Force Control System for an Automotive Semi-Active Suspension Carlos Vivas-Lopez, Diana Hernández-Alcántara, Manh Quan Nguyen, Ruben Morales-Menendez, Olivier Sename

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Force Control System for an Automotive Semi-active Suspension Carlos Vivas-Lopez, Diana Hernández-Alcántara, Manh Quan Nguyen, Ruben Morales-Menendez, Olivier Sename

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Carlos Vivas-Lopez, Diana Hernández-Alcántara, Manh Quan Nguyen, Ruben Morales-Menendez, Olivier Sename. Force Control System for an Automotive Semi-active Suspension. LPVS 2015 - 1st IFAC Workshop on Linear Parameter Varying Systems, Oct 2015, Grenoble, France. hal-01412892

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Force Control System for an Automotive Semi-active Suspension ?

Carlos A. Vivas-Lopez ∗ Diana Hern´andez-Alc´antara ∗ Manh-Quan Nguyen ∗∗ Ruben Morales-Menendez ∗ Olivier Sename ∗∗

∗ Tecnológico de Monterrey, School of Engineering and Sciences Av. E. Garza Sada 2501, Monterrey, NL, México {A00794204, A00469139, rmm}@itesm.mx ∗∗ CNRS-Grenoble INP, GIPSA-lab, 11 rue des Mathématiques38402 St Martin d’Hères cedex, France {manh-quan.nguyen, olivier.sename}@gipsa-lab.grenoble-inp.fr

Abstract: A new semi-active suspension control system is proposed. This control system includes a Linear Parameter Varying (LPV ) controller which was designed to improve the ride comfort. It also incorporates a Force Control System (FCS) to transform the force command from the LPV controller to a input signal for the Electro-Rheological (ER) semi-active damper. This FCS was assessed by its tracking performance of the desired force command, with a 7 % of tracking error. Then the semi-active control system was evaluated in a Quarter of Vehicle (QoV ) model under two tests: a Bump and a Road Proﬁle. The results were a reduction up to 19 % (Bump test) and 29 % (Road Proﬁle test), of the sprung mass position compared with a passive suspension. Additionally, an improvement up to 14 % was obtained when compared with a LPV controller using a simple model inversion Force-Manipulation transformation.

Keywords: LPV systems, Automotive semi-active dampers, Electro-Rheological shock absorber

1. INTRODUCTION 30 CompressionCompress 20 υ The way a suspension system is tuned in the vehicle = 35% design process can aﬀect ride comfort and road holding. 10 υ = 10% Therefore, a passive suspension has to be designed to 0 achieve a good compromise between these goals. −10 (a)

To overcome these passive damper limitations, semi-active [N] Force (b) shock absorbers can be used. These type of devices can on- −20 υ = 20% line change their dissipation characteristics. Technologies −30 (c) such as Electro-Rheological (ER) or Magneto-Rheological −40 Rebound (MR) are the most commercially used because of their −0.3 −0.2 −0.1 0 0.1 0.2 0.3 advantages: fast time response (40 ms), large force range, Velocity [m/s] wide bandwidth of control and cost. ER damper force dynamic is highly non-linear (i.e. satura- Fig. 1. Force-velocity map for different duty cycles. tion, hysteresis, etc.). Figure 1 presents the Force-Velocity (FV ) map of an ER shock absorber. These effects can designed to calculate a force that meets that performance, be well modelled by equations that mimic the damper but a transformation from force to manipulation (υ) is needed. Because of the non-linear damper characteristics, force (FD) as a function of the damper deflection (zdef ), it is possible to achieve the same level of force at different deflection velocity (z ˙def ), and manipulation signal (υ), Guo et al. (2006): conditions. As an example, the three red points in Fig. 1 (a, b, and c) correspond to the same FD = 10 N but, all are FD = cp(z ˙def ) + kp(zdef ) + Fsa (1) achieved at different average velocities and manipulations; (a) 0.12 m/s with 10 %, (b) 0.08 m/s with 20 %, and where Fsa = υ · fc · tanh(a1(z ˙def ) + a2(zdef )) is the semi- (c) 0.02 m/s with 35 %. Moreover, the actuator dynamics active force due to υ. Table 1 resumes the description of cannot be neglected, Priyandoko et al. (2009). This condi- the variables. tion makes the mapping from force to manipulation not a Semi-active suspensions require a control system to main- trivial task hence a tracking control system is needed. tain a desired performance. Normally, those controllers are This problem has been addressed in previous works. Kitch- ? Authors thank CONACyT and CRNS for their partial support in ing et al. (1998) used a cascade control system, the master the bilateral México-France PCP projects 03/10 and 06/13. controller tracks the desired damper force, whereas the

