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 DOI: 10.1177/1077546311421795

2012 18: 1254 originally published online 29 September 2011Journal of Vibration and ControlBo Song and Jian-Qiao Sunapproximation method

Sliding mode control of uncertain dynamical systems with time delay using the continuous time  

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Article

Sliding mode control of uncertaindynamical systems with time delayusing the continuous timeapproximation method

Bo Song and Jian-Qiao Sun

Abstract

A study of sliding mode control of uncertain dynamical systems with time delay is presented in this paper. The systems

are assumed to have constant delay time and uncertain parameters with known upper and lower bounds. The method of

continuous time approximation is applied to formulate the sliding mode control problem. The proposed treatment of the

control delay leads to a higher-order control formulation for a system with additive uncertainties only. An optimal sliding

surface is designed such that on the sliding surface, the controlled state variables act like a feedback control to the

uncontrolled state variables. Examples of non-linear systems are presented to demonstrate the theoretical work.

Keywords

Time delay, continuous time approximation, sliding mode control, uncertain systems, higher order control

Received: 16 July 2010; accepted: 8 July 2011

1. Introduction

In a recent paper (Song and Sun 2011), the authorsproposed a lowpass filter-based continuous timeapproximation (LPCTA) approach for response analy-sis and control design of time-delayed dynamical sys-tems. This paper further applies the LPCTA method tosliding mode control designs of nonlinear time-delayedsystems with uncertainties.

Sliding mode control has been applied to time-delayed systems. Yan et al. (2010) proposed a slidingsurface in an augmented space formed by the systemoutput and the reduced order estimator to stabilizenonlinear systems with mismatched nonlinear termand the nonlinear uncertainty subject to time-varyingdelay. The Lyapunov–Razumikhin approach is used toanalyze the stability of the sliding motion. Song et al.(2001) and Qu et al. (2008) investigated a virtual stateconcept for sliding surface design. Roh and Oh (1999,2000) study sliding mode control for the robust stabili-zation of uncertain linear input-delay systems with non-linear parametric perturbations. The proposed slidingsurface includes a predictor to compensate for the inputdelay of the system. The proposed sliding surfaceincludes a predictor which consists of the current

state and the past control during the period of delay.The predictor compensates for the input delay of thesystem. A state predictor with an adaptation scheme isused. He et al. (2009) presented a delay-dependent slid-ing mode control problem for a class of uncertain time-delay Markov jump hybrid systems. Niu et al. (2004)and He et al. (2008) considered a delay-dependent slid-ing mode control for uncertain neutral delay systems.Xing et al. (2009) applied a delay-independent slidingmode control to a class of parabolic linear uncertaindistributed parameter systems with time-varying delays.Wu and Zheng (2009) studied a sliding mode controlwith passivity of a class of uncertain nonlinear singulartime-delay systems. An integral-type switching surfacefunction was designed by taking the singular matrixinto account. A delay-dependent sufficient conditionwas proposed in terms of linear matrix inequality,which guarantees the stability of the sliding modedynamics. The same integral sliding surface was used

School of Engineering, University of California, Merced, CA, USA

Corresponding author:

Jian-Qiao Sun, School of Engineering, University of California, 5200 North

Lake Road, Merced, CA 95343, USA

Email: jqsun@ucmerced.edu

Journal of Vibration and Control

18(9) 1254–1260

! The Author(s) 2011

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DOI: 10.1177/1077546311421795

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in the sliding mode control for uncertain stochastic sys-tems with time-varying delay by Niu et al. (2005). Thecommon method of linear matrix inequalities is used toderive conditions for the global stochastic stability. Thework of Xu and Cao (2000) makes use of time-delaycontrol during the sliding phase to reduce the switchinggain of the sliding mode control.

