Philip Polack
email: philip.polack@mines-paristech.fr

Sébastien Delprat
email: sebastien.delprat@univ-valenciennes.fr

Brigitte D'andréa-Novel
email: brigitte.dandrea-novel@mines-paristech.fr

Brake and velocity model-free control on an actual vehicle

Keywords: Model-free control, intelligent controllers, longitudinal control, tuning procedure, ALIEN filters

. Several implementation issues of this ALIEN filter, such as the order of the numerical quadrature, are discussed and overcome.

Introduction

Using model-free control can be a risky project since the closed-loop stability can hardly be proven in general. However, it remains an appealing solution when a model is not available due to the system complexity or to its development cost (engineering time and/or required hardware). Proportional-Integral-Derivative (PID) controllers (see [START_REF] Åstrom | PID Controllers: Theory, Design, and Tuning[END_REF]) are among the most popular model-free controller in industrial applications due to their limited number of parameters and quite intuitive tuning for a large class of systems (IFAC Industry Committee, 2015). The model-free control (MFC) approach introduced by [START_REF] Fliess | Model-free control[END_REF] extends the classic PID framework and provides "intelligent"-PID controllers allowing to account for the unknown dynamics of the system. Hence, the considered MFC structure provides a model-free feedforward control that can be used to improve the reference tracking. It is also able to handle some complex nonlinear control problems [START_REF] Fliess | Model-free Control and Intelligent PID Controllers: Towards a Possible Trivialization of Nonlinear Control[END_REF] at a low computational cost as these control laws can be implemented on cheap and small programmable devices [START_REF] Join | Intelligent" controllers on cheap and small programmable devices[END_REF]. Finally, as model-free control does not depend on a nominal model, it can be seen as an implicitely robust control framework. It has been sucessfully applied among others to energy management in build-ings supplied by solar photovoltaic panels [START_REF] Bara | Model-free load control for high penetration of solar photovoltaic generation[END_REF] and to flapping wings [START_REF] Chand | Non-linear model-free control of flapping wing flying robot using iPID[END_REF]. Note that there exists a wide literature about other model-free data-based control techniques, such as model-free adaptive control [START_REF] Hou | Model free adaptive control: theory and applications[END_REF], virtual reference feedback tuning [START_REF] Guardabassi | Virtual reference direct design method: an off-line approach to data-based control system design[END_REF][START_REF] Campi | Virtual reference feedback tuning: a direct method for the design of feedback controllers[END_REF], direct and indirect model reference adaptive control [START_REF] Narendra | Direct and indirect model reference adaptive control[END_REF], iterative feedback tuning [START_REF] Hjalmarsson | Iterative feedback tuning: theory and applications[END_REF], fuzzy control [START_REF] Zimmermann | Fuzzy Control[END_REF] and neuralnetwork based control methods [START_REF] Miller | Neural networks for control[END_REF]. See also [START_REF] Hou | From model-based control to data-driven control: Survey, classification and perspective[END_REF] for a review of existing data-driven control approaches.

In this work, the brake and engine coordinated longitudinal control of a vehicle is addressed. MFC framework is especially relevant in that case since the dynamics of the low-level actuators are complex and unknown. On the one hand, the vehicle propelling is ensured by an internal combustion engine (ICE) controlled by the original engine control unit (ECU) of the vehicle.

On the other hand, the mechanical brakes are actuated using a hydraulic actuator connected to the brake pedal. Therefore, the low-level dynamics is highly complex. Moreover, the brake and throttle dynamics are very different between them.

Existing work on MFC applied to the longitudinal dynamics of a vehicle includes:

(i) simulation results based on actual driver data [START_REF] Menhour | Multivariable decoupled longitudinal and lateral vehicle control: A model-free design[END_REF], [START_REF] Menhour | A new model-free design for vehicle control and its validation through an advanced simulation platform[END_REF], (d'Andréa-Novel et al., 2016) and (d'Andréa-Novel, 2018);

(ii) a combinaison with event-triggered control [START_REF] Wang | Event driven intelligent PID controllers with applications to motion control[END_REF];

(iii) a combination with Stop-and-Go strategies [START_REF] Choi | Model-Free Control of Automotive Engine and Brake control for Stop-And-Go Scenarios[END_REF][START_REF] Teubner | Model-free control techniques for Stop & Go systems[END_REF], (Milanés et al., 2012a).

Reference [START_REF] Join | Etude préliminaire d'une commande sans modèle pour papillon de moteur[END_REF] which applies MFC for controlling the throttle of an engine can also be mentioned. To the best of the author's knowledge, only (Milanés et al., 2012a,b) tested on an actual vehicle the coordinated control of both actuators within a MFC framework. The authors compared MFC, PID and fuzzy control for: (i) controlling the longitudinal lower-level of a Stop-and-Go controller (Milanés et al., 2012a); (ii) for longitudinal control at very low speed (Milanés et al., 2012b). More precisely, they account for the use of two different MFC laws working alternatively: one for the throttle and one for the brake.

In the present work, a cascading structure is suggested. At the high-level, the MFC computes the total force to be applied to the chassis, which is splitted between the two different low-level controllers. At the low-level, another MFC deals with the unknown dynamics of the braking circuit whereas the engine control unit (ECU) remains in charge of the engine torque production.

Compared to (Milanés et al., 2012a), this work relies on ALIEN filters introduced in [START_REF] Fliess | An algebraic framework for linear identification[END_REF] to estimate the vehicle dynamics in order to be robust to noise measurements. Moreover, only the intelligent-Proportional (i-P) controller is investigated and a methodology for the tuning of the control parameters is also proposed.

The purpose of this work is then to discuss an effective way to implement these MFC laws on an actual vehicle. The main contributions are the numerical implementation of the ALIEN filters and the tuning procedure of the model-free controllers. The rest of the paper is then organized as follows: first, in section 2, the theoretical working principle of model-free control is recalled; then, section 3 discusses the practical implementation details mentioned in the previous paragraph. In particular, a simple and efficient tuning procedure that can be applied on an actual system without any prior simulation results is introduced. The application to an internal combustion engine vehicle, shown in Figure 1, is presented in section 4 and experimental results are discussed.

