Sliding Modes Control Design

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Preprint submitted to Automatica 1 9 October 2006

Principles of 2-Sliding Mode Design

Arie Levant*a

This paper was not presented at any IFAC meeting.* Corresponding author. Phone +972-3-6408812. Fax +972-3-6407543.

a School of Mathematical Sciences, Tel-Aviv University, Ramat-Aviv, 69978 Tel-Aviv, Israel, E-mail: [email protected]

Abstract

Second order sliding modes are used to keep exactly a constraint of the second relative degree or just to avoid chattering, i.e.in the cases when the standard (first order) sliding mode implementation might be involved or impossible. Design of a num-

ber of new 2-sliding controllers is demonstrated by means of the proposed homogeneity-based approach. A recently devel-

oped robust exact differentiator being applied, robust output-feedback controllers with finite-time convergence are pro-duced, capable to control any general uncertain single-input-single-output process with relative degree 2. An effective sim-ple procedure is developed to attenuate the 1-sliding mode chattering. Simulation of new controllers is presented.

Keywords: Second-order sliding mode; Robustness; Homogeneity; Chattering; Output feedback.

1. IntroductionSliding mode control (Utkin, 1992; Zinober, 1994; Ed-

wards and Spurgeon, 1998) is considered to be one of themain methods effective under uncertainty conditions. Theapproach is based on keeping exactly a properly chosen

constraint by means of high-frequency control switching,and is known as robust and very accurate. Unfortunately,its standard application is restricted: if an output s is to bezeroed, the standard sliding mode may keep s = 0 only ifthe outputs relative degree is 1 (i.e. if the control appearsexplicitly already in s& ). High frequency control switchingmay also cause the dangerous chattering effect (Fridman,2001, 2003; Lee & Utkin, 2006; Boiko, Fridman & Castel-lanos, 2004; Boiko, 2005).

Consider a smooth dynamic system with a smooth out-put function s, and let the system be closed by some pos-sibly-dynamical discontinuous feedback. Then, providedthe successive total time derivatives s, s& , ..., s

(r-1)are

continuous functions of the closed-system state-space vari-ables, and the set s = s& = ... = s(r-1) = 0 is non-empty andconsists locally of Filippov trajectories (Filippov, 1988),the motion on the set s = s& = ... = s

(r-1)= 0 is said to exist

in r-sliding mode (rth order sliding mode) (Levant, 1993,2003). The rth derivative s

(r)is mostly supposed to be dis-

continuous or non-existent.The standard sliding mode is of the first order ( s& is

discontinuous). Higher-order sliding modes (HOSM) re-move the above-mentioned relative-degree restriction and,

being properly used, can practically eliminate the chatter-ing. Asymptotically stable HOSMs appear in many sys-tems with traditional sliding-mode control (Fridman, 2001,2003) and are deliberately introduced in systems with dy-

namical sliding modes (Sira-Ramrez, 1993). Finite-time

convergent HOSMs preserve the features of the standard(first order) sliding modes and improve their accuracy inthe presence of switching delays and discrete measure-ments (Levant, 1993).

While finite-time-convergent arbitrary-order sliding-mode controllers are mostly still theoretically studied (Le-vant, 2001, 2003; Floquet, Barbot and Perruquetti, 2003),

2-sliding controllers with finite-time convergence have al-ready been successfully implemented for solution of real

problems (Bartolini, Ferrara & Punta, 2000; Bartolini, Pis-ano, Punta & Usai, 2003; Levant, Pridor, Gitizadeh, Yaesh& Ben-Asher, 2000; Sira-Ramrez, 2002; Orlov, Aguilar &Cadiou, 2003; Khan, Goh & Spurgeon, 2003; Shkolnikov& Shtessel, 2002; Shkolnikov, Shtessel, Lianos, & Thies,2000; Shtessel & Shkolnikov, 2003; Shtessel, Shkolnikov& Brown, 2004). There are only few widely used 2-slidingcontrollers: the sub-optimal controller (Bartolini et al.,1998), the terminal sliding mode controllers (Man, Pap-linski & Wu, 1994), and the twisting controller (Levant1993). 2-3 more controllers are presented in (Levant, 1993,

Khan et al., 2003), in particular, the super-twisting control-ler is used in differentiators (Levant, 1998, 2003).Generally speaking, 2-sliding controllers are used to

keep at zero outputs of the relative degree 2 or to avoidchattering while zeroing outputs of the relative degree 1.The main difficulty of their implementation is the necessityto use the first time derivative of the output, which causes

possible sensitivity of the approach to sampling noises.These three subjects are studied in the present paper, andsome standard solutions are proposed.

1. The homogeneity-based approach (Levant, 2005a) isproposed in this paper to regularize the construction of newfinite-time convergent 2-sliding controllers featuring thehighest possible accuracy of 2-sliding control (Levant,

1993). The simplicity of such controller design is demon-


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strated, and a few new controllers are developed, some ofwhich feature new advantageous properties. In particular,the so-called quasi-continuous controllers have better per-formance, being continuous everywhere except the 2-sliding set s = s& = 0. The main features of known 2-

sliding controllers (Levant, 1993; Bartolini et al., 98, 2003)are proved to be valid for general finite-time-convergenthomogeneous 2-sliding controllers.