The current paper introduces a different approach todesign sliding mode controls based on the LPCTAmethod. We consider the sliding mode control in theextended state space within the context of the LPCTAmethod. The system is assumed to be nonlinear withuncertainties. In Section 2 we briefly review the LPCTAmethod. We present a sliding mode control in Section 3together with an optimal design of the sliding surface.Then, we show examples of sliding mode control oflinear and nonlinear uncertain systems with timedelay in Section 4. Section 5 concludes the paper.

2. LPCTA method

Consider an n-dimensional system with time delay

_xðtÞ ¼ fðx tð Þ, xðt� �Þ, tÞ,

xð�Þ ¼ ’ð�Þ, � 2 ½��, 0�, ð1Þ

where x(t)2Rn, f is a nonlinear function of its argu-ments, and � is a time delay. Let �N¼ {�i, i¼ 0, 1,. . .,N} be a mesh of Nþ 1 points in [0, �] such that0¼ �0<�1< ���<�N¼ � and ��i be a sample time of adigital system of an anti-aliasing lowpass filter withbandwidth p> 0. The differentiation of a measuredsignal after passing it through the lowpass filter in dig-ital time domain leads to an estimation of _x tð Þ on themesh �N,

1

2þ ri

� �_xðt� �iÞ þ

1

2� ri

� �_xðt� �iþ1Þ

¼1

��ix t� �ið Þ � x t� �iþ1ð Þ½ �, ð2Þ

where ri¼ 1/(p��i), known as the bandwidth ratio.Equation (1) becomes

HN _y tð Þ ¼ FNðy tð ÞÞ þGNu tð Þ, ð3Þ

where

HN ¼

I 0 � � � � � � 012þ r1� �

I 12� r1� �

I 0 � � � 0

0 � � � . ..

� � � ...

..

. . ..

0

0 � � � 0 12þ rN� �

I 12� rN� �

I

26666664

37777775,

ð4Þ

FN ¼

f y1 tð Þ, yNþ1 tð Þ, t� �1

��1y1 tð Þ � y2 tð Þ� �

..

.

1��N

yN tð Þ � yNþ1 tð Þ� �

266664

377775, GN ¼

B

0

..

.

0

26664

37775, ð5Þ

and

yðtÞ ¼ xðtÞ, xðt� �1Þ, . . . , xðt� �NÞ½ �T

� y0ðtÞ, y1ðtÞ, y2ðtÞ, . . . , yNðtÞ� �T

: ð6Þ

For more discussions on the method, the reader isreferred to Song and Sun (2011).

2.1. Application to systems with control delay

The method discussed above also applies to nonlinearsystems subject to delayed controls,

_x ¼ f x tð Þ, x t� �ð Þ, tð ÞþBu t� �uð Þ, ð7Þ

where �u denotes the delay of the control. Consider amesh �M¼ {�ui, i¼ 0, 1,. . ., M} of Mþ 1 points in [0,�u] such that 0¼ �u0<�u1< ���<�uM¼ �u. The block-vector y(t) can be re-defined as

yðtÞ ¼ xðtÞ, xðt� �1Þ, . . . , xðt� �NÞ,½

uðt� �u1Þ, . . . , uðt� �uMÞ�T

� y1ðtÞ, y2ðtÞ, y3ðtÞ, . . . , yNþMþ1ðtÞ� �T

: ð8Þ

Equation (7) is converted to

_y tð Þ ¼ FCTAðy tð ÞÞ þGCTAu tð Þ: ð9Þ

Consider a special case f(x(t), x(t� �), t)¼Ax(t) as anexample, and apply the forward finite difference schemeto introduce the continuous time approximation of thedelayed control,

_uðt� i��uÞ ¼1

��uu t� ði� 1Þ��uð Þ � u t� i��uð Þ½ �:

ð10Þ

We have then,

FCTA ¼

A 0 � � � � � � B

0 � 1��u

I 0 � � � 0

0 1��u

I � 1��u

I � � � ...

..

. . .. . .

. . ..

0

0 � � � 0 1��u

I � 1��u

I

26666666664

37777777775,

GCTA ¼

0

1��u

I

..

