (27)
Let Y = PK, one obtains (25). Finally, the observer design problem consists in solving the following optimization problem: min γ s.t P > 0 and (25) holds (28)

Then the observer gain K is computed by: K = P -1 Y , and the damper fault is given by α .

V. METHOD 3: A SWITCHED LPV OBSERVER APPROACH This section briefly presents the method, presented in [START_REF] Nguyen | A switched lpv observer for actuator fault estimation[END_REF], to estimate an actuator fault using a switched LPV observer. This method is applied here for the first time to damper fault estimation. For this method, we consider directly the total damper force as the control input u(t) , i.e: u(t) = F sa (t).

Let us now reconsider faulty damper force:

F sa (t) = (1 -α)F sa (t) = λ F sa (t) = λ u(t) ( λ = 1 -α) (29)
Then under the damper fault, the state space representation of the quarter vehicle model ( 1) is given as follows:

ẋ(t) = Ax(t) + B 2 λ u(t) + B 1 w(t) (30) y(t) = Cx(t) x = (z s , żs , z us , żs ) T ∈ R n is the state vector, w = z r is the disturbance input, u ∈ R m is the control input. A =     0 1 0 0 -k s m s 0 k s m s 0 0 0 0 1 k s m us 0 -k s +k t m us 0     , B 1 =     0 0 0 k t m us     , B 2 = 0 -1 m s 0 1 m us T , C =   1 0 -1 0 0 1 0 -1 1 0 0 0  
Let consider now the faulty damper system (30), it can be seen that if the control input u(t) = 0, then the damper fault information λ in (30) becomes unobservable. It may cause the infeasibility problem in the observer design step when the control input equals zero. Thus, an interesting remedy, proposed in [START_REF] Nguyen | A switched lpv observer for actuator fault estimation[END_REF], is to consider a switched model depending on the sign of u(t) in the observer synthesis step. Moreover, since the control input u(t) is known, the system (30) can be represented as an LPV system by choosing u(t) as a scheduling parameter. Let us rewrite u(t) = |u(t)|sign(u(t)), and denote ρ(t) = |u(t)| as a scheduling parameter. Then,

u(t) = ρ(t)sign(u(t)) = ρ(t) i f u(t)) ≥ 0 -ρ(t) i f u(t) < 0 (31)
The scheduling parameter ρ is assumed to satisfy: ε ≤ ρ ≤ ρ m , where ε > 0, ρ m = u m is the bound of the parameter ρ. The system (30) is now rewritten as the following LPV system:

ẋ(t) = Ax(t) + B σ (ρ)λ + B 1 w(t) (32)
where B σ (ρ) = B 2 ρsign(u(t)).

The original system (30) can be augmented in the following form:

ẋ λ = A B σ (ρ) 0 0 A e (ρ) x λ + B 1 0 w (33) y = [C 0] C e x λ
The system (33) is actually a switched system where σ (t) is the switching rule that depends on the value of the function sign(u(t)). Actually, one has B σ (ρ) ∈ {-B 2 ρ, B 2 ρ}. Thus, the system (33) can be rewritten in the following form of a switched LPV system:

ẋ λ = A e,σ (ρ) x λ + B 1 0 w (34) y = C e x λ
where A e,σ (ρ)

switches among 2 subsystems A -B 2 ρ 0 0 , A B 2 ρ 0 0 .
Consider the switched LPV system (34), the following switched LPV extended observer is proposed to estimate the system's state and the effectiveness coefficient λ : is based also on the H ∞ performance while one aims at minimizing the effect of the disturbances w on the estimation error. The following theorem (whose complete proof is in [START_REF] Nguyen | A switched lpv observer for actuator fault estimation[END_REF]) allows to solve this problem: Theorem 2: [START_REF] Nguyen | A switched lpv observer for actuator fault estimation[END_REF] Consider the switched system (34) and the switched extended observer (35), if there exists a collection of matrices P i,k > 0,Y i,k , k = 0, ...K, i = 1, 2, of appropriate dimensions and K is prescribed integer, such that for all i = 1, 2 and j = 1, 2, the following LMIs hold:

ẋ λ = A e,σ (ρ) x λ + K σ (t)(y -ŷ) (35)
    (P i,k+1 -P i,k ) T /K + He[P i,h A ( j) e,σ -Y i,h C e ] B 1e P i,h -γ 2 I I 0 -I     < 0 (36) for k = 0, ...K -1, h = k, k + 1,   A ( j) e,σ P i,K -C e Y i,K + P i,K A ( j) e,σ -Y i,K C e B 1e P i,K -γ 2 I I 0 -I   < 0 (37) P i,K -P l,0 ≥ 0 ∀l = 1, 2 & l = i. (38)
then

K σ (t) (t) = P σ (t) (t) -1 Y σ (t) (t) =      P-1 i,k Ŷi,k i f t ∈ [τ s,k , τ s,k+1 ) P -1 i,K Y i,K i f t ∈ [τ s,K , τ s+1,0 ) P -1 i 0 ,K Y i 0 ,K i f t ∈ [0, τ 1 ) (39)
is the gain of the extended observer (35) and the error estimation asymptotically converges to zero for a dwell time of T , where

Y σ (t) (t) =      Y i,k + (Y i,k+1 -Y i,k ) t -τ s,k T /K := Ŷi,k , i f t ∈ [τ s,k , τ s,k+1 ) Y i,K i f t ∈ [τ s,K , τ s+1,0 ) Y i 0 ,K i f t ∈ [0, τ 1 ) (40)
By designing the switched LPV observer, one obtain the estimation of λ , and the damper fault estimation is given by: α = 1 -λ .

