Controlled Synchronization of a Mechanical System in a Fuzzy Scheme*

Controlled Synchronization of a Mechanical System in a Fuzzy Scheme ? Elsa Rubio ∗ Joaquin Alvarez ∗∗ ∗ Centro de Investigacion en Computacion, I.P.N. U.P. Adolfo Lopez Mateos. Zacatenco, c.p. 07738. Mexico D.F. Mexico. (e-mail: [email protected]). ∗∗ Centro de Investigacion Cientifica y de Educacion Superior de Ensenada. Zona Playitas, c.p. 22860. Ensenada, B.C. Mexico. (e-mail: [email protected])

Abstract: This paper deals with controlled synchronization of a mechanical system in a Takagi-Sugeno fuzzy model scheme. We present a mechanical system formed by two pendulums in a masterslave configuration. The synchronization objective is to match the slave movements to those of the master. We propose a Takagi-Sugeno representation of nonlinear systems, from which the difference between the master and slave positions are represented by local linear input-output relations by fuzzy IF-THEN rules. The performance of the mechanical system in the fuzzy scheme proposed, is illustrated numerically and experimentally. Keywords: Synchronization, Master-slave systems, Fuzzy systems, Fuzzy modelling, Artificial intelligence, Coupling coefficients. 1. INTRODUCTION Nowadays the developments in technology and the requirements on efficiency, flexibility, quality in production processes, ease of implementation, as well as new approaches that include a wide range of applications, have resulted in complex and integrated production systems. Hence, the production processes such as manufacturing, industrial processes, robotic manipulators, automotive applications, etc., provide a motivation to experiment with techniques that allow the combination of variables in a manner that provide reasonable and realistic outcomes. In most of these processes, the use of multi-composed systems allow flexibility and manoeuvrability of the involved components. Meanwhile, an important area of research is the synchronization of mechanisms and at the same time, in the current manufacturing processes, there are tasks that can not be accomplished by a single mechanism but two or more systems must work in sync to make a common task. Accordingly, we propose analyze two pendulums as a fundamental synchronization mechanism. This mechanism can be found it in many industrial and manufacturing systems as well as in benchmarks and educational mechanisms. We install the model and establish the parameters of our proposal: “Controlled Synchronization in a Fuzzy Scheme”, in the didactical equipment of the laboratory. See Fig. 6. In this case, we take advantage of the Artificial Intelligence (AI) to synchronize the pendulums in master-slave scheme. In the context, is well known that the Soft Computing (SC) is an important branch of the AI, this is used to rise any ? This work was supported by CONACYT, COTEPABE-IPN and COFAA-IPN. Mexico.

computing process that includes imprecision into the calculation on one or more levels and, allows this imprecision either to change (decrease) the granularity of the problem. The SC is tolerant of imprecision, uncertainty and partial truth. Is well known that one of the principal constituent of the SC is the fuzzy control, besides their development is directly associated with fuzzy logic, fuzzy systems and in particular fuzzy controllers stand out. Hence, a natural extension of fuzzy logic to control theory is accomplished by using the knowledge from human expert in modelling and control of complex systems, Foulloy et al. (2003), Nguyen et al. (2003), Nguyen et al. (1995), Nguyen and Walker (2006), Takagi and Sugeno (1985), Tanaka and Wang (2001), Yen and Langari (1999). In the mechanical systems domain, we can find the synchronization in manufacturing systems and in automotive industry are clearly their traces; for example, in robots with assembly and jointing tasks, in painting, welding, transport, etc. In this sense, we treat the problem of controlled synchronization of two pendulums in a fuzzy scheme. This problem is established as a problem of synchronization of a fuzzy coupling signal that allow the equality of positions between master and slave. In some situations, the synchronization is a natural phenomenon; however, in other cases we must add an interconnection or controller system to attain it, or to improve its characteristics, Rosas and Alvarez (2006). In many references, the synchronization problem was treated as a problem of control objective and, it is called controlled synchronization. In this sense, several control technique have been used to design the coupling signals which aim to reach a state of synchronization between two or more systems, Rosas and Alvarez (2006), see also

