# American Institute of Mathematical Sciences

2013, 3(1): 175-201. doi: 10.3934/naco.2013.3.175

 1 Boeing Research and Technology Europe, Avenida Sur del Aeropuerto de Barajas, 38, Edificio 4 Planta, 428042 Madrid, Spain 2 School of Aeronautics and Astronautics, Purdue University, W. Lafayette, IN 47907, United States

Received  February 2012 Revised  December 2012 Published  January 2013

The concept of incremental quadratic stability ($\delta$QS) is very useful in treating systems with persistently acting inputs. To illustrate, if a time-invariant $\delta$QS system is subject to a constant input or $T$-periodic input then, all its trajectories exponentially converge to a unique constant or $T$-periodic trajectory, respectively. By considering the relationship of $\delta$QS to the usual concept of quadratic stability, we obtain a useful necessary and sufficient condition for $\delta$QS. A main contribution of the paper is to consider nonlinear/uncertain systems whose state dependent nonlinear/uncertain terms satisfy an incremental quadratic constraint which is characterized by a bunch of symmetric matrices we call incremental multiplier matrices. We obtain linear matrix inequalities whose feasibility guarantee $\delta$QS of these systems. Frequency domain characterizations of $\delta$QS are then obtained from these conditions. By characterizing incremental multiplier matrices for many common classes of nonlinearities, we demonstrate the usefulness of our results.
Citation: Luis D'Alto, Martin Corless. Incremental quadratic stability. Numerical Algebra, Control and Optimization, 2013, 3 (1) : 175-201. doi: 10.3934/naco.2013.3.175
##### References:
 [1] A. B. Açıkmeşe, "Stabilization, Observation, Tracking and Disturbance Rejection for Uncertain/Nonlinear and Time-Varying Systems," Ph.D. thesis, Purdue University, 2002. [2] A. B. Açıkmeşe and M. Corless, Stability analysis with quadratic Lyapunov functions: some necessary and sufficient multiplier conditions, Systems & Control Letters, 57 (2008), 78-94. doi: 10.1016/j.sysconle.2007.06.018. [3] A. B. Açıkmeşe and M. Corless, Observers for systems with nonlinearities satisfying incremental quadratic constraints, Automatica, 47 (2011), 1339-1348. doi: 10.1016/j.automatica.2011.02.017. [4] D. Angeli, A Lyapunov approach to incremental stability properties, IEEE Transactions on Automatic Control, 47 (2002), 410-421. doi: 10.1109/9.989067. [5] V. Balakrishnan, Matrix inequalities in robustness analysis with multipliers, System and Control Letters, 25 (1995), 265-272. doi: 10.1016/0167-6911(94)00087-C. [6] B. R. Barmish, Stabilization of uncertain systems via linear control, IEEE Transactions on Automatic Control, 28 (1983), 848-850. doi: 10.1109/TAC.1983.1103324. [7] B. R. Barmish, Necessary and sufficient conditions for quadratic stabilizability of an uncertain system, Journal of Optimization Theory and Applications, 46 (1985), 399-408. doi: 10.1007/BF00939145. [8] S. Boyd, L. El Ghaoui, E. Feron and V. Balakrishnan, "Linear Matrix Inequalities in System and Control Theory," SIAM, Philadelphia, 1994. doi: 10.1137/1.9781611970777. [9] M. Corless, Robust stability analysis and controller design with quadratic Lyapunov functions, in "Variable Structure