On a stabilizing feedback attitude control

Abstract

The attitude control of a rotating satellite with two control jets leads to a system of four controlled ordinary differential equations of the form

$$dx/dt = X(x) + u_1 Y^1 (x) + u_2 Y^2 (x),x(0) = 0.$$

((S))

Our goal is to derive feedback controlsu ₁,u ₂ which automatically stabilize the system (S), i.e., drive the solution to the (uncontrolled) rest solution zero. Let

$$(ad^0 X,Y) = Y,(adX,Y) = [X,Y],$$

the Lie product of the vector fieldsX, Y, and inductively

$$(ad^{k + 1} X,Y) = [X,(ad^k X,Y)].$$

It is known that, if

$$dim span\left\{ {\left( {ad^j X,Y^1 } \right)\left( 0 \right),j = 0,1,...} \right\} = 4,$$

then all points in some neighborhood of zero can be controlled to zero with just the controlu ₁, i.e.,u ₂≡0. In this problem,Y ¹(0), ..., (ad ³ X, Y ¹)(0) are linearly independent. We give a formula for generating the directions (ad ⁱ X, Y ⁱ)(0) as endpoints of admissible trajectories. Our modified feedback control is then formed as follows. Given an ε>0, if the state of system (S) is measured to beq ¹ ∈ ℝ⁴, we write

$$q^1 = \sum\limits_{i = 1}^4 {\alpha _1 } (ad^{i - 1} X,Y^1 )(0),$$

and choose a controlu(t,q ¹) on the interval 0≤t≤ε to drive the solution in the direction

$$ - \sum\limits_{i = 1}^4 {\alpha _1 } (ad^{i - 1} X,Y^1 )(0).$$

Thus, we assume that the state is measured (say) at time intervals 0, ε, 2ε, ..., while the control depends on the measured state, but then is open loop during a time interval ε until a new state is measured; hence, the terminologymodified feedback control. Numerical results are included for both the case of one control component and the case of two control components.