Fradkov et al. (2000). To date, various techniques have been suggested over a variety of fuzzy synchronization schemes as Kim et al. (2007), Lam and Seneviratne (2008), Meng et al. (2008), Nian and Zheng (2007). Also we can find many examples of synchronization in a social dynamics and in the human activities; like a musical concert, playing sports, choral singing in a soccer stadium, in biology process, in a group of birds under the guidance of a leader, in a church congregation, in a flock of tourists, etc. This paper is organized as follows, in section 2, the TakagiSugeno fuzzy model is presented. In section 3, the statement of the problem for master-slave scheme by a robust synchronization of a lagrangian system is presented. Section 4 addresses a Takagi-Sugeno fuzzy model for the synchronization of the master-slave scheme. The numerical and experimental solutions are presented in section 5. Finally in section 6, the conclusions are given.

This approach was proposed by Takagi-Sugeno (TS), Takagi and Sugeno (1985). It offers a description by IF-THEN rules which represent local linear input-output relations of nonlinear systems. Each rule furnish a local dynamic represented by a linear system model. The overall fuzzy model of the system is achieved by fuzzy “blending” of the linear system models, Tanaka and Wang (2001). Broadly speaking, a TS fuzzy model of a nonlinear system, is structured as an interpolation of linear systems. In facts, it is proved that TS fuzzy models are universal approximators, Tanaka and Wang (2001), Wang et al. (2000). The rules of this fuzzy models are expressed as following: Rule i:

(1)

n

where x(t) ∈ R is the state vector, i = 1, . . . , r, r is the number of IF-THEN rules, Aij are the fuzzy sets, j = 1, . . . , p, u(t) ∈ Rm is the input vector, y(t) ∈ Rq is the output vector, Ai ∈ Rn×n , Bi ∈ Rn×m y Ci ∈ Rq×n ; z1 (t), . . . , zp (t) are the premise variables, they may be functions of the states, external disturbances and/or time. In addition, a weight wi (z(t)) is assigned to each rule; this weight depends on the value of truth (or degree of membership) of the variables zj (t) to the fuzzy set Aij , and on the connector “and” of the premises. On the whole, the system is: r P wi (z(t)){Ai x(t) + Bi u(t)} x(t) ˙ = i=1 (2) r P wi (z(t)) i=1

=

r X

i=1

=

r X

hi (z(t))Ci x(t)

i=1

where z(t) = [z1 (t) z2 (t) . . . zp (t)] p Y wi (z(t)) = Aij (zj (t)) j=1

Aij (zj (t)), is the grade of membership of zj (t) in Aij . wi (z(t)) (4) hi (z(t)) = P r wi (z(t)) i=1

2. TAKAGI-SUGENO FUZZY MODEL

IF zi (t) is Ai1 and . . . and zp (t) is Aip  x(t) ˙ = Ai x(t) + Bi u(t) THEN y(t) = Ci x(t)

by the method of defuzzification barycentric, also called centroid method, Foulloy et al. (2003), Nguyen et al. (2003), Tanaka and Wang (2001). r P wi (z(t))Ci x(t) y(t) = i=1 P (3) r wi (z(t))

hi (z(t)){Ai x(t) + Bi u(t)}

i=1

The output depends on the one hand, by the inference mechanism used and moreover, by the defuzzification procedure chosen. In this model, the output is inferred

moreover, r X

hi (z(t)) = 1 and wi (z(t)) ≥ 0, for all t.

i=1

Since

 r  X

wi (z(t)) > 0, i=1   w (z(t)) ≥ 0, i = 1, 2, . . . , r, i

(5)

we have  r  X

hi (z(t)) = 1, i=1   h (z(t)) ≥ 0, i = 1, 2, . . . , r, i for all t. The free system of (2) is defined as: r P wi (z(t))Ai x(t) · x (t) = i=1 r P wi (z(t))

