Output feedback LMI tracking control conditions with H∞ criterion for uncertain and disturbed T–S models

Information Sciences 179 (2009) 446–457

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Information Sciences journal homepage: www.elsevier.com/locate/ins

Output feedback LMI tracking control conditions with H1 criterion for uncertain and disturbed T–S models B. Mansouri a, N. Manamanni a, K. Guelton a,*, A. Kruszewski b, T.M. Guerra c a b c

CReSTIC, EA3804, Université de Reims Champagne-Ardenne, Moulin de la House BP1039, 51687 Reims Cedex 2, France LAGIS, UMR CNRS 8146, Ecole Centrale de Lille, BP48, 59651 Villeneuve d’Ascq Cedex, France LAMIH, UMR CNRS 8530, Université de Valenciennes et du Hainaut Cambrésis, Le Mont Houy, 59313 Valenciennes Cedex 9, France

a r t i c l e

i n f o

Article history: Received 30 June 2006 Received in revised form 26 August 2008 Accepted 8 October 2008

Keywords: Tracking control Output feedback Fuzzy Takagi–Sugeno models Linear matrix inequality (LMI) Quadratic stability H1 criterion

a b s t r a c t This work concerns the tracking problem of uncertain Takagi–Sugeno (T–S) continuous fuzzy model with external disturbances. The objective is to get a model reference based output feedback tracking control law. The control scheme is based on a PDC structure, a fuzzy observer and a H1 performance to attenuate the external disturbances. The stability of the whole closed-loop model is investigated using the well-known quadratic Lyapunov function. The key point of the proposed approaches is to achieve conditions under a LMI (linear matrix inequalities) formulation in the case of an uncertain and disturbed T–S fuzzy model. This formulation facilitates obtaining solutions through interior point optimization methods for some nonlinear output tracking control problems. Finally, a simulation is provided on the well-known inverted pendulum testbed to show the efficiency of the proposed approach. Ó 2008 Elsevier Inc. All rights reserved.

1. Introduction Design of robust tracking control for uncertain nonlinear systems has attracted great attention in the past few years. Among nonlinear control theory, the Takagi–Sugeno (T–S) fuzzy model-based approach has nowadays become popular since it showed its efficiency to control complex nonlinear systems and has been used for many applications, see e.g. [7,9]. Indeed, Takagi and Sugeno have proposed a fuzzy model to describe nonlinear models [26] as a collection of linear time invariant models blended together with nonlinear functions. A control law, called ‘‘parallel distributed compensation” (PDC), can be synthesized as a collection of feedback gains that are connected using the same nonlinear functions [36]. Stability and stabilization analyses, for several kind of T–S fuzzy model, have been strongly investigated through Lyapunov direct method, see [17,24,25,27] and references therein. The key point of the proposed approaches is to achieve conditions under LMIs (linear matrix inequalities) formulation. This formulation allows obtaining solutions through interior point optimization methods [3]. A lot of works, involving various specifications, are now available for state space feedback: robustness with bounded uncertainties [6,15,23,24,28], time delay models with or without uncertainties [4,37], using pole placement constraints for each linear models [14], including performance specifications H2, H1 [19,20,23,29], and more recently using the circle criterion and its graphical interpretation in the frequency domain [2]. Complementary to these works and with the growing interest on engineering applications of T–S models based stabilization, some studies have been done regarding to output stabilization. These can be considered through two approaches. The first one uses a fuzzy observer

* Corresponding author. Tel.: +33 3 26 91 83 86; fax: +33 3 26 91 31 06. E-mail address: [email protected] (K. Guelton). 0020-0255/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2008.10.007

