On relative degree, chattering and fractal nature of parasitic dynamics in sliding mode control

$: On relative degree, chattering and fractal nature of parasitic dynamics in sliding mode control$
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On relative degree, chattering and fractal natureof parasitic dynamics in sliding mode control

Igor M. Boikon

The Petroleum Institute, P.O. Box 2533, Abu Dhabi, United Arab Emirates

Received 4 August 2012; received in revised form 15 November 2012; accepted 7 January 2013

Available online 16 January 2013

Abstract

With respect to relative degree and chattering in sliding mode (SM) control systems, the notion

of fractal dynamics is introduced, and a conjecture is formulated that the character of parasitic

dynamics of real control systems is fractal. A model of fractal dynamics is proposed. The charac-

teristics of fractal dynamics are studied in the frequency and time domains. It is shown that with

fractal parasitic dynamics SM control systems will always feature chattering and non-ideal closed-

loop performance. An example of analysis is provided.

& 2013 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

1. Introduction

There exist a number of control methods that rely on such characteristics of the process(plant) models as order and relative degree. Sliding mode (SM) control, high gain methods,Pontryagin’s maximum principle, and approaches that use Lyapunov functions usually fallinto this category. Normally these methods use plant dynamic models of relatively loworder (or low relative degree). Control is designed for these low-order models under theassumption that the effect of the mismatch between the model dynamics and the actualplant dynamics is insignificant. These models are built from the fundamental laws ofphysics and represent only the principal dynamics of the process (plant). However, it isknown that beside the principal dynamics, some kind of parasitic dynamics are alwayspresent. This fact is well known from the applications of SM control [1]. The relationshipbetween the principal and parasitic dynamics with respect to SM control is described in

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I.M. Boiko / Journal of the Franklin Institute 351 (2014) 1939–19521940

Ref. [2]. In SM control, the existence of parasitic dynamics is the cause of chattering in notonly first-order SM systems [1,3,4], but also discontinuous control second-order [5–10] andcontinuous first- [11] and second-order SM systems [12,13].The presence of parasitic dynamics has a direct effect on the relative degree of the system

because it is usually attributed to an actuator and a sensor connected in series with theprincipal dynamics of the plant. Account of the dynamics of actuators and sensorsresults in the increase of the relative degree of the plant dynamics. If relative degree isunknown and has to be determined experimentally then the design of respectiveexperiments has a high impact on the quality of the final system. It is demonstrated inthe higher-order SM control design [14]. The SM is, therefore, the principle that is on theone hand insensitive to external matched disturbances and model parametric uncertainties,but on the other hand is very sensitive to the presence of parasitic dynamics in the loop.Other control principles can also be affected by parasitic dynamics, so that some qualitydeterioration due to account of parasitic dynamics can be expected in these systems too.Therefore, adequate account of the parasitic dynamics in the model that is used in thedesign is relatively universal problem, which, however, is revealed in the SM system to ahigher extent.We now take a closer look at the nature of the principal and parasitic dynamics and their

relationship in any control system. Consider an example of the system that is supposed tocontrol water level in a vessel. Usually the dynamics of the level in a vessel can beconsidered an integrator, with the valve dynamics neglected. This would be a veryreasonable approach if the objective is the tank level dynamics. However, for a valveactuator designer the emphasis would be on the actuator dynamics. Most of the controlvalves use pneumatic actuation. At the analysis of the pneumatic dynamics one wouldneglect the electromagnet dynamics, which could not be neglected by an electrical engineerwho develops the electromagnet. Yet this engineer can legitimately ignore the dynamics ofthe electronic amplifier that is used for electromagnet control. For the electronic engineerthe subject of design is the amplifier, and he can disregard the dynamics of transistors.This sequence can be continued to single components, junctions, and particles. To theauthor’s belief, this sequence is really infinite and only limited by our knowledge of nature.The following observations can be made from the example above. (1) At each levelof consideration there are always exist principal dynamics, which provide the maincontribution to the overall dynamics, and parasitic dynamics, the effect of which ismuch smaller, and these dynamics are usually neglected. (2) The connection of theprincipal and parasitic dynamics is serial, which is determined by the control systemdesign principles.Let us call the described property of process (plant) dynamics fractal, which includes

infinite number of levels of consideration and existence of non-neglectable principal andneglectable parasitic dynamics on each level, and the dynamics themselves the fractal

dynamics. The term fractal obviously reflects similarity with the fractal geometry [15]. Infractal dynamics, like in fractal geometry, one can notice the property of self-similarity andthe possibility of scaling at each level of consideration.This paper is devoted to the design of a model of the fractal dynamics that would reflect

these properties, and to the analysis of the properties of this model.The paper is organized as follows: at first, the model of fractal dynamics is proposed.