(6)

(7)

i=1

finally, ·

x (t) =

r X

hi (z(t))Ai x(t)

i=1

Each linear consequent equation represented by Ai x(t), is called subsystem. In summary, the final model is resulting of the fuzzy combination of all linear subsystems. 3. PENDULUMS MODEL Consider the model proposed in Rosas (2005), where a synchronized lagrangian systems with the same number of degree of freedom in the master-slave scheme is developed. There, its consider bounded external disturbances as well as parameters uncertainties affecting the system. In addition, a coupling signal is introduced. Under certain conditions this signal eliminates unwanted terms and adds ones proportional and derivative, as well a discontinuous term that gives robust performance in a closed loop outline. See Fig. 1.

master and slave positions

2.5

master slave

2 1.5

positions

1 0.5 0 −0.5 −1

0

1

2

Fig. 1. Block diagram of the system Coupling signal:

3 time

4

5

6

Fig. 2. Positions

Master equation:     x˙ 1,m x2,m = x˙ 2,m −k1 ψ − k2 x2,m − ξm (·) + k3 τm

master slave

(8) 4

2

(9)

where ψ = sin(x1,m ), xm = [x1,m x2,m ]T is the state vector. ξm (·) is a bounded external perturbation; |ξm (·)| ≤ ρm , where ρm is a constant. Slave equation:     x˙ 1,s x2,s = x˙ 2,s −k1 ϕ − k2 x2,s − ξs (·) + k3 (τs + υ)

master and slave velocities

6

velocities

1 υ = (k1 (sin(x1,m − e1 ) k3 − sin(x1,m )) + kp e1 + kc sign(e1 ))

0

−2

−4

(10)

−6

0

1

2

T

where φ = sin(x1,m − e1 ), e1 = (x1,m − x1,s ), e2 = (x2,m − x2,s ). The numerical results are showed in the Fig. 2, 3, 4 and 5, when kp = 1, kc = 5, ξs = sin(t), ξm = sin(cos(3t)), A = 0.1, ω = 1.2, with the initial conditions: x1,m = −1, x1,s = 1, x2,m = 0, x2,s = 0. The positions are in radians (rad) and the velocities in rad/seg. 4. FUZZY SYNCHRONIZATION SCHEME In this section, we detail a TS fuzzy model for the signal error. This model synchronize the master and slave pendulums represented by (9) and (10). See Fig. 7. 4.1 Takagi-Sugeno Model of the Two Pendulums We propose simplify the model presented in section 3. In this context, (11) is reorganized as:     0 1 0 e˙ = e+ (12) −kp −k2 ξ(·) − kc sign(e1 )

4

5

6

Fig. 3. Velocities coupling signal

2

1.5

1 v

where ϕ = sin(x1,s ), xs = [x1,s x2,s ] is the state vector, ξs (·) is a bounded external perturbation; |ξs (·)| ≤ ρs , where ρs is a constant, ξ(·) = ξs (·) − ξm (·), is a constant, τs = τm = Asin(ωt) and, k1 =67.912, k2 =3.05007, k3 =55.549. The error in the dynamic is given by     e˙ 1 e2 = (11) e˙ 2 −k2 e2 + k1 φ − k1 ψ + ξ(·) − k3 υ

3 time

0.5

0

−0.5

0

1

2

3 time

4

5

6

Fig. 4. Coupling signal where e˙ = [e˙1 e˙2 ]T and e = [e1 e2 ]T . The complete system is shown in the Fig. 7. We use the fuzzy block in order to match the positions x1,m and x1,s of the master and slave respectively. In this block, the antecedent (input) is the error value and the consequences (outputs), are different the values of kp and kc . The dynamic of the error is derived by the fuzzy rules presented below.



response of position error

0.5

e˙ =

−kp asn

(4) IF error is s, THEN,     0 1 0 e˙ = e+ −kp s −k2 ξ(·) + kc s sign(e1 )

