LMI–Based Robust Control of Uncertain Nonlinear Systems via Polytopes of Polynomials

Amato, F., Pironti, A. and Scala, S. (1996). Necessary and sufficient conditions for quadratic stability and stabilizability of uncertain linear time-varying systems, IEEE Transactions on Automatic Control41(1): 125–128.10.1109/9.481616Search in Google Scholar

Apkarian, P. and Gahinet, P. (1995). A convex characterization of gain-scheduled h_∞ controllers, IEEE Transactions on Automatic Control40(5): 853–864.10.1109/9.384219Search in Google Scholar

Barber, C.B., Dobkin, D.P. and Huhdanpaa, H. (1996). The quickhull algorithm for convex hulls, ACM Transactions on Mathematical Software (TOMS)22(4): 469–483.10.1145/235815.235821Search in Google Scholar

Barmish, B.R., Hollot, C.V., Kraus, F.J. and Tempo, R. (1992). Extreme point results for robust stabilization of interval plants with first-order compensators, IEEE Transactions on Automatic Control37(6): 707–714.10.1109/9.256326Search in Google Scholar

Bartlett, A.C., Hollot, C.V. and Lin, H. (1988). Root locations of an entire polytope of polynomials: It suffices to check the edges, Mathematics of Control, Signals and Systems1(1): 61–71.10.1007/BF02551236Search in Google Scholar

Baumann, W. and Rugh, W. (1986). Feedback control of nonlinear systems by extended linearization, IEEE Transactions on Automatic Control31(1): 40–46.10.1109/TAC.1986.1104100Search in Google Scholar

Białas, S. (1985). A necessary and sufficient condition for the stability of convex combinations of stable polynomials or matrices, Bulletin of the Polish Academy of Sciences33(9–10): 473–480.Search in Google Scholar

Białas, S. and Góra, M. (2012). A few results concerning the Hurwitz stability of polytopes of complex polynomials, Linear Algebra and Its Applications436(5): 1177–1188.10.1016/j.laa.2011.07.042Search in Google Scholar

Boyd, S., Ghaoui, L.E., Feron, E. and Belakrishnan, V. (1994). Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia, PA.10.1137/1.9781611970777Search in Google Scholar

Dorf, R. and Bishop, R. (1998). Modern Control Systems, Pearson (Addison-Wesley), Upper Saddle River, NJ.Search in Google Scholar

Ebihara, Y., Peaucelle, D., Arzelier, D., Hagiwara, T. and Oishi, Y. (2012). Dual LMI approach to linear positive system analysis, 12th International Conference on Control, Automation and Systems (ICCAS), JeJu Island, South Korea, pp. 887–891.Search in Google Scholar

Gahinet, P., Nemirovski, A., Laub, A.J. and Chilali, M. (1995). LMI Control Toolbox, Math Works, Natick, MA.Search in Google Scholar

González, T. and Bernal, M. (2016). Progressively better estimates of the domain of attraction for nonlinear systems via piecewise Takagi–Sugeno models: Stability and stabilization issues, Fuzzy Sets and Systems297: 73–95.10.1016/j.fss.2015.11.010Search in Google Scholar

González, T., Bernal, M., Sala, A. and Aguiar, B. (2017). Cancellation-based nonquadratic controller design for nonlinear systems via Takagi–Sugeno models, IEEE Transactions on Cybernetics47(9): 2628–2638.10.1109/TCYB.2016.258954327542189Search in Google Scholar

Guerra, T.M., Estrada-Manzo, V. and Lendek, Z. (2015). Observer design of nonlinear descriptor systems: An LMI approach, Automatica52: 154–159.10.1016/j.automatica.2014.11.008Search in Google Scholar

Guerra, T.M. and Vermeiren, L. (2004). LMI-based relaxed non-quadratic stabilization conditions for nonlinear systems in Takagi–Sugeno’s form, Automatica40(5): 823–829.10.1016/j.automatica.2003.12.014Search in Google Scholar

Henrion, D., Sebek, M. and Kucera, V. (2003). Positive polynomials and robust stabilization with fixed-order controllers, IEEE Transactions on Automatic Control48(7): 1178–1186.10.1109/TAC.2003.814103Search in Google Scholar

Johansen, T.A. (2000). Computation of Lyapunov functions for smooth nonlinear systems using convex optimization, Automatica36(11): 1617–1626.10.1016/S0005-1098(00)00088-1Search in Google Scholar

