# Applied Nonlinear Controls Lab

## Nonlinear Control Theory

### Nonlinear Geometric Observer Design

Student: Y. Wang

This project considers the application of geometric nonlinear control methods to design multi-output continuous-time nonlinear observers. The approach exploits the additional degree of freedom afforded by multiple outputs and focusses on methods with block triangular structure. A main feature of the Block Triangular Form (BTF) coordinates is that they allow single-output designs to be performed in a decentralized manner subsystem-at-a-time on lower dimensional subsystems. The upper states are effectively treated as known measurements given that we have already designed their convergent observer. The reduced dimensionality of the designs can significantly reduce the complexity of the entire multi-output design. In addition to the general BTF, we also consider a transformation to a less broadly applicable Block Triangular Observer Form (BTOF). This form allows for classical exact observer error linearization techniques to be applied in a decentralized manner.

### Flatness-based Control of Underactuated Nonlinear Systems

Student: C. Aguilar

This project considers an attitude trajectory tracking problem for a two-input underactuated rigid spacecraft. The attitude of a spacecraft is its orientation in space, and hence, the attitude control problem involves directing the spacecraft to achieve a specified orientation. Spacecraft attitude control has many applications. For example, it is used to avoid solar or atmospheric damage to sensitive components, to point directional antennas and solar panels, and to orient rockets used for orbit manoeuvres. From a practical and theoretical point of view, it is of interest to consider the attitude control problem when some of the actuators have failed, that is, when the spacecraft is underactuated. The literature on trajectory tracking of the attitude for an underactuated spacecraft is scarce compared to that on stabilization. Differential flatness is used in this project to provide a solution to the tracking problem.

The first video illustrates an open-loop control to drive the satellite between two equilibrium configurations. The non-moving box outline illustrates the desired rest configuration. The second video shows the combination of open- and closed-loop control to obtain asymptotic stability. The control is switched to a stabilizing control once it is sufficiently near it's target attitude. The moving yellow box is the reference attitude trajectory.

### Flatness-based Control of Distributed Parameter Systems

Student: M. Barczyk

While flatness-based open-loop control of infinite-dimensional systems has received significant attention, closed-loop flatness-based control has been less studied. Of course, in practice closed-loop control is essential for reducing the effect of disturbances and model error. Experimental validation of flatness-based control often relies on augmenting the open-loop design with a simple PID structure. However, a rigorous method of PID parameter selection is not available. Recent work by Meurer and Zeitz gives a more systematic basis for designing a closed-loop component of the flatness-based control law. In this work the same series expressions used in the flatness-based open-loop control are used to derive the closed-loop control. The applications considered by Meurer and Zeitz include DPS modeled exclusively by parabolic PDEs. This project makes use of the approach of Meurer and Zeitz for a rotating flexible beam system modeled by a biharmonic PDE. Experimental validation the resulting controller was performed.