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Limit cycles with chattering in relay feedback systems

Johansson, Karl Henrik; Barabanov, Andrey E.; Åström, Karl Johan

Published in:IEEE Transactions on Automatic Control

DOI:10.1109/TAC.2002.802770

2002

Link to publication

Citation for published version (APA):Johansson, K. H., Barabanov, A. E., & Åström, K. J. (2002). Limit cycles with chattering in relay feedbacksystems. IEEE Transactions on Automatic Control, 47(9), 1414-1423. DOI: 10.1109/TAC.2002.802770

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1414 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 9, SEPTEMBER 2002

Limit Cycles With Chattering inRelay Feedback Systems

Karl Henrik Johansson, Member, IEEE, Andrey E. Barabanov, and Karl Johan Åström, Fellow, IEEE

Abstract—Relay feedback has a large variety of applicationsin control engineering. Several interesting phenomena occur insimple relay systems. In this paper, scalar linear systems with relayfeedback are analyzed. It is shown that a limit cycle where partof the limit cycle consists of fast relay switchings can occur. Thischattering is analyzed in detail and conditions for approximatingit by a sliding mode are derived. A result on existence of limitcycles with chattering is given, and it is shown that the limit cyclescan have arbitrarily many relay switchings each period. Limitcycles with regular sliding modes are also discussed. Examplesillustrate the results.

Index Terms—Discontinuous control, hybrid systems, nonlineardynamics, oscillations, relay control, sliding modes.

I. INTRODUCTION

RELAYS are common in control systems. They are usedboth for mode switching and as models for physical

phenomena such as mechanical friction. Relay control is theoldest control principle but is still the most applicable. Anearly reference to on–off control is [1] (as pointed out in [2]),in which Hawkin studied temperature control and noticed thatthe relay controller caused oscillations. Simple mechanicaland electromechanical systems were an early motivation forstudying models with relay feedback [3], [4]. Other applicationswere in aerospace [5], [6]. A self-oscillation adaptive system,which has a relay with adjustable amplitude in the feedbackloop, was tested in several American aircrafts in the 1950s [7].

Recently, there has been renewed interest in relay feedbacksystems due to a variety of new applications. Automatic tuningof proportional-integral-derivative (PID) controllers using relayfeedback is based on the observation that if the controller isreplaced by a relay, there will often be a stable oscillation inthe process output [8]. The frequency and amplitude of thisoscillation can be used to determine PID controller parame-ters similar to the classical approach by Ziegler and Nichols

Manuscript received September 19, 1997; revised November 25, 1998, Oc-tober 31, 2000, and February 13, 2002. Recommended by Associate EditorA. Tesi. The work of K. H. Johansson and K. J. Åström was supported bythe Swedish Research Council for Engineering Science under Contract 95-759.The work of A. Barabanov was supported by Grants from the Russian Funda-mental Research Program and from the Russian Foundation for Basic Research98-01-01009.

K. H. Johansson is with the Department of Signals, Sensors, and Systems,Royal Institute of Technology, SE-100 44 Stockholm, Sweden (e-mail:kallej@s3.kth.se).

A. E. Barabanov is with the Faculty of Mathematics and Mechanics,Saint-Petersburg State University, 198904 Saint-Petersburg, Russia (e-mail:Andrey.Barabanov@pobox.spbu.ru).

K. J. Åström is with the Department of Automatic Control, Lund Institute ofTechnology, SE-221 00 Lund, Sweden (e-mail: kja@control.lth.se).

Publisher Item Identifier 10.1109/TAC.2002.802770.

