M Ben Abdallah

M Ayadi

F Rotella
email: rotella@enit.fr

M Benrejeb
email: mohamed.benrejeb@enit.rnu.tn

Linear time-varying flatness-based control of Anti-lock Brake System (ABS)

In this case, the pole placement problem was solved recently by Marinescu [START_REF] Marinescu | Output feedback pole placement for linear time-varying systems with application to the control of nonlnear systems[END_REF], who proposes some technical methods for factorization of linear time-varying transfert matrices. These key points lead to solve Bezout equation written in the timevarying framework. In order to overcome these two points in LTV framework, namely the choice of desired poles at the outset and the determination of solution for the Bezout equation, we propose in this paper to extend the flatness-based control strategy developed in [START_REF] Rotella | Polynomial controller design based on flatness[END_REF] to the case of time-varying systems. It will be seen that applying the guideline induced by a flatness based control to a LTV system leads to express it in a natural RST form. This control strategy is be compared to the flatness-based control based on the use of a reduced order observer. The paper is organized as follows: in section II, some background notions about SISO LTV systems and flatness-based control strategy are presented. In section III, a reduced order observer for the state vector are presented. In section IV, the polynomial controller design based on exact observer is proposed. The state vector constituted by the flat output and its derivatives and the designed observer is without dynamics. In section V, the proposed strategy is illustrated on the control of an Antilock Brake System (ABS).

II. BACKGROUND NOTIONS

A. SISO linear time-varying systems

For finite-dimensional, several input-output descriptions have been introduced for LTV systems. Here, a time-varying linear system is described by the following state space model of dimension n:

ẋ(t) = A(t)x(t) + B(t)u(t) y(t) = C(t)x(t) (1)
The matrices A(t), B(t) and C(t) whose coefficients depend on the time are of dimensions (n × n), (n × 1) and (1 × n), respectively. If the system (1) is completely controllable and by applying the algorithm presented in [START_REF] Silverman | Transformation of time-variable systems to canonical (phase-variable) form[END_REF], [START_REF] Silverman | Controllability and observability in time-variable linear systems[END_REF], we obtain the controllable form of (1) given by:

Ż(t) = Ā(t)Z(t) + B(t)u(t) y(t) = C(t)Z(t) (2)

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Abstract-In this paper, a flatness-based control strategy for linear time-varying systems is proposed in order to track a desired trajectory. The flatness-based control is designed by using two observers: a reduced order observer with a constant estimator error gain and an exact observer for designing a polynomial two-degrees-of-freedom controller without resolving Bezout equation in time varying framework. The proposed approach is illustrated with the control of an Anti-lock Brake System (ABS) and led to track a given trajectory for the wheel slip.

Index Terms-Linear time-varying systems, trajectory linearization, flatness, path tracking, exact observer, polynomial controller, reduced order observer.

I. INTRODUCTION

In the control theory, the study of linear time-varying (LTV) systems has been important since this situation is encountered not only when some parameters of the system vary with time, but also when the system to be controlled is nonlinear and the problem is approached by linearizing this system around a desired trajectory which leads to an LTV model. For finite-dimensional and time-invariant linear systems, a well-known control design technique is obtained by polynomial two-degrees-of-freedom controllers [START_REF] Franklin | Digital control of dynamic systems[END_REF], [START_REF] Aström | Computer controlled systems. Theory and design[END_REF], [START_REF] Kučera | Analysis and design of discrete linear control systems[END_REF] which were introduced fifty years ago by Horowitz [START_REF] Horowitz | Synthesis of feedback systems[END_REF]. More details are given in the reference therein and in the following these controllers will be denoted as RST controllers [START_REF] Landau | Identification et commande des processus[END_REF]. Whatever the chosen design method, this powerful method is based on pole placement and presents one drawback: it needs to know where to place all the poles of the closed-loop system at the outset. Following [START_REF] Rotella | Polynomial controller design based on flatness[END_REF], by the use of flatness design control principles, the problem of pole placement which consists in imposing closed loop system dynamics can be related to the tracking problem, to design an RST (two-degree-of-freedom) controller with very natural choices of closed loop poles. In this design, a solution of the Bezout equation is obtained depending on the planned trajectories. The RST design controller problem is not easy to transcribe in the case of LTV systems due to the fact that the coefficients do not commute with the time derivative operator. Besides, the structure of the set of the poles of the closed-loop system is more complex. with:

Ā(t) =        0 1 0 . . . . . . . . . 0 0 1 -ψ0(t) -ψ1(t) • • • -ψn-1(t)        B = 0 • • • 0 1 T , C = γ0 (t) • • • γ0 (t) (3)

B. Short survey on flatness

The flatness property, which was introduced by Fliess et al. in (1992) [START_REF] Fliess | Sur les systèmes non linéaires différentiellement plats[END_REF], for continuous-time nonlinear systems, leads to interesting results for control design. This system property was widely introduced and used in literature [START_REF] Rotella | Commande des systèmes par platitude[END_REF], [START_REF] Sira-Ramirez | Differentially flat systems[END_REF], [START_REF] Lévine | Analysis and Control of Nonlinear Systems, A flatness-based Approach[END_REF], [START_REF] Rudolph | Flatness based control of distributed parameter systems[END_REF]. The existence of a variable called a flat output permits to define all other system variables. Let us consider the nonlinear system described by the following differential equation:

ẋ(t) = f (x(t), u(t)) (4)
where x(t) ∈ ℜ n is the state vector and u(t) ∈ ℜ m is the input vector. Roughly speaking, this system is called differentially flat if there exists a variable z(t) ∈ ℜ m of the form:

z(t) = h(x(t), u(t), u(t), ..., u (r) (t)) (5)
such that the state and the input of the system are given by:

x(t) = A(z(t), ż(t), ..., z (α) (t)) (6)
u(t) = B(z(t), ż(t), ..., z (α+1) (t)) (7)
where α is an integer. The variable z(t) is called the flat output of the system. 1) Implication for the LTV systems: Let us consider the controllable state space equation (2) and let us denote by z i (t) the i-th component of Z(t). The variable z 1 (t), denoted as z(t), can be considered for this system as a flat output. Then, the state vector of the controllable form Z(t) is composed by the flat output and its derivatives.

2) Tracking control and pole placement: For a given planned trajectory of the flat output, z d (t), the control law based on flatness is as follows:

u(t) = z (n) d (t) + n-1 i=0 ki(z (i) d (t) -z (i) (t)) + ψi(t)z (i) (t) (8)
and by introducing the polynomial:

K (p) = p n + n-1 i=0 kip i (9)
where the k i are chosen such that K(p) is a Hurwitz polynomial, the control u(t) can be written as:

u(t) = K(p)z d (t) + n-1 i=0 (ψi(t) -ki)z (i) (t) (10)
By applying this control, the tracking error verifies:

lim t→∞ (z d (t) -z (t)) = 0 (11)
and the closed-loop dynamics are given by the roots of K (p). This strategy differs from the usual pole placement for linear time-varying systems obtained by a time-varying state feedback.

By denoting:

ψ -k =    ψ0(t) -k0 . . . ψn-1(t) -kn-1   
the previous control can be written as:

u(t) = K(p)z d (t) + (ψ -k) T Z(t) (12)
where:

Z (t) = z (t) ż (t) • • • z (n-1) (t) T ( 13
)
is the state vector of the controllable form.

To implement the control (12), the vector Z(t) must be estimated with an observer. In the next sections, two type of observers are considered.

III. REDUCED ORDER OBSERVER

Let us consider the observable form of the state equation (1) given by the following relation:

ẋo (t) =         0 • • • 0 -τ0 (t) 1 . . . . . . -τ1 (t) . . . 0 . . . 0 1 -τn-1 (t)         xo (t) +        σ0 (t) σ1 (t) . . . σn-1 (t)        u (t) y (t) = 0 • • • 0 1 xo (t) (14)
As in [START_REF] Fliess | Sur les systèmes non linéaires différentiellement plats[END_REF], the system output is the last component, a reduced order observer is then used to estimate the state vector x o (t).

