Time optimal control of a second order nonlinear plant. (English) Zbl 0547.93040

A model of a nonlinear object is defined as \[ (1)\quad\ddot e+a\dot e|\dot e| +be=-Ku. \] It is assumed that the time-optimal control is of ”bang-bang” type. For \(u=\pm 1\), trajectories of the solution of (1) are defined in the state plane \((\dot e,e)\). It is shown that, for \(u=\pm 1\), the control system is a) totally stable at the origin if \(a/bK>0,\) b) stable in some neighbourhood of the origin if \(b<0\) and \(a/bK>0.\) c) totally unstable at the origin if \(b=0\) and \(a/K<0\). The assumption that the time-optimal control is of ”bang-bang” type hasn’t been justified in the paper. It results from the author’s conviction that time-optimal control of second-order, nonlinear systems (shown in quoted literature) is of ”bang-bang” type. The analysis of the transition of state trajectories in \(E^ 2\), performed for three values of the parameters a and b, is quite rudimentary. The paper must be considered as an elementary excercise of shape examination of trajectories in \(E^ 2\), under the simple assumption that the control is of ”bang-bang” type.
Reviewer: W.Hejmo


93C10 Nonlinear systems in control theory
49J30 Existence of optimal solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.)
93D99 Stability of control systems
93B03 Attainable sets, reachability
93C15 Control/observation systems governed by ordinary differential equations
Full Text: EuDML


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