CONTROLLER SYNTHESIS WITH INPUT AND OUTPUT CONSTRAINTS FOR FUZZY SYSTEMS

Copyright (c) 2005 IFAC. All rights reserved 16th Triennial World Congress, Prague, Czech Republic

CONTROLLER SYNTHESIS WITH INPUT AND OUTPUT CONSTRAINTS FOR FUZZY SYSTEMS1 Miguel Bernal, Petr Husek Department of Control Engeneering, Fa ulty of Ele tri al Engineering, Cze h Te hni al University in Prague, Te hni k a 2, 166 27, Prague 6, Cze h Republi . Phone: +420 22435-7336, Fax: +420 22491-8646, E-mail: xbernal ontrol.felk. vut. z, husek ontrol.felk. vut. z

Abstra t: This work extends the synthesis of ontrollers for Takagi-Sugeno fuzzy systems based on a pie ewise Lyapunov fun tion to in lude onstraints on the input or the output. Extension follows naturally from the existing results based on a

ommon Lyapunov fun tion and an be implemented via linear matrix inequalities, whi h are numeri ally solvable with ommer ial available software. An illustrative example is provided. Copyright

2005 IFAC Keywords: Controller synthesis, fuzzy systems, linear matrix inequality (LMI). 1. INTRODUCTION Fuzzy logi -based ontrol systems have been subje ted to a big growth of industrial appli ations during the re ent years as well as to big eort to study their properties and in rease their reliability, mainly be ause of their satisfa tory results in dealing with highly nonlinear systems with a good

ompromise between simpli ity and a

ura y. Among fuzzy systems, those de ned in (Takagi and Sugeno 1985) have been onsidered as the most onvenient for analysis and design, be ause of their simpli ity and eÆ ien y in modelling nonlinear systems. First attempts on investigation of stability of Takagi-Sugeno fuzzy systems (TSFS) were made employing ommon Lyapunov fun tions as in (Tanaka and Sugeno 1990), (Tanaka

and Sugeno 1992), (Chen and Ying 1993) and (Farinwata and Va htsevanos 1993). Controller synthesis under this s heme was a hieved, in luding a lot of performan e requirements as de ay rate, input or output onstraints, robustness and optimality (Tanaka and Sano 1994), (Wang et al. 1996), (Tanaka et al. 1998) and (Tanaka and Wang 2001). Nevertheless, analysis and design based on ommon Lyapunov fun tions la k exibility be ause of their onservativeness. In order to relax this restri tions, a number of re ent stability analysis pro edures based on pie ewise quadrati Lyapunov fun tions have been developed (Johansson et al. 1999), (Rantzer and Johansson 2000), (Feng 2004). In a very re ent work (Feng 2003),

ontroller synthesis under this s heme was made possible by onstru ting ontrollers in su h a way that a pie ewise ontinuous Lyapunov fun tion

an be used to establish the global stability with H1 performan e. Moreover, this synthesis an be a hieved by means of linear matrix inequalities

1 This work has been supported by the proje t INGO 1P2004LA231 from the Ministry of Edu ation of the Cze h Republi , by the Mexi an Coun il of S ien e and Te hnology (CONACYT) via s holarship 121109 and the proje t GACR 102/04/P050, sponsored by the Grant Agen y of the Cze h Republi .

407

where

(LMIs), whi h an be solved with ommer ially available software. In the present paper, two performan e hara teristi s are added to the ontroller designed in (Feng 2003): onstraints on the input and onstraints on the output. Both of them an be implemented independently via LMIs and follow naturally from the elementary ases in (Tanaka et al. 1998). This paper is organized as follows: se tion 2 introdu es the dynami al fuzzy systems and pie ewise quadrati design this work is based on; se tion 3 shows new results on input and output onstraints; se tion 4 illustrates the previous results with some examples and, nally, se tion 5 draws some on lusions.

Sl = fx j l (x) > i (x); i = 1; 2;


; m; i = 6 lg

and its boundary Sl = fx j l (x) = i (x); i = 1; 2;


; m; i 6= lg:

In addition, let us de ne L as the set of subspa e indexes, so we an des ribe (2) as follows: x_ (t) = (Al +
Al )x(t) + (Bl +
Bl )u(t) +(Dl +
Dl )v(t) z (t) = (Hl +
Hl )x(t)

for x(t) 2 S l , where

2. FUZZY SYSTEMS AND PIECEWISE QUADRATIC DESIGN Consider the following Takagi-Sugeno fuzzy system as in (Feng 2003):


Al =

D() =

l=1

m X l=1

l Dl H () =

m X l=1 m X l=1

i2M

X

i
Bli

l

Pie ewise Lyapunov fun tion is onstru ted as in (Johansson et al. 1999): (4) V (t) = xT Rl x; x 2 S l ; l 2 L with Rl = FlT T Fl ; l 2 L Fl x = Fj x; x 2 S l \ S j ; l; j 2 L:

Employing parallel distributed ompensation (PDC)

(2)

u(t) = Kx(t) =

where l Al B () =

X l

l

where denotes the lth fuzzy rule, m is the sets, x(t) 2 Rn number of rules, Fjl the fuzzy p the state ve tor, u(t) 2 R the ontrol input, z (t) 2 Rr the ontrolled output and Al ; Bl ; Dl ; Hl the lth lo al model of the fuzzy system (1). The previous s heme an be ompa tly rewritten

onsidering membership fun tions l (x)  0, as follows:

m X

X


Bl =


Dl = i
Dli
Hl = i
Hli i2M i2M
Ali = Ai Al
Bli = Bi Bl
Dli = Di Dl
Hli = Hi Hl Ml = fi j i 6= 0; l  i g:

Rl

A() =

i2M

i
Ali

l

Rl : IF x1 is F1l AND


xn is Fnl T HEN x_ (t) = Al x(t) + Bl u(t) + Dl v (t) (1) zl (t) = Hl x(t); l = 1; 2;


; m

x_ (t) = A()x(t) + B ()u(t) + D()v (t) z (t) = H ()x(t)

X

m X l=1

l Kl x(t) x 2 S l ; l 2 L(5)

the system (1) be omes:

l Bl

x_ (t) = A ()x(t) + D ()v (t) z (t) = H ()x(t)

l Hl

(6)

where

Pie ewise quadrati stability is based on state spa e partitioning. In (Feng 2003) a partition's method is proposed in order to a hieve globally stable ontroller with disturban e attenuation. Following similar lines, let us divide the statespa e as follows: S l = Sl [ Sl ; l = 1; 2;


; m (3)

A () = A() + B ()K (x) D () = D() H (t) = H ()

Then, ontroller synthesis an be a hieved a

ording to the following theorem (Feng 2003): 408



: Given a onstant > 0, (6) is globally stable with disturban e attenuation , if there exist onstants l > 0, l = 1; 2;


; m, a symmetri matrix T and a set of matri es Ql; l 2 L su h that with Theorem 1

Pl = (FlT Fl ) 1 FlT T Fl (FlT Fl ) Pl = Rl 1 ; l 2 L

0 < Pl; l 23 L Pl QTl 1 0 5 < 0; l 2 L MPl 0 MQl1

Ql

(7)

(8)

uT (t)u(t) =

where

l = Pl ATl + AlPl + QTl BlT + BlQl T T  +1l (ElA ElA + ElB ElB ) T + 2 1+  DlDlT + 2(1+ l)ElD ElD l 1  3  H T Hl MP = I + 1+  l l

+ 1+2l + 1l

 l

1

(10) (11)

hold, where Pl and Q1 l are de ned as in Theorem 1. Then Kl = QlPl ; l 2 L. Proof: Without loss of generality, suppose that V (0) = xT (0)Rl x(0)  1; l 2 L; x(0) 2 S l : (12) From (7) and (12), we have 1 xT (0)Pl 1x(0)  0, so by S hur omplement we arrive to the LMI (10). Condition ku(t)k2 <  ombined with (5) an be rewritten as follows:

1

the following LMIs are satis ed: 2

l 4 Pl

1 x(0)T   0; l 2 L x(0) Pl   Pl QTl  0; l 2 L Ql 2 I

m X m X l=1 j =1

l (x)j (x)xT (t)KlT Kj x(t)

 2

from whi h m X m 1X l (x)j (x)xT (t)KlT Kj x(t)  1: (13) 2 l=1 j =1

l

Noti e that sin e xT (t)Pl 1x(t)  xT (0)Pl 1x(0)  1 for t > 0, if

T ElH ElH

m X m 1X l (x)j (x)xT (t)K T Kj x(t)

MQl = I l [
Al ()℄[
Al ()℄T  ElA ElAT [
Bl ()℄[
Bl ()℄T  ElB ElBT T [
Dl()℄[
Dl ()℄T  ElD ElD T [
Hl ()℄[
Hl ()℄T  ElH ElH

2 l=1 j =1

l

 xT (t)Pl 1 x(t)

then (13) holds. Therefore, ondition (11) an be obtained from the previous inequality, whi h an be transformed as follows:  1 m X m X 1 T T K Kj P x(t) l (x)j (x)x (t)

with matri es ElA , ElB , ElC , ElD and ElH being upper bounds for the un ertainty terms that an be easily al ulated (see (Feng 2003)). The ontroller gain for ea h lo al subsystem is given by Kl = Ql Pl 1 ; l 2 L: (9)

