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A distributed controller approach for delay-independent stability of networked control systems. (English) Zbl 1185.93120
Summary: This article introduces a novel distributed controller approach for networked control systems to achieve finite gain $${\mathcal L}_2$$ stability independent of constant time delay. The proposed approach represents a generalization of the well-known scattering transformation which applies for passive systems only. The main results of this article are (a) a sufficient stability condition for general multi-input-multi-output input-feedforward-output-feedback-passive nonlinear systems and (b) a necessary and sufficient stability condition for linear time-invariant single-input-single-output systems. The performance advantages of the proposed approach are reduced sensitivity to time delay and improved steady state error compared to alternative known delay-independent small gain type approaches. Simulations validate the proposed approach.

##### MSC:
 93D25 Input-output approaches in control theory 93A15 Large-scale systems
##### Software:
COMPleib; PENNON; YALMIP; SDPT3
Full Text:
##### References:
 [1] Anderson, R.J.; Spong, M.W., Bilateral control of teleoperators with time delay, IEEE transactions on automatic control, 34, 5, 494-501, (1989) [2] Antsaklis, P.; Baillieul, J., Special issue on technology of networked control systems, Proceedings of the IEEE, 95, 1, (2007) [3] Berestesky, B., Chopra, N., & Spong, M. W. (2004). Discrete time passivity in bilateral teleoperation over the internet. In Proceedings of the IEEE international conference on robotics and automation ICRA’04 (pp. 4557-4564) [4] Bonnet, C.; Partington, J., Bezout factors and L1-optimal controllers for delay systems using a two-parameter compensator scheme, IEEE transactions on automatic control, 44, 8, 1512-1521, (1999) · Zbl 0959.93052 [5] Boyd, S.; Ghaoui, L.E.; Feron, E.; Balakrishnan, V., (), (1st ed.). 3600 University City Science Center, Philadelphia, Pennsylvania 19104-2688. SIAM [6] Georgiou, T.; Smith, M., Robust stabilization in the gap metric: controller design for distributed plants, IEEE transactions on automatic control, 37, 8, 1133-1143, (1992) · Zbl 0764.93033 [7] Gu, K.; Kharitonov, V.; Chen, J., Stability of time-delay systems, (2003), Birkhäuser · Zbl 1039.34067 [8] Hale, J.K.; Verduyn Lunel, S.M., Introduction to functional-differential equations, (1993) · Zbl 0787.34002 [9] Hill, D.; Moylan, P., The stability of nonlinear dissipative systems, IEEE transactions on automatic control, 21, 5, 708-711, (1976) · Zbl 0339.93014 [10] Hirche, S., & Buss, M. (2004). Packet loss effects in passive telepresence systems. In Proceedings of the 43rd IEEE conference on decision and control (pp. 4010-4015) [11] Hristu-Varsakelis, D.; Levine, W.S., Handbook of networked and embedded control systems, (2005), Birkhäuser · Zbl 1094.93005 [12] Khalil, H.K., Nonlinear systems, () · Zbl 0626.34052 [13] Kočara, M.; Stingl, M., PENNON: A code for convex nonlinear and semidefinite programming, Optimization methods and software, 18, (2003) · Zbl 1037.90003 [14] Leibfritz, F. (2004). Compleib: Constraint matrix-optimization problem library — a collection of test examples for nonlinear semidefinite programs, control system design and related problems. Technical report. Department of Mathematics, University of Trier. http://www.complib.de [15] Löfberg, J. (2004). YALMIP: A toolbox for modeling and optimization in MATLAB. In Proceedings of the CACSD conference (pp. 284-289). http://control.ee.ethz.ch/ joloef/yalmip.php [16] Lozano, R.; Chopra, N.; Spong, M., Passivation of force reflecting bilateral teleoperators with time varying delay, (), 954-962 [17] Matiakis, T., & Hirche, S. (2006). Memoryless input-output encoding for networked systems with unknown constant time delay. In Proceedings of 2006 IEEE conference on control applications (pp. 1707-1712) [18] Matiakis, T., Hirche, S., & Buss, M. (2005). The scattering transformation for networked control systems. In Proceedings of 2005 IEEE conference on control applications (pp. 705-710) [19] Matiakis, T., Hirche, S., & Buss, M. (2006). Independent-of-delay stability of nonlinear networked control systems by scattering transformation. In Proceedings of the 2006 American control conference, ACC2006 (pp. 2801-2806) [20] Matiakis, T., Hirche, S., & Buss, M. (2008). Local and remote control measures for networked control systems. In Proceedings of the 47th IEEE conference on decision and control, CDC 2008 · Zbl 1185.93120 [21] Miller, D.E.; Davison, D.E., Stabilization in the presence of an uncertain arbitrarily large delay, IEEE transactions on automatic control, 50, 8, 1074-1089, (2005) · Zbl 1365.93452 [22] Munir, S.; Book, W., Internet based teleoperation using wave variable with prediction, ASME/IEEE transactions on mechatronics, 7, 2, 124-133, (2002) [23] Niemeyer, G.; Slotine, J., Stable adaptive teleoperation, International journal of oceanic engineering, 16, 1, 152-162, (1991) [24] Richard, J.-P., Time-delay systems: an overview of some recent advances and open problems, Automatica, 39, 1667-1694, (2003) · Zbl 1145.93302 [25] Secchi, C., Stramigioli, S., & Fantuzzi, C. (2003). Dealing with unreliabilities in digital passive geometric telemanipulation. In Proceedings of the IEEE/RSJ international conference on intelligent robots and systems IROS (pp. 2823-2828). Vol. 3 [26] Stramigioli, S., Modeling and IPC control of interactive mechanical systems: A coordinate free approach, (2001), Springer London · Zbl 1051.93003 [27] Tipsuwan, Y.; Chow, M.Y., Control methodologies in network control systems, Control engineering practice, 11, 1099-1111, (2003) [28] Tütüncü, R.H.; Toh, K.C.; Todd, M.J., Solving semidefinite-quadratic-linear programs using SDPT3, Mathematical programming series B, 95, 189-217, (2003) · Zbl 1030.90082 [29] Willems, J.C., Dissipative dynamical systems, part I: general theory, Archive for rational mechanics analysis, 45, 321-351, (1972) · Zbl 0252.93002 [30] Willems, J.C., Dissipative dynamical systems, part II: linear systems with quadratic supply rates, Archive for rational mechanics analysis, 45, 352-393, (1972) · Zbl 0252.93003 [31] Zames, G., On the input-output stability of time-varying nonlinear feedback systems, part I: conditions derived using concepts of loop gain, conicity, and positivity, IEEE transactions on automatic control, 11, 2, 228-238, (1966) [32] Zames, G., On the input-output stability of time-varying nonlinear feedback systems, part II: conditions involving circles in the frequency plane and sector non-linearities, IEEE transactions on automatic control, 11, 3, 465-476, (1966)
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