RESEARCH ARTICLE Robust Exact Finite-Time...

January 25, 2011 10:36 International Journal of Systems Science paper-final1

International Journal of Systems ScienceVol. 00, No. 00, Month 200x, 1–16

RESEARCH ARTICLE

Robust Exact Finite-Time Output Based Control using

High-Order Sliding Modes

Marco Tulio Anguloa∗, Leonid Fridmana and Arie Levantb

aDepartamento de Control, Facultad de Ingenierıa, UNAM, Mexico; bSchool of Mathematical Sciences,Tel-Aviv University, Ramat-Aviv, Tel-Aviv, Israel.

(Received 00 Month 200x; final version received 00 Month 200x)

Linear time-invariant systems with matched perturbations are exactly stabilized in finite time by means ofdynamic output-feedback control under the assumptions of a permanent complete vector relative degree andbounded perturbations. The approach makes use of global high-order sliding-mode controllers and differentia-tors. A criterion of the differentiator convergence is developed for the detection of a proper time of turning onthe controller. A gain adaptation strategy is proposed for both controller and differentiator. The performancewith noisy discrete sampling is studied.

Keywords: high-order sliding mode; differentiator; finite-time stability;

1. INTRODUCTION

Hybrid systems with strictly positive dwell time can be effectively controlled, provided theconvergence takes place between successive switching times or impulses. Such controllers are tobe robust, and preferably exact (i.e. insensitive) w.r.t. disturbances and model uncertainties.The output measurements should be the only available information in real-time.

A class of controllers with such properties might be sliding mode (SM) controllers, see forinstance Choi (2002), Andrade da Silva et al. (2009) for recent approaches to the output-feedbackdesign problem using linear matrix inequalities. Traditional (first order) SM control can onlyensure the insensitivity w.r.t. any bounded disturbances acting in the control channel (matcheddisturbances, see, for example, Utkin (1992)), and, unfortunately, do not provide for the finite-time exact convergence.

On the other hand, recently introduced High-Order SM (HOSM) controllers (see e.g. Levant(2001), Fridman and Levant (2002), Levant (2003, 2005)) also provide for the exact finite-timeoutput stability. Additionally they also allow chattering attenuation, (see Levant (2007)).

HOSM controllers were originally designed for single-input single-output (SISO) systems. Afamily of predesigned controllers is constructed corresponding to the relative degree of the out-put. They ensure finite-time-stable exact output regulation in spite of the presence of boundedmatched disturbances. Each controller makes use of the successive output derivatives w.r.t. totime of the up-to-the-relative-degree-minus-one order. Using the same HOSM methodology, thederivatives can be real-time evaluated by means of a robust exact differentiator (Levant (2003))converging in finite-time. This way, the HOSM controller is turned on once “enough time” haspassed for the finite-time estimation of the derivatives to be achieved.

∗Corresponding author. Email: [email protected]

ISSN: 0020-7721 print/ISSN 1464-5319 onlinec© 200x Taylor & FrancisDOI: 10.1080/0020772YYxxxxxxxxhttp://www.informaworld.com


2 M. T. Angulo, L. Fridman and A. Levant

The extension of HOSM algorithms to multi-input multi-output (MIMO) systems was done inBartolini et al. (2000), Edwards et al. (2008) and Defoort et al. (2009). In Bartolini et al. (2000)the whole state is assumed to be known, and a design methodology is introduced to replacethe traditional relay control (i.e. first-order sliding modes) with a second order SM controller.Thus, the proposed control is robust and exact, but it does not provide for finite-time statestabilization and the whole state is used in the control, not only the output. An output-basedcontroller is considered in Edwards et al. (2008) and Defoort et al. (2009) in the case when thevector relative degree exists. In Edwards et al. (2008), only asymptotic stability is ensured, thedisturbance needs to be smooth, and the system is to be Bounded-Input Bounded-State stablewith respect to it. In Defoort et al. (2009) the control gains’ matrix is to be approximatelyknown, and a MIMO tracking problem is solved in finite-time. The needed derivatives of theoutput are assumed known, and it is mentioned that they can be calculated by means of therobust exact differentiator (Levant (2003)).

Hence, the problem of global or semi-global output-feedback finite-time exact stabilizationof a disturbed LTI system with matched bounded disturbances is still open. Differentiatorswith variable gains (Levant (2006)) and non-homogeneous arbitrary-order HOSM controllers(Levant and Michael (2008)) are used to provide for global robust observation and stabilization.A mathematically established criterion is developed to detect the time of the differentiatorconvergence in the presence of noises and discrete sampling in order to apply the designedoutput-feedback controller. Some loose restrictions on the initial state are needed to choose theinitial values of the differentiator gains.

2. PROBLEM STATEMENT

Consider a system of the form

x = Ax+B[u+ w(t)]y = Cx

, (1)

where x ∈ Rn, u ∈ R

m, y ∈ Rm, w ∈ R

m are the state, control input, measured output anddisturbance signals, respectively. We assume that only the output y(t) is available for feedbackand that the disturbance satisfies the condition ‖w(t)‖ ≤W+, with W+ being a known constant.In addition, it is assumed that the system has full vector relative degree, i.e. vector relative degree(Isidori 1995, pp. 220) (to be recalled later on) and the sum of the relative degrees of the outputcomponents equals the dimension of the state space.

Since the control objective is to stabilize the system at x = 0 using only the measured outputy(t), both the controllability and the observability of the system in spite of the present dis-turbances are necessary. These requirements are satisfied due to the assumption of full vectorrelative degree. Indeed, it is known that if a vector relative degree exists, the observability in thepresence of “unknown inputs” turns out to be equivalent to the condition of full vector relativedegree, see for instance Fridman et al. (2007). We will recall this in the form of Lemma 4.2 to beintroduced later on. At the same time this condition implies that the system can be transformedinto the standard controllability form.

Note that any nonlinear system with full vector relative degree can be completely linearizedby a static feedback Isidori (1995), taking the output and its derivatives as state variables. Un-certainties matched with the inputs do not interfere with this procedure. Thus, the results of thispaper can be readily extended to the case of nonlinear full-relative-degree systems with uniformlybounded matched disturbancies. Nevertheless, the paper deals with the linear presentation ofthe system, since such models are very common in applications, and some further developmentin the paper is based on it.


