Pole placement in lmi region with takagi-sugeno fuzzy systems

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POLE PLACEMENT IN LMI REGION WITH TAKAGI-SUGENO FUZZY SYSTEMS Elvia Palacios .,1 Andre Titli .,.'

• f,AAS-CNRS. 7, avenue dll Colonel Roche - 31077, TOllloll.~e Ceder 4, FRA NCE. [email protected] •• fNSA de TOlllollM~. 1.'15, avenue de Ranglleil - 31077, TOlllouse Cede:r: 4, FRA NCE. [email protected]

Ahstract: A fuzzy state feedback control law that places the poles of the c1osedloop subsystems in Takagi-Sugeno (TS) fuzzy systems in a convex region is proposed in this paper. This convex region is defined by the intersection of three elementary regions: an Q-st.ahility region, a disk, and a conic sector, mathematically characterized hy linear matrix inequalities (LMls). Sufficient conditions are expressed in terms of LMls, and the resulting formulation is therefore numerically tractahle via LMI optimization. The validity and application of this approach are illustrated on the inverted pendulum and the ma.~s-spring­ damper examples. Gopynqht @20D.9 fFAG I,eywords: Nonlinear Systems, Fuzzy Control, Pole Placement, LMI R.egion, LMls, TS fuzzy system, Parallel Distributed Compensation (PDC)

1. INTR.ODUCTION

hy the Iinearization of the non linear plant in different operating points. Then, original behavior is ohtained by "hlending" the local suhsystems. Using a Takagi-Sugeno model, the design of stahilizing fuzzy controllers is made by the resolution of linear matrix inequalities, applying the PDC technique (Wang et al., 1995). The state feedhack control law is used a~ local controller. There exist several ways to det.ermine t.he gain matrices of the local cont.rollers. The first one is t.o place the poles of each suhsystem and to verify t.he glohal stahility of the closed loop system. The second one is to find glohally stahilizing gains satisfying the contraint.s on pole placement. In (Farinwata et at., 2000) the concept. of LMls pole placement constraints a.~ performance specifications for the synthesis of PDC-type TS fuzzy control system is introduced. This paper ext.ends the work of (.Joh et aI., 1997) and (.Joh et al., 1998). They propose a state feedhack control law with pole placement. This proh-

The minimum requirement for a cont.rol system is the stahility. Moreover, a good controller should deliver sufficiently fast and well-damped time responses. A technique that guarantees satisfactory transients is to place the closed-loop poles in a suitahle region of the complex plane. We refer to this technique a.~ regional pole placement (Chilali and Gahinet, 1996). Because real systems always involve some non Iinearities in their dynamics, it is natural to worry ahout t he design of a non linear controller. Fuzzy controllers are rule-ha~ed nonlinear controllers, therefore their main application should he the control of non linear systems. The Takagi-Sugeno (TS) fuzzy systems have a great impact in the area of nonlinear systems. thanks to their capahility to approximate nonlinear dynamics and the ea.~y way to ohtain them 1 Work SIJPPOrlecJ by CONACyT, Mexico.

243

we can write (l) as:

lem consist to fix the poles of the each linear suhsystem of the TS model in a given convex region of the complex plane. In this article the chosen region is an area as showed in figure (1). This area is characterized hy 3 parameters ((}, p, 0). Defining this parameters ((}, p, 0) of the convex region, the state feedback gain matrices Fi with pole placement can he determined. This region R( (}, p, 0) is the intersection of three elementary regions: an 0stahility region (Re( s) Q), a disk of a radius p ( corresponding to a maximum natural frequency W n = p) and a conic sector 0 ( corresponding to a minimun damping ratio ~ = cos B). The rest of t he paper is organ ized as follows. Section 2 introduces the TS fuzzy systems, and the concept of LMI regions. Section 3 presents pole placement in LMl region for TS fuzzy systems. Two numerical examples are shown in section 4. Finally, conclusions are given in section 5.

x(t) = L

where

Qi

u(t)

i=1

r

=

QAT

QAT

w;(t)

Li=1

(7)


=

:s r.

