International Journal of Control, Automation, and Systems (2011) 9(6):1103-1110 DOI 10.1007/s12555-011-0611-7
http://www.springer.com/12555
Robust Observer-based Fuzzy Control for Variable Speed Wind Power System: LMI Approach Hwa Chang Sung, Jin Bae Park*, and Young Hoon Joo Abstract: In this paper, the output-feedback control for stabilizing the uncertain nonlinear system is proposed. For achieving the robust stability, we deal with the parametric uncertainties of the concerned system which is based on the Takagi-Sugeno (T-S) fuzzy model. Also, we derive the observer-based models preserving the property and structure of the uncertainties. The sufficient conditions for output feedback stabilizing controller designs are given in terms of solutions to a set of linear matrix inequalities (LMIs). The simulation results for variable speed wind power (VSWP) system are demonstrated to visualize the feasibility of the proposed method. Keywords: Linear matrix inequalities (LMIs), observer-based control, parametric uncertainties, Takagi-Sugeno (T-S) fuzzy model, variable speed wind power (VSWP) system.
1. INTRODUCTION Since in many real plants, especially renewable power systems, state variables are often unavailable, output feedback control is necessary and has caused some interest. Also, practical systems are commonly required to have good robustness so that robust control problems become important research topic in control theory. The parametric uncertainty is a principal factor responsible for the degraded stability and the performance of an uncertain control system. In fact, in many cases it is very difficult, if not impossible, to obtain the accurate values of some system parameters. This is due to the inaccurate measurement, inaccessibility to the system parameters, or on-line variation of the parameters. As a result, there have been many researches to deal with the output feedback control problem of uncertain systems. The references [4,6,11,12] have studied various robust control problems for linear continuous or discrete systems with observer. Besides linear systems, the robust control and output feedback control for complex nonlinear systems have also been lively discussions. Among them, fuzzy control approach have attracted investors' attention thanks to its usefulness, for example, Tanaka et al. proposed the fuzzy observer design for __________ Manuscript received October 14, 2010; revised March 31, 2011; accepted July 11, 2011. Recommended by Editorial Board member Poo Gyeon Park under the direction of Editor Jae Weon Choi. This work was supported by the New & Renewable Energy of the Korea Institute of Energy Technology Evaluation and Planning (KETEP) grant funded by the Korea government Ministry of the Knowledge Economy (No. 20103020070070). Hwa Chang Sung and Jin Bae Park are with the School of Electrical and Electronic Engineering, Yonsei University, Seoul 120749, Korea (e-mails: {casfirspear, jbpark}@yonsei.ac.kr). Young Hoon Joo is with the Department of Control and Robotics Engineering, Kunsan National University, Kunsan, Chonbuk 573-701, Korea (e-mail:
[email protected]). * Corresponding author. © ICROS, KIEE and Springer 2011
nonlinear control systems [10] The robustness in TakagiSugeno (T-S) fuzzy control has been extensively studied for some control techniques [1-3,7]. However, except [2], there have been few studies of the output feedback control approach for the fuzzy system with parametric uncertainties. In [2], Tong et al. dealt with the observer-based control for uncertain nonlinear system which was represented by linear matrix inequality (LMI) formats. However, the synthesis conditions in [2] are invalid because of the incorrect use of a property that resulted in neglecting certain terms so that the obtained conditions are not LMI but bilinear matrix inequality (BMI) formulation [3]. This is very important point because the algorithm in [2] is nonconvex so that iterative linear matrix inequality (ILMI) approaches should be required. As a result, the separation principle is not guaranteed for the obtained conditions [3]. The sufficient LMI conditions have the advantage of being linear and, hence, easily tractable by standard optimization techniques [6]. Although the new observer-based control design method was studied in [5], this approach did not deal with the parametric uncertainties problem but the disturbance one. Therefore, to our best knowledge, there are no studies for observer-based wind power system control with parametric uncertainties by LMI approach. Motivated by the above observations, this paper presents a robust control method for both controller and observer designs for a general nonlinear system with parametric uncertainties. The robust stabilization problems are derived for a class of nonlinear system with time-varying but norm-bounded parametric uncertainties even though their state variables are not available for measurement. Also, we derive the observer-based models of the fuzzy control preserving the property and structure of the uncertainties. Its constructive conditions are provided in the LMI formats, and therefore easily tractable by the convex optimization techniques. The
Hwa Chang Sung, Jin Bae Park, and Young Hoon Joo
1104
q xˆ (t ) = ∑ θi ( z (t ))(( Ai xˆ (t ) + Bi u (t ) + Li ( y (t ) − yˆ (t )) i =1 (4) q ˆ y (t ) = ∑ θi ( z (t ))Ci xˆ (t ), i =1
obtained methods are applied to the variable speed wind power (VSWP) system which is constructed as the T-S fuzzy model. The paper is organized as follows: Section 2 reviews T-S fuzzy models and fuzzy observers. The state feedback and output feedback controller designs for robust stabilization of T-S fuzzy systems with parametric uncertainties are presented in Section 3. Section 4 shows a design example of VSWP system and simulation results. Finally, conclusion is given in Section 5.