International Journal of Systems Science 3

3. DIFFERENTIATOR-BASED OUTPUT-FEEDBACK CONTROL

3.1 Convergence criterion for HOSM differentiators

Throughout this paper we assume that all the required derivatives are available for feedbackin real-time by means of a real-time exact robust HOSM differentiator (Levant (2003)). Letf(t) ∈ R be a function to be differentiated, then the k-th order differentiator takes the form

z0 = ν0 = −λkL1

k+1 |z0 − f |k

k+1 sign(z0 − f) + z1,

z1 = ν1 = −λk−1L1k |z1 − ν0|

k−1k sign(z1 − ν0) + z2,

... (2)

zk−1 = νk−1 = −λ1L12 |zk−1 − νk−2|

12 sign(zk−1 − νk−2) + zk,

zk = −λ0L sign(zk − νk−1),

where zi is the estimation of the true signal f (i)(t). The differentiator provides for the finite-time exact estimation under ideal conditions when neither noise nor sampling are present. Theonly needed information is an a-priory known upper bound L for |f (k+1)|. Then a parametricsequence {λi} > 0, i = 0, 1, . . . , k, is recursively built, which provides for the convergence of thedifferentiator of each order k. In particular, the parameters λ0 = 1.1, λ1 = 1.5, λ2 = 2, λ3 = 3,λ4 = 5, λ5 = 8 are enough for the construction of differentiators up to the 5-th order. In thepresence of input noises or discrete sampling, this differentiator provides for the best possibleasymptotic accuracy (Kolmogorov (1962), Levant (2003)).

Nevertheless, the exact estimates of the derivatives are only available after a finite-time tran-sient. For control purposes, the rational solution is first to wait until the finite-time exact estimateof the derivatives is ready, and only then to turn on the controller. Until now this procedurewas performed waiting “enough time” to ensure the differentiator convergence. Is it possible tocheck in real-time whether the HOSM differentiator has converged? This question is answeredin the following theorem.

Theorem 3.1 : Consider the HOSM differentiator (2) of order k, where f(t) is the signal tobe differentiated. Assume that the parameters {λi} provide for the finite-time convergence ofdifferentiator (2) for any k in the absence of noises. Let

f(t) = f0(t) + η(t), |f (k+1)0 (t)| < L, |η(t)| ≤ kηLξk+1, (3)

where f0(t) is an unknown basic signal, η(t) is a Lebesgue-measurable sampling noise, ξ is apositive parameter. Suppose also that f is sampled with the time step τ > 0, and τ ≤ kτξ, withkη, kτ being some positive constants. Then for any positive constants γ0, γ1, ..., γk and any kf ,0 < kf < γ0, there exist kη, kτ , γt > 0, such that if the inequality

|z0 − f(t)| ≤ kfLξk+1 (4)

holds during the time interval of the length γtξ (or only at the sampling instants within the sameinterval) then starting from the beginning of this interval the inequalities

|zi − f (i)0 (t)| ≤ γiLξk−i+1, i = 0, 1, ...k (5)

hold and are kept forever.

Note that in any case the final accuracy of the form (5) is obtained in finite transient time,which is independent of ξ (Levant (2003)). In particular, this shows the applicability of the



Theorem. Obviously, one can arbitrarily increase γt and the time-interval length γtξ and decreasekf , kη, kτ preserving the statement of the Theorem. In practice, this means that one can simplytake a constant length of the observation time interval and a sufficiently small sampling interval,then one has to choose sufficiently small kf , such that keeping (4) during the considered timeinterval implies establishment of (5).

Remark. Exact estimations are obtained in the limit case with ξ = 0 and measurementscontinuous in time Levant (2003). In that case the convergence time from the detected vicinity(5) is proportional to ξ Levant (2005). Therefore, with properly chosen parameters kη, kτ , γt > 0,if (4) is kept during the time γtξ then the exact differentiation is observed further on (i.e. theconvergence time is already included in γtξ).

Proof Denote σi = (zi − f (i)0 )/L, ~σ = (σ0, ..., σk). Subtracting f (i+1)

0 from the both sides of theequation on zi of (2) and dividing by L, obtain the differential inclusion

σ0 = −λk |σ0 − η(t)/L|k

k+1 sign(σ0 − η(t)/L) + σ1,

σ1 = −λk−1 |σ1 − σ0|k−1

k sign(σ1 − σ0) + σ2,

... (6)

σk−1 = −λ1 |σk−1 − σk−2|12 sign(σk−1 − σk−2) + σk,

σk ∈ −λ0 sign(σk − σk−1 ) + [−1, 1].

Excluding derivatives in the right-hand side and using |η(t)| ≤ kηLξk+1, obtain the non-recursiveform

σ0 ∈ −λk∣∣∣σ0 + [−kηξk+1, kηξ

k+1]∣∣∣ k

k+1 sign(σ0 + [−kηξk+1, kηξk+1]) + σ1,

σ1 ∈ −λk−1

∣∣∣σ0 + [−kηξk+1, kηξk+1]

∣∣∣ k−1k+1 sign(σ0 + [−kηξk+1, kηξ

k+1]) + σ2,

... (7)

σk−1 ∈ −λ1

∣∣∣σ0 + [−kηξk+1, kηξk+1]

∣∣∣ 1k+1 sign(σ0 + [−kηξk+1, kηξ

k+1]) + σk,

σk ∈ −λ0 sign(σ0 + [−kηξk+1, kηξk+1]) + [−1, 1].

The right hand side of (7) is minimally enlarged in order to provide for the convexity andupper-semicontinuity of the obtained differential inclusion (Filippov (1960)). The sampling of fcorresponds to the time varying delay of the right-hand side not exceeding ktξ. Denoting (7) by�~σ ∈ Σ(~σ, ξ), obtain that the system with sampling corresponds to

�~σ ∈ Σ(~σ(t− [0, ktξ]), ξ). (8)

The following Lemma is proved after the Theorem proof.

Lemma 3.2: For any positive T and γ, there exist such δ > 0 that for any sufficiently small ξand any solution of (8) it follows from the inequality |σ0| ≤ δ being kept during the time periodT that also the inequalities |σi| ≤ γ, i = 1, ..., k, are kept during that time period.