0

error

−0.5

(5) IF error is asp, THEN,     0 1 0 e˙ = e+ −kp asp −k2 ξ(·) + kc asp sign(e1 )

−1

−1.5

−2

   1 0 e+ −k2 ξ(·) + kc asn sign(e1 )

0

0

1

2

3 time

4

5

6

Fig. 5. Position error

(6) IF error is mp, THEN,     0 0 1 e+ e˙ = ξ(·) + kc mp sign(e1 ) −kp mp −k2 (7) IF error is bp, THEN,     0 0 1 e+ e˙ = ξ(·) + kc bp sign(e1 ) −kp bp −k2 where: · · · ·

error : e1 = (x1,m − x1,s ). bn, bp: big negative, big positive. mn, mp: medium negative, medium positive. asn, asp: almost small negative, almost small positive. · s: small.

Fig. 6. Mechanical pendulum from Mechatronic Systems Inc. (Pendubot)

As such, the dynamic of the fuzzy system is given by: 7  P wi (t) Ai e(t) + Gi e(t) ˙ = i=1 (13) 7 P wi (t) i=1

where:

 Gi =

0



ξ(·) + kc (·) sign(e1 )

5. SIMULATION AND EXPERIMENTAL RESULTS

Fig. 7. Block diagram of the fuzzy system Fuzzy rules: (1) IF error is bn, THEN,     0 1 0 e˙ = e+ −kp bn −k2 ξ(·) + kc bn sign(e1 ) (2) IF error is mn, THEN,     0 1 0 e˙ = e+ −kp mn −k2 ξ(·) + kc mn sign(e1 ) (3) IF error is asn, THEN,

We show the results where the figures of the section 3 and 4, can be compared with the same initial conditions. Fig. 9, shows the incidence of the numerical solution in the triangular fuzzy sets. By the initial conditions, the dynamic of the error begins in the mn set with a minimal membership degree: µmn (error) = 0 and in the bn set, with a maximal membership degree: µbn (error) = 1. An increase of numerical solutions, can be seen in the set close to zero (specifically in the s set); this is due to the syntonization in order to couple the pendulums. If the error is closer to zero, their membership degree is grown in the s set, limt→∞ µs (error) → 1. In Fig. 10, the dynamic of the error is shown. The fuzzy evolution is evince by the smooth shift between fuzzy sets. Fig. 11 illustrates the coupling signal υ in time. This signal is a result of applying the TS approach represented by the seven fuzzy rules listed above. The kp and kc values obtained by the TS approach, are feedback through (8). Broadly speaking, there exist a similarity in the results of applying the methodology proposed in Rosas (2005) and the TS fuzzy model. In order to compare, we present in the Fig. 13, 14 and 15, the dynamics and performances of the systems. In Fig. 14, the position of the master and

fuzzy coupling signal

2

1.5

coupling signal

slave pendulums, represented by (9) and (10) respectively, are shown. The Fig. (15) illustrates the compensation action of the coupling signal υ. This signal rectifies the position error, as we can see in (11) and (13). The compensation action is generated by (8); after varying kp and kc .

1

0.5

0

−0.5

0

1

2

3 time

4

5

6

Fig. 11. Fuzzy coupling signal positions

2.5

salve with fuzzy v master

2 1.5

position

Fig. 8. Membership functions

membership functions

1

0.5 0

0.9 0.8

membership degree

1

−0.5

0.7

−1

0.6

0

1

2

0.5

3 time

4

5

6

0.4

Fig. 12. Master and fuzzy -slave positions

0.3 0.2

response of position error

0.5

0.1 0 −2

−1.5

−1

error

−0.5

0

−comparison with fuzzy v with v

0.5

0

Fig. 9. Incidence in the numerical solution error

−0.5

−1

fuzzy response of position error

0.5

−1.5 0 −2

0

1

2

3 time

4

5

6

error

−0.5

Fig. 13. Position errors: a comparison

−1

6. CONCLUSIONS −1.5

−2

0

1

Fig. 10. Fuzzy error

2

3 time

4

5

6

This paper presented experimental and numerical results of a synchronization of two pendulums in the master-slave scheme. We introduced an approach of a controlled synchronization in a TS fuzzy scheme. A comparison between a lagrangian system and a TS fuzzy model approach is offered. We show divers figures with the experimental and numerical results in order to compare