where xˆ(t ) ∈ » n is the estimation of x(t) and Li is the observer gain matrix. In this paper, the following fuzzy rule for the controller is employed:
2. PRELIMINARIES
R i :IF z1 (t ) is about Γ1i and...and z p (t ) is about Γ pi THEN u (t ) = − K i xˆ (t ),
Consider a nonlinear system of the following form: x (t ) = f ( x(t ), u (t )) y (t ) = h( x(t )),
whose defuzzified output is (1)
where x(t ) ∈ » n constitutes the state vector; u (t ) ∈ » n is the control input; y (t ) ∈ » p is the output. We define compact sets for x(t) and u(t) such that ℘x = {x | x(t ) ≤ ∆ x } ⊂ R n , ℘u = {u | u (t ) ≤ ∆u } ⊂ R m for some ∆ x ∈
» >0 and ∆u ∈ » >0 . Within the scope of the above bound, it is possible to represent the nonlinear system as the T-S fuzzy rules without any approximation on ( x, u ) =℘x ×℘u [9]. The nonlinear systems (1) can be represented as the following T-S fuzzy rules:
Ri :IF z1 (t ) is about Γ1i and... and z p (t ) is about Γ pi x(t ) = ( Ai + ∆Ai ) x(t ) + ( Bi + ∆Bi )u(t ) THEN y(t ) = Ci x(t ),
(2)
where Ri, i ∈ Ι r = {1, 2, , q}, denotes the i th fuzzy rule, zh(t), h ∈ Ip= {1, 2, , p}, is the h th premise variable, Γ hi , (i, h) ∈ I q × I p , is the fuzzy set of zh(t) in Ri, Ai and
Bi are known constant matrices with appropriate dimensions, and ∆Ai and ∆Bi are unknown matrices with appropriate dimensions which represent the parametric uncertainties. Using the singleton fuzzifier, product inference engine, and center-average defuzzification, (2) is inferred as q
x (t ) = ∑ θi ( z (t ))(( Ai + ∆Ai ) x(t ) + ( Bi + ∆Bi )u (t )) i =1 (3) q y (t ) = ∑ θi ( z (t ))Ci x (t ), i =1 q
where θi ( z (t )) = wi ( z (t )) / ∑ i =1 wi ( z (t )) and wi ( z (t )) = p
∏ h =1 µΓ
h
i
( zh (t )), and µΓ i ( zh (t )) : U zh (t ) ⊂ R → R [0,1] h
is the membership function of zh(t) on the compact set U zh ( t ) . To estimate the state, the following full-order observer is utilized;
q
u (t ) = −∑ θi z (t ) K i xˆ (t ),
(5)
i =1
where the control input u(t) is designed by parallel distributed compensation (PDC) approach [10]. Further, we assume that the uncertainties ∆Ai and ∆Bi have the following structure. Assumption 1: We assume that ∆Ai and ∆Bi can be described as follows: [∆Ai ∆Bi ] = Di Fi (t )[ E1i E2i ],
(6)
