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

Incremental quadratic stability

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 & 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, (2002).   Google Scholar

[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.  doi: 10.1016/j.sysconle.2007.06.018.  Google Scholar

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A. B. Açıkmeşe and M. Corless, Observers for systems with nonlinearities satisfying incremental quadratic constraints,, Automatica, 47 (2011), 1339.  doi: 10.1016/j.automatica.2011.02.017.  Google Scholar

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D. Angeli, A Lyapunov approach to incremental stability properties,, IEEE Transactions on Automatic Control, 47 (2002), 410.  doi: 10.1109/9.989067.  Google Scholar

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V. Balakrishnan, Matrix inequalities in robustness analysis with multipliers,, System and Control Letters, 25 (1995), 265.  doi: 10.1016/0167-6911(94)00087-C.  Google Scholar

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B. R. Barmish, Stabilization of uncertain systems via linear control,, IEEE Transactions on Automatic Control, 28 (1983), 848.  doi: 10.1109/TAC.1983.1103324.  Google Scholar

[7]

B. R. Barmish, Necessary and sufficient conditions for quadratic stabilizability of an uncertain system,, Journal of Optimization Theory and Applications, 46 (1985), 399.  doi: 10.1007/BF00939145.  Google Scholar

[8]

S. Boyd, L. El Ghaoui, E. Feron and V. Balakrishnan, "Linear Matrix Inequalities in System and Control Theory,", SIAM, (1994).  doi: 10.1137/1.9781611970777.  Google Scholar

[9]

M. Corless, Robust stability analysis and controller design with quadratic Lyapunov functions,, in, (1993).   Google Scholar

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M. Corless and G. Leitmann, Continuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamic systems,, IEEE Transactions on Automatic Control, 26 (1981), 1139.  doi: 10.1109/TAC.1981.1102785.  Google Scholar

[11]

L. D'Alto, "Incremental Quadratic Stability,", M.S. thesis, (2004).   Google Scholar

[12]

L. D'Alto and M. Corless, "Incremental Quadratic Stability,", Technical report, (2008).   Google Scholar

[13]

B. P. Demidovich, Dissipativity of a nonlinear system of differential equations: Part I,, Vestnik Moscow State University, 6 (1961), 19.   Google Scholar

[14]

B. P. Demidovich, Dissipativity of a nonlinear system of differential equations: Part II,, Vestnik Moscow State University, 1 (1962), 3.   Google Scholar

[15]

B. P. Demidovich, "Lectures on Stability Theory,", Nauka, (1967).   Google Scholar

[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.  doi: 10.1002/(SICI)1099-1239(19990715)9:8<485::AID-RNC417>3.0.CO;2-4.  Google Scholar

[17]

S. Gutman and G. Leitmann, Stabilizing feedback control for dynamical systems with bounded uncertainty,, IEEE Conference on Decision and Control, (1976), 94.   Google Scholar

[18]

A. Isidori, "Nonlinear Control Systems II,", Springer-Verlag, (1999).  doi: 10.1007/978-1-4471-0549-7.  Google Scholar

[19]

H. K. Khalil, "Nonlinear Systems,", 3rd edition, (2002).   Google Scholar

[20]

R. Liu, Convergent systems,, IEEE Transactions on Automatic Control, AC-13 (1968), 384.   Google Scholar

[21]

R. Liu, R. Saeks and R. J. Leake, On global linearization,, SIAM-AMS proceedings, 111 (1969), 93.   Google Scholar

[22]

W. Lohmiller, "Contraction Analysis of Nonlinear Systems,", Ph.D. thesis, (1999).   Google Scholar

[23]

W. Lohmiller and J. J. E. Slotine, On contraction analysis for non-linear systems,, Automatica, 34 (1998), 683.  doi: 10.1016/S0005-1098(98)00019-3.  Google Scholar

[24]

D. G. Luenberger and Y. Ye, "Linear and Nonlinear Programming,", 3rd edition, (2008).   Google Scholar

[25]

A. Megretski and A. Rantzer, System analysis via integral quadratic constraints,, IEEE Transactions on Automatic Control, 42 (1997), 819.  doi: 10.1109/9.587335.  Google Scholar

[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.  doi: 10.1016/j.sysconle.2004.02.003.  Google Scholar

[27]

A. Pavlov, N. van de Wouw and H. Nijmeijer, "Uniform Output Regulation of Nonlinear Systems,", Birkhauser, (2006).   Google Scholar

[28]

I. R. Petersen and C. V. Hollot, A Riccati equation approach to the stabilization of uncertain linear systems,, Automatica, 22 (1986), 397.  doi: 10.1016/0005-1098(86)90045-2.  Google Scholar

[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.  doi: 10.1109/TAC.2003.809795.  Google Scholar

[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.   Google Scholar

show all references

References:
[1]

