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Continuous pole placement for delay equations. (English) Zbl 1034.93026

A controllable system with delay is considered: \[ \dot x(t)= Ax(t)+ Bu(t-\tau),\quad A\in \mathbb{R}^{n\times n},\quad B\in\mathbb{R}^{n\times 1},\tag{1} \] where \(x(t)\in \mathbb{R}^n\) is the state, \(u(t-\tau)\in \mathbb{R}^1\) is the input, \(\tau\geq 0\) represents an input delay. The control is found as a feedback \[ u(t-\tau)= K^T x(t-\tau),\qquad K\in\mathbb{R}^{n\times 1}. \] It is assumed that the pair \((A,B)\) from (1) is controllable. Then there exists a similarity transformation \(z= Tx\) such that the system (1) has the control canonical form \[ \dot z(t)= A_c z(t)+ B_c K^T_c z(t-\tau), \] where \(A_c\) is the Frobenius matrix, \(B_c\) is the unit vector. The characteristic quasi-polynomial of the system has as coefficients \(p_i(\lambda)= a_i+ k_i e^{-\lambda \tau}\). The stabilizing control is found from conditions on the negative real parts of roots of the quasi-polynomial. Examples are provided.

MSC:

93B55 Pole and zero placement problems
93C23 Control/observation systems governed by functional-differential equations
93B10 Canonical structure
93B17 Transformations

Software:

DDE-BIFTOOL
Full Text: DOI

References:

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