2. A number of approaches were developed to avoid thedangerous chattering often featuring standard (1-sliding)modes. In particular, high-gain control with saturation isused to approximate the sign-function in a boundary layeraround the switching manifold (Slotine & Li, 1991), thesliding-sector method (Furuta & Pan, 2000) is suitable tocontrol disturbed linear time-invariant systems.

Another standard way is to avoid chattering by means of2-sliding mode control, treating the input u of the systemas a new state variable, while using its time derivative u& as the actual control ((Emelyanov, Korovin & Levant,1993; Levant, 1993; Bartolini et al., 1998). In that case u& has to dominate in the equation for s&& . Regretfully, in gen-eral, the expression for s&& contains terms with u. Thus, u& is to dominate overu itself, which looks problematic. For-tunately, in the vicinity of the 2-sliding mode u is close tothe so-called equivalent control ueq(t, x) (Utkin, 1992),where t, x denote the time and the state variables. Thefunction ueq is defined from the equation s& = 0 and is in-dependent of u& . Thus, the approach is always valid insome vicinity of the 2-sliding set s = s& = 0, or, in otherwords, in a vicinity of the set s = 0, u = ueq(t,x). Since ueqis typically unknown, the system stability is neverthelessdifficult to guarantee. A simple effective procedure previ-

ously established only for the twisting controller is provedin this paper to resolve the problem by means of all knownand newly-developed 2-sliding controllers.

3. In practice the sampled values of the output s are cor-rupted by noises. The controller robustness with respect tosampling noises is especially important, since most of 2-sliding controllers explicitly use possibly unavailable s& orsign s& . Historically the first way to get the lacking infor-mation was to use finite differences instead of the deriva-tive. That approach is simple and valid for all 2-slidinghomogeneous controllers, but, unfortunately, it is sensitiveto large noises or small sampling intervals. Following Le-vant (2005a), this paper suggests to use a recently devel-oped robust exact differentiator (Levant, 1993, 2003) as astandard part of all 2-sliding homogeneous controllers,

producing robust output-feedback control. The resultingoutput-feedback controllers preserve the ultimate accuracyand finite-time convergence of the original controllers anddo not require any information on the noises. The corre-sponding asymptotic accuracies are estimated.

All contemporary applications are computer-based anduse discrete-time sampling. The corresponding general-case asymptotic accuracies are also calculated.

The results of this paper were partially presented at aconference (Levant, 2002), and can be also considered as ademonstration of the general homogeneity-based HOSMcontroller design (Levant, 2005a) in the simplest case of

the sliding order 2. Simulation demonstrates the feasibilityof new controllers.

2. The problem statementConsider a dynamic system of the form

x& = a(t,x) + b(t,x)u, s = s(t, x), (1)

where x Rn, u R is control, s is the only measured

output, smooth functions a, b, s and the dimension n areunknown. The relative degree of the system (Isidori, 1989)is assumed to be 2. The task is to make the output s vanishin finite time and to keep s 0 by means of a discontinu-ous globally-bounded feedback control. System trajectoriesare supposed to be infinitely extendible in time for any

bounded input. The system is understood in the Filippovsense.

Calculating the second total time derivative s&& along the

trajectories of (1) achieve that under these conditionss&& = h(t,x) +g(t,x)u, h=

0=s

u&& , g= s

&&u

0 (2)

where the functions g, h are some unknown smooth func-tions. Suppose that the input-output termed conditions

0 0. Note that at leastlocally (3) is satisfied for any smooth system (1) with thewell-defined relative degree 2.

Obviously, no continuous feedback controller can solvethe stated problem. Indeed, any continuous control u =

j(s, s&

) providing for s 0, has to satisfy the equalityj(0, 0) = - h(t,x)/g(t,x), whenevers = s& = 0 holds. Theproblem uncertainty prevents it, for the controller will notbe effective for the simple autonomous linear system s&& =c + ku, KmkKM, |c| C, with j(0,0) - c/k. In otherwords, the 2-sliding mode s = 0 is to be established.

Assume now that (3) holds globally. Then (2), (3) implythe differential inclusion

s&& [-C, C] + [Km,KM]u. (4)

Most 2-sliding controllers may be considered as control-lers for (4) steering s, s& to 0 in (preferably) finite time.Since inclusion (4) does not remember the original sys-tem (1), such controllers are obviously robust with respectto any perturbations preserving (3).

Hence, the problem is to find such a feedback

u = j(s, s& ), (5)

that all the trajectories of (4), (5) converge in finite time tothe origin s = s& = 0 of the phase plane s, s& .

Differential inclusion (4), (5) is understood here in theFilippov sense (Filippov, 1988), which means that theright-hand vector set is enlarged in a special way in orderto satisfy certain convexity and semi-continuity conditions(Levant, 2005a). In particular, in the case when j is con-tinuous almost everywhere, a set F(s, s& ) is substituted foru in (4), F being the convex closure of all possible limitvalues of j(s1, 1s& ) obtained when the continuity point


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(s1, 1s& ) approaches (s, s& ). The function j is assumed tobe a locally bounded Borel-measurable function (actuallyall functions used in sliding-mode control satisfy that re-striction). Solution is any absolutely continuous vectorfunction (s(t), s& (t)) satisfying (4), (5) for almost all t.

