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An algorithm of synthesis of \(N\)-positional controls with fixed, minimally admissible intervals between switchings. (Russian) Zbl 0835.93016
We are given a dynamic system \(\dot{\mathbf x}= {\mathbf f}(t, {\mathbf x}, u)\), \(t\in \mathbb{R}\), \({\mathbf x}\in \mathbb{R}^n\), where \({\mathbf f}\) is of class \(C^1[(t, {\mathbf x})]\) and a control parameter \(u\) which is assumed as discontinuous switching between values in a set \(U= \{u_1,\dots, u_N\}\) of finite number with elements \(u_i\), \(i= 1,\dots, N\). The paper deals with a particular case of the above system, i.e. \[ \dot{\mathbf x}= \sum^N_1 \Theta_i{\mathbf f}(t, {\mathbf x}, u_i),\;\Theta= (\Theta_1,\dots, \Theta_N),\;\Theta_i\geq 0,\;i\in [1,\dots, N],\;\sum^N_1 \Theta_i= 1.\tag{\(*\)} \] The problem consists in finding a piecewise constant control function \(u(t)\) for \((*)\) with fixed non-empty minimum intervals between consecutive switching operations. An algorithm for practical solution of the above problem is proposed. Its idea is based on the theoretical solution given by Gamkrelidze in “Bases of optimal control”, Tbilici: isd-wo Tbilic., 321 (1973). The proposed concept may be useful in practical synthesis of an optimal structure.
Reviewer: W.Hejmo (Kraków)
93B50 Synthesis problems
93B40 Computational methods in systems theory (MSC2010)
93C10 Nonlinear systems in control theory