Fix some T > 0 and ξ1 > 0 such that with any ξ < ξ1 all trajectories of (8) starting withinthe region |σi| ≤ γi, i = 0, ..., k, finish within the same region in time T . It is possible due tothe continuous dependence of the solutions on ξ (Filippov (1960)) and the finite-time stability



of the undisturbed system. Then there exist such δ, 0 < δ < γ0, and ξ0, ξ0 < min(δ, ξ1), thatwith any ξ < ξ0 all trajectories of (8) keeping the inequality |σ0| ≤ δ during the time interval[0, T ], also keep |σi| ≤ γ = min(γi), i = 1, ..., k, during the same interval. Thus, keeping theinequality |(z0−f(t))/L| ≤ δ0 = δ−ξ0 implies that |σ0| < δ < γ0 and also implies that |σi| ≤ γi,i = 1, ..., k, are kept during the time interval. The time T was chosen so that if at any momentt the inequalities |σi| ≤ γi, i = 0, ..., k, hold, then they hold also at t+ T . Since the inequalitiesare kept during the whole time interval [0, T ], they are kept forever.

Show now that with any δ1, 0 < δ1 < δ < γ0, and sufficiently small kt it is enough to check|σ0| < δ1 at sampling times only to provide for |σ0| < δ0 during the whole time interval. Indeed,it follows from the discrete-time sampling variant of the above Lemma. In order to prove it,one just needs to replace the division of the time interval [0, T ] considered in the proof with thenatural division of [0, T ] into the sampling intervals of the length not exceeding kτξ.

Due to the homogeneity features (Levant (2005)) under the time-coordinate-parameter trans-formation

Gκ : (t, σ0, σ1, ..., σk, ξ) 7−→ (κt,κk+1σ0,κkσ1, ...,κσk,κξ).

solutions of (8) transfer to solutions of the same inclusion, but with the new disturbance param-eter κξ. Transformation Gκ, κ = ξ/ξ0 transfers solutions of system (8) with the disturbanceparameter ξ0 onto the solutions of system (8) with the disturbance parameter ξ. As a resultget that with arbitrary ξ keeping the inequality |(z0 − f(t))/L| ≤ δ0(ξ/ξ0)k+1 during the timeinterval [0, (ξ/ξ0)T ] implies that also |σi| ≤ γi(ξ/ξ0)k−i+1, i = 1, ..., k, are kept. Multiplying byL and choosing appropriate values of kf , γt, obtain the statement of the Theorem. �

Proof of Lemma 3.2. Suppose that the Lemma statement is not true. Take a sequence δs → 0.Then there exists a sequence ξs → 0, such that for each s there is a solution ~σs(t) of (8) withξ = ξs, for which |σ0| ≤ δ is kept during the time period T , but the inequalities |σi| ≤ γ,i = 1, ..., k, are not kept. For simplicity suppose that each solution is defined on the segment[0, T ]. Note that, due to the convergence of the differentiator, with small ξ solutions for which|σ0| ≤ δ, |σi| ≤ γ, i = 1, ..., k, are kept always exist (Levant (2005)).

Show that ‖~σs(t)‖ remain uniformly bounded. Indeed, divide the segment [0, T ] in 3k equalsub-segments, and fix some δ such that δs ≤ δ. Taking into account |σ0| ≤ δ and applying themean-value Lagrange theorem to the end points of segment triplets, get 3k−1 points where |σ0|is bounded by 3k−1δ/T . It follows now from (7) that also σ1 is bounded at the same points.Once more applying the Lagrange theorem get 3k−2 points, where σ2 is bounded by a constantknown in advance, etc. At the last step get 30 = 1 points, where σk is bounded by a constantknown in advance. Now taking this point as an initial one, and integrating the last equation of(7), get that σk is uniformly bounded over the whole segment [0, T ]. Integrating the last but oneequation starting from anyone of founded 3 points, get that σk−1 is uniformly bounded over thewhole segment [0, T ], etc.

Thus, ‖~σs(t)‖ remain uniformly bounded, which means that also the right-hand side of (7) isuniformly bounded. Therefore, solutions ~σs(t) are bounded, and have a joint Lipschitz constant.Due to the Arcela Theorem there exists a uniformly convergent subsequence ~σsl

, l → ∞. Thelimit function ~σ∗ has to satisfy the undisturbed system

σ0 = −λk |σ0|k

k+1 signσ0 + σ1,

σ1 = −λk−1 |σ0|k−1k+1 signσ0 + σ2,

... (9)

σk−1 = −λ1 |σ0|1

k+1 signσ0 + σk,

σk ∈ −λ0 signσ0 + [−1, 1],



and features the property σ0 ≡ 0 during the time interval [0, T ]. But it follows from the firstequation that σ1 ≡ 0. Now it follows from the second equation that σ2 ≡ 0, etc. In such a wayget ~σ∗ ≡ 0. That contradicts the assumption that none of solutions ~σs(t) keep the inequalities|σi| ≤ γ, i = 1, ..., k. �

3.2 Output-feedback application of the criterion

Consider the output-feedback application of the differentiator. Let yi = cix, i = 1, . . . ,m, bethe outputs of system (1). Assume that a differentiator of the order ri − 1 has been used foreach output, in order to estimate the required derivatives. Denote by zi,0 the variable z0 of thedifferentiator applied to the i-th output. Theorem 3.1 provides an easy way to check whetherthe i-th differentiator has converged by verifying that |z0,i − yi| ≤ kf,iLiτ

ri

i is kept in sometime interval kt,iτ i. It is natural to estimate the constants kf,i and kt,i by simulation. Notice inaddition that this criterion is very robust, since the value of ξ in Theorem 3.1 can be enlargedvoluntarily without changing the values of the noise magnitude or the sampling step. Also thelength of the time interval γtξ can be voluntarily enlarged preserving the theorem statement.

Thus, the resulting control gets the form

u(t) =

u(t) if |z0,i − yi| ≤ kf,iLiτ ri

i in time interval [t− ki,tτ i, t]i = 1, . . . ,m.

0 otherwise, (10)

where u(t) is the control calculated using the estimated derivatives.

4. CONTROL DESIGN

Let us recall the notion of vector relative degree from (Isidori 1995, pp. 220).

Definition 4.1: The output y = Cx of system (1) has vector relative degree (r1, . . . , rm) if

ciAkB = 0, k = 0, 1, . . . , ri − 2, i = 1, 2, . . . ,m,

andrank(Q) = m, Q :=

c1A

r1−1Bc2A

r2−1B...

cmArm−1B

∈ Rm×m.Using this definition, we have the following well-known property.