=

FiQ, i 1,2, ... r, are the new where Q and V; matrix variables of the LMIs.

(1)

The proof of thi.~ theorem is in (Farinwata et al., 2000). To find Q and V; satisfying (7) or to determine if a such Q does not exist (in consequence the matrix P), is a convex prohlem called the LMI fea.<;ihility problem. An LMI is any constraint of the form:

(2)

and where Ilij is the memhership function of the jth fuzzy set in the i th rule. Defining:

= =r'---::"-'---'---

<0

+ AiQ + QA~ + AjQ+ i

j=1

(t )

~~TBT + Bi~~

= 1,2, ... r

VI Br + BYj + ~~T Bj + BjV; < 0,

n

() i

+ AiQ+ i

w;(t)

= IT llij(X.j{t))

j=1

Theorem 1. The equilihrium state of the continuoustime TS fuzzy system (6) (namely, x = 0), where the F;'s are unknown is glohally a<;ymptotically stahle if there exist a common symetric positive definite matrix Q p- 1 > 0 and V; matrices which satisfy:

where Wi is defined as:

lIJi(t)

(5)

The sufficient conditions for the stahility of (6) are the following:

L~-1 Wi(t)C;X(t) Li=l

LQi(t)Fix(t)

r

=

=

=-

x(t) = LLQi(t)Qj(t)(A i - RiFj)x(t) (6)

E Rn is the state vector with components x.j,j 1 to n, i l, ... ,r,r is the numher of rules, A/ij are multivariahle input fuzzy sets, Ai E R nxn , Ri E R nxm , the input vector 1/(t) and y the output. Using singleton fuzzifiers, maxproduct inference and center average defuzzifier, we can write the aggregated model a.<;:

Y

= 1.

The closed-loop TS fuzzy system is descrihed a.<; follows:

X

+ BiU(t))

Qi

i=1

X = AiX(t) + Biu(t) Y = GiX(t)

L~=l w;(t)

L~=1

> 0 and

The overall controller will he:

i lh Plant Rule: IF Xl (t) is M il and ... ,xn(t) is M in THEN

L~=1 wi(t)(Aix(t)

Oi(t)CiX(t)

i lh Controller Rule: IF xJ(t) is M i1 and ... ,xn(t) is M in THEN 11(t) = -Fix(t).

A TS fuzzy system is descrihed hy a set of fuzzy 'IF ... THEN' rules with fuzzy sets in the antecedents and continuous-time LTI systems in the collsequents. A generic TS model rule can he written as follows:

=

(4)

Using the same fuzzy partition for generating TS fuzzy rules for the controller, and a.<;suming that the state is measurahle, we have:

2. 1 Takaqi-Suqeno Fuzzy Systems

X

+ Biu(t))

i=l

2. TS F{jZZY SYSTEMS AND POLE PLACEMENT IN LMI REGION

=

=L

y(t)

:s

where

oi(t)(Aix(t)

i=1

A(x)

= An + Xl A + ... + xNAN 1

< 0 (8)

where x = [Xl"", xNf is the variahle and An, ... , AN are given symmetric matrices. Since

(3)

Wi(t)

244

.,

3. STATE FEEDBACK FUZZY CONTROL LAW AND POLE PLACEMENT

I",

In this section, we translat.e the problem of pole placement in a LMI region int.o the state feedback fuzzy control law and pole placement in a LMI region, We consider the Takagi-Sugeno fuzzy system: r

x=LO';(Aix+Biu)

Fig. 1. The region R(O',p,O).

