Bellman-Gronwall Inequality Approach to Oscillator Design for the Specific Bilinear Systems

In this paper, the existence of limit cycles for the specific bilinear systems is explored. Based on the BellmanGronwall inequality approach, not only the exponentially stable limit cycles phenomenon of such systems can be certified but also the oscillation behaviors of such systems can be correctly predicted. Finally, a numerical example is provided to illustrate the feasibility and effectiveness of the obtained result. INTRODUCTION Nonlinear system can offer oscillations with fixed amplitude and fixed frequency. These oscillations are named limit cycles, e.g., an RLC electrical circuit with a nonlinear resistor and Van der Pol equation. Limit cycles are singular phenomenon of nonlinear systems and have been a main interest of the researchers over the years; see, for example, [1-9], and the references therein. Prediction of limit cycles is very important, because limit cycles can appear in any kind of physical system. Ordinarily, a limit cycle can be desirable. This is the case of limit cycles in the electronic oscillators utilized in laboratories. There are at least two approaches to explore the phenomenon of limit cycles, namely describing function method and Poincare-Bendixson theorem. The disadvantages of the describing function approach are related to its approximate nature, and include the possibility of incorrect predictions. Besides, the Poincare-Bendixson theorem only offers a necessary condition to guarantee the existence of limit cycles. Consequently, if any one of the conditions of the Poincare-Bendixson theorem is not satisfied for some system, it can be guaranteed that there exists no limit cycles in such system. Conversely, even the conditions of the Poincare-Bendixson theorem are satisfied for some system, the existence of limit cycles cannot be guaranteed for such system [9]. In this paper, based on the Bellman-Gronwall inequality approach, the exponentially stable limit cycles for the specific bilinear control systems can be guaranteed. Furthermore, an estimate of the guaranteed convergence rate is also derived for such systems. PROBLEM FORMULATION AND MAIN RESULT In this paper, we consider the following bilinear control systems [10]: ( ) , 0 ), ( ) ( ) ( 0 + = t t t z t ru t Az t z& (1a) ( ) , 20 10 0 = z z t z (1b) *Address correspondence to this author at the Department of Electrical Engineering, I-Shou University, Taiwan, R.O.C.; E-mail: [email protected] where