Lemma 4.2: (See for instance Fridman et al. (2007)) Assume that system (1) has full vectorrelative degree, i.e. r1 + · · ·+ rm = n. Then the following relation is valid

x = M

Yr,1Yr,2

...Yr,m

:= MYr, Yr,i :=

yiyi...

y(ri−1)i

, (11)

where M is full rank matrix.

In fact, matrix M is composed of some rows of the observability matrix of the (A,C) pair.Notice that under the conditions of this last lemma, the control objective of making x = 0 canbe reformulated as the problem of designing u(t) providing for y(t) ≡ 0,∀t ≥ T .



Now, let us introduce the following controller

vi = −αiΦi(Yr)Hri(yi, yi, . . . , y

(ri−1)i ), Φi(Yr) := ki,1‖Yr‖+ ki,2, i = 1, . . . ,m, (12)

where vi is the i-th row of v = Qu, {αi, ki,1ki,2} are positive gains of the controller (to be tuned),Hri

is the ri-th order SM control algorithm with “gain robust parameters” and Φi is the so-called“gain function”, see Levant and Michael (2008) for further details.

Theorem 4.3 : Consider system (1), assume that it has a full vector relative degree, i.e. r1 +· · · + rm = n, and that ‖w(t)‖ ≤ W+, with W+ being a known constant. Then it is finite-timestabilizable to x = 0 by the controller (12), provided that αi is a large enough constant andki,1 > ‖ciAri‖, ki,2 > ‖ciAri−1B‖W+.

Proof Differentiate each output yi until an input appears and group them together as

y(r1)1 = c1A

r1x+ c1Ar1−1B[u+ w(t)]

y(r2)2 = c2A

r2x+ c2Ar2−1B[u+ w(t)]

...

y(rm)m = cmA

rmx+ cmArm−1B[u+ w(t)].

By the assumption of the full vector relative degree, the matrix Q is square and of the full rank.Introduce the input transformation u = Q−1v, so that

y(r1)1 = c1A

r1x+ v1 + c1Ar1−1Bw

...

y(rm)m = cmA

rmx+ vm + cmArm−1Bw.

Now each output yi satisfies the problem formulation of Levant and Michael (2008) with

hi := ciArix+ ciA

ri−1Bw, i = 1, . . . ,m.

Bounds for this last equation are easy to be obtained as

|hi| ≤ ‖ciAri‖‖M‖‖Yr‖+ ‖ciAri−1B‖W+.

Choosing

Φi(Yr) := ki,1‖Yr‖+ ki,2,

and, for instance, selecting ki,1 > ‖ciAri‖, ki,2 > ‖ciAri−1B‖W+ and αi large enough (to com-pensate for M), obtain αiΦi > |hi|. Choosing [Levant and Michael (2008)] the controller

vi = −αiΦi(Yr)Hri(yi, yi, . . . , y

(ri−1)i ),

obtain that in finite-time the following equality is kept

{yi, yi, . . . , y(ri−1)i } ≡ 0, ∀t ≥ T,

for i = 1, . . . ,m, that is {y1, . . . , ym} ≡ 0,∀t ≥ T . Due to the condition r1 + · · · + rm = n andLemma 4.2, the equality y(t) = 0, t ≥ T implies that x(t) ≡ 0, ∀t ≥ T . �



4.1 Differentiator gain adaptation

To evaluate controller (12), a gain Li is needed for each differentiator in order to estimatederivatives of every output yi up to the ri− 1 order. By formulas (5), in the presence of noise ordiscretization, the lower the gain Li the better is the obtained precision.

In addition, if the system is assumed to satisfy the conditions of Lemma 4.2, there is a one-to-one correspondence between the state and the output and its derivatives. Therefore, if theinitial condition of the state is very large, the differentiators’ gains are to be large as well, butonce the trajectories of the system are near the origin, the gains can be significantly reduced.Thus, it is reasonable to make these gains variable in time. It was shown by Levant (2006) thatLi can be any continuous function of time, but in order to guarantee robustness properties, it isrequired in addition that the logarithmic derivative |Li(t)/Li(t)| be uniformly bounded.

Proposition 4.4: Consider system (1) and assume that

a) it has a full vector relative degree i.e. r1 + · · ·+ rm = n,b) ‖w(t)‖ ≤W+ and ‖x0‖ ≤ x+

0 where W+ and x+0 are known constants,

c) ‖u(t)‖ ≤ ρ1‖x‖+ ρ2 for some known constants ρ1 and ρ2,

Introduce the following adaptation algorithm for the gain Li(t) of each differentiator

(1) Set Li(t) = L0,i for 0 ≤ t ≤ t1, with t1 the time instant when the convergence of alldifferentiator is detected and with L0,i a large enough constant.

(2) Set Li(t) = li,1(‖Yr‖+ li,2) for t > t1 with

li,1 >(‖ciAri‖+ ‖ciAri−1B‖ρ1

)‖M‖, li,2 >

‖ciAri−1B‖(ρ2 +W+)li,1

,

and Yr as in (11) constructed using the estimations from the same differentiators.

Then finite-time estimation of the derivatives of each output yi up to the order ri− 1 is ensured,and in addition, the logarithmic derivative Li(t)/Li(t) is uniformly bounded for each i = 1, . . . ,m.

Proof Consider the two steps of the algorithm separately

(1) We need to show that with a large enough gain L0,i, each differentiator converges. In thistime interval the control input is set to u(t) ≡ 0. Since we want to estimate derivatives ofeach output up to the ri − 1 the gain L0,i should satisfy |y(ri)

i (t)| ≤ L0,i in a time interval[0, tf ] and guarantee the convergence of the differentiator.

