On the necessity and sufficiency of the Zames–Falb multipliers for bounded operators

Abstract

This paper analyzes the robust feedback stability of a single-input-single-output stable linear time-invariant (LTI) system against three different classes of nonlinear systems using the Zames–Falb multipliers. The contribution is threefold. Firstly, we identify a class of uncertain systems over which the robust feedback stability is equivalent to the existence of an appropriate Zames–Falb multiplier. Secondly, when restricted to be static (a.k.a. memoryless), such a class of systems coincides with the class of sloped-restricted monotone nonlinearities, and the classical result of using the Zames–Falb multipliers to ensure feedback stability is recovered. Thirdly, when restricted to be LTI, the first class is demonstrated to be a subset of the second, and the existence of a Zames–Falb multiplier is shown to be sufficient but not necessary for the robust feedback stability.

Introduction

The robust stability analysis of the feedback interconnection of a single-input-single-output stable linear time-invariant (LTI) system $G$ and a nonlinear system $\Delta$ belonging to a specified uncertainty class, as depicted in Fig. 1, is a fundamental object of study in the field of control theory. It is often termed absolute stability analysis in the nonlinear systems literature (Popov, 1961, Yakubovich, 1982). When the class of $\Delta$’s considered is static (i.e. memoryless) and sector bounded, a variety of multiplier-based methodologies including the renowned circle criterion and Popov criterion (Khalil, 2002) have been proposed to establish the input–output feedback stability. For static, time-invariant, and monotone $\Delta$’s, the Zames–Falb multipliers are currently the most general class of time-invariant multipliers known for the study of these feedback systems (Carrasco, Heath, & Lanzon, 2013).

The Zames–Falb multipliers were first introduced by O’Shea in O’shea, 1966, O’shea, 1967 (see Carrasco, Turner, & Heath, 2016 for a survey), and formalized later by Zames and Falb in Zames and Falb (1967). Unfortunately, its applicability was limited due to computational constraints. Motivated by the rapid development of computational capability in the 80s, the multiplier-based theory has regained the interests of researchers. Importantly, the seminal work (Megretski & Rantzer, 1997) demonstrates that the Zames–Falb multipliers fit nicely to the framework of integral quadratic constrains (IQCs); see also Khong (2021). A recent research line on numerically searching the Zames–Falb multipliers satisfying a mixture of time and frequency domain conditions can be located in Carrasco et al., 2014, Turner et al., 2009 and Turner and Drummond (2019). The phase limitation of the Zames–Falb multipliers has been studied in Jönsson and Laiou, 1996, Megretski, 1995 and Wang, Carrasco, and Heath (2017), thereby providing a interesting perspective from which to understand the Zames–Falb multipliers.

It is well known (Zames & Falb, 1967) that the robust feedback stability of an LTI system $G$ to static monotone nonlinearities $\Delta$’s with a slope restriction $b$ can be ensured by the existence of an appropriate Zames–Falb multiplier $M\left(j\omega \right)$ satisfying

$$\mathrm{Re}\left\{M\left(j\omega \right)(b^{-1}-G\left(j\omega \right))\right\}>0,\forall \omega \in \mathbb{R}\cup \infty .$$

In Carrasco et al. (2016) and Wang et al. (2017), it is conjectured by Carrasco that if there is no appropriate Zames–Falb multiplier, then the feedback system is not robustly stable. The Carrasco’s conjecture remains unsolved, i.e. it is unclear whether condition (1) is necessary for the robust feedback stability over the class of static monotone $\Delta$’s. Partial discrete-time results on the conjecture can be found in Seiler and Carrasco, 2021, Zhang et al., 2020. The purpose of this work is to investigate both the necessity and sufficiency of the Zames–Falb multipliers in the continuous time over different uncertainty classes of $\Delta$’s, with the goal of enhancing our understanding of the conservatism of the Zames–Falb multipliers in the robust stability analysis of the feedback system in Fig. 1. The necessity of various robust stability conditions has been studied in control over the years, since the pioneering work on that of the small-gain theorem for LTI systems (Doyle, 1984, Zhou et al., 1996). Related converse results for IQCs can be found in Khong and Kao, 2020, Khong and van der Schaft, 2018.

