Fuzzy control of multiplicative noised nonlinear systems subject to actuator saturation and H∞ performance constraints

Neurocomputing 148 (2015) 512–520

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Fuzzy control of multiplicative noised nonlinear systems subject to actuator saturation and H1 performance constraints Wen-Jer Chang n, Ying-Jie Shih Department of Marine Engineering, National Taiwan Ocean University, Keelung 202, Taiwan, ROC

art ic l e i nf o

a b s t r a c t

Article history: Received 4 May 2014 Received in revised form 20 June 2014 Accepted 2 July 2014 Communicated by Cheng-Wu Chen Available online 23 July 2014

In this paper, a H1 performance constrained fuzzy control approach is investigated for a class of nonlinear stochastic systems subject to actuator saturation. The nonlinear stochastic systems considered in this paper are represented by Takagi–Sugeno fuzzy models with multiplicative noises. According to H1 performance constraint and actuator saturation, it is shown that the stabilization of multiplicative noised Takagi–Sugeno fuzzy models can be formulated as a convex optimization problem subject to linear matrix inequalities. The proposed fuzzy control method is accomplished based on the Lyapunov stability theory. Simulation study on a continuous-time nonlinear stochastic ship steering system is given to show the performances of the proposed H1 performance constrained fuzzy control methodology. & 2014 Elsevier B.V. All rights reserved.

Keywords: Takagi–Sugeno fuzzy models Multiplicative noises Actuator saturation H1 performance constraint

1. Introduction The H1 control problem plays a major role in performance constrained control for industrial plants. It becomes one of the most important issues in the fields of robust control techniques and engineering applications [1,2]. The external disturbance effect is often the source of instability and poor performance of systems. Thus, the disturbance attenuation performance investigated via H1 control method is worth to be considered. In recent years, stochastic H1 control theory has made great progress. For example, the stochastic H1 control problems with state multiplicative noises have been extensively studied due to successful applications of stochastic Itô's formula [3,4]. Different from the traditional additive noise, multiplicative noise is more practical since it allows the statistical description of the noise to be unknown a prior but depends on the control and state solution. Some characteristics of nonlinear systems can be closely approximated by models with multiplicative noises rather than by linearized models. For the stochastic systems with multiplicative noises, there have been several approaches developed to deal with the control design [5–7] and filter design [8–10] problems. It is worth emphasizing that the systems considered in [5,6] are represented by the Takagi–Sugeno (T–S) fuzzy models with multiplicative noises. In this paper, the T–S fuzzy model is also employed to represent the considered nonlinear stochastic systems with multiplicative noises.

n

Corresponding author. Tel.: þ 886 2 24622192x7110. E-mail address: [email protected] (W.-J. Chang).

http://dx.doi.org/10.1016/j.neucom.2014.07.012 0925-2312/& 2014 Elsevier B.V. All rights reserved.

The application of the T–S fuzzy models to describe and model nonlinear systems has attracted great attention in nonlinear control engineering [5,6,11–15]. Fuzzy control approach started in the 1970s trying to embed heuristics and reasoning into control loops to mimic human operator actions. In literature, it has been shown that a quite general class of nonlinear systems can be exactly transformed into the T–S fuzzy models [16]. This method is known as sector-nonlinearity fuzzy modeling technique [11]. In order to design a fuzzy controller for the T–S fuzzy models, a popular technique so-called Parallel Distributed Compensation (PDC) was developed in [11]. The main idea of the PDC concept is to derive each fuzzy control rule so as to compensate each plant rule of the fuzzy system. Thus, a global fuzzy controller can be constructed by the aggregation of the local compensators with fuzzy inference systems. In the PDC design, the linear feedback gains for each linear subsystem are first designed. Then, the overall system input can be blended by these linear feedback gains. Based on the PDC concept, some fuzzy control approaches [5,6,11–15, 17–20] have been successfully investigated for several nonlinear systems represented by the T–S fuzzy models. In recent years, many researchers have studied the fuzzy control problem for T–S fuzzy systems subject to actuator saturation [21–25]. In practice, physical capacity of the actuator is limited and the actuator saturation may severely degrade the performance of the closed-loop systems. It can be found that the actuator saturation usually leads to a large overshoot, induces a limit cycle and even makes the closed-loop system unstable. In general, the problem of actuator saturation is considered by either designing a low gain control law [26] or estimating the domain of attraction in the presence of actuator saturation [27]. For nonlinear

W.-J. Chang, Y.-J. Shih / Neurocomputing 148 (2015) 512–520

systems with actuator saturation, it is of great difficulty and challenge to design controllers, in that the saturation will generate a complex nonlinear closed-loop system. In order to overcome the above difficulty, T–S fuzzy models were employed in [21–25] to represent nonlinear systems. Though the fuzzy controller design problem subject to actuator saturation has been studied in [24,25] for T–S fuzzy models with multiplicative noises. To the best of our knowledge, there has been less works on studying the fuzzy control problem of T–S fuzzy models with multiplicative noises subject to actuator saturation and H1 performance constraint, simultaneously. Thus, the motivation of this paper is to develop a novel H1 performance constrained fuzzy control approach subject to actuator saturation for the nonlinear stochastic systems represented by T–S fuzzy models with external disturbance and multiplicative noises. It is worth pointing out that the work of this paper represents the first attempt to consider H1 performance constraint and actuator saturation for nonlinear stochastic systems with external disturbance and multiplicative noises. The contribution of this paper is to combine H1 control scheme with Lyapunov theory to develop novel fuzzy control approach subject to H1 performance constraint and actuator saturation constraint for the nonlinear stochastic systems with external disturbance and multiplicative noises. The H1 control scheme is used to eliminate the effect of external disturbance on the system. The Lyapunov theory is a main tool to analyze the stability and design T–S fuzzy controllers subject to actuator saturation. The Lyapunov-based stability conditions depend on the existence of a common positive definite matrix guaranteeing the stability conditions of all local subsystems. These stability conditions can be transformed to Linear Matrix Inequality (LMI) problem [28–30]. It is known that the LMI problem can be solved easily by using the convex optimization algorithm [28]. By solving the LMI problem developed in this paper, one can find a common positive definite matrix and feedback gains such that the T–S fuzzy model is stabilized subject to H1 performance constraint and actuator saturation. Referring to [31–32], some novel T–S fuzzy modeling techniques were developed in literature, recently. By combining the novel T–S fuzzy modeling techniques [31–32] with proposed fuzzy control approach, an extending fuzzy control approach can be developed to the applications of industrial control in the future. This paper is structured as follows. In Section 2, the problem statements and the descriptions of T–S fuzzy models with external disturbance and multiplicative noises are introduced. In Section 3, the sufficient stability conditions are derived based on Itô's formula and Lyapunov function. Applying the derived fuzzy controller design approach, the simulation results for the nonlinear dynamic ship position systems are stated in Section 4. Finally, some concluding remarks are made in Section 5.