(12)

i=1 r

A(y) < 0 and A(z) < 0 implies A[(y + z)/2) < 0, the LMI (8) is a convex constraint on the variable x. It is well kmown that LMI-based optimization problem as well as LMI feasibility problems can be solved by interior-point algorithms with polynomial time (Boyd et al., 1994), and a toolbox of MATLAB (Gahinet et al., 1995) which is dedicated to convex problems involving LMIs is available.

y= LO'iC;X i=1

and we wish to place the poles of the closed loop system in the following convex region R(O', p, B). So, the LMIs in (13)-(18) can be used as the desired closed-loop pole placement constraints for TS fuzzy control systems. Therefore, combining the LMIs in (13)-( 18) and the TS fuzzy system gives a new design method for TS fuzzy c.ontrollers,

2.2 Pole Placement in LMI Re,qions Theorem 2, The fuzzy state feedbac.k control law for the system (12) is such that. the closed loop poles are in the region R( 0', p, B), if and only if, the symetric matrix S E Rn Xn definit.e positive and the mat.rices AfiE R mxn satisfy t.he following equat.ions:

In this section, the charact.erizat.ion of specific regions by LMI's (Chilali and Gahinet, 1996) is presented, Let D be a subregion of the complex left.-half plane. A dynamic.al system x = Ax is called Dstable if all its poles lie in D (that is, all eigenvalues of the matrix A lie in D), By extension, A is t.hen called D-stable. When D is t.he entire left-half plane, this concept reduces to asymptotic stabilit.y, whic.h is characterized in LMI terms by the Lyapunov theorem, Specifically, A is stable if and only if there exist.s a symmetric matrix X satisfying

AX

+ XA T < 0,

SAT + A iyc;-+:/'v1JBT + BiMj+SA;+ (14) AjS + Af; B j + Bj Mi + 20'S < 0 -pS SAT [ AiS + B;M; -pS

~pS it

X> 0

[ )2

This Lyapunov c.haracterization of stability has been extended to a variety of regions by Gut.man (Gut.man and .Iury, 1981), A region R(n,p,O) c.an be charact.erized in LMI t.erms as follows:

AX

+ X AT + 2QX < 0

~pX AX] [ XA T -pX

+ AiS + Mt BT + BiMi + 2QS < 0(13)

SAT

il

0

(16)

= SAT + MInT + SAT + AftAT

t1 t2

t:J

<

] <0

(15)

h = AiS + D;Mj + AjS + BjM;

(9)

sin O(AX + X AT) [ cosO(-AX + X AT)

T COSO(AX-XA )] sin O(AX + X AT)

-pS

sin B(t,) c.os 0(t 2 )] [ c.os 0(t:J) sin O(t,)

(10)

<0

+ MtBT] < 0

0

(17)

<

= AiS + SAT + Afr BT + B;Af; = A;S - SAT - Mr nT + n;,'"fi

= -A;S + SAT + Mr BT -

B;Mi ( 18)

( 11 )

Cl

= A;S + SAT + flIT BT + niAfj + AiS +

SAT + Afr BT + BjMi C 2 = A;S - SAT - AtT BT + BiAfj + AiSSAT - Atl BT + BjMi C3 = -AiS+ SAT + MJ BT - BiMj - A;S+ SAT + Afr BT - BjAfi

This c.omes from observing t.hat t.his region is the int.ersection of three elementary LMI regions: an n-stability region, a disk, and the conic. sector R(O, 0, 0) (Chilali and Gahinet, 1996).

245

and the gain matrices are Fi = MiS. Proof. According to (Chilali and Gahinet, 1996), with a control law as lI(t) = - L~=l cr;FiX(t) for the system (12), the closed-loop poles are in the region R(cr, p, 0), iff exists a positive matrix P such that

P(A i

+ B;F;) + (Ai + B;Fif P+ 2crP < 0(19) G~P + PGi,j + 2crP < 0 (20)

-pp [ P(A;

+ Bi F;)

(Ai + B;F;)T P] -pP

<0

(21) And finally, defining 5 get the LMls (13-1R).

= p-

1

and Fi

= MiS,

we

4. EXAMPLES sin 0 (P(A • " cos 0 (P( Ai

+ R F) - (A + BF)T p) ] < 0 23 In .this section, two. systems are presented as . .

+ B i Fi ) + (Ai + B i Fi f I

,

COS 0 (PGi j [

"

p)

( ,) deSIgn examples for illustratIOn. The first IS the inverted pendulum on a cart and the second is the famous mass-spring-damper system.

+ G~ p)

sin 0 (G~P - PG ij )

sin 0 (PG ij cos 0 (PGij

G~P)]

4.1 E.rample 1

<0

(24)

This example is the very well known prohlem of the st.ahilization of the inverted pendulum on a cart.. The equations of the mot.ion of the pendulum are:

+ G~p)

with (;i,i = (Ai + /3iFj + A j + /3j F;). Then, pre and post multiplying these equations hy p- l and p-l 0 ] . . [ o p- 1 ,respectIvely, gIves:

p- l AT + p- 1 Ft BT + AiP- l + BiFiP- l +2crP- l < 0

(25)

T +P- 1 A T + P- 1 A T + P-1FTB , J' J l I + BFP- 1 + p- FT /3T + AP%.1 t 1 .1 AjP- 1 + B j Fi P- 1 +2oP- l < 0

(26)

Xl X2

= X2, .qsin(xl) - amlx~ sin(2xl )/2 = 4//3 - am/cos 2 (xtJ

aws(x] )u (31)

where Xl is the angle (in radians) of the pendulum from the vert.ical, X2 is the angular velocity, and 11 is the control force (in Newtons) applied to the cart. The other parameters are as follows: 9

= gravity wnst.ant(9.Rm/s 2 ),

m = mass of the pendulum(2.0k.q), A! = mass of the cart(8.0kg), 21 = length of the pendulum( 1.0m), a = l/(m + A!). The TS fuzzy model of the pendulum in Wang et. al. (Tanaka and Wang, 2001) will he used in this paper. It is composed of two model rules:

(2R)

Model rule 1: If Xl is ahout 0 Then x = A1x + /3 1 11; Model rule 2: If Xl is ahout ±(,,/2)(lxll Then x = A 2 x + 8 2 11;

< ,,/2)

where

.4 1

246

=[

2

4//3 - ami

~].Rl=[ 41/3 -2ami ].

4 ..

--

-12 .. -1.4

Fig. 2. Membership function of inverted pendulum.

=[

A2

B2

=

o

2 Till! (secsl

0

1]

[ 0]

Fig. 3. Plot of the simulated system: inverted pendulum, the PDC approach (dotted line), the Kim's approach (dashed line) and our approach (solid line)

,

and qi(x) is the nonlinear term with respect to the input term. The system can be rewritten as

2g 0' 1I"(4l/3 - aml/P) a/3 4l/3 - aml/32

x = -0.lx3

=

and where /3 cos(88°). Membership functions of 'about 0' and 'about ±11" /2 are shown in Figure (2).

=

=

-

0.02x - 0.67x 3

+U

where x E [-1.5,1.5] and x E [-1.5,1.5]. The TS fuzzy model of this non linear system is

The design goal of this example is to place the closed loop poles of each local model within the desired region shown in fig. 1 as a shaded convex region. Now, we have to solve 10 LMIs in (13-18) since r 2, with Q 1.5, p 950, () 0.7rad. The interior-point method yields

=

2 TIlI!(secsl

Rule i:

If xJ(t) is Mt and X2 (t) is M~ , Then x(t) = A;x(t) + B;u(t), i

=

where x(t)

p = [319.549496.5270] 96.5270 56.2979

= 1, ... ,4.

= [x(t), x(t)jT,

Al

= [-0~01 ~1] , B = [1.4~87] I

and FI F2

= f-3170 -1746.2] = -5030.82646.7]

(32)

P is obviously symmetric and positive definite since the eigenvalues are 0.0047 and 0.003, and the closed-loop poles d l [-4.8575, -153.6891],d 2 [-1.6068, -10.6411] lie in the desired region. The behavior over time of the different variables are given Fig.(3). The simulaton is compared to the PDC approach and the approach in (Kim et al., 1999) as shown in figure (3). The solid line indicate the response with the proposed fuzzy controller.

