0
Research Papers

# A Gain-Scheduling Control Approach for Takagi–Sugeno Fuzzy Systems Based on Linear Parameter-Varying Control Theory

[+] Author and Article Information
Yang Liu

Center for Control Theory and
Guidance Technology,
Harbin Institute of Technology,
Harbin, Heilongjiang 150001, China
e-mail: liuyang5264@163.com

Xiaojun Ban

Center for Control Theory and
Guidance Technology,
Harbin Institute of Technology,
Harbin, Heilongjiang 150001, China
e-mail: banxiaojun@hit.edu.cn

Fen Wu

Department of Mechanical and
Aerospace Engineering,
North Carolina State University,
Raleigh, NC 27695
e-mail: fwu@ncsu.edu

H. K. Lam

Department of Informatics,
King's College London,
Strand, London WC2R 2LS, UK
e-mail: hak-keung.lam@kcl.ac.uk

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received April 27, 2015; final manuscript received October 14, 2015; published online November 16, 2015. Assoc. Editor: Ryozo Nagamune.

J. Dyn. Sys., Meas., Control 138(1), 011008 (Nov 16, 2015) (9 pages) Paper No: DS-15-1194; doi: 10.1115/1.4031914 History: Received April 27, 2015; Revised October 14, 2015

## Abstract

Due to the universal approximation capability of Takagi–Sugeno (T–S) fuzzy models for nonlinear dynamics, many control issues have been investigated based on fuzzy control theory. In this paper, a transformation procedure is proposed to convert fuzzy models into linear fractional transformation (LFT) models. Then, T–S fuzzy systems can be regarded as a special case of linear parameter-varying (LPV) systems which proved useful for nonlinear control problems. The newly established connection between T–S fuzzy models and LPV models provides a new perspective of the control problems for T–S fuzzy systems and facilitates the fuzzy control designs. Specifically, an output feedback gain-scheduling control design approach for T–S fuzzy systems is presented to ensure globally asymptotical stability and optimize $H∞$ performance of the closed-loop systems. The control synthesis problem is cast as a convex optimization problem in terms of linear matrix inequalities (LMIs). Two examples have been used to illustrate the efficiency of the proposed method.

