Robust and Adaptive Higher Order Sliding mode controllers

Mohamed Harmouche∗, Salah Laghrouche∗ and Yacine Chitour∗∗∗SET Laboratory, UTBM, Belfort, France.∗∗L2S Laboratory, Supelec, Paris, France.

mohamed.harmouche@utbm.fr, salah.laghrouche@utbm.fr, yacine.chitour@lss.supelec.fr

Abstract— In this paper, we present Lyapunov-basedrobust and adaptive Higher Order Sliding Mode (HOSM)controllers for nonlinear SISO systems with bounded un-certainty. The proposed controllers can be designed forany arbitrary sliding mode order. The uncertainty boundsare known in the robust control problem whereas they arepartially known in the adaptive control problem. Both theseproblems are formulated as the finite time stabilization of achain of integrators with bounded uncertainty. Saturationfunctions are used for gain adaptation, which do notlet the states exit the neighborhood after convergence.The effectiveness of these controllers is illustrated throughsimulations.

I. INTRODUCTION

Nonlinear dynamic physical systems are difficult tocharacterize due to parametric uncertainty arising fromvarying operating conditions and external perturbationsthat affect the physical characteristics of systems. Thevariation limits or the bounds of this uncertainty mightbe known or unknown. Sliding mode control (SMC) [1]is a well-known method for control of nonlinear systems,renowned for its insensitivity to parametric uncertaintyand external disturbance. This technique is based on ap-plying discontinuous control on a system which ensuresconvergence of the output function (sliding variable)in finite time to a manifold of the state-space, calledthe sliding manifold [2]. In practice, SMC suffers fromchattering; the phenomenon of finite-frequency, finite-amplitude oscillations in the output which appear be-cause the high-frequency switching excites unmodeleddynamics of the closed loop system [3]. Higher OrderSliding Mode Control (HOSMC) is an effective methodfor chattering attenuation [4]. Many HOSMC algorithmsexist in contemporary literature for control of uncertainnonlinear systems, where the bounds on uncertainty areknown. Levant for example, has presented a method ofdesigning arbitrary order sliding mode controllers forSingle Input Single Output (SISO) systems in [5]. Inhis recent works [6], [7], homogeneity approach hasbeen used to demonstrate finite time stabilization of theproposed method. Laghrouche et al. [8] have proposed atwo part integral sliding mode based control to deal withthe finite time stabilization problem and uncertainty re-jection problem separately. Dinuzzo et al. have proposedanother method in [9], where the problem of HOSM

has been treated as Robust Fuller’s problem. Defoortet al. [10] have developed a robust Multi Input MultiOutput (MIMO) HOSM controller, using a constructivealgorithm with geometric homogeneity based finite timestabilization of an integrator chain. Harmouche et al.have also presented their homogeneous controller in [11]based on the works of Hong [12].

The case where the bounds on uncertainty exist, butare unknown, is still an open problem in the field ofarbitrary HOSMC. Huang et al. [13] presented an adap-tation law for first order SMC, which depends directlyupon the sliding variable. This method does not solvethe gain overestimation problem as the gains stabilizeat unnecessarily large values. Plestan et al. [14], [15]have overcome this problem by slowly decreasing thegains once sliding mode is achieved. This method yieldsconvergence to a neighborhood of the sliding surface.However it does not guarantee that the states wouldremain inside the neighborhood after convergence; thestates actually overshoot in a known region around theneighborhood. Shtessel et al. [16] have presented aSecond Order adaptive gain SMC for non-overestimationof the control gains, based on supertwisting algorithm.The states overshoot in this case as well, but unlike [14],the magnitude is unknown. To the best of our knowledge,no contemporary work on adaptive HOSMC has beenpublished for orders greater than two.

In this paper, we present Lyapunov-based robust andadaptive Higher Order Sliding Mode Controllers fornonlinear SISO systems with bounded uncertainty. Thisproblem has been formulated as the stabilization of achain of integrators with bounded uncertainties whichis equivalent to the stabilization of the following rth

differential equation in finite time [5],

s(r) = ϕ(t)+ γ(t)u,

where s(x, t) is a smooth output-feedback function (slid-ing variable) and s(r) is its rth time derivative. The termu is the control input and ϕ(t) and γ(t) are boundeduncertain functions.