Let us group the n-1 first components of x o (t) in χ(t) then 1 :

χ =       0 1 0 . . . . . . 1 0       χ -       τ0 τ1 . . . τn-2       y +       σ0 σ1 . . . σn-2       u, ẏ + τn-1(t)y -σn-1(t)u = 0 • • • 0 1 χ
The observer for this system is then given by:

χ =       0 1 0 . . . . . . 1 0       χ -       τ0 τ1 . . . τn-2       y +       σ0 σ1 . . . σn-2       u +Γ ẏ + τn-1y -σn-1u -0 • • • 0 1 χ
To overcome the output derivation, we are led to propose the following reduced order observer with the introduction of a 1 For space reasons, we dropped the time argument.

new variable

ζ(t) = χ(t) -λ 0 (t) • • • λ n-2 (t)
T y(t):

ζ =       -λ 0 1 -λ 1 . . . . . . 1 -λ n-2       ζ +       σ 0 -σ n-1 λ 0 σ 1 -σ n-1 λ 1 . . . σ n-2 -σ n-1 λ n-2       u +        -τ 0 -λ0 + (τ n-1 -λ n-2 ) λ 0 λ 0 -τ 1 -λ1 + (τ n-1 -λ n-2 ) λ 1 . . . λ n-3 -τ n-2 -λn-2 + (τ n-1 -λ n-2 ) λ n-2        y
As the error dynamics are given by the matrix:

      -λ 0 (t) 1 -λ 1 (t) . . . . . . 1 -λ n-2 (t)      
we choose for all i, λ i (t) as constant parameters to give an asymptotic observer. The observation of x o (t) is then deduced:

xo (t) = ζ(t) + λ 0 (t) • • • λ n-2 (t) T y(t) y(t) (15)
In this solution, the difficulty appears in the choice of the observers' poles in the LTV framework. To overcome this point, an enlightening ideas suggested in [START_REF] Fliess | Sur des pensers nouveaux faisons des vers anciens[END_REF] and applied in [START_REF] Rotella | Polynomial controller design based on flatness[END_REF] can be used. The realization of this controller, using the exact observer, will be the subject of the next part.

IV. EXACT STATE SPACE OBSERVER

Let us consider the model [START_REF] Rotella | Commande des systèmes par platitude[END_REF] where the first component of the state vector Z (t) is the system flat output. By successive derivations of the output plant y(t) until the order (n -1), we get:

Y (t) = O(t)Z(t) + M (t)U (t) (16)
where:

• Y (t) = y(t) • • • y (n-1) (t) T , • U (t) = u(t) • • • u (n-2) (t) T ,
• O(t) is the observability matrix of the pair Ā (t) , C (t) and it is given by:

O(t) = C1 (t) • • • Cn (t) T ( 17
)
such that:

C1 (t) = C(t) Ci (t) = Ċi-1 (t) + Ci-1 (t) Ā(t) for i = 2 to n
• The matrix M (t) has the following expression:

M (t) =           0 0 • • • 0 M 1 . . . . . . . . . . . . . . . 0 0 M n-2 • • • M 1 0 M n-1 M n-1,2 • • • M 1           (18)
with:

-

M 1 (t) = C1 (t) B, -M i (t) = Ṁi-1 (t) + Ci (t) B, for i = 2 to n -1, -M n-1,2 (t) = M n-2 (t) + n-3 i=1 M (n-2-i) i (t) , -M n-1,3 (t) = M n-3 (t) + n-4 i=1 (n -i -2) M (n-3-i) i (t), -etc.
As the pair ( Ā(t), C(t)) is observable, the matrix O(t) is of rank n and the state vector can be written as:

Z(t) = O -1 (t)Y (t) -O -1 (t)M (t)U (t) (19)
Taking into account the state space equation ( 2) and avoiding variable derivations, we get:

Z(t) = p -1 Ā(t)Z(t) + p -1 Bu(t) (20)
where p -1 stands for the integration operator:

p -1 h (t) = t -∞ h (τ )dτ (21)
with h (-∞) = 0. This last hypothesis ensures commutativity between p and p -1 . By rewriting this equation to the order (n-1), the equation [START_REF] Kučera | Analysis and design of discrete linear control systems[END_REF] becomes:

Z(t) = p -1 Ā(t)p -1 ( Ā(t) . . . p -1 ( Ā(t)Z(t)) + p -1 Ā(t) . . . p -1 ( Ā(t) Bp -1 u(t)) + p -1 Ā(t) Bp -1 u(t) + Bp -1 u(t) (22)
If the term Z(t) is replaced in this equation by the one given in [START_REF] Kiencke | Automotive control systems for engine, driveline and vehicle[END_REF], we get:

Z(t) = p -1 Ā(t) . . . p -1 ( Ā(t)O -1 (t)Y (t) -Ā(t)O -1 (t)M (t)U (t)) + p -1 Ā(t) . . . p -1 ( Ā(t) Bp -1 u(t)) + p -1 Ā(t) Bp -1 u(t) + Bp -1 u(t) (23
) To eliminate the terms containing the derivatives of the plant output y(t) in Y (t), we proceed by using successive integrations by parts leading to the following expression of the state vector:

Z(t) = p -n+1 (Θ 1 (t)y(t)) + • • • + p -1 (Θ n-1 (t)y(t)) + (Θ n (t)y(t)) + p -n+1 (∆ 1 (t)u(t)) + • • • + p -1 (∆ n-1 (t)u(t)) + p -1 Ā(t) . . . p -1 ( Ā(t) Bp -1 u(t)) + • • • + p -1 Ā(t) Bp -1 u(t) + Bp -1 u(t) (24)
where

Θ j (t) = θ 1j (t) • • • θ nj (t) T and ∆ j (t) = δ 1j (t) • • • δ nj (t) T . The components θ ij (t) and δ ij (t)
are function of the parameters ψ i (t) and their derivatives. The control law ( 12) can be written in the RST form:

R p -1 , u (t) = K(p)z d (t) -S(p -1 , y(t)) (25)
where:

S(p -1 , y(t)) = (k -ψ) p -n+1 (Θ 1 (t)y(t)) + • • • + Θ n (t)y(t) R(p -1 , u(t)) = u (t) + (k -ψ) p -n+1 (∆ 1 (t)u(t)) + • • • +p -1 (∆ n-1 (t)u(t)) + (k -ψ) p -1 Ā(t) . . . +p -1 ( Ā(t) Bp -1 u(t)) • • • + Bp -1 u(t)
The proposed control design can be seen as an RST controller without resolution of a Bezout identity. Now the design is focused in the choice of the trajectory z d (t) to follow and the traking dynamis with K(p). This regulator-observer permits to the output system to track a desired trajectory without using an observer dynamics then the problem of pole placement, which consists in imposing closed-loop system dynamics, can be related to tracking.

V. APPLICATION TO ANTI-LOCK BRAKING SYSTEM (ABS) IN VEHICLE

As an illustrative example of the proposed strategy, the control of the wheel slip in an Anti-lock Brake System is studied. The considered process is an Anti-lock Brake System (ABS), used to control the slip of each wheel of a vehicle to prevent it from locking such that a high friction is achieved and steerability is maintained. The main objective of this control system is the prevention of wheel-lock while braking and the maintaining of the wheel slip the nearest possible to 0. The problem of wheel slip control is better explained by looking at a quarter car model. A mathematical model of the wheel slip dynamics is given by [START_REF] Freeman | Robust slip control for a single wheel[END_REF], [START_REF] Drakunov | ABS control using optimum search via sliding modes[END_REF]:

λ (t) = -1 v(t) 1 m (1 -λ (t)) + r 2 J F (λ) + 1 v(t) r J T (t) v (t) = -1 m F (λ) (26)
where: ω (t) angular speed of the wheel

(rad/s) v (t) horizontal speed (m/s) T (t)
brake-acceleration torque (N.m) m mass of the quarter car (450 kg) r wheel radius (0.32 m) J wheel inertia (1kg.m 2 ) g acceleration of gravity (9.81 m/s 2 ) and λ (t) is the wheel slip given by:

λ (t) = v (t) -rω (t) v (t) (27)
The input signal T (t) is a brake-acceleration torque applied to the wheel, it is expressed in (N.m), and the output is the vehicle speed v (t). The longitudinal slip λ (t) is defined by the normalized difference between v (t) and the speed of the wheel perimeter ω (t) r. F (λ) is the friction force, which depends on the normal force, steering angle, road surface, tyre

w v F m g z = • F -m v = • T • Fig. 1.
Quarter car forces and torques.

characteristics and velocity of the car. The friction or adhesion coefficient µ (λ) is defined as the ratio of the frictional force acting in the wheel plane F (λ) and the wheel ground contact force F Z :

µ (λ) = F (λ) FZ ( 28
)
The calculation of friction force can be carried out using the Burckhardt method [START_REF] Kiencke | Automotive control systems for engine, driveline and vehicle[END_REF]:

µ (λ) = c1. 1 -e -c 2 .λ(t) -c3λ (t) (29)
The parameters c 1 , c 2 and c 3 are given for various road surfaces.In the case of asphalt and dry road, the friction force is given by:

F (λ) = mg [1.28 × (1 -exp (-24λ (t))) -0.52λ (t)] (30)
To design a control law which maintains the wheel slip the nearest possible to 0, we perform, in the next development, an approximation to a friction force F (λ) by applying the Taylor series for this function with a first-order approximation at λ = 0 to obtain:

F (λ) = aλ (t) (31)
where a = 30.2 × mg. The equation of the system (26) becomes:

λ (t) = - 1 v (t) 1 m (1 -λ (t)) + r 2 J aλ (t) + 1 v (t) r J T (t) v (t) = - a m λ (t) (32)
By analyzing the equation (32), we remark that the input and the output system are function of a finite number of derivatives of the horizontal speed v(t). By denoting z(t) = v(t), we obtain:

T (t) = J r -m a z (t) z (t) -ż (t) 1 + mr 2 J - -m a ż2 (t) λ (t) = - m a ż (t) (33)
Then the vehicle speed is a flat output of the considered nonlinear model. For the considered system, a desired trajectory

(T d (t) , λ d (t) , v d (t)
) is defined and the following variables are given:

δT (t) = T d (t) -T (t), δλ (t) = λ d (t) -λ (t) and δv (t) = v d (t) -v (t).
The drawback of the flatness control is that all system variables are carried out from the flat output trajectory. This situation is critical for the flatness control because the dynamics of the output system can not be well controlled. To deny this critical point, a tyre slip is imposed and all the system variables are designed from this trajectory which is given by:

λ d (t) = 0.04 × (-cos (πt) + 1) if t ∈ [0, 2] (34)
From equation (32), the desired flat output are given by:

z d (t) = t 0 -α m λ d (h) dh = -0.04×α m . -sin(πt) π + t + z d0 (35)
where z d0 = 36.11 m/s is the initial condition for the horizontal speed. The brake-acceleration torque T d (t) is deduced from equation (33). The figures 2 and 3 show the desired trajectories for the input, tyre slip and the flat output of the nonlinear system. The linearized model of (26) around this desired trajectory is given by:

δ λ = -λd v d δv -a 1 -λ d m + r 2 J - aλ d m δλ v d + r Jv d δT δ v = -a δλ m (36)
To design the closed-loop control which allows to track variable reference trajectories, the following state space representation of the system is considered:

ẋ(t) = A(t)x(t) + B(t)δT (t) δλ = C(t)x(t) (37)
with x(t) = δλ δv T is the state vector such that:

A(t) = -a v d 1 m + r 2 J -2λ d m -λd v d -a m 0 B(t) = r Jv d 0 T , C(t) = 0 1 (38)
For the model equation (36), it can be seen that δv is a flat output of the linearized system. The linearization around a reference trajectory leads to a LTV system and its controllability matrix is given by:

K (t) =   r Jv d -r vd Jv 2 d + ar Jv 2 d 1 m + r 2 J - 2λ d m 0 - ar mJv d  
where K (t) has rank 2 because ar 2 mJ 2 v 2 d = 0 ∀ t ≥ 0. Then, the system (37) is completely controllable and following [START_REF] Rotella | Commande des systèmes par platitude[END_REF], the time-varying linearized system (37) is flat. The observability matrix of the pair (A (t) , C (t)) is given by:

O (A(t),C(t)) = 0 1 - a m 0
which has rank 2 ∀ t ≥ 0. The system is then observable and its controllable canonical form is obtained by applying the algorithm presented in section II:

δ Ż(t) = Ā (t) δZ(t) + BδT (t) δλ(t) = CδZ(t) (39)
with Ā (t) = 0 1 -ψ 0 (t) -ψ 1 (t) , B = 0 1 δZ(t) = P (t)x(t) and δZ(t) = δz(t) δ ż(t)

T . The previous control law [START_REF] Silverman | Transformation of time-variable systems to canonical (phase-variable) form[END_REF] can be written as:

T (t) = T d (t) + (k 1 -Ψ 1 (t)) δ ż(t) + (k 0 -Ψ 0 (t)) δz(t)
which leads to:

δT (t) = Λ(t)δZ(t) (40)
with

Λ(t) = (Ψ 0 (t) -k 0 ) (Ψ 1 (t) -k 1 ) .
From equation ( 16), we deduce:

δY (t) = O(t)δZ(t) + M (t)δT (t) (41)
with:

δY (t) = δv(t) δ v(t) T O(t) = C(t) Ċ(t) + C(t) Ā(t) , M (t) = 0 C(t) B
The equation (39) can be written as:

δZ(t) = p -1 Ā(t)δZ(t) + BδT (t) (42)
By replacing the expression of δZ(t), deduced from equation (41), in equation ( 41) in the right side of the equation (42), we get:

δZ(t) = p -1 Ā(t)O -1 (t)δY (t) - p -1 Ā(t)O -1 (t)M (t)δT (t) + Bp -1 δT (t) (43)
with:

Ā(t)O -1 (t) = α1(t) α2(t) α3(t) α4(t) , Ā(t)O -1 (t)M (t) = β1 (t) β2 (t)
By using integration by parts, it leads to the following expression of the state vector:

δZ(t) = α2 (t) α4 (t) δv (t) + p -1 α1 (t) -α2 (t) α3 (t) -α4 (t) δv(t) +p -1 -β1 1 -β2 δT (t) (44
) By rewriting the expression (40), the following form is obtained:

δT (t) = Λ(t) × α2 (t) α4 (t) δv (t) + p -1 α1 (t) -α2 (t) α3 (t) -α4 (t) δv(t) + p -1 -β1 1 -β2 δT (t) (45
) and then:

δT (t) = S p -1 , δv(t) + R p -1 , δT (t) (46)
with:

S p -1 , δv(t) = Λ(t) × α2 (t) α4 (t) δv (t) + p -1 α1 (t) -α2 (t) α3 (t) -α4 (t) δv(t) (47) R p -1 , δT (t) = Λ(t) × p -1 -β1 1 -β2 δT (t) (48)
By applying the previous control strategy to the new state space representation and by considering the tracking model set to be a second order model with a time response of 0.005 s, the simulation results are obtained in figure 2, where the trajectories of the nonlinear Fig. 2. The output v(t) and the slip λ(t) trajectories of the nonlinear system and the traking error for the speed by the use of an exact observer system follow the desired trajectories with a good performance. The observable forme of the state equation ( 38) is given by:

ẋo (t) = 0 -τ0 (t) 1 -τ1 (t) xo (t) + -ra Jmv d 0 δT (t) δv (t) = 0 1 xo (t) (49)
where:

τ0 (t) = a mv 2 d λd v d + 1 + r 2 Jm -2λ d vd τ1 (t) = a mv d 1 -2λ d + mr 2 J (50)
Following section III, the estimated state vector of the observable form is given by:

xo (t
where P (t) is the change of variable from the observable form to the controllable form. With a constant dynamics observer λ0 = 10 and by considering the tracking model set to be a second order model with a time response of 0.005 s, the results are obtained the in figure 3. These results point out the effectiveness of the use of the flatnessbased approach for the LTV systems in a path tracking context.

VI. CONCLUSION

In this paper, we have underlined the advantage of the use of a reduced order observer in order to design a flatness-based control for tracking a desired trajectory in the case of LTV systems. This advantage consists in the calculation of the error estimator gain which is found constant. A method with a direct calculation of the state vector which contains the flat output and its derivatives is proposed and leads to a control law which can be seen as an RST controller but without resolution of the Bezout equation. This regulator-observer permits to the output system to track a desired trajectory without using observer dynamics. The control law applied on an Anti-lock Brake System (ABS) gives a high level performances in terms of the tracking of the wheel slip. Traking error for the speed Fig. 3. The output v(t) and the slip λ(t) trajectories of the nonlinear system and the traking error for the speed by the use of a reduced order observer

  0


F Rotella
email: rotella@enit.fr

I Zambettakis
email: izambettakis@iut-tarbes.fr

An automatic control course without the Laplace transform An operational point of view

Keywords: Transfer operator, automatic control course, operational calculus, Laplace transform, Carson transform, Heaviside, Mikusi ński. Mots-clés-Opérateur de transfert, cours d'automatique, calcul opérationnel, transformée de Laplace, transformée de Carson, Heaviside, Mikusi ński

The question is: Is the Laplace transform needed in an Automatic Control course? The answer is : Obviously, not! Based on an operational standpoint the rst parts are about guidelines for a primer in automatic control. Beyond an undergraduate course, the two last parts, a little bit more technical, are devoted on the one hand to get the model of a computer controlled system and on the other hand to relate the operational standpoint to usual tables used in some cases.

Résumé-Devant les inconvénients pédagogiques engendrés par l'utilisation de la transformée de Laplace en automatique le développement de méthodes basées sur une présentation purement opérationnelle permet de focaliser les raisonnements sur l'aspect pratique de cette discipline. À partir d'un point de vue développé par Heaviside, nous passons en revue, dans les premières parties, quelques résultats de base en automatique. Ces résultats sont bien sûr obtenus à l'aide de la notion de transfert ce qui nous permet d'en préciser l'interprétation et les différentes acceptions. Les deux dernières parties, qui ne sont pas nécessaires au premier abord sont consacrées d'une part à la construction du modèle d'un système commandé par calculateur et d'autre part à la liaison que l'on peut établir entre l'approche proposée et l'utilisation des tables. Cet article n'a pas pour objectif de changer l'automatique mais se propose de fournir les moyens d'en changer la présentation actuelle.