513

a scalar zero mean white noise with variance one. By referring to the reference [33], the properties of EfwðtÞg ¼ 0, EfwðtÞxðtÞg ¼ EfwðtÞgfxðtÞg ¼ 0,EfwðtÞuðtÞg ¼ EfwðtÞgfuðtÞg ¼ 0, and EfwðtÞνðtÞg ¼ EfwðtÞgfνðtÞg ¼ 0 are assumed due to the independent increment property of Brownian motion. Besides, Ai , Ai , Bi , Bi , Ei and Ei are constant matrices with the compatible dimensions; i ¼ 1; 2; :::; r and r is the number of fuzzy model rules, ϕ1 ðtÞ; ϕ2 ðtÞ; :::; ϕp ðtÞ are known premise variables that may be functions of the state variables, M ij is the fuzzy set, p is the premise variable number. Definition 1. The actuator saturating is defined as follows: 8 > < ukL if uk o ukL if ukL o uk o ukH uk ðtÞ ¼ satðuk Þ ¼ uk > :u if ukH ouk kH

ð2Þ

where ukL o 0 o ukH and k ¼ 1; 2; :::; m. Considering the saturating actuator defined in Definition 1, one can formulate the following inequality from relations of uk ðtÞ defined in (2). ð3Þ

jjuðtÞjj Z jjuðtÞjj

Based on the inequality (3) and Remark 1 of [8], one has the following relation. 1ρ 1þρ jjuðtÞjj Z ‖uðtÞ  uðtÞ‖ 2 2

ð4Þ

where 0 o ρo 1. The sector parameter ρ is used to guarantee that the saturation map sat is inside the sector ðρ; 1Þ. With ukH Z ρuk and ukL r ρuk , the inequality (4) can be arranged as follows. ukL u ruk r kH ; ρ ρ

k ¼ 1; 2; :::; m

ð5Þ

In the paper, one can find that if ukH ¼  ukL is set then one has juk j r

ukH ρ

ð6Þ

Expanding the inequality (4), one can obtain the following inequality:  T     1þρ 1þρ 1ρ 2 T uðtÞ uðtÞ r uðtÞ  uðtÞ  u ðtÞuðtÞ ð7Þ 2 2 2 The inequality (7) is an important basis in the following derivations of this paper. Given a pair of ðxðtÞ; uðtÞÞ, the final output of the fuzzy system (1) is inferred as follows: x_ ðtÞ ¼

∑ri ¼ 1 ωi ðϕðtÞÞAi xðtÞþ Bi uðtÞ þEi νðtÞ þ Ai wðtÞxðtÞ þBi wðtÞuðtÞ þ Ei wðtÞνðtÞ ∑ri ¼ 1 ωi ðϕðtÞÞ r

¼ ∑ hi ðϕðtÞÞfAi xðtÞ þBi uðtÞ þ Ei νðtÞþ Ai wðtÞxðtÞ þ Bi wðtÞuðtÞþ Ei wðtÞνðtÞg i¼1

ð8Þ ¼ ∏nj¼ 1 M ij ϕj ðtÞ,

2. System descriptions and problem statements The nonlinear stochastic systems considered in this paper are represented by the T–S fuzzy models with external disturbance and multiplicative noises. The proposed T–S fuzzy models with external disturbance and multiplicative noises can be described as the following IF–THEN form: Plant Rule i: If ϕ1 ðtÞ is M i1 and ϕ2 ðtÞ is M i2 and… and ϕp ðtÞ is M ip Then x_ ðtÞ ¼ Ai xðtÞ þ Bi uðtÞ þEi νðtÞ þ Ai wðtÞxðtÞ þ Bi wðtÞuðtÞ þ Ei wðtÞνðtÞ ð1Þ nx

uy

where xðtÞ A ℜ is the state vector, νðtÞ A ℜ is the external disturbance input vector, uðtÞ ¼ ½u1 ðtÞ; :::; um ðtÞT ¼ ½satðu1 ðtÞÞ; :::; satðum ðtÞÞT ¼ satðuðtÞÞ A ℜm is the saturating control input, wðtÞ is

ωi ðϕðtÞÞ where ϕðtÞ ¼ ½ϕ1 ðtÞ; ϕ2 ðtÞ; :::; ϕp ðtÞ, hi ðϕðtÞÞ Z 0 and ∑ri ¼ 1 hi ðϕðtÞÞ ¼ 1. According to sector parameter ρ, the state Eq. (8) can be rewritten as n r x_ ðtÞ ¼ ∑ hi ðϕðtÞÞ Ai xðtÞ þ Bi uðtÞ þ Ei νðtÞ þ Ai wðtÞxðtÞ i¼1

 þBi wðtÞuðtÞ þ Ei wðtÞνðtÞ      r 1 þρ 1þρ uðtÞ þ Bi uðtÞ  uðtÞ ¼ ∑ hi ðϕðtÞÞ Ai xðtÞ þ Bi 2 2 i¼1   1þρ þEi νðtÞ þ Ai wðtÞxðtÞ þ Bi wðtÞ uðtÞ 2    1þρ uðtÞ þ Ei wðtÞνðtÞ þBi wðtÞ uðtÞ  2

ð9Þ

Considering the above T–S fuzzy model with external disturbance and multiplicative noises, the fuzzy controller employed in