where Di, E1i, and E2i are known real constant matrices of compatible dimensions, and Fi(t) is an unknown matrix function with Lebesgue-measurable elements and with FiT (t ) Fi (t ) ≤ I . There are fruitful research works focusing on the design of the output feedback fuzzy controller [10] and robust fuzzy controller [1-3,5,7]. However, the studies of output feedback control for real wind plants have not been sufficiently achieved. Although reference [2] tried to concern that problem, this approach did not guarantee the LMI, but BMI formulation. Our paper aims at a robust fuzzy control method for a general nonlinear system with parametric uncertainties. 3. NEW ROBUST OBSERVER-BASED FUZZY CONTROL METHODOLOGY Let the state estimation error be e(t ) = x(t ) − xˆ (t ), then the closed-loop system is described as follows: q
q
x (t ) = ∑∑ θi ( z (t ))θ j ( z (t ))(( Ai + ∆Ai ) − ( Bi − ∆Bi ) K j i =1 j =1
q
q
× x(t ) + ∑∑ θi ( z (t ))θ j ( z (t ))( Bi + ∆Bi ) K j e(t ), i =1 j =1
(7) q
q
xˆ (t ) = ∑∑ θi ( z (t ))θ j ( z (t ))(( Ai − Bi K j ) xˆ (t ) i =1 j =1 q
q
+ ∑∑ θi ( z (t ))θ j ( z (t )) Li C j e(t ), i =1 j =1
(8)
Robust Observer-based Fuzzy Control for Variable Speed Wind Power System: LMI Approach q
Remark 1: By Proposition 1, the remain uncertain terms are represented as follows:
q
e(t ) = ∑∑θi ( z(t ))θ j ( z(t ))((∆Ai − ∆Bi K j ) x(t ) i =1 j =1 q
1105
(9)
q
+ ∑∑θi ( z(t ))θ j ( z(t ))( Ai − Li C j − ∆Bi K j )e(t ). i =1 j =1
xT (t )(− K jT ∆BiT P − P∆Bi K j ) x(t ) + e(t )T (− K jT ∆Bi R − R∆Bi K jT )e(t ) + x(t )T P∆Bi K j e(t ) + e(t )T K jT ∆BiT P
Define ΨT (t ) = [ xT (t ) eT (t )],
(10)
× x(t ) + x(t )T ∆AiT Re(t ) + e(t )T R∆Ai x(t )
V (Ψ (t )) = xT (t ) Px(t ) + eT (t ) Re(t ),
(11)
− x(t )T K jT × ∆BiT Re(t ) − e(t )T R∆Bi K j e(t )
where P and R are positive-definite symmetric matrices. The time derivative of V(Ψ(t)) is represented as follows: q
q
V (Ψ(t )) = ∑∑θi ( z (t ))θ j ( z (t )){(xT (t )(GijT P + PGij ) i =1 j =1
×x(t ) + 2 xT (t ) P( Bi + ∆Bi ) K j e(t ) + 2eT (t ) R ×(∆Ai − ∆Bi K j ) x(t ) + eT (t )( HijT R + RHij )e(t )},
(12) where Gij = A i + ∆Ai − ( Bi + ∆Bi ) K j and H ij = Ai − Li C j − ∆Bi K j . Since the uncertainties in (12) may be time-varying, it is not easy to manage. For solving these difficulties, consider the following lemma and proposition: Lemma 1: The LMI Q( y ) + ρ I *
S ( y) ≺0 R( y )
where Q(y)= Q( y )T , R ( y ) = RT ( y ), S(y) depend affinely on y. Proof: This proof is a trivial extension of schur complement of [20]. Proposition 1: Using Lemma 1 and Assumption 1, the uncertain terms in (12) are solved as following: xT (t )(∆AiT P + P∆Ai ) x(t ) ≤ ε1 xT (t ) E1iT E1i x (t ) + ε1−1 PDi DiT Px(t ).
(13)
Proof: Using Assumption 1, the following inequality holds: xT (t )(∆AiT P + P∆Ai ) x(t )
+ ε1−1 xT (t ) Di DiT Px(t ) + ε1 x(t )T E1iT FiT (t ) Fi (t ) E1i x(t ) ≤ ε1 xT (t ) E1iT E1i x(t ) + ε1−1 PDi DiT Px(t ).