3. 2-sliding homogeneity and finite-timestability

The general notion of homogeneous differential equa-tion, review of the corresponding theoretical results andnumerous references can be found in (Bacciotti & Rosier,2001). Introduce a few auxiliary notions and theoremsadopted from (Levant, 2005a), where they are formulatedfor general homogeneous Filippov differential inclusionswith negative homogeneity degrees. The combined time-coordinate transformation

Gk: (t, s, s& ) a ( kt, k2s, k s& ) (6)

transfers solutions of (4), (5) into the solutions of the trans-formed inclusion

s&& [-C, C] + [Km,KM] j(k2s, k s& ).

Inclusion (4), (5) and controller (5) itself are called 2-sliding homogeneous (Levant, 2005a)if these two differen-tial inclusions are equivalent for any s, s& and k > 0 (i.e.have the same solutions). The mapping dk(s, s& ) =(k

2s, k s& ), k > 0 is called the homogeneity dilation (Bac-

ciotti et al., 2001). Obviously, (5) is 2-sliding homogene-ous if

j(k2s, k s& ) j(s, s& ) (7)

holds for any k > 0. Such a function j is called 2-slidinghomogeneous with the homogeneity degree 0. For exam-

ple, the following controllers are 2-sliding homogeneous:

u = - sign s = - sign k2s,

u = (2s - 2s& )/(|s| + 2s& ) =(2 k

2s - (k s& )

2)/(|k

2s| + (k s& )

2).

Surely these controllers do not solve the stated problem.Since the Filippov solutions do not depend on the values

ofj on any set of the zero measure, any changes ofj onsuch a set do not change the homogeneity properties of the

controller. It is assumed in this paper that (7) holds for anys, s& and k > 0. Note that (7) requires global boundednessofj (excluding possibly a zero-measure set), otherwise itis unbounded in any vicinity of 0 and Filippovs definitionis not applicable to (1), (5).1. Differential inclusion (4), (5) is called furthergloballyuniformly finite-time stable at 0 if it is Lyapunov stableand for anyR > 0 exists T> 0 such that any trajectory start-ing within the disk ||(s, s& )|| < R stabilizes at zero in thetime T.2. Differential inclusion (4), (5) is called furthergloballyuniformly asymptotically stable at 0, if it is Lyapunovstable and for anyR > 0 and e > 0 exists T> 0, such thatany trajectory starting within the disk ||(s, s& )|| < R entersthe disk ||(s, s& )|| < e in the time Tto stay there forever.

A set D is called dilation-retractable if 0 D, anddkDD for any k < 1. For example any disk centered atthe origin is dilation-retractable.3. The homogeneous differential inclusion (4), (5) iscalled furthercontractive if there are 2 compact setsD1,D2

and T> 0 such that D2 lies in the interior ofD1 and con-tains the origin; D1 is dilation-retractable; and all trajecto-ries starting at the time 0 within D1 are localized in D2 atthe time moment T.

Most of known 2-sliding controllers (Levant, 1993, Bar-tolini et al., 2003) satisfy these properties. The followingTheorem is actually true for any homogeneous differentialinclusion of a negative homogeneity degree (Levant,2005a).Theorem 1. Let controller (5) be 2-sliding homogeneous,then properties 1, 2and3are equivalent.

Explain the Theorem in few words. Since 1 implies 2and 3, it is sufficient to show that 3 implies 1. The ho-mogeneity of the system (4), (5) means that it is invariantwith respect to the transformation (6). Thus, the behaviorof the system trajectories is geometrically the same in all

points, which can be transferred one into another by meansof a simple linear transformation dk(s, s& ) = (k

2s, k s& ), k

> 0 (the dilation). The only difference is that the corre-sponding motion near the origin requires proportionallyless time, according to (6). Hence, using the contractivity

property 3, a chain of embedded domains can be con-structed, retracting to the origin. The corresponding systemmotion is actually shown to be a finite-time collapse to-wards the origin.

It is natural to call the controller u = f(s, s& ) a smallhomogeneous perturbation of (5) if the difference f j is a

2-sliding homogeneous function with the homogeneity de-gree 0, small in some fixed vicinity of the origin.Corollary 1. Global uniform finite-time stability of the2-sliding homogeneous controller(5) is robust with respectto small homogeneous perturbations of the controller.

Indeed, it follows from the robustness of property 3.The following theorems consider the robustness of homo-geneous controllers with respect to sampling errors, whichare supposed to be some bounded Lebesgue-measurablefunctions of time of any nature. No features of the noisesare assumed to be known.Theorem 2 (Levant, 2005a). Let the noise magnitudes ofmeasurements ofs, s& be less than b0d andb1d

1/2respec-

tively with some positive constants b0

andb1. Then any fi-

nite-time-stable 2-sliding-homogeneouscontroller(5) pro-vides in finite time for keeping the inequalities |s| < g0d,| s& | < g1d

1/2 with the same positive constantsg0, g1 for any

d>0.Controller (5) requires availability of s& . That informa-

tion can be obtained by means of the real-time robust fi-nite-time-convergent exact differentiator (Levant, 1998,2003) as follows:

u = - j(z0, z1), (8)

0z& = - l2L1/2

| z0 - s|1/2

sign(z0 - s) +z1, (9)

1z& = - l1L sign(z0 - s). (10)


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Here z0, z1 are the estimations of s and s& respectively.The parameters of the differentiator li are to be chosen inadvance, in particular l1 = 1.1, l2 = 1.5 is a good choice(Levant, 1998).L is the only differentiator parameter to betuned, and it has to satisfy the only condition | s&& | L, i.e.L

