Attraction Region of Planar Linear Systems with One Unstable Pole and Saturated Feedback
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REGION OF PLANAR LINEAR SYSTEMS WITH ONE UNSTABLE POLE AND SATURATED FEEDBACK J.-Y. FAVEZ, PH. MULLHAUPT, B. SRINIVASAN, and D. BONVIN Abstract. The bifurcation of the attraction region for planar systems with one stable and one unstable pole under a saturated linear state feedback is considered. The attraction region can have either an unbounded hyperbolic shape or be bounded by a limit cycle. An analytical condition, under which either of these boundary shapes occurs, is given with a formal proof. This condition is based on the relationship between the stable and unstable manifolds associated with secondary saddle equilibrium points, whose presence is caused by the saturation on the input.
1. Introduction The study, whose results are presented here, originates from the problem of stabilizing a tokamak plasma reactor [6, 7]. For small excursions around a nominal set point, the model to be controlled is considered as having only one unstable pole and a large number of stable ones. The main difficulty is the presence of the input saturation due to the voltage and current limitation. Additionally, the specificity of the control hardware at the user disposal (at the plant location) is of practical importance. The hardware can implement only linear feedbacks, since no more than matrix multiplications and some extra simple algebra are allowed. The importance of the effect of saturation on the limitation of the stability region is observed experimentally using linear feedbacks, which encourages a sound theoretical treatment of the underlying issues. One key question is the impact of the gains on the shape and size of the stability region. However, such a task (in its full generality) is daunting. Hence, one is naturally conducted to study the simplest system retaining most of the main characteristics, namely a planar system having one stable and one 2000 Mathematics Subject Classification. 34C23, 93C10, 34D99. Key words and phrases. Planar linear system, input saturation, attraction region, bifurcation.
331 c 2006 Plenum Publishing Corporation 1079-2724/06/0700-0331/0
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J.-Y. FAVEZ, PH. MULLHAUPT, B. SRINIVASAN, and D. BONVIN
Table 1. Characteristics of the attraction regions Pole configuration Both stable One stable, one unstable Both unstable
Attraction region R2 bifurcation closed by a limit cycle
unstable pole under a saturated feedback, the results of which are presented hereafter. Nevertheless, and for general planar systems (i.e., nonlinear) under an input saturation, maximum stabilizing feedbacks that guarantee the state convergence in minimum time can also be constructed [2–5,14,15]. However, such feedbacks are not perfectly appropriate in the present context, since, apart from the implementation constraints mentioned above, they are very sensitive to switching time instants. Indeed, inaccurate state measurements and model uncertainty (both of which cannot be avoided) can give a poor performance. This robustness issue is not so crucial when simple linear feedbacks are considered. Two important concepts pe
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