Adaptive fuzzy decentralized control for interconnected MIMO nonlinear subsystems

Automatica 45 (2009) 456–462

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Automatica journal homepage: www.elsevier.com/locate/automatica

Brief paper

Adaptive fuzzy decentralized control for interconnected MIMO nonlinear subsystemsI H. Yousef a , M. Hamdy b,∗ , E. El-Madbouly b , D. Eteim b a

Department of Electrical Engineering, Faculty of Engineering, Alexandria University, Alexandria-21544, Egypt

b

Department of Industrial Electronics and Control Engineering, Faculty of Electronic Engineering, Menofia University, Menof-32952, Egypt

article

info

Article history: Received 27 August 2007 Received in revised form 30 July 2008 Accepted 31 July 2008 Available online 20 December 2008 Keywords: Interconnected nonlinear MIMO subsystems Adaptive fuzzy control Multi-machine power system Stability

a b s t r a c t This paper describes an adaptive fuzzy control strategy for decentralized control for a class of interconnected nonlinear systems with MIMO subsystems. An adaptive robust tracking control schemes based on fuzzy basis function approach is developed such that all the states and signals are bounded. In addition, each subsystem is able to adaptively compensate for disturbances and interconnections with unknown bounds. The resultant adaptive fuzzy decentralized control with multi-controller architecture guarantees stability and convergence of the output errors to zero asymptotically by local output-feedback. An extensive application example of a three-machine power system is discussed in detail to verify the effectiveness of the proposed algorithm. © 2008 Elsevier Ltd. All rights reserved.

1. Introduction A large-scale system is usually constructed from many distributed subsystems, which are interconnected with each other. Such systems are widely employed in practice, for example, multiaxis machinery, electric power systems, chemical reactors, petrochemical systems etc (Mahmoud, 1985). Nonlinear interconnected systems are one of the most difficult to control in the category. The problems of stabilizing control synthesis for interconnected systems were considered in many publications. Decentralized control schemes present practical and efficient means for designing control algorithms based only on local information while computer networks provide an infrastructure for their realization. Real world plants are often dynamical systems with MIMO subsystems. Therefore the MIMO framework might be more appropriate than the SISO one. Fuzzy control has recently found extensive applications for a wide variety of industrial systems. Based on the universal approximation theorem (Wang, 1994), many important adaptive fuzzy control schemes have been developed to incorporate the

I This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Raul Ordonez under the direction of Editor Miroslav Krstic. ∗ Corresponding author. Tel.: +20 12 1374213; fax: +20 48 3660716. E-mail addresses: [email protected] (H. Yousef), [email protected] (M. Hamdy).

0005-1098/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2008.07.018

expert information systematically and guarantee various stable performance criteria (Cho, Park, & Park, 2002; Gao & Feng, 2005). The major advantages in all these fuzzy-based control schemes are that the developed controllers can be employed to deal with increasingly complex systems to avoid the computation of complicated regressor matrix, and to implement the controller without any precise knowledge of the structure of entire dynamic model. However, all these schemes have been applied to classes of nonlinear SISO systems (Chang, 2000; Cho et al., 2002; Díaz & Tang, 2004; Gao & Feng, 2005; Wang, 1994; Yousef, El-Madbouly, Eteim, & Hamdy, 2006) and for MIMO nonlinear systems (Chang, 2000; Xu & Ioannou, 2003). In Tseng’s paper (Tseng & Chen, 2001); the nonlinear interconnected SISO subsystems are presented by an equivalent Takagi–Sugeno fuzzy model. A state feedback decentralized fuzzy control scheme obtained through an observer is developed to override the external disturbances such that the H ∞ model reference tracking performance is achieved. Recently, various advanced nonlinear control technologies have been applied to excitation and steam valve controllers of power systems (Chapman, Ilic, King, Eng, & Kaufman, 1993; King, Chapman, & Ilic, 1994; Lu & Sun, 1989; Lu, Sun, Xu, & Mochizuki, 1996; Wang, Guo, & Hill, 1997). Through careful investigation it is easy to see that most of these are based on differential geometric tools, which cancel the inherent system nonlinearities in order to obtain a feedback equivalent linear system. It has been shown in the literature that the dynamics of power systems can be exactly linearized by employing nonlinear feedback and a state transformation. One can then use the conventional linear control

