Lyapunov approach to frequency response evaluation/optimization,
 and active vibration control

Many engineering systems, such as turbomachines, motors/generators, flexible structures and communication circuits, are exposed to oscillatory input/disturbances of sinusoidal, multi-tone or
general periodic types. The output response of a system under the excitation of such oscillatory input signals is referred to as forced vibration. For linear time-invariant systems, forced vibrations in the steady-state can be well characterized through frequency response, which in turn is easily determined through transfer functions. For nonlinear systems, possibly with time-varying uncertainties, the transfer function does not exist and the relationship between the forced vibration and the frequencies of the input/disturbances is much more complicated because of various nonlinear phenomena, such as jump phenomena, subharmonic oscillations and frequency entrainment. For instance, the ratio between the magnitude of the steady-state output and that of the input depends not only on the frequency of the
input but also on the phase of the input, and even on the initial conditions of the system.

Characterization of frequency response and forced oscillation is a fundamental problem in active vibration control. Can this problem be addressed with a numerically efficient tool? In particular, can frequency response and forced oscillation be  analyzed by a Lyapunov method? Or a Linear Matrix Inequality (LMI) based approach? Answers to these questions can be found in our recent work:

•   T. Hu, A. R. Teel and Z. Lin, ``Lyapunov characterization of forced oscillations," Automatica,    
    41,  October, pp.1723-1735, 2005.

Abstract of the work: This paper develops a Lyapunov approach to the analysis of input-output
characteristics for systems under the excitation of a class of oscillatory inputs. Apart from sinusoidal
signals, the class of oscillatory inputs include multi-tone signals and periodic signals which can be
described as the output of an autonomous system. The Lyapunov approach is developed for linear systems, homogeneous systems (differential inclusions) and nonlinear systems (differential inclusions), respectively. In particular, it is established that the steady-state gain can be arbitrarily closely characterized with Lyapunov functions if the output response converges exponentially to the
steady-state. Other output measures that will be characterized include the peak of the transient
response and the convergence rate. Tools based on linear matrix inequalities (LMIs) are developed
for the numerical analysis of linear differential inclusions (LDIs). This paper's results can be readily applied to the evaluation of frequency responses of general nonlinear and uncertain systems
by restricting the inputs to sinusoidal signals. Guided by the numerical result for a second order LDI,
an interesting phenomenon is observed that the peak of the frequency response can be strictly larger
than the L₂  gain. The following is a powerpoint file for the presentation at ACC04: