Chapter 4 The Maximal Convergence Rate of T-S Fuzzy Control
4.2 On Maximum the Convergence Rate
Consider the feedback fuzzy model (53) under the constraint that U ≤1, we have following definitions.
In here, we give the controller design for maximizing the convergence rate.
The following lemma will illustrate the level set Ω found by Linear Matrix Inequalities (LMIs) [36].
Lemma 1. Consider a T-S fuzzy model (53) with zero input if P>0, α ≥0 and
0 on maximizing the convergence rate.
Proof:
Let V x
( )
=x PxT and P>0. For a positive number α , the level set associatedwith V x
( )
is ellipsoid,(
x,α) {
x n V x( )
x PxT α}
Ω = ∈\ = ≤
Along the trajectory of the system (53),
( )
T(
iT i)
2 T 0V x =x A P+PA x+ x PB U⋅ < , (57)
(
,) { }
\ 0x x α
∀ ∈Ω . From Definition 15, the controller is minimizing (57), we have ( T )
U = −SGN B Px
where SGN
( )
⋅ is sign function. It is clear that the maximal convergence controlproduces the maximal invariant ellipsoid of a given ellipsoid Ω
(
x,α)
. Q.E.D.Remark 12. The system will have no solution if x=0. This is due to the switching plane B PxT = . When the system state close to the switching plane, it is easy to 0 have the chattering.
Remark 13. It becomes obvious that the maximal convergence control is also a bang-bang control.
4.3 Illustrative Examples
In this section, we demonstrate the application of the proposed maximal convergence rate for T-S fuzzy model.
Example 6.
Consider a nonlinear mass-spring-damper mechanical system that can be formulated as
We use the following mass-spring-damper and fuzzy model formulated in [15]:
3 3 the consequent parts are chosen as
1 1
In this example, the system is not solvable therefore there are no information about the number of switching. In this situation, the numerical reachable set is difficult obtained and computation cost is high. We design the controller by purposed controller on maximal convergency rate. With all the ellipsoids satisfying the set invariance condition in Lemma 1, we have
0.0252 -0.0131
In the case, the saturation control in [46] is introduced to compare our results. The initial point is x=[1, -0.5]T. In Fig. 31, the states are converged by saturation control over 35 (sec.). The saturation control input is depicted in Fig. 32.
Case II
Let the initial point as Case I, the maximal convergency rate control is considered in this Case. The states converge at 1.2 (sec.) and depicted in Fig. 33. The corresponded control input is depicted in Fig. 34. The convergence rate of states is expected faster then Case I. We can conclude that the system has faster response by the maximal convergency rate control. Obviously, the sign function is sensitivity when the states approach the original. This phenomenon is called chartering. Since
that the sign function is sensitivity when the states approach the original (switching plane). To overcome this phenomenon, we combine two approach in Case I and II and demonstrated in the following.
Case III
In this case, the mixed control is applied for overcoming the chartering phenomenon.
At first, the he maximal convergency rate control is adopted for fast response and then the saturation control is applied when the states approach the switching plane. In this case, we consider the following control strategy:
( ) ( )
( )
, x 0.01 , .
T
T
SAT B PX U t
SGN B PX other
⎧− <
= ⎨⎪
⎪⎩−
The trajectory is depicted in Fig. 35 and control input is depicted in Fig. 36. We can conclude that the system has fast property by the maximal convergence rate control and smooth when approach the switching plane.
-1.5 -1 -0.5 0 0.5 1 1.5 -1.5
-1 -0.5 0 0.5 1 1.5
x1
x 2
Fig. 30 The level set of Example 6.
0 5 10 15 20 25 30 35
-0.5 0 0.5 1
t (sec.)
X
0 5 10 15 20 25 30 35 -0.2
0 0.2 0.4 0.6 0.8 1 1.2
t (Sec.)
U
Fig. 32 Corresponded control input (Case I).
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-0.5 0 0.5 1
t (Sec.)
X
Fig. 33 The trajectory in phase plane (Case II).
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -1
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
t (Sec.)
U
Fig. 34 Corresponded control input (Case II).
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-0.5 0 0.5 1
t (Sec)
X
0 1 2 3 4 5 0
0.2 0.4 0.6 0.8 1
t (Sec.)
U
Fig. 36 Corresponded control input (Case III).
