SEMI-ACTIVE VIBRATION CONTROL WITH ON-LINE IDENTIFICATION OF THE INVERSE MR DAMPER MODEL

Transcription

1 SEMI-ACTIVE VIBRATION CONTROL WITH ON-LINE IDENTIFICATION OF THE INVERSE MR DAMPER MODEL Jerzy Kasprzyk and Piotr Krauze Institute of Automatic Control, Silesian University of Technology, Gliwice, Poland, The paper presents the semi-active vibration control based on the Skyhook algorithm which benefits from the adaptively tuned inverse magnetorheological (MR) damper model. The structure of the inverse model consists of the atan function and a linear viscous damping component. The proposed algorithm was tested by simulation of the quarter-car suspension model subjected to multiple bump road excitation. The suspension model includes the Bouc-Wen model which serves as a reference model of MR damper dynamics. Additional dynamics in the form of the first order filter is added at the output of the damper model. The key elements of the presented algorithm are iterative estimation of immeasurable signals and identification of the inverse model using the modified FxLMS algorithm. The current controlling MR damper is calculated by the inverse model according to the force determined by Skyhook. Simulation experiments confirmed a good quality of vibration control of the vehicle suspension model traversing a bumpy road profile, as well as the correctness of immeasurable signal estimation and MR damper identification. It can be stated that application of the inverse model adaptation in the semi-active vibration control gradually improves vibration attenuation during the ride. 1. Introduction Semi-active vehicle suspension systems are characterized by such features as low energy consumption, inherent stability and ability to adaptation to road conditions. In most cases these systems take advantage of MR (magnetorheological) dampers, in which viscous damping can be controlled by the current flow. Numerous solutions have been presented in the literature which deal with control of MR dampers in vehicles like, e.g., Skyhook and Groundhook [1, 2, 3], clipped LQ control [4], neural networks [5] and fuzzy control [6]. Assuming that information about the road profile is obtained in advance, modifications of the adaptive LMS algorithm can be applied to suspension MR damper control [7]. However, most of the presented semi-active systems deal with the problem of appropriate damper force generation and require the inverse MR damper model to be evaluated. Inverse modelling is complicated since parameters of MR dampers depend non-linearly on multiple factors, starting from conditions of identification procedure, construction of MR damper, the degree of wear as well as on ambient conditions. Thus, it is strongly recommended to make the inverse model an adaptive component of the Skyhook based controller. Typical approach to identification of the MR damper model is to use inertial and force sensors installed in the vehicle, define an inverse model and tune the model according to the measurement ICSV21, Beijing, China, July 13-17,

2 data, see, e.g., [8, 9]. Application of inertial sensors in the vehicle is straightforward and widely used, but force sensors attached to MR dampers can generate additional costs and substantially increase complexity of the control system. In this paper the semi-active vibration control supported by on-line identification of the inverse MR damper model is considered. However, the authors refer to the identification procedure which requires no force sensors installed in the vehicle suspension as it is based only on inertial measurements. Instead, identification of the inverse model is based on estimation of unmeasured signals. The paper is organised as follows: (1) the model of the vehicle suspension system is presented together with the MR damper model and the road excitation model, (2) the idea of Skyhook control with the adaptively tuned inverse MR damper model is described, (3) the proposed algorithm is tested by simulation experiments, (4) final conclusions are drawn. 2. Modelling of the semi-active suspension system For the purpose of algorithm prototyping, it is recommended to decompose dynamics of the vehicle suspension into four independent quarter-car models [10], which simplifies analysis of the algorithm. The model under consideration includes the well known Bouc-Wen model of the damper [11] which is treated as a reference model. It is assumed that this model is unknown by the control algorithm, what introduces an additional problem of inconsistency in structures of MR damper models used in simulation of the suspension and applied in the controller. This approach resembles somehow the real situation, where the model does not correspond perfectly to the real device. The analysed quarter-car model (Fig. 1) exhibits 2 DOFs and can be treated as a dynamic system with two inputs: the road-induced displacement excitation v r = ż r and the filtered force generated by MR damper F mr, and two outputs: the vertical velocity of the sprung mass v s = ż s and the relative damper piston displacement z mr. Two vibration modes of the suspension model are related to vertical absolute motion of the unsprung and sprung masses denoted as m s and m u, respectively. The linear part of the suspension model, after excluding the MR damper, can be defined as: m u z u = k u (z u z r ) c u (ż u ż r ) F mr + k s (z s z u ) + c s (ż s ż u ), (1) m s z s = k s (z s z u ) c s (ż s ż u ) + F mr, (2) where symbols z u, ż u and z u as well as z s, ż s and z s denote values and derivatives of the absolute vertical displacement of unsprung and sprung masses, respectively. The linear stiffness and damping parameters of the tyre and the suspension part are denoted as k u, c u and k s, c s, respectively. The suspension model is subjected to the filtered vertical force F mr generated by the MR damper. Complex dynamics of the MR damper and its non-trivial response to the piston motion and the control current require a non-linear dynamic model to be used together with the hysteretic behaviour in velocity-force characteristics and dynamic response on the current changes. Among different approaches to MR damper modelling, the Bouc-Wen model is meant to be accurate and appropriate. The Bouc-Wen model consists of two components: mechanical representation of MR damper stiffness and viscous damping, and hysteretic displacement denoted as p. The force generated by the MR damper can be obtained by the following formula: F mr,bw = (c 0,bw ż mr + k 0,bw z mr + α bw p), (3) where k 0,bw and c 0,bw denote damper stiffness and its viscous damping, respectively, whereas α bw is a scaling parameter for p. The hysteresis is defined by the first order non-linear differential equation: ṗ = γ bw ż mr p p n 1 β bw ż mr p n + A bw ż mr, (4) ICSV21, Beijing, China, July 13-17,