After that, the properties of the fractal dynamics are analyzed in the frequency and timedomains. Finally, an example of application is given.

I.M. Boiko / Journal of the Franklin Institute 351 (2014) 1939–1952 1941

2. Fractal dynamics model

The analysis of the level control given above allows one to conclude that on each level ofconsideration there exist non-negligible principal dynamics with higher time constants andnegligible parasitic dynamics with lower time constants, with serial connection between thetwo. The separation line between these two dynamics is drawn based on the system understudy and the problem being solved — as shown by the example above. Therefore, themost complete description of the process dynamics would be a serial connection of thecomponent dynamics, with diminishing dominant time constants. The following model inthe form of a transfer function can be proposed to account for this property:

F T ,l,sð Þ ¼Y1

k ¼ 0

1

l�kTsþ 1, ð1Þ

where T is a time constant, l41 is a rate of decay of the time constants. Formula (1) is,therefore, a serial connection of infinite number of first-order dynamics with diminishingtime constants. Transfer function Eq. (1) can be related with the so-called q-Pochhammersymbol (q-shifted factorial) [16]

ða; qÞ1 ¼Y1

k ¼ 0

ð1�aqkÞ

through setting a¼�Ts and q¼1/l (as a reciprocal of the latter). However, thisrelationship is rather formal and, unfortunately, does not provide a convenient way ofcomputing Eq. (1), because the q-Pochhammer symbol assumes a real quantity, whereas Fis complex. We shall refer to the type of dynamics given by Eq. (1) as the first-order fractal

dynamics. Obviously, we can also design a model of second-order fractal dynamics:

F T ,l,sð Þ ¼Y1

k ¼ 0

1

ðl�kTÞ2s2 þ 2xl�kTsþ 1,

as well as other models of fractal dynamics. Below, let us consider only the first-orderfractal dynamics. Consider some properties of the fractal dynamics. The property of self-

similarity can be formulated as the following expression:

F lT ,l,sð Þ ¼1

lTsþ 1F T ,l,sð Þ, ð2Þ

which means that the increase of the time constant by l times is equivalent to adding onemore multiplier in the product Eq. (1). This property suggests the introduction of thefollowing function:

F0 l,sð Þ ¼F 1,l,sð Þ ¼Y1

k ¼ 0

1

l�ksþ 1: ð3Þ

Then for time constants that are multiples of lk the transfer function can be obtained viaformula (2). Therefore, we can rewrite formula (2) as follows:

F ln,l,sð Þ ¼F0 l,sð ÞYn

k ¼ 1

1

lksþ 1ð4Þ


We shall prove below that despite the infinite character of the formulas for F theproperties of the fractal dynamics are in many ways similar to those of finite-dimensionaldynamics. Let us analyze these properties in the frequency and time domains.

3. Frequency-domain characteristics of fractal dynamics

Assume that the input to the fractal dynamics is a harmonic excitation and find theharmonic response of the fractal dynamics Eq. (1). With the input being harmonic, we canreplace the Laplace variable as follows: s¼ jo. Therefore, F(jo) is a complex frequencyresponse and can be represented by the magnitude component and the phase component asfollows:

FðT ,l,joÞ ¼ jFðT ,l,joÞjexpðjargðFðT ,l,joÞÞÞ: ð5Þ

Let us prove that, despite infinite number of serially connected components, theharmonic response Eq. (5) provides finite attenuation of the amplitude and finite phase lagwith respect to the external harmonic excitation.

Theorem 1. At any finite frequency o, and l41 the magnitude response of the fractaldynamics Eq. (1) is not infinitesimally small (finite attenuation).

Proof. At first prove that 9F0(l,jo)9 provides finite attenuation to any harmonic excitationof finite frequency. Evaluate this function at o¼1.

lnjF0 l,j1ð Þj ¼ lnY1

k ¼ 0

1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ l�2k

p ¼�0:5X1k ¼ 0

lnð1þ l�2kÞ: ð6Þ

Using the following expansion for the logarithmic function [17]

ln 1þ að Þ ¼ a�a2

2þ

a3

3�a4

4þ . . ., jajo1 ð7Þ

we can find the lower estimate of the magnitude function. Assume that a¼l�2. FromEq. (7), it follows that lnð1þ aÞoa. Thus, we can write:X1

k ¼ 0

lnð1þ l�2kÞ ¼

X1k ¼ 0

lnð1þ akÞoX1k ¼ 0

ak: ð8Þ

The last formula provides the infinite geometric series, which gives the sum 1/(1�a).Therefore, considering the minus sign in Eq. (6), we can write the lower estimate for themagnitude of F0.

lnjF0 l,j1ð Þj4�0:5l2

l2�1: ð9Þ

It follows from Eq. (9) that Eq. (3) converges to a finite number at any l41.Next, function 9F0(l,jo)9 is obviously a decreasing function of the frequency o, as the

magnitudes of all the multipliers in Eq. (1) are decreasing functions. Also, it follows fromEq. (4) that for any frequency O there exists integer n such that the following inequality will


hold:

jF 1,l,jOð Þj4jF ln,l,j1ð Þj ¼ jF0 l,j1ð ÞjUYn

k ¼ 1

1

jl�kþ 1

��,

which will be also true for an arbitrary time constant T. &

Now consider the phase response of the transfer function Eq. (1).

Theorem 2. At any finite frequency o, and l41 the phase response of the fractal dynamicsEq. (1) is finite (finite phase lag).

Proof. The phase response for the transfer function Eq. (1) is:

argFðT ,l,joÞ ¼�X1k ¼ 0

arctan ðTol�kÞ: ð10Þ

Considering the well-known expansion of the arctan function [17]

arctan a¼ a�a3

3þ

a5

5�a7

7þ . . ., for a2o1,

from which it also follows that jarctan ajojaj if a2o1, we can write the followinginequality:

argF T ,l,joð Þ ¼�X1k ¼ 0

arctan ðTol�kÞ4�

X1k ¼ 0

ðTol�kÞ ¼�

To

1�l�1: ð11Þ

In formula (11), the negative sign of the phase response is accounted for by using ‘‘4’’,and the formula of geometric series is used to find the sum. Formula (11) is only valid ifToo1. It is not always the case. However, since l41, beginning from a certain k¼kn

inequality Tol�k*o1 will hold, so that we can consider two sums: from k¼0 to kn�1 and

from kn to infinity and treat in the second sum the term Tol�k* as a new To. Both sumswill obviously be finite.

Again, as in the case of the amplitude response, for any l41 and any finite frequencythe phase response of the fractal dynamics Eq. (1) is finite.&

The phase and amplitude characteristics (Bode plot) of the fractal dynamics Eq. (1) fortwo different l are depicted in Fig. 1. The Nyquist plot of Eq. (1) is given in Fig. 2 (Nyquistplot of conventional first-order dynamics is shown for comparison too).

One can see that the dependence of the phase characteristic on the logarithm offrequency for higher frequencies is linear (see Fig. 1). Let us demonstrate that.

Using formula (10), find the value of arg FðT ,l,jloÞ:

argFðT ,l,jloÞ ¼�X1k ¼ 0

arctan ðTlol�kÞ ¼ argFðT ,l,joÞ�arctan ðTloÞ

Fig. 2. Nyquist plots of fractal dynamics (T¼1).

Fig. 1. Bode plots of fractal dynamics (T¼1).


Find the following limit:

limo-1

arg F T ,l,jloð Þ�arg F T ,l,joð Þ½ � ¼ � limo-1

arctan Tloð Þ ¼ �p2:

It follows from the last formula that the slope of the phase characteristic of the fractaldynamics at high enough frequencies is 901=logl per decade. Therefore, for l¼2, it is 2991/decade and for l¼10 it is 901/decade (Fig. 1).

4. Time-domain characteristics of fractal dynamics

Regarding the time-domain characteristics of the fractal dynamics, the most importantwould be the step response of Eq. (1). It would be difficult to obtain analytical formulas for

Fig. 3. Step response of fractal dynamics (T¼1).

Fig. 4. Relay feedback system.


the step response. For that reason, find this function numerically via multiple integrationof the time evolution of the response of first-order dynamics W ðsÞ ¼ 1=ðTsþ 1Þ: hðtÞ ¼

1�expð�t=TÞ. Step response of Eq. (1) for two different values of l is given in Fig. 3(serial connection of 100 1st-order dynamics in the considered example, as per Eq. (1)).