Finally, section 5 concludes this work.

The model-free control paradigm

The ultralocal model

The principle of the model-free control paradigm introduced in [START_REF] Fliess | Model-free control[END_REF] is to replace a complex nonlinear model by an ultralocal model given by (see Appendix A for more details):

y (ν) (t) = F (t) + αu(t) (1)
u is the control input, y is the observed output, ν is the order of the system, F represents both the unmodeled dynamics and the disturbances and α is a constant parameter chosen by the practitioner. In the rest of this work, ν = 1 has been chosen by the practitioner. When there is no ambiguity, the time dependency will be omitted. Furthermore, the sign of α should match the sign of the actual system input gain. In the studied framework, α being a constant, systems with time varying input gain that take values in both R + and R -are not considered.

Estimation of F

At each time step t k = kT s where T s is the sampling time step of the measurements, the dynamic F is estimated from the previous control inputs u applied to the system and outputs y observed. The estimation of F at time t, denoted F , is assumed to remain constant over [t; t + T ]. Therefore, using the model-free control approach requires to have a high measurement frequency compared to the time constant of the system one wants to control.

A direct method for estimating F k at time t k is given by:

F k = ẏk -αu k-1 (2)
However, measurements are usually noisy. Therefore, using a filter is highly recommended. In the rest of the present communication, ALIEN2 filters introduced by [START_REF] Fliess | An algebraic framework for linear identification[END_REF] are used. For a firstorder ultralocal model, one obtains Equation (3), where T denotes the time window of the filter. The computation details can be found in Appendix B.

F k = - 6 T 3 t k t k -T [(T -2τ )y(τ ) + ατ (T -τ )u(τ ))] dτ (3)

Intelligent-PID controllers

The "intelligent" controllers have been introduced within the modelfree control paradigm [START_REF] Fliess | Intelligent PID controllers[END_REF]. For a first-order system, an intelligent-Proportional-Integral (i-PI) or an intelligent-Proportional (i-P) controller given by Equation ( 4) and completed respectively by Equation ( 5)

and Equation ( 6) can be used:

u = - F -ẏr α -K(e) (4)
(i -P I) K(e) = K P e + K I t 0 edτ (5) (i -P ) K(e) = K P e (6)
where e = yy r is the tracking error. K P and K I denote respectively the proportional and the integral gains. Note that the reference y r must be a C 1 signal. This can be enforced, for instance, using a first-order linear low-pass filter on the original reference signal.

Combining Equations ( 1) and (4) for ν = 1 leads to the following equation for the tracking error e:

ė(t) = -αK(e(t)) + (F (t) -F (t)) (7)
Therefore, if the estimation of F is "good", i.e. (F -F ) ≈ 0, "intelligent" controllers ensure the asymptotic stability of the closed-loop system. In particular, an i-P controller is sufficient for α > 0 and K p > 0 as the evolution of the tracking error e is ruled by:

ė(t) = -αK P e(t) (8)
Thus in practice, the core of the i-P controller is the estimation of the unknown signal F . It can be derived, as in (d'Andréa-Novel et al., 2010), using a first order discretization of the output signal (in Equation ( 2)) and its derivative. In that case, the structure of the i-P controller is equivalent to a PI controller. Therefore the benefits of the i-P structure compared to a classic PI comes from a good estimation of the unknown signal F . Hence the discretization of the ALIEN filter needs to be studied carefully. Moreover, the structure allows performing a feedforward control without explicitly needing a model (although it requires a good estimation of the unknown signal F ).

Remark 1. Under the assumption that the ∞-norm of the estimation error ||F -F || ∞ is bounded (which is weaker than the perfect estimation assumption widely encountered in the literature), it is still possible to prove the stability (convergence in a ball centered in 0) as shown in Appendix C.

Applying model-free control on an actual system

In order to apply the model-free control paradigm on an actual system, it is necessary to have a good estimation of the unknown signal F . For that purpose, two main difficulties need to be overcome. First, the numerical implementation of the ALIEN filter is not straightforward but it is crucial if one wants to achieve good performances in particular at steady-state. This will be discussed in section 3.1. Secondly, the performances of model-free control laws are highly influenced by the value of parameter α, in particular during transient phases. Therefore, a tuning procedure is proposed in section 3.2.

Numerical implementation of ALIEN filters

One of the main difficulty for applying the MFC paradigm defined previously on an actual system is to compute numerically the ALIEN filter given by Equation (3). First of all, it is important to provide the actual control u that was applied on the system, and not the one send by the controller as they might differ due to filters or limitations on the actuators.

Theorem 1. In order to have a perfect estimation of F at steady-state, the numerical quadrature to compute Equation ( 3) should be at least of order 2.

Proof. Assume that the system has reached its steady state, denoted by the subscript ∞. In this case, Equation ( 1) becomes:

F ∞ = -αu ∞ (9)
and the expression of the ALIEN filter given by Equation (3) becomes:

F ∞ = - 6 T 3 t t-T P 2 (τ ) dτ ( 10
)
where

P 2 (τ ) = T y ∞ +(-2y ∞ +αT u ∞ )τ -αu ∞ τ 2 is a second order polynomial in τ .
The order of accuracy n of a quadrature rule is defined as the biggest integer value such that the numerical approximation is correct for any polynomial of degree less or equal to n [START_REF] Dahlquist | [END_REF], which concludes the proof. Thus, a classic numerical approximation such as the trapezoidal rule which is of order 1 is not sufficient: in that case the estimation error

F -F is equal to + 1 2 Ts T 2 F (see Appendix D)
. However choosing T >> T s enables to improve the accuracy in this particular case.

