Abstract
Nonlinear systems are typically linearized to permit linear feedback control design, but, in some systems, the nonlinearities are so strong that their performance is called chaotic, and linear control designs can be rendered ineffective. One famous example is the van der Pol equation of oscillatory circuits. This study investigates the control design for the forced van der Pol equation using simulations of various control designs for iterated initial conditions. The results of the study highlight that even optimal linear, time-invariant (LTI) control is unable to control the nonlinear van der Pol equation, but idealized nonlinear feedforward control performs quite well after an initial transient effect of the initial conditions. Perhaps the greatest strength of ideal nonlinear control is shown to be the simplicity of analysis. Merely equate coefficients order-of-differentiation insures trajectory tracking in steady-state (following dissipation of transient effects of initial conditions), meanwhile the solution of the time-invariant linear-quadratic optimal control problem with infinite time horizon is needed to reveal constant control gains for a linear-quadratic regulator. Since analytical development is so easy for ideal nonlinear control, this article focuses on numerical demonstrations of trajectory tracking error.
Highlights
A century ago, in the era of vacuum tube electronics, Balthazar van der Pol sought circuits that oscillated at a fixed frequency for use in signal transmission and receipt [1,2,3,4,5]
The purpose of this paper is to demonstrate the limitation of optimal linear feedback control, and examine effectiveness of idealized, nonlinear feedforward as well combined strategy the of nonlinear feedforward control plus linear optimal control feedback[5],control as as it a combined strategy of nonlinear feedforward control plus linear will optimal feedback control as it pertains to the forced van der Pol equation
These initial forfor evaluation of various control schemes: optimal. These initial conditions conditions(Figure (Figure2)2)are areused used evaluation of various control schemes: optimal feedback control, idealized feedforward control, and combined control using both of the aforementioned
Summary
A century ago, in the era of vacuum tube electronics, Balthazar van der Pol sought circuits that oscillated at a fixed frequency for use in signal transmission and receipt [1,2,3,4,5]. Van der Pol articulated that the oscillatory behavior fit the class of nonlinear equations that are referred to by his name (Equation (1)). The equation exhibits an oscillatory behavior, but the amplitude is not constant, it instead represents an invariant set called a “limit cycle”. Seeking to produce a fixed-amplitude oscillation, forcing functions are added to the nonlinear equation resulting in Equation (2). Typical control design procedures would begin with a linear, time-invariant (LTI) feedback controller based on a linearized version of the system equation. This paper derives such a controller and reveals the difficulties in controlling the nonlinear dynamic
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