Optimality and Passivity of Input-Quadratic Nonlinear Systems

The infinite-horizon optimal control problem with stability in the presence of single-input, <italic>input-quadratic nonlinear systems</italic> is addressed and tackled in this article. In addition, it is shown that similar ideas can be extended to study the property of <italic>passivity</italic> of the underlying input-quadratic system from a given output. The constructive design of the optimal solution revolves around the interesting fact that the property of optimality of the closed-loop underlying system is shown to be locally equivalent to the property that an input-affine system possesses an <inline-formula><tex-math notation="LaTeX">$\mathcal {L}_2$</tex-math></inline-formula>-gain less than one from a virtual disturbance signal. The global version of the statement requires a technical condition on the graph of the storage function of the latter auxiliary plant, and hence leads to the new notion of <italic>graphical storage function</italic>. Finally, the theory is corroborated by the application to the optimal control of the movable plane positioning in micromechanical systems actuators.

[1]  Xiaoming Hu,et al.  Constructive stabilization for quadratic input nonlinear systems , 2008, Autom..

[2]  Wei Lin,et al.  Global asymptotic stabilization of general nonlinear systems with stable free dynamics via passivity and bounded feedback , 1996, Autom..

[3]  A. Isidori,et al.  Passivity, feedback equivalence, and the global stabilization of minimum phase nonlinear systems , 1991 .

[4]  J. Bryzek,et al.  Control issues for MEMS , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[5]  Michael Malisoff,et al.  Further constructions of control-Lyapunov functions and stabilizing feedbacks for systems satisfying the Jurdjevic-Quinn conditions , 2006, IEEE Transactions on Automatic Control.

[6]  D. H. S. Maithripala,et al.  Nonlinear dynamic output feedback stabilization of electrostatically actuated MEMS , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[7]  Van,et al.  L2-Gain Analysis of Nonlinear Systems and Nonlinear State Feedback H∞ Control , 2004 .

[8]  Jian Chen,et al.  A constructive stabilization method via system immersion for multi-input non-affine nonlinear systems , 2016, 2016 IEEE 55th Conference on Decision and Control (CDC).

[9]  Wilfrid Perruquetti,et al.  Stabilization of nonaffine systems: a constructive method for polynomial systems , 2005, IEEE Transactions on Automatic Control.

[10]  P. Kokotovic,et al.  Global asymptotic stabilization of the ball-and-beam system , 1997, Proceedings of the 36th IEEE Conference on Decision and Control.

[11]  Z. Artstein.  Stabilization with relaxed controls , 1983 .

[12]  Jean-Claude Vivalda,et al.  On the stabilization of quadratic nonlinear systems , 2017, Eur. J. Control.

[13]  J. Willems.  Dissipative dynamical systems part I: General theory , 1972 .

[14]  Diego Langarica-Córdoba,et al.  A solution to the problem of transient stability of multimachine power systems , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[15]  Dennis S. Bernstein,et al.  Robust nonlinear control of the electromagnetically controlled oscillator , 2002, Proceedings of the 2002 American Control Conference (IEEE Cat. No.CH37301).

[16]  M. Sassano,et al.  Optimal control of MIMO input-quadratic nonlinear systems , 2019, 2019 IEEE 58th Conference on Decision and Control (CDC).

[17]  A. Schaft.  On a state space approach to nonlinear H ∞ control , 1991 .

[18]  Wei Lin,et al.  Bounded smooth state feedback and a global separation principle for non-affine nonlinear systems , 1995 .

[19]  D. Lukes.  Optimal Regulation of Nonlinear Dynamical Systems , 1969 .

[20]  A. Schaft.  L/sub 2/-gain analysis of nonlinear systems and nonlinear state-feedback H/sub infinity / control , 1992 .

[21]  Eduardo Sontag.  A universal construction of Artstein's theorem on nonlinear stabilization , 1989 .

[22]  Guchuan Zhu,et al.  Improving the Performance of an Electrostatically Actuated MEMS by Nonlinear Control: Some Advances and Comparisons , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

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