[9]. Another application of relay feedback is also in the designof variable-structure systems [10]. The high-gain of the relaymakes it possible to design a control system that is robust toparameter variations and disturbances. Hybrid control systemshave both continuous-time and discrete-event dynamics. An in-teresting class of hybrid systems are switched control systems[11], in which a relay feedback system is the simplest member.Switched controllers have a richer structure than regular smoothcontrollers and can, therefore, often give better control perfor-mance. There exist, however, no unified approach today to de-sign switched controllers. An interesting application of relayfeedback is in the design of delta–sigma modulators in signalprocessing [12], [13]. Delta–sigma modulators have replacedstandard analog-to-digital (AD) and digital-to-analog (DA) con-verters in many applications, because they are often simpler toimplement. The basic setup of a delta–sigma modulator is a filterin a feedback loop with a quantizer, which can be modeled asa relay. Modeling of quantization errors in digital control is an-other motivation to study relay feedback [14].

Limit cycles and sliding modes are two important phenomenathat can occur in relay feedback systems. Research on both theseissues was very active in the former Soviet Union during the1950s and 1960s. Major contributions to the work on oscilla-tions can be found in [3] and [4] (see also [15] on the describingfunction method to analyze these oscillations). While building amathematical framework for sliding modes, interesting proper-ties of differential equations with discontinuous right-hand sideswere found. Uniqueness and existence of solutions and smoothdependency on initial conditions of a solution (all well knownto hold for smooth differential equations) could easily be vio-lated by a nonsmooth system. This was a topic for discussionin which, for example, Filippov and Neimark took part in at thefirst International Federation of Automatic Control (IFAC) con-gress [16]. A standard reference on the concept of solution tononsmooth systems is Filippov’s monograph [17]. Utkin’s defi-nition of sliding modes based on equivalent control is discussedin [10]. A regular (first-order) sliding mode is a part of a tra-jectory on a surface of dimension , where is the systemorder. Higher order sliding modes belongs to a surface of lowerdimension. These sliding modes have many interesting prop-erties, which, for example, can be exploited for control design[18].

A linear system with relay feedback can show several inter-esting phenomena. Local analysis of limit cycles are given in[19]. Few results exist on global stability of limit cycles in higherorder system, but a recent contribution is given in [20]. The re-sponse of a linear system with relay feedback can be compli-cated. Cook showed that a low-order linear system can have a

0018-9286/02$17.00 © 2002 IEEE

JOHANSSONet al.: LIMIT CYCLES WITH CHATTERING 1415

response that is extremely sensitive to the initial condition [21].It was shown in [22] and [23] that there exist trajectories havingarbitrarily fast relay switchings even if an exact sliding mode isnot part of the trajectory. A necessary and sufficient conditionfor this is that the first nonvanishing Markov parameter of thelinear part of the system is positive. It was shown by Anosov[24] that the pole excess is important for the stability of theorigin in relay feedback systems. Systems with pole excess threeor higher are unstable. From a similar discussion, it is possibleto conclude that only systems with pole excess two can have atrajectory with multiple fast relay switchings [22].

The main contribution of this paper is to give conditions forexistence of a new type of limit cycle. If the linear part of therelay feedback system has pole excess one and certain otherconditions are fulfilled, then the system has a limit cycle withsliding mode. Because the sliding mode is exact, it is easy toanalyze this system. If the linear system has pole excess two,there exist limit cycles with arbitrarily many relay switchingseach period. In this case the map that describes one period of thelimit cycle is quite complicated. A simulated example of sucha limit cycle was first shown in [22]. The fast relay switchingsgive rise to (what we call)chatteringor fast oscillations in thestate variables (cf. [18] and [25]). An important step in beingable to analyze the limit cycle is to approximate the chatteringby a second-order sliding mode. An accurate formula is derivedin this paper for how the chattering evolves. It shows that thechattering may be attracted to a second-order sliding mode de-pending on the system parameters. To study the limit cycle withchattering it is shown to be sufficient to study a second-ordersliding mode instead of the complicated map describing thechattering trajectory. The main result of the paper (Theorem 3in Section IV) gives sufficient conditions for the existence ofa limit cycle with chattering. The technique of analyzing chat-tering by sliding mode approximation is related to averaging inperturbation theory [26], [27].

