Parameter tuning and chattering adjustment of Super-Twisting Sliding

Mode Control system for linear plants

Alessandro Pilloni, Alessandro Pisano and Elio Usai

Abstract— A simple procedure for tuning the parameters ofthe Super-Twisting (STW) second-order sliding mode control(2-SMC) algorithm, used for the feedback control of uncertainlinear plants, is presented. When the plant relative degree ishigher than one, it is known [10] that a self-sustained periodicoscillation takes place in the feedback system. The purpose ofthe present work is that of illustrating a systematic procedurefor control tuning based on Describing Function (DF) approachguaranteeing pre-specified frequency and magnitude of theresulting oscillation. The knowledge of the plant’s HarmonicResponse (magnitude and phase) at the desired chatteringfrequency is the only required prior information. By means of asimulation example, we show the effectiveness of the proposedprocedure.

I. INTRODUCTION

Sliding mode control (SMC) is a popular approach to

control system design under heavy uncertainty conditions

[25], and is accurate, robust and very simple to implement.

The main drawbacks of a classical first-order SMC (1-

SMC) are principally related to the so-called chattering effect

[7] [26], undesired steady-state vibrations of the system

variables.

A major cause of chattering has been identified as the

presence of unmodelled parasitic dynamics in the switching

devices [14]. To mitigate the chattering’s effects, many

solutions have been developed over the last three decades.

In particular three main approaches have been developed

• the use of a continuous approximation of the relay (e.g.

the saturation function [16];

• the use of an asymptotic state-observer to confine chat-

tering in the observer dynamics bypassing the plant [14];

• the use of higher-order sliding mode control algorithms

(HOSM) [3] [5] [21] [23] [4] [2].

In this paper we focus our attention on one of most

popular second order sliding mode algorithm (2-SM) known

as Super-Twisting algorithm [20]. Whenever applied to linear

plants with relative greater than one the STW controlled

system always exhibits chattering [10].

In this note, we first recall some known result (see

[10]) concerning the DF-based analysis of the dependence

of frequency and magnitude of chattering from the tuning

parameters of the STW controller. As the novelty, we propose

a procedure for selecting them, in order to assign prescribed

This work was not supported by any organizationA. Pilloni, A. Pisano and E. Usai are with the Department of Electrical

and Electronic Engineering (DIEE), University of Cagliari, Cagliari 09123,Italy

E-mail adresses: alessandro.pilloni@diee.unica.it,pisano@diee.unica.it, eusai@diee.unica.it

values to the frequency and amplitude of chattering. The

ability to affect the frequency of the residual steady state

oscillations may be useful, for example, to avoid resonant

behaviours of the plant.

In the literature there are two main approaches to chat-

tering analysis that provide an exact solution in terms of

magnitude and frequency of the periodic oscillation:

• time-domain analysis (i.e.: Poincare Maps [17] [19]);

• frequency domain techniques (i.e.: Tsypikin Locus [24]

and LPRS Method [8]).

All these approaches require lengthy computations. There-

fore, the application of approximate analysis methods has

been found useful whenever the plant under analysis present

low-pass filtering property [1]. Under this hypothesis, the

well-established DF method has been used to analyze the

characteristics of the periodic motions with 1-SMC [22] [27]

and 2-SMC algorithms, [10] [11] [12] [13], and the results

obtained via the use of exact techniques often feature a

satisfactory similarity to those obtained via the approximate

DF method [6].

To summarize, in this paper such representation is ex-

ploited for design purposes in the frequency domain in order

to provide effective tuning rules for chattering adjustment in

linear plants controlled by STW algorithm.

This paper is organized as follows: Section II presents the

Super-Twisting algorithm and recalls its DF-based analysis

[10]. Section III states the problem under investigation and

presents a simple graphical procedure for setting the param-

eters of the STW algorithm in order to assign prescribed

amplitude and frequency of the chattering motion. In Section

IV the proposed tuning procedure is verified by means of

computer simulations. Section V provides some concluding

remarks and hints for next research.

