Robust sliding control for mismatched uncertain fuzzy time-delay systems using linear matrix inequality approach

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Robust sliding control for mismatched uncertain fuzzy

time-delay systems using linear matrix inequality

approach

Sung-Chieh Liu a & Sheng-Fuu Lin a

a

Department of Electrical Engineering , National Chiao Tung University , Hsinchu 30010 , Taiwan

Published online: 19 Oct 2012.

To cite this article: Sung-Chieh Liu & Sheng-Fuu Lin (2013) Robust sliding control for mismatched uncertain fuzzy time-delay systems using linear matrix inequality approach, Journal of the Chinese Institute of Engineers, 36:5, 589-597, DOI: 10.1080/02533839.2012.734557

To link to this article: http://dx.doi.org/10.1080/02533839.2012.734557

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Robust sliding control for mismatched uncertain fuzzy time-delay systems using

linear matrix inequality approach

Sung-Chieh Liu and Sheng-Fuu Lin*

Department of Electrical Engineering, National Chiao Tung University, Hsinchu 30010, Taiwan (Received 11 July 2011; final version received 25 December 2011)

In this article, we propose a robust sliding control design method for uncertain time-delay systems that can be represented by Takagi–Sugeno fuzzy models. The uncertain fuzzy time-delay systems under consideration have mismatched parameter uncertainties in the state matrix and external disturbances. We derive existence conditions of linear sliding surfaces guaranteeing the asymptotic stability in terms of constrained linear matrix inequalities (LMIs). We present an LMI characterization of such sliding surfaces. Also, an LMI-based algorithm is given to design the switching feedback control term so that a stable sliding motion is induced in finite time. Finally, we give two numerical design examples to show the effectiveness of the proposed method.

Keywords: uncertain time-delay systems; linear matrix inequality; sliding surface; switching feedback control

1. Introduction

Over the past two decades, fuzzy techniques have been widely and successfully exploited in nonlinear system modeling and control. The Takagi–Sugeno (T–S) model (Tagaki and Sugeno 1985) is a popular and convenient tool for handling complex nonlinear sys-tems. Correspondingly, the fuzzy feedback control design problem for a nonlinear system has been studied extensively using T–S model where simple local linear models are combined to describe the global behavior of the nonlinear system (Tanaka et al. 1996, Wang et al. 1996, Ma et al. 1998, Tanaka et al. 1998, Korba et al. 2003, Nguang and Shi 2003). On the other hand, time-delay is often encountered in ical processes. Recently, the feedback stabilization problem for uncertain time-delay systems has also become a problem of interest because the existence of a delay is frequently a source of poor system performance or instability (Jafarov 2005, Peng et al. 2008, Zhang et al. 2009). Delays are sensitive to uncertainty, which directly affects the control systems.

It is known that a sliding mode control (SMC) system has various attractive features, such as fast response, good transient performance, order reduction, insensitivity to parameter variation, and invariance to external disturbances. In the SMC system, the control structure around the plant is intentionally changed using a nonlinear discontinuous controller to obtain a desired system response. Using nonlinear discontinu-ous control, the SMC system drives the system

trajectory onto a specified and user-chosen surface, which is called the sliding or the switching surface, and maintains the trajectory on this sliding surface for all subsequent time. This motion is referred to as the sliding mode. The central feature of the SMC system is the sliding mode on the sliding surface in which the system remains insensitive to internal parameter var-iations and external disturbance. In sliding mode, the order of the system dynamics is reduced. This enables simplification and decoupling design procedures (Utkin 1977, DeCarlo et al. 1988, Walcott and Zak 1988, Choi 2003).