514

W.-J. Chang, Y.-J. Shih / Neurocomputing 148 (2015) 512–520

this paper has the following form. It is constructed by the PDC concept [11]. Controller Rule i: If ϕ1 ðtÞ is M i1 and ϕ2 ðtÞ is M i2 and… and ϕp ðtÞ is M ip Then

Given a positive real number γ, a system is said to have L2 ½0;t f  gain less than γ if Z t p   Z tp  xΤ ðtÞSxðtÞdt r E γ 2 νΤ ðtÞνðtÞdt ; 8 νðtÞ a0 ð18Þ E

ð10Þ

for the continuous time system with zero initial condition for all νðtÞ A L2 ½0; t f  where t f is the terminal time of the control, γ is a prescribed value which denotes the worst case effect of νðtÞ on xðtÞ. Besides, S ¼ ST 4 0 is a positive definite weighting matrix and SA ℜnx nx .

uðtÞ ¼ Ki xðtÞ

where i ¼ 1; 2; :::; r and r is the number of fuzzy model rules, Ki A ℜmnx is a constant matrix. It can be found that the PDC-base fuzzy controller (10) shares the same IF part of the T–S fuzzy model (1). The output of the PDC-based fuzzy controller can be determined by the summation such as r

uðtÞ ¼ ∑ hi ðϕðtÞÞKi xðtÞ

ð11Þ

i¼1

By substituting (11) into (9), one has    r r 1þρ ðKj xðtÞÞ x_ ðtÞ ¼ ∑ ∑ hi ðϕðtÞÞhj ðϕðtÞÞ Ai xðtÞ þ Bi 2 i¼1j¼1   1þρ ðKj xðtÞÞ þEi νðtÞ þ Ai wðtÞxðtÞ þ Bi uðtÞ  2     1þρ 1 þρ ðKj xðtÞÞ þ Bi wðtÞ uðtÞ  ðKj xðtÞÞ þ Bi wðtÞ 2 2  þ Ei wðtÞνðtÞ r

Considering H1 performance constraint for the stochastic T–S fuzzy model represented in (1), the purpose of the paper is to find a PDC-based fuzzy controller (10) subject to actuator saturation defined in (2). To derive sufficient conditions for achieving actuator saturation and H1 performance constraints, some important preliminaries are presented as follows.

XT Y þ Y T X r δXT X þ δ  1 XT X where δ is any positive constant.

r

i¼1j¼1

ð12Þ

where R j ¼ uðtÞ  ðð1 þ ρÞ=2ÞKj xðtÞ, Aij ¼ Ai þ ðð1 þ ρÞ=2ÞBi Kj , and Aij ¼ Ai þ ðð1 þ ρÞ=2ÞBi Kj . Let an ellipsoid ϑ1 and a positive scalar function VðxðtÞÞ be defined as follows, respectively.  ϑ1 ¼ fxðtÞxT ðtÞPxðtÞ r1g and VðxðtÞÞ ¼ xT ðtÞPxðtÞ ð13Þ nx nx

denotes a positive definite matrix. An ellipsoid where P A ℜ ϑ1 , which is inside the domain of attraction, is said to be contractively invariant [8] if the following condition can be satisfied. _ VðxðtÞÞ o0; 8 xðtÞ A ϑ1 n f0g

0

Lemma 1. [36] For any two matrices X and Y, one has

¼ ∑ ∑ hi ðϕðtÞÞhj ðϕðtÞÞfðAij þ Aij wðtÞÞxðtÞ þ ðBi þ Bi wðtÞÞR j ðtÞ þ Ei νðtÞ þ Ei wðtÞνðtÞg

0

ð14Þ

From (6) and (11), the constraint juk j r ðukH =ρÞ can be inferred as follows:    r  u   ðkÞ kH ð15Þ  ∑ hi ðzðtÞÞðKi xðtÞÞ r i ¼ 1  ρ where KðkÞ denotes the kth row of Ki . It is obvious that if (15) holds i with hi ðzðtÞÞ ¼ 1, then one can define the following equation. (   )  T ukH 2 ðkÞ T ðkÞ  ϑ2 ¼ xðtÞ x ðtÞðKi Þ ðKi ÞxðtÞ r ð16Þ ρ

Lemma 2. [12] Given constant matrices Λ1 , Λ2 and Λ3 , where Λ1 ¼ ΛT1 and Λ2 ¼ ΛT2 4 0, then Λ1 þ ΛT3 Λ2 1 Λ3 o 0 if and only if " # ΛT3 Λ1 o0 Λ3  Λ2 or equivalently " #  Λ2 Λ3 o0 ΛT3 Λ1

Using the Lyapunov stability theory, Itô's formula and H1 control scheme, the stability conditions are derived in the next section. By solving these stability conditions, one can obtained feedback gains Ki subject to the constraint of actuator saturation (2).

3. Fuzzy controller design subject to actuator saturation and H1 performance constraint In this section, the stability condition is derived for closed-loop system (12) subject to actuator saturation and H1 performance constraint. First, let us define the Lyapunov function as follows that satisfies xð0Þ A ϑ1  ϑ2 and xðtÞ A ϑ1  ϑ2 , 8 t Z 0 VðxðtÞÞ ¼ xT ðtÞPxðtÞ

ð19Þ

Based on the Lyapunov function (19), one can obtain the following theorem for analyzing the stability of augmented system (12).

In this paper, it is required that xðtÞ A ϑ1  ϑ2 , i.e., ϑ1 is a subset of ϑ2 . The equivalent condition for xðtÞ A ϑ1  ϑ2 can be represented as follows:  2 u ÞP  1 ðKðkÞ ÞT r kH ð17Þ ðKðkÞ i i ρ

Theorem 1. If there exist a positive definite matrix P ¼ PT 4 0, feedback gains Ki , and prescribed value γ satisfying the following sufficient conditions, then the closed-loop T–S fuzzy system (12) with external disturbance is asymptotically stable in the mean square and satisfies H1 performance constraint subject to saturating actuator.

According to the above actuator saturation, this paper tries to find fuzzy controllers such that the T–S fuzzy system (1) satisfies the H1 performance constraint. The H1 performance constraint considered is this paper is described in the following definition.

