Dynamic sliding mode control design

A.J. Koshkouei, K.J. Burnham and A.S.I. Zinober

Abstract: Dynamic sliding mode control and higher order sliding mode are studied. Dynamicsliding mode control adds additional dynamics, which can be considered as compensators. Thesliding system with compensators is an augmented system. These compensators (extra dynamics)are designed for achieving and/or improving the system stability, hence obtaining desired systembehaviour and performance. Higher order sliding mode control and dynamic sliding mode controlyield more accuracy and also reduce and/or remove the chattering resulting from the high frequencyswitching of the control. It is proved that certain J-trajectories reach a sliding mode in a finite time.A sliding mode differentiator is also considered.

1 Introduction

Sliding mode control (SMC) has widely been extended toincorporate new techniques, such as higher-order slidingmode control (HOSMC) [1–3] and dynamic sliding modecontrol (DSMC) [4–9]. These techniques retain the mainadvantages of SMC and also yield more accuracy. Thesetechniques can also be applied to design observers todifferentiate signals achieving robustness in the absence ofnoise.

SMC utilises a high frequency switching control signal toenforce the system trajectories onto a surface, the so-calledsliding surface (or hyperplane), after a finite time andremain within the vicinity of the sliding surface towards theequilibrium point thereafter [8]. The sliding surface isdesigned to achieve desired specifications. SMC is robustwith respect to matched internal and external disturbances.However, undesired chattering produced by the highfrequency switching of the control may be considered aproblem for implementing the sliding mode control methodsfor some real applications. Methods have been presented toreduce the chattering, for instance the continuous approxi-mation technique [8]. Another way is to use HOSMCcontrol. A drawback of continuous approximation methodsis the reduction of the accuracy of the system and the slidingmode stability. SMC techniques are applicable to anyminimum phase system with relative degree less than thesystem order. There are some SMC techniques forstabilisation of non-minimum phase systems including useof DSMC, for stabilising the internal dynamics when theoutput tracking error tends asymptotically to zero in thesliding mode [6, 7].

DSMC has received attention in recent years [4, 6–9].Introducing extra dynamics into a sliding surface helps tosolve many difficulties in practice, such as flight controldesign and timescale separation of control loops in a

multi-loop system [5, 6]; replacement of a state observer toachieve stability under incomplete information aboutactuator dynamics [10]; and, even accommodation ofunmatched disturbances extending the system state spaceinto the exogenous states of an unknown signal modelledby linear dynamics. This method can be applied to the non-minimum phase tracking problem, e.g. stabilisation in adynamic sliding manifold of tracking error dynamicstogether with unstable internal dynamics, plus unmatchedinput exogenous dynamics with insufficient informationabout states of this composite system [6]. DSMC providesstability to the internal states and asymptotic stability to thestates of the tracking error dynamics.

Young and Ozguner [9] and Koshkouei and Zinober [4]have designed compensators using the optimal control andrealisation methods for linear systems in the sliding mode.The sliding system with a compensator (extra dynamics) isan augmented system which is a higher-order systemcompared with the original system. However, the designedcompensators may not only improve the stability of thesliding system but also yield desired performance andcharacteristics. In this paper a non-linear compensator forlinear and non-linear systems is designed.

2 Higher-order sliding mode control

HOSMC is a way to improve the accuracy of the slidingmode and remove chattering. Consider a system of the form

_xx ¼ Aðx; tÞ þ Bðx; tÞu ð1Þwhere x 2 Rn is the state and the scalar control u 2 R: Aðx; tÞand Bðx; tÞ are smooth vector fields. Define the slidingfunction as s ¼ sðx; tÞ: Suppose the system’s relativedegree r with respect to s ¼ sðx; tÞ is constant andknown. Then the system (1) is transferred to newcoordinates

sðrÞ ¼ f ðt; s; _ss; . . .sðr�1Þ; zÞ þ gðt; s; _ss; . . . ; sðr�1Þ; zÞu_zz ¼ fðt; s; _ss; . . . ; sðr�1Þ; zÞ þ cðt; s; _ss; . . . ; sðr�1Þ; zÞ

