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PI stabilization of first-order systems with time delay. (English) Zbl 1023.93052

Consider the problem of stabilizing a first-order plant. If the characteristic quasipolynomial has the form \[ \delta^* (s)=e^{sT_m} d(s)+ e^{s(T_m-T_1)} y_1(s)+e^{s(T_m-T_2)} y_2(s)+\cdots+ e^{ s(T_m-T)}y_m(s), \] where \(d(s)\), \(y_i(s)\), \(i=1,\dots,m\) are polynomials with real coefficients and the following conditions are satisfied:
A1. \(\deg[d(s)]=n\), \(\deg[y_i(s)]<n\), \(i=1,\dots, m\),
A2. \(0<T_1< T_2<\cdots <T_m\),
then stability conditions are known (Pontryagin), (Bellman and Cooke).
Here, an example of a controllable system is investigated where \[ \delta^*(s)= (kk_i+kk_ps)+ (1+T_s)se^{Ls}. \] The following conditions of stabilization are obtained.
Theorem. Under the above assumptions on \(k\) and \(L\), the range of \(k_p\) values for which a solution exists to the PI stabilization-problem of a given open-loop stable plant with transfer function \(G(s)={K\over 1+T_s} e^{-Ls}\) is given by \[ -{1\over k}< k_p <{T_s\over kL}\sqrt {\alpha^2_1+ {L^2\over T^2_s}}, \] where \(\alpha_1\) is the solution of the equation \[ \tan(z)= -{T_s\over L}z \] in the interval \(({\pi \over 2},\pi)\).

MSC:

93D15 Stabilization of systems by feedback
93C23 Control/observation systems governed by functional-differential equations
93C80 Frequency-response methods in control theory
93B51 Design techniques (robust design, computer-aided design, etc.)
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References:

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