B. Mansouri et al. / Information Sciences 179 (2009) 446–457

447

and is interesting when the premise variables are measured [11,21,38]. The second one involves dynamic state feedback [1,10,15,16,23,27,39]. Despite numerous works available, none of them seem able to define a LMI formulation for the problem of robust trajectory tracking for T–S uncertain and/or disturbed models, with H1 performance and in output feedback. Usually, the obtained conditions are only expressed in terms of bilinear matrix inequality (BMI) [22,33]. Moreover, despite abundant literature on stability conditions of T–S fuzzy models, few authors have dealt with the tracking problem recently. Among this literature, some works are concerned with state feedback and H1 performances [27,30,33]. Let us quote that these works correspond to straightforward extensions of previous results. Nevertheless, when dealing with T–S models with external disturbances [34] the results are not more LMI. For the general case of output tracking, the existing approaches are based on variable structure control techniques [40] or on a switching controller using a reference model [18]. The only results available with PDC structure are given under a BMI form and two steps algorithm based on two LMI problems is generally used [22,34]. Moreover, Tong et al. [32] have developed a similar result with parametric uncertainties, i.e. BMI conditions solved in two steps. Nevertheless, the developed conditions proposed in [32] are obtained in spite of the confusion of the state reconstruction error and the tracking error leading to unsuitable conditions for tracking control based on an output feedback [5]. Let us quote that the solution (when it exists) using BMI algorithm is strongly depending on the initial conditions and therefore, no guarantee of convergence is ensured. This can lead to conservative solutions and may leads to less practical applicability of the proposed approaches. This lack of results, understood as the deficiency of LMI formulation, may lead to the aim of this paper. This paper is concerned with uncertain T–S continuous fuzzy model with external disturbances. The goal is to obtain a model reference based robust output feedback tracking control law. This one includes a PDC structure with a fuzzy observer and external disturbances attenuation based on an H1 criterion. The stability of the whole closed-loop model is investigated using the well-known quadratic Lyapunov function. The main contribution of the paper is the proposition of a LMI formulation to derive the proposed robust output feedback control law. The paper is presented as follows: Section 2 presents the T–S fuzzy models with uncertainties. The observer and the fuzzy control design using a reference model is developed in Section 3. The stability conditions, for the whole closed-loop system, derived in LMI formulation are developed in Section 4. Simulation results, showing the tracking performance of the wellknown testbed of the inverted pendulum, with a fuzzy observer are given in Section 5 to show the applicability of the proposed approach. 2. T–S fuzzy models Takagi–Sugeno fuzzy models allow describing nonlinear dynamical models by a set of linear time invariant (LTI) models interconnected by nonlinear functions. Each rule associates a LTI model as a concluding part to a weight function obtained from the premises [26]. In this paper, we focused on the class of uncertain and disturbed T–S fuzzy models [27]. The bounded uncertainties and external disturbances are added, in a classical way, to each nominal LTI models [28]. Thus, the ith rule can be expressed as

If z1 ðxðtÞÞ is F i1 and . . . and zp ðxðtÞÞ is F ip  _ xðtÞ ¼ ½Ai þ DAi ðtÞxðtÞ þ ½Bi þ DBi ðtÞuðtÞ þ uðtÞ; Then yðtÞ ¼ ½C i þ DC i ðtÞxðtÞ;

ð1Þ

where Fij are the fuzzy set and r is the number of If–Then rules, xðtÞ 2 Rn is the state vector, uðtÞ 2 Rm is the input vector, Ai 2 Rnn , Bi 2 Rnm , uðtÞ 2 Rn is a bounded external disturbance and zðtÞ ¼ ½ z1 ðxðtÞÞ    zp ðxðtÞÞ  is the premises vector being state dependent. The Lebesgue measurable uncertainties are defined as DAi(t) = HiDa(t)Eai, DBi(t) = HiDb(t)Ebi, where matrices Hi, Eai and Ebi are constant and the uncertainties Da(t), Db(t) satisfy the classical bounded conditions [41]: DaT(t)Da(t) 6 I, DbT(t)D b(t) 6 I. Given a pair of (x(t), u(t)), the fuzzy system is inferred as follows:

8 r P > > _ ¼ hi ðzðtÞÞ½ðAi þ DAi ðtÞÞxðtÞ þ ðBi þ DBi ðtÞÞuðtÞ þ uðtÞ; > xðtÞ < i¼1

r P > > > : yðtÞ ¼ hi ðzðtÞÞC i xðtÞ;

ð2Þ

i¼1

where wi ðzðtÞÞ ¼ in Fij, and

Qp

j¼1 F ij ðzj ðtÞÞ

wi ðzðtÞÞ hi ðzðtÞÞ ¼ Pr i¼1 wi ðzðtÞÞ

with wi (z(t)) P 0 and

Pr

i¼1 wi ðzðtÞÞ

for i ¼ 1; . . . ; r:

> 0, i = 1, . . . , r. Fij(zi(t)) is the degree of membership of zi(t)

ð3Þ

Therefore, the hi(z(t)), i = 1, . . . , r hold a convex sum property: r X i¼1

hi ðzðtÞÞ ¼ 1;

hi ðzðtÞÞ P 0;

i ¼ 1; . . . ; r:

ð4Þ

448

B. Mansouri et al. / Information Sciences 179 (2009) 446–457

At last, recall that there exists a systematic way to obtain (2) from a nonlinear model called the sector nonlinearity approach [27]. This one allows the T–S model matching exactly the nonlinear one on a compact set of the state space. Two types of uncertainties may occur in the modeling of uncertain nonlinear systems. The first one, called ‘‘structural uncertainty” is referred to as parametric uncertainties that are due to formalized unknown nonlinearities. The second type, known as ‘‘unstructured uncertainty” is often due to non-formalized modeling errors and external disturbances. Let us quote that, taking into account these uncertainties in the control design can be understood as more practical applicability. Indeed, with the growing complexity of nonlinear systems, it is often necessary to make approximations in the dynamical modeling process. Therefore, the main objective is now to provide stability conditions, in terms of LMI, that ensure the tracking performance for uncertain T–S models. 3. Controller synthesis In order to derive an output control law an additional observer is added. This one is based on the nominal model without uncertainties (2) and has the usual form [21,38]:

