## Dissipativity analysis and synthesis of a class of nonlinear systems with time-varying delays.(English)Zbl 1169.93012

Summary: In this paper, new results are established for the delay-independent and delay-dependent problems of dissipative analysis and state-feedback synthesis for a class of nonlinear systems with time-varying delays with polytopic uncertainties. This class consists of linear time-delay systems subject to nonlinear cone-bounded perturbations. Both delay-independent and delay-dependent dissipativity criteria are established as linear matrix inequality-based feasibility tests. The developed results in this paper for the nominal system encompass available results on $$\mathcal H_{\infty }$$ approach, passivity and positive realness for time-delay systems as special cases. All the sufficient stability conditions are cast. Robust dissipativity as well as dissipative state-feedback synthesis results are also derived. Numerical examples are provided to illustrate the theoretical developments.

### MSC:

 93C10 Nonlinear systems in control theory 93B35 Sensitivity (robustness) 93D15 Stabilization of systems by feedback 93B50 Synthesis problems 93C15 Control/observation systems governed by ordinary differential equations

LMI toolbox
Full Text:

### References:

 [1] Anderson, B.O.D.; Vongpanitherd, S., Modern analysis and synthesis: A modern systems theory approach, (1973), Prentice-Hall NJ [2] Basin, M.; Sanchez, E.; Martinez-Zuniga, R., Optimal linear filtering for systems with multiple state and observation delays, International journal of innovative computing, information and control, 3, 5, 1309-1320, (2007) [3] Basin, M.; Perez, J.; Calderon-Alvarez, D., Optimal filtering for linear systems over polynomial observations, International journal of innovative computing, information and control, 4, 2, 313-320, (2008) [4] Basin, M.; Calderon-Alvarez, D., Alternative optimal filter for linear systems with multiple state and observation delays, International journal of innovative computing, information and control, 4, 11, 2889-2898, (2008) [5] Boukas, E.K.; Liu, Z.K., Deterministic and stochastic time-delay systems, (2002), Birkhauser Basel · Zbl 0998.93041 [6] Boyd, S.; El Ghaoui, L.; Feron, E.; Balakrishnan, V., Linear matrix inequalities in control, SIAM studies in applied mathematics, (1994), SIAM Philadelphia [7] Byrnes, C.I.; Isidori, A.; Willems, J.C., Passivity, feedback equivalence and global stabilization of minimum phase nonlinear systems, IEEE transactions on automatic control, 36, 1228-1240, (1991) · Zbl 0758.93007 [8] Cao, Y.; Lam, J., Computation of robust stability bounds for time-delay systems with nonlinear time-varying perturbations, International journal of systems science, 31, 359-365, (2000) · Zbl 1080.93519 [9] Chen, J.L.; Lee, L., Passivity approach to feedback connection stability for discrete-time descriptor systems, (), 2865-2866 [10] Doyle, J.C.; Glover, K.; Khargonekar, P.P.; Francis, B.A., State-space solutions to the standard $$\mathcal{H}_2$$ and $$\mathcal{H}_\infty$$ control problems, IEEE transactions on automatic control, 34, 831-847, (1989) · Zbl 0698.93031 [11] Fridman, E.; Shaked, U., On-delay-dependent passivity, IEEE transactions on automatic control, 47, 664-669, (2002) · Zbl 1364.93370 [12] Fridman, E.; Shaked, U., Delay-dependent stability and $$\mathcal{H}_\infty$$ control: constant and time-varying delays, International journal of control, 76, 48-60, (2003) · Zbl 1023.93032 [13] P. Gahinet, A. Nemirovskii, A.J. Laub, M. Chilali, LMI Control Toolbox, The MathWorks, MA, 1995. [14] Hill, D.J.; Moylan, P.J., Dissipative dynamical systems: basic input – output and state properties, Journal of the franklin institute, 309, 327-357, (1980) · Zbl 0451.93007 [15] Lozano, R.; Brogliato, B.; Egeland, O.; Maschke, B., Dissipative systems analysis and control: theory and applications, (2000), Springer London · Zbl 0958.93002 [16] Mahmoud, M.S.; Xie, L., Stability and positive realness for time-delay systems, International journal of mathematical analysis and applications, 239, 7-19, (1999) · Zbl 0939.93034 [17] Mahmoud, M.S., Robust control and filtering for time-delay systems, (2000), Marcel-Dekker New York · Zbl 0969.93002 [18] Mahmoud, M.S.; Xie, L., Passivity analysis and synthesis for uncertain time-delay systems, Journal of mathematical problems in engineering, 7, 455-484, (2001) · Zbl 1014.93038 [19] Mahmoud, M.S.; Zribi, M., Passive control synthesis for uncertain systems with multiple-state delays, International journal of computers and electrical engineering, 28, 195-216, (2002), (special issue on Time-Delay Systems) · Zbl 1047.93020 [20] Mahmoud, M.S.; Ismail, Abdulla, Passivity and passification of time-delay systems, Journal of mathematical analysis and applications, 292, 247-258, (2004) · Zbl 1084.93014 [21] Mahmoud, M.S.; Nounou, H.N., Dissipative analysis and synthesis of time-delay systems, Mediterranean journal of measurements and control, 1, 97-108, (2005) [22] Mahmoud, M.S.; Shi, Y.; Nounou, H.N., Resilient observer-based control of uncertain time-delay systems, International journal of innovative computing, information and control, 3, 2, 407-418, (2007) [23] Sun, W.; Khargonekar, P.P.; Shim, D., Solution to the positive real control problem for linear time-invariant systems, IEEE transactions on automatic control, 39, 2034-2046, (1994) · Zbl 0815.93032 [24] Willems, J.C., Dissipative dynamical systems—parts I & II, Archive for rational mechanical analysis, 45, 321-393, (1972) [25] Xie, S.; Xie, L.H.; deSouza, C.E., Robust dissipative control for linear systems with dissipative uncertainty, International journal of control, 70, 169-191, (1998) · Zbl 0930.93068 [26] Xu, S.; Van Dooren, P.; Stefan, R.; Lam, J., Robust stability and stabilization for singular systems with state-delay and parameter uncertainties, IEEE transactions on automatic control, 47, 1122-1127, (2002) · Zbl 1364.93723 [27] Xu, S.; Lam, J., Positive real control for uncertain singular time-delay systems via output feedback controllers, European journal of control, 292, 122-127, (2004)
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.