II. SUPER-TWISTING ALGORITHM AND ITS DF

ANALYSIS

We consider a linear SISO system, including actuator,

plant and measurement sensor, described by the following

state-space representation which comprises principal and

parasitic dynamics:

x(t) = Ax(t)+Bu(t), x ∈ Rn, u ∈ R

y(t) = Cx(t), y ∈ R(1)

where A, B, C are matrices of appropriate dimensions, x is

the state vector, u is the actuator’s input, y the plant’s output

and σ = r− y represent the error signal which is used as

sliding variable. We assume that the plant is asymptotically

12th IEEE Workshop on Variable Structure Systems,VSS’12, January 12-14, Mumbai, 2012

978-1-4577-2067-3/12/$26.00 c©2011 IEEE 479

stable with low-pass filtering property. We shall use the plant

description in the form of transfer function as follows

W (s) =Y (s)

R(s)= C (sI −A)B (2)

The super-twisting control u(t) is given as follows [20]

u(t) = u1(t)+u2(t) (3)

u1 = −γsign(σ) , u1 (0) = 0 (4)

u2 = −λ | σ |ρ sign(σ) (5)

where λ , γ and ρ are positive design parameters, with 0.5≤ρ < 1. The control system under analysis can be represented

in the form of the block diagram in Fig. 1. The DF of the

nonlinear function (5) was derived in [10] [15] as follows:

N2 (ay) =2λa

ρ−1y

π

∫ π

0(sinψ)ρ+1dψ

=2λa

ρ−1y

√π

Γ( ρ

2+ 1

)

Γ(ρ

2+ 1.5

)

(6)

where ay is the oscillation amplitude of the sliding variable

σ (i.e., one of the two unknowns of the problem) and Γ is

the Gamma function. Let ρ = 0.5 then the DF formula (6)

specializes to

N2 (ay) =2λ

√ay

ay√

π

Γ(1.25)

Γ1.75≈

1.1128λ√ay

(7)

The DF of the nonlinear integral component (4) can be

written as follows:

N1 (ay,ω) =4γ

πay

1

jω(8)

which is the cascade connection of an ideal relay (with

the DF equal to 4γ/πay [1]), and an integrator with the

frequency response 1/ jω . With the account of both control

components, the DF of the Super-Twisting algorithm (3)-(5)

can be finally written as

N (ay,ω) = N1 (ay,ω)+N2 (ay)

=4γ

πay

1

jω+ 1.1128

λ√ay

(9)

Let us note that the DF of the super-twisting algorithm

depends on, both, the chattering amplitude ay and frequency

ω .

In general, the parameters of the limit cycle can be ap-

proximately found via the solution of the following complex

equation

1 +W ( jω) ·N (ay,ω) = 0 (10)

called the harmonic balance [1]. The harmonic balance

equation (10) can be rewritten as

W ( jω) = −N−1 (ay,ω) (11)

A periodic oscillation of frequency Ω and amplitude ay exists

when an intersection between the Nyquist Plot of the plant

Fig. 1. Block diagram of the system with the Super-Twisting Algorithm.

W ( jω) and the negative reciprocal of the DF, N−1(ay,ω),occurs at ω = Ω. Thus, the parameters of the limit cycle can

be found via solution of the harmonic balance equation (9)-

(10). The negative reciprocal of the DF (9) can be written in

explicit form as

−N−1(ay,ω) = −0.8986

√ay

λ+ j1.0282

γ

ωλ 2

1 + 1.3091ay

( γωλ

)2(12)

It is of interest to plot the negative reciprocal DF (12) in

the complex plane. It depends on the two variables ay and

ω ; which are both nonnegative by construction. It is clear

from (12) that with positive gains λ and γ the locus (12) is

entirely contained in the lower-left quadrant of the complex

plane when the variables ay and ω vary from zero to infinity.

Furthermore, we can see that, fixed ω = ω∗ = const,

every curves −N−1(ay,ω∗) admit an horizontal asymptote

at − j1.0282γ

ω∗λwhen ay → ∞. Also, it is easy to show that

limay→0

arg(−N−1(ay,ω∗) = −

π

2

In Fig. 2, the curves obtained for λ = 0.6 and γ = 0.8,

some values ω = ωi, and by letting ay to vary from 0 to ∞are displayed.

A. Existence of the Periodic Solution

The harmonic balance (10) can be also expressed as

N (ay,Ω) = −W−1( jΩ) (13)

which, considering (9), specializes to

4γ

πay

1

jΩ+ 1.1128

λ√ay

= −W−1( jΩ) (14)

Separating the complex equation (14) in its real and imagi-

nary part it yields

1.1128 λ√ay

= −ReW−1( jΩ)

4γπΩ

1ay

= − ImW−1( jΩ)

(15)

Deriving ay from the first of (15) and substituting this value

into the second of (15), we obtain the next equation

4γ

πΩ

1

ImW−1( jΩ)−

(

1.1128λ

ReW−1( jΩ)

)2

= 0 (16)

12th IEEE Workshop on Variable Structure Systems,VSS’12, January 12-14, Mumbai, 2012