We can get an upper bound for the solution x(t) in the time interval [0, tf ] as follows. Usingthe well-known general solution for a linear system, considering u(t) ≡ 0, ‖w(t)‖ ≤W+ andthat ‖eAt‖ ≤ e‖A‖t (evident from the series definition of the exponential) it yields to

‖x(t)‖ ≤ ‖eAtx0‖+∫ t

0‖eA(t−s)Bw(s)‖ds

≤ e‖A‖t‖x0‖+W+‖B‖e‖A‖t∫ t

0e−‖A‖sds

≤ e‖A‖tx+0 +

W+‖B‖‖A‖

(e‖A‖t − 1

)Let X+ = supt∈[0,tf ] ‖x(t)‖ with tf < ∞. Now each output satisfy y

(ri)i = ciA

rix +ciA

ri−1Bw(t) so

|y(ri)i (t)| = ‖ciAri‖X+ + ‖ciAri−1B‖W+, ∀t ∈ [0, tf ]



Since the gain of each differentiator can be selected large enough to provide any conver-gence time (see Levant (2003)), the gain L0,i can be selected large enough such that the dif-ferentiator converges before tf . For instance, selecting L0,i > qi‖ciAri‖X+ + ‖ciAri−1B‖W+

with qi a large enough constant.(2) Once the differentiators have converged, the signal Yr is available. Since ‖u‖ ≤ ρ1‖x‖ + ρ2

and ‖w(t)‖ ≤W+ the selection of Li(t) = li,1(‖Yr‖+ li,2) with

li,1 >(‖ciAri‖+ ‖ciAri−1B‖ρ1

)‖M‖, li,2 >

‖ciAri−1B‖(ρ2 +W+)li,1

,

ensures that Li(t) is also an upper bound for y(ri)i (t) from t1 and afterward. Now, all that

is missing is to check that its logarithmic derivative is uniformly bounded. Since x = MYrwe can rewrite Li(t) as Li(t) = βi,1(‖x‖ + βi,2) for some constant βi,1, βi,2 > 0. This lastexpression can be written as (

Li(t)βi,1

− βi,2)2

= xTx

Direct differentiation of this last expression gives

2(Li(t)βi,1

− βi,2)Li(t)βi,1

= xT (A+AT )x+ 2xTB(u+ w)

≤ λmax(A)‖x‖2 + 2‖x‖‖B‖(ρ1‖x‖+ ρ2) + 2‖x‖‖B‖W+

≤(λmax(A) + 2ρ1‖B‖

)‖x‖2 + 2‖B‖

(ρ2 +W+

)‖x‖,

where A = A + AT . Noticing that(Li(t)βi,1− βi,2

)= ‖x‖ we can divide both sides by ‖x‖ to

obtain

Li(t)βi,1

≤ 12

(λmax(A) + 2ρ1‖B‖

)‖x‖+ ‖B‖

(ρ2 +W+

).

Now computing

Li(t)Li(t)

=Li(t)/βi,1Li(t)/βi,1

≤12

(λmax(A) + 2ρ1‖B‖

)‖x‖+ ‖B‖ (ρ2 +W+)

‖x‖+ βi,2,

obtain that the logarithmic derivative is uniformly bounded by a suitable constant.

�

Remark. Since the trajectories of the system cannot scape to infinity in finite-time under theconsidered assumptions, at the first stage of the algorithm the constant L0,i can be taken verylarge, providing for the fast and reliable differentiator convergence. Thus there is no problemof choosing the initial value of L0,i. Any large value will suffice. For practical implementation,in the second part of the algorithm it is enough to consider Li(t) = li,1(‖Yr‖ + li,2) with largeenough constants li,1 and li,2. The gain li,2 should be tuned in accordance to the disturbanceamplitude, but, if no information is available, it can be selected as li,2 = 1 and only tune li,1large enough. Also note that the gain of the differentiator decreases as the state approaches zero,which reduces chattering. On the other hand notice that since |Hri

(y, y, . . . , y(ri−1))| ≤ 1, the



nonlinear controller v(t) selected as (12) has a norm of the form considered in assumption c) ofthe proposition.

On the other hand, in the presence of measurement noises and discrete sampling, exact dif-ferentiation is impossible. However, due to the differentiators’ robustness and the correspondingerror asymptotics, the vector Yr is known approximately, and its usage is still possible, due tothe robustness of the whole construction.

4.2 System performance in the presence of noise and sampling

Theorem 4.3 guaranties the semi-global finite-time stabilization in the case when the samplingis continuously performed and no measurement noises are present. Now consider the effects ofsampling step τ and the measurement noises not exceed li,1(‖Yr‖+ li,2)ε in absolute value, thefollowing theorem (an immediate consequence of Levant (2005)) holds

Theorem 4.5 : Let the initial values of x belong to some compact set. Then with sufficientlysmall ξ and ε ≤ kεLξmax(r1,...,rm), τ ≤ kτξ, kε, kτ being some positive constants, the closed loopsystem stabilizes in finite time to a vicinity of the origin defined by the inequalities

|y(j)i | ≤ µi,jξ

ri−j , j = 0, 1, ..., ri − 1, i = 1, 2, ...,m,

where µi,j are some positive constants.

A drawback of the proposed controller is its relatively slow convergence with large initial con-ditions. The restriction is easily removed combining a linear controller with the above nonlinearone. Thus our plan is to wait for the detection of the convergence of the differentiator, then touse a linear controller that guarantees that the trajectories of the system enter a predefined ballcentered at the origin, then we switch to the HOSM controller. This way a faster transient canbe obtained with smaller gains of the controller. This is certainly useful in practical applications.

Let t1 be the moment when the mutual convergence of all differentiators was detected. Fromthat moment on the parameters of the differentiators are variable (see the section above). Now,it is possible to apply a linear controller

u = −Kx = −KYr , (13)

such that the resulting closed loop system is stable, i.e. A−BK is Hurwitz. The linear controllerscan be freely chosen provided that the parameters li,1, li,2 of the differentiator are taken suffi-ciently large with respect to this choice. Since the unperturbed system is exponentially stable,practical stability of the origin is achieved in the presence of a uniformly bounded perturbation,that is, for any chosen R > 0 there exists a (large enough) “control gain” such that in timeinstants t ≥ t2 the inequality ‖x(t)‖ ≤ R is true, or equivalently

‖Yr‖ ≤M−1R = R, ∀t ≥ t2.

From that time on, the nonlinear controller (12) may be applied. Thus, the following theoremholds.

Theorem 4.6 : With sufficiently small ξ and ε ≤ kεξmax(r1,...,rm), τ ≤ kτξ, kε, kτ being some

positive constants, the combined controller, (13) with t1 ≤ t < t2 and (12) afterwards, stabilizes(1) in finite time to a vicinity of the origin defined by the inequalities

|y(j)i | ≤ µi,jξ

ri−j , j = 0, 1, ..., ri − 1, i = 1, 2, ...,m,

for t ≥ t2 and where µi,j are some positive constants.