In Fig. 3, the step input is applied at t¼0. One can see that the effect of the fractaldynamics is similar to time delay (see the plot for l¼2), as there is initial time intervalwhen the output stays almost equal to zero (‘‘acceleration’’ interval). However, fractaldynamics is minimal-phase, which makes it different from the time delay. The length ofthis interval depends on parameter l. One can see that the length of the ‘‘acceleration’’interval for l¼2 is larger than for l¼10. Existence of such initial time interval in a stepresponse of real plants was identified in Ref. [18] many years ago. In fact, the responseprovided in Fig. 3 is often approximated by first-order plus time delay model for thepurpose of process identification and controller tuning.

5. The LPRS of fractal dynamics

The locus of a perturbed relay system (LPRS) was proposed in Ref. [19] as a frequency-domain characteristic convenient for analysis of SM control systems, and was furtherdeveloped in Ref. [20]. The LPRS J(o) is a characteristic of the response of the plant to anunequally spaced square wave control with the asymmetry being an infinitesimally smallvalue. The LPRS, therefore, can be seen as a characteristic similar to the frequency


response (Nyquist plot) of the plant W(jo) with the difference that the Nyquist plot is aresponse to a harmonic signal but the LPRS is a response to the square wave havinginfinitely small asymmetry of the positive and negative pulses.Let, therefore, the SM system be described by the following equations that would

comprise the principal dynamics and the parasitic dynamics (presumably not fractal;however, it will also be extended to fractal dynamics below):

_x ¼Axþ Bu

y¼Cx, ð12Þ

u¼c if s¼ f0�y4b or s4�b, _so0

�c if s¼ f0�yo�b or sob, _s40,

(ð13Þ

where AARn� n, BARn� 1, CAR1� n are matrices, c and b are the amplitude and thehysteresis value of the relay nonlinearity respectively, f0 is a constant input, uAR1 is thecontrol, xARn is the state vector, yAR1 is the system output, sAR1 is the sliding variable (errorsignal), n is finite. Let us call the part of the system described by Eq. (12) the linear part.Alternatively the linear part can be given by the transfer function W(s)¼C(Is�A)�1B.The parasitic dynamics can be present in both: the linear part Eq. (12) or/and in thenonlinearity Eq. (13) as non-zero hysteresis.With the dynamics description in the format of the transfer function, the LPRS can be

computed as follows [19]:

J oð Þ ¼X1k ¼ 1

ð�1Þkþ1Re W koð Þ þ jX1k ¼ 1

1

2k�1Im W 2k�1ð Þo½ �: ð14Þ

State-space matrix form of the LPRS is given in Ref. [20]. Once the LPRS of a givensystem is computed we are able to determine the frequency of the oscillations and theequivalent gain kn, which describes propagation of averaged motions through the relay(Fig.5). It was proved in Ref. [19] that the point of the first intersection of the LPRS and ofthe straight line, which lies at the distance pb/(4c) below (if b40) or above (if bo0) thehorizontal axis and parallel to it (line ‘‘�pb/4c’’) provides the frequency of the oscillationsand the equivalent gain kn of the relay. Therefore, with the LPRS J(o) computed as perEq. (14) the frequency O of the oscillations can be found by solving the following equation:

ImJ Oð Þ ¼ �pb

4c, ð15Þ

Fig. 5. The LPRS and oscillations analysis for b40.

Fig. 6. The LPRS of fractal dynamics (T¼1).


(i.e. y(0)¼�b is the condition of the relay switch) and the equivalent gain of the relaykn,, can be computed as:

kn ¼�1

2ReJðOÞ: ð16Þ

Let us compute the LPRS for the dynamics Eq. (1) as per formula (14). The LPRS plots fortwo different values of l are given in Fig. 6. In calculation of the plot of Fig. 6, 100 products ofEq. (1) and 1000 terms in Eq. (14) were used. It was shown in Ref. [21] that the point of theLPRS for zero frequency for all non-integrating plants is always (0.5;�jp/4). One can see fromFig. 6 that it is also the case for the fractal dynamics. Also, similarly to the Nyquist plots theLPRS plots have spiral shape around the origin with the frequency tending to infinity.