+ ε 3−1eT (t ) RDi DiT Re(t )
(14)
+ ε 3eT (t ) K jT E2iT E2i Ki e(t ) + ε 4 −1 xT (t ) PDi DiT Pe(t ) + ε 3eT (t ) K jT E2iT E2i Ki e(t ) + ε 5−1eT (t ) RDi DiT Re(t ) + ε 5 xT (t ) E1iT E1i x(t ) + ε 6 −1eT (t ) RDi DiT Re(t ) + ε 6 xT (t ) K jT E2iT E2i Ki x(t ).
Before proceeding further, recall the following lemma. Lemma 2 [15]: For any real matrices Λ1 = Λ1T , Λ 2 , Λ3(t), and Λ4 with appropriate dimensions, the following inequality holds:
where Λ3 (t ) satisfies Λ 3T (t )Λ 3 (t ) ≤ I if and only if
R ( y ) < 0, Q( y ) − S ( y ) R ( y ) −1 S ( y )T < − ρ I ,
−1 × ε1 DiT Px(t ) − ε1 Fi (t ) E1i Px(t )
+ ε 2 xT (t ) K jT E2iT E2i Ki x(t )
Λ1 + Λ 2 Λ 3 (t )Λ 4 + Λ 4T Λ 3T (t )Λ 2T ≺ 0,
is equivalent to
−1 = − ε1 DiT Px(t ) − ε1 Fi (t ) E1i Px(t )
≤ ε 2 −1 xT (t ) PDi DiT Px(t )
T
Λ1 + ε −1Λ 4T
ε −1Λ 4 εΛ 2 ≺0 εΛ 2T
for some ε < 0. Now, we propose the following LMI results for the asymptotic stability of (7) and (9). The fuzzy control gain Ki and observer gain Li can be solved by the following LMIs. Theorem 1: Suppose that the matrices P = PT 0, R = RT 0, M i , Ni , nonsingular matrix Pˆ and the scalars ε1, ε2, ε3, ε4, ε5, ε6 are optimal solutions to Φ11 * * * T T Φ 22 * * K j Bi P − ε1 I * 0 DiT T T 0 0 − ε2I K j Di P 0 E2i K jT + DiT R 0 0 T 0 0 E2 i K j T Di T 0 0 0 Di R E K 0 0 DiT R 2i j
Hwa Chang Sung, Jin Bae Park, and Young Hoon Joo
1106
* *
* * * * * * * * * * ≺ 0, (15) −ε 3 I * * * 0 − ε3I * * 0 0 − ε5I * 0 0 0 − ε 6 I
PBi = Bi Pˆ ,
where Φ11 = T
* *
* *
(16)
AiT P + PAi
− K j BiT P − PBi K j ,
T
Φ22 = Ai R +
T
RAi − C j Li R − RL i C j . Then, x(t ) of (7) is globally exponentially stable. The fuzzy gains are given by Ki = Pˆ −1M i , i ∈ I q and fuzzy observer gains are obtained by Li = R −1 Ni , i ∈ I q . Proof: The proof is given in Appendix A. Remark 2: It is notice that (15) is affined with all their respective arguments so that we can use the Theorem 1 to obtain the control and observer gain. Remark 3: In linear system, it is possible to distinguish the conservative problems by selecting the matrix P [4]. However, in nonlinear system, the conservative problems are complex one because the linear independence of the Bi and PBi are not always so. Therefore, it is necessary to maintain the Pˆ as nonsingular. Consider the following observer-based systems which have the another uncertainty ∆Ci: q q x (t ) = ∑∑θi ( z(t ))θ j ( z(t ))(( Ai + ∆Ai ) − Bi K j ) x(t ) i =1 j =1 (17) q q y(t ) = ∑∑θi ( z(t ))θ j ( z(t ))(Ci + ∆Ci ) x(t ), i =1 j =1
