The suspension system is faced with nonlinearities from the spring, damper, and external excitations from the road surface. The objective of any control action provided to the suspension is to improve ride comfort while ensuring road holding for the vehicle. In this work, a robust higher order sliding mode algorithm combining the merits of the modified supertwisting algorithm and the adaptive supertwisting algorithm has been proposed for the nonlinear active suspension system. The proposed controller is robust to linearly growing perturbations and bounded uncertainties. Simulations have been performed for different classes of road excitations and the results are presented.
1. Introduction
The vertical dynamics of a vehicle are governed by the behavior of its suspension and its interaction with the vehicle and tires. The suspension consists of a spring and damper unit with nonlinear properties. The suspension is connected to the vehicle chassis and the tires on either end. The interaction of the suspension unit coupled to the tire with the ground surface in ideal conditions results in minimal vibrations which are dampened out by the tire and the suspension units’ damper. However, in the presence of nonlinearities and road excitations [1], it becomes imperative to provide extra control effort to damp out the effect of these perturbations. The external road excitations arise either from the varying road profiles or road perturbations in the form of ditches or bumps.
In [2] a variable gain control scheme was proposed for an active suspension system faced with road excitations and nonlinear suspension dynamics. Robust control approaches such as integral sliding mode based control and fuzzy sliding mode control for the suspension dynamics were proposed in [3, 4], respectively. The work in [4], which considered the presence of nonlinear suspension stiffness and damping, performed analysis considering the nonlinearities piecewise linear. Further improvisations to the suspension control under influence of actuator faults and delays were done by providing reliable fuzzy control action in [5]. The control of suspension systems employing the dynamics of hydraulic actuators has also been discussed in the literature in [6–8]. In all these approaches the focus was to provide efficient control action to the suspension system ensuring ride quality and road holding at the same time.
Sliding mode [9–11] control has been established over time as a robust technique against perturbations and uncertainties. The works [9, 10] focused on developing sliding mode based observers for nonlinear systems faced with uncertainties and nonlinearities. The applications of sliding mode technique in combination with other high gain techniques were further explored in [11]. Sliding mode has been effectively employed to generate control for active suspension systems as discussed in [3, 8]. In [8] an output feedback control scheme based on first order sliding mode for an active suspension system faced with actuator nonlinearities and road disturbances was proposed. Similarly in [3] an integral sliding mode control was proposed for the linear model of an active suspension system. In [12] a first order sliding mode controller was proposed for the linear active suspension system presenting a comparative analysis between passive and actively controlled suspension system. However these approaches employed first order sliding mode techniques and hence were affected by chattering problems [13, 14]. The emergence of higher order sliding mode (HOSM) and particularly the supertwisting algorithm (STA) for systems with relative degree one has led to the reduction in chattering while providing the necessary control. Applications of higher order sliding mode control [15] in fields of unknown input estimations [16], vehicle control [17], and fuel cell systems [18] amongst others have been successfully explored. By improving on the disturbance rejections attributes of the STA, in [19–21] a generalized STA was proposed to ensure that linearly growing perturbations are also rejected by the control action. In [22] an adaptive STA based technique was proposed which focused on minimizing the chattering effect. Similarly in [23, 24] an adaptive STA based control action for an electropneumatic actuator was proposed. In this work the proposed bounded gains of the STA adapt to reject the linear perturbations and force the convergence of sliding mode in finite time. Higher order sliding mode based controllers for active suspension systems were proposed in [25, 26]. In both these works the second order sliding mode controllers based on STA were employed for the linear suspension system. The works in [5, 12, 25, 26] did not consider the nonlinear dynamics of the active suspension system. Similarly the works in [2, 3, 12, 25, 26] did not consider the presence of road roughness profile.