"

Definition 2. [34,35]

Ξ ij o 0; for i ¼ 1; …; r; j ¼ 1; …; r ℓ1 n

1 þ ρ ðkÞ  1 2 Ki P 1

P

ð20Þ

# Z0

ð21Þ

where n denotes the transposed element in the symmetric position, KðkÞ denotes the kth row of Ki and i

W.-J. Chang, Y.-J. Shih / Neurocomputing 148 (2015) 512–520

" Ξ ij ¼

Ψ ij þ S

PEi

n

T 3Ei PEi γ 2

#

 þ3

ð22Þ

515

 1ρ 2 T T ðKj Bi PBi Kj Þ 2

It is obvious that Eq. (24) can be rewritten as Proof. Consider the Lyapunov function defined in (19). According to the Itô's formula [35] and x_ ðtÞ described in (12), one can obtain the following equation. n r r _ VðxðtÞÞ ¼ ∑ ∑ hi ðϕðtÞÞhj ðϕðtÞÞ ðAij þ Aij wðtÞÞxðtÞ i¼1j¼1

þ ðBi þ Bi wðtÞÞR j ðtÞ þ Ei νðtÞ: T þ Ei wðtÞνðtÞ PxðtÞ þ xT ðtÞPððAij þ Aij wðtÞÞxðtÞ þ ðBi þ Bi wðtÞÞR j ðtÞ þ Ei νðtÞ þ Ei wðtÞνðtÞ   þ ðAij xðtÞ þ Bi R j ðtÞ þ Ei νðtÞÞT P Aij xðtÞ þ Bi Rj ðtÞ þ Ei νðtÞ r r

T ¼ ∑ ∑ hi ðϕðtÞÞhj ðϕðtÞÞ ðAij xðtÞ þBi R j ðtÞ þ Ei νðtÞÞ PxðtÞ

r

r

_ VðxðtÞÞ r ∑ ∑ hi ðϕðtÞÞhj ðϕðtÞÞfxT ðtÞΓ ij xðtÞ þ 2xT ðtÞPðAij wðtÞxðtÞ i¼1j¼1

þ Bi wðtÞR j ðtÞ þ Ei wðtÞνðtÞg h where xT ðtÞ ¼ xT ðtÞ

ð25Þ "

Ψ ij i νT ðtÞ and Γ ij ¼ n

PEi T

3Ei PEi

# . Referring to

the independent increment property of Brownian motion [35], one has EfxðtÞwðtÞg ¼ EfxðtÞgfwðtÞg ¼ 0, EfR j ðtÞwðtÞg ¼ EfRj ðtÞgfwðtÞg ¼ 0 andEfwðtÞνðtÞg ¼ EfwðtÞgfνðtÞg ¼ 0 due to EfwðtÞg ¼ 0. Taking the expectation of (24), the following equation can be thus obtained. EfV_ ðxðtÞÞg r EfLV ðxðtÞÞg r

i¼1j¼1

r

¼ ∑ ∑ hi ðϕðtÞÞhj ðϕðtÞÞ½xT ðtÞΓ ij xðtÞdt

þ xT ðtÞP  ðAij xðtÞ þBi R j ðtÞ þ Ei νðtÞÞ þ ðAij xðtÞ þ Bi R j ðtÞ

ð26Þ

i¼1j¼1

þ Ei νðtÞÞT PðAij xðtÞ þ Bi Rj ðtÞ þ Ei νðtÞÞ

o þ 2xT ðtÞPðAij wðtÞxðtÞ þ Bi wðtÞR j ðtÞ þ Ei wðtÞνðtÞÞ n r r ¼ ∑ ∑ hi ðϕðtÞÞhj ðϕðtÞÞ xT ðtÞðATij P þ PAij ÞxðtÞ þ RTj ðtÞBTi PxðtÞ i¼1j¼1

þ xT ðtÞPBi R j ðtÞ þ νT ðtÞETi PxðtÞ þ xT ðtÞPEi νðtÞ   T  þ Aij xðtÞ þBi R j ðtÞ þ Ei νðtÞ P Aij xðtÞ þ Bi Rj ðtÞ þEi νðtÞ   ð23Þ þ 2xT ðtÞP Aij wðtÞxðtÞ þ Bi wðtÞR j ðtÞ þEi wðtÞνðtÞ

Substituting Rj ¼ uðtÞ  ðð1 þ εÞ=2ÞKj xðtÞ into (23), one can obtain the following relation according to (7) and Lemma 1. n r r V_ ðxðtÞÞ r ∑ ∑ hi ðϕðtÞÞhj ðϕðtÞÞ xT ðtÞðATij P þ PAij ÞxðtÞ þ RTj ðtÞR j ðtÞ i¼1j¼1

þ xT ðtÞPBi BTi PxðtÞ þ νT ðtÞETi PxðtÞ þ xT ðtÞPEi νðtÞ T

T

T

þ 3ðxT ðtÞAij PAij xðtÞ þ R Tj ðtÞBi PBi Rj ðtÞ þ νT ðtÞEi PEi νðtÞÞ o þ 2xT ðtÞPðAij wðtÞxðtÞ þ Bi wðtÞR j ðtÞ þ Ei wðtÞνðtÞ (   r r 1ρ 2 T K j Kj r ∑ ∑ hi ðϕðtÞÞhj ðϕðtÞÞ xT ðtÞ ATij P þ PAij þ 2 i¼1j¼1 !   T 1ρ 2 T T T þ PBi Bi P: þ 3Aij PAij þ 3 ðKj Bi PBi Kj Þ xðtÞ 2

r

r

∑ ∑ hi ðϕðtÞÞhj ðϕðtÞÞ½xT ðtÞΓ ij xðtÞdt

Considering the H1 performance constraint defined in Definition 2, one can define a performance function for nonexternal disturbance, i.e., νðtÞ a 0, such as Z t P  Jðx; ν; tÞ ¼ E ½xΤ ðtÞSxðtÞ  γ 2 νT ðtÞνðtÞdt 0 Z t P  Τ _ ½x ðtÞSxðtÞ  γ 2 νT ðtÞνðtÞ þ VðxðtÞÞdt  Vðxðt P ÞÞ þ Vðxð0ÞÞ ¼E Z