Therefore, Equation (3) should be approximated using one of the two Simpson's rules given by Equations ( 11) and ( 12), which are both of order 3:

• Simpson's 1/3 rule (or three-point Newton-Cotes quadrature rule) b a f (x)dx ≈ b -a 6 f (a) + 4f a + b 2 + f (b) (11) • Simpson's 3/8 rule (or four-point Newton-Cotes quadrature rule) b a f (x)dx ≈ b -a 8 f (a) + 3f a + b 3 + 3f 2(a + b) 3 + f (b) (12)
Newton-Cotes formulas generalize the Simpson rule for interpolation with arbitrary degree polynomials. However, in practice, using high-degree polynomials for interpolation can lead to instability. This is referred as Runge's phenomenon [START_REF] Schlömilch | [END_REF]. Therefore, cutting the computation of the integral into subdivisions is recommended.

In the rest of this work, the time window T = 2kT s (with k ∈ N * ) of the ALIEN filter is chosen such that [0, T ] can be subdivided into subsegments where the rule given by Equation ( 11) can be applied.

Tuning procedure

In this subsection, a tuning procedure for the parameter α will be presented as it also plays an important role in limiting the estimation error, in particular during the transient phases.

Let us write the actual output signal dynamics where F (t) and α(t) are two unknown functions of time:

ẏ = F (t) + α(t)u (13)
In that case:

F = F (t) + (α(t) -α)u (14) u = - F -ẏr α -K(e) (15)
where K(e) is the correction on the tracking error. Combining Equation ( 13) with (15) and choosing K(e) = 0, i.e. considering only the feedforward term of the control law, leads to:

ẏ = α(t) α ẏr + F (t) - α(t) α F (16)
Assuming a perfect estimation of F , i.e. F = F , Equation ( 14) and Equation (16) become Equation ( 17). Therefore, in theory, any value of α should work. Nevertheless, if F = F , the solution of Equations ( 1) and ( 4) is independent of the control: the control u cannot be computed anymore.

ẏ = ẏr (17)

However, in practice, the estimation F is subject to some dynamics, and the closed-loop cannot be reduced to Equation ( 17). As the quality of the controller depends mostly on the estimation error (F -F ), the value of α needs to be choosen carefully, as close to the real gain of the system, to avoid big variations of F .

If α → +∞, the control u given by Equation ( 15) is reduced to K(e). In other words, the controller is reduced to a classic proportional one.

If α → 0 + , the control u depends mostly on the estimation F . As F depends on u, the behavior of the closed-loop mostly depends both on the unknown system dynamics and the ALIEN filter dynamics. The resulting dynamics is quite unpredictable and this situation should be avoided.

It should be noticed that when α is close to the input gain F (t), F is almost independent of u and its estimation becomes easier in practice, leading to a smaller estimation error.

Therefore, tuning α becomes quite straightforward for a stable open-loop system: the practitioner should set K(e) = 0 and observe, for different values of α, the reponse of the system to the control law:

u = - F -ẏr α (18)
Starting with an α large enough, the control will be almost zero. Then, decreasing the value of α will increase the control value until the desired closed-loop response given by Equation ( 17) is achieved. If α is decreased too much, the control amplitude will become too high and the system may oscillate. The tuning guidelines of α are summarized in Table 1.

Once α has been chosen correctly, the gains of K(e) have to be tuned. It can be noticed that the feedforward control only ensures a good tracking of the reference derivative. K(e) is used to track the reference and its tuning is similar to the one of classic P and PI controllers. In practice, an i-P controller is usually sufficient. The disturbances are implicitely embedded within the unknown term F in the model given by Equation ( 1) and are therefore estimated by the ALIEN filter. Thus, they are rejected by the control law of Equation ( 18). The correcting term K(e) is mostly used to overcome the error due to initial conditions and estimation errors of F due to the dynamics of the filter.

Application to the longitudinal control of an actual vehicle: experimental results

The longitudinal control of a vehicle is used as an illustration of the proposed method. The structure of the vehicle prototype used for the experiment is depicted in Figure 2. All the sensors and actuators are connected to the CAN bus and work with a fixed sampling period of 10ms. The vehicle velocity is provided by the original sensor of the vehicle located on the gearbox output shaft. The propelling is ensured by the torque produced by the engine which is controlled by the Engine Control Unit (ECU). The ICE torque, which dynamics is unknown, is transmitted to the wheels through the clutch and the gearbox which are controlled by the Gearbox Control Unit (GCU). This unit is independent and operates using its own unknown rules. The braking circuit comprises the brake pedal, the master cylinder, the Electronic Stability Program (ESP) block and the mechanical brakes.

The brake pedal is actuated using an hydraulic cylinder. Two control inputs are available to control the vehicle velocity: u b that controls the hydraulic cylinder pressures of the brakes and u ice the ICE torque setpoint.

Therefore, the considered control structure, see Figure 3, comprises two parts. First, a high-level velocity controller computes the total force u v to be generated by both the brakes and the engine according to the actual vehicle velocity y v and the reference velocity y v r and acceleration ẏv r . According to the braking capability of the ICE (estimated by the ECU), this high-level control signal is then split into two signals: u ice , the ICE torque setpoint, which is sent to the original vehicle ECU, and y r b , the brake pedal position setpoint, which is sent to the brake model-free controller only when ICE- based braking is not enough to reach the total force u v . A static map is used to approximately convert the braking force into brake pedal position.

In practice, the actual braking force differs significantly from the static map data due to thermal effect, unknown dynamics of the master cylinder and hydraulic braking system.

Brake control

The brake system consists of an hydraulic actuator that pushes or pulls the brake pedal. The position of the latter determines the pressure generated by the master cylinder on the brake disk. The master cylinder and the brake pedal are connected together through a pushrod subject to some stiction which makes the system very difficult to model. Also, the brake pedal dynamics is non-linear due to the master cylinder's resistive force.