.

0

266664

377775: ð11Þ

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With a different interpolation scheme such asChebyshev polynomial or the LPCTA, both FCTA andGCTA would have a different structure. In Equation (9),the control is designed for the current time t for thesystem without time delay. The effect of the controlu(t) is propagated to influence the dynamics of x(t) ata later time. When the numerical simulation or digitalimplementation of the control system uses a fixed timestep that is the same as ��u, the delay of the control willbe exactly �u in M steps. When the integration time stepin simulation or experiment is smaller than ��u, therealized delay can be shorter than �u in M steps. Inother words, the delayed control u(t� �u) is realizedby passing the control u(t) without delay through ahigh-order filter created by the discretization of thecontinuous time approximation scheme. For thisreason, we call such a delayed control the higher-ordercontrol. The higher-order control has been shown torender better performance than lower-order control inthe literature.

It is interesting to point out that the multiplicativeuncertainty in B of the original system with respect tothe control becomes the additive uncertainty in FCTA ofthe extended state space. This makes the sliding modecontrol design simpler in the extended state space.

3. Sliding mode control design

In the following discussions, we assume that we havealready taken care of the control delay in the formula-tion of continuous time approximation. Consider alinear system with uncertain nonlinear elements

_y ¼ FyðtÞ þGuðtÞ þ gðy, tÞ, ð12Þ

where y2Rn(Nþ1), F2Rn(Nþ1)�n(Nþ1), G2Rn(Nþ1)�m,u2Rm and g2Rn(Nþ1) (n(Nþ 1)�m). We assume thatF� F

��� ��� � ~F where F is an estimate of F, ~F is a knownsemi-positive definite matrix, and that g� g

�� �� � ~g,where g is uncertain, g is an estimate of g, and~g 2 RnðNþ1Þ is a known vector with non-negativeelements.

Let s(y)¼Sy be a sliding surface where s2RmandS2Rm�n(Nþ1). Assume that SG is invertible. Whenthe system is sliding on s, we have s(y)¼ 0 and_sðyÞ ¼ S_y ¼ 0: From Equation (12), we have,

S_y ¼ SFyðtÞ þ SGueqðtÞ þ Sgðy, tÞ ¼ 0: ð13Þ

This leads to a nominal control ueq

ueqðtÞ ¼ �ðSGÞ�1ðSFyðtÞ þ Sgðy, tÞÞ: ð14Þ

Applying the estimates of the uncertain componentsand adding a switching term to it, we yield a controlto be implemented

uðtÞ ¼ �ðSGÞ�1ðSFyðtÞ þ Sgðy, tÞÞ � ðSGÞ�1KsignðsÞ,

ð15Þ

where K2Rm�m is a switching gain matrix to be deter-mined. Consider a Lyapunov function,

V ¼1

2sTs � 0: ð16Þ

The time derivative of V reads

_V ¼ sTS_y

¼ sTSðFyþGð�ðSGÞ�1ðSFyþ Sgðt, yÞÞ

� ðSGÞ�1KsignðsÞÞ þ gðt, yÞÞ

� jsT½SðF� FÞyþ Sðg� gÞ�j � sTKsignðsÞ

� jsTS~Fyj þ jsTS~gj � sTKsignðsÞ: ð17Þ

We choose a matrix K such that

sTKsignðsÞ ¼ jsTS~Fyj þ jsTS~gj þ �ðsÞ, ð18Þ

where g(s)> 0 is a positive function of s. Then,_V � ��ðsÞ and the system is stable.Consider a special case K¼kI where k is a scalar and

I is a unit matrix. We have

_V � jsTS~Fyj þ jsTS~gj � �Xmi¼1

jsij � ��Xmi¼1

jsij, ð19Þ

where

� ¼ �þjsTS~Fyj þ jsTS~gjPm

i¼1 jsij, �4 0: ð20Þ

We point out that in the limit s! 0, the ratio in k isfinite.