ð2Þwhere z ¼ ðz1; . . . zn�rþ1Þ [11]. The rth-order sliding modeexists if there is a control u such that the zero dynamicequations

s ¼ _ss ¼ � � � ¼ sðr�1Þ ¼ 0

q IEE, 2005

IEE Proceedings online no. 20055133

doi: 10.1049/ip-cta:20055133

A.J. Koshkouei and K.J. Burnham are with the Control Theory andApplications Centre, Coventry University, Coventry, UK

A.S.I. Zinober is with the Department of Applied Mathematics,University of Sheffield, Sheffield, UK

Paper first received 28th July and in revised form 19th October 2004

IEE Proc.-Control Theory Appl., Vol. 152, No. 4, July 2005392

are satisfied. The control during the sliding mode, theso-called equivalent control, is [2]

ueq ¼ �f ðt ; zÞ=gðt; zÞ ð3Þ

Substituting (3) into (1) yields r-sliding mode motion, whichis the zero dynamics of the system (1). Moreover, the systemis stable if the system is minimum phase. The results ofsecond-order SMC are interesting; in particular, it can besuccessfully applied to real problems. Various methods fordesigning second-order SMC and applications have beenpresented [1, 2].Attention is now focused on the second-order sliding

mode systems. Suppose the system (1) has relative degreeone with respect to s ¼ sðt; xÞ: Then the system (2) is

€ss ¼ f ðt; s; _ssÞ þ gðt; s; _ssÞu ð4ÞThe additional dynamics yields greater accuracy of thecontrol design and suitable response behaviour and perform-ance. These dynamics act as a compensator for the system.The resulting controller has the features of traditional SMCsuch as insensitivity to matched disturbances and non-linearities, and a classical dynamic compensator withaccommodation of unmatched disturbances. For simplicity,consider a system with the following sliding dynamics

€ss ¼ f ðt; sÞ þ u ð5ÞFor the existence of sliding mode control it is sufficient toshow that the trajectories reach the sliding surface s ¼ 0 ina finite time and remain on it thereafter.

3 Dynamic sliding mode control

The system in the sliding mode may need some additionaldynamics to improve the system stability and the slidingmode stability as well as obtaining the desired systemresponse and behaviour.It may also require a controller to be designed such that

the output of an uncertain SISO dynamic system trackssome real-time measured signal.When the output is measurable, the convergence time is

required to be finite so that the tracking is robust withrespect to measurement errors and exact in their absence.In order to solve such a problem some additionalassumptions may still be needed.

3.1 Dynamic sliding surface

A dynamic sliding function s is defined as a linear operator,which has a realisation as a linear time-invariant dynamicsystem

_ww ¼ Fwþ G1eþ G2x

s ¼ Cwþ Heþ Kx

where x is the state of the original system and e ¼ x� xdis the error variable with xd as a desired state. w is astate resulting from realisation of the operator s: F, G1; G2;C, H and K are matrices, with compatible dimensions,which show the relationship between the states. Forminimum-phase systems G2 ¼ 0 and K ¼ 0 whilst fornon-minimum-phase systems they are non-zero [7].

3.2 Dynamic sliding mode

The dynamic sliding mode (DSM) in the dynamicallyextended state space is defined as s ¼ 0: The system outputtracks the desired value if s ¼ 0 so that matcheduncertainties and disturbance do not affect the tracking.

In fact, for a sliding mode linear system (i.e. the systemduring the sliding mode, s ¼ 0) the desired system responseand performance can be achieved by selecting a set ofprespecified eigenvalues.

Non-linear sliding mode dynamics are now introduced.By defining such dynamics, two different sliding surfacesare obtained. However, there is a close relationshipbetween them. A sliding mode control is designed usingthe new component. Define the J non-linear dynamicsliding mode

_ww ¼ ajsj0:5sgnðsÞ � bjwþ sj0:5sgnðwþ sÞJ ¼ wþ s

ð6Þ

with a>0; b>0 and a 6¼ b [5, 6]. w is an error variable of twosliding mode variables s and J. Using the second-ordersliding mode, a sliding mode control u is designed such thats ¼ _ss ¼ 0: The problem is to find a suitable sliding modecontrol to guarantee a J-sliding mode (or s-sliding mode).The following theorem guarantees the existence of thesliding mode with sliding mode control u ¼ �r sgnðJÞwhere r is a suitably large positive real number. In manypractical problems, one needs to know the relationshipbetween the J- and s-dynamics. In fact, it is desired that theJ-sliding mode reaches the sliding surface J ¼ 0 faster thans-dynamics. In this case, lim

t!tJðJÞ=ðsÞ ¼ 0; where tJ is the

reaching time to the sliding surface J ¼ 0: The followingtheorem yields this relationship.