The paper is organized as follows. Notation is introduced inSection II. Sliding modes and limit cycles are defined. Sec-tion III describes the phenomena of fast relay switchings thatwe call chattering. It is proved that chattering takes place closeto a second-order sliding set. An accurate formula for the evo-lution of the chattering is also derived. By using this result, it ispossible in Section IV to prove the main theorem of the paper.It states that there exist limit cycles with chattering. These limitcycles can have arbitrarily many relay switchings each period.An example of a chattering limit cycle is also given. The paperis concluded in Section V.

II. PRELIMINARIES

A. Notation

Consider a linear time-invariant system with relay feedback.The linear system has scalar inputand scalar output and itis described by the minimal state-space representation

(1)

with . Letdenote the transfer function of the system. The relay feedbackis defined by

.(2)

Note that the relay does not have hysteresis. The switching planeis denoted .

An absolutely continuous function is calleda trajectory or a solution of (1) and (2) if it satisfies (1) and (2)almost everywhere. Note that a differential equation with dis-continuous right-hand sides may have nonunique trajectories;see [17]. A limit cycle in this paper denotes the set ofvalues attained by a periodic trajectory that is isolated and notan equilibrium [27]. The limit cycle is symmetricif for every

it is also true that . Let the Euclidean distancefrom a point to a limit cycle be denoted . A limitcycle is then stable if for each there exists such that

implies that for all .

B. Sliding Modes

A sliding modeis the part of a trajectory that belongs to theswitching plane: is a sliding mode for with

, if for all . Sliding modesare treated thoroughly in [17]. Let be thepole excess of , so that but for

. Then, the set

is called the th-order sliding set (cf. [18]). A sliding mode thatbelongs to anth-order sliding set is anth-order sliding mode.We will in particular study first- and second-order sliding modesand the corresponding sets and .

There is an important distinction between first- and second-order sliding modes for (1) and (2). If then a trajectorywith initial condition close to the setwill have a sliding mode. Such first-order sliding modes mayeven be part of a limit cycle, as we will see Section II-C. Ifinstead and , then the set of initial con-ditions that gives a (second-order) sliding mode is of measurezero. What will happen then is that a trajectory with an initialcondition close to will wind aroundthe second-order sliding set. This phenomena give rise to a largenumber of relay switchings and is therefore named chattering.Chattering is described in detail in Section III. In Section IV, it isshown that also chattering can be part of a limit cycle. Existenceof fast relay switchings and their connection to limit cycles arediscussed in [22]. From the analysis therein, it follows that if thelinear part of the relay feedback system has pole excess greaterthan two, then there exists no limit cycle with a large number offast relay switchings.

C. Limit Cycles With Sliding Modes

This paper investigates complex chattering limit cycles. Inorder to present the mechanism generating them, we briefly re-call the simple case when the limit cycle of the relay feedback

1416 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 9, SEPTEMBER 2002

system has an exact (first-order) sliding mode. See [23] for fur-ther discussion and proofs.

Consider the relay feedback system (1) and (2) with state-space representation , where

......

...

...

Note that . This implies that there is a subset ofthe first-order sliding set that is attractive in the sense that theset of initial conditions that gives a sliding mode is of positivemeasure.

There are several ways to derive the sliding mode for a dif-ferential equation with discontinuous right-hand side [17]. Forlinear systems with relay feedback, they all agree. System (1)and (2) with parameterization has sliding set equalto . The equivalent control[10] is . Applying this to (1) givesthe first-order sliding mode as together with the solutionto

where

......

... (3)

Hence, we have the well-known fact that the sliding mode isstable in the sense that as , if all zeros of

are in the open left-half plane.Local stability of limit cycles with first-order sliding modes

can be straightforwardly analyzed by studying a Poincaré mapthat consists of two parts: one part corresponding to the trajec-tory being strictly on one side of and one (sliding mode) partcorresponding to the trajectory belonging to. A limit cyclewith first-order sliding modes is stable if all eigenvalues of

(4)

are in the open unit disc. The limit cycle is unstable if at leastone eigenvalue is outside the unit disc. Here,denotes the pro-jection , the projection

, the projection ,and the unit column vector of length with unity in the

first position. Moreover, denotes theinitial point of the (nonsliding) part of the limit cycle outside,

the final point of this part,the final point of the sliding mode part, and .Sliding limit cycles are further analyzed in [28], where it isshown that limit cycles with several first-order sliding segmentsexist and can be analyzed similarly as previously shown.