The only restrictions on ξ are now that the above accuracy defines a region inside the ball‖x‖ ≤ R and linear controllers (13) should provide for the convergence to that ball. Bothrestrictions can be analytically checked. Note in addition that, since only one switch betweenthe linear controller and the HOSM controller takes place, the previously obtained stability ofthe system is preserved.

5. EXAMPLE

Consider

x =

0 1 0 0 01 1 1 0 00 0 −1 0 10 1 −1 1 10 1 −1 0 0

x+

0 00 01 00 00 1

[u+ w(t)],

y =[

1 0 0 0 00 0 0 1 0

]x

The disturbance is chosen for simulation purposes as

w(t) =[

0.5 + 0.2sq(0.33t)0.2 + 0.7 sin(4t)

],

where 0.2sq(0.33t) is a symmetric square wave of amplitude 0.2 and period of 0.33 seconds.The relative degree vector is (r1, r2) = (3, 2) and it is clear that it is full and well-defined sinceQ = I2×2. Accordingly, Yr = Mx, with Yr := [y1, y1, y1, y2, y2]T and

M =

1 0 0 0 00 1 0 0 01 1 1 0 00 0 0 1 00 1 0 1 1

.

The linear controller u = −Kx = −KYr is designed using the standard LQ methodology withweight matrices Q = 10diag{100, 10−3, 10−2, 10−3, 100} and R = 30I2×2 resulting in

K =[

6.3448 9.8541 4.4102 1.4123 0.5116−1.6018 −0.1048 0.5116 5.3399 7.7657

].

The neighborhood R of the origin in which the HOSM controller is turned on is characterizedby ‖Yr‖1 ≤ 18. The required differentiators are of order 2 and 1 for each output, respectively. Theselected parameters are presented in Table 1. The controllers are selected as non-homogeneousquasi-continuous ones

u1 := −α1ν31Φ1(Yr)H3(y1, y1, y1),

= −α1ν31Φ1(Yr)

y1 + 2ν321y1+ν1|y1|

23 sign y1h

|y1|+ν1|y1|23

i 12

|y1|+ 2ν321

[|y1|+ ν1|y1|

23

] 12

,



Parameter First differentiator (i = 1) Second differentiator (i = 2)order 2 1Li,0 200 100li,1 15 10li,2 15 10kt 100 100kf 7 · 10−2 3τ 10−2 10−2

Table 1. Differentiators parameters

0 5 10 15−120

−100

−80

−60

−40

−20

0

20

40

Figure 1. Closed loop trajectories x(t) without measurement noise.

with α1 = 8, ν1 = 1.2, to regulate the convergence rate, see Levant and Michael (2008), andΦ1(Yr) = ‖Yr‖2 + 1. The second controller is designed as

u2 := −α2ν22Φ2(Yr)H2(y2, y2),

= −α2ν22Φ2(Yr)

y2 + ν2|y2|12 sign y2

|y2|+ ν2|y2|12

,

with α2 = 15, ν2 = 0.9 and Φ2(Yr) := ‖Yr‖2 + 1. The overall controller is constructed as in (10).Two simulations were performed, one with bounded measurement noise y(t) = Cx(t) + η(t)satisfying ‖η(t)‖1 ≤ 2 · 10−3 for all t ≥ 0 (simulated using a uniform random number generator)and the other without measurement noise. Both simulations were carried out using using Eulerintegration with 10−4 time step.

Simulations results without measurement noise are presented in Figures 1 for the state x(t),Figure 2 for the control signal u(t) and differentiators gain Li(t) and Figure 3 for the signal of thedetection of the convergence of all differentiators and the turn on signal of the linear controller.Figure 1 shows that the closed loop state trajectories reach the origin in finite-time in spite ofthe (non vanishing) disturbance.

In Figures 4, 5 and 6 the corresponding simulations with measurement noise are presented.Comparing the closed loop trajectories under measurement noise of Figure 4 with those of Fig-ure 1 without noise, the system’s trajectories are shown to remain in a vicinity of the originalconstraint x = 0 in spite of measurement noise, as Theorem 4.6 claims. The effect of the mea-surement noise in the control signal u(t) and the differentiators’ gains is evident by comparingFigure 5 with the corresponding signals in Figure 2. However, it is worth to remark that in



0 5 10 15−800

−600

−400

−200

0

200

a)

0 5 10 150

200

400

600

800

1000

1200

1400

c)

0 5 10 15−600

−500

−400

−300

−200

−100

0

100

b)

0 5 10 150

200

400

600

800

1000

d)

Figure 2. Control signals and differentiator’s gains without measurement noise: a)u1(t), b) u2(t), c) L1(t), d) L2(t).

0 5 10 15

0

0.2

0.4

0.6

0.8

1

0 5 10 15

0

0.2

0.4

0.6

0.8

1

Figure 3. Differentiators convergence detection signal (top) and signal when the linear controller is active (bottom). Sim-ulation results without measurement noise. Value 0 indicates false, value 1 indicates true.

Figure 5 the effect of noise in the linear controller and in the HOSM controller is comparable inspite of the discontinuous nature of the last one. Finally, Figure 6 shows that the detection ofthe convergence of the differentiator is still possible in spite of noise and, in fact, remains almostthe same as in the simulation results without noise.

6. Conclusions

Linear time-invariant systems with matched perturbations are exactly finite-time stabilized bymeans of global HOSM controllers and differentiators:

• a criterion for the online detection of the convergence time of the differentiators is proposed(section 3.2),

• a semi-global output based controller is designed for linear time invariant systems withmatched bounded uncertainty ensuring finite-time exact state stabilization (section 4),

• an adaptation algorithm for the gains of the differentiators and controllers is suggested (section4.1).



0 5 10 15−100

−80

−60

−40

−20

0

20

40

Figure 4. Closed loop state trajectories x(t) with measurement noise.

0 5 10 15−600

−400

−200

0

200

a)0 5 10 15

−400

−300

−200

−100

0

100

b)

0 5 10 150

200

400

600

800

1000

1200

c)0 5 10 15

0

200

400

600

800

d)

Figure 5. Control signals and differentiator’s gains with measurement noise: a)u1(t), b) u2(t), c) L1(t), d) L2(t).

0 5 10 15

0

0.2

0.4

0.6

0.8

1

0 5 10 15

0

0.2

0.4

0.6

0.8

1

Figure 6. Differentiators convergence detection signal (top) and signal when the linear controller is active (bottom). Bothwith noise measurements. Value 0 indicates false, value 1 indicates true.



