Period-aware asymptotic gain with application to a periodically forced synchronization circuit

Abstract

The classical asymptotic gain (AG) is a concept known from the input-to-state stability theory. Given a uniform input bound, AG estimates the asymptotic bound of the output. Sometimes, however, more information is known about the input than just a bound. In this paper we consider the case of a periodic input. Under the assumption that the system converges to a periodic solution, we introduce a new gain, called period-aware asymptotic gain (PAG), which employs periodicity to enable a sharper asymptotic estimation of the output. Since the PAG can distinguish between short-period ("high-frequency") and long-period ("low-frequency") signals, it is able to rigorously quantify such properties as bandwidth, resonant behavior, and high-frequency damping. We discuss how the PAG can be computed and illustrate it with a numerical example from the field of power electronics.

Figures (3)

• Figure 1: Linear input-output gains: frequency response $\vert G(j\omega) \vert$, derivative $\gamma'$ of the classical asymptotic gain $\gamma$, and period-aware asymptotic gain $\gamma_T$ represented by its AC/DC components $\gamma_\mathrm{ac}(T)$ and $\gamma_\mathrm{dc}$ (Theorem \ref{['th: linear']}). The horizontal axis is both the $\omega$-axis for $\vert G(j\omega) \vert$ and $T$-axis for $\gamma_\mathrm{ac}(T)$ with $T = 2\pi/\omega$. The system has a resonant frequency $\omega_0$.
• Figure 2: Input-output gains of the example system under different magnitudes and AC/DC compositions of the input: $\vert G(j\omega) \vert$ -- frequency response of the linearized system; $\gamma(\cdot)/\cdot$ -- average derivative of the exact nonlinear AG $\gamma$; $\gamma_\mathrm{dc}$ and $\gamma_\mathrm{ac}(T)$ -- components of the exact PAG of the linearized system (Theorem \ref{['th: linear']}); $\mu_\ell(T)$ -- quantities \ref{['eq: example output measure']}. The horizontal axis is both the $T$-axis for $\gamma_\mathrm{ac}(T)$ and $\mu(T)$ and $\omega$-axis for $\vert G(j\omega) \vert$ with $T = 2\pi/\omega$.
• Figure 3: Thin lines: output waveforms in response to randomized $T$-periodic inputs with $T = 0.02$ s and magnitudes (left to right) $\Vert u \Vert_\infty \leq 2\%$, $\Vert u \Vert_\infty \leq 6\%$, and $\Vert u \Vert_\infty \leq 10\%$. Thick lines: amplitude estimations by the exact classical AG (red) and conservative proposed PAG (blue).

Theorems & Definitions (15)

• Definition 1
• Definition 2
• Proposition 1
• Theorem 1
• proof
• Remark 1
• Remark 2
• Proposition 2
• Lemma 1
• proof