To address these issues, in this paper an adaptive higher order sliding mode controller is proposed for the nonlinear active suspension system. The nonlinear model of the active suspension systems for a quarter car is first developed by integrating the suspension nonlinearities due to damper and spring and the road excitations as disturbance inputs to the model. For analysis, the road roughness values based on power spectral density (PSD) values as proposed by International Organization for Standardization (ISO) [1] have been employed. Based on the roughness index different classes or road profiles are generated which serve as the external excitations to the suspension system. The influence of tire damping generally neglected in other works has also been considered in this work. For the nonlinear suspension system, a control based on modified adaptive STA is proposed to reduce the sprung mass acceleration and hence improve ride comfort. The proposed control also ensures that road holding condition is ensured such that the vehicle does not leave the ground and lead to instability. Analysis of the control performance for different classes of road excitations was performed and simulation results have been provided.
2. Problem Formulation
The dynamics of the suspension in the presence of spring and damper nonlinearities and road excitations can be given as [1, 6](1)Msz¨s=-Kszsu-Bsz˙su-φ+Us,Muz¨u=Kszsu+Bszsu-Ktzu-ξ-Btz˙u-ξ˙+φ-Us,where Ms represents the chassis/sprung mass, Mu is the tire/unsprung mass, Ks and Bs are the spring stiffness and damping, respectively, Kt and Bt represent the tire stiffness and damping, respectively, Us represents the active control input to be designed later, and ξ represents the external road excitation. The motions of the suspension system are represented by the sprung mass acceleration z¨s and the unsprung mass acceleration z¨u. Further, zsu represents the suspension stroke/displacement and zu is the unsprung mass displacement. The nonlinearity φ in the suspension dynamics is given as [1](2)φ=Knlzsu3+Bnlz˙susignz˙su,with Knl and Bnl representing the nonlinear percentages of the spring stiffness and damping, respectively. For the suspension system modeled in (1) the objective of the active control input Us in general varies with the specifications of applicability. The schematic representation of the suspension system is shown in Figure 1. The control input can be provided to improve the ride comfort or to ensure proper road holding. It may also be provided to minimize the rattle space and ensure the vehicle static weight is well supported. Ride comfort enhancement can be achieved by the reduction in the sprung mass acceleration while road holding can be ensured by ensuring a reduction in tire deflection, zξ=(zu-ξ). Similarly the rattle space can be limited by fixing the maximum possible suspension stroke with a suitable control effort. However it can be deduced from (1) that the control effort cannot achieve these objectives at the same time. Hence, the problem is formulated as providing control input for an underactuated suspension system with objective of reduction in sprung mass acceleration only. The control action provided abides by the following constraints:
Limiting Suspension Stroke [5]. The provided suspension control while achieving the increase in ride comfort must limit the suspension stroke to a mechanical constraint based on the design of the suspension. Hence, we have(3)zsu≤Zmax.
Road Holding [5]. It is intended to ensure that the control does not cause the vehicle to leave the ground and cause instability. Hence, we have(4)Ktzξ≤Ms+Mug.This ensures that the dynamic tire loading is less than the static vertical load and the tire does not leave the ground vertically.
Under the above control constraints, we propose an active control action based on the adaptive modified supertwisting HOSM algorithm to achieve the control objective of improvement in ride comfort by reduction in sprung mass acceleration. The block diagram representation of the proposed scheme has been shown in Figure 2.
The active suspension system.
The block representation of the proposed control.