3.1. Optimal sliding surface

Now, we discuss how to design the sliding surface s

based on the nominal linear system (Sinha and Miller1995; Pai and Sinha 2007)

_y ¼ FyðtÞ þGuðtÞ: ð21Þ

Consider the null space of the control influencematrix G

NG ¼ fy 2 RnðNþ1Þ : GTy ¼ 0g: ð22Þ

There are in general n(Nþ 1)�m independent vectorsin NG. Let pi denote a set of independent vectors inthe null space NG, and P¼ [p1, p2,. . ., pb>n(Nþ1)�m]

T .Let Z denote a matrix consisting of m independent

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vectors not in the null space NG. Introduce atransformation

q ¼My, ð23Þ

where

M ¼P

Z

: ð24Þ

It can be shown that

MG ¼0

G2

, ð25Þ

where G2 is a non-zero matrix. Applying the transfor-mation to Equation (21), we obtain an equation for q,

_q �_q1_q2

¼

F11 F12

F21 F22

q1q2

þ

0

G2

u, ð26Þ

where we have introduced the notation,

MFM�1 �F11 F12

F21 F22

: ð27Þ

We highlight the first row of Equation (26),

_q1 ¼ F11q1 þ F12q2: ð28Þ

On the sliding surface, s(y)¼Sy¼SM�1 q¼ 0. Hence,

s ¼ S1q1 þ S2q2 ¼ 0, ð29Þ

where we have defined

SM�1 � S1 S2

� �: ð30Þ

In Equation (28), we can consider q2 as a control. Hereq1 is the state to be controlled. The sliding surface con-dition of Equation (29) provides a state feedback con-trol as

q2 ¼ �S�12 S1q1 � �Vq1: ð31Þ

The feedback gain can be determined from a LQRoptimal control problem. Consider a performanceindex

J ¼1

2

Z 1ts

qTQq dt

¼1

2

Z 1ts

ðqT1Q11q1 þ 2qT1Q12q2 þ qT2Q22q2Þ dt, ð32Þ

where

Q ¼Q11 Q12

Q12 Q22

¼ QT � 0, ð33Þ

and ts is the time when sliding begins. The optimal con-trol to minimize J is given by

q2 ¼ �Vq1 ¼ �Q�122 ðG12PþQ12Þq1, ð34Þ

where P is the unique non-negative solution of theRiccati equation (Lewis and Syrmos 1995)

0 ¼ PðF11 � F12Q�122 Q

T12Þ þ ðF11 � F12Q

�122 Q12Þ

TP

� PF12Q�122 F

T12PþD, ð35Þ

and

D � Q11 �Q12Q�122 Q

T12: ð36Þ

Now that we have obtained the gain matrixV ¼ Q�122 ðF12PþQ12Þ, we have

S1 ¼ S2V: ð37Þ

We need to select a matrix S2, say a unit matrix. We canreadily compute S1. The sliding surface is thus deter-mined by S¼ [S1, S2] M.

4. Examples

4.1. First-order system

Consider a first-order system,

_xðtÞ ¼ axðtÞ þ bxðt� �Þ þ uðtÞ þ gðx, tÞ, ð38Þ

where g(x, t)¼ ex3(t)þ d sin t. The system constants areuncertain and fall in known ranges �10� a��1,0� b� 2, �0.1� e� 0.1, and �0.1� d� 0.1. �¼ 1 is aconstant time delay. In the simulation of the slidingmode control, we use the following as the unknownsystem,

_xðtÞ ¼ �5xðtÞ þ 0:5xðt� 1Þ þ 0:05x3ðtÞ

þ 0:05 sin tþ uðtÞ, ð39Þ

while the nominal system used in the sliding mode con-trol design is taken to be

_xðtÞ ¼ �2xðtÞ þ xðt� 1Þ þ uðtÞ: ð40Þ

This implies that the estimates a¼�2, b ¼ 1, andg(x, t)¼ 0.