H. Yousef et al. / Automatica 45 (2009) 456–462

theory to design a controller in order to provide good performance (Chapman et al., 1993; King et al., 1994; Lu & Sun, 1989; Lu et al., 1996; Mak, 1992; Wang et al., 1997; Wang, Xie, Hill, & Middleton, 1992). However, these controllers suffer from some shortcomings. First, it is well known that the amplitude of the control signal is always bounded in the real world. In order to cope with this reality, the simulations in all of these researchers have used the limiting controller, in the form of saturation in the original state feedback controller. But the validity of the saturation of the controller cannot be proved theoretically in these papers. Second, since physical limitation on the system structure makes information transfer among subsystems infeasible, decentralized controllers for multimachine power systems must be used in practice. In this paper, an output-feedback adaptive fuzzy control for an interconnected system composed of N interconnected MIMO subsystems is proposed. The approximation capabilities of fuzzy basis functions have been exploited to provide asymptotic tracking for each subsystem. The adaptive fuzzy-based tracking controllers are constructed to guarantee boundedness of all the states of the closed-loop system. The paper is organized as follows. Section 2 describes the problem under investigation. Section 3 introduces the adaptive fuzzy decentralized control. An extensive application example of a three-machine power system is presented in Section 4, with conclusions given in Section 5. 2. Problem formulation Consider the following large-scale nonlinear system, composed of N interconnected MIMO square subsystems. The ith subsystem can be expressed in the following form: (1)

Yi = Hi (xi ) ni

where xi ∈ R is a state vector of the ith subsystem, Ui = [ui1 , ui2 , . . . , uip ]T ∈ Rp is an input vector, Yi = [yi1 , yi2 , . . . , yip ]T ∈ Rp is an output vector. The vector fields αi (xi ) = [αi1 , αi2 , . . . , αini ]T ∈ Rni , βi (xi ) = [βi1 , βi2 , . . . , βip ] ∈ Rni ×p , where βij (xi ) = [βij1 , βij2 , . . . , βijni ]T ∈ Rni and Hi = [hi1 , hi2 , . . . , hip ]T ∈ Rp are all assumed to be smooth functions and dim (x) denotes the interconnection among subsystems where x = [x1 , x2 , . . . , xN ]T , i = 1, 2, . . . , N, m = 1, 2, . . . , N, m 6= i, and j = 1, 2, . . . , p. The following definition is required. Definition. The system (1) has relative degree vector ri [ri1 , ri2 , . . . , rip ]T if

=

rij −2

Lβij` hij (xi ) = Lβij` Lαij hij (xi ) = · · · = Lβij` Lαij hij (xi ) = 0, rij −1

Now, we can rewrite the input–output equation (2) as: (ri1 )

= fi1 (xi ) + gi11 (xi )ui1 + · · · + gip1 (xi )uip + ∆i1 (x)

yi1

·

(3) (rip )

yip

= fip (xi ) + gi1p (xi )ui1 + · · · + gipp (xi )uip + ∆ip (x).

Denoting Gi (xi ) =

(r )

rij

yij ik = Lαij hij +

Lβij`

∀ xi ∈ Rni , ` = 1, 2, . . . , ni .



 rij −1 Lαij hij (xi ) uij + ∆ij (x)

r



yi1i1

rij

fi1 (xi )

"

 · =

#

ui1

" + Gi (xi )

·

rip

fip (xi )

yip

#

· uip

" # ∆i1 (x) · + . ∆ip (x)

Assumption 2. The plant has relative degree vector [ri1 , . . . , rip ]T ; and its zero dynamics are exponentially attractive (Sastry & Isidori, 1989). For the given reference trajectories yi1m , yi2m , . . . , yimp ; the tracking errors are defined as eio1 = yim1 − yi1 , . . . , eiop = yimp − yip , then, the error vector of the ith subsystem is given by eik = (r −1) [eiok , e˙ iok , . . . , eioki ]T . It is desired that the tracking errors follow r

(r −1)

rip eiop

(rip −1)

(r −2)

i1 eio1 + ci1ri1 eio1i1

+ ci1(ri1 −1) eio1i1

+ · · · + ci11 eio1 = 0 (5)

+ cip(r p −1) eiop

i

i

If coefficients cikr

ik

+ · · · + cip1 eiop = 0.

are chosen such that all the roots of

polynomials in (5) are stable, then limt →∞ |eik (t )| = 0. Control objectives: Determine a robust feedback control uik = u(xi |θik ) based on fuzzy logic systems and adaptive laws for adjusting the parameter vectors such that the tracking error for each subsystem will asymptotically converge to zero and all the states are uniformly bounded. 3. Adaptive fuzzy decentralized control In this section, the adaptive fuzzy decentralized controller is proposed. An approximation of the local control uik will be derived using the fuzzy logic system. The fuzzy logic system considered in this paper has centeraverage defuzzifier, product inference and singleton fuzzifier. This type of fuzzy logic system for the ith subsystem is given by (Wang, 1994):

n M −` P Qi `=1

hi

h¯ =1



M

ni

P

Q

`=1

h¯ =1

µF ` (xih¯ )



ik

 . µF ` (xih¯ )

(6)

ik

The fuzzy logic system in (6) is equivalent to fuzzy basis function (FBF) expansion where the fuzzy basis functions are defined as ni Q

rij −1

Define fij (xi ) = Lαij hij and gijk (xi ) = Lβijk Lαij hij (xi ), where the functions fij (xi ) and gijk (xi ) represent unknown nonlinear dynamics of the system, and ∆ij (x) term accounts for the interconnections among subsystems, k = 1, 2, . . . , p.