1st IFAC LPVS, Grenoble, France, Oct. 7-9, 2015 WePT1.4

slave one controls the opening valve position of the flow. QoV Model Only the damper velocity was considered to compute the control algorithm and the damper force loop had a con- zs zɺɺ ,z siderable delay. A neural network of the inverse model of zr ms s us the damper force was proposed in Chang and Zhou (2002). The desired force and two steps of the damper displace- FD ks ment are needed to obtain the manipulation. Hudha et al. zus Fsa (2005) used a PI controller coupled with conditional rules, des mus however they do not take into account the dynamic of the damper force during the PI tuning. Similarly, Sam and kt Hudha (2006) proposed a PI controller and incorporated z the dynamics of the damper flow valve; but the considered zr def actuator was active thus did not consider the dissipativity ɺ zdef restrictions of the semi-active dampers. Finally, Pellegrini LPV Controller et al. (2011) proposed an inverse model of the damper to obtain the manipulation, but they neglected the dynamic Kc ()ρ1, ρ 2 of the damper. Unlike previous authors, this work proposes a Force Con- Fig. 2. LPV control system. trol System (FCS) that overcomes the non-linear force m z¨ = −F − k (z − z ) s s D s s us (2) constraints of the damper by considering its dynamical m z¨ = k (z − z ) + F − k (z − z ) response. This FCS is designed to operate along with a us us s s us D t us r Linear Parameter Varying (LPV ) controller for a semi- where FD is computed with two equations: 1) a static active suspension control system. model and 2) a dynamical model, Fig. 3. This paper is organized as follows. Section 2 describes the ɺ automotive semi-active suspension and its control system. zdef zdef Section 3 shows the design of the proposed FCS. Section 4 discusses the results. Finally, section 5 concludes the F * F paper. υ ɺ D D FD() z def,, z def υ Gd () s Table 1. Description of variables. Static model Dynamical Variable Description Units model zr Road profile m zs, zus Sprung/Unsprung mass position m Fig. 3. Damper model. zdef , zdeft Damper/Tire deflection m z˙s,z ˙us,z ˙def Sprung/Unsprung mass velocity, damper m/s deflection velocity The static force of the damper is calculated using (1). 2 z¨s,z üs Sprung/Unsprung mass acceleration m/s The dynamical behaviour is represented as a second order FD Damper Force N system, Aubouet (2010): cp Viscous damping coefficient N·s/m FD(s) kd kp Stiffness coefficient N/m Gd(s) = ∗ = 2 2 (3) Fsa Semi-active component of the damper force N FD(s) (1/ωd)s + (2md/ωd)s + 1 υ % of Manipulation % a1 Hysteresis coefficient due to velocity N·s/m 2.2 LPV Semi-Active Suspension Controller a2 Hysteresis coefficient due to displacement N/m fc Damping coefficient N/% A Linear Parameter Varying (LPV ) controller is used. ks, kt Spring/Tire stiffness N/m m , m Sprung/Unsprung mass kg This controller incorporates in its design the characteris- s us tics of saturation and hysteresis of the semi-active damper ωd Natural frequency rad kd Static gain to fulfill its force constraints, Do et al. (2010). md Dynamic damping coefficient Substituting (1) in (2), and using fv = v · fc:

msz¨s = −ks (zs − zus) − cp (z ˙s − z˙us) − kp (zs − zus) 2. AUTOMOTIVE SUSPENSION SYSTEM −fv tanh a1 z˙ + a2 z def def (4) musz¨us = ks (zs − zus) + cp (z ˙s − z˙us) + kp (zs − zus) Figure 2 presents the block diagram of the LPV control +fv tanh a1 z˙def + a2 zdef − kt zdeft system. This control system is designed using the Quarter of Vehicle (QoV ) model. To satisfy the dissipativity constraint of a semi-active damper, fv must be constrained by