Considering these facts above, and utilizing the T–S model, we propose a robust sliding control design method for a mismatched uncertain T–S fuzzy delay-time model with parameter uncertainties and external disturbances. First, we derive a linear matrix inequality (LMI) condition for the existence of the fuzzy control-ler that guarantees a stable sliding motion on the switching surface that is insensitive to norm-bounded uncertainties. We show that the sliding surface param-eter matrix can be characterized in terms of the solution of the LMI existence condition. Second, we design the nonlinear discontinuous term to drive the system trajectories so that a stable sliding motion is induced in finite time on the switching surface and the state converges to zero. Finally, two numerical design examples are given in order to show the effectiveness of the proposed method. The rest of this article is organized as follows. Section 2 describes the T–S

*Corresponding author. Email: sflin@mail.nctu.edu.tw

Vol. 36, No. , 589–597,http://dx.doi.org/10.1080/02533839.2012.734557

ß 201 The Chinese Institute of Engineers

5

2

fuzzy model and reviews some preliminary results. Section 3 presents an LMI existence condition of linear sliding surfaces and an explicit characterization of the sliding surface parameter matrices as well as a sliding control law. Section 4 gives two numerical design examples to demonstrate the validity and effectiveness. Finally, Section 5 offers some concluding remarks.

2. Problem formulation and preliminaries

The T–S fuzzy model is described by fuzzy IF-THEN rules, which represent local linear input–output relations of nonlinear systems. The ith rule of the T–S fuzzy time-delay model is of the following form: Plant Rule i:IF 1 is i1 and. . . and sis is, THEN

_

xðtÞ ¼ AixðtÞ þ Aixðt  d ðtÞÞ þ BiuðtÞ, ð1Þ

xðtÞ ¼ wðtÞ, t 2 ½, 0, ð2Þ where wðtÞ is the initial condition, xðtÞ 2 Rn the state, uðtÞ 2 Rm the control input, Ai2Rnn the state

matri-ces, Ai2Rnnthe delayed state matrices, Bi2Rnmthe

input matrices, jðj ¼1, . . . , sÞ the premise variables, s

the number of the premise variables, ijði ¼1, . . . , r;

j ¼1, . . . , sÞ the fuzzy sets that are characterized by membership function, and r the number of the IF-THEN rules. The time-varying delay d ðtÞ is bounded as d ðtÞ  : The overall fuzzy model achieved by fuzzy synthesizing of each individual plant rule is given by

_ xðtÞ ¼X r i¼1 iðÞ½AixðtÞ þ Aixðt  d ðtÞÞ þ BiuðtÞ, ð3Þ xðtÞ ¼ wðtÞ, t 2 ½, 0, ð4Þ where  ¼ ½1, . . . , s, iðÞ ¼ !iðÞ=Prj¼1!jðÞ,

!i: Rs! ½0, 1, i ¼ 1, . . . , r is the membership

func-tion of the system with respect to plant rule i: The function iðÞ can be regarded as the normalized

weight of each IF-THEN rule and it satisfies that iðÞ 0,

Pr

i¼1iðÞ ¼1: To take into account

param-eter uncertainties and external disturbances, we con-sider the following uncertain T–S fuzzy time-delay model:

_

xðtÞ ¼X

r

i¼1

iðÞ½ðAiþ"AiðtÞÞxðtÞ þ ðAiþ"AiðtÞÞxdðtÞ

þBiðuðtÞ þ hiðt, x, xd, uÞÞ, ð5Þ

xðtÞ ¼ wðtÞ, t 2 ½, 0, ð6Þ

where xdðtÞ ¼ xðt  d ðtÞÞ, "AiðtÞ represents the

parameter uncertainties in Ai, "AiðtÞ the parameter

uncertainties in Ai, hiðt, x, xd, uÞ 2 Rm the external

disturbances. We will assume that the following assumptions are satisfied:

A1 B1¼B2¼    ¼Br:¼ B and rank ðBÞ ¼ m:

A2 The function hiðt, x, xd, uÞ is unknown but

bounded as hiðt, x, xd, uÞik k þu iðtÞ

where iðtÞis a known function and i satisfies

im5 1 for a known constant m.

A3 The time delay d ðtÞ is unknown but bounded as d ðtÞ   and _d ðtÞ  dm5 1 where  and dm are

known constants.