Chapter 5 Conclusion
This dissertation presents a new design of time-optimal controller for controllable Takagi-Sugeno (T-S) fuzzy model in which the maximum principle is applied. In particular, the subsystems of T-S fuzzy model are blended by a set of firing strengths, which leads it to a class of nonlinear system. First, we proposed the proof of the existence of optimal control in T-S fuzzy model, which can be addressed as the compactness of reachable set. The generalized rank condition of accessible Lie algebra is also applied for the proof of the existence of optimal controller for T-S
fuzzy model. This also results in the controllability of the T-S fuzzy model. According to the maximum principle, the time-optimal control of T-S fuzzy model is bang-bang which is determined by switching function. By investigating the singular structure of the switching functions of the controllable T-S fuzzy model, we can yield the conditions for the existence, i.e., if the extremal is normal then there exists the time-optimal controller for the T-S fuzzy model. In other words, the time-optimal control of controllable T-S fuzzy model is bang-bang with finite number of switching over all trajectories for all t . The bounded number of switching is related to the polynomial degree of switching function which is obtained by introducing solvable Lie algebra. Several examples are fully illustrated to show the conditions for the existence of time-optimal controller with their optimal trajectories found by numerical simulation. Further, the feedback controller design of T-S fuzzy model on maximal convergence rate is introduced by level set function. The result of maximizing the convergence rate is characterized from the maximal invariant ellipsoid. The controller is also bang-bang with a simple switching strategy. To handle the chartering phenomenon, a two stages control of saturation and maximizing the convergence rate is also demonstrated. Numerical simulations show the system response is fast and control input is smooth.
References
[1] K. Tanaka and M. Sano, “A robust stabilization problem of fuzzy control systems and its application to backing up control of a truck-trailer” IEEE Transactions on Fuzzy Systems, vol. 2, pp.119-134, May 1994.
[2] K. Tanaka, T. Ikeda, and H. O. Wang, “Robust stabilization of a class of uncertain nonlinear system via fuzzy control: Quadratic stabilizability, H∞ control theory and linear matrix inequalities,” IEEE Transactions on Fuzzy Systems, vol. 4, pp. 1-13, Feb. 1996.
[3] A. E. Gegov and P. M. Frank, “Hierarchical fuzzy control of multivariable systems,” Fuzzy Sets and Systems, vol. 72, no. 3 pp. 299-310, 1995.
[4] C. Lee, “Fuzzy logic in control systems: Fuzzy logic controller Part 1, Part 2,”
IEEE Transaction on Systems, Man, and Cybernetics, Part B, vol. 20, pp.
404-435, 1990.
[5] J. I. Horiuchi and M. Kishimoto, “Application of fuzzy control to industrial bioprocesses in Japan,” Fuzzy Sets and Systems, vol. 128, no. 1 pp. 117-124, 2002.
[6] C. C. Hsiao, S. F. Su, T. T. Lee, and C. C. Chuang, “Hybrid compensation control for affine TSK fuzzy control systems,” IEEE Transaction on Systems, Man, and Cybernetics, Part B, 2004.
[7] Y.G. Leu, T.T. Lee and W.Y. Wang, “Observer-based adaptive fuzzy-neural control for unknown nonlinear dynamical systems,” IEEE Transaction on Systems, Man, and Cybernetics, Part B, vol. 29, no.5, and pp.583-591, Oct.
1999.
[8] H. O. Wang, K. Tanaka, Michael F. Griffin, “An approach to fuzzy control of
nonlinear systems: Stability and design issues,” IEEE Transactions on Fuzzy Systems, vol. 4, no. 1 pp. 14-23, Feb. 1996.
[9] T. Takagi and M. Sugeno, “Fuzzy identification of systems and it’s Applications to modeling and control,” IEEE Transactions on System Man and Cybernetics, vol. 15, pp. 116-132 no. 1, 1985.
[10] S. Sugeno and G. T. Kang, “Structure identification of fuzzy model,” Fuzzy Sets and Systems, vol. 28, no. 10, pp. 15-33, 1988.
[11] T. Takagi and M. Sugeno, “Stability analysis and design of fuzzy control systems,” Fuzzy Sets and Systems, vol. 45, no. 2, pp. 135-156, 1993.
