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Analytical formula of calculating a controller for linear SIMO-system. (Russian. English summary) Zbl 1453.93122

In this article an alternate technique of proving Ackermann’s formula for a SIMO-system is given, which realize the eigenvalues placement of a closed-loop SIMO-system by knowing the coefficients of desirable characteristic polynomial. This alternate approach is “based both on the technique of multilevel decomposition applied to the mathematical model of system while synthesizing its modal control (providing the desirable eigenvalues/poles placement) and on widely known property of inverse matrix of controllability in linear stationary dynamical systems. The main transformations in the mentioned technique are being performed using matrix zero divisors that are rectangular matrices zeroing their products with specified matrices.” To solve this problem two lemmas and a theorem for solving the assigned problem have been formulated and proved. This procedure “allows realizing the search of scalar control bypassing intermediate calculations on each decomposition level and therefore eliminates an occurrence of badly conditioned matrices on the lower decomposition levels if the control object is described by numerical matrices.” To illustrate the efficiency of the new analytical formula to calculate the coefficients of feedback controller matrix, a fourth-order control object at solving the problem of stabilizing a space vehicle longitudinal motion relative to the nominal trajectory in the Earth atmosphere by changing the bank angle is considered.

MSC:

93C35 Multivariable systems, multidimensional control systems
93B52 Feedback control
93C05 Linear systems in control theory
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