3. HOSM Controller Design
In this section we discuss the design of the proposed adaptive modified STA based controller aimed at reducing the sprung mass acceleration. For the design of the controller, we can formulate the following state space model with the states as(5)x=x1x2x3x4T=zsuz˙szξz˙uT;x˙=010-1-KsMs-BsMs0Bsms0001ksmuBsMu-KtMu-Bs+BtMux+0-1Ms01Muφx,t+00-1Btmuξ˙︸fx,t+01Ms0-1mu︸gx,tu.The above dynamics of the suspension can be expressed as(6)x˙=fx,t+gx,tu.For the above system, the following sliding surface is selected:(7)s=z˙su.As system (5)–(7) has a relative degree one for s with respect to the input, Us, the dynamics of the sliding surface can be thus obtained as(8)s˙=z¨s-z¨u=αx,t+Mνs,where M=Ms+Mu/MsMu and ν(s) is the robust sliding mode control input designed based on the adaptive modified STA given as(9)ν=-κ1ϕ1st+ω,(10)w˙=-κ2ϕ2st,with ϕ1(s(t))=s+κ3s1/2sign(s) and ϕ2(s(t))=s+κ32/2signs+3/2κ3s1/2sign(s). The adaptive gains are given as [24](11)κ˙1=ω1μ12signs-δ,if κ1>δmζif κ1≤δm,(12)κ2=2ϵκ1,with κ3, μ1, δ, ζ, ϵ, and ω1 being positive constants. The conditions ζ1≥δ, ζ2>0 hold where δm is an arbitrary small constant. With this control law, the perturbation term α(x,t) for system (8) is now given as(13)αx,t=M-Kszsu-Bsz˙su-φx,t+KtMuzξ+BtMuz˙ξ=α1x,t+α2x,t.The perturbation terms are then given as(14)α1x,t=MBsz˙su,α2x,t=M-Kszsu-2Bsz˙su-φx,t+KtMuzξ+BtMuz˙ξ.The term α2 consists of nonlinear terms and disturbances. The term φ(x,t) is Lipschitz bounded whose Lipschitz constant can be easily computed as discussed in [1]. The quantity ξ refers to the road excitation that serves as a disturbance input for system (5). It can be assumed that this entity is bounded although its bound is not known a priori. The other terms in the disturbance α2 are functions of the system which are bounded for all practical operating conditions. Hence the boundedness of the disturbance terms in (14) can be established as(15)α1x,t≤ρ1s,(16)α˙2x,t≤ρ2.The disturbance term α2(x,t) consists of the suspension system dynamics and the disturbance input ξ, which is bounded but whose exact values are not known. Hence the value of the bound ρ2 cannot be determined a priori.
Theorem 1.
With condition (15) satisfied, controller (9)-(10) will ensure that the sliding surface (s) converges to a practically stable equilibrium in finite time.
Proof.
To establish the convergence, the sliding dynamics (8) can be rewritten along with the control input as(17)s˙1=s2-κ1ϕ1s1+α1x,t,s˙2=-κ2ϕ2s1+α˙2x,t,with s1=s and s2=α2(x,t)+β1(x,t)ω. One could notice that gain κ3 is used to withstand linearly growing perturbations. Since ϕ2s≥κ32/2 one can obtain(18)α˙2<2ρ2κ32ϕ2s.We consider a new state vector as [19](19)ξ=ξ1ξ2=ϕ1s1s2.Thus, system (17) can be rewritten as(20)ξ˙=1+κ32s1-1/2s2-κ1ϕ1s1+α1-κ2ϕ2s1+α˙2=1+κ32s1-1/2A0ξ+α10+0α˙2,where A0 is a Hurwitz matrix selected as(21)A0=-κ11-κ20.To perform the stability analysis of (17), the following Lyapunov candidate function is selected:(22)Vξ,κ=Vξ+12μ1κ1-κ1∗2+12μ2κ2-κ2∗2,where