<>

## References

Ying, H. , 1998, “ Sufficient Conditions on Uniform Approximation of Multivariate Functions by General Takagi-Sugeno Fuzzy Systems With Linear Rule Consequent,” IEEE Trans. Syst., Man Cybern., Part A: Syst. Hum., 28(4), pp. 515–520.
Takagi, T. , and Sugeno, M. , 1985, “ Fuzzy Identification of Systems and Its Applications to Modeling and Control,” IEEE Trans. Syst., Man Cybern., 15(1), pp. 116–132.
Tanaka, K. , Ikeda, T. , and Wang, H. O. , 1998, “ Fuzzy Regulators and Fuzzy Observers: Relaxed Stability Conditions and LMI-Based Designs,” IEEE Trans. Fuzzy Syst., 6(2), pp. 250–265.
Tanaka, K. , and Sugeno, M. , 1992, “ Stability Analysis and Design of Fuzzy Control Systems,” Fuzzy Sets Syst., 45(2), pp. 135–156.
Wang, H. O. , Tanaka, K. , and Griffin, M. F. , 1996, “ An Approach to Fuzzy Control of Nonlinear Systems: Stability and Design Issues,” IEEE Trans. Fuzzy Syst., 4(1), pp. 14–23.
Lam, H. K. , and Leung, F. H. , 2010, Stability Analysis of Fuzzy-Model-Based Control Systems: Linear-Matrix-Inequality Approach, Springer-Verlag, Berlin.
Lam, H. K. , Seneviratne, L. D. , and Ban, X. , 2012, “ Fuzzy Control of Nonlinear Systems Using Parameter-Dependent Polynomial Fuzzy Model,” IET Control Theory Appl., 6(11), pp. 1645–1653.
Lo, J. C. , and Lin, M. L. , 2003, “ Robust H ∞ Nonlinear Control Via Fuzzy Static Output Feedback,” IEEE Trans. Circuits Syst. I: Regular Pap., 50(11), pp. 1494–1502.
Kau, S. W. , Lee, H. J. , Yang, C. M. , Lee, C. H. , Hong, L. , and Fang, C. H. , 2007, “ Robust H ∞ Fuzzy Static Output Feedback Control of T-S Fuzzy Systems With Parametric Uncertainties,” Fuzzy Sets Syst., 158(2), pp. 135–146.
Qiu, J. , Feng, G. , and Gao, H. , 2010, “ Fuzzy-Model-Based Piecewise H ∞ Static-Output-Feedback Controller Design for Networked Nonlinear Systems,” IEEE Trans. Fuzzy Syst., 18(5), pp. 919–934.
Chadli, M. , and Guerra, T. M. , 2012, “ LMI Solution for Robust Static Output Feedback Control of Discrete Takagi-Sugeno Fuzzy Models,” IEEE Trans. Fuzzy Syst., 20(6), pp. 1160–1165.
Lam, H. K. , and Li, H. , 2013, “ Output-Feedback Tracking Control for Polynomial Fuzzy Systems,” IEEE Trans. Ind. Electron., 60(12), pp. 5830–5840.
Lo, J. C. , and Lin, M. L. , 2004, “ Observer-Based Robust H ∞ Control for Fuzzy Systems Using Two-Step Procedure,” IEEE Trans. Fuzzy Syst., 12(3), pp. 350–359.
Tseng, C. S. , and Hwang, C. K. , 2007, “ Fuzzy Observer-Based Fuzzy Control Design for Nonlinear Systems With Persistent Bounded Disturbances,” Fuzzy Sets Syst., 158(2), pp. 164–179.
Lam, H. K. , Li, H. , and Liu, H. , 2013, “ Stability Analysis and Control Synthesis for a Fuzzy-Observer-Based Controller of Nonlinear Systems: A Fuzzy-Model-Based Control Approach,” IET Control Theory Appl., 7(5), pp. 663–672.
Li, J. , Wang, H. O. , Niemann, D. , and Tanaka, K. , 2000, “ Dynamic Parallel Distributed Compensation for Takagi-Sugeno Fuzzy Systems: An LMI Approach,” Inf. Sci., 123(3–4), pp. 201–221.
Han, Z. X. , Feng, G. , Walcott, B. L. , and Ma, J. , 2000, “ Dynamic Output Feedback Controller Design for Fuzzy Systems,” IEEE Trans. Syst., Man, Cybern. Part B, Cybern., 30(1), pp. 204–210.
Nguang, S. K. , and Shi, P. , 2006, “ Robust H ∞ Output Feedback Control Design for Fuzzy Dynamic Systems With Quadratic D Stability Constrains: An LMI Approach,” Inf. Sci., 176(15), pp. 2161–2191.
Yoneyama, J. , 2006, “ Robust H ∞ Control Analysis and Synthesis for Takagi-Sugeno General Uncertain Fuzzy Systems,” Fuzzy Sets Syst., 157(16), pp. 2205–2223.
Guelton, K. , Bouarar, T. , and Manamanni, N. , 2009, “ Robust Dynamic Output Feedback Fuzzy Lyapunov Stabilization of Takagi-Sugeno Systems: A Descriptor Redundancy Approach,” Fuzzy Sets Syst., 160(19), pp. 2796–2811.
Yang, G. H. , and Dong, J. , 2010, “ Switching Fuzzy Dynamic Output Feedback H ∞ Control for Nonlinear Systems,” IEEE Trans. Syst., Man, Cybern. Part B, Cybern., 40(2), pp. 505–516.
Tognetti, E. S. , Oliveira, R. C. L. F. , and Peres, P. L. D. , 2012, “ Reduced-Order Dynamic Output Feedback Control of Continuous-Time T-S Fuzzy Systems,” Fuzzy Sets Syst., 207, pp. 27–44.
Kim, E. , and Lee, H. , 2000, “ New Approaches to Relaxed Quadratic Stability Condition of Fuzzy Control Systems,” IEEE Trans. Fuzzy Syst., 8(5), pp. 523–534.
Teixiera, M. C. M. , Assuncao, E. , and Avellar, R. G. , 2003, “ On Relaxed LMI-Based Designs for Fuzzy Regulators and Fuzzy Observers,” IEEE Trans. Fuzzy Syst., 11(5), pp. 613–623.
Tanaka, K. , Hori, T. , and Wang, H. O. , 2003, “ A Multiple Lyapunov Function Approach to Stabilization of Fuzzy Control Systems,” IEEE Trans. Fuzzy Syst., 11(4), pp. 582–589.
Rugh, W. J. , and Shamma, J. S. , 2000, “ Research on Gain Scheduling,” Automatica, 36(10), pp. 1401–1425.
Balas, G. J. , Fialho, I. , Packard, A. K. , Renfrow, J. , and Mullaney, C. , 1997, “ On the Design of LPV Controllers for the F-14 Aircraft Lateral-Directional Axis During Powered Approach,” American Control Conference, pp. 123–127.
Apkarian, P. , and Adams, R. , 1998, “ Advanced Gain-Scheduling Techniques for Uncertain Systems,” IEEE Trans. Control Syst. Technol., 6(1), pp. 21–32.
Wu, F. , Packard, A. , and Balas, G. , 2002, “ Systematic Gain-Scheduling Control Design: A Missile Autopilot Example,” Asian J. Control, 4(3), pp. 341–347.
Tsourdos, A. , Economou, J. T. , White, B. A. , and Luk, P. C. K. , 2003, “ Control Design for a Mobile Robot: A Fuzzy LPV Approach,” IEEE Conference on Control Applications, pp. 552–557.
Huang, Y. , Sun, C. , Qian, C. , Zhang, R. , and Zhang, J. , 2012, “ Polytopic LPV Gain-Scheduled Control for a Flexible Air-Breathing Hypersonic Vehicle,” 31st Chinese Control Conference, pp. 329–334.
Precup, R. E. , Dragos, C. A. , Preitl, S. , Radac, M. B. , and Petriu, E. M. , 2012, “ Novel Tensor Product Models for Automatic Transmission System Control,” IEEE Syst. J., 6(3), pp. 488–498.
Hanifzadegan, M. , and Nagamune, R. , 2014, “ Smooth Switching LPV Controller Design for LPV Systems,” Automatica, 50(5), pp. 1481–1488.
Packard, A. K. , 1994, “ Gain Scheduling Via Linear Fractional Transformations,” Syst. Control Lett., 22(2), pp. 79–92.
Apkarian, P. , and Gahinet, P. , 1995, “ A Convex Characterization of Gain-Scheduled H ∞ Controllers,” IEEE Trans. Autom. Control, 40(5), pp. 853–864.
Chilali, M. , and Gahinet, P. , 1996, “ H ∞ Design With Pole Placement Constraints: An LMI Approach,” IEEE Trans. Autom. Control, 41(3), pp. 358–367.

## Figures

Fig. 1

Disturbance attenuation of Example 1: (a) measurement output, (b) control input, and (c) L2 gain

Fig. 2

Response to the initial condition x(0) = [85 deg 5.73 deg/s]T of Example 1: (a) measurement output and (b) control input

Fig. 3

Simulation results of Example 2: (a) plant states, (b) output, and (c) control input

## Errata

Some tools below are only available to our subscribers or users with an online account.

### Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related Proceedings Articles
Related eBook Content
Topic Collections