zbMATH — the first resource for mathematics

Stability analysis and synthesis of nonlinear time-delay systems via linear Takagi-Sugeno fuzzy models. (English) Zbl 1002.93051
Consider the time-delay system \[ \dot x(t)= \sum^r_{i=1} h_i(z(t))[A_{1i} x(t)+ A_{2i}x(t- \tau_i(t))],\tag{1} \] \(h_i(z(t))\geq 0\), for \(i= 1,2,\dots, r\) and \(\sum^r_{j=1} h_j(z(t))= 1\) for all \(t\). Having used a quadratic Lyapunov function \(V(x(t))= x^T(t) Px(t)\) and the Razumikhin conditions we have
Theorem 1. The equilibrium of the continuous-time fuzzy system with time delay described by (1) is asymptotically stable in the large if there exist a common matrix \(P> 0\) and \(r\) matrices \(S_i> 0\) such that \[ A^T_{1i}P+ PA_{1i}+ P+ PA_{2i} S_iA^T_{2i}P< 0,\quad P\geq S^{-1}_i,\quad\text{for }i= 1,2,\dots,r. \] Later, more complex time-delay systems are considered.

93D20 Asymptotic stability in control theory
93C42 Fuzzy control/observation systems
93C23 Control/observation systems governed by functional-differential equations
93D30 Lyapunov and storage functions
Full Text: DOI
[1] Boyd, S.; El Ghaoui, L.; Feron, E.; Balakrishnan, V., Linear matrix inequalities in system and control theory, (1994), SIAM Philadelphia, PA · Zbl 0816.93004
[2] Brierley, S.D.; Chiasson, J.N.; Lee, E.B.; Zak, S.H., On stability independent of delay for linear systems, IEEE trans. automat. control, 27, 2, 252-254, (1982) · Zbl 0469.93065
[3] Cao, Y.-Y.; Sun, Y.-X., Robust stabilization of uncertain systems with time-varying multi-state-delay, IEEE trans. automat. control, 43, 10, 1484-1488, (1998) · Zbl 0956.93057
[4] Cao, Y.-Y.; Sun, Y.-X.; Cheng, C., Delay-dependent robust stabilization of uncertain systems with multiple state delays, IEEE trans. automat. control, 43, 11, 1608-1612, (1998) · Zbl 0973.93043
[5] Cheres, E.; Gutman, S.; Palmor, Z.J., Stabilization of uncertain dynamic systems including state delay, IEEE trans. automat. control, 34, 11, 1199-1203, (1989) · Zbl 0693.93059
[6] Ge, J.H.; Frank, P.M.; Lin, C.-F., \(H∞\) control via output feedback for state delayed systems, Int. J. control, 64, 1, 1-7, (1996) · Zbl 0853.93043
[7] Hale, J., Theory of functional differential equations, (1977), Springer New York
[8] Lehman, B.; Bentsman, J.; Lunel, S.V.; Verriest, E.I., Vibrational control of nonlinear time lag systems with bounded delayaveraging theory, stabilizability, and transient behavior, IEEE trans. automat. control, 39, 5, 898-912, (1994) · Zbl 0813.93044
[9] Mahmoud, M.S.; Al-Muthairi, N.F., Design of robust controllers for time-delay systems, IEEE trans. automat. control, 39, 5, 995-999, (1994) · Zbl 0807.93049
[10] Mori, T., Criteria for asymptotic stability of linear time delay systems, IEEE trans. automat. control, 30, 1, 158-161, (1985) · Zbl 0557.93058
[11] Niculescu, S.-I.; Verriest, E.I.; Dugard, L.; Dion, J.-D., Stability and robust stability of time-delay systemsa guided tour, (), 1-71 · Zbl 0914.93002
[12] Stepan, G., Retarded dynamical systems: stability and characteristic functions, Pitman research notes in mathematics, (1989), Longman Scientific and Technical Harlow, UK · Zbl 0686.34044
[13] Takagi, T.; Sugeno, M., Fuzzy identification of systems and its applications to modeling and control, IEEE trans. systems man cybernet., 15, 1, 116-132, (1985) · Zbl 0576.93021
[14] Takagi, T.; Sugeno, M., Stability analysis and design of fuzzy control systems, Fuzzy sets and systems, 45, 2, 135-156, (1992) · Zbl 0758.93042
[15] Tanaka, K.; Ikeda, T.; Wang, H.O., Fuzzy regulators and fuzzy observersrelaxed stability conditions and LMI-based designs, IEEE trans. fuzzy systems, 6, 2, 250-265, (1998)
[16] Tanaka, K.; Sano, M., A robust stabilization problem of fuzzy control systems and its application to backing up control of a truck-trailer, IEEE trans. fuzzy systems, 2, 119-134, (1994)
[17] Wang, H.O.; Tanaka, K.; Griffin, M.F., An approach to fuzzy control of nonlinear systemsstability and design issues, IEEE trans. fuzzy systems, 4, 1, 14-23, (1996)
[18] Zheng, F.; Cheng, M.; Gao, W.B., Feedback stabilization of linear systems with distributed delays in state and control variables, IEEE trans. automatic control, 39, 1714-1718, (1994) · Zbl 0825.93626
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.