κ1∗>0 and κ2∗>0 are some constants. The function Vξ is given as [19](23)Vξ=ξTPξwith P=PT=λ+4ϵ2-2ϵ-2ϵ1, λ>0 and ϵ>0. It is worth noting that matrix P is positive definite if λ and ϵ are any real number. The time derivative of Vξ can be obtained as(24)V˙ξ=1+κ32s1-1/2·ξTA0TP+PA0ξ+2ξTPα10+2ξTP0α˙2.From (18), one can note that(25)2ξTP0α˙2≤2ξTP02ρ2κ32ϕ2=1+κ32s1-1/22ξTP02ρ2κ32ξ1.Denoting 2ρ2/κ32=ρ3, where ρ3 is unknown, we thus obtain(26)V˙ξ≤1+κ32s1-1/2·ξTA0TP+PA0ξ+2ξTPα10+1+κ32s1-1/22ξTP0ρ3ξ1.Using bounds on the perturbation (15), we can obtain(27)V˙ξ≤1+κ32s1-1/2·ξTA0TP+PA0ξ+2ξTPρ1ξ10+1+κ32s1-1/22ξTP0ρ3ξ1.Furthermore, one can obtain, similar to [19],(28)V˙ξ≤-1+κ32s1-1/2ξTQξwith(29)Q=Q1Q2Q2Q3,Q1=2κ1-ρ1λ+4ϵ2-4ϵκ2-ρ3,Q2=κ2-2ϵκ1+2ϵρ1+ρ3-λ+4ϵ2,Q3=4ϵ.In order to guarantee the positive definiteness of matrix Q, we can choose(30)κ2=2ϵκ1.The matrix Q is positive definite if(31)κ1>ρ1λϵ+4ρ1ϵ3-2ϵ2ρ3λϵ+2ϵρ1+ρ3-λ+4ϵ228λϵ.From (19), it can be deduced that(32)ξ2=ξ12+ξ22≥κ32s1.It can be further shown that(33)V˙ξ=-r1Vξ1/2-r2Vξwith(34)r1=-γλminQλmax1/2Pκ322,r2=γλminQλmaxPwith 1>γ>0.
Considering γκ1=κ1-κ1∗ and γκ2=κ2-κ2∗, the time derivative of the Lyapunov candidate function V(ξ1,ξ2,κ1,κ2) can thus be written as(35)V˙ξ,κ≤-r1Vξ1/2-r2Vξ+1μ1γκ1κ˙1+1μ2γκ2κ˙2≤-r1Vξ1/2-r2Vξ-ω12μ1γκ1-ω22μ2γκ2+1μ1γκ1κ˙1+1μ2γκ2κ˙2+ω12μ1γκ1+ω22μ2γκ2.Employing the inequality(36)x2+y2+z21/2≤x+y+z,one could obtain(37)-r1Vξ1/2-ω12μ1γκ1-ω22μ2γκ2≤-γ1Vξ1,ξ2,κ1,κ2,where γ1=min(r1,ω1,ω2). Thus, the following can be written:(38)V˙ξ,κ≤-γ1Vξ,κ-r2Vξ+1μ1γκ1κ˙1+1μ2γκ2κ˙2+ω12μ1γκ1+ω22μ2γκ2.Thus the following can now be written:(39)V˙ξ,κ≤-γ1Vξ,κ-r2Vξ+Δwith Δ=-γκ1(κ˙1/μ1-ω1/2μ1)-γκ2(κ˙2/μ2-ω2/2μ2). Now, considering that s1>δ and κ1>δm∀t>0, we can obtain from (11)(40)κ˙1=ω1μ12,(41)Δ=-γκ2κ˙2-ω22μ2.Taking ϵ=ω2/2μ2/μ1, we can obtain, with Δ=0,(42)V˙ξ,κ≤-γ1Vξ,κ-r2Vξ.For the finite time convergence it is necessary that κ1 satisfies inequality (31). It implies that κ1 increases in accordance with (40) until (31) is met which guarantees the positive definiteness of matrix Q and validity of (42) in finite time to the domain |s|≤δ.
In the condition when s<δ, the adaptive gain κ1 can be obtained as(43)κ˙1=-ω1μ12,if κ1>δmζif κ1≤δmand the term Δ is given as(44)Δ=2κ1-κ1∗12μ1,if κ1>δm-δm-κ1∗+ζ·tζμ1-ω12μ1if κ1≤δm.It can be deduced that second equation in (44) is valid for a finite duration. As κ1 soon becomes less than or equal to δm, its value starts increasing such that κ1=δm+ζ·t. Then the first equation in (44) is valid. The derivative of the Lyapunov function (39) becomes sign indefinite and s may become larger than δ due to the decrease in the adaptive gains. During the adaptation process the sliding variable reaches the domain s<δ in finite time and then may leave the domain in finite time. It is however guaranteed that it always stays in a larger domain s≤ζ1, ζ1>δ in the real sliding mode. It can be shown, similar to [24], that the real sliding mode is defined in the domain(45)W=s1,s˙1:s1≤ζ2,ζ1>δ.