There are two main contributions in this paper. First,a Lyapunov-based approach for arbitrary HOSMC isdeveloped. The controller establishes sliding mode ofany arbitrary order under the condition that the bounds

51st IEEE Conference on Decision and ControlDecember 10-13, 2012. Maui, Hawaii, USA

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of ϕ(t) and γ(t) are known. The advantage of ourmethod is that robust HOSM controllers are developedfrom a class of finite-time controllers for pure integratorchains. The second contribution in this paper is the ex-tension of the robust controller to an adaptive controllerfor the case where the bounds on ϕ(t) are unknown.The proposed adaptive controller guarantees finite timeconvergence to an adjustable arbitrary neighborhood oforigin, i.e. it establishes real HOSM. The advantageof this adaptive controller design, compared to otheralgorithms mentioned before, is that this controller canbe extended to any arbitrary order and the adaptationrates are fast in both directions. Therefore, the amplitudeof the discontinuous control decreases faster as comparedto [14]. In addition, the states are confined inside theneighborhood after convergence and cannot escape. Asa result, there are no state overshoots and no gainoverestimation in this controller, and the neighborhoodof convergence can be chosen as small as possible.

The paper has been organized as follows: the problemformulation has been presented in section 2. The designof the robust controller has been presented in section 3,and that of adaptive controller in section 4. Simulationresults have been presented and discussed in section5 and some concluding remarks have been given insection 6.

II. PROBLEM FORMULATION

Let us consider an uncertain nonlinear system:{x = f (x, t)+g(x, t)u,y = s(x, t), (1)

where x ∈ Rn is the state vector and u ∈ R is the inputcontrol. The sliding variable s is a measured smoothoutput-feedback function and f (x, t) and g(x, t) areuncertain smooth functions. Let us assume:

H1. The relative degree r of the System (1) withrespect to s is constant and known, and the associatedzero dynamics are stable.

The control objective is to fulfil the constraint s(x, t) = 0in finite time and to keep it exact by discontinuousfeedback control. The rth-order sliding mode is definedas follows:

Definition 2.1 [17]. Consider the nonlinear System(1), and let the system be closed by some possibly-dynamical discontinuous feedback. Then, providedthat s, s, ...,s(r−1) are continuous functions, and the setSr = {x|s(x, t) = s(x, t) = ...= s(r−1)(x, t) = 0}, called”rth-order sliding set”, is non-empty and is locally anintegral set in the Filippov sense [18], the motion on

Sr is called ”rth-order sliding mode” with respect to thesliding variable s.

The rth-order SMC approach allows the finite timestabilization to zero of the sliding variable s andits r − 1 first time derivatives by defining a suitablediscontinuous control function. If the system (1) isextended by the introduction of a fictitious variablexn+1 = t, xn+1 = 1, and fe = ( f T 1)

T,ge =

(gT 0)T

(wherethe last component corresponds to xn+1), then the outputs satisfies the equation [5]:

s(r) = ϕ(s)+ γ(s)u, with γ(s) = LgeLr−1fe s and

ϕ(s) = Lrfe s.

For x ∈ X ⊂ Rn, X being a bounded open subset of Rn

containing the origin, the functions ϕ , γ are boundedand it is also customary to assume that γ has a positivelower bound that depends only on X . Then the rth-orderSMC of (1) with respect to the sliding variable s becomesequivalent to the finite time stabilization of the followingsystem, {

zi = zi+1, i = 1, ...,r−1,zr = ϕ(t)+ γ(t)u, (2)

where z = [z1 z2 ...zr]T := [s s... s(r−1)]T . The functions

ϕ and γ are subject to the following hypothesis:H2. The functions ϕ(t) and γ(t) are bounded uncertainfunctions i.e. there exist constants Km,KM > 0 and ϕ0≥ 0such that

0 < Km ≤ γ(t)≤ KM, |ϕ(t)| ≤ ϕ0, ∀t ≥ 0,

This hypothesis implies that results in the proceedingsections of the paper can be only be local to the originwhen applied to the rth-order SMC approach, unless ap-propriate global boundedness assumptions are imposedon LgeLr−1

fe s and Lrfes.

