Keywords

1 Introduction

Recently, controllers which are independent on the mathematical model spread in many filed such as Model Free Control (Fliess and Join 2013) and Active Disturbance Rejection Control (ADRC) (Han 2009).

The ADRC technique used to estimate endogenous and exogenous perturbations with a state observer. This estimated states can be injected in the equation of controller and cancel all unknowns’ phenomena. Unfortunately, for obtaining the optimal control requires a lot of time and needs many essays. The main difficulty of the calibration task is to define properly the parameter that affect the denominator of controller. The range of variation of this scaling parameter is considered un-known and differ from an operator to another one.

This scheme of control was applied to control the passive quarter car system (Hasbullah et al. 2015; Li et al. 2016). Researchers have developed this method with a constant scaling parameter chosen by the operator and are approximate it to the inverse of sprung mass. However, our aim is to find a global approach by changing controller parameter in order to follow the sprung mass variation.

Recently, Alvarez-Sanchez (2013) provided an identification scheme framework to estimate the sprung mass variation based on algebraic rules of Fliess and Sira-Ramírez (2003). Using this approach, the range variation characteristic of body mass can be deduced based on the measured information of sprung mass displacement, un-sprung mass displacement and actuator force. Based on these mentioned principles, ADRC control using real-time identification for time-varying mass is proposed in this paper. A combination between online real-time identification and ADRC control can be used in order to escape the time-consuming in founding the optimal control.

The organization of the paper as follows. Section 2 describes the motion equations of quarter car model. Section 3 gives a simple description of estimator approach. The online estimation of sprung mass is discussed in Sect. 4. Results are presented in Sect. 5 and a conclusion is summarized in the last section.

2 Description of the System and Road Input

The motion equations of passive quarter car system are given below:

$$ m_{s} \ddot{z}_{s} + d_{s} (\dot{z}_{s} - \dot{z}_{u} ) + k_{s} (z_{s} - z_{u} ) = F_{A} $$
(1)
$$ m_{u} \ddot{z}_{u} - d_{s} (\dot{z}_{s} - \dot{z}_{u} ) - k_{s} (z_{s} - z_{u} ) + k_{t} (z_{u} - z_{r} ) = - F_{A} $$
(2)

\( m_{s} \) is the sprung mass which represents the body of the car. \( m_{u} \) is the un-sprung mass. \( k_{s} \) is the suspension stiffness. \( d_{s} \) Represents suspension damping and \( k_{t} \) is the tire stiffness. The actuator force is denoted by \( F_{A} \) (Fig. 1).

Fig. 1.
figure 1

Active quarter car model

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A random road excitation is characterized by a constant of roughness that is given by:

$$ \dot{z}_{r} (t) = - 2\pi Vn_{0} z_{r} (t) + 2\pi \sqrt {G_{0} {\kern 1pt} V\;w(t)} $$
(3)

where \( z_{r} (t) \) is the random road displacement, V is the vehicle speed, the reference spatial frequency \( n_{0} \), \( G_{0} \) is the road roughness coefficient and the white noise signal is given by \( w(t) \).

3 Controller Design

The basic idea of ADRC is using an extended observer ESO without a priori information of the system. The attractiveness of this tool of control results from its capability to estimate on-line and reject unknown perturbations. Significant performances are achieved in the works of Hasbullah et al. (2015) and Pan et al. (2015).

In order to improve the response of suspension system described in Eqs. (1) and (2), the ADRC strategy used to reformulate these equations as,

$$ \ddot{z}_{s} = h(t,\dot{z}_{s} ,z_{s} ) + \beta F_{A} + z_{r} $$
(4)

From practical point of view, the knowledge of h (.) and \( \beta \) is not straightforward. A non trivial approximation is used in other studies. Where, the constant \( \beta \) is chosen empirically and is approximated by \( 1/m_{s} \).

In real conditions, significant uncertainties of sprung mass can affect the behavior of this proposed controller. Estimation of this calibrating parameter can be a source of upgradability in energy consumption.

