Robust flatness-based switching reconfiguration control using state flow machines of electronic throttle valve

In this paper, a robust Fault Tolerant Control reconfiguration approach using State Flow Machines is proposed. Indeed, this reconfiguration strategy is based on robust flatness-based switching control using state machines and flow charts. This approach is developed in discrete time framework in order to track a reference trajectory starting from a flat output variable. For each model, a corresponding flatness-based controller is designed and consequently, a multi controller structure is obtained. The switching flatnessbased control is based on switching between identified Operating Modes (OM) using state flow machines. The Luenberger observer’s gains are determined using LMIs tools in order to identify the corresponding OM. The localization of the current OM is carried out by minimization of a performance test characterizing the distance between the system and the given OM. Study of the stability as well as the use of anti-windup devices related to switching between controllers have been considered in the proposed approach. The proposed approach is applied to the nonlinear system which is in our case of study an Electronic Throttle Valve (ETV) using state flow machines modeling.


Introduction
General multiple model-based control approaches have been developed in last decade.Thus, in the literature, there are many strategies that take in consideration the multi-models methods used for reconfiguration of control law, [1], [2] and [3].Furthermore, Multiple Models (MM) method is proposed in [4].In addition, controller switching approach represents a class of active fault tolerant control detailed in [5].Moreover, Multiple Models Switching and Tuning (MMST) approach that concerns more particularly the reconfigurable control method is proposed in [3], [6] and [7].In this paper, for a nonlinear process with multiple OMs, a robust control reconfiguration strategy, based on switching control, is used to ensure stability and desired performances.The LMI tools which are based on quadratic Lyapunov criteria are used in order to guaranty the stability of the global system.The performances obtained by switching in terms of tracking trajectory and fault rejection are discussed in this paper and the stability of the corresponding switched discrete-time linear systems is given.The case of ETV is studied.

Reconfiguration fault tolerant control approach 2.1 Description
A robust RST multicontroller based on flatness is proposed.The controller selection strategy is based on switching between identified Operating Modes (OMs).A stateflow is proposed in order to select the identified controller.The localization of the current OM is carried out by minimization a e-mail: hajer.gharsallaoui@voila.fr of a performance test characterizing the distance between the system and the given OM.

Stability analysis
The stability analysis of a switching control strategy consists on two steps, [1] stability analysis of each subsystem, i.e., each controller must asymptotically stabilize each process operating mode, stability analysis of the overall system for arbitrary switching signals.
To guarantee the stability of the systems, Linear Matrix Inequalities, represented by 1, must be solved [8].
with P a symmetric positive definite matrix given by L j = P −1 Y j where L j ∈ R n×m is the Lunberger observer gain for the j th model.

RST multicontroller structure
Let us denote the open-loop discrete-time process transfer function by A j (q) and B j (q) are polynomials defined by ( 3) and ( 4) where the parameters a j,i and b j,i are constants, j = 1, ..., m, i = 0, 1, ..., n − 1 and q −1 is the causal operator.
A j (q −1 ) = 1 + a j,n−1 q −1 + . . .+ a j,1 q −n+1 + a j,0 q −n (3) B j (q −1 ) = b j,n−1 q −1 + . . .+ b j,1 q −n+1 + b j,0 q −n (4) The realization of an RST controller based on flatness is carried out by considering the method of direct calculation of the state vector Z j,k = (z j,k z j,k+1 . . .z j,k+n−1 ) T , [9].The state space representation of the system in its controllable form is such that where the matrix A j , B j and C j are given by The flatness control law is given by the following relation where a and k are two constant vectors constituted by the a j,i and k i coefficients of the A j (q) and K(q) polynomials given by The polynomial K(q) is the denominator of the tracking dynamics which is a discrete-time n th order model equivalent of a continuous-time one composed of second order with fixed damping factor ξ, fixed natural frequency ω 0 , given by (11), and a first order model with a fixed time constant τ, given by (12), where s is the Laplace operator.
The structure of RST controller can be then obtained by with and O (A j ,C j ) are the controllability and the observability matrices.

Trajectories planning
The open loop control law can be determined by the following relations [10].
where f and g are vectorial functions.Then, it is sufficient to find a desired continuous flat trajectory t → z d j (t) that must to be differentiable at the (r + 1) order.In order to plan the desired flat trajectory z d j (t), the polynomial interpolation technique is used.Let consider the state vector Z d j (t) = (z d j (t) żd j (t) . . .z d(r+1) j (t)) T containing the desired continuous flat output and its successive derivatives.t 0 and t f are the two moments known in advance.The expression of Z d j (t) can be given as following [10] where M j,1 and M j,2 are such as and the vectors c j,1 and c j,2 defined by After planning a desired flat trajectory, the output desired trajectory y d j,k is defined.In the discrete-time framework, the real output y j,k have asymptotically to track this such as (25), [10].
4 Application to the control of an electronic throttle valve

Electronic throttle valve modelling
The case of the Electronic Throttle Valve (ETV), of the Fig. 1, is studied to test the proposed controller.The electrical part is modeled by ( 26) Fig. 1.Electronic throttle system [11] where L is the inductance, R the resistance, E the electromotive force of its armature, u and i the voltage and the armature current respectively, k an electromotive force constant, θ m the position of the motor shaft and ω the angular velocity of the motor rotor [12] and [13].
The mechanical part of the throttle is modeled by a gear reducer characterized by its reduction ratio η such as ( 27) where C L is the load torque and C g the gear torque.The mechanical part is modeled according to (28), such that, [12] and [13].
J is the overall moment of inertia, Ω = dθ dt the angular velocity of the throttle valve, C e = k e i the electrical torque where k e a constant, C f the torque caused by mechanical friction, C r the spring resistive torque and C a the torque generated by the airflow.By substituting in equation ( 28), the expressions C e , C f and C r and by neglecting the torque generated by the airflow C a , the transfer function of the model becomes such as (29) with k s = (180/π/l 2 )k r , l a constant and s the Laplace operator.
The identified parameters of H(s) are given in Table 1.The default position θ 0 of the ETV creates a discontinuity in the dynamics of the system.The operating mode depends on the throttle position.Two linear models ( j = 1, 2) have been chosen to carry out the study with the sampling period; T e = 0.002s.
-A model representing the position of the plate above the position by default with k s = 1.877 × 10 −4 kg.m 2 .The corresponding discrete-time transfer function H 1 (q −1 ) is given by (30).
-A model representing the position of the plate below the position by default with k s = 1.384 × 10 −3 kg.m 2 .

RST multicontroller based on flatness
The dynamics chosen are those of a fourth-order continuoustime system with a damping factor ξ = 0.8, a natural frequency ω 0 = 20 rad/s and a time constant τ = 0.1 s.The polynomial K(q) is then given by (32).

Stability analysis
The Luenberger observers's gains are given by (37).
The combination of the two controllers with the two process transfer functions, is found to be globally asymptotically stable with P definite positive, is given by (38).

Simulation results
The desired trajectories z 1 d (t) and z 2 d (t) are given by Fig.

2.
The simulation results are given in Fig. 3, 4 and 5.This figures show the effectiveness of the RST multicontroller based on flatness in terms of tracking a desired trajectory with high performance and disturbances rejection.

Conclusion
The proposed approach based on the flatness is applied to an Electronic Throttle Valve (ETV) with multiple Operating Modes (OMs).For each model, a corresponding RST controller based on flatness is designed and consequently, a reconfiguration strategy based on multi-controllers structure obtained.The switching between identified models of an ETV was based on the minimization of a performance test characterizing the distance between the system and the given OM.Simulation results proved the effectiveness of the proposed multicontroller.