State feedback stabilization of uncertain linear time-delay systems: A nonlinear matrix inequality approach. (Q2889391)

A linear time-delay system \( \dot x(t)=A(t)x(t)+A_d(t)x(t-d(t))+Bu(t) \) is considered together with \(x(t)=\phi (t)\quad \forall t\in [-\bar d,0]\), \(\bar d >0\), where \(x(t)\in \mathbb R^n\) is a state vector, \(A(t)\) and \(A_d(t)\) stand for uncertain time-varying matrices, \(\phi \) is a smooth vector-valued initial function and \(d(t)\) represents a time-varying delay satisfying \(0\leq d(t)\leq \bar d\), \(\dot d(t)<\tau <1\;\forall t\geq 0\). It is assumed that the uncertain matrices \(A(t)\) and \(A_d(t)\) can be further decomposed. In detail, \(A(t)=A+D_a F_a(t) E_a\) where \(A\),\(D_a\), and \(E_a\) are constant matrices, and \(F_a^{ T}(t)F_a(t)\leq I\) (the identity matrix); an analogous decomposition is applied to \(A_d(t)\). The objective is to design a stabilizing state-feedback controller in the form of \(u(t)=Kx(t)\).NEWLINENEWLINEBy using two lemmas from the literature, the authors formulate and prove the main theorem on the stabilization of the system with the above-mentioned feedback. Next, a computational algorithm is sketched. Finally, to demonstrate the efficiency of the proposed method, benchmark uncertain time-delay systems are solved and the results are compared with the results from the literature.NEWLINENEWLINEThe paper is written in a rather condensed form and some familiarity with the subject and the relevant techniques is assumed. A less knowledgeable readership would, for instance, appreciate an explicit statement on which of the two common definitions of the inequality between two matrices is used. Moreover, a typo (a missing inequality) seems to occur in the expression labeled (20) where the reader would expect a condition stating that the matrix should be less than the zero matrix.