where [∆Ai ∆Ci ] = Di Fi (t )[ E1i E3i ] with FiT (t ) Fi (t ) ≤ I. In order to estimate the error e(t), we use the full-order observer (4). Then, we obtain the following closed-loop system. q
q
q
q
(18)
+ ∑∑θi ( z (t ))θ j ( z (t )) Bi K j e(t ), q
e(t ) = ∑∑ θi ( z (t ))θ j ( z (t ))((∆Ai − L j ∆Ci ) x(t ) i =1 j =1 q
q
+ ∑∑ θi ( z (t ))θ j ( z (t ))( Ai − Li C j )e(t ). i =1 j =1
(20)
where ΨT (t ) = [ x T (t ) e T (t )], P and R are positivedefinite symmetric matrices. Considering Assumption 1, Lemma 1 and 2, the time derivative of V (Ψ (t )) is computed by q
q
V (Ψ (t )) = ∑∑ θi ( z (t ))θ j ( z (t )){(xT (t )(GijT P + PGij ) i =1 j =1
× x(t ) + 2 xT (t ) PBi K j e(t ) + 2eT (t ) R (∆Ai − L j ∆Ci ) x(t ) + eT (t )( H ijT R + RH ij )e(t )},
where Gij = A i + ∆Ai − Bi K
j
(21) and H ij = Ai − Li C j . By
Proposition 1, the uncertain terms in (21) are solved as follows: xT (t )(∆AiT P − P ∆Ai ) x(t ) + xT (t )(∆Ai − L j ∆Ci )T Re(t ) + e(t )T R(∆Ai − L j ∆Ci ) x(t ) ≤ (ε1 + ε 2 ) xT (t ) E1iT E1i x(t ) + ε 3 xT (t ) E3iT E3i x(t )
(22)
+ ε1−1 xT PDiT Di Px(t ) + ε 2−1eT (t ) RDi DiT Re(t ) + ε 3−1e(t )T RL j Di DiT L jT Re(t ),
where ε1, ε2 and ε3 are positive constants. The new results are summarized as follows: Theorem 2: Suppose that the matrices P = PT 0, R = RT 0, M , N , nonsingular matrix Pˆ and the i
i
scalars ε1, ε2, ε3 are optimal solutions to Φ11 * * * * T T * * K j Bi P Φ 22 * T 0 − ε1 I * * ≺ 0, Di DiT R 0 − ε 2 I * 0 0 − ε 3 I DiT L jT 0 0
(23)
(24)
where Φ11 = AiT P + PAi − K j BiT P − PBi K j , Φ22 = AiT R + RAi − C jT LiT R − RL i C j . Then, x(t ) of (17) is globally
i =1 j =1
q
V (Ψ (t )) = xT (t ) Px(t ) + eT (t ) Re(t ),
PBi = Bi Pˆ ,
x (t ) = ∑∑θi ( z (t ))θ j ( z (t ))(( Ai + ∆Ai ) − Bi K j ) x(t ) i =1 j =1
Define the Lyapunov functional candidate as
(19)
exponentially stable. The fuzzy gains are given by K i = Pˆ −1M i , i ∈ I q and fuzzy observer gains are obtained by Li = R −1 Ni , i ∈ I q . Proof: The proof is given in Appendix B.