I. INTRODUCTION

First level courses in automatic control or basic textbooks of this topic contain rst lessons of Laplace transforms. The Laplace transform approach leads to de ne the transfer function of a system. It is used to get the corresponding response signal of a system with respect to a given input signal. The Laplace transform is also important for the analysis and design of control systems [START_REF] Dorf | Modern Control Systems[END_REF]. This tool appears thus a necessary and unavoidable burden for students participating in automatic control courses. However, the transform has some skeletons-in-the-closet [START_REF] Kailath | Linear Systems[END_REF], [START_REF] Mäkilä | A note on the Laplace transform for initial value problems[END_REF]. In this article, we discuss an alternative approach to the use of Laplace transform in automatic control courses.

Consider a signal x(t); de ned for a positive time t and satisfying some appropriate growth conditions. The Laplace transform of x(t) is

X(s) = Lfx(t)g = Z 1 0 x(t)e st dt; (1)
where s is a complex variable. This de nition can require advanced mathematical machinery [START_REF] Mäkilä | A note on the Laplace transform for initial value problems[END_REF]. This machinery is very demanding, and usually beyond the skills of most undergraduate students. This generates dif culties that lead desertion of students from automatic control classes. Equation [START_REF] Åström | Numerical identi cation of linear dynamic systems from normal operating records[END_REF] assumes that all considerations, diagrams and developments are embedded in a space of transformed signals. Students ask frequently two questions in regards to the usefulness of equation [START_REF] Åström | Numerical identi cation of linear dynamic systems from normal operating records[END_REF]. How can we experimentally exhibit or visualize the transformed signals for example with an oscilloscope? Are there some forbidden signals in automatic control methods? For instance, exp(t 2 ) has no Laplace transform [START_REF] Mikusiński | Sur les fondements du calcul opératoire[END_REF].

Some fundamental theorems of Laplace transform have also provided some misunderstandings about the actual meaning of s. For instance, consider the Laplace transform of a derivative function with the initial condition x(0). Namely, Lf _ x(t)g = sX(s) x(0): When x(0) vanishes the complex variable s is considered as a time derivative operator. However, this can be considered in only the space of transformed signals. Still about this fundamental theorem, the lower limit of integration in equation ( 1) is often replaced by 0 , 0 + , or 1 [START_REF] Dorf | Modern Control Systems[END_REF], [START_REF]The Control handbook[END_REF], Y (s) F (s) Fig. 1. Basic block diagram. The transformed output signal Y (s) is given by F (s)U (s) where U (s) is the transformed input signal. As s is a complex variable the time is a hidden variable while there is no difference in the notation between signals and systems. [START_REF] Ogata | System Dynamics[END_REF] to overcome problems encountered in some particular applications. An attempt to solve this problem and to unify the Laplace formalisms is proposed in [START_REF] Lunberg | Initial conditions, generalized functions, and the Laplace transform[END_REF]. Nevertheless the use of this tool is still an unjusti ed hypothesis. For the 1 case, we are faced with the bilateral Laplace transform, which is questionable also [START_REF] Mäkilä | A note on the Laplace transform for initial value problems[END_REF]. The purpose of this article is to show that the mathematical machinery required by the Laplace transform [START_REF] Schwartz | Théorie des Distributions[END_REF] can be avoided. Moreover, the pedagogic dif culty can be cleared in a natural way.

When the transfer function of a linear system has to be de ned, Laplace transform is applicable. For a single-input single-output system the transfer function is de ned as the quotient of the Laplace transform of the output y(t) to the Laplace transform of the input u(t) with the assumption of zero initial conditions. In other words, the transfer function is de ned by:

F (s) = Lfy(t)g Lfu(t)g = Y (s) U (s) :
Although the name transfer function as a mathematical tool is adequate for s = j!; where j 2 = 1 and ! is the frequency [START_REF] Harris | The frequency response of automatic control systems[END_REF], [START_REF] Bennet | History of Control Engineering[END_REF], this ad-hoc de nition generates some interesting questions. The Laplace transform of signals cannot be obtained in practice, and sometimes we wonder how to determine the transfer function of a system? For instance this question is avoided in identi cation procedures [START_REF] Ljung | Identi cation : Theory for the User[END_REF] where ARMAX models involving recurrence relationships instead of transfer functions are used. In several high quality textbooks for discrete-time systems e.g. [START_REF] Åström | Computer Controlled Systems[END_REF], the complex variable z of the Z-transform [START_REF] Jury | Theory and Application of the z Transform Method[END_REF] and the shift-forward operator q are both used. However, the choice between z and q is not argued in every case. These subtle differences, which are mysterious for students, are useless and can be avoided. In view of our personal experience in teaching automatic control, one desire is to give an experimental meaning of the transfer of a system, irrespective of previous formal de nitions. Rarely, in real occasions students may relate the transfer to the differential operation induced by the system. Indeed the use of the Laplace transform leads to the diagram depicted in Figure 1. This gure describes the relationship between the Laplace transforms of external signals and the transfer of the system. So, the essential meaning is lost. It can be noticed here that we do not use the term transfer function, although we call it transfer as can be seen in the sequel.

As a matter of fact, the use of Laplace transforms is one method among many others [START_REF] Lützen | Heaviside's operationnal calculus and the attempts to rigorize it[END_REF], [START_REF] Kullstram | Heaviside's operational calculus : Oliver's revenge[END_REF] to justify the Heaviside operational calculus (Figure 2). Note that the operational calculus is used to solve differential equations (in most cases linear) rather than automatic control problems. From an historical perspective, the Laplace transform, which has been studied extensively by Deakin [START_REF] Deakin | The development of Laplace transform, 1737-1937 : I[END_REF], [START_REF] Deakin | The ascendancy of the Laplace transform and how it came about[END_REF], was introduced rst in the form as equation ( 1) by Bateman in 1910 to solve the differential equation _

x(t) = x(t) where is a nonzero real number.

In the following, we describe guidelines for starting an automatic control course without using the Laplace transform. The presentation is based on the use of a pure operational point of view that provides an opportunity to link methods developed in automatic control to laboratory applications. The essential proofs based on Laplace transform theorems can be read in standard textbooks of automatic control (e.g. [START_REF] Franklin | Feedback Control of Dynamics Systems[END_REF], [START_REF]The Control handbook[END_REF], [START_REF] Goodwin | Control System Design[END_REF]). Concerning the notation, we consider signals as elements belonging to the set C of integrable real valued functions f = ff (t)g ; supposed to be m times continuously differentiable on [0; 1) except at isolated points where it is assumed that both left limit and right limit exist. As ff (t)g denotes the signal f while f (t) stands for its value at time t we write for two signals a and b in C: for all t 0; a(t) = b(t), or fa(t)g = fb(t)g, or a = b: However, when no confusion is possible the braces or "for all t 0" may be dropped. Nevertheless we do not deal with discrete-time systems except to de ne the transfer of a computer controlled system.

II. TRANSFER OPERATOR

We begin by considering a linearized system around an equilibrium point. We suppose this system can be described by the linear differential equation

y (n) (t) + a n 1 y (n 1) (t) + a n 2 y (n 2) (t) + +a 1 y (1) (t) + a 0 y(t) = b m u (m) (t) + b m 1 u (m 1) (t) + +b 1 u (1) (t) + b 0 u(t); (2)
where y(t) and u(t) stand for the differences of output and input signals with their setpoint values respectively and n and m are two integers. In (2) the coef cients a i and b j are constant parameters.

A. Coding

The aim is to provide a tool with which it is easy to manipulate mathematical expressions, which may be used instead of linear differential equations, to separate input and output variables from the system. Following Heaviside [START_REF] Heaviside | Electromagnetic Theory[END_REF], [START_REF] Petrova | Heaviside and the development of the symbolic calculus[END_REF] or Carson [START_REF] Carson | On a general expansion theorem for the transient oscillations of a connected system[END_REF]

In view of these codings and using the distributivity property for every real numbers and ; [START_REF] Rotella | Du calcul opérationnel à l'opérateur de transfert[END_REF]. Date indicates publication year of a major contribution in operational calculus. Among engineers and mathematicians, Heaviside appears as the focal point. His ideas on the use of the differential operator and on the de nition of the transfer (resistance) operator are the basis of guidelines for an automatic control course without the Laplace transform.

[ p n + p m ]x(t) = x (n) (t) + x (m) (t); equation (2) becomes p n + a n 1 p n 1 + a n 2 p n 2 + + a 1 p + a 0 y(t) = b m p m + b m 1 p m 1 + + b 1 p + b 0 u(t): (4)
-u(t) y(t) F (p) Fig. 3. Operational block diagram. The output signal y(t) is obtained by F (p)u(t) where u(t) is the input signal. Notice that t denotes the time variable and p the derivative operator. The difference between signals and system is due to the use of these notations. The action performed by a system on an input signal is retained. The essential meaning of the transfer F (p) is the linear differential equation that links the output signal and the input signal.