For the LPCTA method, we select the bandwidthratio to be r¼ 0.001 and N¼ 24 based on our previousexperience of computations. The extended vector isy2R(Nþ1) and Nþ 1¼ 17. The dimension of the nullspace NG is 16. Hence, the matrix P2R16�17. Wehave to select a 1� 17 vector Z. In the numerical exam-ple, we have chosen Z¼ [1, 1,. . ., 1]1�17 to complete thetransformation matrix M2R17�17.

Consider a special index for designing the optimalsliding surface,

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J ¼1

2

Z 1ts

yTy dt ¼1

2

Z 1ts

qTQq dt, Q ¼ðM�1ÞTM�1:

ð41Þ

Note that Q can be further partitioned as inEquation (33) based on the dimensions of the sub-matrices P and Z of M in Equation (24). The optimalsliding surface is found to be S¼ [S1, S2] M where wehave picked S2¼ 5. We implement the sliding modecontrol for the special case when K¼ kI. The controlis given by Equation (15) with Z¼ 1.

Figure 1 shows the sliding surface response andsliding mode control of the system starting from theinitial condition x(0)¼ 1. The sliding surface convergesto zero quickly. It should be noted that the slidingmode control is designed based on the knowledgeof upper and lower bounds of the parameteruncertainties and without using knowledge of theparameters in the unknown system (39). The controlperformance is excellent because the control has notime delay. Next, we consider the effect of time delayof the control.

4.2. First-order system with delayed control

Consider a first-order system with delayed control,

_xðtÞ ¼ axðtÞ þ "x3ðtÞ þ � sin tþ buðt� �uÞ, ð42Þ

where g(x, t)¼ ex3(t)þ d sin t. The system constants areuncertain and fall in the known ranges �2� a� 2,0.1� b� 2.1, �0.1� e� 0.1, and �0.1� d� 0.1. Here�u¼ 1 is a constant time delay. In the simulation of

the sliding mode control, we use the following as theunknown system,

_xðtÞ ¼ xðtÞ þ 0:005x3ðtÞ þ 0:005 sin tþ uðt� 1Þ, ð43Þ

while the nominal system used in the sliding mode con-trol design is taken to be

_xðtÞ ¼ �xðtÞ þ 0:5uðt� 1Þ: ð44Þ

This implies that the estimates a¼�1, b ¼ 0:5, andg(x, t)¼ 0. Recall that the uncertainty in b becomespart of the uncertainty of the matrix F in Equation(12) and the matrix G is fully determined. Hence, inthe sliding mode control design, we only deal withthe additive uncertainty in F. We also note that theunknown open-loop system is unstable, while the nom-inal open-loop system is stable.

For the continuous time approximation method, weuse the forward difference approximation in Equation(10), and M¼ 24 based on our previous experience ofcomputations. The extended vector is y2R

(Mþ1) andMþ 1¼ 17. The dimension of the null space NG is 16.Hence, the matrix P2R16�17. We have to select a 1� 17vector Z. In the numerical example, we have chosenZ¼ [1, 1,. . ., 1]1�17 to complete the transformationmatrix M2R17�17, and we have picked S2¼ 5.

To design the optimal sliding surface, we considerthe same index as in Equation (41), and follow thesame steps as in the previous example. We implementthe sliding mode control for the special case whenK¼kI and Z¼ 1.

Figure 2 shows the response and the delayed slidingmode control of the system starting from the initial

0 1 2 3 4 50

1

2

3

x(t)

0 1 2 3 4 5−4

−2

0

2

Con

trol

u(t

−1)

Time (s)

Figure 2. The response x(t) of the first-order nonlinear system

(43) and the delayed sliding mode control u(t� 1). Here x(0)¼ 1.

The uncontrolled system is unstable.

0 0.5 1 1.5 2−1

0

1

2

Con

trol

u(t

)

0 0.5 1 1.5 2

0

2

4

Time (s)

Slid

ing

Surf

ace

s(t)

Figure 1. The sliding control u(t) and sliding function s(t) of the

first-order nonlinear system (39). Here x(0)¼ 1.