The objective is to control the brake pedal position y b by opening the valves (u b ∈ [-80; 80]%) located respectively at the top and at the bottom of the hydraulic actuator. The system is described in Figure 4. The main difficulties to control such a system are: (i) the possible occurence of limit cycles due to pushrod stiction; (ii) reduced stability margin due to communication delays;

(iii) variations of the system gain due to the return spring and the internal pressure of the master cylinder; (iv) unpredictable variations of the system gain over time due to the pressure of the tank and the pump. Note that the system is subject to a known 20ms communication delay. The so-called "position" signal is generated by a control unit on the CAN bus. When the brake pedal is fully released, the "position" signal is 700; when it is fully pressed, the signal is 1600.

The open-loop response of the system to different input steps is depicted in Figure 5 and illustrates the non-linear behavior of the system. The cumulated effect of stiction and communication delays induces a response "delay" between 60ms and 180ms. The MFC law was applied using the tuning method suggested in sec-3.2 while the vehicle is at standstill with the engine turned on. The overall tuning takes only a few minutes. First, the proportional gain K P is set to 0. The system being at the rest position, a reference signal is generated and the open-loop responses are recorded for different values of α (see Figure 6), starting from α = 10000 and decreasing its value until the output and the control signal starts to oscillate (at α = 200, Figure 7). Finally, the value of α is obtained : α = 800. Then, the proportional gain K P is increased such that the dynamics of the output error is acceptable, see Figure 8. The final tuning of the i-P controller is then α = 800, K P = 2 × 10 -3 . The obtained i-P controller is then compared to a PI controller using the same reference signal. The PI controller tuning is K P = 5 × 10 -2 and K I = 5 × 10 -3 . The reference position of the brake pedal is shown in red in Figure 9. The i-P controller performance is sligtly better than the PI one: it has been observed that it provides slightly lower steady-state error.

On a practical point of view, i-P controller has a simpler tuning procedure than the PI one. The α parameter is obtained in a systematic way. Then the K P parameter is tuned empirically whereas for the PI, both parameters needs to be tuned at the same time.

In the rest of this work, the brake system will be controlled the i-P controller with the parameter values obtained in this section.

Longitudinal speed control of the vehicle

In order to control the longitudinal speed of the vehicle, a model-free control approach is also used. The objective is to control the longitudinal speed of the vehicle y v by computing u v , the total wheel force requested. As depicted in Figure 3, the actual total wheel force applied to the vehicle is subject to the dynamics of the low-level controllers. Although the velocity control is not highly challenging, it is still subject to the following difficulties:

(i) at low speed, for small u ice , as the GCU controls the vehicle velocity at 10km/h using the clutch, the system is not controllable; (ii) during propelling, the system becomes non controllable when the clutch is open during a gear shift sequence; (iii) the low-level ICE torque is poorly controlled by the ECU and causes thus disturbances on the control signal.

Tests were performed on the track presented in Figure 10. The track altitude profile is depicted in Figure 11. The disturbance caused by the slope at the beginning of the track forces to use the brakes in order to maintain the vehicle at standstill.

In order to test the control law, the reference speed is composed of suc- cessive steps of ±10m/s triggered every 10s. This raw reference signal is then smoothen using a second-order filter with time constant 0.4s to obtain a C 1 speed reference, denoted y r and shown in red in Figures 12 to 15.

Tuning of the control parameters

The procedure suggested in section 3.2 has been applied for the tuning of α . Throughout this subsection, the time window of the ALIEN filter T is chosen equal to 0.1s which is a good tradeoff between noise filtering and the filter bandwidth. The results obtained for different values of α using only the feedforward control law (K P = 0) are shown in Figure 12. For a better clarity, the absolute value of the tracking error e is displayed in Figure 13.

For large α values such as α ∈ [START_REF] Schlömilch | [END_REF]1000], the control amplitude is too low. In such situations, the gearbox control unit hands over the clutch and the engine to maintain the vehicle speed at approximatively 12km h -1 . This feature is standard for vehicle equipped with an automatic transmission.

The reference tracking is improved by decreasing α. Acceptable values are between 0.005 and 0.01. As expected, when α is too low (α = 1 × 10 -3 ), the closed loop becomes unstable and the experiment is stopped earlier for safety reasons. At this stage of the tuning procedure, α = 0.008 is chosen. The second part of the tuning procedure consists in choosing K P such that the system output smoothly tracks the reference signal. If a perfect estimation for F is assumed, then the closed-loop pole lay at -αK P (see Equation ( 8)). The closed-loop response should be perfectly damped. In practice, as shown in Figure 14, it is not the case as some overshoots for

Comparison with a PI controller

Finally, the i-P controller is compared with a PI controller which is one of the most widespread controller. The main objective of this study being to propose a manual tuning of the controller, a similar approach has been used for the PI controller. Therefore, PID tuning methods have not been used even though there exists some model-free methods in the literature such as Iterative Feedback Tuning [START_REF] Hjalmarsson | Iterative feedback tuning: theory and applications[END_REF], Direct Model

Reference Adaptive Control [START_REF] Pirabakaran | Automatic tuning of pid controllers using model reference adaptive control techniques[END_REF] and Virtual

Reference Feedback Tuning [START_REF] Formentin | Virtual reference feedback tuning for industrial pid controllers[END_REF]. The PI controller has been hand-tuned to obtain a fast closed loop response without too much overshoots over the chosen setpoint profile: the ICE engine should only be used to manage the accelerations whereas the deceleration phases could be manage using only the ICE engine brake and the mechanical brakes. The results are shown in Figure 15. First, between t ∈ [0, 10]s, both the i-P and the PI controllers are able to reject the disturbance due to the initial slope.

Then, the i-P controller provides a faster tracking of the reference than the PI. This is due to the feedforward term ẏr included in the control law, see Equation ( 4). The PI controllers exhibits a 26% overshoots on the first step at t = 15.2s. The i-P controller overshoot is only 7.8% at t = 14.0s. Due to the system non-linearities, the other overshoots are smaller for both controllers.

