modified pole-placement controller via takagi-sugeno fuzzy modelling approach: applied to a load...

8/12/2019 Modified Pole-Placement Controller via Takagi-Sugeno Fuzzy Modelling Approach: Applied to a Load Frequency Sta

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that

su px(t),u(t),w(t)

f(x(t), u(t), w(t))(x(t), u(t), w(t)) (2)

where is an arbitrary small positive number. To brieflyexplain the TS fuzzy concept, let

i = (A i,Bi,D i) (3)

be linear space, where A i,B i andD i are parametric ma-

trices of the ith local linear model, with i = 1,2, , q be-ing an index of triplets, then a TS fuzzy system mapping

function at an instant of time would yield the following.

(x(t), u(t), w(t))= A()x(t)+B()u(t)+D()w(t) (4)

The argument i in the right side of(4) is fuzzy function.A local linear model as output in response to inputs in a

local i th sector of a domain is represented using a rule of

the form Rule i (Ri).

1 if (z1(t) is Mi1 && z2(t) is M

i2 && zs(t) is M

is)

2 then x(t)= Aix(t)+Bi u(t)+Di w(t)

where Ri denotes the ith fuzzy rule, where z(t) =

[z1(t),z2(t), ,zs(t)] is premise input variable, Mij =

[Mi1

,Mi2

, ,Mis] is fuzzy linguistic value quantifyingmembership degree of an input. Many types of func-

tion shapes are used, popular one is the triangular shape

shown in Fig.1.

Fig. 1. A triangular fuzzy membership function

Mathematically the shape is expressed as

M(zi)= max

0, 1+

|zj cj|

0.5j

, j = 1,2, , q (5)

wherecj, j are centre and width of the triangular shape.The overall TS fuzzy model resulting fromR i after com-

bination and aggregation can be written as

x(t)=R

i=1

i (z(t)) (A ix(t)+Bi u(t)+D i w(t)) (6)

where

i (z(t))=i (z(t))R

i=1 i (z(t))(7)

and

i (z(t))=sj

Mij (z(t)) (8)

2.1 State Feedback Control law and Fuzzy Closed

Loop System

In order to evolve the fuzzy based poleplacement con-

troller, we recalled the concept the Parallel Distributed

Compensation (PDC) used in TS fuzzy based control [12].

In PDC a jth control law written as

uj =j (z(t))Kjx(t), j = 1,2, ,R (9)

is design to compensate an ith TS fuzzy rule of Ri.Where Kj is state feedback gain, j is j

th fuzzy scal-

ing factor as defined in (6) and R is number TS fuzzy

rules. If the linear quadratic regulator (LQR) or poleplacement design can be use to determine Kj the systemin (6)with (9)involved is written in a closed loop system

as(10).

x(t)=R

i=1

Rj=1

ij (A i BiKi)x(t) (10)

In this paper, we shall determine the gain in (10)using

fuzzy pole placement in the context of state estimation

technique.

2.2 Feedback Gain Design Through Pole

Placement

Let the i th local linear model inR i behave as define by

the characteristic equation

q(s) = s2 +5s+6

= 0(11)

Based on Ackermanns idea [13], we write the gain as

Kj = qjAj

(12)

Where qi is jth last row of the jth controllability matrix

and

Aj= Anj + an1A

n1j + +a0I2. (13)

Having specified the desired dynamics in (11), we coulduse MATLAB command to generate the appropriate state

feedback gain for system in(10).

3 Application

To test the effectiveness of the control scheme, we con-

sider the problem of maintaining frequency stability in

power system network model shown in Fig. 2.

Fig. 2. Single area power system network

Describing the system in state space form

x = Ax +Bu (14)

y = Cx (15)

where A is n nsystem matrix, B is n pinput matrix,C is m n output matrix and y is system output. Werecalled the approximate parametric linear model of the

network reported in reference [14] as

x1x2 =

0 12n 2n

x1x2+0

1PL (16)

y =

1 0

0 1

x1x2

(17)

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The parametersn and are

n =

foPs

H (18)

=D

2

fo

HPs(19)

The meanings of the system variables and parametersare given in Table1.

Table 1. System variables and parameters description

Variable Description

x1 Rotor angle inr ad

x2 Frequency in radse c

n Natural frequency in radse c

Damping ratio

D Damping constant

H Per unit inertia constant

fo Operating frequency inH z

PL Per unit input load deviationPs Synchronizing power coefficient

3.1 The System TS Fuzzy Model

To set up the TS fuzzy rule base, we first assume that

the inputs to the rule premises are the states(x1,x2). Weselect four operating points x0 =

x01,x02,x03,x04

to es-

tablish the required four local linear models in the out-

put or consequence of the TS fuzzy rules. If we assign

two fuzzy linguistic values of positive small (PS) and pos-

itive (P) to each input with notations

x1: M11 ,M

21 (20)

x2:M12 ,M22

(21)

we then formulate a total of four TS fuzzy rules exclud-

ing the disturbance as follows.