0

 ½xΤ ðtÞSxðtÞ γ 2 νT ðtÞνðtÞ þ LV ðxðtÞÞdt þ Vðxð0ÞÞ 0 Z t P  Nðx; ν; tÞ þ Vðxð0ÞÞ ¼E tP

rE

where r

i¼1j¼1

r

r

r

¼ ∑ ∑ hi ðϕðtÞÞhj ðϕðtÞÞ½xT ðtÞΞ ij xðtÞ

i¼1j¼1

ð29Þ

i¼1j¼1

T þ xT ðtÞPEi νðtÞ þ3νT ðtÞEi PEi νðtÞ þ 2xT ðtÞPðAij wðtÞxðtÞ



¼ LVðxðtÞÞ þ 2xT ðtÞPðAij wðtÞxðtÞ þ Bi wðtÞR j ðtÞ þ Ei wðtÞνðtÞÞ ð24Þ

where Ξ ij is defined in Eq. (22). If condition (20) of Theorem 1 is satisfied, then one can obtain Ξ ij o 0 that implies Nðx; ν; tÞ o 0. Referring to reference [35], by considering the zero initial conditions, i.e. Vðxð0ÞÞ ¼ 0, one has the following relations from Eq. (28) because Nðx; ν; tÞ o0. Jðx; ν; tÞ o 0

r

r

LV ðxðtÞÞ ¼ ∑ ∑ hi ðϕðtÞÞhj ðϕðtÞÞxT ðtÞfΨ ij gxðtÞ þ νT ðtÞETi PxðtÞ i¼1j¼1

T þ xT ðtÞPEi νðtÞ þ3νT ðtÞEi PEi νðtÞ

and Ψ ij ¼ ATij P þ PAij þ

  1 ρ 2 T T Kj Kj þPBi BTi P þ 3Aij PAij 2

ð28Þ

0

r

¼ ∑ ∑ hi ðϕðtÞÞhj ðϕðtÞÞxT ðtÞfΨ ij gxðtÞ þ νT ðtÞETi PxðtÞ

where

ð27Þ

i¼1j¼1

0

¼ xΤ ðtÞSxðtÞ  γ 2 νT ðtÞνðtÞ þ ∑ ∑ hi ðϕðtÞÞhj ðϕðtÞÞ½xT ðtÞΓ ij xðtÞdt

þ 2xT ðtÞPðAij wðtÞxðtÞ þ Bi wðtÞR j ðtÞ þ Ei wðtÞνðtÞÞg

þ Bi wðtÞRj ðtÞ þ Ei wðtÞνðtÞÞg

tP

¼E

Nðx; ν; tÞ ¼ xΤ ðtÞSxðtÞ γ 2 νT ðtÞνðtÞ þ LV ðxðtÞÞ

T

þ νT ðtÞETi PxðtÞ þ xT ðtÞPEi νðtÞ þ 3νT ðtÞEi PEi νðtÞ r

Integrating both sides of Eq. (26) from 0 to t P , one has the following relation: Z t P  _ EfV ðxðt P ÞÞ  Vðxð0ÞÞg r E VðxðtÞÞdt 0 Z t P  LVðxðtÞÞdt rE (Z0 )

or Z E

tp 0

  Z xΤ ðtÞSxðtÞdt r E γ 2

ð30Þ

tp 0

 νΤ ðtÞνðtÞdt ;

8 νðtÞ a 0

ð31Þ

Since the inequality (31) is equivalent to inequality (23) in Definition 2, one can find that the H1 performance constraint is thus achieved.

516

W.-J. Chang, Y.-J. Shih / Neurocomputing 148 (2015) 512–520

2

Next, it is necessary to show that the system is asymptotically stable in mean square. According to (29), if the condition (20) is satisfied, i.e., Ξ ij o 0, one has Nðx; ν; tÞ o 0. By assuming the external disturbance νðtÞ ¼ 0, one can find that LV ðxðtÞÞ o 0 from Eq. (29) due to Nðx; ν; tÞ o 0. Since νðtÞ ¼ 0 and LVðxðtÞÞ o 0, one can obtain the following results from (26). _ EfVðxðtÞÞg r EfLVðxðtÞÞg o0

6 Ω12 ¼ 4

T

P  1 Aij

ð32Þ

2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4

denotes the kth row of Ki . By using Lemma 2, the where following relation can be obtained from (33). " # ρ ðkÞ  1 ℓ1 1 þ 2 Ki P Z0 ð34Þ n P1



P  1 KTj





T

"

0

0

0

0

n

0 0

I 0

0  13P  1

0 0

0 0

n

0

0

0

0

S1

0

0

n

0

0

0

0

 13P  1

0

0

0

0

0

0

0

n

0

0

0

 13P  1

0

0

n

0

0

n

0 0

0 0

0 0

S  1 0

0  13P  1

where " Ω11 ¼

P

1

ATij þ Aij P  1 þ Bi BTi n

#

Ei γ

2

;

3

 13P  1

I

ð35Þ

T

n

0

o0



1ρ 2

0 0

 13P  1

Ω22



n

n

Ω12



1ρ 2

n

0

n



0

n

Ω11

T

γ 2

n

#

7 5;

n

0

Theorem 2. If there exist a positive definite matrix P ¼ PT 40, feedback gains Ki , and prescribed value γ satisfying sufficient conditions (21) and (35), then the closed-loop stochastic T–S fuzzy system (12) with external disturbance is asymptotically stable in the mean square and satisfies H1 performance constraint subject to saturating actuator.

T

Aij

 γ2

From the above derivations, the sufficient conditions of Theorem 1 are developed to analyze the stability of closed-loop T–S fuzzy model (12). However, these conditions are derived in terms of bilinear matrix inequality (BMI) conditions that cannot be calculated by LMI technique. For this reason, the BMI conditions of Theorem 1 are tried to transform into LMI forms in Theorem 2.

Ei

PEi

n

achieved. Therefore, if the conditions (21) and (22) are held then the closed-loop system (12) is asymptotically stable in the mean square and satisfies H1 control performance constraint subject to actuator saturation.