2.1 QoV Model 0 < fvmin ≤ fv ≤ fvmax (5) deﬁning u = f − F , with F = (f + f )/2, The QoV model is used to analyse the vertical dynamics v v 0 0 vmin vmax of a vehicle. It represents a quarter of the vehicle body the dissipativity constraint on fv is recast as a saturation with the wheel, tire, and suspension elements (spring and constraint on uv, i.e. damper) with the following equations: −FI ≤ uv ≤ FI (6)

1st IFAC LPVS, Grenoble, France, Oct. 7-9, 2015 WePT1.4

where FI = (fvmax − fvmin )/2. 3. FORCE CONTROL SYSTEM Denoting k = k + k , z = z − z , z = z − z , f s p def s us deft us r The LPV controller output (F ) cannot go directly andρ ˆ = tanh(a (z ˙ − z˙ ) + a (z − z )), a state space sades 1 s us 2 s us to the semi-active damper as an input, it needs to be representation of (4) is: transformed into a manipulation signal (υ). This signal is considered as a percentage of the possible range manip- ( x˙ s = Asxs + Bsρfˆ v + Bsww ulation input. This transformation becomes complicated P : z = Cszxs + Dszρfˆ v (7) because of the non-linear behaviour of the damper, Fig. 1. y = Csxs

In addition to the LPV controller output (Fsades ), zdef where x =(z , z˙ , z , z˙ )T , w=z , y = (z , z˙ )T . andz ˙def are needed to obtain the corresponding υ. In s s s us us r def def addition, the dynamics of the damper must be considered.  0 1 0 0   0  Hence, a simple inverse model is not enough, then a  kf cp kf cp   1  FCS is proposed. The FCS has two objectives: 1) to − − − ∗  m m m m   m  bound Fsa in a possible force range (F ) and 2) A =  s s s s , B =  s  des sades s  0 0 0 1  s  0      to compute the required υ input to achieve the desired  kf cp kf + kt cp   1  damper force. Figure 4 shows the proposed semi-active − − mus mus mus mus mus suspension control system; the FCS is shown in detail.  0  QoV  0  T   1 0 −1 0 ɺɺ Bsw = 0 , Cs = z zs ,zus   0 1 0 −1 zr s  kt  ms mus −kf −cp kf cp ! −1 ! F(υ ) k D s z Csz = ms ms ms ms , Dsz= ms υ us 0 0 1 0 0 mus F The control input of (7) is parameter-dependent and it can meas kt be rewritten as follows: z zr def ( x˙ = A (ρ1, ρ2) x + Bu + B1w ɺ zdef z = Cz (ρ1, ρ2) x (8) y = Cx F LPV Controller sa des FCS Kc ()ρ1, ρ 2 where: xs As + ρ2Bs1Cs1 ρ1BsCf x = , A (ρ1, ρ2) = , ɺ xf 0 Af zdef ,zdef T 0 Bsw Cs B = , B1 = , C = , zdef Bf 0 0 F* F C (ρ , ρ ) = ( C + ρ D C ρ D C ) D sa z 1 2 sz 2 s1 s1 1 sz f υ 1 υ e sa des des T Gc(s) D F0 F0 g() x(t) Bs1 = 0 − 0 , Cs1 = ( a2 a1 −a2 −a1 ), ms mus Force −F0 Fmeas Force controller clipping Ds1= 0 ms Cf xf F1 tanh(Cs1xs) ρ1 = tanh(Cs1xs) tanh( ) , ρ2 = F1 Cf xf Cs1xs Fig. 4. LPV semi-active control system including the FCS. xf ,Af ,Bf ,Cf are the matrices corresponding to a state space representation of the low-pass filter Wfilter = 3.1 Control System Design wf /(s + wf ) which is added to the system to make the control input matrices parameter-independent. The FCS takes as reference the LPV output (Fsades ) and The LP V controller, scheduled by ρ , ρ , has the form: sends the corresponding manipulation (υ) to the damper, 1 2 Fig. 4. In the first place it is necessary to ensure that F can be delivered by the ER damper. This is done sades x˙c = Ac(ρ1, ρ2)xc + Bc(ρ1, ρ2)y by defining an admissible region (D) which includes the Kc(ρ1, ρ2): H∞ (9) u = Cc(ρ1, ρ2)xc achievable force range of the real damper, Poussot-Vassal et al. (2008). To bound the force a clipping function is This controller minimizes the H -norm of the transfer used: ∞ function between the input disturbances w and controlled ∗ FD if FD ∈ D D(FD, z˙def ) 7→ FD = ⊥ (10) outputs z. The synthesis of the controller is made in FD if FD ∈/ D the LP V/H∞ framework based on the LMI solution, see ⊥ Scherer et al. (1997), for polytopic systems with quadratic where an orthogonal projection of the desired force (FD ) stabilization, Do et al. (2010). over the region D is assumed, this projection is driven