978-1-4577-2067-3/12/$26.00 c©2011 IEEE 480

which allows to compute the frequency Ω. Solution of

(16) cannot be derived in closed form, and a numerical,

or graphical approach is sought. Once obtained Ω , the

−4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0−2

−1.5

−1

−0.5

0

0.5

Re

Im

ay

ωi

ω1

ω2

ω3

ωn

ωi>ω

i+1

Fig. 2. Plots of the negative reciprocal DF (12) for different values of ω .

amplitude of the periodic solution can be expressed as

ay =4γ

πΩ

1

ImW−1( jΩ)(17)

As noticed in [10], a point of intersection between the

Nyquist Plot of the plant and the negative reciprocal of

the DF always exists if the relative degree of the plant

transfer function is higher than one and it is located in the

third quadrant of the complex plane. From Fig. 2, it is also

apparent that the frequency of the periodic solution for the

Super-Twisting algorithm is always lower than the frequency

of the periodic motion for system controlled by conventional

relay.

The orbital asymptotic stability of the periodic solution

can be assessed using the well known Loeb Criterion [1]

[18], that is not mentioned here for the sake of brevity.

III. PROBLEM FORMULATION AND PROPOSED TUNING

PROCEDURE

A. Problem Formulation

Consider the feedback control system in Fig. 1, where

the plant is modeled by an unknown transfer function W (s)having relative degree greater than one. By means od the

previously outlined procedure, once determined the con-

troller gains λ and γ it can be approximately evaluated the

chattering parameters ay and Ω.

The task of this paper is that of giving a simple solution

to the “inverse” problem, reversing the objective from chat-

tering analysis (λ and γ fixed, ay and Ω to be computed) to

chattering adjustment (ay = ady and Ω = Ωd fixed and λ , γto be determined accordingly).

Hence, given the performance requirements in term of

desired frequency Ωd and maximal deviation ady of the chat-

tering motion, we shall define a graphical tuning procedure,

based on the DF method, for computing the parameters of

the algorithm (3).

To begin with, let us substitute (12) into (10) and rewrite

it as:

W ( jω) = −c1

a1.5y

λ

ay + c3

( γωλ

)2− j

c2ayγ

ωλ 2

ay + c3

( γωλ

)2(18)

with

c1 = 0.8986 , c2 = 1.0282 , c3 = 1.3091

Let

A(ω) =γ

ωλ, B(ω) =

γ

ω(19)

Multiplying both sides of (18) by B(ω), we derive

B(ω)W ( jω) = −c1a

1.5y A(ω)

ay + c3A2 (ω)

− jc2ayA

2 (ω)

ay + c3A2 (ω)

(20)

where once considered the design requirements ay = ady and

ω = Ωd , separating (20) in its real and imaginary part as

follows

B(

Ωd)

ReW(

jΩd)

= −c1a

dy

1.5A(Ωd)

ady+c3A2(Ωd)

B(

Ωd)

ImW(

jΩd)

= −c2a

dyA

2(Ωd)ady+c3A

2(Ωd)

(21)

we obtain a well-posed system of equations, where the only

unknown variables are A(Ωd) ≡ Ad and B(Ωd)≡ Bd . In fact

the harmonic response of the plant at ω = Ωd can be achieved

by a simple test on the plant W (s). Therefore, solving (21),

by the change of variables (19) we compute the values of λand γ that satisfy our requirements.

Remark 1: It is important to underline that it cannot be

achieved an arbitrary frequency for the chattering oscillation.

It is easy to conclude that Ωd must be chosen in the interval

Ω1 < Ωd < Ω2 (22)

with the lower and upper bounds such that

argW ( jΩ1) =π

2, argW ( jΩ2) = π (23)

In fact, the intersection between the Nyquist Plot of W ( jω)and the locus −N−1(ay,ω) always lies in the third quadrant

of the complex plane.

Direct solution of the nonlinear equation (20) can be

avoided by resorting to a graphical approach. It is convenient

to refer to the Figure 3, where each curve correspond to the

plot of the right-hand side of (20) in the complex plane, for

a specific constant values of ady and by letting A to vary from

0 to ∞.

Remark 2: It is important to point out the fact that the

right-hand side of (20) is independent of the plant transfer

function. Therefore the set of curves in Fig. 3 define an

abacus very useful to simplify the computation of the STW

control parameters.

12th IEEE Workshop on Variable Structure Systems,VSS’12, January 12-14, Mumbai, 2012

978-1-4577-2067-3/12/$26.00 c©2011 IEEE 481

−0.035 −0.03 −0.025 −0.02 −0.015 −0.01 −0.005 0−0.07

−0.06

−0.05

−0.04

−0.03

−0.02

−0.01

0

Re

Im

0.06

0.05

0.04

0.03

0.02

0.01

ay

0.07

0.08

deg−95deg−105deg−115

deg−125

deg−135

−145deg

deg−155

−165deg

deg−175

Fig. 3. The abacus for STW controller tuning.

B. How to use the Abacus

Now we illustrate step-by-step how to use the abacus in

Fig. 3 and how to compute the parameters of the controller.

1. Select ady and Ωd such that the constraint (22) holds

and determine the harmonic response of the plant through

the parameters∣

∣

∣W

(

jΩd)∣

∣

∣, arg

W(

jΩd)

(24)

2. Identify, among the several curves of the abacus,

(possibly by interpolation) that corresponding to the desired

chattering magnitude ady .

3. Draw in the abacus a segment starting from the origin

and forming an angel of arg

W ( jΩd)

with the positive real

axis (see Fig. 4). Let P the point of intersection between that

segment and the specific curve identified in the step 2.

4. Measure the length of the segment OP and afterwards

apply the next relations to derive A and B.

Bd = B(Ωd) = OP

|W( jΩd)|

Ad = A(Ωd) =c1

√ady

c2tan

arg

W(

jΩd)

(25)

5. According to (19) compute the values of λ and γ as

γ = ΩdBd

λ = γ

ΩdAd= Bd

Ad

(26)

IV. SIMULATION RESULTS

In order to outline the proposed method of tuning, we

consider the cascade connection of a linear plant P(s) and a

dynamic actuator A(s) both with a second-order dynamic.

P(s) = s+1s2+s+1

, A(s) = 10.0001s2+0.01s+1

W (s) = A(s) ·P(s)

(27)

Let us apply the described procedure presented in the previ-

ous section to shape the periodic solution parameters using

the STW controller (3) with ρ = 0.5.

Fig. 5. Step response of the plant W (s) controlled by a STW-2SM withλ ≈ 12.85, γ ≈ 90.5 and ρ = 0.5.

A. Let ady = 0.05 and Ωd = 80 rad/sec;

B. By frequency response test at ω = Ωd on the plant W (s)(27) we obtain:

|W ( jΩd)| ≈ 0.0143 , argW ( jΩd) ≈−155.78deg

C. Draw the segment OP in the abacus until it intersects

the curve associated to ady (see Fig. 4)

D. Calculate the distance OP in Fig. 4:

OP =√

(−0.0147)2 +(−0.0066)2 = 0.0161

E. By (25) we derive the next values for Ad and Bd

Ad = 0.0879 , Bd = 1.1303

and by (19) we compute the gains of the control

algorithm (3) that gives rise to the chattering motion

with the desired characteristics in the feedback control

system represented Fig. 1. One obtains

λ = 12.8565 , γ = 90.4238 . (28)

Figure 5 displays the unit step response of the closed loop

system with parameters (28). The zoomed sub-plot confirms

that the steady-state chattering motion fulfills the given spec-

ification of amplitude and frequency, therefore supporting the

presented analysis results.

V. CONCLUSIONS AND FUTURE WORK

A describing function approach for tuning a feedback

control system with a linear plant driven by STW algorithm

has been presented, which allows to shape the characteristics

of the chattering motion that occurs when the linear plant has

a relative degree greater than one.

A constructive procedure for determining in advance the

periodic solution parameters (frequency and amplitude) has

been developed and tested by means of computer simula-

tions.

Among some interesting directions for improving the

present result, the analysis, and shaping, of the transient

oscillations is of special interest. The mathematical treatment

presented in [9] and the “dynamic harmonic balance” concept

in particular, could be a possible starting point to this end.

12th IEEE Workshop on Variable Structure Systems,VSS’12, January 12-14, Mumbai, 2012

978-1-4577-2067-3/12/$26.00 c©2011 IEEE 482

−10 −5−14.7

x 10−3

−0,012

−0,01

−0,008

−0,004

−0,002

0

−0.0066

Re

Im

−95deg−105deg−115deg

0.01

0.02

0.03

0.04

−175deg

deg−165

−155deg 0.05

−145deg

−125deg−135deg

P

O

Fig. 4. Example of abacus utilization for the system W(s) (27).

REFERENCES

[1] D.P. Atherton. Nonlinear Control Engineering-Describing Function

Analysis and Design. Workingham Berks, U.K., 1975.

[2] G. Bartolini, A. Ferrara, A. Levant, and E. Usai. On second ordersliding mode controllers. Variable structure systems, sliding mode

and nonlinear control, pages 329–350, 1999.