4 “β” On-Line Adaptation

An algebraic estimator used to find the car body mass of the quarter car system based on algebraic identification methods (for more details see Fliess and Sira-Ramirez 2003).

$$ \begin{aligned} m_{s} (t)(2\int {\int {z_{s} } } - 4\int {tz_{s} } + t^{2} z_{s} )dt + d_{s} (2\iint {tz_{s} } + \int {t^{2} } z_{s} - 2\iint {tz_{u} } - \int {t^{2} } z_{u} )dt \hfill \\ + k_{s} (\iint {t^{2} }z_{s} - \iint {t^{2} }z_{u} )dt = \iint {t^{2} F_{A} }dt \hfill \\ \end{aligned} $$
(5)

From the equation, it can be seen that only from the measured responses of vertical displacements and actuator force; we can obtain an approximation of sprung mass variation. This On-line estimation can be easily added to the controller structure where,

$$ \hat{\beta }(t) = \frac{1}{{\hat{m}_{s} (t)}} $$
(6)

The structure of classical controller is changed with the variable \( \hat{\beta }(t) \), the new design is given in this equations:

  1. (1)

    the system’s state:

    $$ \left\{ {\begin{array}{*{20}l} {\dot{x}_{1} = x_{2} } \hfill \\ {\dot{x}_{2} = x_{3} + \hat{\beta }(t)F_{A} } \hfill \\ {\dot{x}_{3} = \dot{f}(t,x_{1} ,x_{2} ,w)} \hfill \\ \end{array} } \right. $$
    (7)

    where \( x_{1} = z_{s} \),\( x_{2} = \dot{z}_{s} \) and \( x_{3} \) represents all the un-known perturbations.

  2. (2)

    The linear Luenberger observer:

    $$ \left\{ {\begin{array}{*{20}l} {\dot{z}_{1} = z_{2} - L_{01} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{e} } \hfill \\ {\dot{z}_{2} = z_{3} - L_{02} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{e} + \hat{\beta }(t)F_{A} } \hfill \\ {\dot{z}_{3} = - L_{03} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{e} } \hfill \\ \end{array} } \right. $$
    (8)

The observer gains \( [L_{01} ,L_{02} ,L_{03} ] \) depends to the location the desired close loop poles. The approximated error is \( \hat{e} = z_{s} - z_{1} \).

The principle of combination between online real-time identification and ADRC control is described in Fig. 2.

Fig. 2.
figure 2

Block diagram of the proposed control

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The gains \( K_{p} \) and \( K_{d} \) are respectively, the proportional gain and the derivative gain, which are needed to implement the proportional-derivative (PD) feedback controller.

5 Results of Simulation

In the simulation algorithm a solver set to ODE5 and fixed integration step of \( 1\;{\text{ms}} \) were used. The parameters of Suspension system are given by Table 1.

Table 1. Parameters of suspension system
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Figure 3 depicts the real value of mass and the identified masse \( \hat{m}_{i} \) when using the “Eq. (5)”. In reality, the on-line estimation process is characterized with few irregularities in the beginning. In order to eliminate this perturbation caused by the singularities; the implementation of identifier process is carried out at t > 0 s. It is observable that the estimation process is achieved after a short time t = 0.004 s. (In the rest of simulation, in the beginning \( \beta \) is chosen constant and independents of load variation. After that the estimation and the adaptive scheme start at t = 10 s).

Fig. 3.
figure 3

On-line estimation of Sprung mass

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Figure 4 shows the suspension deflection with the effect of sprung mass variation \( m_{s} = + 50\% {\kern 1pt} \;m_{{s{\kern 1pt} initiale}} \). The line at t = 10 s represents the instant where the identification started and applied to the controller equation. Before estimation of the constant from measured signals, the \( \beta \) is chosen empirically and it is not calibrated when the body mass change. The ADRC controller with estimated \( \hat{\beta } \) can produce more best tracking position than the ADRC without constant adaptation.

Fig. 4.
figure 4

Suspension deflection

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For quarter car control, the ride comfort is related to the body acceleration. In Fig. 5, we can see that the designed system conserve the best isolation of disturbance with a slight attenuation, for the reason that ADRC with adaptive \( \hat{\beta } \) has the least RMS value among the sprung mass variation Fig. 6.

Fig. 5.
figure 5

Sprung mass acceleration variation

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Fig. 6.
figure 6

RMS of Sprung mass acceleration

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In Fig. 7 we observe a reduction of actuator force when we use the adaptive controller. According to this numerical results, the estimated \( \hat{\beta } \) has an important role in the dynamic behavior of the suspension system. It is quite possible to obtain better sprung mass acceleration attenuation and the best tracking of reference trajectory for the ADRC controller with the lowest power demand.

Fig. 7.
figure 7

The power demand of actuator

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Furthermore, the Integral of the Square of the Error (ISE) performance was calculated with both of controller and with different masses. The results are depicted in Fig. 8.

Fig. 8.
figure 8

ISE criteria was calculated from the power demand

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6 Conclusion

The aim of this paper is to propose an adaptive ADRC controller using an On-line identification of Sprung mass which is intended to extended time-varying observer. This method can overcome time-consuming which are induced by nontrivial calibration. At the same time, this adaptive control provides high performances under sprung mass uncertainty. Lastly, this method applied to quarter car system permits to getting the best ride comfort and the best tracking with the minimum of power demand.