Phase portraits of planar control systems. (English) Zbl 0858.93037

Linear, single input single output control systems in the plane of the form \[ dx/dt=Ax+bu, \qquad y=cx\tag{\(*\)} \] are considered. They are subject to a saturated output feedback \(u=\varphi(y)\). The paper investigates dynamical systems arising as systems \((*)\) with closed feedback loop, and provides a qualitative characterization of phase portraits of these systems under the assumption that the origin is an isolated asymptotically stable equilibrium point. This characterization is made in terms of two parameters of the systems which are the trace and the determinant of the matrix \(A\). The results obtained in the paper establish possible shapes of phase portraits depending on the system parameters. In particular, it is proved that the systems may have two types of closed curve invariant sets: these are either unstable single closed orbits or they consist of a pair of heteroclinic orbits joining two saddle points.


93C15 Control/observation systems governed by ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
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