0

0

0

KTj

KTj Bi

0

I

0

0

Ei

T

7 7 7 7 7 7 7 7o0 7 7 7 7 7 7 5

Pre- and post- multiplying the inequality (36) by the appropriate dimension diagonal matrix diag fP  1 ; I; I; I; I; I; Ig, one has

P  1 KTj Bi 0

0

ATij P þ PAij þ PBi BTi P

P  1 Aij

0

3

P1

  1þρ 1 þ ρ 2 ukH 2 Bi Kj ; ℓ1 ¼ 2 2 ε

Ei

1ρ 2

T

P  1 KTj Bi

0

P  1 ATij þAij P  1 þBi BTi

1ρ 2



1ρ 2

ð36Þ

where ℓ1 ¼ ðð1 þ ρÞ=2Þ2 ððukH Þ=ρÞ2 . Obviously, the inequality (34) is equivalent to condition (21). Hence, if the condition (21) is satisfied then (34) holds and the actuator saturation constraint is 



Proof. Applying Lemma 2, the inequality (20) can be converted as follows:

KðkÞ i

T

 P  1 KTj

  1 1 1 Ω22 ¼ diag  P  1 ;  I;  P  1 ;  S  1 ;  P  1 ; 3 3 3

Referring to Definition 2 of [37], the closed-loop system (12) is asymptotically stable in the mean square. Additionally, the local constraint on xðtÞ A ϑ1  ϑ2 ( 8 t Z 0 for k ¼1, 2, …, m) should be involved in the design. According to (17), the constraint on xðtÞ A ϑ1  ϑ2 is equivalent to  2 u ðKðkÞ ÞP  1 ðKðkÞ ÞT r kH ð33Þ i i ρ

6 6 6 6 6 6 6 6 6 6 6 6 6 6 4

1ρ 2

0

Aij ¼ Ai þ

2



P1

0

0

Ei

T

3 7 7 7 7 7 7 7 7o0 7 7 7 7 7 7 5

ð37Þ

The inequality (37) is equivalent to sufficient condition (35). The proof of this theorem is completed. Applying the conditions of Theorem 2 and using MATLAB Toolbox, the fuzzy control gains can be obtained via LMI technique. Hence, the fuzzy controller for the stochastic T–S fuzzy model (12) subject to saturating actuator can be designed by PDC concept with solving the LMI condition (35) and the saturating actuator condition (21). Then, the closed-loop system (12) is asymptotically stable in the mean square and satisfies the H1 performance constraint and actuator saturation. In the following section, a numerical example is proposed to demonstrate the application and usefulness of the proposed fuzzy controller design approach.

4. Applications to the nonlinear continuous-time dynamic ship positioning systems Consider the dynamic equations of a nonlinear dynamic ship positioning system. It can be described as follows with considering the multiplicative noises and external disturbance. Referring to [37], it is customary to write the dynamic equations using a coordinate frame fixed to the ship as shown in Fig. 1. x_ 1 ðtÞ ¼ cos ðx3 ðtÞÞx4 ðtÞ  sin ðx3 ðtÞÞx5 ðtÞ þ 0:01wðtÞx4 ðtÞþ 0:1wðtÞx5 ðtÞ

ð38aÞ x_ 2 ðtÞ ¼ sin ðx3 ðtÞÞx4 ðtÞ þ cos ðx3 ðtÞÞx5 ðtÞ þ 0:1wðtÞx4 ðtÞ

ð38bÞ

W.-J. Chang, Y.-J. Shih / Neurocomputing 148 (2015) 512–520

517

Rule 3: IF x3 ðtÞ is about π=2 THEN

x3 (t) x6 (t)

x_ ðtÞ ¼ A3 xðtÞ þ B3 uðtÞ þ E3 νðtÞ þA3 wðtÞxðtÞ þ B3 wðtÞuðtÞ þ E3 wðtÞνðtÞ ð39cÞ

x4 (t) x5 (t)

x1 (t)

x2(t) Fig. 1. The coordinate of ship steering system.

where 2

0 6 0 6 6 6 0 A1 ¼ 6 6  0:0358 6 6 4 0 0

0

Fig. 2. Membership function of x3 ðtÞ.

x_ 3 ðtÞ ¼ x6 ðtÞ þ 0:1wðtÞx6 ðtÞ þ 0:1νðtÞ

ð38cÞ

x_ 4 ðtÞ ¼  0:0358x1 ðtÞ  0:0797x4 ðtÞ þ 0:01wðtÞx1 ðtÞ þ0:01wðtÞx4 ðtÞ þ0:9215u1 ðtÞ þ0:01wðtÞu1 ðtÞ

ð38dÞ

x_ 5 ðtÞ ¼  0:0208x2 ðtÞ  0:0818x5 ðtÞ  0:1224x6 ðtÞ þ0:01wðtÞðx2 ðtÞ þ x5 ðtÞ þ x6 ðtÞÞ þ 0:7802u2 ðtÞ þ 1:4811u3 ðtÞ þ0:01wðtÞu2 ðtÞ þ 0:1wðtÞu3 ðtÞ ð38eÞ x_ 6 ðtÞ ¼  0:0394x2 ðtÞ  0:2254x5 ðtÞ  0:2468x6 ðtÞ þ 0:01wðtÞðx2 ðtÞ þx5 ðtÞ þ x6 ðtÞÞ þ 1:4811u2 ðtÞ þ 7:4562u3 ðtÞ ð38f Þ þ0:1wðtÞðu2 ðtÞ þ u3 ðtÞÞ Where x1 ðtÞ and x2 ðtÞ are the earth-fixed positions ðx1 ðtÞ; x2 ðtÞÞ, x3 ðtÞ is yaw angle of the ships, x4 ðtÞ is surge, x5 ðtÞ is sway and x6 ðtÞ is yaw mode. Besides, νðtÞ and wðtÞ are all zero-mean white noises with variance one. The nonlinear terms of the dynamic Eqs. (38) are sin ðx3 ðtÞÞ and cos ðx3 ðtÞÞ. It is necessary to determine the working region of x3 ðtÞ for finding the linearized subsystems for the T–S fuzzy model. In this numerical example, the working region is chosen as x3 ðtÞ A f  π=2; π=2g. In order to minimize the design effort and complexity, we tried to use as few fuzzy rules as possible. Hence, three operating points x3 ðtÞ ¼  π=2, x3 ðtÞ ¼ 0 and x3 ðtÞ ¼ π=2 are chosen to construct corresponding three linear subsystems. The corresponding membership functions are proposed in Fig. 2. The T–S fuzzy model of the nonlinear system (38a)–(38f) can be represented by the following three fuzzy rules. Rule 1: IF x3 ðtÞ is about  π=2 THEN x_ ðtÞ ¼ A1 xðtÞ þ B1 uðtÞ þ E1 νðtÞ þ A1 wðtÞxðtÞ þ B1 wðtÞuðtÞ þ E1 wðtÞνðtÞ ð39aÞ Rule 2: IF x3 ðtÞ is about 0 THEN x_ ðtÞ ¼ A2 xðtÞ þ B2 uðtÞ þ E2 νðtÞ þ A2 wðtÞxðtÞ þ B2 wðtÞuðtÞ þ E2 wðtÞνðtÞ ð39bÞ