A. B. Açıkmeşe, "Stabilization, Observation, Tracking and Disturbance Rejection for Uncertain/Nonlinear and Time-Varying Systems,", Ph.D. thesis, (2002).   Google Scholar

[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.  doi: 10.1016/j.sysconle.2007.06.018.  Google Scholar

[3]

A. B. Açıkmeşe and M. Corless, Observers for systems with nonlinearities satisfying incremental quadratic constraints,, Automatica, 47 (2011), 1339.  doi: 10.1016/j.automatica.2011.02.017.  Google Scholar

[4]

D. Angeli, A Lyapunov approach to incremental stability properties,, IEEE Transactions on Automatic Control, 47 (2002), 410.  doi: 10.1109/9.989067.  Google Scholar

[5]

V. Balakrishnan, Matrix inequalities in robustness analysis with multipliers,, System and Control Letters, 25 (1995), 265.  doi: 10.1016/0167-6911(94)00087-C.  Google Scholar

[6]

B. R. Barmish, Stabilization of uncertain systems via linear control,, IEEE Transactions on Automatic Control, 28 (1983), 848.  doi: 10.1109/TAC.1983.1103324.  Google Scholar

[7]

B. R. Barmish, Necessary and sufficient conditions for quadratic stabilizability of an uncertain system,, Journal of Optimization Theory and Applications, 46 (1985), 399.  doi: 10.1007/BF00939145.  Google Scholar

[8]

S. Boyd, L. El Ghaoui, E. Feron and V. Balakrishnan, "Linear Matrix Inequalities in System and Control Theory,", SIAM, (1994).  doi: 10.1137/1.9781611970777.  Google Scholar

[9]

M. Corless, Robust stability analysis and controller design with quadratic Lyapunov functions,, in, (1993).   Google Scholar

[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.  doi: 10.1109/TAC.1981.1102785.  Google Scholar

[11]

L. D'Alto, "Incremental Quadratic Stability,", M.S. thesis, (2004).   Google Scholar

[12]

L. D'Alto and M. Corless, "Incremental Quadratic Stability,", Technical report, (2008).   Google Scholar

[13]

B. P. Demidovich, Dissipativity of a nonlinear system of differential equations: Part I,, Vestnik Moscow State University, 6 (1961), 19.   Google Scholar

[14]

B. P. Demidovich, Dissipativity of a nonlinear system of differential equations: Part II,, Vestnik Moscow State University, 1 (1962), 3.   Google Scholar

[15]

B. P. Demidovich, "Lectures on Stability Theory,", Nauka, (1967).   Google Scholar

[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.  doi: 10.1002/(SICI)1099-1239(19990715)9:8<485::AID-RNC417>3.0.CO;2-4.  Google Scholar

[17]

S. Gutman and G. Leitmann, Stabilizing feedback control for dynamical systems with bounded uncertainty,, IEEE Conference on Decision and Control, (1976), 94.   Google Scholar

[18]

A. Isidori, "Nonlinear Control Systems II,", Springer-Verlag, (1999).  doi: 10.1007/978-1-4471-0549-7.  Google Scholar

[19]

H. K. Khalil, "Nonlinear Systems,", 3rd edition, (2002).   Google Scholar

[20]

R. Liu, Convergent systems,, IEEE Transactions on Automatic Control, AC-13 (1968), 384.   Google Scholar

[21]

R. Liu, R. Saeks and R. J. Leake, On global linearization,, SIAM-AMS proceedings, 111 (1969), 93.   Google Scholar

[22]

W. Lohmiller, "Contraction Analysis of Nonlinear Systems,", Ph.D. thesis, (1999).   Google Scholar

[23]

W. Lohmiller and J. J. E. Slotine, On contraction analysis for non-linear systems,, Automatica, 34 (1998), 683.  doi: 10.1016/S0005-1098(98)00019-3.  Google Scholar

[24]

D. G. Luenberger and Y. Ye, "Linear and Nonlinear Programming,", 3rd edition, (2008).   Google Scholar

[25]

A. Megretski and A. Rantzer, System analysis via integral quadratic constraints,, IEEE Transactions on Automatic Control, 42 (1997), 819.  doi: 10.1109/9.587335.  Google Scholar

[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.  doi: 10.1016/j.sysconle.2004.02.003.  Google Scholar

[27]

A. Pavlov, N. van de Wouw and H. Nijmeijer, "Uniform Output Regulation of Nonlinear Systems,", Birkhauser, (2006).   Google Scholar

[28]

I. R. Petersen and C. V. Hollot, A Riccati equation approach to the stabilization of uncertain linear systems,, Automatica, 22 (1986), 397.  doi: 10.1016/0005-1098(86)90045-2.  Google Scholar

[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.  doi: 10.1109/TAC.2003.809795.  Google Scholar

[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.   Google Scholar

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