Remark 2.
The adaptive gains κ1 and κ2 are bounded. In the domain δ<s1≤ζ1, a solution to (11) can be obtained as(46)κ1=κ10+ω1μ12·t,0≤t≤t1,where t1 is finite time. Thus the gain κ1 is bounded, and since κ2 is a function of κ1 hence it is also bounded.
4. Results and Discussion
To validate the proposed control scheme, the quarter and active suspension parameters have been considered as Ms=342.5 kg, Mu=40 kg, Ks=20000 N/m, Bs=1000 Ns/m, Kt=268000 N/m, and Bt=14.6 Ns/m. The parameters for the design of adaptive modified STA controller were selected as K3=310, ω1=150, ζ=4.5, δm=4.5, δ=2, μ1=2, and ϵ=0.0021 such that (30)-(31) are complied with. These gains have been selected after repeated trials such that developed controller is robust for different classes of road profiles and ensures high performance. Two cases considering the absence and presence of external road excitation were considered to show the effectiveness of the proposed scheme.
4.1. Case A: No Road Excitation
The analysis was initially focused on the dynamics of the suspension system in the absence of any road profile (i.e., ξ=0). In such a scenario the suspension deflection (zsu), tire deflection (zξ), and sprung mass acceleration (z¨s) were analyzed. In Figures 3(a)–3(c), the variations in suspension deflection, tire deflection, and the sprung mass accelerations have been shown, respectively. A comparison between the above suspension measurements in the passive control (no control input) mode and the active control (control input based on adaptive modified STA) has been also shown. During the passive control, the RMS (root mean square) value of the sprung mass acceleration was obtained as 0.0724 m/s^{2}. In comparison the RMS value of the sprung mass acceleration was obtained as 0.0588 m/s^{2}. It has been further shown in Figure 3(d) that the sliding surface is attained and converges to the origin.
No road profile: (a) suspension deflection, (b) tire deflection, (c) sprung mass acceleration, and (d) sliding surface.
4.2. Case B: External Road Profile
In this case different classes of road profiles were considered with the road conditions varying from good to very poor roads. The vehicle was considered to be traveling at a constant longitudinal speed of 37 km/hr. The road excitation was modeled as a random process with a ground displacement PSD given as(47)ξ˙t=-2πn0vξt+2πσ0vw0,where v is the vehicle longitudinal velocity, σ0 is the road roughness coefficient, n0 is the reference space frequency, and w0 is the Gaussian white noise. Under such driving conditions varying degrees of roughness (σ0) were considered to replicate different classes of road profile. Based on the degree of roughness [1], the road classes were determined as good (σ0=16×10-6), average (σ0=64×10-6), poor (σ0=256×10-6), and very poor (σ0=1024×10-6). The PSD of different classes of roads profile has been shown in Figure 4. It can be clearly seen that with increase in road roughness the PSD of the excitations increases.
PSD for different road classes excitation.
For different classes of road profile discussed earlier, we selected the poor road as an external excitation and the simulation was carried out. For poor class road, the road excitation ξ and the rate of road excitation have been shown in Figure 5.
(a) Road profile for poor road. (b) Rate of road profile for poor road.
To meet the control objective of increase in ride comfort by reducing the sprung mass acceleration, first it has been ensured that the control constraints discussed in (3)-(4) have been met. Thus under the effect of the active control action proposed in this work the reduction in suspension deflection magnitude in comparison to the passive case has been shown in Figure 6. The reduction in suspension deflection caters to the control constraint discussed earlier in (3) and is less than 0.01 m.