0

0:0349

1

0 0

0 0

1 0

0:0349 0

0

0

 0:0797

0

0

0

0:0818

 0:0394

0

0

0:2254

0:2468

0

0

1

0:0349

0

0

0

0:0349

1

 0:0208

0

0

0:0818

0

0

0:2254

0:2468

0

0

0:0349

1

0

0

0

1

0:0349

0

0

0

0

 0:0797

0

0 6 B1 ¼ B2 ¼ B3 ¼ 4 0 0

E1 ¼ E2 ¼ E3 ¼ 0

0

3

 0:0208

0

0

0:0818

7 7 7 7 1 7; 7 0 7 7 0:1224 5

 0:0394

0

0

0:2254

0:2468

0

0

0

0

0

0

 0:0797

0

0

2

3

 0:0394

0 0

0 6 0 6 6 6 0 A1 ¼ A 2 ¼ A 3 ¼ 6 6 0:01 6 6 4 0

0 6 B1 ¼ B 2 ¼ B 3 ¼ 4 0 0

0 1

7 7 7 7 1 7; 7 0 7 7 0:1224 5

2

2

3

0

 0:0208

2

0 6 0 6 6 6 0 A3 ¼ 6 6  0:0358 6 6 4 0

0

7 7 7 7 7; 7 0 7 7 0:1224 5

2

0 6 0 6 6 6 0 A2 ¼ 6 6  0:0358 6 6 4 0

0

0

0:01

0:1

0

0

0:1

0

0

0

0

0

0 0:01

0 0

0:01 0

0 0:01

0 7 7 7 0:1 7 7; 0 7 7 7 0:01 5

0:01

0

0

0:01

0:01

0

0:9215

0

0

0

0

0:7802

0

0

0

1:4811

0

0

0:01

0

0

0

0

0:01

0

0

0

0:1

0:1

0

0

0:01

0

and E1 ¼ E2 ¼ E3 ¼ 0

3

0

0

0

0

0

0

0

0

0

3T

7 1:4811 5 ; 7:4562 3T

7 0:1 5 ; 0:1

T

0

0

T

The actuator saturation of the parameters are chosen as γ ¼ 2, ukH ¼ 1 and ρ ¼ 0:6. By using the LMI toolbox of MATLAB to solving the conditions (21) and (35), the feasible solutions of the control feedback gains can be obtained. The results of the feasible solutions are described as follows: 2

0:2528 6 0:0151 6 6 6  0:0004 P¼6 6 0:5172 6 6 4 0:0874  0:0018

0:0151

 0:0004

0:5172

0:0874

0:2095  0:0048

 0:0048  0:0025

0:0802  0:0020

0:3911  0:0224

0:0802 0:3911

 0:0020  0:0224

2:822 0:4205

0:4205 1:9415

 0:0252

0:0042

 0:0104

 0:1246

 0:0018

3

 0:0252 7 7 7 0:042 7 7  0:0104 7 7 7  0:1246 5 0:0215

ð40Þ

518

W.-J. Chang, Y.-J. Shih / Neurocomputing 148 (2015) 512–520

Using the control feedback gains obtained in (42a)–(42c), one can get the following PDC-based fuzzy controller. 3

uðtÞ ¼ ∑ hi ðx3 ðtÞÞðKi xðtÞÞ

ð43Þ

i¼1

Note that the above fuzzy controller is developed based on the PDC concept. One needs to first design a linear controller for each rule, the final T–S fuzzy controller can be thus made by blending the linear controllers of all rules. In the simulations, the initial h iT condition is chosen as xðtÞ ¼ 0:8 0:5 301 0 0 0 . Applying the fuzzy controller (43) developed by the proposed design method to control the nonlinear system (38), the state responses are shown in Figs. 3–8. From the simulation results, the H1 performance constraint defined in Definition 2 can be checked by the following ratio. nR o t E 0p νΤ ðtÞνðtÞdt nR o ¼ 1163 ð44Þ t E 0p xΤ ðtÞSxðtÞdt

2

Fig. 3. Responses of state x1 ðtÞ∂∂uΩ2 .

The value of (44) is bigger than prescribed value 1=γ 2 ¼ 0:25. One can find that the condition (18) of Definition 2 is satisfied. The proposed design method provides a good performance to confront the multiplicative noise with external disturbance in system subject to the actuator saturation constraint. Therefore, the nonlinear ship steering system (38) controlled by the proposed fuzzy controller (44) is asymptotically stable subject to actuator saturation and H1 performance constraint. In order to show the advantages of proposed fuzzy control approach, some comparisons between present design method and the fuzzy control method of [24] are provided in the simulations. Applying the fuzzy control method developed in [24], the corresponding feedback gains of fuzzy controller described in Eq. (34) of [24] can be obtained as follows: 2

 0:2900 6 K1 ¼ 4  0:0911  0:1017

0:0520  0:2420

0:0009 0:0063

 1:8770  0:1184

 0:1844  1:8441

 0:1456

 0:0008

 0:2711

 0:7537

Fig. 4. Responses of state x2 ðtÞ.

