# 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.

## 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 formwhere 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

andThen 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.

1 | @article{reyhanoglu1997exponential, |