Theorem 1: Consider the sliding dynamics (5) and (6). Leta>0; b>0 and a 6¼ b: Then the following statements areimplied.

(i) The J-dynamics sliding mode exists.(ii) The J-sliding mode occurs if and only if the s-slidingmode exists.(iii) The J-dynamics reaches and remains on the slidingsurface J=0 before the trajectories of the s-dynamics hitthe sliding surface s ¼ 0 if and only if

jJð0Þj � b

a

� �2

jsð0Þj

Proof:(i) From (5) and (6) one can obtain

€JJþ 1

2

b

jJj0:5� a

jsj0:5� �

_JJ ¼ Fð�Þ þ u ð7Þ

where

Fð�Þ ¼ � a2

2

� �sgnðsÞ þ ab

2

J

s

��������0:5sgnðJÞ þ f ðt; sÞ:

Assume that jFð�Þj � L: Consider the sliding mode controlu ¼ �r sgnðJÞ with r>L: Then (7) yields

€JJ ¼ � 1

2

b

jJj0:5� a

jsj0:5� �

_JJþ Fð�Þ � r sgnðJÞ ð8Þ

Let R ¼ r� Fð�ÞsgnðJÞ: Then

R 2 ½r� L rþ L�sgnðJÞand substituting into (8) implies

€JJ ¼ � 1

2

b

jJj0:5� a

jsj0:5� �

_JJ� R sgnðJÞ ð9Þ

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In the Fillipov sense,

€JJ 2 � 1

2

b

jJj0:5� a

jsj0:5� �

_JJ� ½r� L rþ L�sgnðJÞ ð10Þ

where the right-hand side is a differential inclusion €JJ:Suppose ajsð0Þj0:5 6¼ bjJð0Þj0:5: Consider the trajectory G1

r� L if ðbjsð0Þj0:5 � ajJð0Þj0:5Þ _JJð0Þ>0

rþ L if ðbjsð0Þj0:5 � ajJð0Þj0:5Þ _JJð0Þ< 0

�ð11Þ

and the trajectory G2

rþ L if ðbjsð0Þj0:5 � ajJð0Þj0:5Þ _JJð0Þ>0

r� L if ðbjsð0Þj0:5 � ajJð0Þj0:5Þ _JJð0Þ< 0

�ð12Þ

Any trajectory G between the two trajectories G1 and G2;crosses the _JJ-axis (i.e. when bjsð0Þj0:5 ¼ ajJð0Þj0:5) at_JJ1 _JJ2; . . . so that for any i � 0; j _JJiþ1j � j _JJij: Therefore,fj _JJijg1i¼1 is a non-negative decreasing sequence. So lim

t!1_JJ¼ 0

and the second-order J-sliding mode occurs.(ii) From (6)

_ssþ ajsj0:5sgnðsÞ ¼ _JJþ bjJj0:5sgnðJÞTherefore, _ssþ ajsj0:5sgnðsÞ � 0 if and only if _JJþ bjJj0:5sgnðJÞ � 0: This is equivalent to _sss � �ajsj1:5 if and onlyif _JJJ � �bjJj1:5: So _sss � 0 if and only if _JJJ � 0:(iii) Assume that _ssþ ajsj0:5sgnðsÞ � 0 then _ss ! 0 ands ! 0 in a finite time. So

ts ¼jsð0Þj0:5

2að13Þ

Consequently, from _JJþ bjJj0:5sgnðJÞ � 0 one can see that_JJ ! 0 and J ! 0 in a finite time and

tJ ¼jJð0Þj0:5

2b

This completes the proof of (iii).

