• 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 $M$ and the damping matrix $D$ are constant and diagonal:

Remark: Rewrite $m_{22} \dot{v}_y + m_{11} v_x \omega_z + d_{22}v_y = 0$

## Global Coordinate Transformation

Define the state variables

so that the state equations are given by

where $\alpha = d_{22}/m_{22},\beta = m_{11}/m_{22}$ and

## Discontinuous feedback control laws

Consider the subsystem and letting $(x_5, x_6)$ be the control variables $(v_1, v_2)$:

## Stabiliztion of the reduced system

Consider the reduced system, restricting considering to $x_1 \not = 0$, apply the $\sigma-$process

to obtain

The feedback control law

where $z = (z_1, z_2, z_3)^T$, and $k > 0$ and $l = (l_1, l_2, l_3)$ are the gains, yields the reduced closed-loop system

The $z-$dynamics can be rewritten as

where

Remark: $\dot y = -ky \Rightarrow y = y_0 e^{-kt}$

If $k \not = \alpha$, the spectrum of the matrix $A_1$ can be assigned arbitrarily through the gain matrix $l$.

The matrix $A_2(t)$ goes to zero as $t \to \infty$ and

The $z-$dynamics can also be rendered globally exponentially stable at the origin $z=0$ by selecting $l$ such that the matrix $A_1$ 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 $A_1$ is the constant and Hurwitz(i.e., has all eigenvalues strictly in the left-half plane), and the time-varying matrix $A_2(t)$ is such that $A_2(t) \to 0$ as $t \to \infty$
and

Then the system is globally expoentially stable.

## Stabilization of the complete system

Restrict consideration $x_1 \not = 0$ 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: $\dot w_1 = -K w_1$

where

Remark: $\dot x_2 = x_6 + x_3 x_5$

where

The $(y, \omega_1, \omega_2)-$dynamics is globally exponentially stable at $(y, w_1, w_2)= (0,0,0)$. Moreover, it can be easily shown that if $y_0\omega_{10}\geq 0$, then $A_2(t)$ and $h(t)$ go to zero as $t \to \infty$ and

Thus, for any initial condition$(y_0, z_0, \omega_{10}, \omega_{20})$ satisfying $y_0 \not= 0$ and $y_0 \omega_{10} \geq 0$, both the trajectory $(y(t),z(t),\omega_1(t),\omega_2(t))$ and the control $(u_1(t), u_2(t))$ are bounded for all $t \geq 0$ and converge exponentially to zero.