The overshoots of the PI controller can be reduced at the price of a slower tracking. Finally, the overall controller performances can be measured using the root mean square (RMS) of the tracking error. Data are given in Table 2.

The i-P performs 36% better than the PI.

According to the authors' experience, the tuning of the PI controller was less intuitive than the i-P and did require more time and experimental trials.

The system dynamics is strongly affected by the engaged gear. As the chosen PI controller has fixed gain, the final tuning has been chosen to average the performances over the gears 1 to 3 used in this experiment. Better tuning can be obtained on individual gears at the price of worse performance on the other gears. On the contrary, the tuning of the i-P controller was relatively straightforward with the procedure proposed in section 3.2. Moreover, it is less sensible to gear changes since the system dynamics is implicitly captured through the estimation of the unknown F .

During acceleration phases, gear shifts occur as shown in Figure 15. They are engaged by the GCU whose control law is unknown. During this period, the clutch is opened. Therefore, the system is not controllable as the engine torque generated is not transmitted to the wheels. During deceleration phases, gear shifts happen too but do not impact the controllability of the system as the brakes can still be applied.

Conclusion and perspectives

This portant implementation details such as the required order for the numerical quadrature used to compute the ALIEN filters and the of the control law parameters are presented. While the order of the numerical quadrature used to implement the ALIEN filter impacts the steady-state error, the tuning of the α parameter affects the transient behavior. Consequently, this paper provides general guidelines for an efficient implementation of modelfree controllers. It has to be pointed out that this tuning procedure can be performed directly on the actual system without any prior simulations.

Nevertheless, such a procedure should be followed with sufficient care as the order of magnitude of α and K P are unknown in practice.

The MFC laws were used to control both the brake system and the longitudinal speed of an actual vehicle. The results illustrated the ability of this approach to control a system with unknown or complex dynamics. Due Starting from Equation ( 1) with ν = 1, assuming that F is a piecewise constant function, apply the Laplace transform:

sY (s) = F s s + αU (s) + y(0) (B.1)
Then, get rid of the initial condition y(0) by multiplying by d ds on both sides. In order to obtain an equation only composed of terms in the form of Equation (B.2) and Equation (B.3) for which the inverse Laplace transforms are known, multiply both side by s -2 . This leads to Equation (B.4). Set A = (1 -T s αK P ), B = T s and f k = (F k -F k ). Iterating from t = 0, one obtains (see [START_REF] Kermani | Predictive energy management for hybrid vehicle[END_REF]):

c s α , α ≥ 1, c ∈ C ⇒ c t α-1 (α -1)! , t > 0 (B.2) 1 s α d n Y ds n ⇒ (-1) n (α -1)! T 0 (T -τ ) α-
e k = A k e 0 + k-1 i=0 A i Bf k-1-i (C.2)
Assume that the ∞-norm of the estimation error is bounded. In this case,

- 6T s T 3 N -2 k=0 (T -2(k + 1 2 )T s ) Y k + Y k+1 T s - 6T s T 3 N -2 k=0 (k + 1 2 )T s (T -(k + 1 2 )T s ) α k+1 U k+1 + α k U k T s
At steady-state, ẏ = 0 and u is constant, thus F exact = -αu. The analog expression of the ALIEN filter given by Equation (3) leads to:

F f ilter = - 6 T 3 T 0 [(T -2τ )y + τ (T -τ )αu)] dτ = -αu (D.2)
However, the numerical quadrature of the integral using the trapezoidal rule leads to an estimation error F -F at steady-state. More precisely:

F trap = -αu - 1 2 T s T 2 αu (D.3)

Figure 1 :

 1 Figure 1: The experimental vehicle.

1 :

 1 run the experiment; while the average value of | ẏẏr | decreases, decrease α and repeat rule n • 1; Rule n • 2: if the system output oscillates around the setpoint, increase α; Feedback Tuning Rule n • 1: increase K p progressively until (i) tracking error | yy r | is low and (ii) disturbances are rejected.

Figure 2 :

 2 Figure 2: Structure of the vehicle powertrain to be controlled.

Figure 3 :

 3 Figure 3: Considered cascading control structure.

Figure 4 :Figure 5 :

 45 Figure 4: Description of the brake system: the brake pedal (1) pushes the master cylinder (4) through the pushrod (2); the pressure of the fluid increases and activates the brakes on the brake disk (5). (3) is the fluid reservoir. The hydraulic actuator is used to control the brake pedal position.

Figure 6 :

 6 Figure 6: Tuning of parameter α for the brake system.

Figure 7 :Figure 8 :

 78 Figure 7: Feedforward response for α = 200 of the brake system.

Figure 9 :

 9 Figure 9: Comparison between PI and i-P controllers for the brake system

Figure 10 :Figure 11 :

 1011 Figure 10: Top view of the open road test field track.

Figure 12 :

 12 Figure 12: Tuning of the α parameter for the high-level controller.

Figure 13 :Figure 14 :

 1314 Figure 13: Absolute value of the tracking error e = y -y r for different values of α (K P = 0)for the high-level controller.