6. Chattering and closed-loop performance of SM systems affected by parasitic fractal

dynamics

In SM systems, the hysteresis of the relay in Eq. (13) is always zero. Therefore, inaccordance with the LPRS method, the frequency O of chattering in a SM system can becomputed via solving the equation:

ImJðOÞ ¼ 0: ð17Þ

As a result, in SM systems, for analysis of chattering and of the closed-loop performance(propagation of external signals and disturbances) we should look for the point of the fistintersection of the LPRS and of the real axis. This point would determine both effectscharacterizing performance deterioration due to parasitic dynamics: the chattering effectand the closed-loop performance deterioration.

In accordance with Ref. [20], the averaged motions are described by the following modelof the original order (non-reduced order model):

_x0 ¼Ax0 þ Bu0,

y0 ¼Cx0 ð18Þ


u0 ¼ knUðf�y0Þ, ð19Þ

where subscript ‘‘0’’ stands for averaged value of the respective variable, f(t) is externalinput or disturbance to the system (Fig. 4).It is known that chattering in SM control systems is caused by the inevitable

existence of parasitic dynamics, which exist along with the principal dynamics of the

plant. The principal dynamics in a SM system are the dynamics of the plant model that

is used for the SM controller design and of the sliding surface. However, to implement

the designed control algorithms such devices as actuators and sensors are needed,

which brings into the system certain parasitic dynamics that were not accounted for at

the SM control design. Due to the SM design principle, the relative degree of principal

dynamics is always one [22]. The existence of only principal dynamics would result in

the LPRS shape in the vicinity of the origin similar to the one depicted in Fig. 6 for the

1st order dynamics (dashed line). In this case, the LPRS would not have any points of

intersection with the real axis except the origin. The origin is the only point of

intersection that corresponds to infinite frequency of chattering and, therefore, to the

ideal sliding. However, in any actual application parasitic dynamics always exist along

with the principal one.Assuming that the parasitic dynamics are fractal, as given by Eq. (1), we can come to the

following conclusions: in the case of fractal parasitic dynamics, the principal dynamics areconnected in series with the parasitic dynamics. This arrangement results in the LPRS ofthe overall dynamics similar to the one depicted in Fig. 6 for l¼2 and l¼10 (spiral shapein the vicinity of the origin). In particular, the point of intersection of the LPRS and of thereal axis exists, which corresponds to a solution of Eq. (17). Moreover, the point ofintersection is not the origin, which means that finite-frequency oscillations (chattering)occur in a SM system.Another observation is that due to the spiral shape of the LPRS the point of the first

intersection with the real axis will always be located in the left half-plane and the following

equality will hold: ReJðOÞo0. As a result, the equivalent gain of the relay is a finite

positive value: 0oknoN. As it follows from Eqs. (18), (19), with a finite equivalent gain,

the SM system would necessarily have deteriorated closed-loop performance in the form of

non-ideal disturbance rejection and non-ideal tracking of external signals.Therefore, with fractal parasitic dynamics, SM systems will always feature chattering and

non-ideal closed-loop performance.

In Refs. [5,12], a frequency-domain interpretation of second-order SM control wasgiven. It was shown that the second-order SM algorithms shift the point of intersection ofthe Nyquist plot and of the negative reciprocal of the describing function of the respectivealgorithm to the second quadrant — in comparison with the relay control, for which thispoint is located on the real axis (like for the LPRS analysis presented above; however, theLPRS analysis is exact, while the describing function analysis is approximate). Due to thespiral shape of the Nyquist plot (and of the LPRS) of fractal dynamics, there will always bea point of intersection of the Nyquist plot and of the negative reciprocal of the describingfunction of a second-order SM algorithm, corresponding to a finite frequency. Therefore,with fractal parasitic dynamics, chattering elimination and ideal closed-loop performance

become impossible even if a high-order SM algorithm is used.An example below illustrates the methodology of analysis of chattering and closed-loop

performance in a SM system with fractal parasitic dynamics.


