Pole assignment for second-order system by acceleration feedback

Pole assignment problem for a linear stationary dynamical system with static output feedback belongs to the hard-to-solve problems of mathematical control theory. In general case this problem is reduced to solving systems of nonlinear algebraic equations. In this paper pole assignment problem for a second-order system with acceleration feedback is solved. The system, the behavior of which is described by the equation 0 1 2 A y  A y  A y  Bu , is considered, where u, y are 2-vectors of input and output, 0A , 1A , 2 , AB are 2 × 2-matrices with real elements. It is proposed to construct control law in the form of feedback with respect to the second derivative vector, acceleration vector: u  F y , where F is a feedback 2 × 2-matrix with real elements. The spectrum of open-loop system is the roots of the polynomial   2 4 3 2 0 1 2 0 1 2 3 4 a(s)  det A s  As  A  a s  a s  a s  a s  a . The spectrum of closed-loop system is the roots of the polynomial   2 0 1 2 b(s)  det (A  BF)s  As  A . Pole assignment problem is to selecting a feedback matrix F under which the roots of the polynomial b(s) coincide with a given set of complex numbers 1 2 3 4 S  {s ; s ; s ; s } . It is shown that the spectrum of a closed-loop system have to satisfy the relation 3 1 2 3 4 4 1 1 1 1 a s s s s a      . If this relation is satisfied, then the solution of pole assignment problem reduces to the sequential solution of the square equation and the system of linear algebraic equations. The conditions under which there exist real feedback matrices providing a given spectrum of a closed-loop system are determined. The results of the paer can be applied in solving the problems of vibration control of mechanical systems based on the signals of acceleration senso that is confirmed by the example of solving pole assignment problem of a two-mass mechanical system.

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