Remark 1: rth-order sliding mode can also be es-tablished on a system with a relative degree ρ < rby increasing the length of integrator chain by r− ρ

integrators [19]. In other words, rth-order sliding modecan be established by applying the discontinuous controlon the (r−ρ)th time derivative of the control input. Forthe sake of clarity, we will consider r = ρ in all furthersections.In the following sections, a robust controller is developedfirst for the system (2) under hypothesis H2, consideringthat the uncertainty bounds presented in H2 are known.Then, an adaptive controller is developed to extend thefunctionality of the robust controller to the case wherethe bounds of ϕ are not known.

III. DESIGN OF ROBUST HIGHER ORDER SLIDINGMODE CONTROLLER

In this section, we present a robust controller whichstabilizes System (2), considering that the bounds on ϕ

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and γ are known. This controller has been derived froma class of Lyapunov-based controllers that guaranteefinite time stabilization of pure integrator chains, andsatisfy certain additional geometric conditions. The pureintegrator chain (ϕ ≡ 0 and γ ≡ 1) is represented asfollows: {

zi = zi+1, i = 1, ...,r−1,zr = u. (3)

Let us recall the theorem:Theorem 1: [20] Consider System (3). Suppose there

exists a continuous state-feedback control law u = u0(z),a positive definite C1 function V1 defined on a neighbor-hood U ⊂ Rr of the origin and real numbers c > 0 and0 < α < 1, such that the following condition is true forevery trajectory z of system (3),

V1 + cV1α(z(t))6 0, if z(t) ∈ U , (4)

where V1 is the time derivative of V1(z).Then all trajectories of System (3) with the feedbacku0(z) which stay in U converge to zero in finite time.If U = Rr and V1 is radially unbounded, then System(3) with the feedback u0(z) is globally finite time stablewith respect to the origin.Based on this theorem, we now present a robust con-troller for System (2).

Theorem 2: [21] Consider System (2) subject to Hy-pothesis H2. Then the following control law establishesHigher Order Sliding Mode with respect to s in finitetime:

u =1

Km(u0 +ϕ0sign(u0)), (5)

where u0(z) is any state-feedback control law that satis-fies Theorem 1 and obeys the following further condi-tions:

∂V1

∂ zr(z)u0(z)≤ 0, and u0(z) = 0⇒ ∂V1

∂ zr(z) = 0, ∀z ∈ U .

(6)If U =Rr and V1 is radially unbounded, then System (2)with the feedback u(z) is globally finite time stable withrespect to the origin.Proof of this theorem can be found in [21].For a ∈ R, the function sign(a) is defined as follows,

sign(a)

{=

a|a|

, a 6= 0,

∈ [−1,1] , a = 0.(7)

Therefore the solutions of System (2) with the feedbacklaw (5) are the solutions of a differential inclusionand are understood in Filippov’s sense. Note that u isdiscontinuous only when u0 = 0.This result becomes non empty if controllers satisfyingTheorem 1 and Condition (6) can be identified. It canbe verified that the controllers proposed by Hong [22]and Huang [23] fulfill these conditions. Let us consider

Hong’s controller as an example.For simplicity, let us introduce baeκ to denote|a|κ sign(a) for a ∈ R and κ > 0. Then Hong’scontroller [22] is defined as follows.

Let k < 0 and l1, · · · , lr positive real numbers. Forz = (z1, · · · ,zr), we define for i = 0, ...,r−1:

pi = 1+(i−1)k,v0 = 0, vi+1 =−li+1bbzi+1eβi −bvieβie(αi+1/(βi),

(8)

where αi =pi+1

pi, for i = 1, ...,r, and, for k < 0 suffi-

ciently small,

β0 = p2, (βi +1)pi+1 = β0 +1 > 0, i = 1, ...,r−1.(9)

Consider the positive definite radially unbounded func-tion V1 : Rr→ R+ given by

V1 =r

∑j=1

z j∫v j−1

bseβ j−1 −⌊v j−1

⌉β j−1ds. (10)

It has been proved in [22] that for a sufficiently small k,there exist li > 0, i = 1, ...,r, such that the control lawu0 = vr defined above stabilizes System (3) in finite timeand there exists c > 0 and 0 < α < 1 such that u0 andV1 fulfill the conditions of Theorem 1. Moreover,

∂V1

∂ zr= bzreβr−1 −bvr−1eβr−1 ,

u0(z) = vr =−lr⌊bzreβr−1 −bvr−1eβr−1

⌉ αrβr−1 .