VI. SIMULATION RESULTS

The physical parameters characterize the considered quarter car model is given in the Table I The following scenario is used to test the performance of three different observers. It is assumed that the vehicle runs at 30km/h over a sinusoid road profile w(t) = 0.03sin(2πt). Then, the nominal semi-active damper force is given by: F sa = cż de f (blue line in Fig. 2), where c is controlled damping of semi-active damper, satisfies 700 ≤ c ≤ 5000.

The vehicle has a faulty semi-active damper because of a oil leakage. In the detail, from t = 0 -5s, the damper looses 20% of its total force (α = 0.2). From t = 5 -8s, an higher fault occurs and the damper looses 60% of its efficiency (α = 0.6).

It is noted that all observers are activated from t = 0.6s. Fig. 2 plots the behavior of the nominal damper force F sa (blue liine), the faulty damper force F sa (= (1α)F sa = F sa + f ) (green line) and the estimation of the additive fault f (t) (red line). 3 shows the comparison of the damper fault estimation using 3 different approaches: fast adaptive fault estimation approach (FAFE), Adaptive Observer approach (AO) and switched LPV observer approach (LPVO). The AO approach gives the estimation of α. For the FAFE approach, the additive faut f (t) is estimated and for LPVO approach, λ is estimated. Then, the corresponding α is computed from f (t), λ by α = f /F sa or α = 1 -λ . It can be seen from the estimation results that, despite of the presence of road disturbance, the estimation errors still converge to zeros. All three methods give a good estimation of damper fault. Moreover, it seems that the AO approach provides a better estimation than LPVO and FAFE approaches. Now, let us consider the second scenario. It is assumed that the damper oil leak slowly from t = 0 -6s, one has a loss of efficiency from 0 → 80%. Fig. 4 demonstrates once again that the damper fault can be well estimated using the different observer approaches.

VII. CONCLUSIONS

The paper provides a comparison of different methods for actuator fault estimation in semi-active suspension systems using FAFE observer, AO approach and a switched LPV observer approach. The damper fault is firstly modeled in different ways: additive model or multiplicative model. An H ∞ performance is used to attenuate the effect of the road disturbance on the estimation error. The simulation results show that all three methods can give a good estimation results for damper fault. For the future work, fault tolerant control can be designed thanks to the fault estimation, and the implementation on a real testbed will be done.

Fig. 1 .

 1 Fig. 1. Quarter-car vehicle model

Fig. 2 .Fig. 3 .

 23 Fig. 2. Damper force behaviors

Fig. 4 .

 4 Fig. 4. Fault estimation α scenario 2

  .

	Parameter m s [kg]	m us [kg]	k s [N/m]	k t [N/m]	c[Nm/s]
	Value	315	37.5	29500	210000	{700, 5000}

TABLE I PARAMETERS

 I OF RENAULT M ÉGANE COUP É QUARTER CAR MODEL


M Q Nguyen

M Canale

O Sename

L Dugard
email: luc.dugard@gipsa-lab.grenoble-inp.fr

A Model Predictive approach for semi active suspension control problem of a full car

Keywords: Semi-active suspension, road disturbance estimation, MPC control, Input Constraints

A suspension controller aims at enhancing the ride comfort and the handling of vehicle which are evaluated by the acceleration at center of gravity and the roll motion resspectively. In this paper, a semi-active suspension Model Predictive Control (MPC) is designed for a full vehicle system equipped with 4 semi-active dampers. The main challenge in the semi-active suspension control problem is to tackle with the dissipativity constraints of the semi-active dampers , here recasted as input and state constraints. The controller is designed in the MPC framework where the effects of the unknown road disturbances are taken into account. An observer approach allows to estimate the road disturbance information to be used by the controller during the prediction step. Then, the MPC control law with road estimation (but without road preview) is computed by minimizing a quadratic cost function, giving a trade-off between the comfort and the handling, while guaranteeing some phyiscal constraints of the semi-active dampers. Some simulation results performed on a nonlinear full car model are presented in order to illustrate the effectiveness of the proposed approach.

