Oscillating system design applying universal formula for control

Abstract

Abstract-The problem of oscillating system design applying the homogeneity approach is studied. The Anti-control Lyapunov Function (ALF) is introduced as a counterpart of Control Lyapunov Function (CLF) for the control design that destabilizes a nonlinear system. A universal anti-control formula is proposed. Next, the universal control formulas based on ALF and CLF are used to design an oscillating system. Efficiency of the proposed approach is demonstrated on example.

On an extension of homogeneity notion for differential inclusions

“…Extensions to local homogeneity have been proposed recently [18], [20]. Let us note that this notion was also used in different contexts: polynomial systems [21] and switched systems [22], self-triggered systems [23], control and analysis of oscillations [20], [24].…”

On conditions of oscillations and multi-homogeneity

“…The procedure can also be applied in a reverse way for a complex oscillating system design and for the control synthesis that provides oscillating behavior for a nonlinear system [37]. Let us test this procedure on several academic examples.…”

On necessary conditions of instability and design of destabilizing controls

“…Such a Chetaev function V is closely related with the ALF introduced in [1] for the case of globally repulsing equilibrium. As for the case of the autonomous system (1), a Lyapunov function can be used for a repulsing steady-state analysis (in this caseV > 0 in a vicinity of the equilibrium) and a Chetaev function covers a more general case, when the repulsing trajectories belong to a part of the vicinity.…”

“…All previously mentioned results for instability were given as sufficient conditions only, the first attempt to show that for the case of a purely repulsing equilibrium the Lyapunov's conditions are also necessary, has been performed in [11], some necessary conditions for the instability results from [9] were derived in [12]. Next, the CLF approach has been extended in [1] for a destabilizing control design via the ALF framework. The utility of ALF and the design of destabilizing controls themselves was shown also in [1] on the example of control design for generation of oscillations in nonlinear systems.…”

“…Next, the CLF approach has been extended in [1] for a destabilizing control design via the ALF framework. The utility of ALF and the design of destabilizing controls themselves was shown also in [1] on the example of control design for generation of oscillations in nonlinear systems. Another possible motivation for the interest in destabilizing controls is related with a recently introduced notion of the norm controllability [2], which is a property dual to input-to-state stability characterizing how far the state may go applying a bounded control.…”