3 0.1 (a) Displacement [m] vertical displacement vertical velocity vertical acceleration Time [s] Figure 1: Simulation environment of the semi-active vehicle suspension: a) mechanical representation of the quarter-car suspension model; b) vertical displacement and velocity of the bumpy road profile where γ bw and β bw are parameters in (4). Relationship between model parameters and a current is assumed to be linear: κ bw = κ a,bw + κ b,bw i mr, (5) where κ bw denotes any parameter of the model. The values of the parameters of the 2 DOFs model together with the parameters of the Bouc- Wen model used in this research are presented in [7]. Additionally, the first order filter S Fmr with time constant t mr equal to 6.6 ms is placed at the output of the Bouc-Wen model giving filtered MR damper force Fmr, see fig. 3. This filter is used to model dynamics between the current changes and the force response. As it was shown in [12], the average response time is equal to 20 ms apart from cases when the piston velocity is close to zero. 3. Semiactive vibration control with adaptive inverse modelling Vibration control algorithms applied in vehicles can be distinguished with respect to types of road-induced excitation: periodic or aperiodic. The presented analysis is focused on influence of aperiodic excitation, strictly road bumps, on vehicle body vibrations. To make continuous all three derivatives of excitation, the velocity signal of the road bumps is defined and filtered in order to estimate both road-induced vertical displacement and acceleration (presented in Fig. 1b). The vehicle suspension is subjected to the series of resultant displacement smooth bumps with height equal to approximately eight centimetres. In case of a vehicle traversing a single road bump, two response phases can be distinguished. The first phase is associated with the direct response to the road impulse excitation, in the second phase damped vibrations of the vehicle body with the resonant frequency can be noticed. The Skyhook control is meant to be distinctively robust in case of such aperiodic impulse road excitations assuming the MR damper inverse model is accurate [1]. Therefore, the algorithm of adaptive identification of the MR damper model is included in the presented semi-active control problem. Because the Bouc-Wen model can be inverted only numerically, thus for the purpose of algorithm implementation, the approach based on MR damper modelling by means of the atan function is proposed [13]. This model consists of viscous damping component and the atan function which maps saturation behaviour. The velocity-force characteristics can be described by the following formula: F mr,at = {α at atan[β at ż mr ] + c at ż mr }, (6) (b) ICSV21, Beijing, China, July 13-17,