I. INTRODUCTION

The suspension system plays a key role in enhancing the vehicle performances with regard to ride comfort and road handling. Semi-active suspensions are today more and more used in automotive industry because of their efficiency, while being less expensive and comsumming less energy than pure active suspensions. However, the main challenge of semi-active suspension control problems is to handle the dissipativity constraints of semi-active dampers. Several control design problems have then been tackled with many different approaches. [START_REF] Tseng | State of the art survey: active and semiactive suspension control[END_REF], [START_REF] Poussot-Vassal | Survey and performance evaluation on some automotive semi-active suspension control methods: A comparative study on a single-corner model[END_REF] give extensive surveys on semi active suspension control. In more detail, [START_REF] Karnopp | Vibration control using semi-active force generators[END_REF] proposed a classical control method based on the Skyhook control to improve the ride comfort. Then, several extensions of skyhook control have been presented in the past decades as in [START_REF] Ahmadian | No-jerk skyhook control methods for semiactive suspensions[END_REF], or [START_REF] Savaresi | Mixed sky-hook and add: Approaching the filtering limits of a semi-active suspension[END_REF] where Mixed Skyhook-ADD is proposed. More recently, some modern control approaches have been suggested. [START_REF] Unger | Application of lq-based semi-active suspension control in a vehicle[END_REF] proposed an LQ-based clipped optimal control. Some LPV techniques for semi-active suspension control problems have been presented as in [START_REF] Poussot-Vassal | A new semi-active suspension control strategy through LPV technique[END_REF], [START_REF] Savaresi | Semi-active suspension control design for vehicles[END_REF] or [START_REF] Nguyen | A state feedback input constrained control design for a 4-semi-active damper suspension system: a quasi-LPV approach[END_REF] used an LPV approach with sector condition to deal with massimo.canale @polito.it the dissipativity constraints of semi-active dampers.

Regarding the actuator saturation control problem, Model Predictive Control allows to explicitly take into account the effect of input and state constraints in the control design step. Several works have employed the MPC approach for semiactive suspension systems. Nevertheless most of the studies considered a quarter-car model only. Let us mention here [START_REF] Giorgetti | Hybrid model predictive control application towards optimal semi-active suspension[END_REF] where the constrained quarter-car semi-active suspension is modelled as a switching affine system and the MPC controller is computed thanks to a mixed-integer quadratic programming. In [START_REF] Canale | Semi-active suspension control using fast model predictive control[END_REF], a fast MPC is designed for a half car model, but the MPC controller is still designed based on a quarter car suspension model. Moreover, the effects of disturbances are not taken into account in the prediction horizon. [START_REF] Poussot-Vassal | A methodology for optimal semi-active suspension systems performance evaluation[END_REF] proposes a methodology for optimal semiactive suspension system based on MPC approach for a quarter car model while assuming that the road disturbance is measured in advance and used in the prediction horizon. However, the quarter car model equipped with one semiactive damper which is not enough to express the full dynamic of the vehicle with four semi-active dampers. In order to deal with the full car model case, one possibility is to design four seperated controllers at the four corners. However, by this way, the effects of the coupling and the load transfer distribution between the corners during various driving situations (cornering, steering, accelerating, and braking...) may not be considered, which could lead to lower performance. Concerning the full car dynamics, up to the authors knowledge, very few studies have been proposed to develop MIMO MPC semi-active control techniques. [START_REF] Gohrle | Model predictive control of semi-active and active suspension systems with available road preview[END_REF] employs a nonlinear programming approach (not suitable for implementation). To overcome such a problem, [START_REF] Gohrle | Model predictive control of semi-active and active suspension systems with available road preview[END_REF] introduces an approximate descripition of the constraints as well as the clipping of the control action. On the other hand, taking into account the disturbances during the prediction horizon gives a better efficiency for the predictive control. [START_REF]Road profile estimation and preview control for low-bandwidth active suspension systems[END_REF] uses the road profile preview by means of expensive and not standard sensors (e.g. camera) for the case of active suspension. Disturbance estimation is not used, except in a case, to verify the preview obtained by the camera and also in the context of active suspension.

In this work, a semi-active suspension MPC controller is designed for a full vehicle model equipped with 4 semi-active dampers. The proposed solution integrates a state feedback control with an observer of the vehicle state variables and of the road disturbance. The paper contributions are twofolds:

• An observer approach is proposed to estimate both the system state (needed anyway by all MPC approaches) and road disturbances. It is woth mentioning that while the estimation of the road inputs allow to improve the efficiency of the predictive controller, it is here obtained using standard sensors and then differs from the preview approach.

• A MPC suspension control with road disturbance estimation (but without road preview) is obtained by optimimizing a quadratic cost function. This cost describes the ride comfort and road holding performances, while ensuring the dissipativity constraints of the semi-active dampers. The controller solution is derived within the mixed integer quadratic programming (MIQP). Such an approach allows online implemention of the solution.

The results are compared to those obtained by MPC with disturbance preview and by MPC without taking into account the disturbance during the prediction; they show the interest of the proposed approach. The structure of the paper is given as follows. Section II describes a full vertical vehicle model and the problem formulation. Section III presents the semi-active suspension control design using MPC. Section IV presents the obsever design for state and road disturbance estimation. Some simulation results are given in the section V. Finally, some conclusions are drawn in the section VI.

II. A FULL CAR MODEL EQUIPPED WITH 4 SEMI-ACTIVE SUSPENSIONS

A. Full car model A full car vertical model is used for the analysis and control of the vehicle dynamic behavior. This is a classical 7 degree of freedom (DOF) suspension model, obtained from a nonlinear full vehicle model (referred in [START_REF] Poussot-Vassal | Attitude and handling improvements through gainscheduled suspensions and brakes control[END_REF]). This model not only involves the chassis dynamics (vertical (z s ), roll (θ ) and pitch(φ )), but it also figures out (z us i j ) the vertical displacements of the wheels at the front/rear (i = ( f , r))-left/right corner ( j = (l, r)). The vertical 7DOF full-car model is governed by the following equations:

       m s zs = -F s f l -F s f r -F srl -F srr I x θ = (-F s f r + F s f l )t f + (-F srr + F srl )t r I y φ = (F srr + F srl )l r -(F s f r + F s f l )l f m us zus i j = -F s i j + F tz i j (1)
where I x , I y are the moments of inertia of the sprung mass around the longitudinal and lateral axis respectively, h is the height of center of gravity (COG). l f , l r ,t f ,t r are COG-front, rear, left, right distances respectively. F tz i j are the vertical tire forces, given as:

F tz i j = -k t i j (z us i j -z r i j ) (2)
where k t i j are the stiffness coefficients of the tires, and z r i j the road profiles.