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condition x(0)¼ 1. Before the control kicks in, thesystem is unstable and the response grows. The slidingmode control stabilizes the system. Note that the delayof the control is shorter than �u¼ 1. This is due to thefact that the numerical integration in the simulationuses the time step smaller than ��u¼ 1/24, whichhelps propagate the control u(t) to u(t� �u) over ashorter time delay. In an experimental setting or afully discretized simulation, we can establish a definiterelation between the discretization time step ��u andthe sample time of the controller so that we can realizethe exact time delay of the control.

4.3. Second-order systems

Consider a second-order system,

_xðtÞ ¼0 1

�k �c

xðtÞ þ

0 0

�kp �kd

xðt� �Þ

þ gðx, tÞ þ0

1

uðtÞ, ð45Þ

gðx, tÞ ¼0

"x31ðtÞ þ � sin t

, ð46Þ

where x¼ [x1(t), x2(t)]T, k¼ 4, c¼ 0.2 and �¼ p/2.

Other system parameters are assumed to be unknown,and to fall in a range with known upper and lowerbounds: �1� kd�0, �1� kp� 0, �0.1� e� 0.1, and�0.1� d� 0.1. In the simulation of the sliding modecontrol, we use the following as the unknown system,

_xðtÞ ¼0 1

�4 �0:2

xðtÞ þ

0 0

0:5 0:5

xðt� �=2Þ

þ gðx, tÞ þ0

1

uðtÞ, ð47Þ

gðx, tÞ ¼0

0:05x31ðtÞ þ 0:05 sin t

, ð48Þ

while the nominal system used in the sliding mode con-trol design is taken to be

_xðtÞ ¼0 1

�4 �0:2

xðtÞ þ

0 0

0:9 0:7

xðt� �=2Þ

þ0

1

uðtÞ: ð49Þ

This implies that the estimates kp ¼ �0:9, kd ¼ �0:7,and g(x, t)¼ 0.

We select r¼ 0.0001 and N¼ 24 for the LPCTAmethod. We use the same index for the optimal sliding

mode surface, and find the transformation matrix M todetermine the matrix Q in the same manner as in theprevious examples. The dimension of the null space NG

is 33. Hence, the matrix P2R33�34. We have to select a1� 34 vector Z. In the numerical example, we havechosen Z¼ [1, 1,. . ., 1]1�34 to complete the transforma-tion matrix M2R34�34, and we have picked S2¼ 400.We implement the sliding mode control for the specialcase when K¼ kI and Z¼ 1.

Figure 3 shows the system response in the x1� x2state space under the sliding mode control with initialcondition x(0)¼ 1 and _xð0Þ ¼ 1. Figure 4 shows thecontrol u(t) and the sliding function s(t).

−0.5 0 0.5 1−1

−0.5

0

0.5

1

x(t)

dx/d

t(t)

Figure 3. The response trajectory in the x1� x2 state space of

the second-order system (47).

0 2 4 6 8 10−4

−2

0

2

Con

trol

u(t

)

0 2 4 6 8 10−500

0

500

1000

Time (s)

Slid

ing

Surf

ace

s(t)

Figure 4. The sliding control u(t) and sliding function s(t) of the

second-order system (47).

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5. Concluding remarks

In this paper, we have reviewed the previously pro-posed method of continuous time approximation.Sliding mode controls of uncertain nonlinear systemsare formulated for the cases of system delay and controldelay. The continuous time approximation treatment ofthe control delay leads to a higher-order control formu-lation for a system with additive uncertainties only. Theoptimal sliding surface design problem has also beendiscussed. Numerical examples have been presented toshow the effectiveness of the proposed sliding modecontrol design.

Funding

This research received no specific grant from any funding

agency in the public, commercial, or not-for-profit sectors.

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