8 r P > _ > ^ðtÞÞ > < ^xðtÞ ¼ hi ðzðtÞÞ½Ai ^xðtÞ þ Bi uðtÞ þ Li ðyðtÞ  y i¼1

r P > > > ^ðtÞ ¼ hi ðzðtÞÞC i ^xðtÞ :y

for i ¼ 1; . . . ; r;

ð5Þ

i¼1

n

where ^ xðtÞ 2 R is the estimated state and Li is the observer gain for the ith LTI model. At last, note also that we are in the special case where the premises are supposed measurable, i.e. z(t) instead of ^zðtÞ in the general case. The former allows in various cases a separation principle [38]. The latter case remains in a more complicated design [11]. To specify the desired trajectory, consider the following reference model [34]:

x_ r ðtÞ ¼ Ar xr ðtÞ þ rðtÞ;

ð6Þ

where xr(t) is the reference state, Ar is a specified asymptotically stable matrix, and r(t) is a bounded reference input. The attenuation of external disturbances is guaranteed considering the H1 performance related to the tracking error xr(t)  x(t) as follows [8,13,34]:

Z

tf

ð½xr ðtÞ  xðtÞT Q ½xr ðtÞ  xðtÞÞdt 6 g2

0

Z

tf

ðrðtÞT rðtÞ þ uðtÞT uðtÞÞdt;

ð7Þ

0

where tf denotes the final time, Q is a positive definite weighting matrix, and g is a specified attenuation level. At last, the control law is based on the classical structure of a PDC law [36] sharing the same nonlinear functions as the T–S model:

uðtÞ ¼ 

r X

hi ðzðtÞÞK i ½xr ðtÞ  ^xðtÞ;

ð8Þ

i¼1

where Ki are gain matrices with appropriate dimension. Let us consider the estimation error eo ðtÞ ¼ xðtÞ  ^ xðtÞ, the tracking xðtÞ ¼ ½ eo ep xr T . error ep(t) = x(t)  xr(t) and the state reference xr(t). The state vector for the global closed loop is ~ Then,combining the control law (8), the system (2) and the observer (5), one obtains, after some easy manipulations, the following closed-loop model:

~x_ ðtÞ ¼

r X r X i¼1

e ij ~xðtÞ þ e ~ ðtÞ hi ðzðtÞÞhj ðzðtÞÞ A Su

ð9Þ

j¼1

with

2

Ai  Li C j  DBi K j e ij ¼ 6 A 4 Bi K j  DBi K j 0

DAi þ DBi K j Ai þ Bi K j þ DAi þ DBi K j 0

Note that with the state vector ~ xðtÞ, ~ ~ T /ðtÞ: rðtÞT rðtÞ þ uðtÞT uðtÞ ¼ /ðtÞ

Z 0

tf

e ~xðtÞdt 6 g2 ~xT ðtÞ Q

Z

tf

~ ~ T ðtÞ/ðtÞdt: /

(7)

3 DA i 7 A i  A r þ DA i 5 ; Ar

can be

written

3 I 0 6 7 e S ¼ 4 I I 5; 0 I

with

2

e ¼ diag½ 0 Q

  uðtÞ ~ /ðtÞ ¼ : rðtÞ Q

0

and

the

disturbances

ð10Þ

0

e ij described in (9) to ensure the asymptotic stability of the closedThe objective is now to compute the gains Ki and Li from A ~ A straightforward result is summarized in the loop model (9) guaranteeing the H1 tracking performance (10) for all /ðtÞ. following theorem. e ,e e eT Theorem 1. For t > 0 and hi(z(t))hj(z(t)) – 0, with A ij S defined in (9), if there exist a matrix P ¼ P > 0, and a positive constant g such that the following matrix inequalities are satisfied, i, j 2 {1, . . . , r}:

B. Mansouri et al. / Information Sciences 179 (2009) 446–457

(

 ii < 0; 2  r1 ii

"

þ  ij þ  ji 6 0;

eT P e ij þ Q e eþP eA A ij e e ST P

449

ð11Þ

i–j #

ee P S

. Then the asymptotic stability of the closed-loop fuzzy system (9) is ensured and the H1 g2 I tracking control performance (10) is guaranteed with an attenuation level g. with  ij ¼

Proof. Consider the following candidate Lyapunov function:

e ~xðtÞ with P e¼P e T > 0: Vð~x; tÞ ¼ ~xT ðtÞ P

ð12Þ

The stability of the closed-loop model (9) is satisfied under the H1 performance (10) with the attenuation level g if [7]:

e ~xðtÞ  g2 / _ ~x; tÞ þ ~xT ðtÞ Q ~ ~ T ðtÞ/ðtÞ Vð 6 0:

ð13Þ

The condition (13) leads to

~xT ðtÞ

" r X r X i¼1

# Te e e e e e ~xðtÞ þ ~xT ðtÞ P ee ~  g2 / ~ 60 ~ T ðtÞ/ðtÞ hi hj ð A ij P þ P A ij þ Q Þ ~xðtÞ þ /T ðtÞe ST P S /ðtÞ

ð14Þ

j¼1

or equivalently



~xðtÞ ~ /ðtÞ

T X r X r i¼1

" hi hj

eT P e e ee A ij þ P A ij þ Q

ee P S

e e ST P

g2 I

j¼1

#

 ~xðtÞ 6 0; ~ /ðtÞ

ð15Þ

which considering the work of Tuan et al. [35], is satisfied if conditions (11) hold. h The goal is now to obtain a tractable LMI problem that allows searching the gain matrices (both for the control Ki and the e > 0Þ ensuring the prescribed attenuation level (g). observer Li) and to prove the closed-loop stability (finding P 4. LMI formulation of the stability conditions To propose LMI conditions for T–S models tracking control, the following lemmas are needed to put the further provided conditions into LMI. Lemma 1 [41]. For real matrices X, Y and S = ST > 0 with appropriate dimensions and a positive constant c, the following inequalities hold:

X T Y þ Y T X 6 cX T X þ c1 Y T Y

ð16Þ

X T Y þ Y T X 6 X T S1 X þ Y T SY:

ð17Þ

and

Lemma 2. For real matrices A, B, W, Y, Z and a regular matrix Q with appropriate dimensions one has

"

Y þ BT Q 1 B WT W Z þ AQAT

"

# <0)

Y

W T þ BT AT

W þ AB

Z

# < 0:

ð18Þ

Proof of Lemma 2. For real matrices A, B, W, Y, Z and a regular matrix Q with appropriate dimensions, the inequality:       Y W T þ BT AT < 0 can be rewritten as: Y W T þ 0 BT AT < 0. W Z W þ AB Z AB 0 From the inequality (17), it exists a matrix Q > 0 such that

" # " #     0 0 BT BT T T ½0 A  6 Q 1 ½ B 0  Q½ 0 A  þ ½B 0þ A A 0 0

ð19Þ

that leads to (18) and ends the proof. h Lemma 3 [11]. Let a matrix X < 0, a matrix X with appropriate dimension such that XTXX 6 0, and a scalar a, the following inequality holds:

X T XX 6 aðX T þ XÞ  a2 X1 :

ð20Þ

450

B. Mansouri et al. / Information Sciences 179 (2009) 446–457

Proof of Lemma 3. X is a negative definite matrix, then if XTXX 6 0, hence

9a 2 R such that : ðX þ aX1 ÞT XðX þ aX1 Þ 6 0 i:e: X T XX þ aðX T þ XÞ þ a2 X1 6 0:



As usual, () will indicate a transpose quantity in a symmetric matrix. The main result is given in the following theorem. Theorem 2. For all t > 0 and hi(z(t))hj(z(t)), if there exist matrices P1 ¼ P T1 > 0, P 3 ¼ PT3 > 0, N = NT > 0, Yi, Zi, positive constants l1, l2, l3, l4, l5, l6, l7, l8 and g, such that the following LMI conditions are satisfied i, j 2 {1, . . . , r}:

(

 ii < 0; 2  r1 ii

þ  ij þ  ji 6 0;

ð21Þ

i–j

with

and

2

0 ðÞ ðÞ ðÞ 0 2aN 6 0 a N 0 0 0 0 2 6 6 6 Ebi Y j 0 l1 0 0 0 1 I 6 6 E Y 1 0 0  l I 0 0 6 bi j 5 Cij ¼ 6 6 aI 0 0 0 P1 Ai  Z i C j þ ATi P1  C Tj Z Ti ðÞ 6 6 T 1 1 1 6 0 0 0 0 Hi P 1 ðl1 þ l1 2 þ l3 Þ I 6 6 T 4 0 0 0 0 0 Hi P 1 aI 0 0 P1 0 0 2 Wij ð1; 1Þ ðÞ ðÞ ðÞ ðÞ ðÞ ðÞ 0 1 6 N Q 0 0 0 0 0 0 6 6 1 6 Ebi Y j 0  l I 0 0 0 0 0 2 6 6 0 0 l1 0 0 0 0 6 Ebi Y j 7 I 6 1 Wij ¼ 6 N 0 0 0  l I 0 0 0 E ai 3 6 6 0 0 0 0 l1 0 0 6 Eai N 6 I 6 T T 6 AT  AT 0 0 0 0 0 Ar P3 þ P3 Ar þ l4 Eai Eai ðÞ 6 i r 6 4 l1 0 0 0 0 0 0 Eai 8 I