=

=

and the membership fuctions of the fuzzy terms are

= 1 - (xi!2.25), = (xi;2.25),

MI (XI) M~(xJ) Mi (X2) = M~(X2) =

=

=

=

P

In this example, we consider the mass-springdamper system.

= qi(x)u

Mi, M~.

with Q 1, p 3 and () 0.2rad. Now, we have to solve 31 LMIs in (13 - 18) since r 4. We obtain the solution as follows

4.2 Example 2

Mx+g(x,x)+f(x)

= MI, = M~, (34)

Mf(xJ) M~(xJ) 1 - (xV6.75), Mi(X2) = (xV6.75), Mi(X2) =

3.45203.1626]

= 1.0e + 11 * [ 3.16263.3390 FI F3

(33)

=

= [-2.6090 = [-4.7953

- 2.5294] - 4.7063]

.

(35)

= F2 =F 4

The closed loop poles are d l = [-1.0480,-3.5910] and d2 [-1.1973, -2.4443]. The performance of the proposed controller is checked by simulation.

where M is the mass and u is the force; f(x) and g( x, x) are non linear or incertain terms with re-

=

spect to the spring and the damper, respectively;

247

0.5

1.5

....................

--o.

..............

.:

.

~5L---""'···....·'--·~~-~~

o

4

6

10

Tme(il!CS)

4

6

10

Tme(secs)

Fig. 4. Plot of the simulated system: mass-springdamper, the PDC approach (dotted line), the Kim's approach (dashed line) and our approach (solid line) The simulaton is compared to the PDC approach and the approach in (Kim et al., 1999) as shown in figure (4). The solid line indicate the response with the proposed fuzzy controller.

5. CONCLUSIONS The LMI characterization of a region R( Q, p, B) that is the intersection of three elementary LMI regions has been introduced in this paper, and used with the Takagi-Sugeno model for the control of nonlinear systems. The resulting method is simple and the LMIs to solve are feasible, place the closed loop poles in the convex region and guarantee the global stability of the TS fuzzy system.

REFERENCES Boyd, S., L. El Ghaoui, E. Feron and V. Balakrishnan (1994). Linear matrix inequalities in system and control theory. SIA M series in Applied Mathematics. Chilali, M. and P. Gahinet (1996). hinf design with pole placement constraints: An Imi approach. IEEE Trans. on Aut. Contr. 41, 358-367. Farinwata, S.S, D. Filev and R. Langari (2000). Fuzzy Control, Synthesis and Analysis. Wiley & Sons. Gahinet, P., A. Nemirovski, A. Laub and M. Chilali (1995). The LMI Control Toolbox. The Math Works, Inc. Gutman,S. and E.I. Jury (1981). A general theory for matrix root clustering in su bregions of the complex plan. IEEE Trans. Automat. Contr. AC-26, 853-863. Joh, J., R. Langari and W.J. Chung (1997). A new design method for continuous takagisugeno fuzzy controller with pole placement constraint: An Imi approach. Proc. on IEEE Int. Conf. on Systems, Man and Cybernetics pp. 2969-2974.

248

Joh, J., S.K. Hong, Y. Nam and W.J. Chung (1998). On the systematic design of takagisugeno fuzzy control systems. Proc. Int. Symp. on Engineering of Intelligent Systems, EIS '98 pp. 113-119. Kim, K., J. Joh, R. Langari and W. Kwon (1999). Lmi-based design of takagi-sugeno fuzzy controllers for nonlinear dynamic systems usign fuzzy estimators. International Journal of Fuzzy Systems. Tanaka, K. and H.O. Wang (2001). Fuzzy Control,Systems Design and Analysis. John Wiley & Sons. Wang, H.O., K. Tanaka and M.F. Griffin (1995). Parallel distributed compensation of nonlinear systems by takagi-sugeno fuzzy model. IEEE Trans. on Fuzzy Syst. pp. 531-537.