ISA Transactions 53 (2014) 305–316

Contents lists available at ScienceDirect

ISA Transactions journal homepage: www.elsevier.com/locate/isatrans

Research Article

On observer-based controller design for Sugeno systems with unmeasurable premise variables Hoda Moodi, Mohammad Farrokhi n Department of Electrical Engineering, Iran University of Science and Technology, Tehran 16846-13114, Iran

art ic l e i nf o

a b s t r a c t

Article history: Received 6 August 2012 Received in revised form 29 September 2013 Accepted 3 December 2013 Available online 3 January 2014 This paper was recommended for publication by Prof. A.B. Rad

This paper considers the design of observer-based controller for a class of continuous-time nonlinear systems presented by Takagi–Sugeno (T–S) model with unmeasurable premise variables. This T–S structure can represent a larger class of nonlinear systems as compared to the measurable premise variable case but its analysis is more complicated. To reduce the design complexity, a common output model for subsystems is employed by the use of local nonlinear rules. As a result, the proposed T–S structure reduces the number of rules in the Sugeno model as well as the analysis complexity. The proposed controller guarantees exponential convergence of states based on the fuzzy Lyapunov function analysis and Linear Matrix Inequality (LMI) formulation. Simulation results illustrate effectiveness of the proposed method. & 2013 ISA. Published by Elsevier Ltd. All rights reserved.

Keywords: Fuzzy observer-based controller Unmeasurable premise variable LMI Nonlinear subsystems

1. Introduction Takagi–Sugeno (T–S) fuzzy models are well-known tools for nonlinear system modeling with increasing interests in recent years. Since T–S models are universal approximator, they have the potential to model any smooth nonlinear system with desired accuracy [1]. Furthermore, local linear subsystems of these models allow one to use powerful linear system tools such as Linear Matrix Inequalities (LMIs) to analyze and synthesize the T–S fuzzy systems. Fuzzy observers were first introduced by Tanaka and Sano and ever since have been an active research issue [2]. Different types of fuzzy observers have been discussed in literatures. The separation property of fuzzy observers and controllers was first discussed by Xiao-Jun et al. in [3] and later on, more completely by Yan and Sun in [4]. One major problem in fuzzy observer based controller design is that when uncertainties exist in the model of the system, the separation property does not hold in general. In this regard, robust fuzzy observers have been discussed in different papers such as [5,6]. Adaptive fuzzy observers also have attracted researchers’ attention in recent years [7,8]. Sliding-mode fuzzy observers were first presented by Palm and Driankov in [9]. Robust sliding-mode observers for systems with unknown input have been considered in several papers such as [10]. For estimating the states of T–S systems, two cases for the premise variables can be distinguished. In the first case, the premise variables do not depend on the estimated states while in the second

n

Corresponding author. Tel.: þ 989123939087; fax: þ982173225777. E-mail addresses: [email protected] (H. Moodi), [email protected] (M. Farrokhi).

case, the premise variables depend on some of the estimated states. Although most researchers assume that the weighting functions depend on the measurable premise variables, relaxing this assumption can represent a larger class of nonlinear systems. The motivations for using unmeasured states as premise variables have been discussed in several papers; some of them are briefly reviewed here. To get a T–S model out of a nonlinear system, one of the most common methods is the well-known sector nonlinearity approach [11], which allows obtaining an exact T–S representation of the system. It is shown by Yoneyama that if the system output is affected by disturbances (which cannot be avoided in practical situations) and this output is considered as a decision variable, then the obtained T–S system does not represent precisely the system [12]. It is also pointed out in this reference that if the output is nonlinear with respect to the states of the system, then it is difficult or even impossible to obtain a T–S model by nonlinear sector transformation using the output as a premise variable. The author has given some examples of systems for which the output cannot be used as premise variables and hence, the states are required for modeling of the system [12,13]. Ichalal et al. state that in the field of diagnosis there is a force to design observers with weighting functions depending on the input, for the detection of the sensors faults, and on the output, for the detection of actuator faults [14]. However, modeling with unmeasurable premise variables allows developing only one model of the system behavior to detect and isolate actuator and sensor faults. These facts motivate researchers to take the state of the system as premise variable in order to describe a wider class of nonlinear systems. Unfortunately, the developed methods for fuzzy observer design with measured premise variables are not directly applicable for the systems with unmeasured premise variables [15]. Most works in

0019-0578/$ - see front matter & 2013 ISA. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.isatra.2013.12.004

306

H. Moodi, M. Farrokhi / ISA Transactions 53 (2014) 305–316

literatures are based on the first case. The second case was first discussed by Bergsten [16]. Later, Bergsten et al. employed the Luenberger observer by using Lipschitz weighting functions [17]. The stability conditions of the observer are formulated in the form of LMIs. However, in their method, the Lipschitz constant appears in the LMIs, which needs to be solved and hence, reduces the applicability of the method. The same authors employed the sliding mode in T–S observer to compensate for the undesirable terms of the system [18]. However, the considered perturbed term depends on the input u(t) and the state variables x(t). This means that the larger the bound on the input, the larger the value of the constant, which could lead to an infeasible set of LMIs. Some recent papers have also tried to obtain less conservative conditions for this case such as [15,19]. Fewer works are devoted to observer-based controller design for this class of systems, which is the subject of this paper. The main contribution of this paper is the design of an observer-based controller for fuzzy systems with unmeasured premise variables in presence of local nonlinear rules, which are employed to design T–S models with common output structure. The proposed method reduces the complexity of the problem of designing observers with unmeasured premise variables. This paper is organized as follows: Section 2 provides the problem of the observer-based controller design for T–S systems with unmeasurable premise variables and the challenges of this design. In Section 3, the case of common output structure for all subsystems, as a special case for the design, is discussed and a simple method for the controller design is proposed. In Section 4, as a generalization of Section 3, nonlinear T–S model with unmeasurable premise variables is discussed. As it will be shown, nonlinear subsystems can be used to model more complicated systems with a common output model. Numerical examples are given in Section 5 to show effectiveness of the proposed method. Section 6 concludes the paper.

2. Problem statement In this section, the main problem is described and challenges of the design and available solutions are stated. Consider the following fuzzy system: Plant Rule i :

IF z1 ðtÞ is M i1 ðzÞ and…and zp ðtÞ is M ip ðzÞ; _ ¼ Ai xðtÞ þ Bi uðtÞ THEN xðtÞ

ð1Þ

yðtÞ ¼ Ci xðtÞ where i ¼1, …, r is the number of rules, z1 ðtÞ; …; zp ðtÞ are the premise variables, Ai A Rnn ,Bi A Rnnu and Ci A Rny n are constant matrices, and M ij (j ¼ 1; …; p) denote the fuzzy sets. In this case, the overall fuzzy system can be represented as r

_ ¼ ∑ ωi ðzÞðAi xðtÞ þ Bi uðtÞÞ xðtÞ i¼1 r

yðtÞ ¼ ∑ ωi ðzÞðCi xðtÞÞ

ð2Þ

i¼1

where

ωi ðzÞ ¼

hi ðzÞ r

;

∑ hk ðzÞ

p

r

∑ ωi ðzÞ ¼ 1;

ωi ðzÞ Z 0 and hi ðzÞ ¼ ∏ μij ðzÞ:

i¼1

j¼1

k¼1

ð3Þ The controller used in this paper has the following structure: Controller Rule i :

IF z1 ðtÞ is M i1 ðzÞ; …; and zp ðtÞ is M ip ðzÞ; ^ þ B uðtÞ þL ðyðtÞ  yðtÞÞ ^ ^_ ¼ A xðtÞ THEN xðtÞ i

^ ¼ Ci xðtÞ ^ yðtÞ ^ uðtÞ ¼  Ki xðtÞ

i

i

ð4Þ

For analysis of the controller stability, two cases can be distinguished: (1) the scheduling vector z does not depend on the estimated states and (2) z depends on some of the estimated states. The second case is the subject of this paper and will be discussed here. In this case, controller (4) becomes r

