Feedback linearization

We study the problem of observer linearization for single-output dynamical systems in the presence of output-dependent time-scaling changes. An alternative algorithm for the solution of the observer linearization problem is introduced. The algorithm employs an exterior calculus approach that provides a simple procedure for the solution of the observer linearization problem by means of an output dependent time-scale transformation.

Observer linearization by output-dependent time-scale transformations

Brief Nonlinear observer design via passivation of error dynamics

This thesis explores the paradigm of two degree of freedom design for nonlinear control systems. In two degree of freedom design one generates an explicit trajectory for state and input around which the system is linearized. Linear techniques are then used to stabilize the system around the nominal trajectory and to deal with uncertainty. This approach allows the use of the wealth of tools in linear control theory to stabilize a system in the face of uncertainty, while exploiting the non-linearities to increase performance. Indeed, this thesis shows through simulations and experiments that the generation of a nominal trajectory allows more aggressive tracking in mechanical systems.

The generation of trajectories for general systems involves the solution of two point boundary value problems which are hard to solve numerically. For the special class of differentially flat systems there exists a unique correspondence between trajectories in the output space and the full state and input space. This allows us to generate trajectories in the lower dimensional output, space where we don't have differential constraints, and subsequently map these to the full state and input space through an algebraic procedure. No differential equations have to be solved in this process. This thesis gives a definition of differential flatness in terms of differential geometry, and proves some properties of flat systems. In particular, it is shown that differential flatness is equivalent to dynamic feedback linearizability in an open and dense set.

This dissertation focuses on differentially flat systems. We describe some interesting trajectory generation problems for these systems, and present software to solve them. We also present algorithms and software for real time trajectory generation, that allow a computational tradeoff between stability and performance. We prove convergence for a rather general class of desired trajectories. If a system is not differentially flat we can approximate it with a differentially flat system, and extend the techniques for flat systems. The various extensions for approximately flat systems are validated in simulation and experiments on a thrust vectored aircraft. A system may exhibit a two layer structure where the outer layer is a flat system, and the inner system is not. We call this structure outer flatness. We investigate trajectory generation for these systems and present theorems on the type of tracking we can achieve. We validate the outer flatness approach on a model helicopter in simulations and experiment.

Trajectory generation for nonlinear control systems

Lagrangian control systems that are differentially flat with flat outputs that only depend on configuration variables are said to be configuration flat. We provide a complete characterisation of configuration flatness for systems with n degrees of freedom and n-1 controls whose range of control forces only depends on configuration and whose Lagrangian has the form of kinetic energy minus potential. The method presented allows us to determine if such a system is configuration flat and, if so provides a constructive method for finding all possible configuration flat outputs. Our characterisation relates configuration flatness to Riemannian geometry.

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Configuration flatness of Lagrangian systems underactuated by one control

We discuss the design of observers for autonomous nonlinear systems with one output. New necessary and sufficient conditions for existence of a solution to the observer error linearization problem are given. In the case when an exact linearization of the observer error dynamics is impossible, a simple least-squares scheme is proposed that yields an observer whose error dynamics is approximately linear. An algorithm for observer construction is provided. An example of observer design for a mass-spring system with nonlinear spring and dynamic friction forces is given.

https://ieeexplore.ieee.org/iel4/4827/13355/00612110.pdf

Willem M. Sluis

Papers

On nonlinear observers with approximately linear error dynamics

A Test for Differential Flatness by Reduction to Single Input Systems

A homotopy algorithm for approximating geometric distributions by integrable systems

On Measures of Non-Integrability of Pfaffian Systems