[12] M. C. M. Teixerira and S.H. Zak, “Stabilization controller design for uncertain nonlinear systems using fuzzy models,” IEEE Transactions on Fuzzy Systems, vol. 7, pp. 133-142, Apr. 1999.
[13] S. G. Cao, N. W. Rees, and G. Feng, “H-infinity control of nonlinear continuous-time system based on dynamical fuzzy models,” Int. J. Syst. Sci., vol.
27, pp. 821-830, 1996.
[14] K. Tanaka, T. Ikeda, and H. O. Wang, “Fuzzy regulators and fuzzy observers:
Relaxed stability conditions and LMI-based designs,” IEEE Transactions on Fuzzy Systems, vol. 6, pp. 250-265, 1998.
[15] B.S. Chen, C.S. Tseng, H.J. Uang, “Mixed H2/H∞ fuzzy output feedback control design for nonlinear dynamic systems: An LMI approach,” IEEE Transactions on Fuzzy Systems, vol. 8, pp. 249-265, 2000.
[16] H. J. Lee, J.B. Park, and G. Chen, “Robust fuzzy control of nonlinear system with parametric uncertainties,” IEEE Transactions on Fuzzy Systems, vol. 9, pp.
369-379, 2001.
[17] G. Feng, “Approaches to quadratic stabilization of uncertain continuous time
pp. 369-379, 2001.
[18] K. R. Lee, E.T. Jeung, and H. B. Park, “Robust fuzzy H∞ control for uncertain nonlinear systems via state feedback: An LMI approach,” Fuzzy Sets and Systems, vol. 120, pp. 123-134, 2001.
[19] Piotr Kulczycki, “Fuzzy controller for mechanical systems,” IEEE Transactions on Fuzzy Systems, vol. 8, no. 5, pp. 645-652, Oct 2000.
[20] Pontryagin, L. S., V. Boltyanskii, T. Gamkrelideze, and E. Mishchenko, “The mathematical theory of optimal processes,” Interscience Publishers, Inc., New York, 1962.
[21] Michael Athans and Peter L. Palb, Optimal Control, McGraw-hill Inc., 1966 [22] Henry Hermes and Joseph P. Lasalle, “Functional Analysis and Time Optimal
Control,” Academic Press, 1969.
[23] Jack Macki and Aaron Strauss, Introduction to Optimal Control Theory, Springer-Verlag, New York, 1982.
[24] Donald E. Kirk, Optimal Control Theory: An Introduction, Prentice-Hall, 1970.
[25] Bronislaw Jakubczyk and Witold Respondek, Geometry of Feedback and Optimal Control, Marcel Dekker Inc., 1998.
[26] H. J. Sussmann, Differential Geometric Control Theory, Birkhäuser, Boston, MA, 1983.
[27] H. J. Sussmann, The Maximum Principle of Optimal Control Theory, J. Baillieul, J.C. Willems(Eds), Springer-Verlag, pp. 140-198, 1998.
[28] H. J. Sussmann and V. Jurdjevic, “Controllability of nonlinear system,” Journal of Diff. Equations, 12,95, 1972.
[29] E.D. Sontag and H. J. Sussmann, “Time-optimal control of manipulators,” Proc.
IEEE Int. Conf. on Robotics and Automation, pp. 1692-1697, 1986.
[30] E.D. Sontag, “Remarks on the time-optimal control of a class of Hamiltonian
systems,” IEEE Conf. on Decision and Control, pp. 317-221, 1989.
[31] M. Chyba, N.E. Leonard and E.D. Sontag, “Singular trajectories in multi-input time-optimal problems: Application to controlled mechanical,” Journal on Dynamics and Control Systems, vol. 9, pp. 73-88, 2003.
[32] A.D. Lewis, The Geometry of the Maximum Principle for Affine Connection Control Systems, Preprint, 2000.
[33] A. A. Agrachev, Mathematical Control Theory, A. A. Agrachev (Eds), Steklov Mathematical Institue, Moscow and SISSA, Trieste.
[34] R. Vinter, Optimal Control, Birkhäuser, Boston, MA, 2000.
[35] R. W. Brockett, “Nonlinear systems and differential geometry,” Proc. IEEE, vol.
64, no. 1, pp. 61-71, 1976.
[36] Boyd, S., El Ghaoui, L., Feron, E., and Balakrishnan, V., Linear Matrix Inequalities in System and Control Theory,” SIAM, Philadelphia, PA, 1994.