positions −comparison

2.5

slave with fuzzy v master slave

2

position

1.5 1 0.5 0 −0.5 −1

0

1

2

3 time

4

5

6

Fig. 14. Positions: a comparison coupling signals

2

−comparison fuzzy v v

1.5

signals

1

0.5

0

−0.5

0

1

2

3 time

4

5

6

Fig. 15. Coupling signals: a comparison the two approaches; we also have to highlight the similarity of the results. Once again has been demonstrated the efficiency of a fuzzy model, now in the case of the system synchronization. This fuzzy model, syntonize the parameters kp and kc of the coupling signal. Furthermore, is was shown that this fuzzy synchronization proposal renders the closed-loop system more robust with respect to matched bounded disturbances. With a set of suitable membership functions, this syntonization improved the performance of the slave response. REFERENCES Foulloy, L., Galichet, S., and Titli, A. (eds.) (2003). Commande Floue 1- de la stabilisation a la supervision. Hermes Science. Fradkov, A., Nijmeijer, H., and Pogromsky, A.Y. (2000). Adaptive observer-based synchronization. In G. Chen (ed.), Controlling Chaos and Bifurcations in Engineering Systems, 417–438. CRC Press, Boca Raton, FL. Kim, J., Hyun, C., Kim, E., and Park, M. (2007). Adaptive synchronization of uncertain chaotic systems based on T-S model. IEEE Transaction on Fuzzy Systems, 15(3). Lam, H. and Seneviratne, L. (2008). Chaotic synchronization using sampled-data fuzzy controller based on fuzzymodel-based approach. IEEE Transaction on Circuits and Systems-I, 55(3).

Meng, X., Yu, Y., Wen, G., and Chen, R. (2008). Chaos synchronization of unified chaotic system using fuzzy logic controller. In Proc. IEEE Int. Conf. on Fuzzy Systems FUZZ-IEEE’08, 544–547. Hong Kong, China. Nguyen, H., Prasad, N., Walker, C., and Walker, E. (2003). A First Course in Fuzzy and Neural Control. Chapman & Hall/CRC. Nguyen, H., Sugeno, M., and et.al. (1995). Theoretical Aspects of Fuzzy Control. Jhon Wiley & Sons Inc. Nguyen, H. and Walker, E. (2006). A First Course in Fuzzy Logic. Chapman & Hall/CRC, third edition. Nian, Y. and Zheng, Y. (2007). Synchronization for Rossler chaotic systems using fuzzy impulsive controls. In Proc. of IIH-MSP’07, volume 2, 179–182. Kaohsiung, Taiwan. Rosas, D. (2005). Sincronizacion robusta de sistemas lagrangianos utilizando controladores discontinuos. Ph.D. thesis, CICESE, Mexico. Rosas, D. and Alvarez, J. (2006). Robust synchronization of nonlinear SISO systems using sliding mode control. Nonlinear Dyn., 46, 293–306. Takagi, T. and Sugeno, M. (1985). Fuzzy identification of systems and its applications to modeling and control. IEEE Trans. Syst. Man Cyber., 116–132. Tanaka, K. and Wang, H. (2001). Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality. Jhon Wiley & Sons Inc. Wang, H., Li, J., Niemann, D., and Tanaka, K. (2000). T-S fuzzy model with linear rule consequence and PDC controller: A universal framework for nonlinear control systems. In Proc. IEEE Int. Conf. on Fuzzy Systems FUZZ-IEEE’00, 549–554. San Antonio, TX. Yen, J. and Langari, R. (1999). Fuzzy Logic, Intelligence, Control and Information. Prentice-Hall, Upper Saddle River, NJ.