The suspension deflection, zsu.
While ensuring that the suspension stroke is limited, the effective tire load has been shown in Figure 7(a). It can be seen that the dynamic load is less than the static load which meets the control constraint discussed in (4). With the control constraints met, the control effort that ensures the control objective is met is shown in Figure 7(b). In Figure 8 the adaptive gains K1 and K2 of the proposed adaptive control scheme in (10) have been shown.
(a) Effective tire loading. (b) Control input provided.
(a) Adaptive gain, K1. (b) The dynamic gain K2.
It can be seen that the gains change based on the perturbation and are not unbounded. Under the effect of this control action, the sliding surface which represents the rate of suspension stroke movement has been shown in Figure 9. It can be seen that the sliding surface has converged to a finite bound, thus ensuring practical stability.
The sliding surface, σ.
With the convergence of the sliding surface, the control objective of reduction in sprung mass acceleration is met and shown in Figure 10. It can be seen that in comparison to the passive suspension system there is a reduction in magnitude of sprung mass acceleration. During the passive control, the RMS value of the sprung mass acceleration was obtained as 0.0724 m/s^{2}. In comparison the RMS value of the sprung mass acceleration was obtained as 0.0588 m/s^{2} during active control. In Figure 11 the PSD of the sprung mass acceleration has been shown. It can be seen that under the effect of control action the performance of active controlled suspension is better in the frequency range of 1 Hz to 5 Hz which corresponds to the ride comfort frequency range. Thus it can be concluded that the proposed control action achieves the desired control objective by enhancing ride comfort.
The sprung mass acceleration, z¨s.
The PSD of sprung mass acceleration.
To ensure that the provided control does not reduce the ride quality or lead to instability, the PSD of the unsprung mass acceleration should also be reduced and stay within definite limits. In Figure 12 the PSD of the unsprung mass acceleration has been shown.
The PSD of unsprung mass acceleration.
For further analysis purposes the proposed controller scheme was applied to the suspension system under the effect of various classes of road profiles. The RMS values of the sprung mass acceleration, suspension deflection, and tire deflection for these cases under the influence of active and passive control are shown in Table 1. It can be deduced from the results shown in Table 1 that employing the proposed modified adaptive STA based controller improves the performance considerably. The proposed control strategy in this paper counters the effect of Lipschitz nonlinearities of the suspension system and the linearly growing perturbations represented by the road profile. Hence a controlled performance is provided enhancing the ride comfort while simultaneously maintaining road holding and constraining suspension stroke to specified mechanical limits.
RMS values of suspension deflection, tire deflection, and vertical acceleration.
Road class
zsu (m)
zξ (m)
(z¨s) (m/s^{2})
Passive
Active
Passive
Active
Passive
Active
B (good)
0.000919
0.000272
0.000729
0.000682
0.5440
0.5253
C (average)
0.0024
0.0019
0.0019
0.0013
1.4068
0.9462
D (poor)
0.0051
0.0028
0.0040
0.0019
2.9747
1.4017
E (very poor)
0.0103
0.0032
0.0081
0.0040
6.0613
2.9376
5. Conclusion
In this work an active suspension control scheme was proposed based on the modified adaptive STA for a nonlinear suspension system faced with nonlinearities and road excitations. The proposed control provided improvement in ride comfort for various classes of road profiles while ensuring that road holding capabilities of the vehicle were not compromised. In comparison to previous works in the literature the proposed controller was designed for linearly growing perturbations and was suitably employed for the suspension system. Simulation results were performed for different classes and presented results show the effectiveness of the proposed scheme.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant 2014R1A1A2A10056145).