[37] J.-P. Aubin and A. Cellina, Differential Inclusions: Set-Valued Maps and Viability Theory, Springer-Verlag, Berlin, 1984.
[38] Hans Samelson, Note on Lie Algebras, Van Nostrand Reinhold, New York, 1969 [39] Olver, P.J., Application of Lie Groups to Differential Equations,” 2nd ed.,
Springer-Verlag, 1993
[40] Lastman, G. J., “A Shooting Method For Solving Two-Point Boundary-Value Problems Arising From Nonsingular Bang--bang Optimal Control Processes,”
International Journal of Control, vol. 27, pp. 513-524, 1978.
[41] Zhendong Sun1, S.S. Gee and T.H. Lee, “Controllability and reachability criteria for switched linear systems,” Automatica, 38, pp. 775-786, 2002.
[42] Guangming Xie, Dazhong Zheng and Long Wang, “Controllability of switched linear systems,” IEEE Transactions on Automatic Control, vol. 47, no. 4, pp.
[43] Daizhan Cheng, “Controllability of switched bilinear systems,” IEEE Transactions on Automatic Control, vol. 50, no. 4, pp. 511-515, 2005.
[44] Alberto Bemporad, Giancarlo Ferrari-Trecate, and Manfred Morari,
“Observability and controllability of piecewise affine and hybrid Systems,”
IEEE Transactions on Automatic Control, vol. 45, no. 4, pp. 1864-1876, 2000.
[45] L. Habets and J.H. van Schuppen, “A controllability result for piecewise-linear hybrid systems,” Proceedings of European Control Conference, 2001.
[46] Tingshu Hu and Zongli Lin, Control Systems with Actuator Saturation, Birkhäuser, Boston, MA, 2001
Vita
Name: Pao-Tsun Lin
Personal:
Place of Birth: Hsinchu County, Taiwan, R.O.C.
Day of Birth: January 14, 1976 Gender: Male
Education background:
Degree Date School
B.S. E.E. 1997/9 - 1999/6 National Taiwan University of Science and Technology, Department of Electronic Engineering
M.S. E.E. 1999/9 - 2001/6 National Taiwan University of Science and Technology, Department of Electronic Engineering
Ph.D. E.C.E. 2001/9 - 2008/7 National Chiao Tung University,
Department of Electrical and Control Engineering
Advisor(s):
M.S. -
Dr. Shun-Feng Su, National Taiwan University of Science and Technology Ph.D. –
Dr. Chi-Hsu Wang, National Chiao Tung University
Dr. Tsu-Tian Lee, National Taipei University of Technology
Publication List
Accepted Journal Paper:
[1] Pao-Tsun Lin, Chi-Hsu Wang and Tsu-Tian Lee, “Time-Optimal Control of T-S Fuzzy Models via Lie Algebra” accepted to be published in IEEE Transactions on Fuzzy Systems.
International Conference Papers:
[1] Pao-Tsun Lin, T. T. Lee and Chi-Hsu Wang,” Accessibility of T-S Fuzzy Models via Lie Algebra,” accepted to be published in 2008 National Symposium on System Science and Engineering.
[2] Pao-Tsun Lin, T. T. Lee and Chi-Hsu Wang,” Analysis of Time-Optimal Problem in T-S Fuzzy Model via Lie Algebra,” Proc. of The 26th IASTED International Conference on Modelling, Identification, and Control, pp. 749-755 2007
[3] Pao-Tsun Lin, Shun-Feng Su, and Tsu-Tian Lee, “Time-optimal fuzzy control based on computation method,” in Proc. of 2005 IEEE International Conference on Systems, Man and Cybernetics, Vol. 1. pp. 600- 605, Oct. 2005.
[4] Pao-Tsun Lin, Shun-Feng Su, and Tsu-Tian Lee, “Support vector regression performance analysis and systematic parameter selection,” in Proc. of 2005 IEEE International Joint Conference on Neural Network, Vol. 2. pp. 877- 882, Aug.
2005.
[5] Pao-Tsun Lin, Shun-Feng Su, and Tsu-Tian Lee, “Time-optimal control via fuzzy approach,” in Proc. of 2004 IEEE International Conference on Systems, Man and Cybernetics, Vol. 4. pp. 3817- 3821, Oct. 2004.