1st IFAC LPVS, Grenoble, France, Oct. 7-9, 2015 WePT1.4

by the current measure of the damper velocity. Figure 5 Table 2. Design of Experiments for the FCS. presents the admissible region D and the simulated force Displacement of 5 mm. for diﬀerent percentages of manipulation. Test Displacement Manipulation 40 Signal f [Hz] Signal Characteristics 1 Sin 1 f = 2 Hz, 30 2 Chirp [1-10] Square A∈[10,30] % 3 Triangular 1 20 D performance index. Table 3 presents the RMS index of 10 the tracking error and its normalization against the force

0 range (FDmax − FDmin ) corresponding to each test.

Force [N] Force −10 0 % a) Test #1 10 % 15 −20 20 % 30 % −30 −40 0

−0.3 −0.2 −0.1 0 0.1 0.2 0.3 [N] Force Velocity [m/s] −15 Fig. 5. Admissible force region (D) and damper force at b) Test #2 diﬀerent manipulation levels. 30 Considering the ER damper model, the following control law is proposed: υ = g(x(t))−1υ¯ (11) 0

with [N] Force υ¯(s) G (s) = (12) c e(s) −30 c) Test #3 where g(x(t)) = fc · tanh(a1z˙def + a2zdef ) and e(t) = 8

Fsades (t) − Fsa(t).

The controller Gc(s), (12), is designed using the dynamic model (3), and classical control techniques, considering the following specifications: bandwidth around 100 rad/s, gain 0 and phase margin greater than 12 dB and 45◦ respectively. [N] Force A controller which fulfil these specifications is given by: 86(s + 120) −6 Gc(s) = (13) 0 1 2 3 4 5 s(s + 80) Time [s] 4. RESULTS Fig. 6. Comparison of reference force (solid red) and the The evaluation of the semi-active suspension control sys- achieved force (dashed blue) at different displacement tem is made in two steps: 1) Assessment of the FCS using inputs to the damper model. the tracking error of the force as the performance index, A qualitative analysis can be made based on Fig. 6. It and 2) evaluation of the control system at different tests, can be seen that the proposed FCS was able to follow this evaluation is held in time and frequency domains. the reference signal, even for the Chirp signal, where the frequency of the wave increased up to 10 Hz. 4.1 Force Control System Assessment Table 3. Tracking error of different tests. To evaluate the performance of the FCS, a reference of Test #1 #2 #3 force (Fsades ) that mimics the characteristics and behaviour of an automotive semi-active damper under nor- RMS 1.63 2.01 1.42 mal operating conditions is needed. For this purpose an Normalized RMS 5.66 % 3.96 % 13.14 % ER shock absorber was simulated under different dis- placements (z ), velocities (z ˙ ) and manipulations (υ) A quantitative analysis was made by using Table 3. The def def RMS errors from the different experiments show that the inputs to generate a force reference (Fsades ). The selected signals are summarized in Table 2. FCS was able to track the reference for the different signals. In the test # 2 with the Chirp signal, the error Figure 6 shows the response of the FCS. The Root-Mean- is higher, but not considerably. The error in Test # 3 Square (RMS) value of the tracking error is used as is considerably high, but in this case the force reference