A4 "A_ðtÞand "AiðtÞare of the form Ti&iðtÞwhere

&iðtÞ is a known time-varying matrix but

bounded as &iðtÞ



   1:

Using the above assumptions, the uncertain T–S fuzzy model (5) can be written as follows:

_

xðtÞ ¼X

r

i¼1

iðÞ½ðAiþTi&iðtÞÞxðtÞ þ ðAiþTi&iðtÞÞxdðtÞ

þBhiðt, x, xd, uÞ þ BuðtÞ, ð7Þ

xðtÞ ¼ wðtÞ, t 2 ½, 0, ð8Þ A large number of examples in the literature and various mechanical systems, such as motors and robots, fall into special cases of the above model (7), as reported in Lin et al. (2005), Xia and Jia (2003), Choi (2008), El-Khazali (1998), and Oucheriah (1995, 2003). The above model (7) also involves the uncertain time-delay system models considered in the previous SMC design methods (Oucheriah 1995, El-Khazali 1998, Oucheriah 2003, Xia and Jia 2003, Lin et al. 2005, Choi 2008). The symbol  will be used in some matrix expressions to induce a symmetric structure. For given symmetric matrices, F and G of appropriate dimensions, the following holds: F þ X þ   Z G   ¼ F þ X þ X T _ZT Z G " # ð9Þ

When no confusion arises, the arguments t, x, xd, ,

etc. can be omitted for brevity.

3. Sliding control design via LMI approach

In this section, we demonstrate the problem of designing a robust sliding controller via LMI approach.

2 S.-C. Liu and S.-F. Lin

590

3.1. LMI characterization of linear sliding surfaces The SMC design is decoupled into two independent tasks of lower dimensions. The first is concerned with the design of a sliding surface for the sliding mode such that the reduced-order sliding mode dynamics satisfies the design specifications such as stabilization, tracking, regulation, etc. The second involves choosing a switch-ing feedback control for the reachswitch-ing mode so that it can drive the system’s dynamics into the switching surface (Utkin 1977). We first design a sliding surface that guarantees asymptotic stability of the reduced-order sliding mode dynamics using LMIs.

Theorem 1: Consider the following LMIs: Y4 0, F 0, N11    N21 N22   sXi sZi sKTYK 0 N41 N42 0 sKTYK 2 6 6 6 4 3 7 7 7 55 0, 8i, ð10Þ where

N11¼F þ KTðAiþTi&iðtÞÞYK þ Xiþ , ð11Þ

N21¼KTYðAiþTi&iðtÞÞTK  XiþZTi, ð12Þ

N22¼ ð1  dmÞF  ZiZTi, ð13Þ

N41 ¼KTðAiþTi&iðtÞÞYK, ð14Þ

N42 ¼KTðAiþTi&iðtÞÞYK: ð15Þ

The matrix K 2 RnðnmÞ is any full rank matrix such that BTK ¼ 0, KTK ¼ I: The matrices Y 2 Rnn, F 2 RðnmÞðnmÞ, Xi2RðnmÞðnmÞ and Zi2RðnmÞðnmÞ

are decision variables. Suppose that the LMIs equation (10) have a solution ðY, F, Xi, ZiÞ for given Ai, Ai,

B, dm, , then there exists a linear sliding surface

parameter matrix S and using a solution matrix Y to Equation (10), S can be parameterized as follows:

rðxÞ ¼ Sx ¼ ðBTY1BÞ1BTY1x ð16Þ Proof: Defining a nonsingular transformation matrix Mand the associated vector v ¼ Mx such that

M ¼ ðK T YKÞ1KT ðBTY1BÞ1BTY1 " # ¼ V S   , ð17Þ v ¼ v1 v2   ¼ Vx Sx   ¼Mx, ð18Þ

where v12Rnm, v22Rm:Then we can easily see that

M1¼ ½YK, B and v2¼r: Let the positive definite

matrix P0be P0¼KTYK, where Y is a solution to the

LMIs equation (10). By the above transformation we can obtain, we can transform Equation (7) into the following regular form:

_v ¼ A11 A12  A21 A22 " # v þ A11 A12  A21 A22 " # vd þ 0 I   u þX i ihi ! , ð19Þ where vd¼vðt  d ðtÞÞ and  A11 ¼ Xr i¼1 iP10 K T_ðA iþTi&iðtÞÞYK, ð20Þ  A12¼ Xr i¼1 iP10 KTðAiþTi&iðtÞÞB, ð21Þ  A21 ¼ Xr i¼1

iBTY1ðAiþTi&iðtÞÞYK, ð22Þ

 A22¼ Xr i¼1 iBTY1ðAiþTi&iðtÞÞB, ð23Þ  A11 ¼ Xr i¼1

iP10 KTðAiþTi&iðtÞÞYK, ð24Þ

 A12¼ Xr i¼1 iP10 KTðAiþTi&iðtÞÞB, ð25Þ  A21 ¼ Xr i¼1

iBTY1ðAiþTi&iðtÞÞYK, ð26Þ

 A22 ¼

Xr i¼1

iBTY1ðAiþTi&iðtÞÞB: ð27Þ

Thus, from the above regular form, by setting _r ¼ r ¼ 0, we can obtain the following sliding mode dynamics:

_a ¼ Aoa þ Adad, ð28Þ

where a ¼ v1, ad¼v1ðt  d ðtÞÞ, A0¼ A11, and Ad¼



A11: Let us define a Lyapunov–Krasovskii function

(LKF) as VðtÞ ¼ aTðtÞP0aðtÞ þ Z t td aTðsÞFaðsÞds þ Z 0  Z t tþ _aTðsÞP0_aðsÞds d , ð29Þ

where P0¼KTYK 2 Rnn and F 2 Rnn are solution

matrices for the LMIs equation (10). It should be noted that a large number of previous methods such as the methods given in Zhang et al. (2009) and Peng et al. (2008) have used similar LKFs to obtain less-conservative stability conditions by exploiting infor-mation on the upper bounds of delay and its time derivative. None of the previous SMC design methods (Oucheriah 1995, El-Khazali 1998, Oucheriah 2003, Xia and Jia 2003, Lin et al. 2005, Choi, 2008) have used the term R_0 R_tþ t _aT_ðsÞP

0_aðsÞds d in stability analysis.

The time derivative of the LKF is given by _ V ¼2aTP0ðA0a þ AdadÞ þaTFa  ð1  _d ÞaTdFad þ _aTP0_a  Zt t _aTðsÞP0_aðsÞds: ð30Þ

Using Equation (28) and the Newton–Leibniz formula a  ad Rt td_aðsÞds ¼ 0, we have _ V ¼2aT_P 0ðA0a þ AdadÞ þaTFa  ð1  _d ÞaTdFad þðA0a þ AdadÞTP0ðA0a þ AdadÞ  Z t t _aT_ðsÞP 0_aðsÞds þ 2ðaTXTþaTdZTÞ  a  ad Zt t _aðsÞds   , ð31Þ

where X ¼PiXiand Z ¼PiZi:Using the

inequal-ity 2xT_{y  x}T_{Hx þ y}T_H1_{Y, where x and y are any}

vectors with appropriate dimensions and H 4 0, we can obtain 2 a TðtÞXTþaT_dðtÞZT Z t t _aðsÞds  a TðtÞXTþaT_dðtÞZTP1₀ ½XaðtÞ þ ZadðtÞ þ Zt t _aT_ðsÞP 0_aðsÞds, ð32Þ which leads to _ V 2aTðP0A0a þ P0AdadÞ þaTFa  ð1  dmÞaTdFad þ½aTXTþa_dTZTP1₀ ½Xa þ Zad þ2ðaTXTþaT_dZTÞða  adÞ þðP0A0a þ PAdadÞT P1₀ ðP0A0a þ P0AdadÞ: ð33Þ

By applying the Schur complement formula (Boyd et al. 1994) to Equation (10), we can obtain

N11  N21 N22   þ X T i ZT_i " # P1₀ X T i ZT_i " #T þ K T YðAiþTi&iðtÞÞTK KTYðAiþTi&iðtÞÞTK " # P1_o K T YðAiþTi&iðtÞÞTK KTYðAiþTi&iðtÞÞTK " #T 5 0: ð34Þ

This implies that V  ð a_ k k2þkadk2Þ for some

4 0: After all, we can conclude that the sliding mode dynamics equation (28) is stable.

3.2. Sliding control law design

After the switching surface parameter matrix S is designed so that the reduced-order sliding mode dynamics has a desired response, the next step of the SMC design procedure is to design a switching feedback control law for the reaching mode such that the reachability condition is met (Utkin 1977, DeCarlo et al. 1988, Choi 2008). If the switching feedback control law satisfies the reachability condition, it drives the state trajectory to the switching surface r ¼ Sx ¼ 0 and maintains it there for all subsequent time. In this section, we design a sliding fuzzy control law guaran-teeing that r converges to zero. We will use the following nonlinear sliding switching feedback control law as the local controller:

Control Rule i: IF 1is i1and. . . and sis is, THEN

uðtÞ ¼ SðAiþTi&iðtÞÞx  SðAiþTi&iðtÞÞxdiðtÞ

r r k k, ð35Þ where iðtÞ ¼ 1 1  v_mðdiðtÞ þ vm SðAiþTi&iðtÞÞx   þSðAiþTi&iðtÞÞxd   þ iÞ ð36Þ

and i4 0: The final controller inferred as the weighted

average of the each local controller is given by

uðtÞ ¼ X r i¼1 ðÞ  SðAiþTi&iðtÞÞx þSðAiþTi&iðtÞÞxdþiðtÞ r r k k  , ð37Þ

and we can establish the following theorem.

Theorem 2: Suppose that the LMIs equation (10) is feasible and the sliding surface is given by Equation(16). Then, the switching feedback control law equation (37) induces an ideal sliding motion on the sliding surface r ¼ o in finite time and the state converges to zero. Proof: Since Theorem 1 implies that the sliding mode dynamics restricted to r ¼ Sx ¼ 0 is stable, we only have to show that the reachability condition 4 S.-C. Liu and S.-F. Lin

592

rT_{_r 5   rk k is satisfied for some  4 0: Using}

SB ¼ Iand the assumption A2, we can obtain

rT_r ¼ rTX r i¼1 iðSðAiþTi&iðtÞÞx þSðAiþTi&iðtÞÞxdþhiÞ þrTu X r i¼1 iðiik k u iÞk kr  X r i¼1 ik k:r

After all, we can conclude that r converges to zero.

4. Numerical example

In this section, two examples are used to illustrate the effectiveness of the proposed method and to compare with the existing method.

Example 1: To illustrate the performance of the proposed sliding fuzzy control design method, consider the following T–S fuzzy time-delay model (Wu and Li 2007) without mismatched parameter uncertainties and external disturbances. _ xðtÞ ¼X 2 i¼1 iðÞ½AixðtÞ þ AixdðtÞ þ BuðtÞ, ð38Þ where xðtÞ ¼ x1ðtÞ x2ðtÞ  T and A1¼ 0 0:6 0 1   , A1¼ 0:5 0:9 0 2   , ð39Þ A2¼ 1 0 1 0   , A2¼ 0:9 0 1 1:6   , B ¼ 1 1   ð40Þ 1¼ 1 1 þ e2x1ðtÞ, 2¼1  1: ð41Þ We assume that d ðtÞ ¼  ¼0:4, i¼0, i¼1,

m¼0, hi¼0 and i¼0:5: Figure 1 shows the

control results for system equation (38) via the proposed controller equation (37) under the initial condition ’ðtÞ ¼ ½2 0T. In Figure 1, it should be noted that since it is impossible to switch the input u instantaneously, oscillations always occur in the sliding mode of an SMC system.