_^ ¼ ∑ ω ðz^ Þ½A xðtÞ ^ xðtÞ þ Bi uðtÞ þLi ðyðtÞ  yðtÞÞ i i^ i¼1 r

^ ¼ ∑ ωi ðz^ ÞðCi xðtÞÞ ^ yðtÞ i¼1 r

^ uðtÞ ¼ ∑ ωi ðz^ Þð  Ki xðtÞÞ

ð5Þ

i¼1

For the sake of notation, the following definitions are used: r

Xz^ :¼ ∑ ωi ðz^ ÞXi

ð6Þ

Xzz^ :¼ Xz  Xz^

ð7Þ

i¼1

where X A fA; B; C; L; K; Gx ; Ka ; Kb g. Hence, the augmented system dynamic can be written as " # " #" # _ Az  Bz Kz^ Bz Kz^ xðtÞ xðtÞ ¼ ð8Þ Azz^  Bzz^ Kz^  Lz^ Czz^ Az^ þ Bzz^ Kz^  Lz^ Cz^ eðtÞ e_ ðtÞ ^ where eðtÞ ¼ xðtÞ  xðtÞ. Stabilization analysis of this system has been studied in different papers such as [20]. However, stating stability conditions as an LMI, which is more complicated, has been discussed only in a few papers. Sing Kiong and Peng designed an H 1 controller based on the well-known Lyapunov functional approach, which results in convex LMIs for the observer parameters [21]. Substituting these parameters into the controller equations yields convex LMIs, which can be efficiently solved. For this procedure, a single parameter is selected arbitrary and then 2r 2 LMIs are solved. In this line of research, Guerra et al. have shown stabilization conditions for the global loop in terms of LMIs [22]. However, in their method, four parameters must be selected carefully in advance and then r 3 LMIs are solved. Tseng et al. have proposed an L1 gain controller formulation with 3r 3 LMIs, in which six parameters must be determined in advanced in order to obtain strictly LMIs [23]. The authors have used the genetic algorithm to determine the best values for these six parameters. Lian et al. have used a different observer and tracking controller, where two diagonal matrices must be carefully selected in order to achieve LMIs [24]. In [25] a method based on Finsler Lemma is introduced to handle the problem where uncertainties exits in the model however; r 3 LMIs are used and the solution is too conservative. Yoneyama has derived output feedback controller for such systems by rewriting the system as an uncertain system [13]. Some known terms of the dynamics are considers as uncertainties to make the analysis simpler. A common problem in the aforementioned papers is that the stability analysis of the general system (8) yields non-strictly LMI conditions and the difficulty of selecting proper parameters before solving LMIs. Moreover, in many practical situations, the output is given by a set of sensors measuring a subset of the state variables. Assuming that the location of these sensors does not depend on the operating point, the output of the system is common for all subsystems which means that [26] C1 ¼ C2 ¼ … ¼ Cr ¼ C

ð9Þ

This assumption, which is the subject of the next section, will simplify the design drastically.

H. Moodi, M. Farrokhi / ISA Transactions 53 (2014) 305–316

3. Common output model case In this section, the case of common output matrix, C, is discussed and it will be shown how this will reduce the design complexity. For this case, the augmented dynamic of the system reduces to " # " #" # _ Az  Bz Kz^ Bz Kz^ xðtÞ xðtÞ ¼ ð10Þ _ Azz^  Bzz^ Kz^ Az^ þ Bzz^ Kz^ Lz^ C eðtÞ eðtÞ Designing observer for estimation of states of the system in the common output matrix case is discussed broadly by Ichalla et al., who have studied the case of common observer gains in [27] and later in their extended work in [14], where the estimation error is considered like a perturbed system. In order to attenuate the effects of this perturbation on the state estimation error, the conditions for convergence of the observer are obtained using the Lyapunov theory and L2 design. These conditions are expressed in terms of LMIs. This method makes it possible to synthesize an observer for T–S systems without any Lipschitz weighing function. Moreover, no knowledge on the input bound of the system is required in order to find the gains of the observer. The same authors have estimated the states and the unknown inputs simultaneously such that the H 1 performance of the system is guaranteed [26]. Stability of the closed-loop system (10) is also discussed in [28], where the authors have assumed a common B matrix. In the followings, the conditions for asymptotic convergence of (10) are discussed. To this end, the following lemmas are needed in the sequel. Lemma 1. [29] Let M be a symmetric matrix of the form " # M11 M12 M¼ T M12 M22

ð11Þ

where M11 and M22 are assumed to be invertible. Then, M is positive definite if and only if either of the following conditions holds: M11 40

and

M22 MT12 M11  1 M12 4 0

M22 40

and

1 T M11 M12 M22 M12 4 0

1

Y

Π40

ð13Þ

1 o i or

i¼1j¼1

where αi ð1 r i o rÞ satisfies 0 r αi r1, Σ i ¼ 1 αi ¼ 1. r

8 ρ A ½1; …; r  i

P2ρ þX  P2i Z 0 2 6 6 4

ð17Þ ð18aÞ

P~ 2ϕ  MAρ  AT MTρ þ Y ρ C þ CT Y ρ þ μI

n

P2ρ þ MT  MAρ þ Yρ C

M þ MT

M

T

M

n T

3 7

n 7 5o0

ρ ¼ 1; …; r

I

where Θij ¼

and P~ 2ϕ ¼ 7

μ¼μ ϕi X þ Σ rρ ¼ 1 ϕρ ðP2ρ þ X  P2i Þ. The 7sign means that the LMIs must ðAi Q 1 þQ 1 ATi Þ ðBi Xj þ Xj T BTi Þ,

2 1,

be checked for both positive and negative signs. Then, the controller and observer gains would be Li ¼ M  1 Y i ;

i ¼ 1; …; r

ð19Þ

ð20Þ

plus a zero term would be as follows.

To discuss the stability of (10) first, it is rewritten as below. Proposition 1. The augmented dynamic (10) can be rewritten as follows: " # " #" #   _ Bz Kz^ Az  Bz Kz^ xðtÞ xðtÞ 0 ¼ ð14Þ þ _ 0 Az^  Lz^ C eðtÞ eðtÞ Δ

r

r

_ ρ ðz^ ÞPρ xa þ2xTa ∑ ωρ ðz^ ÞPρ x_ a V_ ¼ xa ∑ ω ρ¼1 r

_ ¼ xa ∑ ω ρ¼1

ρ¼1

T ρ ðz^ ÞPρ xa þ 2xa

"

r

∑ ωρ ðz^ ÞPρ x_ a þ 2½eT ðtÞM1

ρ¼1

r

þ e_ T ðtÞM2   e_ ðtÞ  ∑ ωj ðz^ ÞðAj  Lj CÞeðtÞ  Δ

# ð21Þ

j¼1

where

i¼1

Θii o 0; 1 oi o r Θii þ 12ðΘij þ Θji Þ o 0; 1 o ia j o r

and the Lyapunov function as V ¼ xTa ∑rρ ¼ 1 ωρ ðz^ ÞPρ xa , where " # εP1 0 Pρ ¼ . Then, the derivative of the Lyapunov function 0 P2ρ

r

∑ ∑ αi αj Mij o 0

r

1 r1

ð16Þ

Proof. Consider the following augmented variables: " # " #   Bz Kz^ Az Bz Kz^ xðtÞ 0 ; Δa ¼ xa ¼ ; Ga ¼ 0 Az^  Lz^ C eðtÞ Δ

1 o i a j or

Δ ¼ ∑ ½ðωi ðzÞ  ωi ðz^ ÞÞðAi xðtÞ þ Bi uðtÞÞ

∥Δ∥ r μ1 ∥eðtÞ∥

Ki ¼ Xi Q 1 1 ;

then, the following inequality holds: r

_ ρ ðzÞj r ϕρ for known positive real numbers Theorem 1. Assume jω ϕρ and ρ A ℜ. The augmented dynamic (14) is asymptotically stable if there exist positive definite and symmetric matrices P2i A Rnn ði ¼ 1; …; rÞ and Q 1 A Rnn , and matrices X A R nn , M A R nn , Y i A R nny ; and Xi A R nu n ði ¼ 1; …; rÞ, and scalar μ1 4 0 such that the following inequalities hold:

ð18bÞ

Lemma 3. [30] If the following conditions hold: Mii o 0 1 1 M þ ðM þ M ii ij ji Þ 4 0 r 1 2

In the following theorem, sufficient conditions for the stability of the augmented dynamic (14) are given. Here, for the controller design we use a common quadratic Lyapunov function while for the observer design a fuzzy Lyapunov function is employed in order to reduce the conservativeness. Fuzzy Lyapunov functions have been of increasing interest in recent years. A complete review of recent Lyapunov functions for discrete fuzzy systems is presented in [31]. For continuous systems, it is more difficult to obtain LMI conditions by using fuzzy Lyapunov function, because the stability conditions depend on the time derivative of the membership functions. In [32], LMI conditions are derived for state feedback controller design by adding some slack matrices. In [33], a method is proposed, which does not depend on the derivative of the membership functions. More recent works have been performed on employment of non-quadratic Lyapunov functions for continuous-time TS models by providing local asymptotic conditions at the price of computationally demanding. For a complete review of them, the reader may refer to [34] and references therein. The method employed in this paper is based on [35], which has tried to decrease conservativeness seen in other works.

ð12Þ

Lemma 2. [18] For any positive-definite matrix II with appropriate dimensions, the following property holds: XT Y þ Y T X r XT ΠX þY T Π

307

ð15Þ

In order to be able to state the results as LMIs, it is needed to select M2 ¼ αM1 ¼ M, where α is an arbitrary constant for which a

308

H. Moodi, M. Farrokhi / ISA Transactions 53 (2014) 305–316

simple possible choice would be α ¼ 1. According to Lemma 2

μ

2 T 1 e ðtÞeðtÞ 

Δ Δ40 T

ð22Þ

Then, based on S-procedure [36] if r

r

∑ ∑ ωi ðzÞωj ðz^ Þ½ xT ðtÞ

i¼1j¼1

where 2

Ξ ij ¼ 4

eT ðtÞ

2

xðtÞ

Using the following theorem is helpful when some constraint on the input signals can be assumed.

3

6 eðtÞ 7 T 6 7 e_ T ðtÞ Δ Ξ ij 6 _ 7 o 0 4 eðtÞ 5

ð23Þ

Δ Ξ 11 ij

n

Ξ 21 Ξ 22 ij ij

3 5

T Ξ 11 ij ¼ ε½ðAi  Bi Kj Þ P1 þ P1 ðA i  Bi Kj Þ

ðBi Kj ÞT P1 0 0 T Ξ 21 ij ¼ ε½ 2

6 6 Ξ 22 ij ¼ 4

_ ρ P2ρ  MAj  ATj MT þ Y j C þ CT Y j þ μ21 I ∑rρ ¼ 1 ω T

n

P2j þ M  MAj þ Y j C

MþM

 MT

 MT

3

n

T

7 n 7 5 I

then V_ o 0. Based on, (18) and considering Theorem 4.3 in [35], it 22 results that Ξ ij o0. By selecting ε small enough, and pre- and post-multiply (17) by Q 1 1 ¼ P1 , and based on Lemma 3, it can be written

Ξ11 ij o0 12 T 11  1 12 Ξ22 Ξij  o 0 ij  ε½ðΞij Þ ðΞij Þ

Remark 2. Note that although Theorem 1 is rather conservative because of assumption (16), it states a simple solution. Moreover, the observer and controller gains can be determined separately if μ1 does not depend on the control input. It should be noted that assumption (16) is a mild condition when trying to design an observer for fuzzy systems with unmeasurable premise variables. According to [7], this condition is satisfied if ωi ðzÞ is differentiable w.r.t x(t) almost everywhere and has a bounded first derivative for almost all x that is satisfied by most membership functions in practice. Note that for ∥Δ∥ we have r

∥Δ∥ r ∑ ∥ðωi ðzÞ  ωi ðz^ ÞÞðAi x þ Bi uÞ∥ ð25Þ

i¼1

States of a system are usually bounded, so based on the Lipschitz condition of ωi ðzÞ and if ∥uðtÞ∥ r μ2 8 t 4 0 then r

∥Δ∥ r ∑ μ3 ðμ4 ∥Ai ∥ þ μ2 ∥Bi ∥Þ∥ðz  z^ Þ∥

ð26Þ

i¼1

Hence, it is always possible to assume a value for μ1 independent of the control input such that (16) is satisfied. In order to obtain the best value for μ1 , Theorem 1 can be stated as an eigenvalue problem as follows: maximize μ1 subject to ð17Þ and ð18Þ

Proof. See [11]. When the initial condition xð0Þ is not known, Ref. [11] provides a solution. The interested reader may refer to this reference for more details. It is obvious that modeling the system with a common output matrix (C) is not always a proper choice, because linearizing output nonlinearities may yield different output models. In the following section, a solution will be proposed to this problem by keeping the output nonlinearities in the fuzzy model.

4. Nonlinear consequent case

Remark 1. In Theorem 1, there are r 2 LMIs for the controller and 2r LMIs for the observer.

i¼1 r

Theorem 2. [11] Assume that the initial condition xð0Þ is known. Then, the constraint ∥uðtÞ∥ r μ2 8 t 4 0 is enforced if the following LMIs hold: " # 1 xð0ÞT Z0 xð0Þ Q1 " # ð28Þ Q 1 XTi Z0 i ¼ 1; …; r Xi μ22 I

ð24Þ

According to Lemma 1, it can be concluded that Ξ ij o 0. This shows that V_ o 0, which implies that the augmented dynamic is asymptotically stable.

r ∑ ∥ðωi ðzÞ  ωi ðz^ ÞÞ∥ð∥Ai ∥∥x∥ þ ∥Bi ∥∥u∥Þ

If the solution of (27) satisfies (16), then the obtained μ1 is acceptable; otherwise, there is no solution for Theorem 1 and hence, the states of the observer, designed based on (27), may not converge to the system states.

ð27Þ

As complexity of the system increases, the number of rules in the fuzzy model and hence, the number and dimensions of LMIs (used for the stability analysis) increases and becomes harder to solve. One possible solution is to use nonlinear local subsystems for the T–S model. This will decrease the number of rules while increasing the model accuracy. Researchers have made fewer attempts in this area. A very simple form of these nonlinear T–S model is used by Rajesh and Kaimal in [37], where they have used linear form plus a sinusoidal term for the consequence part. A more advanced work is performed by Dong et al. [38,39], where they employed sectorbounded functions in the subsystems. In [40,41], Tanaka et al. have proposed a T–S model with polynomial subsystems. For stability analysis, they used Sum of Squares (SOS) approach. This was the first use of SOS instead of LMI in fuzzy systems analysis. Sala and Ario [42] and Sala [43] have represent a similar form of the Sugeno model and have used the Taylor series expansion of the system for construction of the polynomial subsystems. The authors state that the nonlinear consequent part in T–S model not only reduces the number of rules but also reduces the conservativeness in the controller design. The observer design for continuous-time systems, with unmeasured premise variables, with and without uncertainties is proposed in [44,45], respectively. However, the controller design is not considered in these papers. A robust observer-based controller for this class of systems is considered in [46]. Nevertheless, the premise variables must be measurable. In contrast to the aforementioned papers, in this manuscript, an observer-based controller for fuzzy systems with unmeasured premise variables in presence of local nonlinear rules is designed based on the fuzzy Lyapunov function. To achieve this goal, local nonlinear rules are employed to design T–S models with common output structure. Moreover, some conditions on the control input are derived for this model. The proposed method reduces the complexity of the problem of designing observer-based controller with unmeasured premise variables drastically. In the followings, the main contributions of this paper will be given.