1st IFAC LPVS, Grenoble, France, Oct. 7-9, 2015 WePT1.4

behaves as a square signal (i.e. the worst case). It can be Figure 7 shows the displacement of the sprung mass and concluded that the controller is able to follow the reference the deflection of the tire for the Bump test. This test signal and the tracking error can be neglected. was used to evaluate the response of the system against a highly uncomfortable situation. Figure 7a shows that 4.2 Semi-active Suspension Control System Evaluation the LPV + FCS control system was able to compensate the effect of the Bump with a smooth transition, while the Passive case presents amplification of the Bump effect To evaluate the performance of the proposed control sys- and oscillations in its transient response. For the LPV tem, three cases were compared: 1) A passive suspension + SMI control system, the simplistic transformation of system where the ER damper manipulation was fixed at the force to manipulation introduces a negative impact 20 % (Passive), 2) the proposed control system (LPV + in the performance. Figure 7b shows, in both cases an FCS) and 3) a LPV controller coupled with a Simple improvement on the tire deflection and in their transient Model Inversion (LPV + SMI ) function. The comparison behaviour, meaning less tire bounce. The Passive case was based on: ride comfort index, and road-holding index. presents higher oscillations in its transient response. In the LPV + SMI control system, the Fsades command was transformed from force to manipulation by using the next simple inverse model function: a) zs  35% if F ≥ f · 35% sades c 20  F LPV + FCS υ(F ) = sades % if f · 10% < F < f · 35% (14) sades f c sades c  c 10 10% if Fsades ≤ fc · 10% 0 this function substitutes the FCS block in Fig. 4.

Time Domain Evaluation. Figure 7 shows the results of −10 a test with a Bump of 5 mm hight, and Fig. 8 a test with Passive LPV + SMI a Road Proﬁle input signal. Table 4 summarizes the RMS [mm] Displacement −20 index. The reported indexes are computed as improvement z of each variable against the Passive case, as: b) def t RMS(X ) 3 % of Improvement = 1 − iControlled (15) RMS(XiP assive ) 2 where Xi is the corresponding controlled variable (zs or 1 z ) for the control systems. deft 0

a) zs −1 Displacement [mm] Displacement 6 Passive −2 1 2 3 4 5 6 7 8 4 LPV + SMI Time [s] 2 Fig. 8. Comparison of control systems for the Road Proﬁle 0 test. Displacement [mm] Displacement −2 Figure 8 compares the control systems for the Road Proﬁle z test. This test evaluates the performance of a control sys- b) def t tem in a common automotive operation condition during 4 riding. Figure 8a shows how both control systems LPV + 3 SMI and LPV + FCS reduce the movement of the sprung 2 LPV + FCS mass, having better performance the LPV + FCS case. Figure 8b shows the response of z ; it can be observed 1 deft in the detail view that the LPV + FCS control system 0 has a better performance. −1 −2 Table 4 shows a quantitative comparison, it can be seen that the LPV + FCS controller has the better suspension

Displacement [mm] Displacement −3 performance. Regarding comfort, in the Bump test the −4 improvement is considerably higher (19.14 %) compared 0 0.2 0.4 0.6 0.8 with the LPV + SMI case (5.49 %), in both cases they Time [s] were better than the Passive case. The same occurs in the Road Proﬁle test where the LPV + FCS has a 29.38 % Fig. 7. Comparison of control systems for the Bump test. of improvement compared with the Passive case, against