C+KM sup|j| (recall that j is bounded).Theorem 3 (Levant, 2005a). Suppose that controller(5) is2-sliding homogeneous and finite-time stable, thenthe out-

put-feedback controller(8) - (10)provides in finite time forkeepings = s& = 0. If s is sampled with a constant sam-

pling intervalt > 0, the inequalities |s| < g0t2, | s& | < g1t

are established with some positive constantsg0, g1. Ifs issampled (continuously) with a noise being a Lebesgue-measurable function of time of the magnitude e > 0, theinequalities |s| < m0e, | s& | < m1e

1/2are established with

some positive constantsm0, m1.There is another way to estimate s& with discrete sam-

pling. Indeed, let

u = j(s, Dsi/t) = j(t2s, Dsi), (11)

where si = s (ti,x(ti)), Dsi = si- si-1, t [ti, ti+1).Theorem 4 (Levant, 2005a). Let controller(5) be 2-slidinghomogeneous and finite-time stable, then in the absence ofmeasurement noises controller (11) provides in finite time

for keeping the inequalities |s| < g0t2, | s& | < g1t with some

positive constantsg0, g1.Note that 1-sliding mode provides only for the accu-

racy proportional to t. The accuracy described in Theo-rems 3, 4 is the best possible with discontinuous controland the relative degree 2 (Levant, 1993).

Due to the finite-time convergence of the controllers,Theorems 2 - 4 have obvious local analogues in the casewhen (3) holds only locally. Recall that 2-sliding point is a

point where s = s& = 0. Then in the absence of noises alltrajectories of (1), (5) (respectively (1), (8) (10) or (1),(11)) starting from some vicinity of a 2-sliding point withwell-defined relative degree 2 converge in finite time to the2-sliding mode s 0, or the corresponding inequalities areestablished in the case of noisy measurements or discretesampling. The long-term motion is determined by the sys-tem properties, especially by its zero dynamics (Isidori,1989). Note that in the case, when s

&&u

is negative thesame controller (5) is to be used, but with the oppositesign.

The popular sub-optimal controller (Bartolini et al.,

1998, 2003) is defined by the formula

u = - r1 sign (s - s*/2) + r2 sign s*, r1 > r2 > 0,

where s* is the value ofs detected at the closest time inthe past when s& was 0. The initial value ofs* is 0. Thecorresponding convergence conditions are

2[(r1 + r2)Km - C] > (r1 - r2)KM + C, (r1 - r2)Km > C.

Usually the moments when s& changes its sign aredetected using finite differences. The control u dependsactually on the whole history of measurements of s& and s,and does not have the feedback form (5). Nevertheless, thehomogeneity transformation (6) preserves its trajectories,

and it is natural to call it 2-sliding homogeneous in the

broad sense. Also the statements of Theorems 2 - 4 remainvalid for this controller.

Results similar to Theorems 1 - 4 are formulated andproved in (Levant, 2005a) for any relative degree r andr-sliding homogeneous controllers. At the same time de-

sign of new r-sliding controllers is rather difficult with r>2 due to the complicated geometry of R

r(Levant, 2003,

2005b, Floquet et al., 2003).

4. Design of 2-sliding controllersAs follows from the previous Section it is sufficient to

build a 2-sliding-homogeneous contractive controller. De-sign of such 2-sliding controllers is greatly facilitated bythe simple geometry of the 2-dimensional phase plane withcoordinates s, s& : any smooth curve locally divides the

plane in two parts.A number of known 2-sliding controllers may be con-

sidered as particular cases of a generalized 2-sliding ho-mogeneous controller

u = - r1 sign(m1 s& +l1|s|1/2

sign s)

- r2 sign(m2 s& + l2|s|1/2

sign s), r1, r2 > 0. (12)

Drawing the two lines mi s& + li|s|1/2

sign s = 0, mi2

+ li2

> 0, mi, li 0, i = 1, 2, m12

+ m22

> 0, l12

+ l22

> 0, in the

phase plane, and considering various possible cases, one

can readily check that r1, r2 can always be chosen so that

controller (12) be finite-time stable. Indeed, if, for exam-

ple, m1, l1 > 0, then a 1-sliding mode can easily be organ-ized on the line m1 s& + l1|s|

1/2sign s = 0. Sliding mode

conditions being fulfilled at one point of the line, they

automatically hold along the whole line due to the homo-

geneity properties. If for each i one of the coefficients is

zero, the twisting controller (Levant, 1993)

u = - (r1 sign s + r2 sign s& ),

is built. Its convergence condition is

(r1 + r2)Km - C> (r1 - r2)KM + C, (r1 - r2) Km > C.