• the performance with discrete noisy sampling is presented (section 4.2).

Marco Tulio Angulo received the M.S. in 2009 from the Department of Control at UNAM,Mexico. Currently, he is pursuing a Ph.D in the same institution. His research interests includethe existing tradeoffs between robustness and exactness in observation and control, particularlyin the real-time differentiation problem, and the observability properties of disturbed systems.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination.

FERREIRA et al.: ROBUST CONTROL WITH EXACT UNCERTAINTIES COMPENSATION 7

Fig. 4. Precision of [rad] using OISMC applying a (left) first-orderHOSM differentiator and a (right) second-order HOSM differentiator.

• two robust output feedback control strategies were com-pared:— continuous compensation control based on the es-

timated states and the compensation of identifiedunknown inputs (EOFS);

— output integral sliding mode control based on estimatedstates (OISMC).

• a methodology is suggested for the selection of an ap-propriate controller based on the comparison of both con-trol strategies considering the accuracy of observation andidentification algorithms as well as the actuator time con-stant;

• the proposed methodology is experimentally validated inan inverted rotary pendulum system.

REFERENCES

[1] V. I. Utkin, Sliding Modes in Control and Optimization. Berlin, Ger-many: Springer Verlag, 1992.

[2] V. I. Utkin, J. Guldner, and J. Shi, Sliding Modes in ElectromechanicalSystems. London, U.K.: Taylor and Francis, 1999.

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[7] I. Boiko, Discontinuous Control Systems: Frequency-Domain Analysisand Design. Boston, MA: Birkhäuser, 2009.

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[11] J. Barbot and T. Floquet, “A canonical form for the design of unknowninputs sliding mode observers,” in Advances in Variable Structure andSliding Mode Control, C. Edwards, E. Fossas-Colet, and L. Fridman,Eds. Berlin, Germany: Springer, 2006, vol. 334, pp. 103–130.

[12] A. Pisano and E. Usai, “Globally convergent real-time differentiationvia second order sliding modes,” Int. J. Control, vol. 38, pp. 833–844,2007.

[13] L. Fridman, A. Levant, and J. Davila, “Observation and identificationvia high-order sliding modes,” in Modern Sliding Mode ControlTheory New Perspectives and Applications, G. Bartolini, L. Fridman,A. Pisano, and E. Usai, Eds. Berlin, Germany: Springer Verlag,2008, pp. 293–320.

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Alejandra Ferreira was born in Mexico in 1976.She received the B.Sc. degree from National Au-tonomous University of Mexico (UNAM), MexicoCity, Mexico, in 2004, where she is pursuing thePh.D. degree in automatic control.

In 2000–2005, she was with the InstrumentationDepartment, Institute of Astronomy, UNAM. Herprofessional interests include electronics design,observation, and identification of linear systems,sliding mode control, and its applications.

Francisco Javier Bejarano received the Masterand Doctor degrees in automatic control from theCINVESTAV-IPN, Mexico City, Mexico, in 2003and 2006, under the direction of Prof. A. Poznyakand Dr. L. Fridman.

He stayed one year at the ENSEA, France, and twoyears at UNAM, Mexico with respective posdoctoralpositions. He has published nine papers in interna-tional journals.

Leonid M. Fridman (M’98) received the M.S. de-gree in mathematics from Kuibyshev State Univer-sity, Samara, Russia, in 1976, the Ph.D. degree in ap-plied mathematics from the Institute of Control Sci-ence, Moscow, Russia, in 1988, and the Dr.Sci. de-gree in control science from Moscow State Universityof Mathematics and Electronics, Moscow, Russia, in1998.

From 1976 to 1999, he was with the Departmentof Mathematics, Samara State Architecture and CivilEngineering Academy. From 2000 to 2002, he was

with the Department of Postgraduate Study and Investigations at the ChihuahuaInstitute of Technology, Chihuahua, Mexico. In 2002, he joined the Departmentof Control, Division of Electrical Engineering of Engineering Faculty, NationalAutonomous University of Mexico (UNAM), México. He is an Editor of threebooks and five special issues on sliding mode control. He has published over200 technical papers. His research interests include variable structure systemsand singular perturbations.

Dr. Fridman is an Associate Editor of the International Journal of System Sci-ence and Conference Editorial Board of IEEE Control Systems Society, Memberof TC on Variable Structure Systems and Sliding mode control of IEEE ControlSystems Society.

Leonid Fridman (M’98) received the M.S. degree in mathematics from Kuibyshev State Uni-versity, Samara, Russia, in 1976, the Ph.D. degree in applied mathematics from the Institute ofControl Science, Moscow, Russia, in 1988, and the Dr.Sci. degree in control science from MoscowState University of Mathematics and Electronics, Moscow, Russia, in 1998. From 1976 to 1999,he was with the Department of Mathematics, Samara State Architecture and Civil EngineeringAcademy. In 2002, he joined the Department of Control, Division of Electrical Engineering ofEngineering Faculty, National Autonomous University of Mexico (UNAM), Mexico. He is anEditor of three books and five special issues on sliding mode control. He has published over200 technical papers. His research interests include variable structure systems and singular per-turbations. Dr. Fridman is an Associate Editor of the International Journal of System Scienceand Conference Editorial Board of IEEE Control Systems Society, Member of TC on VariableStructure Systems and Sliding mode control of IEEE Control Systems Society.

LEVANT: CHATTERING ANALYSIS 1389

sign stage, and are taken into account only during the simu-lation-based parameter adjustment of high-order sliding-modecontrollers. It is proved that the faster the actuators and sen-sors the less is their influence on the system. At the same timethe sliding controller parameters together with the plant deter-mine whether the actuators and sensors can be considered reallyfast. Indeed, faster controllers require faster actuator and sensorresponses.

An obvious drawback of the proposed approach is that theworst-case noises and disturbances are considered. Thus, evenunbounded chattering may vanish with appropriate noises anddisturbances. At the same time such an approach leads to thesimplest chattering classification. Infinitesimal mathematicalchattering presumably corresponds to negligible chattering inreal systems.

The author did not consider theoretically discretization issuesand small delays, which can also produce dangerous chatteringin systems with large gains (Fig. 3). It follows from Section IVthat such imperfections do not produce additional chattering inthe systems with homogeneous discontinuous control.