III. CHATTERING

If and , then thesetof initial conditions thatgives a second-order sliding mode is of measure zero. Instead tra-jectories close to may give rise tochattering. In this section, a detailed analysis of the chattering isgiven and a formula is proved that shows that in many cases thechattering can be approximated by a second-order sliding mode.In Section IV, this result is used to show existence of limit cy-cles with chattering. “Chattering” discussed here should not bemixed up with fast relay switchings occurring in systems withrelay imperfections such as hysteresis. The system descriptionhere is exact. Chattering is a trajectory with a finite number ofrelay switchings close to a second-order sliding mode.

Consider the relay feedback system (1) and (2) with state-space representation , where

......

...

...

Note that this parameterization corresponds to a linear systemwith pole excess two, such that and .Since , trajectories close to the set

will give fast relay switchings [22]. Due to the choice ofparameterization, the fast behavior takes place in the variables

and . Therefore, they are called thechattering variables,as opposed to thenonchattering variables .

The second-order sliding mode can be derived similarly to thefirst-order sliding mode in Section II. It is given byand the solution of

where

......

... (5)

JOHANSSONet al.: LIMIT CYCLES WITH CHATTERING 1417

A trajectory starting at a point with ,sufficiently small, and will wind around the set

. This follows from Theorem 1 given next, which states afirst-order approximation for the amplitude of the chattering.

Theorem 1: Consider (1) and (2) with parameterizationand order over a time interval , .

Let the initial state be ,where is a variable but are fixed. Letthe switching times be denoted by , , i.e., let

be the time instances such that . Iffor all , then the chattering variable satisfies

as (6)

and the envelope of the peaks of the chattering variableisgiven by

(7)

where as uniformly for all.

Proof: See the Appendix.Remark 1: The nonchattering variables

are close to the corresponding sliding mode. This follows from that the solution of a linear system

depends continuously on the initial data. Hence

for , where as andis given by (5).

Remark 2: The variable is almost constant overcompared to the chattering variable . Therefore, (7) givesthat the sign of determines if the chattering in has anincreasing or decreasing amplitude.

The following result is a formula for the number of switchingson a chattering trajectory.

Theorem 2: Given the assumptions of Theorem 1, thenumber of switchings on the interval is equal to

(8)

where as uniformly for.Proof: See the Appendix.

Remark 3: Equation (7) captures the behavior of chatteringquite well. Consider a chattering solution that starts withand

small and smaller than one. Since changes rapidlyin comparison with , (7) tells that oscillates with expo-nentially decreasing amplitude if . The length of theswitching intervals will decrease as decreases. As ap-proaches one, however, it follows from (8) that the intervals be-

Fig. 1. Chattering for a fourth-order system (solid) together with accurateenvelope estimate from Theorem 1 (dashed). Note that the chattering endswhenx becomes greater than one. Furthermore, as predicted, the length ofthe switching intervals decreases untilx becomes close to one and then theintervals increase.

tween the switchings increase again. Note that (7) and (8) arenot proved for and that the expressions are singularfor . This case needs further research.

Example 1: Consider

with state-space representation

and relay feedback. Let . Fig. 1 shows a simulation ofthe system starting in (solid line)together with the continuous estimate of the envelope ofob-tained from Theorem 1 (dashed lines). We see that the estimatefrom the theorem is accurate. The chattering ends whenbe-comes greater than one. Note that the switching periods increaseclose to the end point of the chattering, as was mentioned in Re-mark 3. The estimated number of switchings from Theorem 2 is

151, while the true number is 152.