7. Example

Consider the equations of the spring-loaded cart with viscous output damping on theinclined plane:

_x1 ¼ x2

_x2 ¼�x1�x2 þ ua þD,

(

where x1 is the linear displacement of the cart, x2 is the linear velocity, ua is the forcedeveloped by the actuator, and D is the disturbance (projection of the gravity onto theinclined plane). The goal is to stabilize the cart in the point corresponding to zerodisplacement. Let us design the switching surface (line) as follows:

x1 þ x2 ¼ 0

and the control as a relay control that can make the point x¼ [x1 x2]T¼0 an asymptotically

stable equilibrium point of the closed-loop system under the applied disturbance D¼�1:

u¼�4 signðx1 þ x2Þ:

Suppose that the force ua is developed by an actuator with the dominant time constantTa¼0.01s, which can be considered parasitic dynamics in SM control. Assume that thenature of these parasitic dynamics is fractal with parameter l¼5. Therefore, the transferfunction of the actuator is:

Wa sð Þ ¼F 0:01,5,sð Þ ¼Y1

k ¼ 0

1

5�k0:01sþ 1:

Clearly, the system should exhibit oscillations due to the actuator presence. Finding thefrequency and the amplitude of those oscillations is one of the goals of this analysis.Another goal is an assessment of the disturbance effect. In the case of ideal sliding, even ifthe disturbance were applied to the system the trajectory would tend to the origin. In thecase of non-ideal sliding (due to the actuator presence) the trajectory does not tend to theorigin. Let us show that.

Write an expression for the transfer function of the linear part:

W ðsÞ ¼ ðsþ 1ÞWaðsÞWpðsÞ,

where

Wp sð Þ ¼1

s2 þ sþ 1,

Compute the LPRS for W(s) and plot it on the complex plane (Fig. 7). Find the point ofintersection of the LPRS and of the real axis. This point corresponds to the frequencyO¼189.4 rad/s, which is the frequency of chattering in the system. The real part of theLPRS in this point is Re J(O)¼�0.00194 and the equivalent gain of the relay (according toformula (9)) is kn¼257.7. As a result, the non-reduced order model of the slow motions

Fig. 7. The LPRS of the linear part (actuator, plant and sliding surface).


can be written as follows (subscript ‘0’ denotes the slow component of respectivevariables):

_x01 ¼ x02

_x02 ¼�x01�x02 þ u0a þD

u0 ¼�kns0s0 ¼ x01 þ x02

u0aðsÞ ¼WaðsÞUu0ðsÞ

,

8>>>>>><>>>>>>:

where the last formula is written in the Laplace domain (due to infinite-dimensionalcharacter of the dynamics).The reduced-order model can be obtained as a limiting case: if the time constant is set to

zero: Ta-0 and consequently the equivalent gain approaches infinity: kn-N then s0¼0and, consequently, x01¼x02, which is the condition of the ideal sliding. One can notice thatthe actual value of the equivalent gain is finite. Since the transient processes in both: thereduced model and the non-reduced model look alike, the accuracy of the non-reducedorder model can be best demonstrated, if an external disturbance is applied to the system,and the effect of this disturbance is of interest.In the example being considered, the equivalent gain kn does not vary. For that reason,

the effect of the applied disturbance is identical in the transient and the steady state modes,and the analysis of disturbance attenuation can be carried out with the use of thetechniques relevant to linear systems. Analyze the disturbance attenuation. In a steadystate, there exists oscillations of frequency O with the center (x01,0) where x01¼D/(1þkn)¼�0.0039, which can be considered a disturbance rejection measure. This meansthat in a steady state, the cart exhibits oscillations around the point (�0.0039,0), with thefrequency O¼189.4 rad/s. This measure of non-ideal closed-loop performance can be seenonly due to the use of the non-reduced order model (model of the original order). The useof the reduced-order model would not allow one identify non-ideal disturbance rejection.

Fig. 8. Simulations: motion from zero initial point; inset plot provides system output at tA[9.0, 9.2]s.


Simulations of the system are provided in Fig. 8. They show an excellent match to theLPRS analysis.

8. Conclusion

A conjecture is formulated that the nature of parasitic dynamics in control systems isfractal. A model of the fractal dynamics is proposed, and the properties of the fractaldynamics are analyzed in the frequency and time domains. Characterization of the fractaldynamics by parameter l is suggested. It is shown that the larger l the closer the effects ofthe parasitic dynamics to the effects of conventional first-order dynamics (non-fractal).Properties of fractal dynamics are analyzed in the frequency and time domains. It is shownthat in SM control systems chattering and non-ideal closed-loop performance areunavoidable if parasitic dynamics present in the system are fractal. This conclusion alsoapplies to high-order SM control systems. An example of analysis that supports the aboveconclusions is given.

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On relative degree, chattering and fractal nature of parasitic dynamics in sliding mode control

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