A typical trajectory in the plane s, s& is shown in Fig. 1a.Controller (12) may be considered as a generalization ofthe twisting controller, when the switching takes place on

parabolas m1 s& + l1|s|1/2

sign s = 0 instead of the coordi-

nate axes.Thus, the resulting controller satisfies Theorem 1, andits discrete-sampling version

u = - r1 sign(m1 Dsi + l1t |si|1/2

sign si)- r2 sign(m2 Dsi + l2t |si|

1/2sign si). (13)

provides for the accuracy described in Theorem 4, i.e. s ~t

2, s& ~ t. Similarly, the noisy measurements lead to the

accuracy provided by Theorems 2, 3.Consider special interesting cases of controller (12),

(13). With m1 = l2 = 0, r2 > r1 > 0, achieve the driftcontroller (Levant, 1993) from (13) (it does not convergewith continuous measurements). A homogeneous form ofthe controller with prescribed convergence law (Levant1993) arises when m1 = m2, l1 = l2:


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u = - a sign( s& + b|s|1/2

sign s), (14)

where a = r1 + r2, b = l1/m1. This controller is a 2-slidinghomogeneous analogue of the terminal sliding mode con-troller originally featuring a singularity at s = 0 (Man et

al., 1994). The complete elementary proof of the followingsimple proposition was actually never published.

Fig. 1. Convergence of various 2-sliding

homogeneous controllers

Proposition 1. With aKm - C> b2/2 controller (14) pro-

vides for the establishment of the finite-time stable2-sliding modes 0.Proof. Differentiating the function S = s& + b|s|

1/2sign s

along the trajectory, obtain

S& [-C, C] - a [Km,KM] sign S + 21 b s& |s|

-1/2.

Checking the condition S& sign S < const < 0 in a vicin-ity of each point on the curve S = 0, obtain using s& =-b|s|

1/2sign s that the 1-sliding-mode existence condition

holds at each point except of the origin, ifaKm - C> b2/2.

The trajectories of the inclusion inevitably hit the curveS = 0 due to geometrical reasons. Indeed, each trajectory,starting with S > 0, terminates sooner or later at the semi-axis s = 0, s& < 0, ifu = -a sign S keeps its constant value-a (Fig. 1b). Thus, on the way it inevitably hits the curve S= 0. The same is true for the trajectory starting with S < 0.Since that moment the trajectory slides along the curve S =0 towards the origin and reaches it in finite time. Obvi-ously, each trajectory starting from a disk centered at theorigin comes to the origin in a finite time, the convergencetime being uniformly bounded in the disk.

Consider the region We confined by the lines s& = eand the trajectories of the differential equations s&& = -C+

Km

a with initial conditions s = e2/b

2, s& = - e, and s&& =

C - Kma with initial conditions s = - e2/b2, s& = e

(Fig. 1b). No trajectory starting from the origin can leaveWe. Since e can be taken arbitrarily small, the trajectorycannot leave the origin. The same reasoning proves theLyapunov stability of the origin. n

The 2-sliding stability analysis is greatly simplified by

the fact that all the trajectories in the plane s, s& whichpass through a given continuity point of u = j(s, s& ) areconfined between the properly chosen trajectories of thehomogeneous differential equations s&& = C+ KMj(s, s& )and s&& = C+Kmj(s, s& ). These border trajectories cannot

be crossed by other paths, if j is locally Lipschitzian, andmay be often chosen as boundaries of appropriate dilation-retractable regions. The rule is to take trajectories satisfy-ing s&& = C+ KMj(s, s& ) and s&& = - C+ Kmj(s, s& ) withj(s, s& ) > 0, and s&& = - C + KMj(s, s& ) and s&& = C +

Kmj(s, s& ) when j(s, s& ) < 0. Recall that a region is dila-tion-retractable iff, with each its point (s, s& ), it containsall the points of the parabolic segment (k

2s, k s& ), 0 k

1.As follows from Corollary 1 a small violation of the

conditions of Proposition 1 preserves the finite-time stabil-ity of controller (14), if aKm - C< b

2/2, but still aKm > C.

Fig. 2. Contraction property of some 2-slidinghomogeneous controllers

Proposition 2. Let C< aKmC+ b2/2. Then with suffi-

ciently smallC+ b2/2 - aKmcontroller (14) provides for

the twisting-like convergence to the finite-time stable2-sliding modes 0.Proof. The corresponding trajectories and dilation-retractable sets are shown in Fig. 2b. Define once more S =s& + b|s|

1/2sign s. The boundaries of the regions are the

trajectories of the differential equation s&& = -C+Kma withS > 0 and s&& = C-Kma with S < 0. Consider the combinedcurve 1-2-3-4 (Fig. 2b). With sufficiently small C+ b

2/2 -

aKm point 4 is closer to zero than point 1. This requirement


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forms the convergence condition. Due to the homogeneityof the system, this geometric condition does not depend onthe placement of point 1 on the axis s& . Direct calculationwill replace it with an algebraic condition. The absolutevalue | s&& | is separated from zero by Kma - C. Therefore,

the convergence time to the smaller set is estimated by3 Ms& /(Kma - C), where Ms& is the maximal value of | s& |in the corresponding dilation-retractable set. Hence, thefinite time stability is obtained due to Theorem 1. n

A new controller is obtained when (l1, m1) (l2, m2)and all components are not zero, especially interesting arethe cases when

r1 = r2 , 4r1Km - 2C > max(l12/ m1

2, l2

2/ m2

2).

In that case the trajectory is confined between two para-bolic segments (Fig. 1c, 2a). The control vanishes in thatregion. The corresponding dilation-retractable sets areshown in Fig. 2a. The convergence time to the smaller set

is estimated by 2 Ms&

/(2Kmr1 - C), where Ms&

is the maxi-mal value of | s& | in the corresponding dilation-retractableset. The idea of that controller is close to that of the slidingsector control (Furuta et al., 2000). The corresponding

proposition as well as its proof are obvious and are omit-ted.