(11)It can be verified that∂V1

∂ zru0(z)≤ 0 and u0(z) = 0⇒ ∂V1

∂ zr= 0.

The feedback law of [22] can be simplified by choosingall βi = 1 in (8) as explained below:

Proposition 1: [21] For System (3), there exist asufficiently small k < 0 and real numbers li > 0, such thatthe control law u0 = vr defined below stabilizes System(3) in finite time.For i = 0, ...,r−1,

v0 = 0, vi+1 =−li+1bzi+1− vie1+(i+2)k1+(i+1)k . (12)

IV. ADAPTIVE CONTROLLER

We shall now consider the case for System (2) wherethe uncertainty bound on ϕ is unknown. More precisely,according to the definitions given in H2, the controldesign requires the a-priori knowledge of Km onlywhereas ϕ0 and KM need not be known.

For any a ∈ R, let σ(a) be the standard saturationfunction defined by

σ(a) =a

max(1, |a|). (13)

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For ε > 0, a ∈ R, we define νε(a) by

νε(a) =12+

12

σ

(|a|− 3

4 ε

14 ε

). (14)

Our main result for the adaptive case is given by thefollowing theorem.

Theorem 3: Consider System (2), subject to assump-tion H1 and H2, and let u0, V1 be the control law and theLyapunov function respectively, provided by Theorem1.Then the following holds true,∃M > 0 such that ∀ε > 0 small enough and 0 < η < 1,∃kε > 0 such that, for every non zero initial conditionz0 ∈ U ; if uε is the controller defined by:

uε =1

Km(u0 + ϕε sign(u0)) , (15)

where (z, ϕε) is the solution of the Cauchy problemdefined by (2) starting at (z0,0) (i.e. z(0) = z0 andϕε(0) = 0) with the controller uε and the dynamics

˙ϕ = kε νε(V1(z(t)))− (1−νε(V1(z(t))))bϕεeη +σ(V1),(16)

then we have:

(i) limsupt→∞

V1(z(t))≤ ε;

(ii) limsupt→∞

|ϕε | ≤Mmε ϕ0

εα, where mε =max

V1≤ε

∣∣∣∣∂V1

∂ zr

∣∣∣∣.Proof of this theorem can be found in [21].

Remark 2: As uε is discontinuous when u0 = 0, thesolutions of System (2) with the controller uε are solu-tions of a differential inclusion and must be understoodin Filippov’s sense.

Remark 3: Inequality (i) of Theorem 3 is equivalentto Levant’s concept of real Higher Order Sliding Mode,defined as

∃t1 > 0 : ∀t > t1, |zi(t)| ≤ µi, i = 1, · · · ,r−1,

where µi is an arbitrarily small positive number. Thisis equivalent to practical stability of z1, · · · ,zr. Detailson real sliding mode and real HOSM can be found inSection 2 of [17].

Remark 4: For both choices of controllers from [22]or Proposition 1, it is not difficult to show that thereexists C0,γ > 0 such that for sufficiently small ε > 0, wehave mε ≤ C0ε

γ with γ < α . Then, both upper boundsfor the gain value kε and the upper bound on ϕε tend toinfinity as ε tends to zero.

V. SIMULATION RESULTS

The performance of the control laws presented in theprevious sections has been evaluated through simulation.Let us consider an academic kinematic model of a car[5] (see Fig.1), given by:

x2

x1

x4

x3

l

Fig. 1. Kinematic car model.

x1x2x3x4

=

wcos(x3)wsin(x3)w/

L tan(x4)0

+

0001

u, (17)

where x1 and x2 are the cartesian coordinates of therear axle middle point, x3 is the orientation angle, x4is the steering angle and u is the control input. w isthe longitudinal velocity (w = 10ms−1), and L is thedistance between the two axles (L = 5m). All the statevariables are assumed to be measured and the velocityis assumed to have an uncertainty of δw = 5% .

The goal is to steer the car from a given initial positionto the trajectory x2re f = 10sin(0.05x1) + 5 in finitetime. Considering the sliding variable s = x2−x2re f : therelative degree of the system w.r.t. s is 3. The 3rd timederivative of s is s(3) = ϕ + γu, where

ϕ = [1

800cos( x1

20

).(cos(x3))

2

− 140L

sin( x1

20

).sin(x3) .tan(x4)]w3 cos(x3)

+[− 120

sin( x1

20

)cos(x3)sin(x3) ,

γ =w2

L

[12

cos( x1

20

)sin(x3)+ cos(x3)

].[1+ tan2 (x4)

].

