On the equivalence between dissipativity and optimality of nonlinear hybrid controllers

In this paper we derive guaranteed hybrid gain, sector, and disk margins for nonlinear optimal and inverse optimal hybrid regulators that minimize nonlinear-nonquadratic hybrid performance functional. Furthermore, we develop a hybrid return difference inequality to provide connections between dissipativity and optimality of nonlinear hybrid controllers. Specifically, we show that optimal hybrid controllers imply dissipativity with respect to a quadratic supply rate.

[1]  Wassim M. Haddad,et al.  Nonlinear impulsive dynamical systems. I. Stability and dissipativity , 1999, Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304).

[2]  D. Baĭnov,et al.  Systems with impulse effect : stability, theory, and applications , 1989 .

[3]  V. Lakshmikantham,et al.  Theory of Impulsive Differential Equations , 1989, Series in Modern Applied Mathematics.

[4]  V. Chellaboina,et al.  Nonlinear impulsive dynamical systems. II. Feedback interconnections and optimality , 1999, Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304).

[5]  P. Moylan.  Implications of passivity in a class of nonlinear systems , 1974 .

[6]  Wassim M. Haddad,et al.  Stability margins of nonlinear optimal regulators with nonquadratic performance criteria involving cross-weighting terms ☆ , 2000 .

[7]  B. Anderson,et al.  Nonlinear regulator theory and an inverse optimal control problem , 1973 .

[8]  Sanjay P. Bhat,et al.  An invariance principle for nonlinear hybrid and impulsive dynamical systems , 2003, Proceedings of the 2000 American Control Conference. ACC (IEEE Cat. No.00CH36334).

[9]  A. Samoilenko,et al.  Impulsive differential equations , 1995 .

[10]  P. Moylan,et al.  The stability of nonlinear dissipative systems , 1976 .

[11]  Wassim M. Haddad,et al.  Stability margins of discrete-time nonlinear nonquadratic optimal regulators , 1998, Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No.98CH36171).

[12]  P. Kokotovic,et al.  Inverse Optimality in Robust Stabilization , 1996 .

[13]  W. Haddad,et al.  Stability margins of nonlinear optimal regulators with nonquadratic performance criteria involving cross-weighting terms , 1998, Proceedings of the 1998 American Control Conference. ACC (IEEE Cat. No.98CH36207).

[14]  Wassim M. Haddad,et al.  Non-linear impulsive dynamical systems. Part II: Stability of feedback interconnections and optimality , 2001 .

[15]  P. S. Simeonov,et al.  Impulsive Differential Equations , 1989 .