Robust Observer-based Fuzzy Control for Variable Speed Wind Power System: LMI Approach
4. SYSTEM MODELLING AND SIMULATION RESULTS 4.1. Wind power system and its fuzzy modelling In this section, we introduce the VSWP system and its fuzzy modelling. A three bladed horizontal-axis 60KW wind turbine is considered and Fig. 1 is the schematic diagram of a concerned system. The dynamics of the VSWP system can be represented as following [16,17]: QA − A = J T ω t , Q − QE = J G ω G , t
Q = Qs + QD = K s ∫ (ωt − ω g ) dt + Bs (ωt − ω g ),
KS
QA
BS
JT
ω
1107
QE
JG
Grid
ωg
t
Generator
Gearbox Aeroturbine
Fig. 1. Schematic diagram of the wind power system. (25)
0
where QA is the input wind torque, QE is the generator torque, ωt is the turbine angular velocities, ωg is the generator angular velocities, Ks is shaft compliance, Bs is shaft damping, JT is turbine inertia, JG is generator inertia, and Q is shaft torque. The power produced by the wind is expressed as [18] 1 PT ( w, λ ) = πρ R 2C p (λ ) w3 , 2
(26)
where ρ is air density, R is blade length, w is wind velocity, and Cp denotes power coefficient of the wind turbine. It is presented as a nonlinear function of the tip speed ratio λ λ=
Rωt . w
(27)
Fig. 2 shows the trajectory of the VSWP system with x(0)=[10 –10 10]T. In order to construct the fuzzy model for the VSWP system, it is necessary to analyze the variable torque coefficient function Cp(λ). This specific function plays an important role of converting kinetic energy of the moving air to mechanical torque QA. For optimum energy capture, the wind turbine must operate at the point of maximum power coefficient Cpmax when λ=λopt [18,22,23]. In the view of above equations (26) and (27), we can obtain the following expression of the aerodynamic torque: QA =
1 ρ ARCq (λ ) w2 , 2
(28)
where Cq(λ) is variable torque coefficient function C p (λ ) = λ Cq (λ ).
(29)
From (28) and (29), we know that QA has a nonlinear function of two variables (w and ωt). Denote xT = [ x1 x2 x3 ] = [ωt ω g Qs ], then the dynamics of the VSWP system can be represented as the following equations: x1 =
(γ − Bs ) B 1 x1 + s x2 + x3 , JT JT JT
(30)
x2 =
Bs B 1 1 x1 − s x2 + x3 − u, JG JG JG JG
(31)
x3 = K s x1 − K s x2 ,
(32)
Fig. 2. Trajectory of the VSWP system when u(t)=0. Cq ( λ ) 1 ρ AR3 x1. In order to construct the 2 λ2 T-S fuzzy model for VSWP system, the nonlinear term has to be represented as convex combination of appropriate vertices. Assume x1 ∈ [ M1 M 2 ] and consider the following equations:
where γ =
η1 ( x1 (t )) = Γ11 ( x1 (t )) x1 (t ) + Γ 21 ( x1 (t )) x1 (t ), 1=
Γ11 ( x1 (t )) + Γ 21 ( x1 (t )),
(33) (34)
where Γ11 =
x − Di − x1 + Di , Γ 21 = 1 . Di − M1 Di − M 1
(35)
Then, the identified T-S fuzzy rules of the VSWP system are represented as follows: R1 :IF x1 (t ) is Γ11 , THEN x (t ) = A1 x(t ) + B1u (t ), R 2 :IF x1 (t ) is Γ 21 , THEN x (t ) = A2 x(t ) + B2u (t ),
where γ 2 − Bs Bs γ 1 − Bs Bs 1 J J JT JT JT T T B B A1 = Bs 1 , A2 = Bs − s − s J J J J J G G G G G K s − K s K s − K s 0 1 B1 = B2 = 0 − JG
T
0 .
1 JT 1 , JG 0
Hwa Chang Sung, Jin Bae Park, and Young Hoon Joo
K1 = [−120.6573 − 48.9923 − 8.5947],
Value 40000 kg·m2 65 kg·m2 100 N·ms/rad 1800 N·ms/rad 35 m 1.225 kg/m3
−6000
0
0.5
1 t (sec)
1.5
2
In this paper, the observer-based robust fuzzy control has been proposed. Our main contribution consists in showing that the complex nonlinear control problem can be transformed to find the fuzzy gains stabilizing the whole control systems. Also, the robust stability condition has been successfully incorporated in the proposed method. Since the proposed conditions are LMI-based ones, they can be easily extended to other control problems (output feedback control, decentralized control, H∞ control, etc.). The simulation results for VSWP system have demonstrated that it is possible to obtain the excellence performance through the proposed method.
2.5 2 1.5
1
−5000
5. CONCLUSIONS
3
x (t)
−4000
As shown in Figs. 3-5, all system trajectories converge to zero despite of the system uncertainties. The simulation results show that the T–S fuzzy control through fuzzy observer is robust against norm-bounded parametric uncertainties.