0

0

0

0

0

0

0 0 0 ðÞ 0

3

ðÞ 7 7 7 0 7 7 0 7 7 7; P1 7 7 7 0 7 7 7 0 5

l4 I 0 g2 I 3 ðÞ 0 7 7 7 0 7 7 7 0 7 7 0 7 7; 7 0 7 7 ðÞ 7 7 7 0 5 g2 I

0

P3

0

T 1 1 1 Wij ð1; 1Þ ¼ Ai N þ Bi Y j þ NT ATi þ Y Tj BTi þ ðl1 5 þ l6 þ l7 þ l8 ÞH i H i ;

then the asymptotic stability of the closed-loop fuzzy system (9) is ensured and the H1 tracking control performance (10) is guaranteed with an attenuation level g. Moreover, if a solution exists, the gains Ki and Li are obtained using: Ki = YiN1 and Li ¼ P 1 1 Zi . e ¼ diag½ P1 Proof. For a convenient design, let us assume that P r X r X i¼1

P2

P 2 . Eq. (15) can be rewritten as

e ij þ D P e ij Þ 6 0 hi ðzÞhj ðzÞð P

ð22Þ

j¼1

with

2 6 6 6 e ij ¼ 6 P 6 6 6 4

P1 ðAi  Li C j Þ þ ðAi  Li C j ÞT P1

ðÞ

0

ðÞ

P2 Bi

P 2 ðAi þ Bi K j Þ þ ðAi þ Bi K j ÞT P2 þ Q

ðÞ

ðÞ

0

ðATi  ATr ÞP2

ATr P3 þ P3 Ar

0

P1

P2

0

g2 I

0

P2

P3

0

0

3

7 ðÞ 7 7 7 ðÞ 7 7 7 0 5 2 g I

451

B. Mansouri et al. / Information Sciences 179 (2009) 446–457

and

2

P1 DBi K j  K Tj DBTi P1

6 6 P2 DBi K j þ K T DBT P1 þ DAT P1 6 i j i e ij ¼ 6 T DP 6 D A P 6 i 1 6 4 0

ðÞ 0 0

ðÞ P 2 ðDAi þ DBi K j Þ þ ðDAi þ DBi K j ÞT P2 þ Q

DATi P2 0 0

0

3

7 ðÞ 0 0 7 7 7 : 0 0 07 7 7 0 0 05 0

0 0

Then using the uncertainties structure defined in (2) and the well-known property given in Lemma 2, can be bounded as follows: r X r X i¼1

e ij 6 hi ðzÞhj ðzÞD P

j¼1

r X r X i¼1

 hi ðzÞhj ðzÞdiag d1ij

d2ij

d3i

0 0

Pr Pr i¼1

j¼1 hi ðzÞhj ðzÞD



e ij P

ð23Þ

j¼1

with T 1 1 1 d1ij ¼ ðl1 þ l5 ÞK Tj ETbi Ebi K j þ ðl1 1 þ l2 þ l3 þ l4 ÞP 1 H i H i P 1 ; T T 1 1 1 d2ij ¼ ðl2 þ l7 ÞK Tj ETbi Ebi K j þ ðl1 5 þ l6 þ l7 þ l8 ÞP 2 H i H i P 2 þ ðl3 þ l6 ÞEai Eai ;

d3i ¼ ðl4 þ l8 ÞETai Eai : Then, the inequality (22) holds if

2 r X r X i¼1

j¼1

Hij ð1; 1Þ

6 P B K 2 i j 6 6 hi hj 6 0 6 6 4 P1

ðÞ

Hij ð2; 2Þ ðATi



ATr ÞP 2

ATr P3

P2

0

P 2

0

ðÞ

ðÞ

ðÞ

þ P3 Ar þ d3i

0

0

g2 I

P3

0

0

3

ðÞ 7 7 7 ðÞ 7 7 6 0; 7 0 5

ð24Þ

g2 I

T

Hij ð1; 1Þ ¼ P1 ðAi  Li C j Þ þ ðAi  Li C j Þ P 1 þ d1ij ; Hij ð2; 2Þ ¼ P2 ðAi þ Bi K j Þ þ ðAi þ Bi K j ÞT P2 þ Q þ d2ij : 2

I 60 6 In order to rearrange the matrices involved in (24) a congruence with the full-rank matrix 6 60 40 0 (24) is equivalent to