Furthermore, if for the system (4), the conditions

(i) s ! 0 and _ss ! 0 when t ! 1(ii) lim

t!1f ðt; 0; 0Þ ¼ lim

t!1gðt; 0; 0Þ ¼ 0

are satisfied, then limt!1

€ss ¼ 0:

4 Estimation of differentiation using DSMC

A DSMC differentiator is presented using the theory inSection 3. A SMC differentiator was introduced by Levant[3]. The Levant differentiator is as follows: let y ¼ xþ �where � is a Gaussian noise and y is a measurable variable.An estimation of _xx is required. An estimate of _xx is defined by

_xxxxðtÞ ¼ ajeðtÞj0:5sgnðeðtÞÞ þ b

Z t

0sgnðeðsÞÞds

eðtÞ ¼ yðtÞ � xxðtÞð14Þ

where xx is an estimate of x. Select x1 ¼ _xxxx: It is possible toestimate a differentiator for x1 as

_xxxx1ðtÞ ¼ aje1ðtÞj0:5sgnðe1ðtÞÞ

þ b

Z t

0sgnðe1ðsÞÞds

e1ðtÞ ¼ y1ðtÞ � xx1ðtÞ

ð15Þ

where xx1 is an estimate of x1: In fact, (15) yieldsan estimation of the second-order differentiation of x.

This process can be applied a finite number of times toobtain the desired higher-order differentiation.

Shtessel and Shkolnikov [5, 6] have introduced thefollowing differentiator

_ww ¼ ajej0:5sgnðeÞ � bjJj0:5sgnðJÞJ ¼ wþ e

e ¼ y� xx

€xxxx ¼ �K sgnðJÞ

ð16Þ

where xx is an estimate of x. Let y1 ¼ _xxxx: The newdifferentiator can be defined as

_ww1 ¼ a1je1j0:5sgnðe1Þ � b1jJ1j0:5sgnðJ1ÞJ1 ¼ w1 þ e1

e1 ¼ y1 � yy1€yyyy1 ¼ �K1 sgnðJ1Þ

ð17Þ

which can be considered as a filter for the estimation of _xx: Inthis way, a finite series of filters can be produced. Theorem 1implies that for any i � 0; the ei- and Ji-sliding modetrajectories converge to the sliding surfaces ei ¼ 0 and Ji ¼0 in finite time and if an appropriate condition is satisfied,the Ji-dynamics converge faster than ei-dynamics.

5 Example

Consider a system that can be converted to the slidingdynamics (5)

€ss ¼ f ðt; sÞ þ u

where

f ðt; sÞ ¼ 2s2 � s� 2 sinð2t � 0:5Þand

u ¼ �K sgnðJÞThis may arise from a system with tracking signal�2 sinð2t � 0:5Þ: Consider the non-linear dynamic slidingmode (6)

_ww ¼ ajsj0:5 sgnðsÞ � bjwþ sj0:5 sgnðwþ sÞJ ¼ wþ s

with a>0; b>0 and a 6¼ b: According to theorem 1, thesliding mode J- and s-dynamics exist. Assume that a>0;b>0 a 6¼ b and jJð0Þj � ððbÞ=ðaÞÞ2jsð0Þj: Then theJ-dynamics trajectories reach the sliding surface J ¼ 0before the s-dynamics trajectories hit the sliding surfaces ¼ 0: For simulation select a ¼ 1; b ¼ 2; Jð0Þ ¼ 0:1;sð0Þ ¼ 1 and _ssð0Þ ¼ 0:5: The condition (iii) of theorem 1 issatisfied:

jJð0Þj ¼ 0:1 � b

a

� �2

jsð0Þj ¼ 4

Therefore, the J-dynamics are faster than s-dynamics.The simulations also show this result. The reaching time ofthe J-sliding mode is less than 0.4 whilst the s-sliding modereaching time is larger than 1.7 (see Fig. 1). Figure 1 alsoshows that _ss converges to zero at finite time t _ss � 2; and alsoillustrates the behaviour of the sliding mode control and thesliding mode reaching time; thereafter the s-trajectoriesremaining on the sliding surface s ¼ 0: Figure 2 illustratesthe behaviour of the non-linear function f ðt; sÞ with respectto time, which shows that the non-linear function f ðt; sÞ