H. Moodi, M. Farrokhi / ISA Transactions 53 (2014) 305–316

Consider the following nonlinear system: _ ¼ f a ðxðtÞÞ þ f b ðxðtÞÞφðxðtÞÞ þgðxðtÞÞuðtÞ xðtÞ yðtÞ ¼ f ya ðxðtÞÞ þ f yb ðxðtÞÞφðxðtÞÞ

ð29Þ

where xðtÞ is the state vector, uðtÞ is the control input vector, yðtÞ is the measurable output and φðxðtÞÞ ¼ ½φ1 ðxðtÞÞ; …; φs ðxðtÞÞ is a vector of nonlinear functions satisfying the Lipschitz condition ^ ^ ∥φi ðxðtÞÞ  φi ðxðtÞÞ∥ r θi ∥R i ðxðtÞ  xðtÞÞ∥

1rirs

ð30Þ

where θi are Lipschitz constants and Ri are constant matrices with appropriate dimensions and s is the number of nonlinear functions. 2 2 2 T T Define R :¼ ½R T1 R 2 … R s  and θ :¼ diag½ θ1 I1 θ2 I2 … θs Is , where Ii is an identity matrix whose dimension (ni  ni ) is equal to the number of rows of R i . Moreover, it is assumed that these nonlinear functions are sector bounded, i.e.,

φi ðxðtÞÞ A cof0; Ei xðtÞg;

1 r i rs

ð31Þ

where Ei are constant vectors with appropriate dimensions and define T T T E : ¼ ½ E1 E2 … Es  for later use. For this system, a fuzzy model with common output model can be represented as _ ¼ Az xðtÞ þ Gxz φðxðtÞÞ þ Bz uðtÞ xðtÞ yðtÞ ¼ CxðtÞ þ Gy φðxðtÞÞ

ð32Þ

stable if for some fixed ξ A ½1; …; r, there exist positive definite and symmetric matrices P2i A Rnn ði ¼ 1; :::; rÞ and Q A Rnn and matrices M A Rnn , Xai A Rnu n , Xbi A Rsn , Y i A Rnny ð1 r ir rÞ and X A Rnn 1 and scalar   η 4 0, Λ ¼ diag½γ 1 ⋯ γ s ss ; γ i 40 and Γ ¼ diag λ1 ⋯ λs  1 , λi 4 0 such that the following inequalities hold: ‖ΨðtÞ‖ r η1 ‖eðtÞ‖ 8 ρ A ½1; …; r  ξ

P2ρ þX  P2ξ Z 0

Ξj o 0; 1 r1

ð40Þ

ð33Þ

and the augmented dynamic would be #" # " # " _ Az þ Bz Kaz^  Bz Kaz^ xðtÞ xðtÞ ¼ _ 0 Az^  Lz^ C eðtÞ eðtÞ " #" # # "  Bz Kbz^ Gxz þBz Kbz^ φðxðtÞÞ 0 þ 0 Gxz^  Lz^ Gy φe ðxðtÞÞ þ ΨðtÞ

Θii o 0; 1oior Θii þ 12ðΘij þ Θji Þ o 0; 1 o ia j o r

where "

Θij ¼ 2 6 6 Ξj ¼ 6 6 6 4

HeðAi Q 1 þ Bi Xaj Þ

n

ΓGTxi þ XTbj BTi þ EQ 1

 2Γ

^ eðtÞ ¼ xðtÞ  xðtÞ ΨðtÞ : ¼ Azz^ xðtÞ þ Gxzz^ φðxðtÞÞ þ Bzz^ uðtÞ ^ φe ðxðtÞÞ ¼ φðxðtÞÞ  φðxðtÞÞ Defining Γ

φðxðtÞÞ Γ T

1

h

φðxðtÞÞ ¼ λ

: ¼ diag½λ1 ⋯λs 

E xðtÞ  φðxðtÞÞ Γ T

1 1

s

1 s

φ1 ðtÞ⋯λ 1

¼ ∑ λi i¼1

λ

then,

T

E1 xðtÞ

3

1

ðφi ðtÞEi xðtÞ  φ2i ðtÞÞ ¼ ∑ λi i¼1

T

3

n

n

n

n 7 7

P2j þ M  MAj þ Y j C

MþM

 GTxj MT þ GTy Y j T

 GTxj MT þ GTy Y j T

Λ

 MT

 MT

0

7 7

n 7 5

I

ð44Þ

ρ ¼ 1 ϕρ ðP2ρ þ X  P2ξ Þ in which ρaξ

R and θ are the same as in (30), ΨðtÞ is defined in (35), the observer gains are 1rirr

Kai ¼ Xai Q 1  1 ; Kbi ¼ Xbi Γ

ð45Þ

1

1 o i or

ð46Þ

Proof. For the augmented dynamic (34), consider the Lyapunov   x r where xa ¼ and function V ¼ xTa Σ ρ ¼ 1 ωρ ðz^ ÞPρ xa , e " # εP1 0 Pρ ¼ . Then, the derivative of the Lyapunov function is 0 P2ρ

7 i6 s 6 E2 xðtÞ 7 1 2 7 φs ðtÞ 6 6 ⋮ 7  ∑ λi φi ðtÞ 4 5 i¼1 Es xðtÞ s

n

r

r

ρ¼1

ρ¼1

r

r

_ ρ ðz^ ÞPxa þ 2xTa ∑ ωρ ðz^ ÞPx_ a V_ ¼ xTa ∑ ω

1

2

ð43Þ

and the controller gains are ð34Þ

ð35Þ 1 1 1 ⋯ s 

¼ diag½λ

ð42Þ

#

P~ 2ϕ  MAj  ATj MT þ Y j C þ CT Y j þ RT θΛR þ ηI

Li ¼ M  1 Y i

where

1

ð41:bÞ

where η ¼ η21 , and P~ 2ϕ ¼ 7 ϕξ X þ∑r

^ þ Kbz^ φðxðtÞÞ ^ uðtÞ ¼ Kaz^ xðtÞ

1

ð41:aÞ

1ojor

Suppose that the premise variable z is unmeasurable. Then, the fuzzy controller would be _^ ¼ A xðtÞ ^ ^ þ Gxz^ φðxðtÞÞ xðtÞ þ Bz^ uðtÞ þ Lz^ ðyðtÞ  yðtÞÞ z^ ^ ^ ^ ¼ CxðtÞ ^ þ Gy φðxðtÞÞ yðtÞ

309

_ ρ ðz^ ÞPxa þ 2xTa ∑ ωρ ðz^ ÞPx_ a ¼ xTa ∑ ω ρ¼1

h

ρ¼1

þ 2 e ðtÞM1 þ e_ T ðtÞM2 "

fφi ðtÞðEi xðtÞ  φi ðtÞÞg

T

i

#

r

 e_ ðtÞ  ∑ ωj ðz^ ÞðAj  Lj CÞeðtÞ ðGxz^  Lz^ Gy Þφe ðxðtÞÞ  ΨðtÞ j¼1

ð36Þ

ð47Þ

According to (31), it can be written

φi ðtÞ½Ei xðtÞ  φi ðtÞ Z 0 for 1 r i rs

ð37Þ

Combining (37) and (36), it yields

φðxðtÞÞ Γ T

1

ExðtÞ  φðxðtÞÞ Γ T

1

φðxðtÞÞ Z 0

η

ΨT Ψ 4 0

ð48Þ

ð38Þ

On the other hand, since φðxðtÞÞ satisfies the Lipschitz condition (30) and Λ ¼ diag½γ 1 ⋯γ s  ss ; γ i 4 0, it immediately results that eT ðtÞR T θΛReðtÞ  φTe ðtÞΛφe ðtÞ Z 0

According to (40) 2 T 1 e ðtÞeðtÞ 

ð39Þ

The following theorem provides conditions for stability of system (34) based on fuzzy Lyapunov function. _ ρ ðzÞj r ϕρ for known positive real numbers Theorem 3. Assume jω ϕρ and ρ A ℜ. Then, the augmented dynamic (34) is asymptotically

Then, based on (38), (39) and (48) and S-Procedure Lemma if 2 3 xðtÞ 6 7 6 φðxðtÞÞ 7 6 7 6 7 r r 6 eðtÞ 7 T ∑ ∑ ωi ðzÞωj ðz^ Þ½ xT ðtÞ φT ðxðtÞÞ eT ðtÞ e_ T ðtÞ φTe ðxðtÞÞ Ψ ðtÞΞ ij 6 7o 0 _ 6 eðtÞ 7 i¼1j¼1 6 7 6 φ ðxðtÞÞ 7 4 e 5

ΨðtÞ

310

H. Moodi, M. Farrokhi / ISA Transactions 53 (2014) 305–316

where 2

Ξij ¼ 4

n

Ξ

Ξ22 ij

2

12 ij

4 Ξ11 ij ¼ ε "

Ξ ¼ε 12 ij

2 6 6 22 Ξij ¼ 6 6 6 4

Remark 6. As in the linear case, a constraint can be defined on the input. This will help to satisfy (40) and to reduce the controller gains. The following theorem is useful in this regard.

