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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

MSC:

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
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References:

[1] M. Athans, P. L. Falb: Optimal Control. McGraw Hill, New York 1966. · Zbl 0196.46303
[2] E. B. Lee, L. Markus: Foundations of Optimal Control Theory. John Wiley, New York 1967. · Zbl 0159.13201
[3] J. L. G. Almuzara, I. Flugge-Lotz: Minimum time control of a nonlinear system. J. Differential Equations 4 (1968), 1, 12-39. · Zbl 0162.14102
[4] A. Boettiger, V. B. Haas: Synthesis of time-optimal control of a second order nonlinear process. J. Optim. Theory Appl. 4 (1969), 1, 22-39. · Zbl 0167.09102
[5] E. M. James: Time optimal control and the Van der Pol oscillator. J. Inst. Math. Appl. 13 (1974), 1, 67-81. · Zbl 0275.49039
[6] I. Vakilzadeh: Bang-bang control of a plant with one positive and one negative real pole. J. Optim. Theory Appl. 24 (1978), 2, 315- 324. · Zbl 0349.93027
[7] I. Vakilzadeh, A. A. Keshavarz: Bang-bang control of a second order nonlinear stable plant with second order nonlinearity. Kybernetika 18 (1982), 1, 66-71. · Zbl 0487.93030
[8] I. Vakilzadeh, A. A. Keshavarz: Bang-bang control of a second order nonlinear unstable plant with third order nonlinearity. Internat. J. Control 34 (1981), 3, 457-463. · Zbl 0465.93038
[9] V. G. Boltyanskii: An example of nonlinear synthesis. (in Russian). Differencial’nye Uravneniya 6 (1970), 4, 644-649; (in English) Differential Equations 6 (1970), 4, 495-498.
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