The vertical suspension forces F s i j at the 4 corners of the vehicle are modeled by a spring and a damper with non linear characteristics for simulation, and linear ones for control design. The equation ( 3) allows to model the suspension force used in the control design step:

F s i j = k i j (z s i j -z us i j ) + F d i j (3)
where k i j is the nominal spring stiffness coefficient, and z s i j is the chassis position at each corner, and F d i j is the semiactive controlled damper force given by:

F d i j = c i j (.)ż de f i j = c i j (.)(ż s i j -żus i j ) (4)
where c i j (.) the damping coefficient is assumed to be varying for control purpose. To ensure the dissipativity constraint of each semi-active damper, the following constraint must be considered: 0 c min i j c i j (.) c max i j (5)

Now, let us rewrite the damper force (4) as follows:

F d i j = c nom i j żde f i j + u i j (6)
where c nom i j = (c max i j + c min i j )/2 is the nominal damping coefficient, u i j is the incremental force and is considered as the control input. Then the equation (3) becomes:

F s i j = k i j (z s i j -z us i j ) + c nom i j (ż s i j -żus i j ) + u i j (7)
where the sprung mass positions z s i j at each corner of the vehicle can be easily derived from the vehicle equations of motions, and given by:

       z s f l = z s -l f sin φ + t f sin θ , z s f r = z s -l f sin φ -t f sin θ , z s rl = z s + l r sin φ + t r sin θ , z s rr = z s + l r sin φ -t r sin θ , (8)
Assuming that the roll and pitch angles are small enough, these nonlinear equations are linearized by:

       z s f l = z s -l f φ + t f θ , z s f r = z s -l f φ -t f θ , z s rl = z s + l r φ + t r θ , z s rr = z s + l r φ -t r θ , (9)
By substituting the tire force equations ( 2) and the suspension force equations ( 7) into the vehicle equations ( 1), then the following state-space representation with 14 state variables is given by:

ẋ(t) = Ax(t) + B 1 w(t) + B 2 u(t) y(t) = Cx(t) + D 1 w(t) + D 2 u(t) (10)
where

           x = [z s , θ , φ , z us f l , z us f r , z usrl , z usrr , żs , θ , φ , żus f l , żus f r , żusrl , żusrr ] u = [u f l , u f r , u rl , u rr ] w = [z r f l , z r f r , z rrl , z rrr ] y = [z s f l , zs f r , zsrl , zsrr , z us f l , z us f r , z usrl , z usrr ]
are the state, the control input vector, the disturbance inputs, and output measurements vectors respectively. A, B 1 , B 2 ,C, D 1 , D 2 are matrices of the state space representation. Since in the MPC approach, the optimization problem must be described in discrete time domain, then for this purpose, the continuous time model [START_REF] Giorgetti | Hybrid model predictive control application towards optimal semi-active suspension[END_REF] has been discretized with a sampling time T s using for example the zero order hold method. The obtained discrete time model is denoted as follows:

x k+1 = A d x k + B 1d w k + B 2d u k (11) y k = C d x k + D 1d w k + D 2d u k ( 12
)
where A d , B 1d , B 2d ,C d , D 1d , D 2d are matrices of the state space representation.

B. Input constraints

The dissipativity conditions of the semi-active damper given in ( 5) is here transformed into input constraints. Note that from [START_REF] Savaresi | Mixed sky-hook and add: Approaching the filtering limits of a semi-active suspension[END_REF][START_REF] Unger | Application of lq-based semi-active suspension control in a vehicle[END_REF], it follows that:

c min i j żde f i j ≤ F d i j ≤ c max i j żde f i j if żde f i j ≥ 0 (13) c max i j żde f i j ≤ F d i j ≤ c min i j żde f i j if żde f i j < 0
The dissipativity constraint is now recast into:

c min i j żde f i j ≤ c nom i j żde f i j + u i j ≤ c max i j żde f i j if żde f i j ≥ 0 c max i j żde f i j ≤ c nom i j żde f i j + u i j ≤ c min i j żde f i j if żde f i j < 0 Since c nom i j = (c max i j + c min i j ) 2
, we must have:

|u i j | ≤ (c max i j -c min i j ) 2 |ż de f i j | (14)
Or equivalently, for i corresponds to front/rear and j to left/right:

if żde f i j ≥ 0, then    u i j ≥ (c min i j -c max i j ) 2 (ż s i j -żus i j ) u i j ≤ (c max i j -c min i j ) 2 (ż s i j -żus i j ) if żde f i j < 0, then    u i j ≥ (c max i j -c min i j ) 2 (ż s i j -żus i j ) u i j ≤ (c min i j -c max i j ) 2 (ż s i j -żus i j ) (15)
The dissipativity constraint [START_REF] Poussot-Vassal | Attitude and handling improvements through gainscheduled suspensions and brakes control[END_REF] can be expressed as a set of linear inequalites between the control input u i j and the state variables x. Actually, żde f i j = żs i jżus i j is a linear combination of the system state x, i.e żde f i j = żs i jżus i j = C in x, where C in is an appropriate matrix. Thus, in the discrete time domain, the control input u i j k must satisfy the following constraints:

if C in x k ≥ 0,    u i j k ≥ (c min i j -c max i j ) 2 C in x k u i j k ≤ (c max i j -c min i j ) 2 C in x k if C in x k < 0,    u i j k ≥ (c max i j -c min i j ) 2 C in x k u i j k ≤ (c min i j -c max i j ) 2 C in x k (16)
III. SEMI-ACTIVE SUSPENSION CONTROL USING MPC