The robust control law can hence be expressed as

u =1

Km(u0 +ϕ0sign(u0)),

where u0 is determined below,

v1 = −l1bbz1eβ0 −0eα1/β0 ,

v2 = −l2bbz2eβ1 −bv1eβ1eα2/β1 ,

u0 = v3 =−l3bbz3eβ2 −bv2eβ1eα3/β2 .

(18)

Figures 2 to 4 illustrate the simulation results of therobust controller on the car model. Fig.2 shows that thesliding variable s and its first two time derivatives s and sconverge exactly to zero. Fig.3 shows the discontinuouscontrol u. Fig.4 shows that x2 converges to the desired

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trajectory. These results show the applicability and ro-bustness of the controller. The adaptive control law canbe expressed as:

u =1

Km(u0 + ϕε sign(u0)) ,

where u0 is determined in the same way as in (18). Fig.5shows the convergence of s, s and s under the adaptivecontroller. Fig.7 shows the dynamics of ϕε . It can be seenthat the states converge to zero in 4 seconds, and thenϕε starts to decrease rapidly. During this time, the statesare kept equal to zero, ensuring an ideal HOSM. This isbecause the gains are overestimated in this duration. InFig.6, it can be seen that real HOSM is established after11 sec because the gains are just sufficient to counteractthe uncertainty. The states converge to neighborhoodof zero defined by V1 ≤ ε , i.e. V1 ≤ 1. Fig.8 showsthe control u where the amplitude of the discontinuouspart of u adapts to counteract the uncertainty. Fig.9shows the convergence of x2 to the desired trajectoryx2re f . Comparing Figures 3 and 8, it can be seen thatafter convergence, the amplitude of the control signal issignificantly smaller when adaptive gains are applied.

VI. CONCLUSIONS

In this paper, we have presented two HOSMC controllersfor the stabilization of nonlinear uncertain SISO systems.The first controller has been developed from a class ofLyapunov based controllers for the stabilization of a pureintegrator chain. This controller is robust and dependsupon the knowledge of the perturbation bounds. Thesecond controller is adaptive, as it does not require anyquantitative knowledge of the perturbation bounds of theuncertain function ϕ , while the bounds of the uncertainfunction γ need to be known. The simulation resultsillustrate the good performance and effectiveness of boththese controllers. In future works, we aim to addressthis issue and develop adaptive controllers which arecompletely independent of the knowledge of the bounds.

0 1 2 3 4 5 6 7 8 9 10−100

−50

0

50

100

150

200

250

Time (sec.)

s (m)

s(1)

(m/s)

s(2)

(m/s2)

Fig. 2. s(m), s (ms−1) and s (ms−2) versus time (s).

0 1 2 3 4 5 6 7 8 9 10−8

−6

−4

−2

0

2

4

6

8

10

12

Time (sec.)

Control u

Fig. 3. control law u versus time (s).

0 1 2 3 4 5 6 7 8 9 10−10

−5

0

5

10

15

20

Time (sec.)

x2 (m)

x2ref

(m)

Fig. 4. x2 (m) and x2re f (m) versus time (s)

0 2 4 6 8 10 12 14 16 18 20−50

0

50

100

150

200

Time (sec.)

5 10 15 20−1

0

1

s (m)

s(1)

(m/s)

s(2)

(m/s2)

Fig. 5. s(m), s (ms−1) and s (ms−2) versus time (s).

0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

8x 10

5

Time (sec.)

V1

5 10 15 200

0.5

1

Fig. 6. V1 versus time (s).

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0 2 4 6 8 10 12 14 16 18 200

20

40

60

80

100

120

140

160

Time (sec.)

ϕε

Fig. 7. ϕε versus time (s).

0 2 4 6 8 10 12 14 16 18 20−4

−3

−2

−1

0

1

2

3

4

5

6

Time (sec.)

Control u

Fig. 8. control law u versus time (s).

0 2 4 6 8 10 12 14 16 18 20−10

−5

0

5

10

15

20

Time (sec.)

x2 (m)

x2ref

(m)

Fig. 9. x2 (m) and x2re f (m) versus time (s)

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