1 if (x1 is M11 && x2 is M

21)

2 then x=A1x+B1 u

3 if (x1 is M11 && x2 is M

22)

4 then x=A2x+B2 u

5 if (x1 is M21 && x2 is M

12)

6 then x=A3x+B3 u

7 if (x1 is M21 && x2 is M

22)

8 then x=A4x+B4 u

The overall T-S fuzzy model would take the form of(6).

4 Simulation

To set up the closed loop system for numerical evalua-

tion, we state the following remarks.

1. The universe of discourse of the state is

X= [0.001,1].

2. Membership function is triangular expressed in (5).

In Table2 are the states, centres of membership func-

tion shapes and operating points as in 2011,Shehu and

Dan-Isa proposed [15] here with slight modification in

range and notations. The parameters specifications are

given in Table3.

Using the operating points we obtain the four distinct

state matrices A =A i

in(22), (23), (24) and(25); and

B =Bi

in(26) respectively.

Table 2. Fuzzy system set variables and parameters values

States CTMF OP

x1 = [ 110

1] M11 = [0 1

814

]

M21 = [18

14

12

]

x11 = 11100

x21 = 14

x31 = 920

x41 = 12

x2 = [ 3

10009

1000] M12 = [0

1400

1200

]

M

2

2 = [

1

500

1

200

1

100 ]

x12 =

11000

x22 =

1200

x

3

2 =

7

1000 x

4

2 =

1

100

CTMF: Centres of Triangular Membership Functions

OP: Operating Points

Table 3. Power network parameter constants

V= 1 p.u. X= 1320

p.u. fo = 50H z

D = 69500

H= 8 89100

M JMV A

E o = 1 720

p.u.

o = 775

ra d cos = 45

Ps = 1 99500

p.u.

maximum generator electromagnetic field

A1 = 0 1

1000

3 864710000

31250

(22)

A2 =

0 1

200

8 39175000

615000

(23)

A3 =

0 7

1000

15 810110000

17110000

(24)

A4 =

0 1

17 14172500

612500

(25)

B1 =B2 =B3 =B4 =

0

1

(26)

4.1 Simulation Results

We may evaluate the system performance in cases as

follows.

Case 1: Openloop with nominal parameter values

(D = 0.138,Ps = 1.988 p.u.).

Case 2:Openloop with damping reduced (D = 0.0138,Ps = 1.988 p.u.).

Case 3: Openloop with negative power coefficient(D = 0.0138,Ps =1.988 p.u.).

Case 4: Closedloop with the designed controller

under worst parameter condition (D = 0.0138, Ps =

1.988 p.u.).

Applying1 p.u.load demand, we obtain the system re-

sponses in Fig.3.

5 Discussion

To understand the capability of the designed controller

we first obtain results of the networks frequency when

in open loop condition at nominal and varied parameter

conditions. At nominal parameter values, the networkresponse of the rotor angle is shown in Fig.3(a) and the

load frequency in the steady state settles at the pre-set

value of50 H zas shown in Fig.3(b).

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IJAEPR 2(1):15, 2014

Fig. 3. Openloop network response of the (a) rotor angle and (b) frequency at nominal D and Ps; openloop network response of the(c) rotor angle and (d) frequency at varying parameter values of D = 0.0138 (D reduced to 10%) and Ps = 1.988 (nominal); openloopnetwork response of the (e) rotor angle and (f) frequency at varying parameter values ofD = 0.138 (nominal) and Ps = 1.988 (varied worst case); and closedloop system response of the (g) rotor angle and (h) frequency at worst parameter settings of D= 0.0138 (Dreduced to 10%) and Ps =1.988(worst case).

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International Journal of Applied Electronics in Physics & Robotics

This performance was also observed when damping

is reduced however, with severe oscillation during the

transient phase of the rotor angles response as shown

in Fig. 3(c), and the response of the frequency is shown

in Fig.3(d). At negative power coefficient the rotor angle

and frequency results are unbounded (unstable) as seen

in Fig.3(e) and Fig.3(f).

Unlike in the case of PDC in the frame work of Lin-ear Matrix Inequality result reported by Shehu and

Dan-Isa [15], here the fuzzy pole placement based con-

troller even at the combine reduced damping and nega-

tive power coefficient parameter values it stabilizes the

power network, providing good transient and excellent

steady state performances (see Fig.3(g) and Fig.3(h)).

6 Conclusion

In the paper we presented a fuzzy pole placement based

controller design. We applied the designed controller for

the control of load frequency in a single machine and sin-

gle area power generation network. First we represent

the network with its equivalent TS fuzzy model. Basedon the fuzzy model, we formulate a PDC design deter-

mining the state feedback gain using a conventional pole

placement technique. Simulation results obtained shows

that the controller stabilizes the power network despite

severe changes in the power networks parameters val-

ues.

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modified pole-placement controller via takagi-sugeno fuzzy modelling approach: applied to a load...

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