IEE Proc.-Control Theory Appl., Vol. 152, No. 4, July 2005394

does not tend to zero when t ! 1: However, since s ¼ 0;for t>0:4; f ðt;sÞ ¼ �2 sinð2t � 0:5Þ: When a ¼ b; and theinitial conditions are sð0Þ ¼ Jð0Þ; then the behaviour of theJ-dynamics and s-dynamics coincide and the sliding systemis marginally stable (see Fig. 3). In Fig. 3, the lower plot ofthe second column, depicts the control action when a ¼ band sð0Þ ¼ Jð0Þ: The switching between two control valuesis not very fast in comparison with the case when thediscontinuous control, u ¼ �15 signðJÞ with a ¼ 1; b ¼ 2is applied (see Fig. 1). In fact, the control is retained as aconstant value,�K; for a while and then switches to anothervalue, �K: This process repeatedly occurs. For example, ifthe control is K for a certain time, then it is switched to �Kand after another certain period of time, the control isswitched to K again. The control is a rectangular signal withconstant amplitude and a different width. Its width dependson the J-dynamics behaviour periods. The behaviour of the

function f ðt; sÞ with these conditions is shown in Fig. 4,which is completely different from Fig. 2. In this case

f ðt; sÞ ¼ 2s2 � s� 2 sinð2t � 0:5Þ

6 Conclusions

Dynamic and higher-order sliding mode controls have beenstudied in this paper. DSMC is a technique for improvingand=or achieving the system stability or desired behaviour,by designing compensators. This paper has presented someconditions for reaching trajectories to the appropriatesliding surfaces. The prediction of the behaviour of differentsliding mode dynamics is important for designing a sliding

Fig. 2 Behaviour of function f (s, t) for a ¼ 1 and b ¼ 2

Fig. 4 Behaviour of function f (s, t) for a ¼ b, and the initialconditions s(0) ¼ J(0)

Fig. 3 Response of sliding dynamics with discontinuous control,u ¼ �15 sign (J) when a ¼ b (¼ 2), and the initial conditions are s(0) ¼ J(0)

a Sliding function sb Sliding function Jc Function sd Action of the discontinuous control u

Fig. 1 Response of sliding dynamics with discontinuous control,u ¼ �15 sign (J) when a ¼ 1 and b ¼ 2

a Sliding function sb Sliding function Jc Function _ssd Action of the discontinuous control u

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mode control and for achieving the sliding mode stabilityand furthermore, the system stability. Using the maintheorem in this paper, the DSMC differentiator has also beenstudied.

7 References

1 Bartolini, G., Pisano, A., Punta, E., and Usai, E.: ‘A survey ofapplications of second-order sliding mode control to mechanicalsystems’, Int. J. Control, 2003, 76, pp. 875–892

2 Levant, A.: ‘Higher-order sliding modes, differentiation and output-feedback control’, Int. J. Control, 2003, 76, pp. 924–941

3 Levant, A.: ‘Sliding order and sliding accuracy in sliding mode control’,Int. J. Control, 1993, 58, pp. 1247–1263

4 Koshkouei, A.J., and Zinober, A.S.I.: ‘Robust frequency shapingsliding mode control’, IEE Proc. Control Theory Appl., 2000, 147,pp. 312–320

5 Shtessel, Y., and Shkolnikov, I.: ‘Aeronautical and space vehiclecontrol in dynamic sliding manifolds’, Int. J. Control, 2003, 76,pp. 1000–1017

6 Shkolnikov, I., and Shtessel, Y.: ‘Tracking a class of non-minimumphase systems with nonlinear internal dynamics via sliding modecontrol using method of system center’, Automatica, 2002, 38,pp. 837–842

7 Shtessel, Y., Zinober, A.S.I., and Shkolnikov, I.: ‘Sliding mode controlof boost and buck-boost power converters using the dynamic slidingmanifold’, Int. J. Robust Nonlinear Control, 2003, 13, pp. 1285–1298

8 Zinober, A.S.I. (Ed.): ‘Variable Structure and Lyapunov Control’,‘Lecture notes in control and information sciences’ (Springer-Verlag,Berlin, 1994)

9 Young, K.D., and Ozguner, U.: ‘Frequency shaping compensator designfor sliding mode’, Int. J. Control, 1993, 57, pp. 1005–1019

10 Krupp, D., and Shtessel, Y.: ‘Chattering-free sliding mode control withunmodeled dynamics’. Proc. American Control Conf., San Diego, CA,1999, pp. 530–534

11 Isidori, A.: ‘Nonlinear control systems’ (Springer Verlag, New York,1995, 3rd edn.)

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