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Table 4. RMS index performance improve- non-linearities, the damper can deliver wrong output ma- ment. nipulations in different conditions, the FCS adjusts the manipulation to reach the force reference, regardless the Case Variables uncontrolled variables in the force control loop (zdef , and zs zdeft z˙def ). Bump Test LPV + FCS 19.14 % 12.09 % In order to validate the proposal, the LPV + FCS control LPV + SMI 5.49 % 5.31 % system was compared with a LPV plus a Simple Model In- Road Profile Test version (SMI ) mapping function, taking the Passive case LPV + FCS 29.38 % 21.92 % as reference. The LPV + FCS control system proved its LPV + SMI 15.67 % 11.87 % effectiveness by maintaining the original control objectives the 15.67 % obtained by the LPV + SMI control system. with better performance in comfort and road-holding. These results are consistent with the qualitative ones. REFERENCES Frequency Domain Evaluation. Figure 9 presents the fre- Aubouet, S. (2010). Semi-active SOBEN suspensions mod- quency response of zs/zr, and zdeft /zr functions, Poussot- eling and control. Thesis, Institut National Polytech- Vassal et al. (2012). It can be seen that the LPV + FCS nique de Grenoble - INPG. control system has better performance in comfort (zs/zr) Chang, C. and Zhou, L. (2002). Neural Network Emulation than both Passive and LPV + SMI cases, specially in the of Inverse Dynamics for a Magneto-Rheological Damper. resonance frequencies. For the road-holding (zdeft /zr) the J. of Structural Eng., 128(2), 231–239. LPV + FCS control system has also better performance in Do, A., Sename, O., and Dugard, L. (2010). An LPV the range of 0-5 Hz. It can be seen that the use of the FCS Control Approach for Semi-Active Suspension Control improves the performance of the LPV control system. with Actuator Constraints. In American Control Conf., 4653–4658. USA. a) zs /z r Guo, S., Yang, S., and Pan, C. (2006). Dynamical Mod- eling of Magneto-Rheological Damper Behaviors. J. of 4 Intell. Mater., Syst. and Struct., 17, 3–14. Passive Hudha, K., Jamaluddin, H., Samin, P., and Rahman, R. 3 (2005). Effects of Control Techniques and Damper Con- straint on the Performance of a Semi-Active Magneto- Rheological Damper. Int. J. Vehicle Autonomous Sys-

Gain 2 tems, 3, 230–252. LPV + SMI Kitching, K.J., Cole, D.J., and Cebon, D. (1998). Perfor- 1 mance of a Semi-Active Damper for Heavy Vehicles. J. Dyn. Sys., Meas., Control, 122(3), 498–506. 0 Pellegrini, E., Koch, G., Spirk, S., and Lohmann, B. b) z /z (2011). A Dynamic Feedforward Control Approach for a def t r Semi-Active Damper based on a New Hysteresis Model. 0.4 In 18th IFAC World Congress, 6248–6253. Italy. LPV + FCS Poussot-Vassal, C., Sename, O., Dugard, L., Gáspár,P., Szabó,Z., and Bokor, J. (2008). A New Semi-Active 0.3 Suspension Control Strategy through LPV Technique. Control Eng. Practice, 16, 1519–1534.

Gain 0.2 Poussot-Vassal, C., Spelta, C., Sename, O., Savaresi, S., and Dugard, L. (2012). Survey and Performance Eval- 0.1 uation on Some Automotive Semi-Active Suspension Control Methods: A Comparative Study on a Single- Corner Model. Annual Reviews in Control, 36(1), 148– 0 2 4 6 8 10 12 14 160. Priyandoko, G., Mailah, M., and Jamaluddin, H. (2009). Frequency [Hz] Vehicle Active Suspension System using Sky-Hook Adaptive Neuro Active Force Control. Mechanical Sys- Fig. 9. Frequency responses of the variables of interest. tems and Signal Processing, 23, 855–68. Remark 1. In the frequency analysis the range beyond 15 Sam, Y. and Hudha, K. (2006). Modelling and Force Hz was not take into account due to unmodeled dynamics Tracking Control of Hydraulic Actuator for an Active in the mathematical model beyond that point. Suspension System. In IEEE Conf. on Ind Electronics and Applications, 1–6. Singapure. 5. CONCLUSIONS Scherer, C., Gahinet, P., and Chilali, M. (1997). Multiob- jective Output-Feedback Control via LMI Optimization. IEEE Trans on Automatic Control, 42(7), 896–911. A Force Control System (FCS) was proposed to improve a Linear Parameter Varying (LPV ) control system. The FCS considers the non-linear dynamic behaviour of an Electro-Rheological (ER) damper, Fig. 1. Due to these