(rip −2)

+ cipr p eiop

(2)

and L2αij hij (xi ) = Lαij (Lαij hij (xi ))).

(4)

Assumption 1. Gi (xi ) is bounded away from singularity over a compact set Si ⊂ Rni specifically kGi (xi )k2 = tr(Gi (xi )T Gi (xi )) ≥ b21 > 0, where b1 represents the smallest singular value of the matrix Gi (xi ).

Z ( xi ) =

where Lαij hij (xi ) is the Lie derivative of hij (xi ) with respect to αij (e.g. Lαij hij (xi ) =

· ∈ Rp×p gi1p (xi ) . . . gipp (xi ) ·



j =1

∂ hij α (x ) ∂ xi ij i

#

then (3) can be written as

Using feedback linearization (Sastry & Isidori, 1989), one can differentiate the output equation rik times to get p X

gi11 (xi ) . . . gip1 (xi )

"

·

x˙ i = αi (xi ) + βi (xi )Ui + dim (x)

Lβij` Lαij hij (xi ) 6= 0,

457

ζ (Xi ) =

h¯ =1 M



µF ` (xih¯ )

ni

P

Q

`=1

h¯ =1

ik

 µF ` (xih¯ ) ik

(7)

458

H. Yousef et al. / Automatica 45 (2009) 456–462

where M is the number of if–then rules in the fuzzy rule base for each subsystem Si . The if–then rules take the following form: `

`

`

Ri : IF xi1 is Fi1 and . . . xini is Fini THEN `

hi is Gi ,

ui1 (xi |θi1 )

#

"

·

·  

`

× − 

ik

and µG` respectively, and hi is the linguistic variables which can be i

−`

considered as output of the fuzzy logic system. The parameter hi −`

is the point at which µG` ( hi ) achieves its maximum value where i

−`

µG` ( hi ) = 1. i

Consider the system (4), if fik (xi ), gijk (xi ) are known, |Gi (xi )| 6= 0, and ∆ik (x) = 0, the ideal feedback control given by: ui1

#

· uip

fi1 (xi )

" #! vi1 · = Gi (xi ) − + · fip (xi ) vip "

−1

#

(8)

r

" # vi1  · = · . rip vip yip yi1i1



(9)

The reference trajectory can be asymptotically tracked if the external inputs are chosen as: (r −1)

r

vi1 = yimi11 + ci1ri1 eio1i1 vip =

(rip −1)

+ cipri p eiop

r

yi1i1



rip yip

+ · · · + cip1 eiop .

(11)

T gˆijk (xi |θijk ) = θijk ζ (xi ).

(12)



· uip (xi |θip )

r

yi1i1



 · = rip yip

#−1

gˆi11 (xi |θi11 ) . . . gˆi1p (xi |θi1p )

" =

·

·

gˆip1 (xi |θijk ) . . . gˆipp (xi |θipp ) fˆi1 (xi |θi1 )

" # v¯ i1 + ·  · × −  v¯ ip fˆip (xi |θip )  



(13)

where the new external inputs [¯vi1 , ., v¯ ip ]T are chosen as:

v¯ ik = vik + `ik (t )sgn(zik (t )) + ηik (t )zik (t )/2

(16)

(14)

where the signals `ik (t ), zik (t ), and ηik (t ) are functions to be defined later (below we will drop the time index (t )). The terms `ik (t )sgn(zik (t )), (where sgn(x) = 1 if x > 0, sgn(x) = 0 if x = 0, sgn(x) = −1 if x < 0) are used to attenuate the fuzzy approximation errors, while the terms ηik (t )zik (t )/2 are used to compensate for unknown effects from the interconnections between subsystems.

fi1 (xi )

"

#

1 · + (Gi (xi ) + Gˆ i (xi ) − Gˆ i (xi ))Gˆ − i (xi ) fip (xi )    " # " # vi1 `i1 sgn(zi1 ) fˆi1 · × −  ·  + · + vip `ip sgn(zip ) fˆip " # " # ηi1 zi1 /2 ∆i1 + · · + ηip zip /2 ∆ip

(17)

which gives

Choosing an equivalence control as

#

(15)

Using (16), we can write (4) as

 · =

fˆik (xi |θik ) = θikT ζ (xi )

ui1 (xi |θi1 )

" # vi1 · vip

where b2 represents the smallest singular value of the matrix ˆ i (xi |θijk ). G

In this case the error dynamics take the form of (5). Unfortunately, the nonlinear functions fik (xi ), gijk (xi ) are unknown and ∆ik (x) 6= 0, so obtaining control law (8) is impractical. However, in this situation, one can approximate fik (xi ), gijk (xi ) with fuzzy logic systems fˆik (xi |θik ), gˆijk (xi |θijk ) defined as:

"

· fˆip (xi |θip )