Table 1. Nominal parameters of the VSWP system
1 0.5 0 −0.5
1 t (sec)
−3000
Fig. 5. System states of x3 when T= 0.1s.
L2 = [−71.9981 22.6239 0.1323].
0.5
−2000
−9000
L1 = [78.7233 30.0912 − 3.0913],
0
−1000
−8000
Utilizing Theorem 1 yields the observer gain matrix
−1
0
−7000
K 2 = [−89.9836 − 39.6006 − 10.3913].
Parameter JT JG KS BS R pi
1000
3
4.2. Simulation result This section presents simulation result of the VSWP system. The nominal values of system parameters are presented as Table 1. During the simulation time, all system parameters are randomly varied within the bounds of 10% of their nominal values - JT and JG. Applying Theorem 1 yields the total fuzzy system control gain matrices, for the sampling period T=0.1s as
x (t)
1108
1.5
2
APPENDIX A To summarize the equations (12)-(14), we have q
q
V (Ψ (t )) = ∑∑ θi ( z (t ))θ j ( z (t )){(xT (t )( AiT P + PAi
Fig. 3. System states of x1 when T= 0.1s.
i =1 j =1
8000
− K j BiT P − PBi K j ) x(t ) + e(t )T ( AiT R + RAi
6000
− C jT × LiT R − RLi C j )e(t ) + 2 x(t )T PBi K j e(t ) + (ε1 + ε 5 ) xT (t ) E1iT E1i x(t ) + (ε 3 + ε 4 )eT (t ) K j
2000
× E2iT E2i K j e(t ) + (ε 3−1 + ε 6−1 )eT (t ) RD1T Di
x2(t)
4000
0
× Re(t ) + ε1−1 xT (t ) PDi DiT Px(t ) + ε 2−1 xT (t )
−2000
× PDi K j K jT DiT Px(t ) + ε 5−1eT (t ) RDi DiT R × e(t ) + ε 6 xT (t ) K jT E2iT E2i K j x(t )}
−4000
−6000
0
0.5
1 t (sec)
Fig. 4. System states of x2 when T= 0.1s.
1.5
2
q q Φ11 Φ12 = ∑∑θi ( z (t ))θ j ( z (t )){ψ T (t ) T Φ12 Φ 22 i =1 j =1
×ψ (t ) + (ε 3 + ε 4 )eT (t ) K j E2iT E2i K j e(t )
Robust Observer-based Fuzzy Control for Variable Speed Wind Power System: LMI Approach
+ (ε 3−1 + ε 6 −1 )eT (t ) RDi DiT Re(t ) + ε1−1 xT (t ) P
− C jT LiT R − RLi C j )e(t ) + 2 x(t )T PBi K j e(t )
× Di DiT Px (t ) + ε 2 −1 xT (t ) PDi K j K jT DiT Px(t )
+ (ε1 + ε 2 ) xT (t ) E1iT E1i x(t ) + ε 3 xT (t ) E3iT E3i
+ ε 4 −1 xT (t ) PDi DiT Px(t ) + ε 5−1eT (t ) RDi DiT
× x(t ) + ε1−1 xT (t ) PDi DiT Px(t ) + ε 2−1eT (t ) R
× Re(t ) + ε 6 xT (t ) K jT E2iT E2i K j x(t )},
× Di DiT Re(t ) + ε 3−1eT (t ) RL j Di DiT Re(t ) q q Φ11 Φ12 = ∑∑θi ( z (t ))θ j ( z (t )){ψ T (t ) T i =1 j =1 Φ12 Φ 22
(A.1) where Φ11 = Ai P + PAi − K j Bi P − PB i K j + (ε1 + ε 5 ) × T
T
E1iT E1i + ε 2 E2iT E2i , Φ12 = PB i K j , Φ 22 = AiT R + R × Ai T
1109
×ψ (t ) + ε1−1 xT (t ) PDi DiT Px(t ) + ε 2−1eT (t ) R
T
−C j Li R − RLi C j .