4 where parameters α at and c at are linearly related to the control current according to the similar formula as presented in (5). The parameter, β at, is invariant with respect to the current. The important feature of this model is a lack of component which maps hysteretic behaviour. Here, the difference between reference and inverse models is intentionally introduced to validate an inaccurate MR damper model applied in the control scheme. The inverse atan based MR damper model can be evaluated directly from (6) with the assumption about linear relation between model parameters and the current feeding the damper: i mr,at = { F mr (α a,at atan(β at ż mr ) + c a,at ż mr }/{α b,at atan(β at ż mr ) + c b,at ż mr }. (7) The block diagram of the complete adaptive vibration control system is shown in Fig. 2. The algorithm consists of two layers: one responsible for inverse modelling of the damper, and the second consisting of the concurrently executed control and identification algorithms. The reference suspension model which is simulated in quasi continuous time is marked by dashed-dotted frame. The feedback Skyhook algorithm benefits from the measured sprung vertical velocity v s (n) and gives an optimal force that should be generated by MR damper as follows: F alg = c sh v s, (8) where c sh = 1800 Nsm 1 is the Skyhook parameter. The identification algorithm is running for determining the model parameters, supported by iterative estimation of the sprung mass velocity component generated by the damper v s,fmr. This estimation is performed in each control cycle. Hereafter, the measured signals will be marked by the solid line frames, unknown signals by dotted lines, whereas signals unknown but estimated by dashed lines. Figure 2: Skyhook control including the adaptive inverse MR damper model 3.1 Iterative estimation of immeasurable sprung velocity components In the further study it is assumed that only measurements of vertical absolute velocity of the sprung mass and derivatives of the damper piston relative motion are available. It is also assumed that unsprung and sprung masses as well as stiffness and damping parameters of the vehicle suspension and tyres are known. According to the block diagram presented in Fig. 3, the linear discrete-time suspension model can be decomposed into four models defined for the following signal paths: road ICSV21, Beijing, China, July 13-17,

5 excitation velocity v r to sprung velocity v s denoted as P vr,v s, velocity v r to the damper piston relative displacement z mr denoted as P vr,z mr, damper force F mr to sprung velocity v s denoted as S Fmr,v s and damper force F mr to damper piston relative velocity v mr denoted as S Fmr,v mr. In order to estimate parameters of the atan model, the response of the sprung mass on the damper force F mr denoted as v s,fmr needs to be estimated, which cannot be done directly. Therefore, the iterative estimation method was developed. Figure 3: Block diagram for estimation of immeasurable signals In order to estimate v s,fmr at time instant n, the procedure needs to be executed counter clockwise starting from the initial value of vs,f 0 mr (n) = 0. Models of the primary and secondary paths are ordered in pairs what results in cancellation of unstable poles of the neighbouring discrete-time inverted models. Consequently, two resultant models are constituted: T vs,z mr arising from P vr,z mr and inverted P vr,v s, and T vmr,v s arising from S Fmr,v mr and inverted S Fmr,v s. In subsequent iterations, vs,v i r (n) is obtained by subtracting previously estimated v i 1 s,f mr (n) from the measured sample v s (n). The superscript i indicates the number of the current iteration. Subsequently, filtering vs,v i r (n) by the primary path T vs,z mr an estimate of zmr,v i r (n) is obtained, which subtracted from the measurement sample z mr (n) gives vmr,f i mr (n) after differentiation. Next, vmr,f i mr (n) filtered by the secondary path T vmr,vs gives ṽs,f i mr (n) as a result of the current iteration. However, additional processing is performed at the end of iteration. The final result of the loop vs,f i mr (n) is calculated by low-pass filtration of the previous value v i 1 s,f mr (n) and new ṽs,f i mr (n) using adaptation coefficient µ e according to the following relation: v i s,f mr (n) = (1 µ e )v i 1 s,f mr (n) + µ e ṽ i s,f mr (n) for abs(ṽ i s,f mr (n) v i 1 s,f mr (n)) > δ e, (9) where the stop condition is assumed as δ e = 10 5 and the adaptation coefficient µ e = On-line identification of the MR damper model using modified FxLMS The modified Normalized FxLMS algorithm (i.e. Least Mean Square algorithm with filtered reference signal and normalization [14]) was used for the atan model identification. The resultant dynamic model T Fmr,v s built of the previously estimated damper model S Fmr and suspension model S Fmr,v s is used to calculate the response of the identified atan model ˆv s,fmr as follows: ˆv s,fmr (n) = T Fmr,v s (z 1 ) F mr,at [θ at (n), i mr (n), v mr (n)], (10) where F mr,at is calculated according to (6). Here, previously estimated parameters are grouped in a vector θ at = [α a,at, α b,at, β at, c a,at, c b,at ]. The reference signal g(n) used by the Normalized FxLMS algorithm is defined as a gradient of the atan model with respect to the vector θ at. The gradient vector g(n) is filtered using the secondary path model T Fmr,v s giving r(n) signal which is then used in the adaptation algorithm as follows: θ at (n + 1) = θ at (n) + (p on/off (n) > δ i ) µ i r(n) e mr (n) r T (n)r(n) + ζ i, (11) ICSV21, Beijing, China, July 13-17,