3

Ξ

11 ij

5

ðAi Bi Kaj ÞT P1 þ P1 ðAi Bi Kaj Þ

n

ðGxi þ Bi Kbj ÞT P1

0

 ðBi Kaj ÞT P1

0

0

0

 ðBi Kbj ÞT P1 0

_ ρ P2ρ  MAj  A ∑ρ ¼ 1 ω r

T

MTj

6 6 þ6 6 4

Γ  1 E 2Γ  1

n

n

M þ MT

n

 GTxj MT

T

þ GTy Y j T

M

R θΛR þ ηI

0

0

0

0

0

0 0

0 0

Λ 0

T

Theorem 4. Assume that the initial condition xð0Þ is known. Then, the constrained ∥uðtÞ∥ r μ2 8 t 4 0 is enforced if the following LMIs hold: " # 1 xð0ÞT Z0 ð51Þ xð0Þ Q1



n

0

#T

þ Yj C þ C Yj

þ GTy Y j T

M

2

5þ



0 T

P2j þ MT  MAj þ Y j C  GTxj MT

0

3

0

T

n

3

0

7 n7 7 7 n7 5

0

0

3

7 0 7 7 0 7 5 I

Then, V_ o 0. Based on (44) and based on the proof of Theorem 22 4.3 in [35] it results that Ξ ij oh0. By selecting i ε small enough and 1 pre- and post-multiply (43) by Q  1 Γ , and based on Lemma 3, it can be written

Ξ11 ij o 0 12 T 11  1 12 Ξ22 Ξij  o0 ij  ε½ðΞij Þ ðΞij Þ

ð49Þ

Based on Lemma 1, it results that Ξ ij o 0. This shows that V_ o 0, which implies that the augmented dynamic is asymptotically stable.□ Remark 3. Note that the nonlinearity φðxðtÞÞ should be selected such that it yields constant output matrices.

2

n

Q1 6 EQ 1 4 Xai

3

n

2Γ Xbi

n 7 5 Z 0 i ¼ 1; …; r

Proof. Assume that V ¼ xT P1 x is a Lyapunov function. Based on (51) and Schur complement Lemma, it yields xð0ÞT P1 xð0Þ o 1. Then, based on ∥uðtÞ∥ r μ2 , it can be written " T # r r Kai uðtÞT uðtÞ ¼ ∑ ∑ hi ðzðtÞÞhj ðzðtÞÞ½ xT φðxÞT  KTbi i¼1j¼1 " # x ½ Kaj Kbj  ð53Þ r μ22 φðxÞ For V_ o 0 we must have xðtÞT P1 xðtÞ o xð0ÞT P1 xð0Þ o 1. Hence, if " T # Kai 1 r r T T x φ ðxÞ ∑ ∑ h ðzðtÞÞh ðzðtÞÞ½  j μ22 i ¼ 1 j ¼ 1 i KTbi " # x ½ Kaj Kbj  ð54Þ r xðtÞT P1 xðtÞ φðxÞ then (53) holds. Therefore, we have r

r

∑ ∑ hi ðzðtÞÞhj ðzðtÞÞ½ x

T

i¼1j¼1

"

Remark 4. Although Theorem 3 is rather conservative, because of assumption (40), it states a simple solution. Moreover, the observer and controller gains can be determined separately if η1 does not dependent on the control input.



As mentioned for the linear case, condition (40) is satisfied if

ωi ðzÞ is differentiable w.r.t x(t) almost everywhere and have a

bounded first derivative for almost all x(t), which can be satisfied by most membership functions in practice. It is possible to find a proper value for η1 independent of the control input such that (40) is satisfied; however, LMI (41) may not be feasible with this η1 . Therefore, for the observer design, a fuzzy Lyapunov function is considered to reduce the conservativeness and to obtain larger values for η1 . In order to find the best value for η1 , Theorem 1 can be stated as an eigenvalue problem as follows: maximize η1 subject to ð41Þ and ð42Þ

ð50Þ

If the solution of (50) satisfies (40), then the obtained η1 is acceptable; otherwise, there is no solution for Theorem 3 and hence, the states of the observer designed based on (50) may not converge to the system states. Remark 5. It is also possible to use the same Lyapunov function for all subsystem so that the terms containing the derivative of membership functions are omitted from LMIs. Using a common Lyapunov function, it gives _ ρ P2 ¼ P2 ∑rρ ¼ 1 ω _ρ ¼0 ∑rρ ¼ 1 ω Hence, In this case, P~ 2ϕ in (44) should be replaced by zero.

ð52Þ

μ22 I

φðxÞ  T

1

x

 ½ Kaj

Kbj  

r0

P1

0

0

0

!

ð55Þ

r

∑ ∑ hi ðzðtÞÞhj ðzðtÞÞ½ x

i¼1j¼1



#

#

φðxÞ

"

KTai

μ22 KTbi

According to (38), if r

"

0

P1 Γ

1

E

2Γ

1

T

φðxÞ 

#!"

T

x

φðxÞ

#

1

"

KTai

μ22 KTbi

# ½ Kaj

r0

Kbj  ð56Þ

then (55) holds. Based on the same procedure as in the proof of Theorem 2: " T # r r 1 Kai K ½ aj Kbj  ∑ ∑ hi ðzðtÞÞhj ðzðtÞÞ½ xT φðxÞT  2 μ2 KTbi i¼1j¼1 # " #!" x P1 0  1 1 φðxÞ  Γ E 2Γ " T # r 1 Kai T T φðxÞ  2 T ½ Kai Kbi  r ∑ hi ðzðtÞÞ½ x μ2 Kbi j¼1 # " #!" x P1 0  1 1 φðxÞ  Γ E 2Γ Therefore, if " T # " P1  1 Kai  K K  ai bi 1 T 2 μ2 Kbi Γ E

0 2Γ

1

# o0

ð57Þ

H. Moodi, M. Farrokhi / ISA Transactions 53 (2014) 305–316

311

Table 1 Parameter values of the Buck converter.

