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.
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:
Define the state variables
so that the state equations are given by
Consider the subsystem and letting be the control variables :
Consider the reduced system, restricting considering to , apply the process
The feedback control law
where , and and are the gains, yields the reduced closed-loop system
The dynamics can be rewritten as
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
Then the system is globally expoentially stable.
Restrict consideration and consider the following controller:
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
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.