2

Hij ð1; 1Þ ðÞ 6 P g2 I 1 6 r X r X 6 P2 hi hj 6 6 P 2 Bi K j 6 i¼1 j¼1 4 0 0 0

0

ðÞ

0

ðÞ

0

Hij ð2; 2Þ ðATi



ATr ÞP2

P2

ðÞ ATr P3

þ P3 Ar þ d3 P3

0

0 0 0 I 0

0 I 0 0 0

3 0 07 7 07 7 is made. Thus 05 I

0 0 I 0 0

3

0 7 7 7 ðÞ 7 7 6 0: 7 ðÞ 5

ð25Þ

g2 I

Then, we proceed a bijective change of variables followed by a pre-post multiply of the inequality (25) by diag½ N N N I I  with N ¼ P 1 2 , Yi = KiN and Zi = P1Li, one obtains

ð26Þ

with

"

X1ij ¼

T 1 1 1 P1 Ai  Z i C j þ ATi P1  C Tj Z Ti þ ðl1 1 þ l2 þ l3 þ l4 ÞP 1 H i H i P 1

P1

P1

g2 I

# ð27Þ

;

T T 1 1 1 X2ij ¼ NAi þ Bi Y j þ ATi N þ Y Tj BTi þ NQN þ ðl2 þ l7 ÞY Tj ETbi Ebi Y j þ ðl1 5 þ l6 þ l7 þ l8 ÞH i H i þ ðl3 þ l6 ÞNEai Eai N: ð28Þ

Looking to expressions (26)–(28) shows that the major point to allow an LMI formulation is the product Now, applying Lemma 3 to the first diagonal block of (26), it yields



N 0

  N 0 X1ij 0 N

 0 . N

452

B. Mansouri et al. / Information Sciences 179 (2009) 446–457



N 0

  T T 0 þ ðl1 þ l5 ÞY j Ebi Ebi Y j N 0

  0 N X1ij N 0

     0 6 2a N 0  a2 X1 þ ðl1 þ l5 ÞY Tj ETbi Ebi Y j 0 1ij 0 N 0 0 0   T T 0 ¼ 2aN þ ðl1 þ l5 ÞY j Ebi Ebi Y j  a2 X1 1ij : 0 2aN

ð29Þ

Then, applying the Schur complement, (29) becomes

2 6 6 6 4

2aN þ ðl1 þ l5 ÞY Tj ETbi Ebi Y j

0

ðÞ

0

2aN

0

aI

0 aI

Nij ð3; 3Þ

0

P1

3

0

ðÞ 7 7 760 ðÞ 5

ð30Þ

g2 I

T 1 1 1 with Nij ð3; 3Þ ¼ P1 Ai  Z i C j þ ATi P1  C Tj Z Ti þ ðl1 1 þ l2 þ l3 þ l4 ÞP 1 Hi H i P 1 . Substituting (30) in (26), we obtain the following inequality:

ð31Þ

with

Hij ð1; 1Þ ¼ 2aN þ ðl1 þ l5 ÞY Tj ETbi Ebi Y j ; T 1 1 1 Hij ð3; 3Þ ¼ P 1 Ai  Z i C j þ ATi P 1  C Tj Z Ti þ ðl1 1 þ l2 þ l3 þ l4 ÞP 1 Hi H i P 1 ;

Hij ð5; 5Þ ¼ NAi þ Bi Y j þ ATi N þ Y Tj BTi þ NQN þ ðl2 þ l7 ÞY Tj ETbi Ebi Y j T T 1 1 1 þ ðl1 5 þ l6 þ l7 þ l8 ÞHi Hi þ ðl3 þ l6 ÞNEai Eai N;

Hij ð6; 6Þ ¼ ATr P3 þ P3 Ar þ ðl4 þ l8 ÞETai Eai : Applying Schur’s complement on the diagonal blocks Hij(1, 1), Hij(3, 3), and Hij(6, 6), the conditions of Theorem 2 hold.

h

Remark. The proposed approach provides quasi-LMI conditions where only the 4 scalars a, l1, l2 and l3 are required to obtain exact LMI conditions. Note that, in the previous literature, several BMI solutions are available [22,33] that can be solved by iterative algorithms using two general eigenvalue problems (GEVP). Thus, these ones are strongly dependent on the initialization and on the different variables set in each problem (number of epochs, feasibility radius, etc.) for the GEVP formulation. Providing LMI conditions allows preventing this problem. 5. Example and simulation 5.1. System modeling To illustrate the proposed approach, consider the angular position tracking control of an inverted pendulum on a cart (Fig. 1). The system’s dynamical equations are expressed as

x1 (t )

m

2l u (t )

M

Fig. 1. Inverted pendulum on a car.