IV. L IMIT CYCLES WITH CHATTERING

In this section, the main result of the paper is presented. Wewill show that limit cycles with chattering can be analyzed, sim-ilar to limit cycles with first-order sliding modes, by using The-orem 1. This will lead to conditions for existence of chatteringlimit cycles. A chattering limit cycle consists of two symmetrichalf-periods, each of them has one chattering part (which has afinite, typically large, number of relay switchings) and one non-chattering part.

To prove existence of a chattering limit cycle, we need to con-firm that the chattering is sufficiently close to a second-ordersliding mode. The analysis of chattering in Section III showedthat the chattering variable can be approximated to a high ac-

1418 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 9, SEPTEMBER 2002

curacy by a product of one time-dependent factor and one factordepending only on the nonchattering variable, see (7) of The-orem 1. The variables are almost constant comparedto the chattering variables and , so the second factor of (7)is almost constant during chattering. If the first factor, which isan exponential function in, is decreasing, then there is contrac-tion in the chattering variable . It then follows from (6) thatthere is also a contraction in the chattering variable. Duringthe chattering, the variables can be approximated bythe differential equation for the sliding mode with an accuracyproportional to the amplitude of . If also this differential equa-tion gives a contraction, then the two contractions form a con-tracting mapping for the full system. Such a system has a limitcycle containing one chattering part and one nonchattering part.This is formulated in the following theorem.

Theorem 3: Consider (1) and (2) with and let. Assume

with and let bedefined as in (5) but with replaced by . If the followingconditions hold:

1) matrix is Hurwitz and the eigenvalue of with largestreal part is unique;

2) ;3) the solution of

reaches the hyperplane at , it holdsthat for , and ;

4) for all , whereand .

Then there exists such that for every , (1)and (2) have a symmetric limit cycle with chattering.

Proof: See the Appendix.Remark 4: The number of relay switchings each period

can be made arbitrarily large by choosing sufficientlysmall. This follows from that a second-order sliding mode forthe system is long if the unstable zeros of are close to theorigin (i.e., if is small). Therefore, the number of fast relayswitchings each period of a chattering limit cycle increasesas the distance to the origin for the unstable zeros decreases.Note also that if the unstable zeros are close to the origin then

, because and is Hurwitz. It then followsfrom Theorem 1 that the variables and have decayingamplitudes during the chattering. Hence, the chattering bringsthe trajectory close to the second-order sliding set.

Remark 5: The location of the zeros of has a nice geo-metric interpretation. First note that the assumptions

and Hurwitz imply positive steady-state gain of, i.e., . The stable equi-

librium point of is equal to . Hence,gives that . Therefore, a relay switching is

guaranteed to occur for any trajectory with such that. It is easy to see that belongs to the hyperplane

. A Taylor expansionof shows that is small, if all zeros of areclose to the origin (compared to the zeros of ), i.e., if is

Fig. 2. Limit cycle with chattering for a system in Example 2. The dashed lineis the second-order sliding setS .

small. The trajectory of will thus approach a pointclose to . The assumption of Theorem 1is thus fulfilled if is sufficiently small.

Remark 6: The Jacobian of the Poincaré map consistingof one part outside and one (exact) second-order sliding modepart is given by

where the notation is similar to (4).Remark 7: A ball with center in

and radius proportional to is invariant underthe system dynamics, as is shown in the proof of the theorem.Note that although this ball captures the recurrence of the limitcycle (and although the ball can be made arbitrarily small), itdoes not follow that the limit cycle is stable.

The key condition of Theorem 3 is that the zeros ofshould be close to the origin. The other four conditions are, forexample, always fulfilled in the following fourth-order case.

Proposition 1: Suppose the dimension of (1) and (2) is. If all poles of are real and stable,

all zeros are real and unstable, and , then Conditions1)–4) of Theorem 3 are satisfied.