Khalil, H. (2002). Nonlinear Systems, 3rd Edn., Prentice Hall, Upper Saddle River, NJ.Search in Google Scholar

Kharitonov, V. (1978). Asymptotic stability of an equilibrium position of a family of systems of linear differential equations, Differential Equations14(11): 1483–1485.Search in Google Scholar

Kwiatkowski, A., Boll, M. and Werner, H. (2006). Automated generation and assessment of affine LPV models, 45th IEEE Conference on Decision and Control, San Diego, CA, USA, pp. 6690–6695.10.1109/CDC.2006.377768Search in Google Scholar

Pan, J., Guerra, T., Fei, S. and Jaadari, A. (2012). Nonquadratic stabilization of continuous T–S fuzzy models: LMI solution for a local approach, IEEE Transactions on Fuzzy Systems20(3): 594–602.10.1109/TFUZZ.2011.2179660Search in Google Scholar

Rhee, B. and Won, S. (2006). A new fuzzy Lyapunov function approach for a Takagi–Sugeno fuzzy control system design, Fuzzy Sets and Systems157(9): 1211–1228.10.1016/j.fss.2005.12.020Search in Google Scholar

Robles, R., Sala, A., Bernal, M. and González, T. (2017). Subspace-based Takagi–Sugeno modeling for improved LMI performance, IEEE Transactions on Fuzzy Systems25(4): 754–767.10.1109/TFUZZ.2016.2574927Search in Google Scholar

Sala, A. and Ario, C. (2009). Polynomial fuzzy models for nonlinear control: A Taylor series approach, IEEE Transactions on Fuzzy Systems17(6): 1284–1295.10.1109/TFUZZ.2009.2029235Search in Google Scholar

Sala, A. and Ariño, C. (2007). Asymptotically necessary and sufficient conditions for stability and performance in fuzzy control: Applications of Polya’s theorem, Fuzzy Sets and Systems158(24): 2671–2686.10.1016/j.fss.2007.06.016Search in Google Scholar

Sanchez, M. and Bernal, M. (2017). A convex approach for reducing conservativeness of Kharitonov’s-based robustness analysis, 20th IFAC World Congress, Toulouse, France, pp. 855–860.Search in Google Scholar

Shamma, J.S. and Cloutier, J.R. (1992). A linear parameter varying approach to gain scheduled missile autopilot design, American Control Conference, Chicago, IL, USA, pp. 1317–1321.10.23919/ACC.1992.4792317Search in Google Scholar

Sideris, A. and Barmish, B.R. (1989). An edge theorem for polytopes of polynomials which can drop in degree, Systems & Control Letters13(3): 233–238.10.1016/0167-6911(89)90069-8Search in Google Scholar

Sturm, J.F. (1999). Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones, Optimization Methods and Software11(1–4): 625–653.10.1080/10556789908805766Search in Google Scholar

Tanaka, K., Hori, T. and Wang, H. (2003). A multiple Lyapunov function approach to stabilization of fuzzy control systems, IEEE Transactions on Fuzzy Systems11(4): 582–589.10.1109/TFUZZ.2003.814861Search in Google Scholar

Tanaka, K. and Wang, H. (2001). Fuzzy Control Systems Design and Analysis. A Linear Matrix Inequality Approach, John Wiley&Sons, New York, NY.10.1002/0471224596Search in Google Scholar

Taniguchi, T., Tanaka, K. and Wang, H. (2001). Model construction, rule reduction and robust compensation for generalized form of Takagi–Sugeno fuzzy systems, IEEE Transactions on Fuzzy Systems9(2): 525–537.10.1109/91.940966Search in Google Scholar

Tapia, A., Bernal, M. and Fridman, L. (2017). Nonlinear sliding mode control design: An LMI approach, Systems & Control Letters104: 38–44.10.1016/j.sysconle.2017.03.011Search in Google Scholar

Wang, H., Tanaka, K. and Griffin, M. (1996). An approach to fuzzy control of nonlinear systems: Stability and design issues, IEEE Transactions on Fuzzy Systems4(1): 14–23.10.1109/91.481841Search in Google Scholar

Xu, S., Darouach, M. and Schaefers, J. (1993). Expansion of det(a + b) and robustness analysis of uncertain state space systems, IEEE Transactions on Automatic Control38(11): 1671–1675.10.1109/9.262036Search in Google Scholar