+

2

ˆ

Gi (xi |θijk ) = tr(Gˆ i (xi |θijk ))T Gˆ i (xi |θijk ) ≥ b22 > 0

(10) rip yimp



" # " # `i1 sgn(zi1 ) ηi1 zi1 /2  · · + + `ip sgn(zip ) ηip zip /2    " # " # ui1 (xi |θi1 ) fˆi1 vi1 1    · · · = Gˆ − ( x |θ ) − + i ijk i uip (xi |θip ) vip fˆip " # " # `i1 sgn(zi1 ) ηi1 zi1 /2 . · · + + `ip sgn(zip ) ηip zip /2

 + · · · + ci11 eio1

·

fˆi1 (xi |θi1 )

ˆ i (xi |θijk ) is bounded away from singularity over a Assumption 3. G compact set Si ⊂ Rni specifically

with external input vik and yields the linearized systems



·

gˆip1 (xi |θijk ) . . . gˆipp (xi |θipp )

where Fik and Gi are the fuzzy sets with membership functions µF `

"

# −1

gˆi11 (xi |θi11 ) . . .gˆi1p (xi |θi1p )

=

uip (xi |θip )

for ` = 1, 2, . . . , M , i = 1, 2, . . . , N `

Using (14) in (13) yields:

"

fi1 (xi )

"

#

1 · + (Gi (xi ) − Gˆ i (xi ))Gˆ − i ( xi ) fip (xi )    " # " # fˆi1 vi1 `i1 sgn(zi1 )    · × − · + · + vip `ip sgn(zip ) fˆip    " # " # ηi1 zi1 /2 fˆi1 vi1 1  + Gˆ i (xi )Gˆ − −  ·  + · · + i ( xi ) ηip zip /2 vip fˆip  " # " # " # `i1 sgn(zi1 ) ηi1 zi1 /2 ∆i1 + · . · · + + (18) `ip sgn(zip ) ηip zip /2 ∆ip

Using (15) in (18) one can write the following equation



r

yi1i1





fi1 − fˆi1

 · = rip

yip

· fip − fˆip

  + (Gi (xi ) − Gˆ i (xi ))

ui1 (xi |θi1 )

"

#

· uip (xi |θip )

" # " # " # " # `i1 sgn(zi1 ) ηi1 zi1 /2 vi1 ∆i1 · · + + + · + · . (19) `ip sgn(zip ) ηip zip /2 vip ∆ip

H. Yousef et al. / Automatica 45 (2009) 456–462

Using (10) in (19) one can obtain the following error dynamics: ri1 eio1

 

(ri1 −1)

+ ci1r1 eio1

+ · · · + ci11 eio1

·



(ri1 −1)

rip eiop



(r −1)

(r −2)

+ cik(rik −1) eiokik

(20)

" e˙ ik = Aik eik + Bik (fˆik (xi |θik ) − fˆik (xi |θik∗ ))

+

p X ∗ (ˆgijk (xi |θijk ) − gˆijk (xi |θijk ))uij j=1

= (fˆik − fik ) + (ˆgi1 − gi1 )ui1 + · · · + (ˆgip − gip )uip − `ik sgn(zik ) − ηik zik /2 − ∆ik .

#

(21)

− `ik sgn(zik ) − ηik zik /2 + wik − ∆ik .

The state space equation of (21) can formulated as e˙ ik = Aik eik + Bik (fˆik (xi |θik ) − fik (xi )) +

(28)

From the universal approximation property, the minimum approximation errors wik is bounded by some finite Wik > 0, i.e., |wik | ≤ Wik . Then the tracking error dynamic equation (22) can be rewritten as:

+ · · · + cik1 eiok

"

p X ∗ (ˆgijk (xi |θijk ) − gijk (xi ))uij . j =1

The error dynamic equation of the kth error (k = 1, . . . , p) for the ith subsystem (i = 1, . . . , N) can be rewritten from (20) as: r

The minimum approximation errors can be written in terms of the optimal parameter estimates as:

wik = (fˆik (xi |θik∗ ) − fik (xi )) +

+ ci1rp eio1 + · · · + cip1 eiop   " # fˆi1 − fi1 ui1 (xi |θi1 ) ˆ   · · = + (Gi (xi ) − Gi (xi )) uip (xi |θip ) fˆip − fip " # " # " # `i1 sgn(zi1 ) ηi1 zi1 /2 ∆i1 · · − − − · . `ip sgn(zip ) ηip zip /2 ∆ip

ik + cikrik eiokik eiok

459

p X (ˆgijk (xi |θijk )

Using (11) and (12), the tracking error dynamics take the following form:

j=1

" #

− gijk (xi ))uij − `ik sgn(zik ) − ηik zik /2 − ∆ik

(29)

(22)

e˙ ik = Aik eik + Bik ΦikT ζ (xi ) +

p X

T Φijk ζ (xi )uij

j =1

# where

− `ik sgn(zik ) − ηik zik /2 + wik − ∆ik

 Aik = 



0 0

1 0

0 1

−cik(rik −1)

.

·

−cikrik

 · 0 · 0  , ·  ·−cik1

0

.

Bik =  ..  .