Fig. 1. Equivalent circuit of a basic buck converter.

then (55) holds. Pre- and post-multiplying (52) by ½ Q 1 1 and its transpose, it gives 2 3 Q1 1 n n 6 7 6  Γ  1 E 2Γ  1 n 7 4 5 Z0 Kai Kbi μ22 I

Γ1 I 

Parameter

Value

Unit

Input voltage, Vin(t) Inductance current, iL(t) Inductance, L Parasitic resistance of L, RL Capacitance, C Parasitic resistance of C, Rc Resistance of switch, RM Diode voltage, VD Load resistance, R

30 [  8, 8] 98.58 48.5 202.5 162 0.27 0.82 6

V A mH mΩ mF mΩ Ω V Ω

where 2

ð58Þ

A1 ¼ A 2 ¼ 4 "

Based on the Schur complement, it yields " T # " # 0 P1  1 Kai  Kai Kbi  1 1 o0 μ22 KTbi  Γ E 2Γ

B1 ¼ ð59Þ

h i c  1L RL þ ðRRR þ Rc Þ R CðR þ Rc Þ

h  1L RM i L  V in  V D

C1 ¼ C2 ¼

h

0 RRc ðR þ Rc Þ

R ðR þ Rc Þ

i

R  LðR þ Rc Þ

 CðR 1þ Rc Þ i# " ;

B2 ¼

3 5; h i#  1L RM iL  V in  V D 0

;

:

The membership functions are selected as follows: This completes the proof.

h1 ¼

5. Simulation results In this section, some practical examples will be provided to show the effectiveness of the proposed method. First, a DC voltage regulator will be considered using the linear T–S system. Next, the ball and plate system is considered for the proposed nonlinear T–S system.

5.1. Linear case Example 1. In this example, the proposed observer-based controller design is applied to a DC voltage regulator (Fig. 1) to illustrate the feasibility of the controller design and to demonstrate its good performance. Along with rapidly grown electronic technology, DC–DC switching converters are widely used in DC power supplies and DC motor drive applications. The output voltage of the DC–DC converters must be regulated to a desired level in the presence of output load and input voltage fluctuations. The dynamic equation of the basic buck converter can be obtained as follows [47,48]: i 3" # " # 2 1h R c  LðR þ  L RL þ ðRRR x ðtÞ x_ 1 ðtÞ Rc Þ þ Rc Þ 4 5 1 ¼ R x2 ðtÞ x_ 2 ðtÞ  CðR 1þ Rc Þ CðR þ Rc Þ " þ y¼

h

RRc ðR þ Rc Þ

 1L ½RM x1 ðtÞ V in  V D  0

#

"

uþ

 VLD

#

0

i R ðR þ Rc Þ xðtÞ

where xðtÞ ¼ ½ iL ðtÞ vC ðtÞ  and u(t) is the duty ratio of the power MOSFET and R and Vin are uncertain parameters satisfying R A ½R; R and V in A ½V in ; V in . It is assumed that the inductor current belongs to the compact set iL A ½iL ; i L . Hence, this system can be exactly represented by the following two-rule fuzzy model: Plant Rule i :

IF x1 is M i1 _ ¼ Ai xðtÞ þ Bi uðtÞ THEN xðtÞ yðtÞ ¼ Ci xðtÞ

 iL ðtÞ  iL iL  i L

; h2 ¼ 1  h1 ; ϕ1 ¼ ϕ2 ¼ 100

Table 1 gives the parameters value of the buck converter. In this example, the objective is to make the output voltage of the buck converter to follow a desired signal. This reference signal is as follows: 8 > < rðtÞ ¼ 10 0 r t o10 rðtÞ ¼ 5 10 rt o 20 > : rðtÞ ¼ 20 20 rt o 30 First, for each set point, x and u should be found that are the equilibrium points of the state and control input, respectively. Then, the total control input to the system would be ut ¼ u þ u, where u is designed based on the proposed methods. To design a tracking controller for this system, different methods are given in literature. In many cases such as [47], a Hall sensor is used to measure the load current while here, this sensor is ommited. When the load current is assumed unmeasurable, to avoid the difficulty of observers with unmeasured premise, an output feedback controllers is used such as in [48]. Here, the observer and controller gains are obtained based on Theorem 1. Theorem 2 is also used to constraint the input. Hence, the gains are obtained as      2209:7 2206:7 L1 ¼ ; L2 ¼ 3549:5 3550:8 K1 ¼ ½  0:2985  0:1703; K2 ¼ ½  0:5876  0:3471; P21 ¼



3571:7  2206:3

 ;

 2206:3 7121:1    3565:6  2244:5 0:4717  0:3646 ; Q1 ¼ P22 ¼  2244:5 6880:8 0:3646 0:5900     8765:7  675:1  0:2218 0:3423 X¼ ; M¼  675:1 9543:6  0:3643  0:1082 

Figs. 2 and 3 show the simulation results of the state variables of the method proposed in this paper and the method in [48]. The output feedback controller gains based on [48] are K1 ¼ ½  6:0943; K2 ¼ ½ 7:1963:

312

H. Moodi, M. Farrokhi / ISA Transactions 53 (2014) 305–316

Fig. 2. State trajectory by method proposed here (Solid line) and method proposed in [48] (Dotted line) and its desired value (Dashed line) for DC–DC converter.

Fig. 3. State trajectory by method proposed here (Dashed line) and method proposed in [48] (Dotted line) and its desired value (Solid line) for DC–DC converter.

Fig. 4. State estimation error for DC–DC converter.

In order to show the performance of the buck converter using the proposed observer-based controller in this paper and compare it with the method in [48], some variations in the load and the

input voltage are considered. The input voltage is changed from 30 to 25 at time instant 2 s and back to 30 at 2.02 s. The load is changed from 6 to 5 at time instant 5 s and back to 6 at 5.02 s. As

H. Moodi, M. Farrokhi / ISA Transactions 53 (2014) 305–316

313

Fig. 5. Control input of method proposed here (dashed line) and method proposed in [48] (dotted line) for DC–DC converter.

Fig. 6. Satisfaction of condition (16) for DC–DC converter.

Fig. 2 shows, the proposed method exhibits shorter settling times, smaller overshoots, and nearly-zero steady-state errors as compared to the method in [48]. Moreover, the method in [48] creates very large spikes in state variable x1 (i.e., the current in inductance L), which may damage it. However, the method in [48] is more robust against changes in the input voltage and the load. Fig. 4 shows the state estimation error of the proposed observer in this paper. Fig. 5 shows comparison of the control input for the method proposed here and that of [48]. As this figure shows, the method in [48] creates control inputs with very large spikes when the desired value changes while the proposed method in this paper has reasonable changes in the control input with constant value for the steady states. Fig. 6 shows that condition (16) is also satisfied during the entire time of simulations. 5.2. Nonlinear case Example 2. As another practical case, the ball-and-plate system is considered here [39]. For simplicity, it is assumed that the mutual interactions between two coordinate axes are negligible. This assumption is made just to simplify the design procedure. However, the results are applied to the main coupled system in the

simulations. Due to the symmetry of x and y directions, only the x direction is discussed here. The procedure for the y-axis is exactly the same. The state-space model of the system can be expressed as 3 2 3 2_ 3 2 x1 0 x2 6 x_ 2 7 6 bðx x2  g sin x Þ 7 6 0 7 1 4 3 7 6 7 6 7 6 7 þ 6 7ux þ ω 6_ 7¼6 5 405 4 x3 5 4 x4 x_ 4 0 1  y ¼ ½x1 ; x3 T þ 1

0:1

T

υ

where, b¼0.7143, g¼ 9.81 m/s2 and x ¼ ½ x1 x2 x3 x4 T ¼ ½ x x_ θ θ_ T , in which x is the position of the ball along the xaxis and θ is the angle of the plate measured from the x-axis, ω represents unmolded dynamics, which in this case is ω ¼ ½ 0 Bðx4 x1y x4y Þ 0 0 T , where xiy represents the corresponding state in y-axis, and υ represents the measurements noise, which in this case is a band-limited white noise with the maximum value of 0.01. Assume that x1 x4 A ½  d1 ; d1 , then this system can be modeled by a T–S model with at least four linear rules or by only two nonlinear rules. It is known that sin ðx3 Þ A cof2=π x3 ; x3 g. If we select φðxðtÞÞ ¼ sin ðx3 Þ  2=π x3 , then φðxðtÞÞ A cof0; 2=π x3 g.

