Slow peaking and low-gain designs for global stabilization of nonlinear systems

This paper presents an analysis of the slow-peaking phenomenon, a pitfall of low-gain designs that imposes basic limitations to large regions of attraction in nonlinear control systems. The phenomenon is best understood on a chain of integrators perturbed by a vector field up(x, u) that satisfies p(x, 0) = 0. Because small controls (or low-gain designs) are sufficient to stabilize the unperturbed chain of integrators, it may seem that smaller controls, which attenuate the perturbation up(x, u) in a large compact set, can be employed to achieve larger regions of attraction. This intuition is false, however, and peaking may cause a loss of global controllability unless severe growth restrictions are imposed on p(x, u). These growth restrictions are expressed as a higher order condition with respect to a particular weighted dilation related to the peaking exponents of the nominal system. When this higher order condition is satisfied, an explicit control law is derived that achieves global asymptotic stability of x = 0. This stabilization result is extended to more general cascade nonlinear systems in which the perturbation p(x, v) v, v = (ξ, u) T, contains the state ξ and the control u of a stabilizable subsystem ξ = a(ξ, u). As an illustration, a control law is derived that achieves global stabilization of the frictionless ball-and-beam model.

Dissolving and aligning carbon nanotubes in thermotropic liquid crystals.

It has been widely recognized that the combination of carbon nanotubes (CNTs) and low molar mass thermotropic liquid crystals (tLCs) not only provides a useful way to align CNTs, but also dramatically enhances the tLC performance especially in the liquid crystal display technology. Such CNT-tLC nanocomposites have ignited hopes to address many stubborn problems within the field, such as low contrast, slow response, and narrow view angle. However, this material development has been limited by the poor solubility of CNTs in tLCs. Here, we describe an effective strategy to solve the problem. Prior to integrating with tLCs, pristine CNTs are physically "coated" by a liquid crystalline polymer (LCP) which is compatible with tLCs. The homogeneous CNT-tLC composite obtained in this way is stable for over 6 months, and the concentration of CNTs in tLCs can reach 1 wt %. We further demonstrate the alignment of CNTs at high CNT concentrations by an electric field with a theory to model the impedance response of the CNT-tLC mixture.