Nonsmooth Measurement Feedback Control of Genuinely Nonlinear Systems with Applications to Biologically-Inspired Systems

Case Western Reserve University, Cleveland OH

Investigators

Abstract

This proposal aims to address some of the most fundamental and challenging issues in the field of nonlinear control. The main objectives are to develop nonsmooth feedback design approaches achieving global stabilization, adaptive regulation and output tracking, via measurable signals, for a significant class of nonlinear systems with unstabilizable/undetectable linearization, which cannot be dealt with by any smooth feedback, even locally. Nonsmooth feedback designs ofer the potential to overcome certain topological obstacles that cannot be addressed by smooth controls, for example, to globally stabilize a nonlinear system that is not stabilizable by any linear or smooth state feedback, because the linearized system has uncontrollable modes associated with eigenvalues on the right-half plane. The basic research components of this proposal are to design nonsmooth partial state or output feedback controllers that achieve global strong stability and do so in a robust fashion, that is, in a manner which is not sensitive to perturbations and parametric uncertainties in the system. The other focal points of this proposal will be the development of a new output tracking theory for inherently nonlinear systems by measurement feedback. The objective is to make the controlled plant track a desired reference signal. A key feature of the classical servomechanism theory is the ability to absorb the effect of modeling errors and to secure asymptotic tracking in the presence of uncertainties. A nonlinear analogue of these robustness properties was developed via center manifold theory, under the assumption that the linearized system is controllable and observable, at least partially. However, little result is known on the output regulation of nonlinear systems whose linearization is unobservable and uncontrollable. Our preliminary analysis suggests that a new formulation to the nonlinear reg- ulator problem is needed. Indeed, some dramatic examples in the proposal indicate that while asymptotic output tracking is usually not possible, even locally, practical output tracking is achievable. One of important components of this project will be to establish a general theory for global practical output tracking of nonlinear systems with uncontrollable/unobservable linearization, in the presence of nonlinear parameterization. The theoretical research will be complemented by several novel applications. In particular, the nonsmooth control theory to be developed will be used to address diffcult control problems including output feedback stabilization of underactuated mechanical systems and output tracking/regulation of antagonistic biomimetic actuation systems, by nonsmooth measurement feedback. Intellectual Merits: This program will not only provided a general framework for the control of nonlinear systems with uncontrollable/unobservable linearization by nonsmooth measurement feedback, but also make a fundamental contribution to a number of important and challenging open problems such as output feedback stabilization, output tracking and regulation, for genuinely nonlinear systems that cannot be controlled by existing smooth synthesis tech- niques. The proposed research will yield new insight, advance the state-of-the-art of nonsmooth control theory, and further facilitate the development of nonsmooth design methods and their applications for a large class of inherently nonlinear systems. Broader Impact: The proposed research will have enormous impact on developing a new generation of nonlinear control technologies, with emphasis on the development of computationally efficient and practically feasible nonsmooth control tools and algorithms, for critical biomedical control applications and for achieving high-performance operation of engineering systems (e.g. mobile robots, underactuated mechanical systems, and biomimetic or biologically-inspired systems). Applications of the nonsmooth control methods to biomimetic systems have the potential to overcome some fundamental diffculties in designing biologically-inspired robots (e.g. cockroach-like hexapod robots), functional elec- trical stimulation muscles, and other man-made devices whose embedded control systems can function autonomously in the real world, thus improving significantly performance of the first generation biomimetic systems. The outcomes of this program are envisioned to provide new avenues for a significant improvement of operability and efficiency of engineering and biologically-inspired systems, to create commercial or biomedical application opportunities, and eventually to result in economic benefits. 1