To separate input and output signals from the system we divide equation (4) by the polynomial factor p n + a n 1 p n 1 + +a 0 ; which yields to code the input-output relationship as y(t) = F (p)u(t) with

F (p) = b m p m + b m 1 p m 1 + + b 1 p + b 0 p n + a n 1 p n 1 + a n 2 p n 2 + + a 1 p + a 0 : (5)
We must insist here that y(t) = F (p)u(t) cannot be considered as the solution of the differential equation [START_REF] Åström | Computer Controlled Systems[END_REF]. Indeed, the initial conditions are not known. y(t) is determined with this writing as with the writing (2). In equation [START_REF] Carson | On a general expansion theorem for the transient oscillations of a connected system[END_REF], F (p) stands for the transfer operator. In essence it is the transfer, which represents the operation induced by the system to transform the input signal into the output signal. The operational approach provides an opportunity to relate the transfer operator [START_REF] Carson | On a general expansion theorem for the transient oscillations of a connected system[END_REF] to the differential equation [START_REF] Åström | Computer Controlled Systems[END_REF]. The diagram, depicted in Figure 3 corresponds to an experimental situation.

B. Operational calculus as polynomial calculus

Operational calculus is understood as algebraic methods for solving differential or recurrence equations, speci cally in a linear time-invariant framework. In our point of view solving a differential equation for a given input, is not the aim of automatic control but is a mathematical exercise [START_REF] Mäkilä | A note on the Laplace transform for initial value problems[END_REF]. In automatic control operational calculus means rules for transfer connections or decompositions through polynomial calculus. Thus the coding (4) is useless when we are not allowed to associate transfer operators. From the operational standpoint the connection rules can be demonstrated through the following steps. The transfers of the connected system provide differential equations. The connections and the elimination of intermediate signals lead to a differential equation between the output and input signals. The encoding of this differential equation with p ensures the results. Although we can use this procedure in every case it is suf cient to exemplify with respect to series or parallel connections for two rst-order systems.

Let us consider two linear systems described by y 1 (t) = F 1 (p)u 1 (t) and y 2 (t) = F 2 (p)u 2 (t) where u 1 (t) and u 2 (t) are the input signals, F 1 (p) and F 2 (p) the transfers of the systems, and y 1 (t) and y 2 (t) are the corresponding output signals. In this regard we have

F 1 (p) = b 1 p + b 0 a 1 p + a 0 and F 2 (p) = 1 p + 0 1 p + 0 ;
where a 0 ; a 1 ; b 0 ; b 1 ; 0 ; 1 ; 0 , and 1 are constant parameters. The series connection is de ned as u 2 (t) = y 1 (t); u(t) = u 1 (t); and y(t) = y 2 (t): The application of the procedure yields

y(t) = 1 b 1 p 2 + ( 0 b 1 + 1 b 0 )p + 0 b 0 1 a 1 p 2 + ( 0 a 1 + 1 a 0 )p + 0 a 0 u(t);
where we recognize the product F 1 (p)F 2 (p): The parallel connection is de ned as u 2 (t) = u 1 (t) = u(t) and y(t) = y 1 (t) + y 2 (t); which yields to

y(t) = 1 1 a 1 p 2 + ( 0 a 1 + 1 a 0 )p + 0 a 0 ((a 1 1 + 1 b 1 )p 2 + (a 1 0 + b 1 0 +a 0 1 + b 0 1 )p + (a 0 0 + b 0 0 ))] u(t);
where we recognize the sum F 1 (p) + F 2 (p): These results can be extended to any order by induction. It can be seen that the transfer of the series connection of two systems is the product of their transfer and the transfer of the parallel connection of two systems is the sum.

The parallel and series rules give a meaning to the decompositions and the handling of transfer operators with polynomial calculus. These operations on transfer operators are the basis of operational calculus in automatic control. We can apply usual techniques as Mason's rule associated to the signalow graph [START_REF] Mason | Feedback theory -some properties of signal ow graphs[END_REF]. This operational calculus can be applied also for multiple-input multiple-ouput systems with the difference that commutativity does not occur anymore. Let us note that the polynomial formalism is used in several textbooks on multivariable systems [START_REF] Kailath | Linear Systems[END_REF], [START_REF] Ku Cera | Analysis and Design of Discrete Linear Control Systems[END_REF] with no need of the Laplace transform.

C. The delay operator

A pure time delay of T between input and output signals induces y(t) = u(t T ): This particular linear system cannot be associated to a differential equation as [START_REF] Åström | Computer Controlled Systems[END_REF]. A special treatment must be used for delay equations. Following an idea of Euler [START_REF] Dugowson | Les Différentielles Métaphysiques[END_REF], the Taylor expansion of u(t T ) yields

u(t T ) = u(t) _ u(t)T + • u(t) T 2 2 +u (n) (t) ( T ) n n! + ;
which is encoded to give

y(t) = u(t) pT u(t) + p 2 T 2 2 u(t) + p n ( T ) n n! u(t) + ; = 0 @ X n 0 ( pT ) n n! 1 A u(t) = (exp( pT )) u(t):
We obtained the transfer operator for the time delay T as F (p) = e pT :

III. SYSTEM RESPONSES

System analysis is often the study of some particular responses of the system. Of special interest are the step and frequency responses.

A. Step response

Let us consider the Heaviside or step signal fH(t)g de ned by H(t) = 1 for t 0 and 0 elsewhere. The step response of a system is the solution of the differential equation of the system to a step input signal with zero initial conditions. With operational calculus, we can expand transfer operator as a linear combination of simple transfers a n (p + a) n or

1 p n ;
where a stands for a nonzero complex number and n for an integer. which to the the relationships c 0;n+1 = 1;

for i = 1 to n : c i;n+1 = a i c i 1;n = a i i! :
We deduce the well known result that, for the Strejć model

a n (p + a) n ; the step response s n (t) is s n (t) = 1 e at n 1 X i=0 a i t i i! ! :
So the step response of a given system de ned by a transfer operator can be calculated using polynomial calculus.

B. Stability

Due to the fact that stability analysis does not use the Laplace transform there is no novelty in this paragraph. However, we may insist again on the link between the transfer operator, the differential equation and the transient behavior.

Let us consider the transfer operator a n (p + a) n where a and n have the same meaning as before. From the previous paragraph we can see that the step response is composed of a constant term and a time-dependent term. The rst term is the forced response and the second term is the transient behavior. The transient behavior tends asymptotically to zero if and only if the real part of a is strictly negative: More generally, consider the n-th order transfer

F (p) = b m p m + + b 1 p + b 0 (p p 1 ) 1 (p p 2 ) 2 (p p r ) r ;
where p 1 ; p 2 ; : : : ; p r are the r complex poles of F (p) and i are the respective multiplicities with n = P r i=1 i . For the transient response the poles generate terms associated with the signals e p1t ; e p2t ; : : : ; e prt weighted by time polynomials of order i 1 respectively. If all the poles have strictly negative real part, the transient behavior tends asymptotically to zero. Namely, the system is asymptotically stable.

C. Frequency response

For asymptotically stable systems the frequency response is deduced from the steady-state output response corresponding to a given sinusoidal input signal u(t) = e j!t where ! is the frequency and j 2 = 1. Consider a system de ned by For the sinusoidal input u(t) = e j!t ; we obtain B(p)u(t) = jB(j!)j e j(!t+arg(B(j!))) ;

where jB(j!)j and arg(B(j!)) denote the module and the argument of the complex number B(j!) respectively. The output y(t) is the sum of a particular solution of the differential equation and the general solution of the differential equation without second member. The general solution characterizes the transient response that vanishes in case of asymptotically stable systems. For a particular solution, we look for the steady-state behavior as the form y(t) = Y e j(!t+') where Y and ' are constant parameters. Replacing this expression for y(t) in the differential equation yields

Y = jB(j!)j jA(j!)j = jF (j!)j ; ' = arg(B(j!)) arg(A(j!)) = arg(F (j!)):
The frequency response is de ned by the evolution of (jF (j!)j ; arg(F (j!))) as the frequency ! varies from 0 to +1: We can notice that F (j!) is the transfer function of the system such as Harris de ned it [START_REF] Harris | The frequency response of automatic control systems[END_REF]. In our standpoint, this transfer function must not be confused with the transfer operator F (p). Nevertheless, F (j!) such as a function of the frequency is the only actual transfer function. Graphic representations such as Bode, Black-Nichols, or Nyquist loci may be used to analyze the frequency response [START_REF]The Control handbook[END_REF]. For unstable systems the loci are valid as calculated representations only. But for stable systems, experiments cannot allow to obtain the frequency response.

IV. ANALYSIS

A. Poles and zeros

The names of poles and zeros come from the interpretation of a transfer operator F (p) as a function of a complex variable p: This interpretation is a consequence of the formulation of Laplace transform and it misunderstands the physical meaning of these notions. The consideration of a transfer operator as a coding of a differential equation provides an immediate physical interpretation. Namely, let us consider the transfer operator

F (p) = p + a p + b ; (7)
where a and b are constant parameters. In the operational standpoint the transfer (7) corresponds to the input-output differential equation _ y(t)+by(t) = _ u(t)+au(t) where y(t) and u(t) are the output and input signals. First consider u(t) = 0 for t > 0 and a nonzero initial condition y(0) we obtain y(t) = y(0)e bt for t > 0: Second consider a zero initial condition for the output and the input signal u(t) = e at for t > 0 we obtain _ u(t) + au(t) = 0 so y(t) = 0 for t > 0:

In general poles correspond to signals generated by the system with zero input. Zeros correspond to signals absorbed or blocked by the system. Let us write the transfer operator (5) as

F (p) = k (p z 1 ) 1 (p z 2 ) 2 (p z d ) d (p p 1 ) 1 (p p 2 ) 2 (p p r ) r ;
where k = b m ; z i ; i = 1; : : : ; d and p i ; i = 1; : : : ; r are complex numbers, and i ; i = 1; : : : ; d and i ; i = 1; : : : ; r are integers. For i = 1; : : : ; r, e pit is solution of the coded differential equation

(p p 1 ) 1 (p p 2 ) 2 (p p r ) r y(t) = 0;
and for i = 1; : : : ; d, e zit is solution of the coded differential equation

(p z 1 ) 1 (p z 2 ) 2 (p z d ) d u(t) = 0:
On the one hand we can use the correspondence between e pit and the transfer denominator roots p i that characterizes the transient rate in the linear constant parameter framework only. The same remark can be said about the correspondence between e zit and the transfer numerator roots z i . On the other hand this signal approach for the pole and zeros meaning can be extended to time-varying or nonlinear multivariable systems with an algebraic standpoint [START_REF] Bourlès | Finite poles and zeros of linear systems : an intrinsic approach[END_REF], [START_REF] Fliess | An algebraic approach to linear and nonlinear control, Essays on Control : Perspectives in the Theory and its Applications[END_REF].

In order to underline and to exemplify the important problem of pole/zero cancellation let us consider the series

y(t) = 1 p 1 u(t); z(t) = p 1 p + 1 y(t):
The pole 1 induces, in the transient behavior or in the initial conditions effect for the rst system, an e t signal. This signal is blocked by the second system, which has 1 as zero. As lim t!1 e t = 1; this fact forbides such a connection. Indeed, while the input and output signals are zero, there exists in the system a non observed and non controlled unbounded signal.

The conclusion is different if we consider the series connection with the systems :

y(t) = 1 p + 1 u(t); z(t) = p + 1 p 1 y(t):
Due to the pole/zero cancellation at 1, y(t) has an e t component that vanishes at 1: Except during the transient behavior, the pole/zero cancellation is acceptable for asymptotically stable cancelled zeros.

B. DC gain

Let us keep in mind that the transfer operator F (p) in equation ( 5) is just a coding of the differential equation ( 2). In the case of an asymptotically stable system, with the input taking a constant value U; the step response analysis indicates that the output tends to a constant value Y as t goes to +1

given by the relationship a 0 Y = b 0 U: The ratio Y U de nes the DC gain of the system G DC : The stability condition implies a 0 6 = 0; and from [START_REF] Carson | On a general expansion theorem for the transient oscillations of a connected system[END_REF] we obtain G DC = F (0).

C. Steady-state error analysis

In all this part systems are supposed to be asymptotically stable, namely the transient behavior vanishes and only the permanent behavior remains. For the reference inputs r i (t) de ned as, for t > 0; r i (t) = t i i! ; and for t < 0; r i (t) = 0; the corresponding outputs are y i (t) = F (p)r i (t): The inputoutput error " i (t) = r i (t) y i (t) is called the system error of order i: Two notions can be pointed out here. First a norm of the instantaneous system error " i (t) characterizes the system performance. Second the value " i (1) = lim t!1 " i (t) during the permanent behavior characterizes the steady-state error. In basic lecture of automatic control this last notion is usually considered. We detail it according to our formulation, namely without the use of the nal value theorem.

A steady-state error of order N is ensured if " i (1) = 0; for i = 0 to N; and " N +1 (1) 6 = 0: Consider the transfer operator F (p) in equation ( 5) of an asymptotically stable system. The corresponding permanent step response value is given by the DC gain b 0 a 0 : It can be seen that " 0 (1) = 0 if and only if b 0 = a 0 : We can conclude that a steady-state error of zero order is ful lled whether the DC gain is equal to 1. In other words since the input-error transfer is 1 F (p); we obtain a steady-state error of zero order when the input-error DC gain is zero. This is a fundamental remark for the following.

If we notice that r 1 (t) is the integral of r 0 (t); namely r 1 (t) = 1 p r 0 (t); we have

" 1 (t) = r 1 (t) y 1 (t); = 1 p r 0 (t) F (p) 1 p r 0 (t); = 1 F (p) p r 0 (t):
Clearly, from the previous result for " 0 (1); " 1 (1) vanishes if and only if the DC gain of the transfer operator 1 F (p) p is equal to zero. Since

1 F (p) p = (a 0 b 0 ) + (a 1 b 1 )p + (a 2 b 2 )p 2 + p(a 0 + a 1 p + a 2 p 2 + + a n p n )
we obtain " 1 (1) = 0 if and only if a 0 = b 0 and a 1 = b 1 : It can be seen that when a 0 6 = b 0 ; we have " 0 (1) 6 = 0 and with "

1 (1) = lim p!0 a 0 b 0 pa 0 = 1;
when a 0 = b 0 ; we obtain " 0 (1) = 0 and with "

1 (1) = a 1 b 1 a 0 : Moreover " 1 (1) = 0 when a 1 = b 1 .
In the same way we can show by recurrence that the system can have a steady-state error of order N if and only if its transfer F (p) in equation ( 5) is such that, for i = 0 to N; a i = b i . In this case the steady-state error of order N + 1 is

" N +1 (1) = a N +1 b N +1 a 0 ;
and the next ones have an in nite module. The degree of the steady-state error can be obtained by just a visual inspection of the transfer operator of the system.

V. COMPUTER CONTROLLED SYSTEMS The last question we wish to deal with concerns the construction of the model of a linear system controlled by a numerical algorithm (e.g. [START_REF] Åström | Computer Controlled Systems[END_REF], [START_REF] Franklin | Digital Control of Dynamic Systems[END_REF]). The problem is to nd the discrete-time model corresponding to the structure presented in Figure 4 where DAC denotes a digital-analog converter and it is usually modelled as the series connection of an ideal sampler and a zero-order hold. ADC denotes an analogdigital converter usually modelled by an ideal sampler and the discrete output signal of the ADC device is y k = y(kT s ) for k in N. Both are supposed to be synchronized with the sampling period T s : In this section we use the notations fx k g for the discrete-time real valued signal de ned for k in N and q 1 for the delay operator q 1 fx k g = fx k 1 g. All the considered discrete-time signals are supposed to be zero for negative values of k: For the discrete input signal fu k g the output of the DAC device is where H(t kT s ) stands for the delayed step signal :

u(t) = X k 0 u k (H(
The discrete-time transfer of the system (Fig. 4) is obtained through the formula

F (q 1 ) = (1 q 1 )Z L 1 F (p) p ; (9)
where L 1 stands for the inverse Laplace transform and Z f:g stands for the Z-transform [START_REF] Jury | Theory and Application of the z Transform Method[END_REF]. The expression in the square bracket denotes the sampling of the signal with a period T s .

Although the Z-transform doesn't suffer the same drawbacks as the Laplace transform, for instance discrete impulse is really a discrete signal, the Laplace transform appears once more time here. To carry on we obtain the discrete-time transfer operator F (q 1 ) in the operational framework.

Denoting by S(t) the step response of F (p); the response of

F (p) to u k (H(t kT s ) H(t (k + 1)T s ) is u k (S(t kT s ) S(t (k + 1)T s ):
The sampling of this response at the time period T s leads to a value for t = lT s ; l in N; given by u k (S l k S l k 1 ) where S k stands for S(kT s ): Denoting l the independent integer variable the corresponding discrete signal is fu k (S l k S l k 1 )g = (1 q 1 ) fu k S l k g for a given k: Because of linearity we deduce from (8) that the sampled response corresponding to the input signal fu k g is

fy l g = (1 q 1 ) 8 < : X k 0 u k S l k 9 =
; :

Denoting n P k 0 u k S l k o
= fv l g ; fu k g and fv k g are linked by a discrete convolution operation. Consequently [START_REF]The Control handbook[END_REF], they are linked by a difference equation such as

v k + d 1 v k 1 + + d n 0 v k n 0 = n 0 u k + n 1 u k 1 + + n m 0 u k m 0 ;
where the d i and the n i are real numbers and n 0 and m 0 are integers. In the same idea that for continuous systems, this equation can be coded by means of the delay operator q 1

(1

+ d 1 q 1 + + d n 0 q n 0 ) fv k g = (n 0 + n 1 q 1 + + n m 0 q m 0 ) fu k g ;
which leads to the discrete transfer operator

G(q 1 ) = n 0 + n 1 q 1 + + n m 0 q m 0 1 + d 1 q 1 + + d n 0 q n 0 :
Let us denote the numerator and the denominator of G(q 1 ) by N (q 1 ) and D(q 1 ) respectively. The division of N (q 1 ) by D(q 1 ) leads to

G(q 1 ) = X l 0 g l q l ;
where g l ; l 0; are real numbers. With fv k g = G(q 1 ) fu k g we obtain

fv l g = 8 < : X k 0 g k u l k 9 = ; = 8 < : X k 0 u k S l k 9 =
; :

Since for k < 0; u k = S k = 0; we have for k 0; g k = S k : So, G(q 1 ) is directly obtained from the sampling of the step response of the continuous-time transfer operator. Finally, we conclude that the discrete-time transfer operator is de ned by fy k g = F (q 1 ) fu k g = (1 q 1 )G(q 1 ) fu k g :

As an example let us consider the rst order model

F (p) = K 1 + p ; (10)
where K and are real numbers: The sampling, with period T s , of the step response of F (p) gives

g k = K(1 D k );
where D = e T s : We deduce

G(q 1 ) = K X k 0 (1 D k )q k ; = K(1 D)q 1 (1 q 1 )(1 Dq 1 )
; so the discrete-time transfer of ( 10) is

F (q 1 ) = K (1 D)q 1
(1 Dq 1 ) :

VI. WHAT'S ABOUT TABLES?

The main reason of using the Laplace transform is the tables we have at our disposal. First they contain information to determine the response of a system with respect to a given input signal. Second to obtain the discrete transfer operator of a computer controlled system with the formula [START_REF] Deakin | The ascendancy of the Laplace transform and how it came about[END_REF]. Although transforms are not used in our presentation, we show that these tables can be used without any change. To do that, let us introduce the notion of generator of a continuous signal, which consists in writing the time expression of this signal by means of the operator p.

The previous parts show that the transfer operator allows to link input u(t) and output y(t) signals of a linear system by a differential equation coded as y(t) = F (p)u(t): Until now we obtained the step response by solving this differential equation when initial conditions are all zero. In case of no input and non zero initial conditions such a transfer operator produces an output signal solution of the associated homogeneous differential equation. The coding of this differential equation with the p operator de nes then the generator of this signal. Two ways can be considered to take into account initial conditions in this coding. Namely, on the one hand the Mikusiński operational calculus and on the other hand the integral form of a differential equation.

A. The Mikusiński operational calculus

All the previous developments can be rigorously proved by means of operational calculus of Mikusiński [START_REF] Mikusiński | Operational Calculus[END_REF], which is based on convolution algebra of operators. Let us brie y describe this operational calculus whereas keeping in mind that the considerations below are not needed in a rst level course. Convolution product is a fundamental tool in dynamic systems eld [START_REF] Rugh | Nonlinear System Theory[END_REF], [START_REF] Schetzen | The Volterra and Wiener Theories of Nonlinear Systems[END_REF], speci cally in case of linear systems [START_REF] Gohberg | Theory and Applications of Volterra Operators in Hilbert Space[END_REF], [START_REF] Desoer | Feedback Systems : Input-Output Properties[END_REF]. This tool is de ned by

(f; g) 7 ! gf = Z t 0 f ( )g(t )d ;
while the Heaviside function H = f1g is of great importance due to the fact that we have for every f in the set of integrable function C

Hf = Z t 0 f (x)dx :
Consequently, H appears as the integration operator. The successive powers of H with respect to the convolution product are for all n in N; n 1; H n = t n 1 (n 1)! :

To distinguish a constant signal f g with the operator de ned by the constant gain we denote it by [ ] : For all f in C

f g f = Z t 0 f ( )d
and [ ] f = f f (t)g :

[1] is the unit element for the convolution and we can give a meaning to H 0 as H 0 = [START_REF] Åström | Numerical identi cation of linear dynamic systems from normal operating records[END_REF] . We de ne the derivative operator as the solution of the convolutional equation pH = [START_REF] Åström | Numerical identi cation of linear dynamic systems from normal operating records[END_REF] ; and then we write p = H 1 : With the understanding p 0 = H 0 = [1], we have p n = H n for n in N:

Mikusiński [START_REF] Mikusiński | Operational Calculus[END_REF] has proved the two results below, which are essential for our purpose.

Theorem 1 For every continuous function f in C, f (1) (t) = pf [f (0)] : More generally, for every integer

k n f (k) (t) o = p k f k 1 X i=0 h f (i) (0) i p k i 1 : (11)
Theorem 2 For every f in C such that

R 1 0 e tp f (t)dt exists f = Z 1 0 e tp f (t)dt:
The rst theorem allows to write the generator of a signal ff (t)g when is known the differential equation whose this signal is solution. Indeed, let us suppose that this differential equation can be written

n X i=0 i f (i) (t) = 0; (12)
with initial conditions f (0) = f 0 ; _ f (0) = f 1 ; : : : ; f (n 1) (0) = f n 1 ; where n is an integer and the i are real numbers. With [START_REF] Dimovski | Convolutional Calculus[END_REF] the coding of [START_REF] Doetsch | Theorie und Anwendung der Laplace-Transformation[END_REF]

The second theorem indicates that when the one-sided Laplace transform of a signal exists, its expression is identical to the generator of the signal, the complex variable s of Laplace transform being changed into the derivative operator p (to avoid any confusion). A major consequence is that the tables [START_REF] Doetsch | Theorie und Anwendung der Laplace-Transformation[END_REF], [START_REF] Spiegel | Formules et Tables de Mathématiques[END_REF], can be used. Since H = p 1 we remark that the generator can be written indifferently with the operators H or p.

For example when we look for a generator, say for sin(!t); we observe that sin(!t) is the solution of the differential equation

• x(t) + ! 2 x(t) = 0; x(0) = 0; _ x(0) = 1: (14)
Using [START_REF] Dimovski | Convolutional Calculus[END_REF] the coded form is obtained as

p 2 x(t) 1 + ! 2 x(t) = 0;
which leads to the generator of sin(!t) fsin(!t)g = M 1 p 2 + ! 2 ; where the symbol "M" denotes "in the Mikusiński sense". Indeed, we see in the next section an alternative to the Mikusiński approach. The Mikusiński operational calculus has been extended recently by the convolutional calculus [START_REF] Dimovski | Convolutional Calculus[END_REF].

B. The integral form

A second point of view can be used to introduce initial conditions in differential equations. In particular, let us consider the result below [START_REF] Rouche | Équations Différentielles Ordinaires[END_REF] f _

x(t) = f (t); x(0) = x 0 g if and only if x(t) = x 0 + R t 0 f ( )d : (15)
Denoting the integral operator by H, the derivative operator by p; and using the fact that for zero initial conditions, pH = Hp = I where I stands for the identity operator, we can code the differential equation ( 15) by x(t) x 0 = Hf (t) or px(t) px(0) = f (t) = _ x(t): In general, from the differential equation

x (n) (t) = f (t);
with the initial conditions x(0) = x 0 ; _ x(0) = x 1 ; : : : ; x (n 1) (0) = x n 1 ; we obtain In particular for ff (t)g = 0 and the initial conditions

x(t) = n 1 X k=0 x k t k k! + Z Z t 0 | {z } n times f ( )d : (16)
x 0 = x 1 = = x k 1 = x k+1 =
x n 1 = 0 and x k = 1 we obtain x(t) = t k k! : As the corresponding differential equation can be coded p k x(t) = 1 we get for every integer k;

t k k! = 1 p k :
Moreover for zero initial conditions and for all integer k we have p k H k = H k p k = I: These remarks induce for [START_REF] Fliess | An algebraic approach to linear and nonlinear control, Essays on Control : Perspectives in the Theory and its Applications[END_REF] the coding

x (n) (t) = p n x(t) n 1 X k=0 x k p n k : (17)
The generator of a signal ff (t)g can be obtained in this framework by coding with [START_REF] Fliess | Une interprétation algébrique de la transformation de Laplace et des matrices de transfert[END_REF] the differential equation ( 12) whose this signal is solution. We obtain the coding

" n X i=0 i p i # f (
However, if for a given function ff (t)g we compare the generators ( 13) and ( 18) it can be seen that P IC (p; f 0 ; : : : ; f n 1 ) = pP IC (p; f 0 ; : : : ; f n 1 ). We have the following relationship between the generators : the second one is the rst one multiplied by p: The generator [START_REF] Fliess | Closed-loop parametric identi cation for continuous-time linear systems via new algebraic techniques[END_REF] is the generator of f (t) in the Carson sense. The Carson transform was introduced in 1926 [START_REF] Carson | Electric Circuit Theory and the Operational Calculus[END_REF] and it differs from the Laplace transform by a factor p: The Carson tables can be used in this framework.

For example when we evaluate this generator for sin(!t); we obtain from the differential equation ( 14) the coding p 2 x(t) p + ! 2 x(t) = 0; which leads to the generator of sin(!t)

sin(!t) = C p p 2 + ! 2 ;
where the symbol "C" denotes "in the Carson sense". It is obvious that the integral form approach appears more intuitive than the Mikusiński approach but the results are quite similar.

C. Consequences

The generator of a signal allows to perform both of the two points below. First, for a system de ned by the transfer operator F (p) we can calculate the response y(t) to an input u(t) by using Carson or Laplace transform tables. Indeed when U (p) is a generator of u(t) we obtain the generator of the output y(t) = F (p)E(p):

We must remark here that the generators can be obtained in any sense as de ned above (Mikusiński or Carson). However, we must keep the coherence in using tables. For instance, when we want to know the beginning of the response we can develop y(t) in power of p 1 . This procedure was used by Heaviside [START_REF] Heaviside | Electromagnetic Theory[END_REF]. However, different functions may be associated to p k according to the adopted generator sense. Second, we can obtain the discrete-time model of a computer controlled system. Since we see it above, this model uses the step response S(t) of a system de ned by the transfer operator F (p): Incidentally, we can remark that the step generator in the Mikusiński sense for S(t) is F (p) p . In this framework we obtain the formula (9) for the discrete-time model. Whatever the used generator sense the tables can be used also. Moreover, in order to see the importance of the generator for operational calculus, let us consider the following example where two signals y 1 and y 2 are de ned by the differential equations :

(p 1)y 1 (t) = u(t);

(p 1)y 2 (t) = u(t); (19)
and the initial conditions y 0 1 and y 0 2 respectively. If we consider the operational calculus, the parallel connection y(t) = y 1 (t) y 2 (t) yields to :

y(t) = 1 p 1 u(t) 1 p 1 u(t) = 0:
This conclusion is obviously wrong. Indeed our setting and the suggested proof of the operational calculus indicate that we consider formal differential equations, namely without initial conditions. When we write :

y(t) = 1 p 1 u(t);
it is just a coding of the differential equations ( 19) and [START_REF] Franklin | Feedback Control of Dynamics Systems[END_REF] only. The initial conditions can be taken into account by mean of the generator notion. We can write [START_REF] Franklin | Digital Control of Dynamic Systems[END_REF] and [START_REF] Franklin | Feedback Control of Dynamics Systems[END_REF] as, respectively : This point indicates that y(t) is solution of the differential equation :

y 1 (t) = M
(p 1)y(t) = 0; y(0) = y 0 1 y 0 2 ; or, equivalently, y(t) = (y 0 1 y 0

2 )e t : This standpoint can also be explained in a more algebraic framework as the Fliess'module-theoretic approach [START_REF] Fliess | Une interprétation algébrique de la transformation de Laplace et des matrices de transfert[END_REF]. Nevertheless, let us remind a sentence of a recent paper [START_REF] Fliess | Closed-loop parametric identi cation for continuous-time linear systems via new algebraic techniques[END_REF] where is used this point of view for the design of a new identi cation procedure : "Let us add we tried to write the examples in such a way that they might be grasped without the necessity of reading the sections on the algebraic background. Our standpoint on parametric identi cation should therefore be accessible to most engineers."

VII. CONCLUSION

We show in this short survey that from the use of the differential operator we obtain all the usual results derived by means of the Laplace formulations. To teach a basic lecture in automatic control using the operational method offers some advantages. The integral or derivative operators allow to link every notion to its physical meaning. We keep in mind that a transfer operator is always related to only a differential equation or a difference equation. Mathematical background is minimized, however, when a rigorous justi cation is needed the Mikusiński operational calculus may be used. This operational calculus is based on the convolution operator, which is a natural tool for linear equations.

Moreover, we meet here through a pedagogical step the operational standpoints adopted directly in some advanced textbooks to modelize the input-output relationship induced by a linear system. For instance, the discrete-time autoregressive moving-average model A(q 1 )y k = B(q 1 )u k + C(q 1 ) k where A(q 1 ); B(q 1 ); and C(q 1 ) are polynomials in the delay operator and f k g a noise signal is used in [START_REF] Cochrane | Application of least squares regression to relationships containing autocorrelated error terms[END_REF], [START_REF] Åström | Numerical identi cation of linear dynamic systems from normal operating records[END_REF] for identi cation purposes to describe the difference equation between the input u k and the output signals y k of a given system. For multivariable continuous-time linear systems [START_REF] Rosenbrock | State-space and Multivariable Theory[END_REF], [START_REF] Wolowich | Linear Multivariable Systems[END_REF], [START_REF] Willems | Paradigms and puzzles in the theory of dynamical systems[END_REF] introduce the model where P (p); Q(p); R(p); and W (p) are matric polynomials in the differential operator and (t) is a vector-valued function of time called the partial state. More recently, [START_REF] Blomberg | Algebraic Theory for Multivariable Linear Systems[END_REF] de ne the generator of a multivariable system as the polynomial matrix M (p) in the derivative operator; which allows to write the relationship between input and output signals as

M (p) y(t) u(t) = 0.
The generator in the sense de ned in [START_REF] Blomberg | Algebraic Theory for Multivariable Linear Systems[END_REF] must not be confused with signal generators. The interested reader can see the quoted literature.

However, the operational pedagogic point of view has one drawback. When one of our students meets an automatic control engineer or reads an automatic control textbook, the student must be warned that he or she might nd some not up-to-date operational thoughts using the Laplace transform. But, this gap will be lled in the future.

  e-STA copyright 2008 by SEE Volume 5, N°3, pp 18-28

  e-STA copyright 2008 by SEE Volume 5, N°3, pp 18-28

  the transfer operator F (p) = B(p) A(p) where A(p) and B(p) are two polynomials. From the operational approach we have the encoded input-output differential equation as A(p)y(t) = B(p)u(t).

  e-STA copyright 2008 by SEE Volume 5, N°3, pp 18-28 connection with the systems :

  e-STA copyright 2008 by SEE Volume 5, N°3, pp 18-28

  e-STA copyright 2008 by SEE Volume 5, N°3, pp 18-28

1 ;

 1 in the Mikusiński's generator sense. It yields for y(t) = y 1 (t) y 2 (t) the generator :

  e-STA copyright 2008 by SEE Volume 5, N°3, pp 18-28e-STA copyright 2008 by SEE Volume 5, N°3, pp 18-28

P

  (p) (t) = Q(p)u(t); y(t) = R(p) (t) + W (p)u(t);

  , we introduce the derivative operator

	p ,	d dt	;
	which upon acting on x(t) in C gives the codings
	_ x(t) = px(t); • x(t) = p 2 x(t); : : : ; x (n) (t) = p n x(t); : : :

  The step response of a multiple integrator with the

	transfer the Strejć system [41] with the transfer 1 p n is t n : Let us consider the step response s n (t) of n! a n (p + a) n : For n = 1 the associated differential equation to the transfer a is p + a ay(t) + _ y(t) = au(t) and we obtain by usual methods [51] the
	corresponding response to a given input u(t) with the initial
	condition y(0)
	Z t
	y(t) = e at y(0) + a	e
	0

a u( )d : For u(t) = 1 and zero initial conditions the step response becomes s 1 (t) = 1 e at : For n = 2 we have s 2 (t) = a p + a s 1 (t) from which follows s 2 (t) = ae at Z t 0 (e a 1) d = 1 e at (1 + at): (6) For the general case n 1; let us suppose s n (t) = 1 e at n 1 X i=0 c i;n t i ! ; and s n+1 (t) = 1 e at n X i=0 c i;n+1 t i ! ; where the coef cients c i;n and c i;n+1 are constant parameters. These signals are linked by the differential equation _ s n+1 (t) + as n+1 (t) = as n (t);

  Fig. 4. Linear computer controlled system. Equipped with digital-analog (DAC) and analog-digital (ADC) converters, a continuous-time model (F (p)) leads to a discrete-time model. The discrete-time model is a coding of the recurrence equation between the discrete input signal u k and the sampled output y k = y(t k ) where t k are the sampling times. The puzzle is to obtain this discrete-time model with the operational standpoint.

	u k	-	DAC	-u(t)	F (p)	-y(t)	ADC	-	y k

t kT s ) H(t (k + 1)T s ) ; (8) e-STA copyright 2008 by SEE Volume 5, N°3, pp 18-28