An important class of HOSM controllers comprisesrecently proposed so-called quasi-continuous controllers,featuring control continuous everywhere except of theHOSM s = s& = ... = s

(r-1)= 0 itself. Since the r-sliding

condition has the codimension r, with the sliding orderr>1 the general-case trajectory does never hit the r-slidingmanifold. Hence, the condition s = s& = ... = s

(r-1)= 0 is

never fulfilled in practice with r 2, and the control re-mains continuous function of time all the time. As a result,the chattering is significantly reduced. Following is the 2-sliding controller from such a family of arbitrary-ordersliding controllers (Levant, 2005b):

u = - a2/1

2/1

||||

sign||

sb+s

ssb+s

&

&. (15)

This control is continuous everywhere except of the ori-gin. It vanishes on the parabola s& + b|s|

1/2sign s = 0. With

sufficiently large a there are such numbers r1, r2, 0 < r1 0, aKm - C > 0 (16)

and the inequality

aKm - C - 2 aKm b+rb

-21 r

2> 0 (17)

hold for some positiver > b (which is always true with suf-ficiently largea), thencontroller(15) provides for the es-tablishment of the finite-time stable 2-sliding modes 0.

Conditions of the proposition can be solved for a, butthe resulting expressions are redundantly cumbersome.

Proof. Denote r = - s& /|s|1/2

. Calculations show that u =a (r - b)/(|r| + b) and

r& ([-C, C] - [Km,KM]a b+r

b-r

||+

21 r

2sign s) |s|

-1/2.

Due to the symmetry of the problem, it is enough toconsider the case s > 0, - < r < . With negative orsmall positive r the rotation velocity r& is always positivedue to (16), thus there is such positive r1 < b that the tra-

jectories enter the region r > r1. It is needed to show nowthat there is r2 > b such that in some vicinity ofr = r2 theinequality r& < 0 holds. That is exactly condition (17).Thus, conditions (16), (17) provide for the establishmentand keeping of the inequality r1 < r < r2.n

Following is another example of a quasi-continuous2-sliding controller:

u=

=ssa-

ssb+ssg-a-a

0,sign

0,)}sign||/(,max[,min{ 2/1

&

&

,(18)

where a, g, b > 0, aKm - C> b2/2 , gb > a. The definition

of the control u with s = 0 is made by continuity and doesnot influence the system trajectories, since it only influ-ences values on a zero-measure set. Enlarging g one com-

pels the trajectories to get closer to the parabola s& +b|s|

1/2sign s = 0 without increasing the control magnitude

(Fig. 1d). Also here the discontinuity is concentrated at s =s& = 0. Thus, in the presence of measurement errors themotion takes place in some vicinity of the mode s = s& = 0without entering it, and the control signal turns out to becontinuous.

Moreover, let z(y) be any monotonously growing posi-tive continuous function of a non-negative argument, thenthe following controller generalizes (18)

uz = min{a,max[- a,-gz(| s& |/|s|1/2

)sign s& +z(b)sign s)]},

u =

=ssa-

sssV0,sign

0),,(

&

&u. (19)

Proposition 4.Leta, b, g > 0, aKm - C> z(b)2/2 , gz(b) >

a andg be sufficiently large, then controller(19)providesfor the establishment of the finite-time stable 2-slidingmodes 0.

The proof is very similar to that of Proposition 3.

Enlarging g one compels the trajectories to get closer to theparabola s& + z(b) |s|1/2

sign s = 0 without increasing thecontrol magnitude (Fig. 1d). Like other listed controllers,this one also provides for the finite-time convergence tothe 2-sliding mode and the accuracies corresponding toTheorems 2 - 4.Chattering attenuation. The standard problem of classi-cal (first order) sliding-mode control is attenuation of thechattering effect (Slotine et al, 1991; Furuta et al., 2000;Fridman, 2003). 2-sliding mode control provides effectivetools for the reduction or even practical elimination of thechattering without compromising the benefits of the stan-dard sliding mode (Boiko & Fridman, 2005; Boiko, Frid-man, Iriarte, Pisano & Usai, 2006). Let the relative degree

of the system (1) be 1, i.e. (2) is replaced by


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s& = h(t,x) +g(t,x)u, 0 0 (21)

is assumed. Consider u& as a new control, in order to over-come the chattering. Differentiating (20) achieve

s&& = h1(t,x,u) +g(t,x) u& ,h1 = th + xh (a + bu) + ( tg + xg (a + bu)) u .