(23)

1

Assumption 4. The interconnection function ∆ik (x1 , x2 , . . . , xN ) is assumed to satisfy the following condition

∆ik (x1 , x2 , . . . , xN ) ≤ dik (t ) + δik (x1 , x2 , . . . , xN ) (24)

PN Pp ik

where |δik (x1 , x2 , . . . , xN )| ≤ j=1 s=1 qjs ejs 2 (k.k2 is the j6=i

Euclidean vector norm). The scalars qik js (i = j = 1, . . . , N, and k = s = 1, . . . , p) quantify the strength of the interconnections. Let `∗ik = sup[dik (t )] − inf[dik (t )]/2 be a measure of the variation of dik (t ) and d¯ ik = sup[dik (t )] + inf[dik (t )]/2 be a measure of the center of dik (t ). We may thus let dik (t ) = position

d¯ ik + d˜ ik (t ), where d˜ ik ≤ `∗ik , with `∗ik assumed to be bounded. Also, if d¯ ik is nonzero, one can include d¯ ik into the first vector of

the right hand side of (4) as an unknown constant. Within the adaptation algorithm, the bound `∗ik will be estimated by `ik , with the corresponding parameters error defined as ϕ`ik = `ik − `∗ik . ∗ Also define Hi∗ = [ηi1 , . . . , ηip∗ ] as a vector of desired feedback gains. Each ηik will be adapted to achieve these feedback gains with ∗ parameter errors ϕηik = ηik − ηik . Define the optimal parameter estimates θik ∗ , θijk ∗ as follows (Wang, 1994):

θik ∗ = arg min



θik ∈Ωik

θijk = arg min ∗

θijk ∈Ωijk





sup fˆik (xi |θik ) − fik (xi )

(25)

xi ∈Uc



sup gˆijk (xi |θijk ) − gijk (xi )

where Mki , Mijk are design parameters

θ˙ik = −γik eTik Pik Bik ζ (xik )

(31)

θ˙ijk = −γijk eTik Pik Bik ζ (xik )uik 1 T e Pik Bik `˙ ik = ik

(32)

q`ik

η˙ ik =

1 2qηik

(eTik Pik Bik )2

(33) (34)

where γik , γijk , q`ik , and qηik are positive constants and Pik is a solution of the following Lyapunov matrix equation Pik Aik + ATik Pik = −Qik .

(35)

The updating laws (31)–(32) are used to estimate the dynamics of the subsystem under control using fuzzy systems, while (33)–(34) are used to attenuate the fuzzy approximation error and to stabilize the subsystem by estimating the effects of the interconnections. Theorem 1. For the large-scale nonlinear system (1), the adaptive fuzzy decentralized control is chosen as (16) with parameter updating laws for each subsystem defined by (31)–(34) and the interconnections satisfy (24), then the tracking error for each subsystem will asymptotically converge to zero and all the states of each subsystem will be uniformly bounded. 

(26) 4. Simulation results

where Ωik , Ωijk are the compact sets defined as

Ωijk = {θijk : kθijk k ≤ Mijk }

where Φik = (θik − θik ∗ ), Φijk = (θijk − θijk ∗ ). The parameter adaptation laws are chosen as:

Proof. See Appendix.



xi ∈Uc

Ωik = {θik : kθik k ≤ Mik } ,

(30)

 

(27)

In this section, we shall concentrate on enhancing the transient stability of power systems by means of adaptive fuzzy decentralized nonlinear control.

460

H. Yousef et al. / Automatica 45 (2009) 456–462

Generator #2. xd = 2.36 p.u. x0d = 0.319 p.u. xq = 0.70 p.u. x0q = 0.20 p.u. xT = 0.110 p.u., Tdo = 7.96 p.u., H = 5.1 s, kc = 1 and D = 3 p.u. It is easy to verify that this system has relative degrees of ri1 = 3, ri2 = 1. Seven Gaussian fuzzy membership functions are defined for each state variable: For δi 1

µF 1 =

1 + exp(5(xi1 + π /2))     µF 3 = exp −(xi1 + π /4)2 , µF 2 = exp −(xi1 + 3π /8)2 , i1 i1     µF 5 = exp −(xi1 − π /4)2 , µF 4 = exp −(xi1 )2 , i1 i1   µF 6 = exp −(xi1 − 3π /8)2 , i1

Fig. 1. The rotor angles for generators #1 and #2.