6 where e mr (n) = v s,fmr (n) ˆv s,fmr (n) defines an error between estimated value of the damper response (see Section 3.1) and response of the identified atan model (10). Convergence speed of the algorithm is influenced by the vector of adaptation step parameters µ i = [0.06, 0.5, 0.025, 0.04, 0.15 ]. FxLMS algorithm is normalized using r(n) being a vector of filtered reference samples g(n). In order to avoid division by zero in the case of zero reference signal, an offset value ζ i = is added in the denominator. Road profile excitation is composed of single bumps and periods of a smooth profile between them. In the absence of stimulation identification is highly sensitive to numerical errors. Hence, the process of parameters update is activated only if the level of excitation is significant enough, see (11), i.e., if the root mean square value denoted as p on/off obtained for the estimated signal v s,fmr is greater than a certain identification on/off threshold δ i, here assumed as Figure 4: Identification procedure for the atan model using the modified FxLMS algorithm 4. Results of simulation experiments The simulation environment consists of two elements: quasi continuous-time simulation of the suspension model and discrete-time control. The first component covers numerically solved differential equations which describe quarter-car model (eqs. 1 and 2), dynamics of the Bouc-Wen model (eqs. 3 and 4) and additional output dynamics S Fmr (s). Solution of the differential equations is obtained in every simulation cycle (T s,model = 0.5 ms) by the fourth order Runge-Kutta method with the varying step size. Identification and control algorithms presented in Section 3 are executed in the discrete-time at the sample rate equal to 500 Hz (T s,control = 2 ms). Each test was carried out for 200 seconds. The vehicle suspension model was subjected to series of road bumps defined in Section 2. It was observed that the identification process was stable and adaptation of all parameters included in θ at converged to steady-state values which are listed in Table 1. The identification procedure was initialized with the following initial values of parameters: θ 0 at = [ 20 ; ; 75 ; 200 ; 4500 ]. It was observed that the parameter α 1 has a crucial influence on the model response. Thus, activation of other parameter estimation was delayed with respect to parameter α 1. Fig. 5a presents an example of the adaptation process of α 1 plotted in time. For validation of the identification procedure, both the reference Bouc-Wen model and the identified atan model were subjected to damper piston velocity of amplitude equal to 0.2 metres per second and frequency 2 herz. The strong compliance between model responses for higher piston velocities was observed. For lower velocities the atan model properly averages the hysteretic behaviour of the Bouc-Wen model. ICSV21, Beijing, China, July 13-17,

7 Table 1: Parameters of the identified atan model Atan based MR damper model {θ a,at, θ b,at } i mr ( ) A α at = {65, 530} N β at = 75 sm 1 c at = {418, 5030} Nsm 1 Comparison of the road vertical displacement with the sprung displacement response is plotted in Fig. 5. Improvement of the Skyhook vibration control can be noticed a few seconds after initiation of the algorithm, when the value of α 1 fell from initial to approximately 700, that is when the inverse model was tuned. Parameter α Time [s] (a) Vertical displacement [m] road excitation 0.02 sprung response Time [s] Figure 5: Results for the Skyhook control with the inverse model identification: a) on-line estimation of the parameter α 1 ; b) road profile excitation and the vertical sprung displacement. (b) 5. Conclusions Quality of the most semi-active vibration control algorithms significantly depends on the accuracy of the inverse MR damper model that can be questionable as dynamics of the MR damper depends on many different factors, including level of the control current, frequency and amplitude of piston excitation or degree of wear of the MR damper. In this paper the FxLMS based on-line identification procedure is introduced for the inverse MR damper model updating in the Skyhook control scheme. It was assumed that only inertial sensors are available. Hence, in order to estimate MR damper parameters, the algorithm of iterative estimation of immeasurable signals was developed. The proposed algorithm was validated in simulation experiments, where the quarter-car 2 DOFs vehicle suspension model was subjected to bump road profile excitation. The results proved the efficiency of vibration control in this case. Future research will be focused on extension of the algorithm to a half-car vehicle suspension model and its implementation in the experimental vehicle suspension system. 6. Acknowledgements The work reported in this paper has been partially financed by the National Science Centre, decision no. DEC-2011/01/B/ST7/ The second author received a scholarship under the project "DoktoRIS - Scholarship Program for Innovative Silesia" co-financed by the European Union under the European Social Fund. REFERENCES 1 Karnopp, D., Crosby, M. J., Harwood, R., A., Vibration control using semi-active force generators, Journal of Engineering for Industry, 96, , May, (1974). ICSV21, Beijing, China, July 13-17,

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