REFERENCES

[1] Bacciotti and L. Rosier, Liapunov Functions and Stability in ControlTheory. London, U.K.: Springer Verlag, 2005.

[2] G. Bartolini, “Chattering phenomena in discontinuous control sys-tems,” Int. J. Syst. Sci, vol. 20, pp. 2471–2481, 1989.

[3] G. Bartolini, A. Ferrara, and E. Usai, “Chattering avoidance by secondorder sliding-mode control,” IEEE Trans. Autom. Control, vol. 43, no.2, pp. 241–241, Feb. 1998.

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[7] I. Boiko, L. Fridman, A. Pisano, and E. Usai, “Performance analysis ofsecond-order sliding-mode control systems with fast actuators,” IEEETrans. Autom. Control, vol. 52, no. 6, pp. 1053–1059, Jun. 2007.

[8] I. Boiko, L. Fridman, A. Pisano, and E. Usai, “Analysis of chattering insystems with second order sliding modes,” IEEE Trans. Autom. Con-trol, vol. 52, no. 11, pp. 2085–2102, Nov. 2007.

[9] M. Djemai and J. P. Barbot, “Smooth manifolds and high order slidingmode control,” in Proc. 41st IEEE Conf. Decision Control, 2002, pp.335–339.

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[11] A. Ferrara and L. Giacomini, “Output feedback second order slidingmode control for a class of nonlinear systems with non matched un-certainties,” ASME J. Dyn. Syst., Meas. Control, vol. 123, no. 3, pp.317–323, 2001.

[12] A. Ferrara, L. Giacomini, and C. Vecchio, “Control of nonholonomicsystems with uncertainties via second-order sliding modes,” Int. J. Ro-bust Nonlin. Control, vol. 18, no. 4/5, pp. 515–528, 2008.

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[16] A. T. Fuller, “Relay control systems optimized for various performancecriteria automation and remote control,” in Proc. First IFAC World Con-gress, Moscow, Russia, 1960, vol. 1, pp. 510–519.

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[18] A. Isidori, Nonlinear Control Systems, second ed. New York:Springer Verlag, 1989.

[19] A. Levant, “Sliding order and sliding accuracy in sliding mode control,”Int. J. Control, vol. 58, pp. 1247–1263, 1993.

[20] A. Levant, “Robust exact differentiation via sliding mode technique,”Automatica, vol. 34, no. 3, pp. 379–384, 1998.

[21] A. Levant, A. Pridor, R. Gitizadeh, I. Yaesh, and J. Z. Ben-Asher, “Air-craft pitch control via second-order sliding technique,” AIAA J. Guid.,Control Dyn., vol. 23, no. 4, pp. 586–594, 2000.

[22] A. Levant, “Higher order sliding modes, differentiation and output-feedback control,” Int. J. Control, vol. 76, no. 9/10, pp. 924–941, 2003.

[23] A. Levant, “Homogeneity approach to high-order sliding mode de-sign,” Automatica, vol. 41, no. 5, pp. 823–830, 2005.

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[27] A. Levant and L. Alelishvili, “Integral high-order sliding modes,” IEEETrans. Autom. Control, vol. 52, no. 7, pp. 1278–1282, Jul. 2007.

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Arie Levant (M’08) (formerly L.V. Levantovsky) re-ceived the M.S. degree in differential equations fromMoscow State University, Moscow, Russia, in 1980,and the Ph.D. degree in control theory from the In-stitute for System Studies (ISI), USSR Academy ofSciences, Moscow, Russia, in 1987.

From 1980 to 1989, he was with ISI, Moscow.From 1990 to 1992, he was with the Mechan-ical Engineering and Mathematical Departments,Ben-Gurion University, Beer-Sheva, Israel. From1993 to 2001, he was a Senior Analyst at the

Institute for Industrial Mathematics, Beer-Sheva. Since 2001, he has been aSenior Lecturer at the Applied Mathematics Department, Tel-Aviv University,Tel-Aviv, Israel. Since January 2009 he has been an Associate Professor. Hisprofessional activities have been concentrated in nonlinear control theory,stability theory, singularity theory of differentiable mappings, image processingand numerous practical research projects in these and other fields. His currentresearch interests are in high-order sliding-modes and their applications tocontrol and observation, real-time robust exact differentiation and nonlinearrobust output-feedback control.

Authorized licensed use limited to: CNR Area Ricerca Genova. Downloaded on July 20,2010 at 12:33:11 UTC from IEEE Xplore. Restrictions apply.

Arie Levant (M’08) (formerly L.V. Levantovsky) received the M.S. degree in differential equa-tions from Moscow State University, Moscow, Russia, in 1980, and the Ph.D. degree in controltheory from the Institute for System Studies (ISI), USSR Academy of Sciences, Moscow, Russia,


16 REFERENCES

in 1987. From 1980 to 1989, he was with ISI, Moscow. From 1990 to 1992, he was with theMechanical Engineering and Mathematical Departments, Ben-Gurion University, Beer-Sheva,Israel. From 1993 to 2001, he was a Senior Analyst at the Institute for Industrial Mathematics,Beer-Sheva. Since 2001, he was a Senior Lecturer at the Applied Mathematics Department, Tel-Aviv University, Tel-Aviv, Israel. Since January 2009 he is Associate Professor. His professionalactivities have been concentrated in nonlinear control theory, stability theory, singularity theoryof differentiable mappings, image processing and numerous practical research projects in theseand other fields. His current research interests are in high-order sliding-modes and their appli-cations to control and observation, real-time robust exact differentiation and nonlinear robustoutput-feedback control.

References

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Bartolini, G., Ferrara, A., Usai, E., and Utkin, V. (2000), “On multi-input chattering-free secondorder sliding mode control,” IEEE Transactions on Automatic Control, 45(9), 1711–1717.

Choi, H.H. (2002), “Variable structure output feedback control design for a class of uncertaindynamic systems,” Automatica, 38(2), 335 – 341.

Defoort, M., Floquet, T., Kokosy, A., and Perruquetti, W. (2009), “A novel higher order slidingmode control scheme,” Systems and Control Letters, 58(2), 102 – 108.

Edwards, C., Floquet, T., and Spurgeon, S. (2008), “Circumventing the Relative Degree Condi-tion in Sliding Mode Design,” Modern Sliding Mode Control Theory New Perspectives andApplications, Lecture Notes in Control and Information Sciences, 375, 137–158.

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