Consider an N-machine power system which can be modeled by the following equation (Bergen, 1986; Wang et al., 1992; Xi, Feng, Cheng, & Lu, 2003):

i1

 x˙ i1 = xi2   n  X    ˙ x = − a x + c + b x − b x Bij xj4 sin(xi1 − xj1 )  i2 i i2 i i i3 i i4 

µF 7 = i1

j =1

(36)

x˙ i3 = −eei xi3 + ki + ui1    n X    ˙  x = − h x + d Bij xj4 cos(xi1 − xj1 ) + ui2 i4 i i4 i 

as state variables, ui1 =

CHi u THSi Hi

and ui2 =

kci T0

d0i

Efi as controls,

yi1 = δi (t ) and yi2 = Vti (t ) as outputs and denote ai = ci =

w0 CMi Pm0i Hi

, di =

(xdi −x0di ) , T0 d0i

eei =

1 , THSi

ki =

CHi Pm0i , THSi

Di , Hi

hi =

bi =

1 0 Td0i

w0 Hi

,

where

δi is the power angle between the q-axis electrical potential vector EEqi and a reference bus voltage vector VEREF in the system in rad; wi is a rotating speed of the ith generator, in rad/s; PHi is a mechanical 0

power of high-pressure (HP) turbine, in per unit; Eqi is a q-axis internal transient electric potential of the ith generator, in per unit; Pm0i is the initial mechanical power of the ith generator, in per unit; Hi is the moment of inertia in second; CHi is the power fraction of HP turbine; Bij represents ith row and jth column element of the nodal susceptance matrix, which is symmetric, at the internal nodes after eliminating all physical buses, in per unit; Pei is the electric power; THSi = THgi + THi is the equivalent time constant of HP turbine where THgi is the time constant of oil-servomotor of regulated valve of HP turbine and THi is the time constant of HP turbine; uHi is the electrical control signal from the controller for the regulated valve; Eqi is the EMF in the quadrature axis in per 0 unit; Efi is the equivalent EMF in the excitation coil in per unit; Tdoi is the direct axis transient short circuit time constant in seconds; xdi is the direct axis reactance of the ith generator in per unit; x0di is the direct axis transient reactance of the ith generator, in per unit; Qei is the reactive power, in per unit; Idi is the direct axis current, in per unit; Iqi is the quadrature axis current, in per unit; kci is the gain of the excitation amplifier, in per unit; ufi is the input of the SCR amplifier of the ith generator, in per unit; xadi is the mutual reactance between the excitation coil and the stator coil of the ith generator, in per unit; xTi is the transformer reactance; xij is the transmission line reactance between the ith generator and the jth generator; and Vti is the terminal voltage of the ith generator. A three-machine example system is chosen to demonstrate the effectiveness of the proposed adaptive fuzzy decentralized controller. The system parameters used in the simulation are as follows (Wang et al., 1997).

ω0 = 314.159,

XL12 = 0.55,

XL13 = 0.53,

1

µF 4 µF 6

  = exp −(xiz − 0.6)2 ,

µF 2

iz

iz

iz

1 + exp(5(xiz + 0.8))

iz

µF 7 = iz

1 1 + exp(−5(xiz − 0.8))

.

0 For Eqi (t )

µF 1 =

1

µF 4

,     = exp −(xi4 + 3)2 , µF 3 = exp −(xi4 + 2)2 , i4     = exp −(xi4 )2 , µF 5 = exp −(xi4 − 2)2 ,

µF 6

  = exp −(xi4 − 3)2 ,

i4

µF 2

i4

i4

i4

1 + exp(5(xi4 + 4))

i4

µF 7 = i4

1 1 + exp(−5(xi4 − 4))

for i = 1, 2, and z = 2, 3. For each subsystem, we define seven fuzzy rules, in the following format: R`i : IF xi1 is Fi1` and . . . xi4 is Fi4` THEN hi is G`i ,

` = 1, 2, . . . , 7, i = 1, 2, and k = 1, . . . , 4. P7 Q 4 Denoting Di1 = ` (xik ), then the FBF used to generate `=1 k=1 µFik approximations for both fik (xi ) and gijk (xi ) can be written as: " #T 4 4 Y Y ζ (Xi ) = µF 1 (xik )/Di1 , . . . , µF 7 (xik )/Di1 . i=1

ik

i=1

ik

Then, we can construct the fuzzy systems as follows. Choose c113 = c213 = 5, c112 = 15, c111 = c212 = c211 = 20, c121 = c221 = 25, q`ik = 10, and qηik = 50. Let Qi1 = diag(1, 1, 1), Qi2 = 1 and solve the Riccati-like equation (35) to get the following positive definite matrices and scalar values:

"

4.8983 −0.5000 −0.8746

−0.5000 0.8746 −0.5000

"

4.5380 −0.5000 −0.6595

−0.5000 0.6595 −0.5000

P11 =

XL23 = 0.6.

Generator #1. xd = 1.863 p.u. x0d = 0.257 p.u. xq = 1.46 p.u. x0q = 0.546 p.u. xT = 0.129 p.u., Tdo = 6.9 p.u., H = 4 s, kc = 1 and D = 5 p.u.

.

,     = exp −(xiz + 0.6)2 , µF 3 = exp −(xiz + 0.4)2 , iz     = exp −(xiz )2 , µF 5 = exp −(xiz − 0.4)2 ,

iz

0 where xi1 = δi , xi2 = wi (t ) − wo , xi3 = PHi (t ), and xi4 = Eqi (t )

1 + exp(−5(xi1 − π /2))

For (wi (t ) − wo (t )) and PHi (t )

µF 1 =

j=1

1

P21 =

and Pi2 = 0.02.