Assume that with |u| k1, k1 > k, the function h1(t,x,u) isbounded:

1||

supku

|h1(t,x,u)| C1 . (22)

Any listed controlleru = f(a, s, s& ) can be used here inorder to overcome the chattering and improve the slidingaccuracy of the standard sliding mode. Indeed, define

u& =

ssaf

>-

.||),,,(

||,

ku

kuu

&(23)

Theorem 5.Letfbe anyone of controllers (14), (15), (18),(19) and the controller parameters be chosen in accor-dance with the corresponding Propositions 1, 2, 3 or 4.Then with sufficiently large a controller(23) provides forthe establishment of the finite-time-stable 2-sliding modes 0. Also the statements of Theorems 2 - 4 are valid with

sufficiently small noises or sampling intervals.Controller (23) keeps |u| k, and on certain time inter-

vals u k or u -k is kept in 1-sliding mode. Note thatthough Theorem is not formulated for arbitrary 2-slidinghomogeneous controllers, it is valid for all standard con-trollers (Levant, 1993, Bartolini et al., 1998).Proof. It follows from (20), (21) that the inequality | s& | kor |u| k. The remark that u = -ksign scan be established only with s s& < 0 proves the Lemma. nLemma 2. With sufficiently large a any trajectory of the

system (1), (23) hits in finite time the manifolds = 0.Proof. Denote S the set defined by the inequalities | s& | 0, |u| < k. Within this regionu& = -a sign s. Hence, the control u moves towards thevalue - ksign s, and on the way the trajectory hits the sets& = 0, which still features |u| < k. As follows from (20), |u|kimplies the global bound | s& | kKM + C. That restric-tion is true also at the initial point on the axis s = 0. Simplecalculation shows that the inequality |s|

21 (kKM + C)

2

/(aKm - C1) takes place at the moment when s& vanishes.With sufficiently large a that point inevitably belongs to

W. nOnce the trajectory enters W, it continues to converge to

the 2-sliding mode according to the corresponding2-sliding homogeneous dynamics. This proves the conver-gence to the 2-sliding mode. In the presence of smallnoises and sampling intervals the resulting motion willtake place in a small vicinity of the 2-sliding mode s = s& =0. Thus, if this motion does not leave W, the homogeneousdynamics is still in charge, and the statements of Theorems1 - 4 are true. n

5. Simulation resultsA number of new controllers from the previous Section

are demonstrated here. Consider an academic example of avariable-length pendulum with motions restricted to somevertical plane. A load of a known mass m moves withoutfriction along the pendulum rod (Fig. 3a). Its distance fromO equals R(t) and is not measured. An engine transmits atorque u, which is considered as control. The task is totrack some function xc given in real time by the angularcoordinatex of the rod.

The system is described by the equation

x&& = - 2R

R&x& -g

R

1sin x +

2

1

mRu, (24)


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where m = 1 andg= 9.81 is the gravitational constant. Let0 < Rm RRM, R& , R&& , cx& and cx&& be bounded, s =x-xc

be available. Following are the "unknown" functionsR andxc considered in the simulation:

R = 0.8 + 0.1 sin 8t+ 0.3 cos 4t,xc= 0.5 sin 0.5t+ 0.5 cos t.

Let s = x-xc. The relative degree of the system equals 2.The assumptions (3) are fulfilled here only locally, and thecontrollers to be applied are effective only for some

bounded set of initial conditions. Choosing the controllerparametera the convergence region can be made arbitrar-ily large.

Both main applications of 2-sliding modes are demon-strated: the straight-forward implementation of 2-sliding-mode controllers leading to discontinuous control and pos-sibly dangerous chattering, and the standard practical re-moval of chattering by means of 2-sliding mode. In the lat-

ter case a redefinition of the output and input are needed.Note that the 3-sliding-mode controllers (Levant, 2003,2005b) are probably more effective in that case, but are outof the scope of this paper.

The Euler integration method was used, being the onlymethod valid for sliding-mode simulation. The parametersof the controllers were found by simulation, since the di-rect calculation of the constants C,Km,KM is difficult and,inevitably, not precise.

Discontinuous control

The controllers include a real-time differentiator andhave the form

u = j(z0,z1), s =x-xc ,0z& =- 10.61 | z0 - s|

1/ 2sign(z0 - s)+z1, (25)

1z& = - 55 L sign(z0 - s), (26)

wherez0,z1 are real-time estimations ofs, s& respectively.Differentiator (25), (26) is exact for input signals s withsecond derivative not exceeding 50 in absolute value.

The initial conditionsx(0) = x& (0) = 0 were taken, z0(0)=x(0) - xc(0) = - 0.5, z1(0) = 0, the sampling step t and theintegration steps being the same, t = 0.0001.

Consider a controller of the form (14)

u = - 10 sign(z1 + 11| z0|1/2

signz0 ). (27)

The magnitude of the control is not sufficiently large hereto establish a 1-sliding mode on the curve s& +11|s| sign s= 0, nevertheless the 2-sliding mode s = s& = 0 is estab-lished here in finite time according to Proposition 2. Notethat it is the first time when such a twisting-type conver-

gence is demonstrated for this controller. The phase trajec-tories in the plane s, s& and the first 0.1 seconds of the dif-ferentiator convergence are shown in Fig. 3b,c respec-tively, the corresponding accuracies being |s| = | x -xc | 4.210

-5, | s& | = |x& - cx& | 2.710

-2. After the sampling step t

was reduced from 10-4

to 10-5

the resulting accuracieschanged to | x - xc | 4.810

-7, | s& | = |x& - cx& | 3.110

-3

which corresponds to Theorem 4. The differentiator con-

vergence is demonstrated in Fig. 3f.

Fig. 3: Pendulum and performance of controllers(25), (26), (27) (b, f) and (25), (26), (28) (c, d, e)

Fig. 4: Performance of the controller (28), (25), (26)with noisy measurements

Consider the quasi-continuous controller

u = - 10 2/1

2/1

||||

sign||

ss

sss

+

+

&

&

. (28)

The trajectory and 2-sliding tracking performance in theabsence of noises are shown in Figs. 3d, and 3c respec-tively, the corresponding accuracies being |s| = |x - xc| 5.410

-6, |x& - cx& | 1.010

-2with t = 0.0001. Control (28) is

demonstrated in Fig. 3e. It is seen from the graph that thecontrol remains continuous until the entrance into the 2-sliding mode s = s& = 0.

