On a stabilizing feedback attitude control

AbstractThe attitude control of a rotating satellite with two control jets leads to a system of four controlled ordinary differential equations of the form(S) $$dx/dt = X(x) + u_1 Y^1 (x) + u_2 Y^2 (x),x(0) = 0.$$ Our goal is to derive feedback controlsu1,u2 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 controlu1, i.e.,u2≡0. In this problem,Y1(0), ..., (ad3X, Y1)(0) are linearly independent. We give a formula for generating the directions (adiX, Yi)(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 beq1 ∈ ℝ4, we write $$q^1 = \sum\limits_{i = 1}^4 {\alpha _1 } (ad^{i - 1} X,Y^1 )(0),$$ and choose a controlu(t,q1) 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.