# −0.87460 −0.5000 0.6186 # −0.6595 −0.5000 0.6899

H. Yousef et al. / Automatica 45 (2009) 456–462

Fig. 2. The bus terminal voltages at generators #1 and #2.

461

Fig. 3. The control signals u11 , u21 for generators #1 and #2.

Suppose the operating point is given as (0.9076, 0, 0.246, 1.05, 0.9076, 0, 0.236, 1.12), whereas θik (0) = θijk (0) = 0 and the adaptation gains are selected as γik = γijk = 0.01 (i = k = j = 1, 2). The control input limitations are supposed as:

−0.05 ≤ u11 ≤ 0.05, −0.05 ≤ u21 ≤ 0.05, −0.582 ≤ u12 ≤ 0.582, −1.229 ≤ u22 ≤ 1.229. The system response under a three-phase short circuit fault is tested. The following temporary fault sequence is used in the simulation studies: Stage 1: The system is in a pre-fault steady state. Stage 2: A three-phase short circuit fault occurs at t = 2.1 s. Stage 3: The fault is removed by opening the circuit breakers at t = 2.2 s. Stage 4: The transmission line is restored at t = 3.4 s. Stage 5: The system is in a post-fault state. The fault we will consider in this work is a symmetrical threephase to ground short circuit fault, which occurs at the transmission lines. The fraction of the faulted line to the left of the fault is denoted by β . If β = 0 the fault is at bus 4. If β = 0.5 puts the fault in the middle of the line between bus 4 and bus 5, and so on. The simulation results concerning the dynamic behavior of the three-machine system under the proposed control law (16) and updating parameters (31)–(34) are shown in Figs. 1–4, where a typical symmetrical three-phase short-circuit fault occurring on the transmission line near bus 4 is considered. It can observe from Figs. 1 to 4 that with the proposed adaptive fuzzy decentralized control, the dynamic system is stabilized under the fault occurring at 2.1 s, cleared at 2.2 s, and restored at 3.4 s. We notice that the resulting operating point can be restored if the transmission line is restored. This indicates that restoring transmission line is useful for dynamical performance of closed loop system. The simulation results show that, the proposed adaptive fuzzy decentralized control can enhance the transient stability, enrich damping and, simultaneously achieve good postfault generator terminal voltage. 5. Conclusion An output-feedback decentralized adaptive fuzzy tracking control has been proposed for a class of large-scale nonlinear system with MIMO subsystems. The adaptive fuzzy-based tracking controllers are constructed to guarantee boundedness performance. The scheme can not only guarantee closed-loop stability but also asymptotic zero tracking errors. Finally, a three-machine power system has been presented to illustrate the proposed adaptive fuzzy decentralized control can enhance the transient stability, enrich damping and simultaneously achieve good post-fault generator terminal voltage.

Fig. 4. The control signals u12 , u22 for generator #1 and #2.

Acknowledgments The authors would like to thank Prof. O. A. Sebakhy, of Alexandria University, for his help, as well as the Editor and the reviewers for their inspiring encouragement and constructive comments, which have contributed much to the improvement of the clarity and presentation of this paper. Appendix. Proof of Theorem 1 Consider the Lyapunov function candidate V = V1 + V2 + · · · + VN ,

and

Vi = Vi1 + Vi2 + · · · + Vip ,

i = 1, . . . , N ,

where each Vik , k = 1, . . . , p, is defined as: Vik = eTik Pik eik +

1

γik

ΦikT Φik +

p X 1 T Φ Φijk + q`i ϕ`2i + qηi ϕη2i . γijk ijk j =1

(A.1) Differentiating V , Vik , we obtain V˙ ik = (˙eTik Pik eik + eTik Pik e˙ ik ) +

+

2

γik

˙ ikT Φik Φ

p X 2 T ˙ ijk Φijk + 2q`i ϕ`i ϕ˙ `i + 2qηi ϕηi ϕ˙ ηi . Φ γ ijk j =1

(A.2)

462

H. Yousef et al. / Automatica 45 (2009) 456–462

˙ ik = Using (24), (30), (33) and (34), with ϕ˙ `ik = `˙ ik , ϕ˙ ηik = η˙ ik , Φ ˙θik , Φ ˙ ijk = θ˙ijk , we get: V˙ ik = eTik (Pik Aik + Aik Pik )eik + 2eTik Pik Bik [−d˜ ik (t )

− δik (x1 , x2 , . . . , xN ) − `ik sgn(zik ) − ηik zik /2] + 2ϕ`i eTik Pik Bik + ϕηi (eTik Pik Bik )2 + 2eTik Pik Bik wik +

2

γik

(γik eTik Pik Bik ζ (xik ) + θ˙ikT )Φik

p X 2 T (γijk eTik Pik Bik ζ (xik )uik + θ˙ijk )Φijk . (A.3) γ ijk j =1 Choosing zik (t ) = eTik Pik Bik , `∗ik = d˜ ik + Wik , and if the parameter

+

updating laws are chosen as (33) and (34), and use (35) then from (A.3) we get ∗ T V˙ ik ≤ −eTik Qik eik − ηik (eik Pik Bik )2 − 2eTik Pik Bik δik

+ 2eTik Pik Bik wik − 2 eTik Pik Bik (`∗ik − |d˜ ik (t )| − wik ).