A. Performance index

The main objective of the suspension in the vehicle system is to isolate the body (comfort performance) from the road disturbances, without deteriorating the road holding (handling performance). Comfort and handling performance can be described through vehicle center of gravity heave acceleration zs and roll angle θ respectively ( [START_REF] Kiencke | Automotive control systems: for engine, driveline, and vehicle[END_REF]). In particular, the following performance indices can be considered:

J com f ort = T 0 z2 s (t)dt (17)
J handling = T 0 θ 2 (t)dt (18)
However, it is a well known fact that ( 17) and ( 18) are conflicting objectives. For this reason, a control law has to be designed in order to optimize the overall performance through a suitable trade-off between ( 17) and ( 18) and taking into account the dissipativity constraint [START_REF] Poussot-Vassal | Attitude and handling improvements through gainscheduled suspensions and brakes control[END_REF]. The semiactive suspension control problem can be formulated as a constrained optimization problem that can be casted in the well known framework of Model Predictive Control (MPC), see e.g [START_REF] Mayne | Constrained model predictive control: Stability and optimality[END_REF].

In this regard, by defining N p as the prediction horizon, the following cost function is chosen as the performance index to be minimized:

J(U, N p , x k|k ) = N p -1 ∑ i=0 (1 -ρ)(z s k+i|k ) 2 + ρ(θ k+i|k ) 2 (19)
where zs k+i|k , θ k+i|k denote the chassis acceleration and roll angle predicted by using the model [START_REF] Canale | Semi-active suspension control using fast model predictive control[END_REF], given the initial state x k|k , and U = [u k|k , u k+1|k , ..., u k+N p -1|k ] is the vector of the control moves to be optimized. ρ ∈ [0 1] is a weighting coefficient which can be tuned to achieve a suitable trade-off between comfort and handling performance. It is worth mentionning that the tuning of the design parameter ρ can be usually obtained through a sequence of trial and error steps. However, in this work, an alternative approach is proposed which consists in considering ρ as the Load Transfer Ratio (LTR) of the vehicle. Actually, LTR can be computed by evaluating of the roll load transfer while the vehicle is running. As soon as there exists a load transfer from the left to the right or vice-versa, it means that the vehicle is faced to roll motion. By defining the vertical forces acting on the left and right sides by F zl and F zr respectively, we have:

F z l = m s g 2 + m s h a y l f F z r = m s g 2 -m s h a y l r (20)
that allows us to introduce the LTR as:

ρ := F z l -F z r F z l + F z r (21)
where a y is the lateral acceleration of the vehicle at center of gravity. Note that the LTR ratio can be evaluated online through the measurement of the lateral acceleration a y . Since ρ ∈ [0 1], when ρ → 0, there is neither lateral load transfer, nor roll motion, i.e the cost function (19) minimizes the chassis acceleration, aiming at improving the comfort performance. On the other hand, when ρ → 1, the vehicle is within a critical situation caused by the roll motion. In this case, the roll motion in (19) needs to be weighted in order to improve the road holding performance.

B. Optimisation problem setup

Following the previous definitions, the optimization problem of the MPC design can be defined as:

min U J(U, N p , x k ) s.t x k+1 = A d x k + B 1d w k + B 2d u k dissipativity constraint (16) (22)
In order to compute the control action in the MPC framework, the cost function J in (19) has to be evaluated along the state trajectory, within the prediction horizon, using the state equation [START_REF] Canale | Semi-active suspension control using fast model predictive control[END_REF]. In this regard, given the available measurements described in section II at each sampling time, the system state has to be estimated using a suitable observer.

Regarding the disturbance contribution, differently from [START_REF] Gohrle | Model predictive control of semi-active and active suspension systems with available road preview[END_REF], it is not assumed that road profile preview measurements can be obtained using a camera. Thus, to be able to account for the disturbance effect during the prediction, an extended state observer is designed, considering exisiting standard sensors, allowing to estimate the road input and the state variables simultaneously. In this way, the overall state equation employed for both prediction and state estimation.

