Nonlinear Control for Unstable Nonlinear Non-Minimum Phase Vehicles

The fundamental challenge in controlling non-minimum phase phenomena is that typical nonlinear feedback methods have fundamental stability limits when the internal dynamics of the system are unstable. This limit arises because prevailing feedback designs fail to address the existence of a continuum of internal state solutions (known as the nonstandard behaviour). This research synthesizes a globally stable nonlinear autopilot command structure to maximize performance under instabilities. For the first time the nonstandard behaviour is addressed without making any modifications to the output definition by inducing an inherent time scale separation in the closed-loop dynamics. Unlike previous time scale control techniques, this synthesis is based on recent theoretical advancements made in control of nonlinear singularly perturbed systems by Prof. Narang (published in the form of a book in 2014). It is shown that this procedure provides means to enable (i) control of aerospace vehicles without modifying the location of the sensor (ii) means to simplify implementation through use of model reduction, and (iii) construction of a rigorous design tool for tuning feedback gains. Simulation results indicate that arbitrary feedback gain selection is not feasible (has a 70% chance of inducing instability) and the developed design tool is crucial. 

Findings of this research have resulted in a two publications in the Guidance, Navigation & Control Conference.

Working with me on this research is MS student, 2D Lt.Thomas McKenna.

Constructive Feedback Methodology for Control Nonaffine Applications

U.S Air Force Office of Scientific Research

This research pursues to find a constructive feedback control strategy to address the inherent nonlinearties and complexities of underlying physical systems that form the basis of today’s heterogeneous applications (for example: flapping-wing systems). Dynamics of such systems are nonlinear in the control input, and are more commonly referred as control-nonaffine systems in the literature. This special structure, however violates collinearity; a key property essential to employ nonlinear control techniques such as feedback linearization, backstepping and Lyapunov-based redesign methods. As a consequence, the current state-of-art feedback methodologies for control-nonaffine system are either descriptive in nature or are restrictive to stable and/or special class of nonlinearities. Thus, the primary goal of this research is to synthesize constructive concepts that will lead to a system with desired properties without making approximations to the intrinsic nonlinearities of the system.

Approach: Motivated by the results of Sontag for affine-systems, this research explores a universal stabilization formula for an unstable non-affine system by developing a novel composite control law structure. The intuitive idea behind this control form is to introduce stiffness and damping into the system. The major contribution comes in identifying principles that converts an open-loop unstable system into stable in the Lyapunov sense closed-loop system through static-feedback alone. Most importantly, when is a general nonlinear system passive? Under what conditions can a nonlinear system can be rendered passive through static state-feedback?  Using these sufficiency conditions a novel method for construction of control laws for single-input non-affine system without making any assumptions about the nature of the control influence is designed.

Contributions:

1. The result published in 2013 American Control Conference Proceedings gives a generalization of the famous Kalman-Yakubovich-Popov lemma for non-affine systems under mild restrictions. This new result helps to determine whether or not an input-output description of a nonlinear system is passive. It is expected that this generalization will play a vital role in developing adaptive control laws for nonlinear systems based on Lyapunov's direct method analogous to its linear counterpart.
2. The 2013 American Control Conference paper also presents for the first time, a general control law design procedure for asymptotic stabilization of unstable non-affine systems with application to magnetic levitation concept applications.
3. Results in 2013 SIAM Conference on Control & Its Applications (recipient of NSF-SIAM Early Career Travel Award) provides conditions under which a nonlinear system can be made passive through state-feedback and forms the basis for stabilization of general non-affine systems. This paper presents application of developed concepts to a continuously-stirred tank reactor.

Working with me on this research is PhD student, Adam Tahir.