2 l

l=1 j =1

0

and by S hur omplement m X

The pie ewise Lyapunov fun tion an be then expressed as in (4).

l

l=1

l (x)



Pl 1 KlT Kl 2 I



 0:

Congruen e with the full rank matrix   Pl 0 0I

3. CONSTRAINTS ON INPUT AND OUTPUT Results in (Feng 2003), brie y des ribed in the previous se tion, an be extended by onsidering

onstraints on input and output. Theorem 2 Assume that the initial ondition x(0) in system (6) is known. The onstraint ku(t)k2 <  is enfor ed at all times t  0 if the LMIs

leads to (11), where Kl = QlPl 1; l 2 L, whi h

ompletes the proof. Theorem 3 Assume that the initial ondition x(0) in system (6) is known. The onstraint kz(t)k2 <  is enfor ed at all times t  0 if the LMIs 409



1 x(0)T   0; l 2 L x(0) Pl Pl Pl HlT Hl Pl 2 I



 0; l 2 L

(14)

S1

(15)

0.8

hold, where Pl and Q1 l are de ned as in Theorem 1. Then Kl = QlPl ; l 2 L. Proof: Proof follows the same lines as that for Theorem 2.

0.6

0.4

0.2

0 −1

4. EXAMPLE In order to illustrate the in uen e of the input and output onstraints, onsider the following example taken from (Feng 2003), orresponding to a ball and beam system:

−0.2

0 x1

0.2

0.4

0.6

0.8

1

Output signal x1

−0.02

−0.03

−0.04

−0.05

−0.06 0.5

1

1.5

2

2.5 Time (s)

3

3.5

4

5

4.5

Fig. 2. Comparison of output signal x1 with and without onstraint

2 3 0 66 0 77 B1 = 4 5 0

1

00 0 0 1 T D1 = [0 0 0 1℄ ; H1 = [1 0 0 0℄ 2 2 3 01 0 0 3 0 66 0 0 bg 2b 77 66 0 77 A2 = 4 0 0 0 1 5 ; B2 = 4 0 5 00 0 0 1 T D2 = [0 0 0 1℄ ; H2 = [1 0 0 0℄ = 0:01; b = 0:7143; g = 9:81

0.5 0 −0.5

Control signal

−1 −1.5 −2 −2.5 −3 −3.5

and x(t) = [x1 x2 x3 x4 ℄T is the state ve tor, where x1 represents the ball position, x2 the ball velo ity, x3 the beam angle and x4 the beam's angular velo ity. Noti e also that z1 = z2 = x1 . Fig. 1 shows the membership fun tions employed in the example. A

ording to the state-spa e partition (3), we will have two subspa es for whi h the following hara terizing and bounding matri es

an be taken:     1 0 0 0 1 0 0 0 ; F2 = F1 = I44 E1A = E2A = 0:5(A2 E1B = E2B = 0:5(B2

−0.4

−0.01

0

2 01 0 0 3 6 0 0 bg 2b 7 A1 = 6 4 0 0 0 1 75 ;

−0.6

0

1

where

−0.8

Fig. 1. Membership fun tions and state-spa e partition

: IF x1 > 0 T HEN x_ (t) = A1 x(t) + B1 u(t) + D1 v (t) z1 (t) = H1 x(t) 2 R : IF x1 < 0 T HEN x_ (t) = A2 x(t) + B2 u(t) + D2 v (t) z2 (t) = H2 x(t) (16)

R

S2

1

Membership values



−4 −4.5 0

0.5

1

1.5

2

2.5 Time (s)

3

3.5

4

4.5

5

Fig. 3. Comparison of input signal with and without onstraint E1D = E2D = 0:5(D2 D1 ) E1H = E2H = 0:5(H2 H1 ) Employing the synthesis pro edure des ribed in Theorem 1 with = 100; 1 = 2 = 10, we have a feasible solution for LMIs (8) giving ontroller gains K1 = [ 8.0334 10.4772 40.0059 11.2510℄ and K2 = [ 7:9824 10:4272 39:8936 11:2299℄ whi h stabilize the system output x1 as is shown

I44 A1 ) B1 )

410

107.0897 127.2083 14.0323℄ and K2 = [347.2602 120.0060 139.7417 15.0352℄. Fig. 4 shows the output signal z(t) = x1 while Fig. 5 exhibits the orresponding ontrol input.