Figure 15 :

 15 Figure 15: Comparison of the speed profile between the i-P and PI controllers.

  Stability proof for bounded estimation errorsSet x k = x(t k ) = x(kT s ) where x is a time-dependent variable and consider a discrete version of Equation (7) with K(e) = K P e (i-P controller).The derivative of the error at time k, denoted ėk , is approximated by a forward Euler scheme: ėk = (e k+1e k )/T s . The following equation on the error is thus obtained:e k+1 = (1 -T s αK P )e k + T s (F k -F k ) (C.1)

∀

  ∀k ∈ N, |f k | ≤ f max = |F -F | max . Thus, Equation (C.2) becomes:|e k | ≤ |A k e 0 | + k-1 i=0 |A| i |B|f max = |A k e 0 | + (1 -|A|) -1 (1 -|A| k )|B|f max (C.3)If K P and α are well chosen (i.e. 2 Ts > αK P > 0), for sufficiently large k, |A k | ≈ 0, thus for any δ > 0 sufficiently small (see Proof below for further details):|e k | ≤ (1 + δ) 1 αK P |F -F | max (C.4)Therefore, the system is stable as the tracking error e k remains in a ball B 0, fmax αK P of radius fmax αK P centered in 0.Proof. Let δ > 0. Choosing K P and α such that |A| < 1, i.e. 2 Ts ≥ αK P ≥ 0, leads to|A| k -→ > 0, ∃K ∈ N * , ∀k ∈ N * , k ≥ K ⇒ |A| k < (C.5)Hence, Equation (C.3) leads to:∀ > 0, ∃K ∈ N * , ∀k ≥ K, |e k | < |e 0 | + (1 is defined as f max = 0leads to the presented results. Thus:∀δ > 0, ∃K ∈ N * , ∀k ∈ N * , k ≥ K, |e k | ≤ (1 + δ) 1 αK P f max (C.7) Appendix D. Numerical quadrature of ALIEN filters with trapezoidal rule Equation (3) is an analog equation which needs to be approximated numerically. A common approach used to discretize an integral is the trapezoidal rule. Let's define N = T /T s + 1. Set Y k = y(t k ) and U k = u(t k ) with t k = kT s , for k ∈ 0; N -1 . The trapezoidal rule applied to Equation (-2τ )y(τ ) + ατ (Tτ )u(τ ))] dτ ≈ (D.1)

Table 1 :

 1 Model-free controller tuning guidelines

Table 2 :

 2 Comparison of the root mean square of the tracking error between PI and i-P controllers.

ALgebra for Numerical Identification and Estimation

Acknowledgement

The authors would like to thank the engineering school ENSIAME for enabling them to test algorithms on its autonomous vehicle. The present research was supported by the international Chair MINES ParisTech -Peugeot-Citroen -Safran -Valeo on ground vehicle automation, the Regional Delegation for Research and Technology, the French Ministry of Higher Education and Research, and the National Center for Scientific Research. It was also supported by the ELSAT2020 project, co-funded by the European Regional Development Fund, the French state and the Hauts de France Region Council. The authors greatly acknowledge these institutions for their support.

to the model-free feedforward action, the tuning becomes easier than for a classic PID.

Future works include a generalization of the tuning method to systems of any order (ν > 1) and the development of a procedure to estimate the order ν when almost no information about the system is available. Moreover, more challenging applications such as the lateral control of an actual vehicle using model-free control are thought.

Appendix A. Deriving the ultralocal model

Consider a system where y is the output and u the input. It can be written as an input-output differential equation in the form of Equation (A.1) where E is a function not necessarily linear but assumed to be sufficiently smooth. E(t, y, ẏ, ..., y (m) , u, u, ..., u (n) ) = 0 (A.1)

If there exist ν ∈]0; m[ such that ∂E ∂y (ν) = 0, the implicit function theorem yields then locally to: 1) , y (ν+1) , ..., y (m) , u, u, ..., u (n) ) (A.2)

In practice, ν is always equal to 1 or 2.

Appendix B. ALIEN filters

ALIEN filters were introduced in [START_REF] Fliess | An algebraic framework for linear identification[END_REF]. They are based on operational calculus (see [START_REF] Yoshida | Operational Calculus, A Theory of Hyperfunction 1[END_REF]). In order to obtain Equation (3), the following operations were made.


Masataka Sawayama

Yoann Lemesle

Pierre-Yves Oudeyer

WATCHING ARTIFICIAL INTELLIGENCE THROUGH THE LENS OF COGNITIVE SCIENCE METHODOLOGIES

. These remarkable abilities enable us to read out someone's subtle feelings and facilitate social communications in our daily life. However, they can get in the way when you interpret the behavior of an unknown system, e.g., when you are an AI researcher and would like to understand what representations are acquired in a machine learning model. You may unconsciously interpret a machine's behavior as human-like from the similarity to daily human behaviors. But, it may be an overestimation due to our automatic prediction. We have to keep in mind that human interpretations tend to include a lot of expectations.

Recent machine learning models are trained using big data with massive parameters and achieve incredible performances for diverse tasks. Interpreting what representations are learned in these models is challenging due to their computational complexity. Even when you have a metric to evaluate system performance, quite different systems might output similar results for the metric. For example, in her ICLR 2022 keynote, Dr. Been Kim explained a clear example of the difficulty in interpreting machine learning models (the talk is available from the link: https://www.youtube.com/ watch?v=Ub45cGEcTB0&t=6s) [START_REF] Adebayo | Sanity checks for saliency maps[END_REF][START_REF] Kim | Beyond interpretability: Developing a language to shape our relationships with ai[END_REF]. The middle and right panels of Figure1B show saliency maps, which are metrics to explain which pixels machine learning models focus on when achieving an image classification task. Both saliency maps look similar and detect the main object of the photo, i.e., the bird. However, the right panel is actually the saliency map from an untrained network, while the middle one is from a trained network. This observation suggests that even when you see a visually meaningful output from a metric, you have to be cautious about interpreting the system.

Figure 1: (A) "Summer" by Giuseppe Arcimboldo (B) Saliency maps by trained and untrained networks (figure credit: Been Kim, 2022, Beyond interpretability: developing a language to shape our relationships with AI [START_REF] Kim | Beyond interpretability: Developing a language to shape our relationships with ai[END_REF], https://medium.com/@beenkim/beyond-interpretability-4bf03bbd9394) Considering the difficulty in interpreting systems, how can we understand the learned representations of machine learning models? We think the literature on perceptual and cognitive sciences in humans and animals offers clues to this problem. No magical methodologies exist in these research fields to answer whether a machine learning model acquires human-like intelligence. Even so, these fields are sensitive to interpretation bias and have focused on trying to avoid over-interpretation or mis-interpretation. We believe these methodologies in the perceptual and cognitive science litterature could also help us understand the mechanisms, properties and functions of machine learning models.