This paper is concerned with the robust stability analysis of the feedback interconnection shown in Fig. 1 over three uncertainty classes of $\Delta$’s using the Zames–Falb multipliers. While one might be inclined to yearn for a condition that guarantees the robust feedback stability over as large a class of $\Delta$’s as possible, this is unwise from the perspective of establishing the necessity of the condition. In fact, there is a subtle trade-off between the size of the class of $\Delta$’s and the strictness of the condition on $G$ when it comes to robust stability, the latter of which obviously affects its necessity. Understandably, a larger class of $\Delta$’s leads to a stricter condition on $G$ in order to ensure robust stability. This is well illustrated by our results involving classes of $\Delta$’s of different sizes. Firstly, we identify a set of nonlinear dynamic $\Delta$’s and show that the existence of a Zames–Falb multiplier satisfying condition (1) is equivalent to the robust stability against this uncertainty set. This is established using the S-procedure lossless theorem from Jönsson, 2001, Megretski and Treil, 1993 and Yakubovich (2000). Secondly, when this set is restricted to be static, it coincides with the class of static monotone nonlinearities, and the existence of a Zames–Falb multiplier is sufficient for the robust stability whereas its necessity remains unknown. Thirdly, when the same set is restricted to consist of only LTI dynamics, it is shown to be a subset of the class of static monotone nonlinearities, and the existence of a Zames–Falb multiplier is sufficient but not necessary to establish the uniform stability. For the ease of exposition, the main results centered around monotone nonlinearities $\Delta$’s is presented in Section 2, and Section 3 is dedicated to its extension to the general two-sided slope restrictions. Specifically, in Section 3, the sufficiency direction considered in Zames and Falb (1967) is importantly generalized in order to make the condition consistent with the necessity direction. A crucial part of our results shows that the ‘monotonicity’ in the set of monotone static time-invariant nonlinearities is completely characterizable via the Zames–Falb IQCs. Therefore, while there exists a larger class of IQCs that such nonlinearities satisfy (Kulkarni & Safonov, 2002), using them does not deliver additional benefits as far as establishing robust feedback stability is concerned.

The remainder of this section sets up the notation and mathematical preliminaries to the rest of the paper. Some final remarks are described in Section 4.

Let $\mathbb{R}$ and $\mathbb{N}$ denote the sets of real numbers and non-negative integers, respectively. For a vector $v$, its Euclidean norm is denoted by $\left|v\right|$. Given a matrix $M$, the transpose and conjugate transpose are denoted respectively as $M^T$ and $M^{\ast }$. We use $\mathrm{Re}\left\{\lambda \right\}$ to denote the real part of a complex number $\lambda$.

Define ${\mathcal{L}}_{\mathrm{1}}(-\infty ,\infty )≔\left\{z:(-\infty ,\infty )\to \mathbb{R}\mid {\Vert z\Vert }_1≔{\int }_{-\infty }^{\infty }\left|z\left(t\right)\right|dt<\infty \right\}$. We use ${\mathcal{L}}_{\mathrm{1}}^{+}(-\infty ,\infty )$ to denote the set of all $z\left(t\right)\in {\mathcal{L}}_{\mathrm{1}}(-\infty ,\infty )$ satisfying $z\left(t\right)\ge 0,\forall t$. Given a signal $z\left(t\right)\in {\mathcal{L}}_{\mathrm{1}}(-\infty ,\infty )$, denote by $Z\left(j\omega \right)$ its Fourier transform and $\mathbf{Z}$ the convolution operator whose kernel is $z\left(t\right)$. Let ${\mathcal{L}}_{\mathrm{2}}[0,\infty )≔\left\{x:[0,\infty )\to \mathbb{R}\mid {\Vert x\Vert }^2≔{\int }_0^{\infty }{\left|x\left(t\right)\right|}^{2}dt<\infty \right\}$. Given signals $x\left(t\right),y\left(t\right)\in {\mathcal{L}}_{\mathrm{2}}[0,\infty )$, define their inner product $〈x,y〉≔{\int }_0^{\infty }x\left(t\right)y\left(t\right)dt$. The Fourier transform of $x\left(t\right)\in {\mathcal{L}}_{\mathrm{2}}[0,\infty )$ is denoted as $\hat{x}\left(j\omega \right)$. Define the truncation operator $\left(P_{T}x\right)\left(t\right)≔x\left(t\right)$ for $t\in [0,T]$ and $\left(P_{T}x\right)\left(t\right)≔0$ for $t>T$, and the extended space ${\mathcal{L}}_{\mathrm{2e}}≔\left\{x:[0,\infty )\to \mathbb{R}\mid P_{T}x\in {\mathcal{L}}_{\mathrm{2}}[0,\infty ),\forall T\in [0,\infty )\right\}$.