The tracking results of (25), (26), (28) and the differ-entiator performance in the presence of a noise with themagnitude 0.01 are demonstrated in Figs. 4c,d respec-tively, the tracking accuracy being |s| = | x - xc | 0.036.

The noise was a periodic nonsmooth function with nonzero


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average.The performance does not significantly change,

when the frequency of the noise varies from 10 to 100000.These results correspond to Theorem 3.

Chattering attenuation

The controller includes a real-time second-order differ-entiator (Levant, 2003)

0z& = v0 =- 13.4 | z0 - s|2/ 3

sign(z0 - s)+z1, (29)

1z& = v1 = - 26.0 | z0 - v0|1/ 2

sign(z0 - v0)+z2, (30)

2z& =- 330 sign(z2 - v1), s =x-xc , (31)

where z0, z1, z2 are real-time estimations of s, s& , s&& re-spectively. Differentiator (29) - (31) is exact for input sig-nals s with the third derivative not exceeding 300 in abso-lute value. The initial valuesz0(0) = s(0), z1(0) =z2(0) = 0were taken.

Consider a controller of the form (19), (23) which estab-

lishes in finite time the 2-sliding mode S = S&

= 0, whereS = s + s& . Substituting the estimations z0,z1,z2 of the de-rivatives s, s& , s&& obtain the controller

u& = min{30, max[-30, - 40 (ln(|s1|/|s0|1/2

+ 1) signs1 +

ln 2 signs0)]} with |u| < 10, |s1| < 100|s0|1/2

, (32)

u& = -30 signs1 with |u| < 10, |s1| 100|s0|1/2

, (33)

u& = - u with |u| 10, s0 = z0 +z1, s1 =z1 +z2, (34)

where the function z(*) = ln(* + 1) is chosen, u(0) = 0.Without changing the control values, (33) is constructed sothat the overflow be avoided during the computer simula-tion (or practical implementation).

In the absence of noises the controller provides in finitetime for keeping s + s& 0. As a result the asymptoticallystable 3-sliding mode s = s& = s&& = 0 is established. Notethat since (19) is quasi-continuous, the applied controller(29) (34) produces the control u, whose derivative u& remains continuous until the entrance into the 2-slidingmode S = S& = 0. The graph of u& isvery similar to Fig. 3eand is omitted.The graphs of 3-sliding deviations s, s& , s&& and the control u are demonstrated in Figs 5a,c respec-tively. The 2-sliding convergence in the plane S, S& isdemonstrated in Fig. 5b. It is seen from Fig. 5d that s + s& 0 is kept in 2-sliding mode. Due to this equality the accu-racies achieved at t = 10 are of the same order: |s| =

| x -xc | 4.010-4

, | s& | = |x& - cx& | 4.010-4

, | s&& | = |x&& - cx&& | 4.010-4

.

6. Conclusions2-sliding homogeneity and contractivity are shown to

provide for all needed features of 2-sliding mode control-lers (Theorems 1-4). Construction of controllers is not dif-ficult due to the simplicity of the plane geometry. New fi-nite-time stable 2-sliding controllers were obtained in sucha way, significantly increasing the choice of known 2-sliding controllers. In particular, the quasi-continuous con-trollers (15), (18), (19) have probably better performancethan the standard 2-sliding controllers popular today.

Fig. 5: Chattering attenuation

The number of such 2-sliding homogeneous controllersis obviously infinite, and one can adjust a controller to hisneeds. Unfortunately, design of higher-order sliding con-trollers is much more difficult due to the higher dimensionof the problem (Levant, 2003, 2005b).

A simple efficient procedure was developed for thechattering attenuation in the systems with standard (firstorder) sliding modes based on their replacement by2-sliding modes (Theorem 5). The resulting continuouslipschitzian control can be used to keep auxiliary con-straints, and, when solving practical control problems, isreadily combined with different control technique (Levant

et al., 2000).The main results known for the known 2-sliding con-

trollers are extended to any finite-time-stable 2-sliding-homogeneous controllers.

A real-time robust exact differentiator having been usedas a standard part of the 2-sliding controllers, the full sin-gle-input-single-output control is achieved based on theinput measurements only. The resulting controllers are lo-cally applicable to any uncertain smooth process of relativedegree 2, and they are globally applicable, if the bounded-ness conditions (3) hold globally.

The resulting robust output-feedback controllers pre-serve the ultimate accuracy of the original 2-sliding con-trollers with direct measurements of the input derivative(Theorem 3). In the absence of noises the tracking accu-racy proportional to t

2is provided, t being a sampling pe-

riod. That is the best possible accuracy with discontinuoussecond output derivative (Levant, 1993). In the presence ofa measurement noise the tracking accuracy is proportionalto the unknown noise magnitude. That result does not de-

pend on the noise features.The differentiator is to be used whenever the sampling

step can be taken small. At the same time in the practicallyimportant case, when the sampling step is sufficiently largecompared with the noises, the differentiator is successfullyreplaced by the first finite difference (Theorem 4).


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