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

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This proposal aims to address some of the most fundamental and challenging issues in the field of nonlinearcontrol. 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 withunstabilizable/undetectable linearization, which cannot be dealt with by any smooth feedback, even locally. Nonsmoothfeedback designs ofer the potential to overcome certain topological obstacles that cannot be addressed by smoothcontrols, for example, to globally stabilize a nonlinear system that is not stabilizable by any linear or smooth statefeedback, 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 controllersthat achieve global strong stability and do so in a robust fashion, that is, in a manner which is not sensitive toperturbations and parametric uncertainties in the system.The other focal points of this proposal will be the development of a new output tracking theory for inherentlynonlinear systems by measurement feedback. The objective is to make the controlled plant track a desired referencesignal. A key feature of the classical servomechanism theory is the ability to absorb the effect of modeling errors andto secure asymptotic tracking in the presence of uncertainties. A nonlinear analogue of these robustness properties wasdeveloped 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 linearizationis 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 outputtracking is usually not possible, even locally, practical output tracking is achievable. One of important componentsof this project will be to establish a general theory for global practical output tracking of nonlinear systems withuncontrollable/unobservable linearization, in the presence of nonlinear parameterization.The theoretical research will be complemented by several novel applications. In particular, the nonsmooth controltheory to be developed will be used to address diffcult control problems including output feedback stabilization ofunderactuated mechanical systems and output tracking/regulation of antagonistic biomimetic actuation systems, bynonsmooth measurement feedback.Intellectual Merits: This program will not only provided a general framework for the control of nonlinear systemswith uncontrollable/unobservable linearization by nonsmooth measurement feedback, but also make a fundamentalcontribution to a number of important and challenging open problems such as output feedback stabilization, outputtracking 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, andfurther facilitate the development of nonsmooth design methods and their applications for a large class of inherentlynonlinear systems.Broader Impact: The proposed research will have enormous impact on developing a new generation of nonlinearcontrol technologies, with emphasis on the development of computationally efficient and practically feasible nonsmoothcontrol tools and algorithms, for critical biomedical control applications and for achieving high-performance operationof engineering systems (e.g. mobile robots, underactuated mechanical systems, and biomimetic or biologically-inspiredsystems). Applications of the nonsmooth control methods to biomimetic systems have the potential to overcome somefundamental 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 autonomouslyin the real world, thus improving significantly performance of the first generation biomimetic systems. The outcomesof this program are envisioned to provide new avenues for a significant improvement of operability and efficiencyof engineering and biologically-inspired systems, to create commercial or biomedical application opportunities, andeventually to result in economic benefits.1