B. Mansouri et al. / Information Sciences 179 (2009) 446–457

8 < x_ 1 ¼ x2 ; : x_ 2 ¼

ðmþMÞg sinðx1 Þmlx22 sinðx1 Þ cosðx1 Þcosðx1 Þu lð13mþ43Mþm sin2 ðx1 ÞÞ

453

ð32Þ

;

where x1(t) and x2(t) are respectively the angular position and velocity of the pendulum, u(t) is the force applied to the cart, m = 0.1 kg and M = 1 kg are respectively the masses of the pendulum and the cart, 2l = 1 m is the length of the pendulum and g = 9.8 m s2. Note that m sin2(x1(t)) is small regarding to 13 m þ 43 M. Then, it will be assumed to be neglected. Therefore, the system (32) becomes

8 < x_ 1 ¼ x2 ;

ð33Þ

2

mlx2 sinðx1 Þ cosðx1 Þ 1 Þcosðx1 Þu : x_ 2 ¼ ðmþMÞglsinðx þ : l ð4MþmÞ ð4MþmÞ 3

3

Using the well-known sector nonlinearity approach [27], the goal is now to derive a T–S model from (33). Indeed, the above model is constituted by three nonlinearities to be splitted: g1(x1(t)) = sin(x1(t)), g2(x1(t)) = cos(x1(t)) and g3 ðx2 ðtÞÞ ¼ x22 ðtÞ. Let us consider that the velocity signal x2(t) is not available from measurements. Then, the nonlinear function x22 ðtÞ is re 3 . Then, moved from the certain part of the T–S model. We assume that this nonlinear function is bounded as g3 ðx2 ðtÞÞ 2 ½0; g we write it as an uncertainty and (33) can be written as



x_ 1 ¼ x2 ; x_ 2 ¼ dðx1 ; uÞ þ Ddðx; uÞ

ð34Þ

g1 ðx1 Þg2 ðx1 Þu 1 ðx1 Þg2 ðx1 Þ  with dðx1 ; uÞ ¼ ðmþMÞgl ð4MþmÞ , Ddðx1 ; x2 ; uÞ ¼ mllgð4MþmÞ g3 f ðx2 Þ and f ðx2 Þ ¼ g3gðx3 2 Þ. 3

3

The T–S membership functions are obtained following the same procedure as presented in [12]. Let x1 2 [h0, h0], then the nonlinear functions can be written as follows:

sinðh0 Þ x1 ðtÞ; h0 1 2 g2 ðx1 ðtÞÞ ¼ x2 ðx1 ðtÞÞ þ x2 ðx1 ðtÞÞ cosðh0 Þ;

g1 ðx1 ðtÞÞ ¼ x11 ðx1 ðtÞÞx1 ðtÞ þ x21 ðx1 ðtÞÞ

where for i = 1, 2 and j = 1, 2 we have 0 6 xji 6 1 and

h0 sinðx1 ðtÞÞ  x1 ðtÞ sinðh0 Þ ; x1 ðtÞðh0  sinðh0 ÞÞ h x ðtÞ  h0 sinðx1 ðtÞÞ x21 ðx1 ðtÞÞ ¼ 1  x11 ðx1 ðtÞÞ ¼ 0 1 ; x1 ðtÞðh0  sinðh0 ÞÞ 1  cosðx1 ðtÞÞ x22 ðx1 ðtÞÞ ¼ 1  x12 ðx1 ðtÞÞ ¼ : 1  cosðh0 Þ

x11 ðx1 ðtÞÞ ¼

x12 ðx1 ðtÞÞ ¼

cosðx1 ðtÞÞ  cosðh0 Þ ; 1  cosðh0 Þ

A way to reduce the computational complexity of the LMI conditions is to minimize the number of rules used to model the system [31]. With the previous nonlinear splitting, the obtained fuzzy model should have 4 rules. Nevertheless, it is still possible to reduce this fuzzy model noticing that x11 and x12 , respectively x21 and x22 , are very closed [12]. Thus, for i = 1, 2 we assume that xi ¼ x1i ¼ x2i and a 2 rules fuzzy model representing the nonlinear uncertain model (34) can be proposed as

8 2 P > < xðtÞ _ ¼ xi ðtÞ½ðAi þ DAi ðxðtÞÞÞxðtÞ þ Bi uðtÞ; i¼1 > : yðtÞ ¼ CxðtÞ " # " # # " # 1 0 1 0 0 ðMþmÞg sinðh0 Þ cosðh0 Þ , B ¼ ¼ ¼ , A , B 1 2 1 2  l ð4MþmÞ  l ð4MþmÞ , C ¼ ½ 1 0 0 l ð4MþmÞ h0 3 3 3 # " # 0 0 0 sinðh0 Þ cosðh0 Þ ml 0 and DA2 ðxðtÞÞ ¼ l ð4MþmÞ g 3 f ðx2 ðtÞÞ 0 . h0 "

with xðtÞ ¼ ½ x1 ðtÞ "