−3

8

x 10

6

Output signal x1

4

5. CONCLUSION Controller synthesis for Takagi-Sugeno fuzzy systems (TSFS) based on pie ewise Lyapunov fun tion remains a hallenging task, whi h has been developed just re ently. Extensions from those results available for ommon Lyapunov fun tion based TSFS are ne essary in order to in rease the

apabilities of this approa h. In this paper, two new results regarding performan e requirements have been added to the existing disturban e reje tion theorem. Considering

onstraints on the input and output signals is a typi al demand whi h in reases ontrol quality. Both of them have been established in this paper and an be implemented via LMIs, whi h

an be easily solved with ommer ially available software. The examples provided illustrate the ee tiveness of the developed te hniques.

2

0

−2

−4 0

0.5

1

1.5

2

2.5 Time (s)

3

3.5

4

4.5

5

Fig. 4. Constrained output signal x1 3

2

Control input

1

0

−1

−2

−3

−4 0

0.5

1

1.5

2

2.5 Time (s)

3

3.5

4

4.5

REFERENCES Chen, G. and H. Ying (1993). On the stability of fuzzy ontrol systems. Pro . 3rd. IFIS, Houston. Farinwata, S. S. and G. Va htsevanos (1993). Stability analysis of the fuzzy logi ontroller. Pro . IEEE CDC, San Antonio. Feng, G. (2003). Controller synthesis of fuzzy dynami systems based on pie ewise lyapunov fun tions. IEEE Transa tions on Fuzzy Systems 11, 605{612. Feng, G. (2004). Stability analysis of dis rete time fuzzy dynami systems based on pie ewise lyapunov fun tions. IEEE Transa tions on Fuzzy Systems 12, 22{28. Johansson, M., A. Rantzer and K. Arzen (1999). Pie ewise quadrati stability of fuzzy systems. IEEE Transa tions on Fuzzy Systems 7, 713{722. Rantzer, A. and M. Johansson (2000). Pie ewise linear quadrati optimal ontrol. IEEE Transa tions on Automati Control 45, 629{ 637. Takagi, T. and M. Sugeno (1985). Fuzzy identi ation of systems and its appli ations to modeling and ontrol. IEEE Transa tions on Systems, Man and Cyberneti s 15, 116{132. Tanaka, K. and H. O. Wang (2001). Fuzzy Control Systems Design and Analysis. John Wiley & Sons. New York. Tanaka, K. and M. Sano (1994). A robust stabilization problem of fuzzy ontrol systems

5

Fig. 5. Constrained input signal with a solid line in Fig. 2. Corresponding ontrol signal is also shown with a solid line in Fig. 3. In order to redu e the magnitude of the ontrol input signal, LMIs (10-11) should be added to those of (8). Choosing  = 3 under initial

onditions x(0) = [0 0:1 0:1 0:1℄T , these LMIs proved to be feasible with ontroller gains K1 = [1.3398 1.9901 9.4936 3.6913℄ and K2 = [1.3248 1.9847 9.5022 3.6899℄. In Fig. 3,

ontrol signal is shown with a dashed line to make

lear the dieren e between non- onstrained and

onstrained ase. Constraint ku(t)k2   has been satisfa tory a

omplished. Constraints on the output an be satis ed by adding LMIs (14-15) to the original design in (8). With  = 0:009 under initial onditions x(0) = [0 0:1 0:1 0:1℄T , these LMIs proved to be feasible with ontroller gains K1 = [ 5020.3 710.9 690.1 47.3℄ and K2 = [ 6439.1 878.3 847.8 55.2℄. In Fig. 2 output signal z(t) = x1 is shown with a dashed line so an be ompared with the original one. Constraint kz(t)k2   holds. Finally, ombining all the previous s hemes under the initial onditions x(0) = [0 0:1 0:1 0:1℄T to a hieve kz(t)k2  0:009 and ku(t)k2  4:2, LMIs (10-11), (14-15) and (8) proved to be feasible giving ontroller gains K1 = [ 284.5527 411

and its appli ation to ba king up ontrol of a tru k-trailer. IEEE Transa tions on Fuzzy Systems 2, 119{134. Tanaka, K. and M. Sugeno (1990). Stability analysis of fuzzy systems using lyapunov's dire t method. Pro . NAFIPS'90 pp. 133{136. Tanaka, K. and M. Sugeno (1992). Stability analysis and design of fuzzy ontrol systems. Fuzzy Sets and Systems 45, 135{156. Tanaka, K., T. Ikeda and H. O. Wang (1998). Fuzzy regulators and fuzzy observers: Relaxed stability onditions and lmi-based designs. IEEE Transa tions on Fuzzy Systems 6, 250{264. Wang, H. O., K. Tanaka and M. GriÆn (1996). An approa h to fuzzy ontrol of nonlinear systems: Stability and design issues. IEEE Transa tions on Fuzzy Systems 4, 14{23.

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