An operator $H:{\mathcal{L}}_{\mathrm{2e}}[0,\infty )\to {\mathcal{L}}_{\mathrm{2e}}[0,\infty )$ is said to be causal if $P_{T}H{P}_T=P_{T}H$ for all $T>0$ and anti-causal if $(I-P_T)H(I-P_T)=(I-P_T)H$ for all $T>0$. It is said to be static (a.k.a. memoryless) if it is simultaneously causal and anticausal. A causal operator $H:{\mathcal{L}}_{\mathrm{2e}}[0,\infty )\to {\mathcal{L}}_{\mathrm{2e}}[0,\infty )$ is said to be bounded if

$$\Vert H\Vert ≔\underset{T>0;\Vert P_{T}u\Vert \ne 0}{sup}\frac{\Vert P_{T}Hu\Vert }{\Vert P_{T}u\Vert }=\underset{0\ne u\in {\mathcal{L}}_{\mathrm{2}}}{sup}\frac{\Vert Hu\Vert }{\Vert u\Vert }<\infty .$$

Denote by ${\mathcal{L}}_{\infty }$ the set of transfer functions that are essentially bounded on the imaginary axis. Denote ${\mathcal{RH}}_{\infty }$ as the space of proper real-rational transfer functions with no poles in the closed right half plane. Every element $G\in {\mathcal{RH}}_{\infty }$ is associated with a causal bounded LTI operator $G:{\mathcal{L}}_{\mathrm{2e}}[0,\infty )\to {\mathcal{L}}_{\mathrm{2e}}[0,\infty )$, which we do not differentiate for notational convenience.

The main object of study in this work is the feedback interconnection of a $G\in {\mathcal{RH}}_{\infty }$ and a causal bounded $\Delta :{\mathcal{L}}_{\mathrm{2e}}\to {\mathcal{L}}_{\mathrm{2e}}$, as illustrated in Fig. 1. Denote the feedback system as $[G,\Delta ]$.

Definition 1

$[G,\Delta ]$ is said to be well-posed if the map $\left[\begin{array}{c}\hfill u_1\hfill \\ \hfill u_2\hfill \end{array}\right]\mapsto \left[\begin{array}{c}\hfill d_1\hfill \\ \hfill d_2\hfill \end{array}\right]$ in Fig. 1 has a causal inverse on ${\mathcal{L}}_{\mathrm{2e}}[0,\infty )$. It is said to be stable if it is well-posed and the inverse is bounded, in which case $[G,\Delta ]$ is also used to denote the map $\left[\begin{array}{c}\hfill d_1\hfill \\ \hfill d_2\hfill \end{array}\right]\mapsto \left[\begin{array}{c}\hfill u_1\hfill \\ \hfill u_2\hfill \end{array}\right]$.

For the feedback system $\left[G,\Delta \right]$ as shown in Fig. 1, its norm is defined as

$$\Vert \left[G,\Delta \right]\Vert ≔\underset{T>0;\Vert P_{T}d\Vert \ne 0}{sup}\frac{\Vert P_{T}u\Vert }{\Vert P_{T}d\Vert },$$

where $d≔\left[\begin{array}{c}\hfill d_1\hfill \\ \hfill d_2\hfill \end{array}\right]$ and $u≔\left[\begin{array}{c}\hfill u_1\hfill \\ \hfill u_2\hfill \end{array}\right]$.

We define uniform feedback stability as follows.

Definition 2

The feedback system $[G,\Delta ]$ is said to be uniformly stable over $\hat{\mathbf{\Delta }}$ if $[G,\Delta ]$ is stable for all $\Delta \in \hat{\mathbf{\Delta }}$, and there exists $\gamma$ such that

$$\underset{\Delta \in \hat{\mathbf{\Delta }}}{sup}\Vert [G,\Delta ]\Vert <\gamma .$$

This work is concerned with the uniform stability analysis of the interconnection system shown in Fig. 1 over three classes of $\Delta$’s using the Zames–Falb multipliers.

Definition 3

A pair $v,w\in {\mathcal{L}}_{\mathrm{2}}[0,\infty )$ is said to satisfy the IQC defined by $\Pi \in {\mathcal{L}}_{\infty }$ if

$${\int }_{-\infty }^{\infty }{\left[\begin{array}{c}\hfill \hat{v}\left(j\omega \right)\hfill \\ \hfill \hat{w}\left(j\omega \right)\hfill \end{array}\right]}^{\ast }\Pi \left(j\omega \right)\left[\begin{array}{c}\hfill \hat{v}\left(j\omega \right)\hfill \\ \hfill \hat{w}\left(j\omega \right)\hfill \end{array}\right]d\omega \ge 0.$$

A bounded causal system $\Delta$ is said to satisfy the IQC defined by $\Pi \in {\mathcal{L}}_{\infty }$, denoted by $\Delta \in \text{IQC}\left(\Pi \right)$, if (2) holds for all $v\in {\mathcal{L}}_{\mathrm{2}}[0,\infty )$ and $w=\Delta v$.