(A.4)

Note that the last term in (A.4) is negative and if each ηik > 0, we simply obtain ∗

V˙ ik ≤ −eTik Qik eik +

1

ηik∗

δik2 (x1 , . . . , xN ).

(A.5)

The composite ith subsystem Lyapunov candidate Vi : V˙ i ≤

p X

 −eTik Qik eik +

k=1

Since

Pp

s=1

1

p N X X

ηik∗

j=1 s=1

js

2

!2  .

(A.6)



Pp ik

ej , where qik = maxs qik and ej = q e ≤ qik j j js

s= 1 js js 2

ejs , Eq. (A.6) can be rewritten as 2

p

V˙ i ≤

qik ejs

X

−λik keik k22 +

k=1

1

T T ∗ Θ Γik Γik Θ

ηik

 (A.7)

where λik is the real part of the

valueT of Qik with the PN ik eigen

= Θ Γik , where Θ := minimum magnitude, and q e j j =1 j

[ke1 k , . . . , keN k]T and Γik = [qi1k , . . . , qiNk ]T . ∗ There exist some Hi∗ = [ηi1 , . . . , ηip∗ ] such that (A.7) is negative ∗ semi-definite. To show this, let ηik = P ηi , k = 1, . . . , p for some ηi > 0. Define λi := mink λik , Mi = pk=1 Γik ΓikT , Eq. (A.7) may be written as V˙ i ≤ −λi kei k22 +

1

ηi

Θ T Mi Θ .

(A.8)

PNNow consider the composite system Lyapunov candidate V =

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H. Yousef received the B.S. and M.S. degrees in Electrical Engineering from Alexandria University, Alexandria, Egypt, and the Ph.D. degree in Electrical Engineering from the University of Pittsburgh, Pittsburgh, PA, USA, in 1979, 1983 and 1989 respectively. He is currently a Professor with Electrical Engineering department, Faculty of Engineering, Alexandria University, Alexandria, Egypt. His research interests include intelligent and adaptive control, fuzzy control applications to electrical drive systems, large scale systems, and nonlinear control. M. Hamdy received the B.S., M.S. and Ph.D. degrees in Automatic Control Engineering from Menofia University, Egypt, in 1995, 2002 and 2007 respectively. He is currently an Assistant Professor with Industrial Electronics and Control Engineering department, Faculty of Electronic Engineering, Menof, Menofia University, Egypt. His research interests include adaptive control, intelligent control systems, large scale systems, and nonlinear control.

References

E. El-Madbouly received the B.S. and M.S. degrees in Nuclear Engineering from Alexandria University, Alexandria, Egypt, and the Ph.D. degree in Electrical Engineering from the University GH-Duisburg, Germany, in 1968, 1975 and 1983 respectively. He is currently a Professor with Industrial Electronics and Control Engineering department, Faculty of Electronic Engineering, Menof, Menofia University, Egypt. His research interests include automatic control and its industry applications, and measurements.

Bergen, A. R. (1986). Power systems analysis. Englewood Cliffs, NJ: Prentice-Hall. Chang, Y. C. (2000). Robust tracking control for nonlinear MIMO systems via fuzzy approaches. Automatica, 36, 1535–1545. Chapman, J. W., Ilic, M. D., King, C. A., Eng, L., & Kaufman, H. (1993). Stabilizing a multimachine power system via decentralized feedback linearizing excitation control. IEEE Transaction on Power Systems, 8, 830–839. Cho, Y., Park, C. W., & Park, M. (2002). An indirect model reference adaptive fuzzy control for SISO Takagi–Sugeno model. Fuzzy Sets and Systems, 131, 197–215. Díaz, D. V., & Tang, Y. (2004). Adaptive Robust fuzzy control of nonlinear systems. IEEE Transactions on Systems, Man, and Cybernetics, Part B, 34, 1596–1601.

D. Eteim received the B.S., M.S. degrees in Automatic Control Engineering from Menof, Menofia University, Egypt, and the Ph.D. degree in Automatic Control Engineering from Slovak Technical University, Bratislava, Czech, in 1978, 1984 and 1991 respectively. He is currently an Assistant Professor with Industrial Electronics and Control Engineering department, Faculty of Electronic Engineering, Menof, Menofia University, Egypt. His research interests include automatic control and its industry applications, and large scale systems.

Vi and using the same arguments as in Spooner and Passino d kek2 = eT e˙ / k˙ek2 ≤ kek2 ∈ L∞ . Using (1996), we conclude that dt Barbalat’s lemma, we thus establish that limt →∞ Θ = 0 ∈ RN , thus we are guaranteed asymptotically stable tracking for each subsystem. i=1

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