Proof: See the Appendix.The following example illustrates a chattering limit cycle.Example 2: Consider the system in Example 1 again. The

parameter gives zeros that are sufficiently close to theorigin to give a limit cycle with chattering. Fig. 2 shows thelimit cycle in the subspace . The fast oscillationsclose to during the chatteringare magnified in Fig. 3. Fig. 4 shows the four state variablesduring the chattering. In agreement with the analysis above, thechattering starts at , ends at , and isalmost constant. By approximating the chattering with a second-order sliding mode, it is possible to get a rough estimate of thebehavior. For the example, this leads to a nonsliding time of

and a sliding time of , while for the chatteringlimit cycle simulations give the times 7.4 and 4.2. The Jacobianin Remark 6 is . Note that the existence of thelimit cycle in this example is not formally proved by Theorem

JOHANSSONet al.: LIMIT CYCLES WITH CHATTERING 1419

Fig. 3. A closer look on the winding around the second-order sliding setS

(dashed line) for the limit cycle in Fig. 2.

Fig. 4. Chattering part of the limit cycle in Example 2. The behavior is welldescribed by the presented theory. Note the chattering inx andx , how thischattering starts whenx = �1 and ends whenx = 1, and thatx is almostconstant.

3, because the theorem does not provide a bound on how closeto the origin the zeros have to be.

V. CONCLUSION

A large number of fast relay switchings can appear in linearsystems with relay feedback if the linear part has pole excesstwo. This chattering was analyzed in detail in this paper and asufficient condition for existence was derived. It was also shownthat chattering may be part of a limit cycle. The limit cycle canhave arbitrarily many relay switchings each period. The mainresult of this paper stated that the chattering in the limit cyclecan be approximated by a sliding mode.

Chattering occurs in systems with pole excess two. Manyconsecutive fast switchings can, however, not occur in systemswith higher order pole excess. This can be understood intu-itively, since a system whose first nonvanishing Markov param-eter is of order have fast behavior similar to . Adouble integrator gives a periodic solution with arbitrarily shortperiod, while higher-order integrators are unstable under relay

feedback. It is shown in [22] that for systems with pole excessthree or higher there exist limit cycles with only a few extraswitchings each period.

APPENDIX

A. Proof of Theorem 1

Consider with , small, and .For up to next switching instant, it holds that

where is constant and

Note that it follows from that there will bea next switching if is sufficiently small. For the sake ofsimplicity, introduce the notation

where the last equation holds if the order . If , thisequation and the following still holds, but with . Notethat and that . Now, assume thatis the next switching instant, i.e., . Then, itholds that

(9)

and

(10)

for small . Introduce as an approximation ofto the accuracyof through the equation

(11)

Then, since for small

we get

(12)

1420 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 9, SEPTEMBER 2002

as . It is obvious from this expression thathas the same order as as . For this reason, the

expressions and are equivalent for every. In particular, from we have that

as for all , whichproves (6).

In the following, it will be shown that is proportionalto and the relation (7) will be derived. Letbe the starting point for the next part of the trajectory in thechattering mode between two successive switchings. The map

describes the envelope of in the chatteringmode. By substituting with and taking into account that

at any switching point, we get from (10) and (11)that

Then, (12) gives

(13)

where the last equality follows from (9). The chattering variablethus shifts sign in successive switching points. After ne-

glecting these sign shifts, the last equation looks very similarto a one-step iteration of a numerical solution of a differentialequation. Next, we show that such a differential equation existsand that it describes the envelope of at the switching in-stants . It is surprising that this equation can be analyticallyintegrated.

Consider three successive switching points at the time in-stants 0, , and . The relay output has opposite sign inthe intervals and . This influences and , sothat they show a gap in two successive switching points, whereasthey are close with a step of two switchings. Denotein threesuccessive switching points by , , and . Denote byand the corresponding values for and . It was previ-ously proven that

Therefore, after two successive switching points

Straightforward calculations using (12), (13), andshow that

(14)

Furthermore

This gives the differential equation associated with the peakvalues of the chattering variable as

where is the solution to the sliding modeequation with given by (5). We have

Therefore, the associated differential equation can be rewrittenas

Integration of this equation leads to the formula forand theproof is completed.