Example 2: Consider a well-studied example of a continuous-time truck-trailer with time-delay pro-posed in Chen et al. (2009). The time-delay model is given by _ x1ðtÞ ¼ a vT Lt0 x1ðtÞ  ð1  aÞ vT Lt0 x1ðt  d Þ þvT lt0 ½uðtÞ þ hðtÞ, ð42Þ 0 5 10 15 20 -5 0 5 x 1 Time (sec) 0 5 10 15 20 -2 0 2 x 2 Time (sec) 0 5 10 15 20 -10 0 10 £m Time (sec) 0 5 10 15 20 -10 0 10 u Time (sec) Figure 1. Control results for the system.

_ x2ðtÞ ¼ a vT Lt0 x1ðtÞ þ ð1  aÞ vT Lt0 x1ðt  d Þ, ð43Þ _ x3ðtÞ ¼ vT t0 sin x2ðtÞ þ a vT 2Lx1ðtÞ þ ð1  aÞ vT 2Lx1ðt  d Þ   , ð44Þ where x1ðtÞ is the angle difference between truck and

trailer (in radians), x2ðtÞthe angle of trailer (in radians),

x3ðtÞthe vertical position of rear of trailer (in meters),

uðtÞ the steering angle (in radians), T ¼ 2:0, l ¼ 2:8, L ¼5:5, v ¼ 1:0 and t0¼0:5: The constant

param-eter a is the retarded coefficient satisfying a 2 ½0, 1: The limits 1 and 0 correspond to a no-delay term and to a completed-delay term. We assume that the disturbance input hðtÞ is unknown but bounded as hðtÞ  1: Using the fact that sinðxÞ x if x 0, we can represent the above model as the following two-rule T–S fuzzy model, including parameter uncertainties and external disturbances:

Plant rule 1: IF ðtÞ is about 0, THEN

_

x ¼ ðA1þT1&1ðtÞÞx þ ðA1þT1&1ðtÞÞxdþBu þ Bh1

ð45Þ

Plant rule 2: IF ðtÞ is about , THEN

_

x ¼ ðA2þT2&2ðtÞÞx þ ðA2þT2&2ðtÞÞxdþBu þ Bh2,

ð46Þ where ðtÞ ¼ x2ðtÞ þ avTx1ðtÞ=2L þ ð1  aÞvTx1ðt  d Þ=2L ð47Þ A1¼ avT Lt0 0 0 avT Lt0 0 0 av 2_T2 2Lt0 vT t0 0 2 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 5 , ð48Þ A1¼ ð1  aÞvT Lt0 0 0 ð1  aÞvT Lt0 0 0 ð1  aÞv 2_T2 2Lt0 0 0 2 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 5 , ð49Þ A2¼ avT Lt0 0 0 avT Lt0 0 0 a10v 2_T2 2L 10vT  0 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 , ð50Þ A2 ¼ ð1  aÞvT Lt0 0 0 ð1  aÞvT Lt0 0 0 ð1  aÞ10v 2_T2 2L 0 0 2 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 5 , ð51Þ B ¼ vT lt0 0 0 2 6 6 6 4 3 7 7 7 5, ð52Þ T1¼T2¼ 0:1 0:1 0:1 2 6 4 3 7 5, ð53Þ

&1ðtÞ ¼ &2ðtÞ ¼ sin t 0 0

  , ð54Þ 1¼ 1  1=ð1 þ e2ð0:5ÞÞ 1 þ e2ðþ0:5Þ , ð55Þ 2¼1  1, ð56Þ h1¼h2¼hðtÞ: ð57Þ

We assume that d ðtÞ ¼  ¼ 0:1: Considering LMI optimization with the data equation (45)–(57), a ¼0,  ¼ 0:1 and dm¼0, we can obtain the sliding

surface r ¼ Sx: Since hiðtÞ

 1, we can set i¼0, i¼1, m¼0

and i¼0:2: We can obtain the following fuzzy

controller:

Controller rule 1: IF ðtÞ is about 0, THEN uðtÞ ¼ SðA1þT1&1ðtÞÞx  SðA1þT1&1ðtÞÞxd

1:2sgnð Þ: ð58Þ

Controller rule 2: IF ðtÞ is about , THEN uðtÞ ¼ SðA2þT22ðtÞÞx

SðA2þT22ðtÞÞxd1:2sgnð Þ: ð59Þ

6 S.-C. Liu and S.-F. Lin

594

The final controller inferred as the weighted aver-age of each local controller is given by

uðtÞ ¼ X

2

i¼1

i½SðAiþTiiðtÞÞx

þSðAiþTiiðtÞÞxdþ1:2sgnð Þ: ð60Þ

To demonstrate the controller ability, we apply the above fuzzy controller equation (60) to the system model equations (45)–(57) with hðtÞ ¼ sin t and d ðtÞ ¼  ¼0:1: Figure 2 shows the closed-loop system responses of equations (45)–(57) and the proposed controller Equation (60) with the initial condition wðtÞ ¼ ½0:4, 0:8,  4T: Moreover, the closed-loop

0 10 20 30 40 50 -2 0 2 x 1 (r ad) Time (sec) 0 10 20 30 40 50 -5 0 5 x 2 (r ad) Time (sec) 0 10 20 30 40 50 -20 0 20 x 3 (m) Time (sec) 0 10 20 30 40 50 -5 0 5 u Time (sec)

Figure 3. Simulation results with the proposed method on model equations (42)–(44).

0 10 20 30 40 50 -2 0 2 x 1 (r ad) Time (sec) 0 10 20 30 40 50 -5 0 5 x 2 (r ad) Time (sec) 0 10 20 30 40 50 -50 0 50 x 3 (m) Time (sec) 0 10 20 30 40 50 -5 0 5 u Time (sec)

Figure 2. Simulation results with the proposed method on model equations (45)–(57).

system responses of the truth model equations (42)–(44) and the proposed controller equation (60) with the initial condition wðtÞ ¼ ½0:4, 0:8,  4T are also shown in Figure 3.

From Figures 2 and 3, the proposed controller is applicable to low-order fuzzy control synthesis for uncertain fuzzy time-delay systems with mismatched parameter uncertainties in the state matrix and exter-nal disturbances and the nonlinear truth model. The control performances of the two-rule T–S fuzzy model equations (45)–(57) and the nonlinear truth model equations (42)–(44) are satisfactory.

5. Conclusions

A robust sliding fuzzy control design method was developed for the uncertain T–S fuzzy time-delay model which includes mismatched parameter uncer-tainties and external disturbances. As the local con-troller, an SMC law with a nonlinear switching feedback control term is used. We gave an LMI condition for the existence of linear sliding surfaces guaranteeing the asymptotic stability of the reduced-order equivalent sliding mode dynamics. An explicit formula of the switching surface parameter matrix is derived in terms of the solution of the LMI existence condition and an LMI-based algorithm is developed to design the nonlinear switching feedback control term guaranteeing the reachability condition. Finally, using two numerical design examples, we have shown the effectiveness of our method.

Nomenclature

wðtÞ, xðtÞ, uðtÞ initial condition, state, and control input

Ai, Ai, Bi state matrices, delayed state

matrices, and input matrices j, ij, s, r premise variables, fuzzy sets,

number of premise variables, and number of IF-THEN rules d ðtÞ, iðÞ time-delay unknown function

and membership function , dm known constants

xdðtÞ, "AiðtÞ, "AiðtÞ,

hiðt, x, xd, uÞ

delayed state, parameter uncertainties in Ai, parameter

uncertainties in Ai, and

exter-nal disturbances

iðtÞ, i known function and known

constant

Ti, &iðtÞ constant matrix and

time-vary-ing matrix

S linear sliding surface parame-ter matrix

m, i known constants

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