Van der Pol Oscillator, its Control Theoretical Analysis Using Dissipative Canonical Equations and its Lyapunov Function

In this research paper, beginning with the Lagrangian and generalized velocity proportional (Rayleigh) dissipation function of a physical/engineering system, the Lagrange-dissipative model ( {L,D}-model briefly) of the system is initially developed. Upon satisfying the prerequisite condition for a Legendre transform, the Hamiltonian function can be obtained. With the Hamiltonian function and the generalized velocity proportional (Rayleigh) dissipation function, dissipative canonical equations can be derived. Using these dissipative canonical equations as the state-space equations of the system allows for the investigation of observability, controllability, and stability properties. In addition to the equilibrium (or critical or fixed) points of the system, stability properties can also be verified through a Lyapunov function as a residual energy function (REF). Since the proposed method is valid for both linear and nonlinear systems, it has been applied to the Van der Pol oscillator/equation.