B. Proof of Theorem 2

Introduce a slower time associated with the number ofswitchings on a trajectory. The monotonous functionindicates the time instants of switchings for an integer argu-ment: is switching instant number. Equation (14)in the proof of Theorem 1 states that the increments of thisfunction can be approximated as

Since the increments are small as , the functioncan be approximated by the solution of the differential equation

The inverse function satisfies

It remains now only to substitute with the expression givenin Theorem 1 and integrate over.

C. Proof of Theorem 3

We will show that a trajectory starting close to the second-order sliding set has one part outside and one chatteringpart. By application of Theorem 1 and the fixed point theorem,it will be proved that two such parts form a limit cycle.

JOHANSSONet al.: LIMIT CYCLES WITH CHATTERING 1421

Consider the trajectory defined in Condition 3) and let. Then it holds that . Let be a solution

of (1) and (2) with

It follows from the assumption in Condition 3)that . With and ,this implies that and for small .Thus, the solution can, for small , be written as

where

......

......

and

...

...

Denote by the first instant when . This instantexists because as by Con-ditions 1) and 2). Note that

The first entry of the first term of the right-hand side is positivefor all , because by Condition 4). Denote by

the eigenvalue of with the largest real part. Thenis realand negative by Condition 1). The corresponding eigenvector of

is

...

All entries are positive, because of the following argument.Since is a stable polynomial and , it holds that

for some stable polynomial. Obviously, , for

, and . We prove by mathematical inductionthat for all . It then follows that ispositive. First, note that . Assume thatfor . Then

Since as , it holds that, where as . The scalar follows

from the equation and is, hence, given by.

Define the vector-valued function bythe equation

Then, . All entries, , are proportional to and negative for suffi-

ciently small , since .For small , it holds that and . A

quick jump occurs, because . The motion issimilar to the one described in the proof of Theorem 1. It followsthat it takes the time to reach

, where . Therefore,since , we have

...

...

For small , all the entries , , remain negative.For small , is positive and . It

follows from the (1) and (2) that the function satisfies

whenever . The structure of the matrix indicates thatif for and then for .Hence, the values of , , decrease, the valueof becomes negative, and reaches zero at

. All the entries of , , andare proportional to . Therefore, the time length of themotion with does not tend to zero as , in contrastto . It is easy from the relation to derivethat for . The same argument provesthat these “nonchattering” variables decrease on the next switchintervals provided that is not close to one. All conditionsof Theorem 1 hold on the interval forany fixed . Hence, the chattering mode starts at the point

1422 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 9, SEPTEMBER 2002

, where we recall that is defined as the first time instantwhen .

The trajectory , , can be approximated accordingto Theorem 1 and Remark 1. The nonchattering part of the state

is close to the solution of

Make the following change of variables: andfor . It is easy to see that

the new state vector satisfies the equation

According to Condition 3), this trajectory reaches the hy-perplane at time , i.e., .The condition implies that the sliding modefor a trajectory that starts in is not brokenon the interval . This gives that the end point of thesliding mode corresponds to or equivalently

, where thesliding time is given by .

According to Theorem 1, the chattering variable is pro-portional to the initial value . Thus, startingfrom the point the trajectory reaches the point

by passing through the nonchat-tering part and the chattering part. Denote the correspondingmap by , i.e.,

Next, it will be proved that the mapping can be defined ina neighborhood of the point and that this mapping is acontraction. The existence of a symmetric limit cycle followsby the fixed-point theorem.

Let be the ball in the hyperplane with a centerat and with the radius . Consider a trajectorystarting from a point . Similarly to the trajectory ,the first part of is nonchattering and lies in the set .A switch to occurs at the time instant , andas uniformly in . Hence, it holds that

Since the dominant parts of the values and areequal, the next chattering parts of these trajectories are close. Inthe normalized time the trajectory of the statevector is close to defined in Condition 3). Inparticular, there exists an instant where the sliding mode isbroken. The vector is the value of the function

on the vector . Thus, the function is well-defined on. It holds that . Therefore, the

mapping transforms the ball into itself for small . Themapping is continuous, because the time interval is uniformlybounded and the vector field of the relay system generates tra-jectories which continuously depend on the initial states, wherethe latter follows from Theorem 1. Any continuous mapping of aball into itself has a fixed point, hence, a fixed point ofexists

in . The fixed point defines the limit cycle. This concludesthe proof.