× Di DiT Re(t ) + ε 3−1e(t )T RL j Di DiT Re(t )},
By Lemma 1, the LMI condition (15) implies Xˆ 11 *
(B.1) T
where Φ11 = Ai P + PAi − K j Bi P − PB i K j + (ε1 + ε 2 ) ×
Bi K j Xˆ 22
E1iT E1i + ε 3 E3iT E3i , Φ12 = PB i K j , Φ 22 = AiT R + R × Ai
PDi PDi K j 0 0 K j T E2 i T PDi − 0 0 K j E2iT + RDi K j E2iT RDi RDi − ε1 I 0 0 − 0 0 0
0
0
0
−ε 2 I
0
0
0
− ε3I
0
0
0
− ε4I
0
0
0
0
0
0
DiT P T T K j Di P 0 × T Di P 0 E2i K j
0
0 0 0 0 0 0 0 − ε5 I 0 0 − ε6I
−C jT LiT R − RLi C j . By Lemma 1, the LMI condition (23) implies X 11 *
−1
−1
(B.2) DiT P 0 T 0 Di R ≺ − ρ I . 0 DiT L jT
2
≤ V (Ψ (t )) ≤ max[1, λ max ( R )] ⋅ Ψ (t )
(A.2)
2 V (Ψ (t )) ≤ − ρ ⋅ Ψ (t ) .
2
[1]
[2]
(A.3) [3]
and 2 V (Ψ (t )) ≤ − ρ ⋅ Ψ (t ) .
(A.4)
APPENDIX B The system (21) is represented as follows: q
V (Ψ (t )) = ∑∑ θi ( z (t ))θ j ( z (t )){(xT (t )( AiT P + PAi i =1 j =1
− K j BiT P − PBi K j ) x(t ) + e(t )T ( AiT R + RAi
2
(B.3)
and
2
≤ V (Ψ (t )) ≤ max[1, λ max ( R )] ⋅ Ψ (t )
0
0 0 − ε 3 I
min[1, λ min ( R )] ⋅ Ψ (t )
[λmax ( P ), λmax ( R )])]. By (12), (13), and above equation we have min[1, λ min ( R)] ⋅ Ψ (t )
0 −ε 2 I
By (21) and (22) we have
0 0
E2i K jT + DiT R ≺ −ρ I. E2 i K j T DiT R DiT R
0 Bi K j PDi 0 − T 0 RD L D X 22 i j i
− ε1 I − 0 0
The matrix is equal to the matrix Фij with P = I and L = R −1 Lˆ , and the system convergence rate is [ ρ /(2 ⋅
q
T
[4]
[5]
(B.4)
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Jin Bae Park received his B.E. degree in Electrical Engineering from Yonsei University, Seoul, Korea, and his M.S. and Ph.D. degrees in Electrical Engineering from Kansas State University, Manhattan, in 1977, 1985, and 1990, respectively. Since 1992, he has been with the Department of Electrical and Electronic Engineering, Yonsei University, Seoul, Korea, where he is currently a professor. His research interests include robust control and filtering, nonlinear control, intelligent mobile robot, fuzzy logic control, neural networks, and genetic algorithms. He served as Editor-in-Chief for the International Journal of Control, Automation, and Systems (IJCAS) (2006-2010) and is serving as the Vice-President for the Institute of Control, Robotics and Systems (ICROS) (2009-present). Young Hoon Joo received his B.S., M.S., and Ph.D. degrees in Electrical Engineering from Yonsei University, Seoul, Korea, in 1982, 1984, and 1995, respectively. He worked with Samsung Electronics Company, Seoul, Korea, from 1986 to 1995, as a project manager. He was with the University of Houston, Houston, TX, from 1998 to 1999, as a visiting professor in the Department of Electrical and Computer Engineering. He is currently a professor in the School of Electronic and Information Engineering, Kunsan National University, Korea. His major interest is mainly in the field of intelligent robot, intelligent control, and human-robot interaction. He serves as President for Korea Institute of Intelligent Systems (KIIS) (20082009) and Editor for the International Journal of Control, Automation, and Systems (IJCAS) (2008-present).