To this aim, a disturbance model is needed. One of the most common assumption in MPC design is that the disturbance is considered to be constant during the prediction horizon, i.e w k+i = w k , i = 0...N p -1. In this way, the following augmented state space model can be considered in the optimization problem [START_REF] Gohrle | Active suspension controller using mpc based on a full-car model with preview information[END_REF]:

       x k+1 w k+1 = A d B 1d 0 I x k w k + B 2d 0 u k y k = C d D 1d x k w k + D 2d 0 u k (23)
The optimzation problem [START_REF] Gohrle | Active suspension controller using mpc based on a full-car model with preview information[END_REF] now becomes: min

U J(U, N p , x k )
subject to (23) and constraint [START_REF] Kiencke | Automotive control systems: for engine, driveline, and vehicle[END_REF] (24)

The MPC control law is then computed by applying the receding horizon strategy, where only the first element of the sequence U is applied as the actual control action u k = u k|k . On the other hand, the dissipativity constraint ( 16) is defined in a way where the control input signal depends on the sign of the predicted state, especially the suspension deflection speed C in x k . Therefore, the switching between the constraints according to the sign of C in x k must be satisfied. To this aim, the optimization problem (24) can be formulated as a quadratic problem involving logic constraints. In this regard, the optimization procedure is a mixed integer quadratic programming (MIQP) problem ( [START_REF] Bemporad | Control of systems integrating logic, dynamics, and constraints[END_REF]). Thanks to YALMIP [19] and using GUROBI optimisation solver [START_REF] Gurobioptimization | Gurobi optimization solver[END_REF], the optimal control law can be computed.

IV. STATE AND ROAD DISTURBANCE ESTIMATION

This section presents the observer design methodology. Within available measurements described in section II, the observability condition of the augmented system (23) is satisfied. Then, both the system state and road disturbances can be estimated by using an extended observer with the following structure:

       xk+1 ŵk+1 = A o xk ŵk + B 2o u k -L(y k -ŷk ) ŷk = C o xk ŵk + D 2d 0 u k ( 25
)
where L is the observer gain to be designed and

A o = A d B 1d 0 I ; B 2o = B 2d 0 ; C o = C d D 1d .
Let us define the estimation error of the augmented system:

e k = x k w k -xk ŵk
Then, the estimation error can be inferred from ( 23) and ( 25) as:

e k+1 = A o x k w k -xk ŵk -LC o x k w k -xk ŵk (26)
Finally, one has:

e k+1 = (A o -LC o )e k (27)
Therefore, the observer design consists in calculating observer gain L that ensures that the matrix A o -LC o is Hurwitz. This can be calculated thanks to the pole placement approach within the poles of the extended observer are chosen around 5 to 10 times faster than the closed-loop system, so that the state estimation is good as early as possible.

First simulation results are here presented to assess the observer performance. In Figure 2, the estimated road profiles are a different standardized roads (ISO 8608). Note that here only the result of the estimated road profile under the front left wheel is shown, but the road profiles at four corners have been estimated correctly.

Remark: Note that, due to the presence of the state observer (25) a suitable robust MPC design method that explicitly takes into account the state estimation error should be adopted. In this regard, a possible solution is made up by the method introduced in [START_REF] Mayne | Robust output feedback model predictive control of constrained linear systems[END_REF], where a constraint tightening approach is adopted to deal with the state uncertainty induced by the observer. However, such an approach leads to a more conservative design procedure that may carry to unfeasibility issues of the optimization problem as well as a slight performance degradation. For this reason, similarly as done in [START_REF] Canale | Semi-active suspension control using fast model predictive control[END_REF] and [START_REF] Gohrle | Active suspension controller using mpc based on a full-car model with preview information[END_REF], the MPC design will be performed without taking explicitly into account the effects of estimation errors, thus exploiting its inherent robustness properties. As it will be seen in section V, the proposed approach is able to provide quite good overall performance while satisfying the dissipativity constraint.

V. SIMULATION RESULTS

To assess the proposed observer-controller strategy, simulations are performed on a full non linear vehicle model [START_REF] Poussot-Vassal | Attitude and handling improvements through gainscheduled suspensions and brakes control[END_REF] with non linear suspension forces, validated on a Renault Mégane Coupé. The simulations are performed with a sampling time T s = 0.005s, and a prediction horizon N p = 10.

The following scenario is used to test the effectiveness of the proposed hybrid MPC controller:

• The vehicle runs at 120km/h in a straight line on dry road ( µ = 1, µ the adherence to the road). • A 5cm bump occurs simultaneously on the left and right wheels (from t = 0.5s to t = 1s) to excite the bounce motion and chassis vibration. • A 5cm bump on the left wheels (from t = 2s to t = 2.5s) causes the roll motion. Firstly, Fig. 3 shows that the road profiles are well estimated using the proposed observer approach. This estimated road profiles are used for MPC design. To demonstrate the efficiency of the proposed approach, we show a comparison results of the following control strategies:

• MPC with road disturbance estimation, called Proposed MPC • MPC without taking into account the road disturbance during the prediction horizon, called MPC without w • MPC with road disturbance preview, called MPC preview w • Uncontrolled passive suspensions (i.e u i j = 0), called Nominal damper Note that for MPC preview w, it is to be assumed the road disturbances are known in advance during the prediction horizon N p . shows that when the bump occurs at t = 0.5s, the suspension is set to be soft to reduce the shock and then, after the bump, the suspension is harder to limit the movement of the vehicle. This allows to enhance the behavior of the vehicle. Now, a deeper analysis is provided. Simulations are carried out using benchmark road profiles employed in standard industrial tests. In particular, the following road profiles (see fig. 9) are taken into account:

• ISO road A (smooth runway), vehicle runs at 130km/h • ISO road D (rough runway), maximum amplitude of 0.015 m and run at 90 km/h • A random road profile for comfort test, run at 60 km/h. The simulation is performed during 14s. To evaluate the effectiveness of each approach, the RMS (root mean square) of the chassis acceleration (z s ) is computed and results are presented in table I: As shown, in the context of semi-active suspensions it seems that MPC road preview does not introduce significant improvements with respect to MPC using road disturbance estimation , moreover, the feedforward action obtained by the MPC disturbance estimation introduces improvements over the case of absence of disturbance compensation (MPC without w). This demonstrates one again the interest of the proposed approach.