D. Proof of Proposition 1

Conditions 1) and 2) are obviously satisfied. To show Con-dition 3), first assume that where

. Then

which has the solution

Hence, there exists such that . It is easy tosee that for . Furthermore

Thus, Condition 3) holds if . A similar calculationshows that Condition 3) also holds if .

Finally, to check Condition 4), note that is equal tothe impulse response of a system with transfer function

where are the eigenvalues of. The impulse responseis equal to

where denotes the inverse Laplace transform andconvo-lution. This completes the proof.

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Karl Henrik Johansson (S’92–M’98) received theM.S. and Ph.D. degrees in electrical engineering fromLund University, Lund, Sweden, in 1992 and 1997,respectively.

He was an Assistant Professor at Lund Universityfrom 1997 to 1998, and a Visiting Research Fellowat the University of California at Berkeley from 1998to 2000. He is currently an Assistant Professor in theDepartment of Signals, Sensors, and Systems, theRoyal Institute of Technology, Stockholm, Sweden.His research interests are in theory and applications

of hybrid and switched systems, network control systems, and embeddedcontrol.

Dr. Johansson has received awards for his research, including the Young Au-thor Prize at the IFAC World Congress, San Francisco, CA, for a paper (coau-thored with A. Rantzer) on relay feedback systems, in 1996. He was awardedthe Peccei Award from the International Institute for Applied Systems Anal-ysis (IIASA), Laxenburg, Austria, 1993, and a Young Researcher Award fromSCANIA, Södertälje, Sweden, in 1996.

Andrey E. Barabanov was born in 1954 inLeningrad, Russia. He graduated the LeningradState University in 1976 and received the Ph.D. andDr.Sci. degrees in 1980 and 1997, respectively.

Since 1980, he has been a Member of the Mathe-matics and Mechanics Faculty of the Leningrad (St.Petersburg) State University. He is the author of morethan 100 papers and of the bookDesign of MinimaxRegulators. His main scientific interest include H-in-finity optimization, L1 control, adaptive control, andoptimal filtering, as well as TV image processing,

vocoder design, speed modem design, radar tracking systems, navigation, andship control.

Dr. Barabanov is a Member of the St. Petersburg Mathematical Society.

Karl Johan Åström (M’71–SM’77–F’79) was ed-ucated at the Royal Institute of Technology, Stock-holm, Sweden.

During his graduate studies, he worked on inertialguidance for the Research Institute of National De-fense, Stockholm, Sweden. After working five yearsfor IBM in Stockholm, Sweden, Yorktown Heights,PA, and San Jose, CA, he was appointed Professor ofthe Chair of Automatic Control at Lund University,Lund, Sweden, in 1965, where he built the depart-ment from scratch. Since 2000, he has been Professor

Emeritus at the same university. Since January 2002, he has been a part-timeProfessor in Mechanical and Environmental Engineering at the University ofCalifornia, Santa Barbara. He has broad interests in automatic control, includingstochastic control, system identification, adaptive control, computer control, andcomputer-aided control engineering. He has supervised 44 Ph.D. students, andhas written six books and more than 100 papers in archival journals. He hasthree patents, including one on automatic tuning of PID controllers which hasled to substantial production in Sweden.

Dr. Åström is a Member of the Royal Swedish Academy of Engineering Sci-ences (IVA) and the Royal Swedish Academy of Sciences (KVA), and a ForeignMember of the U.S. National Academy of Engineering, the Russian Academyof Sciences, and the Hungarian Academy of Sciences. He has received manyhonors, among them four honorary doctorates, the Quazza Medal from the Inter-national Federation of Automatic Control (IFAC), the Rufus Oldenburger Medalfrom the American Society of Mechanical Engineers (ASME), the IEEE FieldAward in Control Systems Science, and the IEEE Medal of Honor.