and Lyapunov Control," (ed. A. Zinober), Springer-Verlag, 1993. [10] M. Corless and G. Leitmann, Continuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamic systems, IEEE Transactions on Automatic Control, 26 (1981), 1139-1144. doi: 10.1109/TAC.1981.1102785. [11] L. D'Alto, "Incremental Quadratic Stability," M.S. thesis, Purdue University, 2004. [12] L. D'Alto and M. Corless, "Incremental Quadratic Stability," Technical report, Purdue University, 2008. [13] B. P. Demidovich, Dissipativity of a nonlinear system of differential equations: Part I, Vestnik Moscow State University, Ser. Mat. Mekh., (Russian), 6 (1961), 19-27. [14] B. P. Demidovich, Dissipativity of a nonlinear system of differential equations: Part II, Vestnik Moscow State University, Ser. Mat. Mekh., (Russian), 1 (1962), 3-8. [15] B. P. Demidovich, "Lectures on Stability Theory," Nauka, Moscow, (Russian), 1967. [16] V. Fromion, G. Scorletti and G. Ferreres, Nonlinear performance of a PI controlled missile: an explanation, International Journal of Robust and Nonlinear Control, 9 (1999), 485-518. doi: 10.1002/(SICI)1099-1239(19990715)9:8<485::AID-RNC417>3.0.CO;2-4. [17] S. Gutman and G. Leitmann, Stabilizing feedback control for dynamical systems with bounded uncertainty, IEEE Conference on Decision and Control, Clearwater, Florida, (1976), 94-99. [18] A. Isidori, "Nonlinear Control Systems II," Springer-Verlag, London, 1999. doi: 10.1007/978-1-4471-0549-7. [19] H. K. Khalil, "Nonlinear Systems," 3rd edition, Prentice-Hall, Inc., 2002. [20] R. Liu, Convergent systems, IEEE Transactions on Automatic Control, AC-13 (1968), 384-391. [21] R. Liu, R. Saeks and R. J. Leake, On global linearization, SIAM-AMS proceedings, 111 (1969), 93-102. [22] W. Lohmiller, "Contraction Analysis of Nonlinear Systems," Ph.D. thesis, Massachusetts Institute of Technology, 1999. [23] W. Lohmiller and J. J. E. Slotine, On contraction analysis for non-linear systems, Automatica, 34 (1998), 683-696. doi: 10.1016/S0005-1098(98)00019-3. [24] D. G. Luenberger and Y. Ye, "Linear and Nonlinear Programming," 3rd edition, Springer, 2008. [25] A. Megretski and A. Rantzer, System analysis via integral quadratic constraints, IEEE Transactions on Automatic Control, 42 (1997), 819-830. doi: 10.1109/9.587335. [26] A. Pavlov, A. Pogromsky, N. van de Wouw and H. Nijmeijer, Convergent dynamics, a tribute to Boris Pavlovich Demidovich, Systems and Control Letters, 52 (2004), 257-261. doi: 10.1016/j.sysconle.2004.02.003. [27] A. Pavlov, N. van de Wouw and H. Nijmeijer, "Uniform Output Regulation of Nonlinear Systems," Birkhauser, Boston, 2006. [28] I. R. Petersen and C. V. Hollot, A Riccati equation approach to the stabilization of uncertain linear systems, Automatica, 22 (1986), 397-411. doi: 10.1016/0005-1098(86)90045-2. [29] R. Shorten and K. S. Narendra, On common quadratic Lyapunov functions for pairs of stable LTI systems whose system matrices are in companion form, IEEE Transactions on Automatic Control, 48 (2003), 618-621. doi: 10.1109/TAC.2003.809795. [30] V. A. Yacubovich, The matrix-inequality method in the theory of the stability of nonlinear control systems: 1. The absolute stability of forced vibrations, Automation and Remote Control, 25 (1964), 905-916.

