Multiplier analysis of Lurye systems with power signals

TL;DR

This work analyzes Lurye systems with memoryless slope-restricted nonlinearities and shows that OZF multipliers can bound the system's power gain for nonzero-mean exogenous signals. It proves that if a suitable multiplier $M$ exists in the full class $\mathcal{M}$ (but not necessarily in $\mathcal{M}_{\text{odd}}$), then the system attains finite-gain offset stability (FGOS), ensuring small output power when inputs include power signals or constant biases. The paper derives power and bias-specific stability results and extends them to discrete time, illustrated by a saturating-nonlinearity example where the gain, and hence robustness to noise, depends on the chosen multiplier class and gain $g$. The findings offer a practical framework for assessing noise insensitivity in Lurye systems via multipliers, with clear limits when exogenous signals are non-steady or when high gains are involved.

Abstract

Multipliers can be used to guarantee both the Lyapunov stability and input-output stability of Lurye systems with time-invariant memoryless slope-restricted nonlinearities. If a dynamic multiplier is used there is no guarantee the closed-loop system has finite incremental gain. It has been suggested in the literature that without this guarantee such a system may be critically sensitive to time-varying exogenous signals including noise. We show that multipliers guarantee the power gain of the system to be bounded and quantifiable. Furthermore power may be measured about an appropriate steady state bias term, provided the multiplier does not require the nonlinearity to be odd. Hence dynamic multipliers can be used to guarantee Lurye systems have low sensitivity to noise, provided other exogenous systems have constant steady state. We illustrate the analysis with an example where the exogenous signal is a power signal with non-zero mean.

Figures (4)

• Figure 1: Lurye system.
• Figure 2: Deterministic step signals $r_2^a$ and $r_2^b$. Both are bias signals, with $r_2^b$ a delayed version of $r_2^a$. Their difference $r_2^a-r_2^b\in\ell_2$.
• Figure 4: Respective responses $Hr_2^a$ and $Hr_2^b$ to deterministic step signals $r_2^a$ and $r_2^b$ when the gain is $g=0.8$. There is a suitable multiplier $M\in\mathcal{M}^d$ for $G$. It follows that $H$ is FGOS and the difference $Hr_2^a-Hr_2^b\in\ell_2$.
• Figure 6: Exogenous signals $r_1$ and $r_2$ for the step response with noise. Both are power signals.

Theorems & Definitions (26)

• Definition 1
• Remark 1
• Remark 2
• Definition 2
• Remark 3
• Definition 3: Zames68
• Definition 4: Zames68
• Definition 5
• Theorem 1: Zames68desoer75
• Remark 4