L. Giovanardi

On Optimal Stabilization of Periodic Orbits via Time Delayed Feedback Control

The paper considers the problem of designing time delayed feedback controllers to stabilize unstable periodic orbits of a class of sinusoidally forced nonlinear systems. This problem is formulated as an absolute stability problem of a linear periodic feedback system, in order to employ the well-known circle criterion. In particular, once a single test is verified by an unstable periodic orbit of the chaotic system, our approach directly provides a procedure for designing the optimal stabilizing controller, i.e. the one ensuring the largest obtainable stability bounds. Even if the circle criterion provides a sufficient condition for stability and therefore the obtained stability bounds are conservative in nature, several examples, as the one presented in this paper, show that the performance of the designed controller is quite satisfactory in comparison with different approaches.

LMI-based synthesis for controlling periodic solutions in a class of nonlinear systems

The paper considers the problem of designing controllers to stabilize periodic orbits in a class of sinusoidally forced nonlinear systems. This problem is formulated as an absolute stability problem of a linear periodic feedback system, in order to employ the well-known circle criterion. In this setting, we provide an LMI-based synthesis of the optimal stabilizing controller, i.e., the one ensuring the largest obtainable stability bounds.

LOWER BOUNDS FOR THE STABILITY DEGREE OF PERIODIC SOLUTIONS IN FORCED NONLINEAR SYSTEMS

In this paper the problem of local exponential stability of periodic orbits in a general class of forced nonlinear systems is considered. Some lower bounds for the degree of local exponential stability of a given periodic solution are provided by mixing results concerning the analysis of linear time-varying systems and the real parametric stability margin of uncertain linear time-invariant systems. Although conservative with respect to the degree of stability obtainable via the Floquet-based approach, such lower bounds can be efficiently computed also in cases where the periodic solution is not exactly known and the design of a controller ensuring a satisfactory transient behavior is the main concern. The main features of the developed approach are illustrated via two application examples.

On stabilizing periodic orbits of a chaotic system via feedback control

The paper deals with the problem of designing feedback controllers to stabilize unstable periodic orbits of chaotic systems. Moving from Pyragas time-delayed controllers (1992), classical frequency-domain stability criteria are exploited in order to select an optimal solution. An example illustrates the efficacy of the proposed method.

Stability Analysis of Periodic Solutions via Integral Quadratic Constraints

The paper considers stability of periodic solutions in a class of periodically forced nonlinear systems depending on a scalar parameter and subject to disturbances. A result concerning local existence of a family of periodic solutions for such systems is also given. The stability analysis — based on a combined use of linearization techniques and frequency-domain stability criteria expressed via Integral Quadratic Constraints — can be efficiently performed in terms of Linear Matrix Inequalities. An application example is carried out for illustrative purposes.

Stabilizing periodic orbits in forced nonlinear systems: an LMI-based synthesis technique

The paper considers the problem of designing controllers to stabilize periodic orbits in a class of sinusoidally forced nonlinear systems. This problem is formulated as an absolute stability problem of a linear periodic feedback system, in order to employ the circle criterion. In this setting, we provide an LMI-based synthesis of the optimal stabilizing controller, i.e., the one ensuring the largest obtainable stability bounds.

Analysis of a chaos control scheme through simple frequency domain criteria

In this paper, we revisit an existing chaos control scheme by using simple frequency-domain analysis criteria. In particular, we highlight some interesting links with other proposed chaos control techniques and envisage the possibility of applying well-known sufficient stability criteria, such as the circle criterion and generalizations thereof, to evaluate its performance. The Duffing equation is chosen as a simple test-bench for the proposed analysis.

Controller design for improving the degree of stability of periodic solutions in forced nonlinear systems

The problem of local exponential stability of periodic orbits in a general class of forced nonlinear systems is considered. A criterion for designing a finite dimensional linear time invariant controller that improves the degree of local exponential stability with respect to the uncontrolled case is provided. Such a criterion is based on a well-known condition for exponential stability of linear time varying systems, that is suitably tailored to our periodic setting. An application example is presented to illustrate the improvement on the transient behavior of the periodic orbits that is provided by the designed controller.