RathJ.VeluvoluK.DefoortM.Simultaneous estimation of road profile and tyre road friction for automotive vehicleTahboubK.Active nonlinear vehicle-suspension variable gain controlProceedings of the 13th IEEE International Mediterrean Symposium on Control and Automation: Intelligent ControlJune 2005569574SamY. M.OsmanJ. H.GhaniM. R.A class of proportional-integral sliding mode control with application to active suspension systemYagizN.HaciogluY.TaskinY.Fuzzy sliding-mode control of active suspensionsLiH.LiuH.GaoH.ShiP.Reliable fuzzy control for active suspension systems with actuator delay and faultRajamaniR.HedrickJ.Performance of active automotive suspensions with hydraulic actuators: theory and experiment2Proceedings of the American Control Conference1994IEEE1214121810.1109/ACC.1994.752251MaM.-M.ChenH.Disturbance attenuation control of active suspension with non-linear actuator dynamicsXiaoL.ZhuY.Sliding-mode output feedback control for active suspension with nonlinear actuator dynamicsVeluvoluK. C.SohY. C.CaoW.LiuZ. Y.Observer with multiple sliding modes for a class of nonlinear uncertain systemsProceedings of the American Control Conference (ACC '05)June 2005244524502-s2.0-23944501384VeluvoluK. C.SohY. C.Multiple sliding mode observers and unknown input estimations for Lipschitz nonlinear systemsVeluvoluK. C.LeeD.Sliding mode high-gain observers for a class of uncertain nonlinear systemsChamseddineA.NouraH.RaharijaonaT.Control of linear full vehicle active suspension system using sliding mode techniquesProceedings of the IEEE International Conference on Control Applications (CCA '06)October 2006Munich, GermanyIEEE1306131110.1109/cacsd-cca-isic.2006.47768312-s2.0-74049091439VeluvoluK. C.SohY. C.Fault reconstruction and state estimation with sliding mode observers for Lipschitz non-linear systemsBoikoI.FridmanL.Analysis of chattering in continuous sliding-mode controllersKamalS.ChalangaA.MorenoJ. A.FridmanL.BandyopadhyayB.Higher order super twisting algorithmProceedings of the 13th International Workshop on Variable Structure Systems201415DefoortM.VeluvoluK. C.RathJ. J.DjemaiM.Adaptive sensor and actuator fault estimation for a class of uncertain Lipschitz nonlinear systemsImineH.FridmanL. M.MadaniT.Steering control for rollover avoidance of heavy vehiclesEvangelistaC.PulestonP.KunuschC.Feasibility study of variable gain Super-Twisting control in fuel cells based systemsProceedings of the 13th International Workshop on Variable Structure Systems (VSS '14)July 2014Nantes, FranceIEEE1610.1109/vss.2014.68811222-s2.0-84906815094MorenoJ. A.OsorioM.Strict Lyapunov functions for the super-twisting algorithmMorenoJ. A.On strict Lyapunov functions for some non-homogeneous super-twisting algorithmsMorenoJ. A.Lyapunov analysis of non homogeneous Super-Twisting algorithmsProceedings of the 11th International Workshop on Variable Structure Systems (VSS '10)June 201053453910.1109/vss.2010.55446722-s2.0-77956581874UtkinV. I.PoznyakA. S.OrdazP.Adaptive super-twist control with minimal chattering effectProceedings of the 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC '11)December 20117009701410.1109/cdc.2011.61607202-s2.0-84860674377PlestanF.ShtesselY.BrégeaultV.PoznyakA.New methodologies for adaptive sliding mode controlShtesselY.TalebM.PlestanF.A novel adaptive-gain supertwisting sliding mode controller: methodology and applicationToyamaS.IkedaF.SorimachiY.A second order sliding mode controller for active suspension systemsProceedings of the International Conference on Control, Automation and SystemsOctober 20081823Sánchez-TorresJ. D.LoukianovA. G.GaliciaM. I.RuizJ.RiveraJ.ABS and active suspension control via high order sliding modes and linear geometric methods for disturbance rejectionProceedings of the 8th International Conference on Electrical Engineering, Computing Science and Automatic ControlOctober 20111610.1109/iceee.2011.61065732-s2.0-84855765636