314

H. Moodi, M. Farrokhi / ISA Transactions 53 (2014) 305–316

Therefore, the T  S model would be as follows: Plant Rule i :

IF x1 x^ 4 is μi1

_ ¼ Ai xðtÞ þ Gxi φðxðtÞÞ þ Bi uðtÞ; THEN xðtÞ

i ¼ 1; 2

yðtÞ ¼ Ci xðtÞ where 2

0

60 6 A1 ¼ 6 40 0

1

0

0

 2π bg

0 0

0 0

2

0

3

0

2

0

60 bd1 7 7 6 7; A 2 ¼ 6 40 1 5 0

3

6 bg 7 6 7 Gx1 ¼ Gx2 ¼ 6 7; 4 0 5

1

0

0

 2π bg

0 0

0 0

 bd1 7 7 7 1 5

0 0 2 3 0  607 1 0 6 7 B1 ¼ B2 ¼ 6 7; C1 ¼ C2 ¼ 405 0 0

0

54:0650

Kb1 ¼ 3:0714;  Ka2 ¼ 32:6825

111:9915

Kb2 ¼ 3:0714; 2 345:9332 6  40:8586 6 Q1 ¼ 6 4 12:7238

3

0

 Ka1 ¼ 17:2825

 15:2236

0

0

1

0

Γ ¼ 3:7915 2 95:5399 6  0:1567 6 P21 ¼ 6 4 0:2668



1

2

h1 ¼ ðx1 x4 þ d1 Þ=2d1 h2 ¼ 1  h1

6 0:3664 6 M¼6 4 0:2818

Based on Theorem 3, the observer and controller gains are 2 3 2 3 95:5633 0:0360 112:9353  0:1569 6 229:5124 7 6 3:1249 7 3:5172 7 6 6 270:5307 7 L1 ¼ 6 7L 2 ¼ 6 7; 4  31:0331 4  36:5934 228:9965 5 238:8560 5  103:1856

 121:7986

741:2249

 0:0017  0:9884

  16:8372 ;

 317:5357

  30:4810 ;

 566:5426 12:7238

 15:2236

106:1189

17:0386

14:3696

17:0386

5:7529

7 7 7;  18:2923 5

14:3696

 18:2923

377:6688

 0:1567

0:2668

0:8765  0:2013

 0:2013 190:1011

0:0011

 0:0017

0:0011 7 7 7;  0:1306 5

 0:1306

 0:0040

 0:0005

 0:1535

0:0159

 0:2294

 0:8217

 0:0002

η ¼ 0:3; ϕ1 ¼ ϕ2 ¼ 2

0:0033  0:0002

1.1

7 7 7  0:0025 5

 0:0014

 0:2998

0:0195

states of x direction

0.2

X1

X3

1 0.1

0.8

0.9

X1

0.8

0.6

0

0.1

0.2

0.3

0.4

0

0

0.1

0.2

0.3

0.4

0.4 0.2

X3

0

X4

-0.2 -0.4

X2 0

5

10

15

time Fig. 7. States (solid line) and their estimation (dashed line) for ball and plate system without input constraint.

1.2 1.1 1

states of x direction

X1

0.1

X3

1

0.8

0.05

0.9

X1

0.8

0.6

0

0.1

0.2

0.3

0.4

0

0

0.1

0.2

0.3

0.4

0.4 0.2

X3 0

X4 -0.2 -0.4

X2 0

3

0:0001

1.2 1

3

0:0048  0:0034 0:3611  0:1118 2 3 113:4836  0:1589 0:1139  0:0002 6  0:1589 0:8963  0:2256  0:0014 7 6 7 P22 ¼ 6 7; 4 0:1139  0:2256 199:6315  0:2998 5

766:8715

5

3

40:8586

10

time Fig. 8. States (solid line) and their estimation (dashed line) for ball and plate system with input constraint.

15

H. Moodi, M. Farrokhi / ISA Transactions 53 (2014) 305–316

315

10

Control Input

Control input without constraint Control input with constraint

5

0

-5

0

5

10

15

time Fig. 9. Control input with (dashed line) and without (solid line) constraint in Example 2.

0.5 ||

0.4

(t)||

|| 1e(t)||

A

0.3 0.2 0.1 0

0

1

2

3

4

5

6

7

8

9

10

0.5 ||

0.4

(t)||

|| 1e(t)||

B

0.3 0.2 0.1 0

0

1

2

3

4

5

6

7

8

9

10

Time Fig. 10. Satisfaction of condition (40) by designed observer of Example 2. (A) without input constraint, (B) with input constraints.

2

20:4758

6 0:0169 6 X ¼6 4 0:0988

 0:0009

0:0169

0:0988

 0:0009

 0:0043

 0:3671

0:0018

 0:3671

28:1530

0:0945

0:0018

0:0945

 0:0076

3 7 7 7 5

h i In this example, R ¼ ½ 0 0 1 0 , θ ¼ 1, E ¼ 0 0 π2 0 , and d1 ¼ 2.Fig. 7 shows convergence of the system states to zero in the x direction and their estimation based on the designed observer and controller. Fig. 8 shows the states and their estimation after applying a bound on the control input. This constraint control input is shown in Fig. 9. Fig. 10 shows how condition (40) is satisfied. As a comparison, this system is modeled by T–S with linear consequents based on [11]. In that case, the model has six rules with one measured and one non-measured premise variable. Based on Theorem 1 in [22], 33  22 ¼ 108 LMIs must be solved simultaneously. Authors could not find any feasible solution for this example with the method proposed in [22]. On the other hand, based on Theorem 3, proposed in this paper, only four LMIs for the controller and four LMIs for the observer are used, which clearly shows drastic reduction in the design complexity. Moreover, the number of fuzzy rules is reduced from four to just two.

6. Conclusion A Sugeno-type fuzzy observer-based controller with nonlinear local subsystems and unmeasurable premise variables for a class of continuous-time non-linear systems was proposed in this paper. By assuming unmeasurable premise variables, one can model

larger class of non-linear systems. Moreover, the use of nonlinear consequent for the T–S system reduces the number of rules in the model while allows to have constant output matrixes, which drastically simplifies the design. The controller stabilization and convergence of the estimation error were shown using Lyapunov stability theory and LMI formulation with much fewer numbers of LMIs in contrast to the existing methods in literature. Furthermore, to reduce conservativeness in the observer design, fuzzy Lyapunov function was employed. Simulation results showed effectiveness of the proposed method as compared to the recent methods proposed in well-established literature.

References [1] Castro JL, Delgado M. Fuzzy systems with defuzzification are universal approximators. IEEE Trans Syst Man Cybern Part B Cybern 1996;26:149–52. [2] Tanaka K, Sano M. On the concepts of regulator and observer of fuzzy control systems. In: Third IEEE international conference on fuzzy systems, IEEE world congress on computational intelligence, Orlando, Florida, USA; 1994. p. 767–72. [3] Xiao-Jun M, Zeng-Qi S, Yan-Yan H. Analysis and design of fuzzy controller and fuzzy observer. IEEE Trans Fuzzy Syst 1998;6:41–51. [4] Yan S, Sun Z. Study on separation principles for T–S fuzzy system with switching controller and switching observer. Neurocomputing 2010;73: 2431–8. [5] Chadli M, El Hajjaji A. Comment on Observer-based robust fuzzy control of nonlinear systems with parametric uncertainties. Fuzzy Sets Syst 2006;157: 1276–81. [6] Changa W-J, Wua W-Y, Ku C-C. H1 constrained fuzzy control via state observer feedback for discrete-time Takagi–Sugeno fuzzy systems with multiplicative noises. ISA Trans 2011;50:37–43.

