• ## Notes-Global uniform asymptotic stabiliztion of an underactuated surface vessel

• This is a Time-varying Continuous Approach.

The dynamics of an underactuated ship is

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## Nonlinear Systems

A nonlinear dynamic system can usually be represented by a set of nonlinear differential equations in the form

It is directly applicable to feedback control systems. The reason is that the equation can represent the close-loop dynamics of a feedback control system, with the control input being a function of state $x$ and time $t$, and therefore disappearing in the closed-loop dynamics.

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• ## Notes-Exponential Stabilization of an Underactuated Autonomous Surface Vessel

• Time-invariant discontinous feedback laws are constructed to asymptotically stabilize the system to the desired configuration with exponential convergence rates.
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• Uncertainties

Structured(or parametric) uncertainties
Corresponds to inaccuracies on terms actually included in the model
Unstructured uncertainties(or unmodeled dynamics)
Corresponds to inaccuracies on(i.e., underestimation of) the system order

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• ## Backstepping Technique

Consider the system

where, $x,y \in \mathbb{R}$ are the states, $u \in \mathbb{R}$ is the control input, and $f,g$ are assumed to be known.

Objective: $x \to 0$ as $t \to \infty$

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