VI. CONCLUSION

In this work, a single MIMO state feedback control was designed for the semi-active suspension system of a full vertical vehicle using a MPC strategy. An observer was designed to estimate the system states and the road disturbances. The effects of the disturbances were taken into account in the control design step. The simulation results demonstrate the effectiveness of the presented approach. Thanks to predictive control techniques, multi-objective problems were considered where the control laws were computed to improve the passenger comfort and the road handling, while ensuring dissipativity constraints. For the future works, the implementation of this strategy on a testbed, available at Gipsa-lab Grenoble, will be made. It consists of a vehicle equipped with four controllable Electro-Rheological dampers, and of 4 DC motors generating separately different road profiles on each wheel.

Fig. 1 .

 1 Fig. 1. A full vehicle model

Fig. 2 .

 2 Fig. 2. Road estimation at front left conrner

Fig. 3 .

 3 Fig. 3. Road profiles and their estimations

Fig. 4 Fig. 5 .Fig. 6 .

 456 Fig. 4. Chassis acceleration

Fig. 7 .

 7 Fig. 7. Damper force versus the deflection speed

Fig. 8 .

 8 Fig. 8. Front left suspension force

Fig. 9 .

 9 Fig. 9. Different benchmark road profiles

Dipartimento di Automatica e Informatica, Politecnico di Torino, Corso Duca degli Abruzzi 24 -10129 Torino -Italy.


C Hamouda

B Poussot

M Villegas

J.-M Laheurte

Easily fabricated 60 GHz WR15/MSL transition to measure apparent reflection coefficient of differentially fed patch antennas

A new procedure to measure the apparent reflection coefficient of differential patch antennas at 60 GHz is described. The measurement method makes use of two WR15/microstrip-line (MSL) transitions which are simple to realise. The experimental results are compared with simulated results. This comparison emphasises the limited effect of transition losses on measurement accuracy.

Introduction: To keep the benefits of differential amplifiers in radiofrequency (RF) receivers and transmitters, the antenna feeds should be directly compatible with differential input receiver circuits and/or differential output transmitter circuits [START_REF] Lee | Active integrated antenna design using a contact-less, proximity coupled, differentially fed technique[END_REF]. Compared with single-ended circuits, differential amplifiers have the advantages of high immunity from cross-talk, improved common mode susceptibility, supply noise suppression, higher dynamic range and compression of even-order harmonic distortion [START_REF] Karki | Fully Differential Amplifier[END_REF]. Differential antennas are widely used in the 57-66 GHz unlicenced band [START_REF][END_REF] and intense RF engineering efforts have been made for a decade due to the growing demand for high data rates [START_REF] Wang | An on-board differential bunny-ear antenna design for 60 GHz applications[END_REF][START_REF] Wang | An on-board differential patch array antenna and interconnects design for 60 GHz applications[END_REF][START_REF] Hirano | 60 GHz on-chip dipole antenna with differential feed[END_REF][START_REF] Yang | Performance analysis of a novel LTCC UWB 60 GHz semi-shielded aperture stacked patch antenna with differential feeding[END_REF].

The return loss measurement of most types of published differential antennas (dipole, loop, Yagi, Vivaldi and so on) of about 60 GHz makes use of baluns to avoid differential measurements. However, baluns can either be lossy and prevent a correct de-embedding of the antenna S-parameters [START_REF] Wang | An on-board differential bunny-ear antenna design for 60 GHz applications[END_REF] or create spurious radiations thus altering the actual radiation pattern [START_REF] Wang | An on-board differential patch array antenna and interconnects design for 60 GHz applications[END_REF]. In [START_REF] Yang | Performance analysis of a novel LTCC UWB 60 GHz semi-shielded aperture stacked patch antenna with differential feeding[END_REF], the balun suffered from a ±30°p hase imbalance between 55 and 65 GHz. This phase error resulted in common mode excitation which altered the antenna radiation. This Letter presents a 60 GHz patch antenna with a differential interface which can be connected directly to the amplifier without a dedicated balun. A simple and accurate procedure is proposed to measure the apparent reflection coefficient of differential patch antennas (DPAs). This procedure does not require a probe station as it is based on two WR15/microstrip-line (MSL) transitions designed to excite the DPA ports. The apparent reflection coefficient is extracted from the S-parameters of the dual-port DPA once the thru, reflect and line (TRL) de-embedding method has been applied. In spite of the 1.7 dB loss in each transition, good agreement between measurements and simulation is achieved.

Description of WR15/MSL transition:

The principle of the WR15/ MSL transition was first carried out in [START_REF] Iizuka | Millimeter wave microstrip line to waveguide transition fabricated on a single layer dielectric substrate[END_REF]. The TE10 mode of a WR15 waveguide is transformed into the quasi-TEM mode of the MSL thanks to a parasitic patch resonator located at the waveguide opening (Fig. 1). The parasitic patch is centred on a slot etched in the MSL ground plane. The slot dimensions are identical to those of the waveguide aperture. In [START_REF] Iizuka | Millimeter wave microstrip line to waveguide transition fabricated on a single layer dielectric substrate[END_REF], vias reducing the radiation losses connected the ground plane to the metallic pattern on the upper side of the substrate. The same principle of transition is optimised at 60 GHz with vias replaced by two metallic skews to simplify the designs. The picture of two back-to-back WR/MSL transitions is shown in Fig. 2. The transition is etched on a 254 µm-thick Duroid 5880 substrate (ε r = 2.24 and tan δ = 0.004 at 60 GHz [START_REF] Lafond | Substrates characterisation (ε r , tan δ) up to millimetre-wavelength[END_REF]). To improve the rigidity of the Duroid substrate, a 1.6 mm-thick frame of epoxy substrate is placed on the top of the Duroid substrate. This epoxy frame is screwed through the Duroid substrate to the waveguides. The transition geometry is optimised with the commercial electromagnetic simulator ANSOFT high-frequency structure simulator (HFSS). The S-parameters of the transition estimated by the TRL algorithm are depicted in Fig. 3. The minimum transmission loss is 1.7 dB which is much more than that (0.4 dB at 76.5 GHz) observed in [START_REF] Iizuka | Millimeter wave microstrip line to waveguide transition fabricated on a single layer dielectric substrate[END_REF]. These losses are mainly radiation losses due to the use of screws instead of vias. However, the results are encouraging with regard to the mechanical simplicity of the structure. S 11 and S 22 roughly remain below -10 dB between 62 and 67 GHz. Their values are slightly different because the transition is not fully symmetrical. Measurement of differential reflection coefficient of DPA: When a differential feed network is used to excite a differential antenna, the key parameter at the interface between the circuit and the antenna is the odd mode reflection coefficient. This is the apparent reflection coefficient at port 1, when the excitation at port 2 is constrained to be equal in amplitude but perfectly out of phase with the excitation at port 1. The apparent reflection coefficient Γ odd of a DPA can be determined as follows [START_REF] Lee | Active integrated antenna design using a contact-less, proximity coupled, differentially fed technique[END_REF]:

G odd = S 11 DPA -S 12 DPA (1)
where S 11 DPA and S 12 DPA are extracted from the DPA scattering matrix (S DPA ) defined by applying in-phase excitations at ports 1 and 2. Let us call T T the chain matrix of the transition

T T = T 11 T 12 T 21 T 22 (2)
The relationship between the chain matrix (T DPA ) of the DPA alone and the chain matrix (T DAT ) of the DPA including the WR15/MSL transitions measured previously is given by

T DAT = T 11 T 12 T 21 T 22 • T DPA • T 11 -T 21 -T 12 T 22 (3)
T DPA is calculated by inverting matrix (3). Then, the DPA scattering matrix (S DPA ) is calculated from (T DPA ) and Γ odd is deduced from [START_REF] Lee | Active integrated antenna design using a contact-less, proximity coupled, differentially fed technique[END_REF]. The dimensions of the DPA were adjusted through HFSS simulations to cover the upper part of the unlicensed band. The simulated resonant frequency 63.5 GHz was obtained with good DPA matching (Γ odd < -8 dB) in the bandwidth 62-65 GHz. The DPA was then measured with WR15/ MSL transitions (Fig. 4). In Fig. 5, the simulated Γ odd is compared with the experimental value extracted by the TRL calibration. The reference planes are indicated in Fig. 4. Excellent agreement is obtained with the WR15/MSL transitions both for the resonant frequency and the level predictions.

reference planes

Fig. 1

 1 Fig. 1 WR15/MSL transition showing waveguide aperture, parasitic patch and upper metallic pattern

Fig. 2

 2 Fig. 2 Back-to-back WR15/MSL transitionTransition measurement: The S-parameters of the back-to-back transition were first measured. Then, the TRL method[START_REF] Marks | A multiline method of network analyzer calibration[END_REF] was applied to determine the S-matrix of the single transition between the reference plane indicated in Fig.2and the coaxial connector. The TRL calibration set makes use of two different line lengths. The first length 1 mm ensures a shift phase lower than 90°at the lower frequency. It is used to determine the MSL propagation constant. The second length 20 mm allows an accurate estimation of the MSL attenuation constant. As the transition is lossy, it is necessary to have a much longer MSL length showing high losses to extract meaningful data.The S-parameters of the transition estimated by the TRL algorithm are depicted in Fig.3. The minimum transmission loss is 1.7 dB which is much more than that (0.4 dB at 76.5 GHz) observed in[START_REF] Iizuka | Millimeter wave microstrip line to waveguide transition fabricated on a single layer dielectric substrate[END_REF]. These losses are mainly radiation losses due to the use of screws instead of vias. However, the results are encouraging with regard to the mechanical simplicity of the structure. S 11 and S 22 roughly remain below -10 dB between 62 and 67 GHz. Their values are slightly different because the transition is not fully symmetrical.

Fig. 3

 3 Fig. 3 Scattering parameters of WR15/MSL transition

Fig. 4 Fig. 5

 45 Fig. 4 DPA connected to WR15/MSL transitions