DA1 ðxðtÞÞ ¼

x2 ðtÞ T , A1 ¼ 0

ml l ð4MþmÞ 3

g 3 f ðx2 ðtÞÞ

ð35Þ

0

ðMþmÞg l ð4MþmÞ 3

0 ,

3

Considering the uncertainties structure used to obtain Theorems 1 and 2, we can write DA1(t) = H1F1(t)Ea1 and " # " # 0 0  3 0 , H2 ¼  3 0 . sinðh0 Þ cosðh0 Þ , Ea2 ¼ ½ g , Ea1 ¼ ½ g DA2(t) = H2F2(t)Ea2 with H1 ¼ ml ml l ð4MþmÞ 3

l ð4MþmÞ 3

h0

5.2. Simulation results  3 ¼ 0:01. Note that when x1 = ±p/ The simulation was performed with a maximum angular velocity set at 0.1 rad/s2, then g 2, the system presents a singularity and so it is locally uncontrollable. To overcome this problem, the modeling space has been reduced to x1(t) 2 [h0, h0] with h0 = 22p/45.

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Fig. 2. Tracking performances: (a) position, (b) zoom in the transient state of the position, (c) velocity, (d) zoom in the transient state of the velocity.

Fig. 3. (a) Position estimation error, (b) velocity estimation error, (c) control signal, (d) quadratic error tracking.

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455

Fig. 4. Tracking performances with high disturbances: (a) position, (b) zoom in the transient state of the position, (c) velocity, (d) zoom in the transient state of the velocity.

Fig. 5. Simulation with high disturbances: (a) position estimation error, (b) velocity estimation error, (c) control signal, (d) quadratic error tracking.

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After trials, the presented simulations are performed with the following tuning:   0 1 Hurwitz to set a desired dynamics to follow. – The reference model was arbitrary chosen with Ar ¼ 6 5   2:7 0 . – The dynamics of the closed-loop system was fixed by choosing Q ¼ 106 0 2 – The value a = 100, l1 = 100, l2 = 100, l3 = 0.01 was arbitrary chosen (note that these value will be balanced by the computed value of l4, l5, l6, l7, l8). – The solution P1, N, Yi and Zi are computed (if feasible) by solving the set of LMI conditions (21) given in Theorem 2 with classical LMI toolbox. – Finally, the gains Ki and Li are obtained from the bijective change of variables Ki = Yi N1 and Li ¼ P 1 1 Zi . Therefore, for the proposed example of the inverted pendulum on a cart, the solution of Theorem 2 is obtained using the Matlab LMI toolbox and is given by the gains K1 = [294.3233 126.2873], K2 = [294.3233 126.2873], L1 = [30.0 332.8878]T, the scalars l4 = 2.7013, l5 = 2.7978  109, l6 = 2.7978  109, l7 = 2.7978  109, L2 = [30.0 330.1325]T,       0:1921 0:4056 94 2 14 8 , P 3 ¼ 104 , N ¼ 107  . l8 = 2.7978  109 and the matrices P1 ¼ 104 0:4056 1:2086 2 10 8 7 Fig. 2a and c shows the tracking trajectory position and velocity with respectively initial system states xð0Þ ¼ ½ 0:2 0 T and observed state ^ xð0Þ ¼ ½ 0 0:2 T for rðtÞ ¼ ½ 0 2:46 sinðtÞ T . Note that the system is subject to the external disturbances uðtÞ ¼ ½ 0:03 sinðtÞ 0 T that are set in simulation with amplitude about 10% of the tracking trajectory to test the efficiency of its attenuation. Fig. 2b and d illustrates the controller and observer efficiency in the transient state. The input signal and the position tracking quadratic error are represented in Fig. 3. To show the effectiveness of the disturbance attenuation by the H1 criterion, a high external disturbance is applied to the inverted pendulum uðtÞ ¼ ½ 0:15 sinðtÞ 0 T that is about 50% of the tracking trajectory. The obtained results are depicted in Fig. 4. Fig. 4a and b shows the tracking trajectory position and velocity with the same initial states as the previous simulation. Note that despite the huge disturbance amplitude, the system does not have an unstable behavior. Even if the system position seems to follow the reference position, in this simulation the tracking velocity performances are lost showing the limits of such control law synthesis. In this case, the input signal and the position tracking quadratic error are presented in Fig. 5. 6. Conclusion In this paper, a fuzzy tracking control has been designed for an uncertain nonlinear dynamic system with external disturbances using a T–S fuzzy model and a state observer design. A control scheme based on an augmented structure with a guaranteed H1 performance and model reference tracking is proposed. The main result of the paper is the quasi-LMI formulation that can be applied for tracking control design of uncertain and disturbed T–S fuzzy model. This can be considered as an improvement of previous theoretical studies on T–S fuzzy model-based output tracking control design and it constitutes a starting point to further applicative studies on complex industrial plants. At last, a design example has illustrated the efficiency of the proposed approach on the well-known testbed of an inverted pendulum. 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