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

Mathematical model

The kinematic model which describes the geometrical relationship between the earth-fixed(I-frame) and vehicle-fixed(B-frame) motion is given as

The dynamic equations of motion of the vehicle can be expressed in thr B-frame as

The following simplified model can be obtained by assuming that both inertia matrix and the damping matrix are constant and diagonal:

Remark: Rewrite

Global Coordinate Transformation

Define the state variables

so that the state equations are given by

where and

Discontinuous feedback control laws

Consider the subsystem and letting be the control variables :

Stabiliztion of the reduced system

Consider the reduced system, restricting considering to , apply the process

to obtain

The feedback control law

where , and and are the gains, yields the reduced closed-loop system

The dynamics can be rewritten as

where

Remark:

If , the spectrum of the matrix can be assigned arbitrarily through the gain matrix .

The matrix goes to zero as and

The dynamics can also be rendered globally exponentially stable at the origin by selecting such that the matrix is a Hurwitz matrix.
(Applied Nonlinear Control, Slotine and Li, Section 4.2.2)

Perturbed linear systems
Consider a linear time-varying system of the form

where the matrix is the constant and Hurwitz(i.e., has all eigenvalues strictly in the left-half plane), and the time-varying matrix is such that as
and

Then the system is globally expoentially stable.

Stabilization of the complete system

Restrict consideration and consider the following controller:

where

while avoiding the set

Consider the coordinate transformation

It can be shown that in the above coordinates the closed-loop system can be written as

Remark:

where

Remark:

where

The dynamics is globally exponentially stable at . Moreover, it can be easily shown that if , then and go to zero as and

Thus, for any initial condition satisfying and , both the trajectory and the control are bounded for all and converge exponentially to zero.

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@article{reyhanoglu1997exponential,
title={Exponential stabilization of an underactuated autonomous surface vessel},
author={Reyhanoglu, Mahmut},
journal={Automatica},
volume={33},
number={12},
pages={2249--2254},
year={1997},
publisher={Elsevier}
}