Our recent paper at ICLR 2022 [START_REF] Lemesle | Language-biased image classification: Evaluation based on semantic representations[END_REF] follows this approach, aiming to evaluate machine learning models through the lens of cognitive science methodologies. Specifically, we conduct cognitive science experiments for machine learning models [START_REF] Lupker | The semantic nature of response competition in the picture-word interference task[END_REF][START_REF] Rosinski | Picture-word interference is semantically based[END_REF] and analyze the models' internal representation using neuroscience methodologies [START_REF] Kriegeskorte | Representational similarity analysis-connecting the branches of systems neuroscience[END_REF]. By applying this approach to a recent machine learning model, CLIP [START_REF] Radford | Learning transferable visual models from natural language supervision[END_REF], jointly trained with texts and images, we find that the vision and language do not appropriately share semantic representations in the model. This finding is unexpected, given that the model has been successful in many recent language-vision tasks [START_REF] Bau | Paint by word[END_REF][START_REF] Patashnik | Styleclip: Text-driven manipulation of stylegan imagery[END_REF]. Researchers may achieve to develop machine learning models with remarkably higher performances for some distributions of tasks in the future, but they also have to consider how to interpret them deeply.

A key feature of recent advances in machine learning models is to utilize a large size of language models, and many researchers attempt to apply the models to various tasks. In this blog post, we first introduce how language processing contributes to acquiring various abilities in humans and machine learning models. Then, we explain a phenomenon, picture-word interference, observed both in humans and models. We argue how we evaluate the mechanisms underlying picture-word interference in models in our ICLR paper. By importing cognitive science methodologies, we evaluate models in a controlled way. We suggest that such a strategy prevents us from overestimating artificial intelligence due to partial and surface similarity between humans and models.

Language as a tool to shape perception and cognition

Humans use language not only as a tool to communicate with each other but also to shape fundamental aspects of perception and cognition. For example, color categorization emerges even before infants' language acquisition [START_REF] Yang | Cortical response to categorical color perception in infants investigated by near-infrared spectroscopy[END_REF], but language acquisition reorganizes these color categorical representations [START_REF] Franklin | Lateralization of categorical perception of color changes with color term acquisition[END_REF]. Humans also share common semantic representations for vision and language in some brain regions. For example, [START_REF] Quiroga | Explicit encoding of multimodal percepts by single neurons in the human brain[END_REF] recorded single-cell activities from human patients, implanted with intracranial electrodes for clinical reasons. They show that single neurons in the medial temporal lobe respond selectively to representations of the same individual across the visual portrait and its written name.

Similar to advantages in humans, language has benefits in building autonomous agents capable of acquiring diverse skills in open-ended environments. For the design of Vygotskian autotelic AIs, a novel perspective suggested by [START_REF] Colas | Vygotskian autotelic artificial intelligence: Language and culture internalization for human-like ai[END_REF], agents are immersed into and interact with rich socio-cultural worlds, and then they internalize physical and sociocultural interactions within themselves (for the details, see also the website: https://vygotskian-autotelic-ai.github.io/). They use language as a cognitive tool that mediates stimulus and actions to imagine their own goals, and for planning, reasoning, and learning about them. For example, in the IMAGINE approach [START_REF] Colas | Language as a cognitive tool to imagine goals in curiosity driven exploration[END_REF], a Vygotskian autotelic AI approach, agents receive language descriptions from a social partner about their behaviors, internalize these descriptions, and create new goals they have never experienced by leveraging the compositionality of the internalized language information.

In addition, many joint learning models of language and vision have been reported recently. One prominent example is the CLIP (Contrastive Language-Image Pre-training) model reported by the OpenAI team [START_REF] Radford | Learning transferable visual models from natural language supervision[END_REF], consisting of joint learning language and vision, which we evaluated in our paper [START_REF] Lemesle | Language-biased image classification: Evaluation based on semantic representations[END_REF]. It efficiently learns visual concepts from natural language supervision and can be applied to various visual tasks in a zero-shot manner.

The CLIP architecture consists of image and text encoders (Figure 2). These encoders are trained on a large dataset of image-text pairs with contrastive objectives, where they learn to align text and image representations for each pair (Figure 2, left). The pre-trained CLIP model can be used for various visual tasks. When applying the model for an image classification task (Figure 2, right), one uses the pre-trained text encoder to create a set of the embedding representations for classification labels by combining each label with a text prompt like "a photo of [label]"(Figure 2, [2]). For instance, for the labels "plane," "car," and "dog," the text encoder outputs each embedding representation from the prompt "a photo of plane," "a photo of car," or "a photo of dog." The pre-trained image encoder output the embedding representation for the input image (Figure 2,[3]). By computing the similarity of the image representation and each text label representation and selecting the highest similarity pair (e.g., the similarity between the image dog and text "a photo of dog"), the CLIP model can solve the image classification task in a zero-shot manner. Many applications of the CLIP model exist while combining the pre-trained model with other networks [START_REF] Bau | Paint by word[END_REF][START_REF] Patashnik | Styleclip: Text-driven manipulation of stylegan imagery[END_REF].

Picture word interference in humans and machines

While language contributes to acquiring general visual abilities, its abstraction can produce biased recognition for humans. Picture word interference is such a phenomenon. When humans observe an image coupled with a word having a different meaning from the image, the image categorization is disturbed by the word (Figure 3) [START_REF] Lupker | The semantic nature of response competition in the picture-word interference task[END_REF][START_REF] Rosinski | Picture-word interference is semantically based[END_REF]. For example, when a human observes a "dog" image with the written word "cat," the written word "cat" interferes with the reaction time to answer the image category "dog" and increases the error rate.

In particular, the interference effect is strong when the image category is similar to the superimposed word. The finding suggests that shared semantic representations of images and words disrupt image categorization. It has been considered that multiple mechanisms mediate this effect in humans. First, when a participant observes a word-superimposed image, an activation process synthesizes semantic representations corresponding to the superimposed word and picture. This semantic representation is shared for pictures and superimposed words, so when the word is semantically similar to the picture, these activations are similar to each other. Based on the representations, a selection process decides which possible activation is the answer for the current task. Since the activated representation is shared across words and pictures, these dual activations confuse the decision in the selection process. Similar to humans, recent joint models of language and vision also show recognition interference. For example, while the CLIP model can have high image classification performances for natural images in a zero shot manner (Figure 2), it can also be disrupted by superimposing a written word on the image, called a typographic attack (Figure 3) [START_REF] Goh | Multimodal neurons in artificial neural networks[END_REF]. In the case of Figure 3, the CLIP model recognizes the word-superimposed image as the "cat" image category.

Joint multimodal representations are bound to be very useful in many contexts. Even when you obtain noisy information from one modality (e.g., finding parking by car on a rainy day), information from another modality (e.g., the written word 'parking") helps what you see. Picture-word interference is a special case of the functionality showing negative side effects. Therefore, showing similar interference to humans in machine learning models might look like a good sign that they also acquire the generic skill. However, it's unclear what interference in machines means. If picture-word interference in machine models is due to a simple bias toward language information while ignoring visual information, the interference can result only in negative side effects.

Surface performance similarity does not always mean identical underlying mechanisms

Both humans and the CLIP model show picture-word interference. Observing a few similar biases in a picture-word interference task may lead to believing the mechanisms are the same. But, the surface performance similarity by partial observations may be due to largely different underlying mechanisms. The lens of cognitive science methodologies tells us that we should conduct a systematic, structured examination of the biases. Our study investigates this and shows that different functional mechanisms mediate picture-word interference in humans and the CLIP model (Figure 3).

We first focused on the fact that human picture-word interference depends on the semantic relationship between images and written words. Cognitive science works show that the interference effect of written words on the image classification is larger when the image category is semantically similar to the word one (e.g., a written word "cat" for a "dog" image) [START_REF] Rosinski | Picture-word interference is semantically based[END_REF]. We imported this experimental paradigm for machine learning models and created the picture-word image dataset, in which we controlled the semantic categories between images and written words (Figure 4). Our dataset consists of a combination of natural image datasets and hierarchical superordinate/basic word labels. Our benchmark test is a 2 x 2 block design. One condition is the classification task type, indicating what is the image category level to be answered. The category level is superordinate (e.g., "animal," "furniture") or basic ("dog," "cat"). The other condition is the superimposed word category level, superordinate or basic. In total, there are four conditions as follows.

1. The superordinate image classification for the superordinate word embedding (S/S).

2. The basic image classification for the superordinate word embedding (B/S).

3. The superordinate image classification for the basic word embedding (S/B).

The basic image classification for the basic word embedding (B/B).

We evaluated the CLIP model using the benchmark test. We showed that language-biased classification in the CLIP model does not depend on the semantic relationship between images and written words, although superimposing written words on images strongly biased the image classification to the written word category. The finding suggests that even when we observe picture-word interference in the CLIP model, the way to process the semantics of images and written words is different from humans.

To further explore what representations are acquired in the CLIP, we imported a neuroscience methodology tool, Representational Similarity Analysis (RSA) [START_REF] Kriegeskorte | Representational similarity analysis-connecting the branches of systems neuroscience[END_REF], and investigated what is represented in the CLIP image encoder for word-superimposed images. RSA refers to an image-by-image similarity assessment of intermediate representations in a brain or model. By computing the representational similarity of images, one can understand which internal activations are similar. Through the analysis of images and written words, we found that the CLIP image encoder represents the neural representation of written words different from that of visual images (For example, the neural representation of a written dog is different from a visual dog image). Consistent results are also recently reported in another study [START_REF] Materzyńska | Disentangling visual and written concepts in clip[END_REF].

Our study shows that the CLIP model does not have a common representation of language and vision, and the image classification is strongly biased toward written words (Figure 3). In this study, we imported cognitive science methodology to evaluate the models in a controlled manner. This approach prevents us from overestimating artificial intelligence due to partial performance similarity between humans and models.

Analogous to our picture-word interference example, previous works in comparative studies of humans and machines also see that interpretation from partial observations can lead to misunderstanding the underlying mechanisms of machine learning models and that we need to resist human interpretation bias (reviewed by [START_REF] Funke | Five points to check when comparing visual perception in humans and machines[END_REF]. For example, feedforward convolutional neural networks show high accuracies comparable to human performance in an image classification task. Although the classification accuracy is similar between humans and models, [START_REF] Geirhos | Imagenet-trained cnns are biased towards texture; increasing shape bias improves accuracy and robustness[END_REF] show that the models rely on texture features of an image, whereas humans rely on shape features, by conducting psychophysical experiments for humans and models.

In the future, people will expect more from artificial intelligence capabilities. The expectation can cause overestimations when interpreting the functional mechanisms of artificial intelligence. We may achieve to develop artificial intelligence showing remarkably higher performances for some distributions of tasks in the future. However, we cannot judge how generic they are without a strict way of interpreting them. We will need to further explore methodologies about how to interpret them.

Figure 2 :

 2 Figure 2: Overview of the CLIP model. (figure credit: Alec Radford, et al., 2021, Learning Transferable Visual Models From Natural Language Supervision (Radford et al., 2021), https://openai.com/blog/clip/)

Figure 3 :

 3 Figure 3: Picture-word interference in humans and machines. Summary of Yoann Lemesle, et al., (2022); Languagebiased image classification: evaluation based on semantic representations (Lemesle et al., 2022).

Figure 4 :

 4 Figure 4: Overview of our benchmark test.

This blog is posted at https://developmentalsystems.org/watch_ai_through_cogsci