The explicit synthesis of stabilizing (time optimal) feedback controls for the attitude control of a rotating satellite
The following equation is chosen as a model of a rotating satellite: (*) \(x=X(x)+u_ 1(x)X^ 1(x)+u_ 2(x)X^ 2(x)\), \(x(0)=0\), where \(x=(x_ 1,x_ 2,x_ 3,x_ 4)\), \(X^ 1(x)=(-bx_ 2\), \(bx_ 1\), \(-x_ 2(1- x^ 2_ 3-x^ 2_ 4)^{1/2}+cx_ 4\), \(x_ 1(1-x^ 2_ 3-x^ 2_ 4)^{1/2}-cx_ 3)\), \(X^ 1(x)=(1,0,0,0)\), \(X^ 2(x)=(0,1,0,0)\) are column vectors. The linearization technique used in the paper involves choosing a feedback control of the form: \(u_ 1(x)=g_ 1(x)+sgn h_ 1(x)\), \(u_ 2(x)=g_ 2(x)+sgn h_ 2(x)\), where \(g_ 1,g_ 2,h_ 1,h_ 2:R^ 4\to R^ 1\), \(g_ 1(0)=g_ 2(0)=h_ 1(0)=h_ 2(0)=0\). With respect to (*) these controls are not time- optimal ones which should be of 'bang-bang' type. However, in a sufficiently small neighbourhood of the origin \(g_ 1\), \(g_ 2\) are close to zero and the controls become 'bang-bang'. The synthesis procedure shown in the paper may be used for any n-dimensional system of the form (*) with arbitrary control vector \(u=(u_ 1,u_ 2,...,u_ m)\), which satisfies the necessary and sufficient conditions to be feedback equivalent to a linear system. The work has an essential meaning for the practical solution of sub- optimal problems.
scientific article; zbMATH DE number 3920310