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##### References:
 [1] A. B. Açıkmeşe, "Stabilization, Observation, Tracking and Disturbance Rejection for Uncertain/Nonlinear and Time-Varying Systems," Ph.D. thesis, Purdue University, 2002. [2] A. B. Açıkmeşe and M. Corless, Stability analysis with quadratic Lyapunov functions: some necessary and sufficient multiplier conditions, Systems & Control Letters, 57 (2008), 78-94. doi: 10.1016/j.sysconle.2007.06.018. [3] A. B. Açıkmeşe and M. Corless, Observers for systems with nonlinearities satisfying incremental quadratic constraints, Automatica, 47 (2011), 1339-1348. doi: 10.1016/j.automatica.2011.02.017. [4] D. Angeli, A Lyapunov approach to incremental stability properties, IEEE Transactions on Automatic Control, 47 (2002), 410-421. doi: 10.1109/9.989067. [5] V. Balakrishnan, Matrix inequalities in robustness analysis with multipliers, System and Control Letters, 25 (1995), 265-272. doi: 10.1016/0167-6911(94)00087-C. [6] B. R. Barmish, Stabilization of uncertain systems via linear control, IEEE Transactions on Automatic Control, 28 (1983), 848-850. doi: 10.1109/TAC.1983.1103324. [7] B. R. Barmish, Necessary and sufficient conditions for quadratic stabilizability of an uncertain system, Journal of Optimization Theory and Applications, 46 (1985), 399-408. doi: 10.1007/BF00939145. [8] S. Boyd, L. El Ghaoui, E. Feron and V. Balakrishnan, "Linear Matrix Inequalities in System and Control Theory," SIAM, Philadelphia, 1994. doi: 10.1137/1.9781611970777. [9] M. Corless, Robust stability analysis and controller design with quadratic Lyapunov functions, in "Variable Structure and Lyapunov Control," (ed. A. Zinober), Springer-Verlag, 1993. [10] M. Corless and G. Leitmann, Continuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamic systems, IEEE Transactions on Automatic Control, 26 (1981), 1139-1144. doi: 10.1109/TAC.1981.1102785. [11] L. D'Alto, "Incremental Quadratic Stability," M.S. thesis, Purdue University, 2004. [12] L. D'Alto and M. Corless, "Incremental Quadratic Stability," Technical report, Purdue University, 2008. [13] B. P. Demidovich, Dissipativity of a nonlinear system of differential equations: Part I, Vestnik Moscow State University, Ser. Mat. Mekh., (Russian), 6 (1961), 19-27. [14] B. P. Demidovich, Dissipativity of a nonlinear system of differential equations: Part II, Vestnik Moscow State University, Ser. Mat. Mekh., (Russian), 1 (1962), 3-8. [15] B. P. Demidovich, "Lectures on Stability Theory," Nauka, Moscow, (Russian), 1967. [16] V. Fromion, G. Scorletti and G. Ferreres, Nonlinear performance of a PI controlled missile: an explanation, International Journal of Robust and Nonlinear Control, 9 (1999), 485-518. doi: 10.1002/(SICI)1099-1239(19990715)9:8<485::AID-RNC417>3.0.CO;2-4. [17] S. Gutman and G. Leitmann, Stabilizing feedback control for dynamical systems with bounded uncertainty, IEEE Conference on Decision and Control, Clearwater, Florida, (1976), 94-99. [18] A. Isidori, "Nonlinear Control Systems II," Springer-Verlag, London, 1999. doi: 10.1007/978-1-4471-0549-7. [19] H. K. Khalil, "Nonlinear Systems," 3rd edition, Prentice-Hall, Inc., 2002. [20] R. Liu, Convergent systems, IEEE Transactions on Automatic Control, AC-13 (1968), 384-391. [21] R. Liu, R. Saeks and R. J. Leake, On global linearization, SIAM-AMS proceedings, 111 (1969), 93-102. [22] W. Lohmiller, "Contraction Analysis of Nonlinear Systems," Ph.D. thesis, Massachusetts Institute of Technology, 1999. [23] W. Lohmiller and J. J. E. Slotine, On contraction analysis for non-linear systems, Automatica, 34 (1998), 683-696. doi: 10.1016/S0005-1098(98)00019-3. [24] D. G. Luenberger and Y. Ye, "Linear and Nonlinear Programming," 3rd edition, Springer, 2008. [25] A. Megretski and A. Rantzer, System analysis via integral quadratic constraints, IEEE Transactions on Automatic Control, 42 (1997), 819-830. doi: 10.1109/9.587335. [26] A. Pavlov, A. Pogromsky, N. van de Wouw and H. Nijmeijer, Convergent dynamics, a tribute to Boris Pavlovich Demidovich, Systems and Control Letters, 52 (2004), 257-261. doi: 10.1016/j.sysconle.2004.02.003. [27] A. Pavlov, N. van de Wouw and H. Nijmeijer, "Uniform Output Regulation of Nonlinear Systems," Birkhauser, Boston, 2006. [28] I. R. Petersen and C. V. Hollot, A Riccati equation approach to the stabilization of uncertain linear systems, Automatica, 22 (1986), 397-411. doi: 10.1016/0005-1098(86)90045-2. [29] R. Shorten and K. S. Narendra, On common quadratic Lyapunov functions for pairs of stable LTI systems whose system matrices are in companion form, IEEE Transactions on Automatic Control, 48 (2003), 618-621. doi: 10.1109/TAC.2003.809795. [30] V. A. Yacubovich, The matrix-inequality method in the theory of the stability of nonlinear control systems: 1. The absolute stability of forced vibrations, Automation and Remote Control, 25 (1964), 905-916.
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