2

4:0112e  004 6 2:3504e  005 6 6 6  1:3902e  005 S¼6 6 2:0023e  004 6 6 4  7:7166e  006

2

9:2554 6 K1 ¼ 4 1:0235 0:5213

ð45aÞ

2:3891e  005

1:3643e 005

2:0759e  004

6:6535e  006

4:0482e  004

3:2766e  005

6:7384e 006

1:9861e  004

3:2862e  005

4:3304e  004

2:1765e  006

7:0463e  006

6:6337e 006

3:5642e  006

0:0013

1:1882e  006

1:8245e  004

7:5276e 006

2:7954e  006

0:0013

1:5153e  005

1:4099e  004

1:1052e  006

8:3831e  006

 7:9098e  006

 1:3957  4:1998  1:4589

0:0367 0:2024  0:0409

 42:5920  4:7879  2:2100

7:3600 20:8917 7:1318

3 0:1831 7 1:1770 5  0:1133

2

 0:3389 6 K2 ¼ 4  0:0049  0:0392

 7:5539e  006

3

1:4607e  005 7 7 7 1:3671e  004 7 7 1:9058e  006 7 7 7  1:0640e  005 5

ð41Þ

0:0010

 0:0230

0:0006

 2:0128

 0:2015

 0:2005  0:1999

0:0066  0:0089

 0:0632  0:2475

 1:8492  0:8375

ð42aÞ 2

 9:3868 6 K2 ¼ 4  1:1736  0:9217

 1:3881

0:0365

 43:1885

 7:3164

 4:9544

0:2201

 5:6067

 24:6073

 3:5618

0:0085

 4:4137

 17:5406

0:1820

3

2

 0:3219 6 K3 ¼ 4 0:0647 0:0140

7 1:2707 5 0:1480

 9:2596 6 K3 ¼ 4  0:9477  0:3001

 1:4077

0:0370

 42:5979

 7:3941

 4:1854  1:4268

0:2020  0:0417

 4:6260  1:7148

 20:8161  7:0163

0:1845

3

7 1:1753 5  0:1171

ð42cÞ

0:0181

3

7 0:3160 5  0:0674

ð45bÞ  0:1136

0:0003

 1:9163

 0:2400

 0:2247

0:0063

 0:0094

 1:8176

 0:1354

 0:0088

 0:2185

 0:7211

ð42bÞ 2

3 0:0073 7 0:3170 5  0:0775

0:0329

3

7 0:3133 5  0:0803

ð45cÞ h

According 0:8

0:5

30

to 1

0

the 0

same initial condition, i.e., xðtÞ ¼ iT 0 , the state responses driven by the

fuzzy controller of [24] with the above feedback gains (45) are also shown in Figs. 3–8. Besides, the H1 performance constraint

W.-J. Chang, Y.-J. Shih / Neurocomputing 148 (2015) 512–520

Fig. 5. Responses of state x3 ðtÞ.

519

Fig. 8. Responses of state x6 ðtÞ.

in Definition 2 cannot be satisfied via the fuzzy controller of [24] with feedback gains of (45). Moreover, form the simulation results of Figs. 3–8, one can find that the transient responses and steadystate responses of proposed fuzzy design approach are better than that developed in [24]. Thus, the proposed actuator saturation constrained fuzzy controller design approach can be successfully used to stabilize T–S fuzzy model with external disturbance and multiplicative noises subject to H1 performance constraint.

5. Conclusions In this paper, a novel fuzzy control approach has been developed to deal with performance constrained control problem for nonlinear stochastic systems subjected to H1 performance constraint and actuator saturation. The T–S fuzzy model with external disturbance and multiplicative noises was employed to represent the considered nonlinear stochastic systems. According to the above T–S fuzzy model, the sufficient stability conditions were derived via Lyapunov theory and Itô's formula. By transforming the derived BMI conditions into LMI forms, a PDC-based fuzzy controller can be solved to achieve actuator saturation and H1 performance constraints, simultaneously. At last, a simulation example of the nonlinear dynamic ship positioning system was applied to demonstrate the effectiveness and usefulness of the proposed fuzzy controller design method.

Fig. 6. Responses of state x4 ðtÞ.

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Fig. 7. Responses of state x5 ðtÞ.

can be checked by calculating the following ratio: nR o t E 0p νΤ ðtÞνðtÞdt nR o ¼ 0:000216 t E 0p xΤ ðtÞSxðtÞdt

ð46Þ

It can be found that the value of (46) is less than prescribed value 1=γ 2 ¼ 0:25. Hence, the H1 performance constraint defined

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Wen-Jer Chang received the B.S. degree from National Taiwan Ocean University, Taiwan, ROC, in 1986. The Marine Engineering is his major course and the Electronic Engineering is his minor one. He received the M.S. degree in the Institute of Computer Science and Electronic Engineering from the National Central University in 1990, and the Ph.D. degree from the Institute of Electrical Engineering of the National Central University in 1995. Since 1995, he has been with National Taiwan Ocean University, Keelung, Taiwan, ROC. He is currently the Vice Dean of Academic Affairs, Director of Center for Teaching and Learning and a full Professor of the Department of Marine Engineering of National Taiwan Ocean University. He is now a life member of the CIEE, CACS, CSFAT and SNAME. Since 2003, Dr. Chang was listed in the Marquis Who's Who in Science and Engineering. In 2003, he also won the outstanding young control engineers award granted by the Chinese Automation Control Society (CACS). In 2004, he won the universal award of accomplishment granted by ABI of USA. In 2005 and 2013, he was selected as an excellent teacher of the National Taiwan Ocean University. Dr. Chang has over 100 published journal papers. His recent research interests are fuzzy control, robust control, performance constrained control.

Ying-Jie Shih received the B.S. degree from the Department of Mechanical Engineering of Chung Hua University, Taiwan, ROC, in 2008. In 2014, He received the M.S. degree from the Department of Marine Engineering of the National Taiwan Ocean University, Taiwan, ROC. His research interests focus on fuzzy control and the applications of intelligent control.