316

H. Moodi, M. Farrokhi / ISA Transactions 53 (2014) 305–316

[7] Lendek Z, Lauber J, Guerra TM, Babuška R, De Schutter B. Adaptive observers for TS fuzzy systems with unknown polynomial inputs. Fuzzy Sets Syst 2010;161:2043–65. [8] Liu Y-J, Zhou N. Observer-based adaptive fuzzy-neural control for a class of uncertain nonlinear systems with unknown dead-zone input. ISA Trans 2010;49:462–9. [9] Palm R, Driankov D. Towards a systematic analysis of fuzzy observers. In: Eighteenth international conference of the North American Fuzzy Information Processing Society, New York, NY, USA; 1999. p. 179–83. [10] Akhenak A, Chadli M, Ragot J, Maquin D. Design of sliding mode unknown input observer for uncertain Takagi–Sugeno model. Mediterranean Conference on Control & Automation, Athens, 2007. p. 1–6. [11] Tanaka K, Wang HO. Fuzzy control systems design and analysis: a linear matrix inequality approach. Wiley; 2001. [12] Yoneyama J. H 1 filtering for fuzzy systems with immeasurable premise variables: an uncertain system approach. Fuzzy Sets Syst 2009;160:1738–48. [13] Yoneyama J. H 1 output feedback control for fuzzy systems with immeasurable premise variables: discrete-time case. Appl Soft Comput 2008;8:949–58. [14] Ichalal D, Marx B, Ragot J, Maquin D. Design of observers for Takagi–Sugeno systems with immeasurable premise variables: an L2 approach. In: Seventeenth IFAC World Congress. Elsevier, COEX, South Korea, 2008. p. 2768–2773. [15] Ichalal D, Marx B, Ragot J, Maquin D. State estimation of Takagi–Sugeno systems with unmeasurable premise variables. IET Control Theory Appl 2010;4:897–908. [16] Bergesten P. Observrs and controllers for Takagi–Sugeno fuzzy systems. PhD dissertation. Sweden: Orebro University; 2001. [17] Bergsten P, Palm R, Driankov D. Fuzzy observers. In: Tenth IEEE International Conference onFuzzy Systems, Melbourne; 2001. p. 700–3 vol. 3. [18] Bergsten P, Palm R, Driankov D. Observers for Takagi–Sugeno fuzzy systems. IEEE Trans Syst Man Cybern Part B Cybern 2002;32:114–21. [19] Ichalal D, Marx B, Maquin D, Ragot J. On observer design for nonlinear Takagi– Sugeno systems with unmeasurable premise variable. In: International Symposium on Advanced Control of Industrial Processes, Hangzhou, China; 2011. p. 353–8. [20] Yoneyama J, Nishikawa M, Katayama H, Ichikawa A. Design of output feedback controllers for Takagi–Sugeno fuzzy systems. Fuzzy Sets Syst 2001;121:127–48. [21] Sing Kiong N, Peng S. H 1 fuzzy output feedback control design for nonlinear systems: an LMI approach. IEEE Trans Fuzzy Syst 2003;11:331–40. [22] Guerra TM, Kruszewski A, Vermeiren L, Tirmant H. Conditions of output stabilization for nonlinear models in the Takagi–Sugeno's form. Fuzzy Sets Syst 2006;157:1248–59. [23] Tseng C-S, Chen B-S, Li Y-F. Robust fuzzy observer-based fuzzy control design for nonlinear systems with persistent bounded disturbances: a novel decoupled approach. Fuzzy Sets Syst 2009;160:2824–43. [24] Lian K-Y, Hui-Wen T, Chi-Wang H. Current sensorless regulation for converters via integral fuzzy control. IEICE Trans Electron 2007:507–14 (E90-C). [25] Asemani MH, Majd VJ. A robust H1 observer-based controller design for uncertain T–S fuzzy systems with unknown premise variables via LMI. Fuzzy Sets Syst 2013;212:21–40. [26] Ichalal D, Marx B, Ragot J, Maquin D. State and unknown input estimation for nonlinear systems described by Takagi–Sugeno models with unmeasurable premise variables. In: Seventeenth Mediterranean conference on control and automation, Thessaloniki, Greece; 2009. p. 217–222. [27] Ichalal D, Marx B, Ragot J, Maquin D. Design of observer for Takagi–Sugeno discrete-time systems with unmeasurable premise variables. In: Fifth workshop on advanced control and diagnosis, Grenoble, France; 2007.

[28] Lian K-Y, Che-Hsueh C, Hui-Wen T. LMI-based sensorless control of permanent-magnet synchronous motors. IEEE Trans Ind Electron 2007;54: 2769–78. [29] Anderson BDO, Moore JB. Optimal control. Englewood Cliffs, NJ: Prentice-Hall; 1989. [30] Tuan HD, Apkarian P, Narikiyo T, Yamamoto Y. Parameterized linear matrix inequality techniques in fuzzy control system design. IEEE Trans Fuzzy Syst 2001;9:324–32. [31] Guerra TM, Kruszewski A, Lauber J. Discrete Tagaki–Sugeno models for control: where are we? Annu Rev Control 2009;33:37–47. [32] Mozelli LA, Palhares RM. Avellar GSC. A systematic approach to improve multiple Lyapunov function stability and stabilization conditions for fuzzy systems. Inf Sci 2009;179:1149–62. [33] Rhee B-J, Won S. A new fuzzy Lyapunov function approach for a Takagi– Sugeno fuzzy control system design. Fuzzy Sets Syst 2006;157:1211–28. [34] Guerra TM, Bernal M. Strategies to exploit non-quadratic local stability analysis. Int J Fuzzy Syst 2012;14:372–9. [35] Faria FA, Silva GN, Oliveira VA. Reducing the conservatism of LMI-based stabilisation conditions for TS fuzzy systems using fuzzy Lyapunov functions. Int J Syst Sci 2012;0:1–14. [36] Yakubovich VA. S-procedure in nonlinear control theory. Vestn Leningr Univ; 1973. [37] Rajesh R, Kaimal MR. T–S fuzzy model with nonlinear consequence and PDC controller for a class of nonlinear control systems. Appl Soft Comput 2007;7:772–82. [38] Dong J, Wang Y, Yang GH. Output feedback fuzzy controller design with local nonlinear feedback laws for discrete-time nonlinear systems. IEEE Trans Syst Man Cybern Part B Cybern 2010;40:1447–59. [39] Dong J, Wang Y, Yang GH. Control synthesis of continuous-time T–S fuzzy systems with local nonlinear models. IEEE Trans Syst Man Cybern Part B Cybern 2009;39:1245–58. [40] Tanaka K, Ohtake H, Wada M, Wang HO, Ying-Jen C. Polynomial fuzzy observer design: a sum of squares approach. In: Proceedings of the 48th IEEE conference on decision and control, & 28th Chinese Control Conference, Shanghai, China; 2009. p. 7771–6. [41] Tanaka K, Yoshida H, Ohtake H, Wang HO. A sum-of-squares approach to modeling and control of nonlinear dynamical systems with polynomial fuzzy systems. IEEE Trans Fuzzy Syst 2009;17:911–22. [42] Sala A, Ario C. Polynomial fuzzy models for nonlinear control: a Taylor series approach. IEEE Trans Fuzzy Syst 2009;17:1284–95. [43] Sala A. On the conservativeness of fuzzy and fuzzy-polynomial control of nonlinear systems. Annu Rev Control 2009;33:48–58. [44] Moodi H, Farrokhi M. State estimation of fuzzy Sugeno systems with local nonlinear rules and unmeasurable premise variables. In: Seventeenth international conference on methods and models in automation and robotics Miedzyzdroje, Poland; 2012 p. 558–63. [45] Moodi H, Farrokhi M. Robust observer design for T–S systems with nonlinear consequent parts. In: Second international conference on computer and knowledge engineering, Mashad, Iran; 2012. [46] Moodi H, Farrokhi M. Robust observer-based controller design for T–S systems with nonlinear consequent Parts. In: Twenty-first Iranian conference on electrical engineering. IEEE, Mashhad, Iran; 2013. [47] Lian KY, Liou J-J, Huang C-Y. LMI-based integral fuzzy control of DC–DC converters. IEEE Trans Fuzzy Syst 2006;14:71–80. [48] Nachidi M, El Hajjaji A. Output tracking control for fuzzy systems via staticoutput feedback design. In: Iqbal S, Boumella N, Figueroa Garcia JC, editors. Fuzzy controllers, recent advances in theory and applications. InTech Open Access Publisher; 2012.