[3] G. Bartolini, A. Ferrara, and E. Usai. Output tracking control of un-certain nonlinear second-order systems* 1. Automatica, 33(12):2203–2212, 1997.

[4] G. Bartolini, A. Ferrara, and E. Usani. Chattering avoidance bysecond-order sliding mode control. Automatic control, IEEE Transac-

tions on, 43(2):241–246, 1998.

[5] G. Bartolini, A. Pisano, E. Punta, and E. Usai. A survey of appli-cations of second-order sliding mode control to mechanical systems.International Journal of Control, 76, 9(10):875–892, 2003.

[6] I. Boiko. Analysis of sliding mode control systems in the frequencydomain. In American Control Conference, 2003. Proceedings of the

2003, volume 1, pages 186–191. IEEE, 2003.

[7] I. Boiko. Frequency domain analysis of fast and slow motions insliding modes. Asian Journal of Control, 5(4):445–453, 2003.

[8] I. Boiko. Discontinuous control systems: frequency-domain analysis

and design. Springer Verlag, 2009.

[9] I. Boiko. Dynamic harmonic balance and its application to analysisof convergence of second-order sliding mode control algorithms. InAmerican Control Conference, 2011. San Francisco, CA, USA, June29-July 01, 2011.

[10] I. Boiko and L. Fridman. Analysis of chattering in continuoussliding-mode controllers. Automatic Control, IEEE Transactions on,50(9):1442–1446, 2005.

[11] I. Boiko, L. Fridman, and MI Castellanos. Analysis of second-ordersliding-mode algorithms in the frequency domain. Automatic Control,

IEEE Transactions on, 49(6):946–950, 2004.

[12] I. Boiko, L. Fridman, R. Iriarte, A. Pisano, and E. Usai. Parametertuning of second-order sliding mode controllers for linear plants withdynamic actuators. Automatica, 42(5):833–839, 2006.

[13] I. Boiko, L. Fridman, A. Pisano, and E. Usai. On the transfer propertiesof the generalized sub-optimal second-order sliding mode controlalgorithm. Automatic Control, IEEE Transactions on, 54(2):399–403,2009.

[14] Bondarev S. A. Kostyleva N. E. Bondarev, A. G. and Utkin V. I.Sliding modes in systems with asymptotic state observers. Automationand Remote Control, pages 679–684, 1985.

[15] R. S. Burlington. Handbook of Mathematical Tables and Formulas.New York: McGraw-Hill, 1973.

[16] J. Burton. Continuous approximation of variable structure control.International journal of systems science, 17(6):875–886, 1986.

[17] M. Di Bernardo, K.H. Johansson, and F. Vasca. Self-oscillationsand sliding in relay feedback systems: Symmetry and bifurcations.International Journal of Bifurcations and Chaos, 11(4):1121–1140,2001.

[18] A. Gelb and W.E. Vander Velde. Multiple-input describing functions

and nonlinear system design. McGraw-Hill, New York, 1968.

[19] J.M. Goncalves, A. Megretski, and M.A. Dahleh. Global stability ofrelay feedback systems. Automatic Control, IEEE Transactions on,46(4):550–562, 2001.

[20] A. Levant. Sliding order and sliding accuracy in sliding mode control.Int. J. Control, 58(6):1247–1263, 1993.

[21] A. Levant. Higher-order sliding modes, differentiation and output-feedback control. International Journal of Control, 76, 9(10):924–941,2003.

[22] Y.B. Shtessel and Y.J. Lee. New approach to chattering analysisin systems with sliding modes. In Decision and Control, 1996.,

Proceedings of the 35th IEEE, volume 4, pages 4014–4019. IEEE,1996.

[23] H. Sira-Ramirez. Dynamic second-order sliding mode control of the

12th IEEE Workshop on Variable Structure Systems,VSS’12, January 12-14, Mumbai, 2012

978-1-4577-2067-3/12/$26.00 c©2011 IEEE 483

hovercraft vessel. Control Systems Technology, IEEE Transactions on,10(6):860–865, 2002.

[24] Y. Z. Tsypkin. Relay control systems. Cambridge University Press,1984.

[25] V.I. Utkin and J. Guldner. Relay control systems. Springer-Verlag,Berlin, 1992.

[26] V.I. Utkin, J. Guldner, and Shi J. Sliding mode control in electrome-chanical systems. 1999.

[27] K. K. Zhilcov. Approximate methods of variable structure systems.Mahsinostroeniye, Moscow (in Russian), 1974.

12th IEEE Workshop on Variable Structure Systems,VSS’12, January 12-14, Mumbai, 2012

978-1-4577-2067-3/12/$26.00 c©2011 IEEE 484