Maximizing degree of stability of two-mass system with PID-control
Object of research. Degree of stability single-channel PID-contorl of a two mass systems is studied. Operating force affects the same weight which deviation from position of balance is adjustable size. Research technique. Research was carried out according to the following scheme. At first object and control parametrization at which the problem of management in both cases (depending on a choice of the operated and observable weight) can be studied from uniform positions (as one task) is found. Then (within this task) the description of a set of all characteristic polynoms is found: it is shown that this set of all polynoms of the fifth degree with two linear communications of factors, and these communications are found. At the expense of time scale selection these communications were simplified. Now the problem of search of the controls providing the greatest degree of stability, is reduced to search of the steadiest polynoms in a class of polynoms of the fifth degree with two linear communications. In a class of polynoms with these communications polynoms with no more than three roots on the right vertical (that is the sum of frequency rates of roots with the greatest material part no more than 3) are considered, and for them is shown, it is possible to change continuously factors of a polynom so that the polynom remained in this class, and its right vertical continuously moved to the left until on it there will be no one more root. So on the right vertical not less than four roots can be collected. After that it is possible, remaining in a class of polynoms, continuously to shift to the left the right vertical so that at least two of these four roots on the right vertical "stuck together". Result. The greatest degree of stability of a two mass systems is provided by controls for which on the right vertical of the characteristic polynom is or at least a double real root and a complex pair, or at least a double complex pair.