Phase limitations of Zames-Falb multipliers

TL;DR

This paper investigates phase limitations of Zames-Falb multipliers in continuous and discrete time, relating these limitations to the Kalman conjecture, and demonstrates their implications through classical and novel counterexamples.

Contribution

It generalizes a known phase limitation for continuous-time multipliers, introduces a new phase limitation for discrete-time multipliers, and explores their impact on stability criteria and the Kalman conjecture.

Findings

Existence of fourth-order plants without suitable Zames-Falb multipliers showing instability.

A new discrete-time phase limitation for Zames-Falb multipliers is established.

Discrete-time limitations imply no direct off-axis circle criterion counterpart.

Abstract

Phase limitations of both continuous-time and discrete-time Zames-Falb multipliers and their relation with the Kalman conjecture are analysed. A phase limitation for continuous-time multipliers given by Megretski is generalised and its applicability is clarified; its relation to the Kalman conjecture is illustrated with a classical example from the literature. It is demonstrated that there exist fourth-order plants where the existence of a suitable Zames-Falb multiplier can be discarded and for which simulations show unstable behavior. A novel phase-limitation for discrete-time Zames-Falb multipliers is developed. Its application is demonstrated with a second-order counterexample to the Kalman conjecture. Finally, the discrete-time limitation is used to show that there can be no direct counterpart of the off-axis circle criterion in the discrete-time domain.

Tables

Table 1. TABLE I: Various slope restrictions discussed in the text.

${\hat{k}}_{ZF}$	Maximum slope for which a Zames-Falb multiplier is known
$k_{ZF}$	Maximum slope for which there exists a Zames-Falb multiplier
$k_S$	Maximum slope for which the Lur’e system is absolutely stable
$k_{PL}$	Minimum slope for which phase limitation implies there is no
	Zames-Falb multiplier
${\hat{k}}_C$	Minimum slope for which a counterexample to absolute
	stability is known
$k_O$	Slope for direct discrete-time counterpart off-axis circle criterion
	(which is false)
$k_{RO}$	Slope for Reduced Off-axis circle criterion in [42]
$k_N$	Nyquist value

Table 2. TABLE II: Results of different slope restrictions (non-odd nonlinearity)

$k$	$k_{RO}$	${\hat{k}}_{ZF}$	${\hat{k}}_C$	$k_{PL}$	$k_O$	$k_N$
Result	0.8962	1.3028	1.3666	1.4603	3.61	3.61

Equations

$$\text{mbox}Re\{M(1+kG)\}>0,$$

$$\{\begin{array}{ll}v=f+Gw,& \\ w=-(\varphi v)+g.& \end{array}$$

$$0\le \frac{N(x_1)-N(x_2)}{x_1-x_2}\le k,\forall x_1,x_2\in \text{mathds}R,x_1\ne x_2.$$

$$k_N=\sup \{k>0:\underset{\omega }{\inf }\{|1+\tau kG|\}>0),\forall \tau \in [0,1]\},$$

$$0\le \frac{dN(x)}{dx}\le S,\forall x\in \text{mathds}R.$$

$$M(s)=1-H(s)$$

$${\int }_{-\infty }^{\infty }|h(t)|dt<1.$$

$$\text{mbox}Re\left\{M(j\omega )(1+kG(j\omega ))\right\}\ge \delta \text{mbox}forall\omega \in \mathbb{R}.$$

$$h(t)=h_a(t)+\sum _{i=1}^{\infty }{h}_i\delta (t_i).$$

$$M(s)=1-H(s)$$

$${\int }_{-\infty }^{\infty }h(t)dt<1.$$

$$M(s)=1-H(s)$$

$${\int }_{-\infty }^{\infty }|h(t)|dt<1.$$

$$M(z)=1-H(z)$$

$$\sum _{i=-\infty }^{\infty }|h_i|<1.$$

$$Re\left\{M(z)(1+kG(z))\right\}>0,\forall |z|=1.$$

$$\theta \le -\frac{1}{2}{\omega }_0+\frac{\pi }{2},$$

$$\theta \ge \frac{1}{2}{\omega }_0-\frac{\pi }{2}.$$

$$f(t)=g(t)+O(t^n),\text{mbox}ast\to 0.$$

$$|f(t)-g(t)|\le M{t}^n\text{mbox}on[0,t_0].$$

$$\lfloor \nu \rfloor =\max \{m\in \mathbb{Z}\mid m\le \nu \}.$$

$${\rho }^c=\underset{t>0}{\sup }\frac{|\psi (t)|}{\varphi (t)},$$

$${\rho }_{odd}^c=\underset{t>0}{\sup }\frac{|\psi (t)|}{\tilde{\varphi }(t)},$$

$$\psi (t)$$

$$\varphi (t)$$

$$\tilde{\varphi }(t)$$

$${\varphi }_1(t)=\frac{\lambda \sin (at)}{t}-\frac{\lambda \sin (bt)}{t}+\frac{\kappa \mu \sin (ct)}{t}-\frac{\kappa \mu \sin (dt)}{t}.$$

$$\frac{\lambda }{\mu }=\frac{d^2-c^2}{b^2-a^2},$$

$$\frac{\psi (t)}{\varphi (t)}=\gamma t+O(t^3)\text{mbox}and\frac{\psi (t)}{\tilde{\varphi }(t)}=\gamma t+O(t^3)\text{mbox}ast\to 0,$$

$$\gamma =-\frac{1}{4}\frac{\lambda (b^4-a^4)-\mu (d^4-c^4)}{\lambda (b^3-a^3)+\kappa \mu (d^3-c^3)}.$$

$$\text{mbox}Im(M(j\omega ))$$

Full text

I Introduction

The absolute stability of a negative feedback interconnection between an LTI system $G$ and a nonlinearity $\phi$ with a slope restriction $k$ has aroused the interests of many researchers. The stability tests include the circle criterion, Popov criterion [1], [2], and off-axis circle criterion [3], [4] in continuous time and the circle criterion [5], Tsypkin criterion [6] and Jury-Lee criterion [7], [8] in discrete time. For a recent discussion, see [9] and [10]. Apart from these, loop transformation and multiplier theory are both important tools to establish the stability of feedback interconnections. The Zames-Falb multipliers are a class of multipliers with the property of preserving the positivity of monotone and bounded nonlinearities, and hence of slope-restricted nonlinearities after loop transformation. The class of Zames-Falb multipliers can be defined in either continuous time [11, 12] or discrete time [13, 14]. Specifically, after loop transformation, the stability of the negative interconnection between an LTI system $G$ and a nonlinearity $\phi$ with a slope restriction $k$ is guaranteed if there exists a Zames-Falb multiplier $M$ such that

[TABLE]

with $M$ and $G$ evaluated over all frequencies. That is to say, at $j\omega$, $\omega\in\mathds{R}$ for continuous-time systems and at $e^{j\omega}$, $\omega\in[0,2\pi]$ for discrete-time systems.

The Zames-Falb multipliers may be considered a classical tool [15]. Nevertheless, there has been considerable recent interest, largely sparked by the availability of numerical searches ([16], [17], [18], [19], [20], [21], [22], for continuous time; [23], [24] for discrete time) and their encapsulation within an IQC (integral quadratic constraint) framework [25], [26], [27], [28]. There has also been interest in generalising the class, both to MIMO (multi-input, multi-output) nonlinearities [29], [30], [31], [32], [33] and to nonlinearities outside the original classes considered by Zames and Falb [34], [35], [28], [36]. In addition to determining stability conditions, they can be used to analyse performance [37], [38]; further, they can be used to obtain tighter versions of the Popov criterion [39]. Applications of Zames-Falb multipliers range from input-constrained model predictive control [40] to first order numerical optimisation algorithms [41].

Although both continuous-time and discrete-time Zames-Falb multipliers are defined with similar conditions, there are clear distinctions between their properties. In discrete time the Zames-Falb multipliers are the full set of multipliers preserving the positivity of monotone and bounded nonlinearities, besides direct phase substitutions [14], [43]. In continuous time matters are more nuanced, but the class of Zames-Falb multipliers remains the widest known class of multipliers preserving the positivitiy of monotone and bounded nonlinearities, up to phase equivalence [39, 44]. For a tutorial introduction to the phase properties of continuous-time Zames-Falb multipliers, phase-equivalence results and the issues associated with causality, see [45]. Phase properties are essential to our understanding of Zames-Falb multipliers.

For example, if $k<k_{N}$ (see Table I for various slope restrictions discussed in this paper), then the phase of $(1+kG)$ lies between $-180^{\circ}$ and $180^{\circ}$. Meanwhile multipliers must be positive so are restricted to lie between $-90^{\circ}$ and $90^{\circ}$. But as the Kalman conjecture is false, any set of suitable multipliers must be restricted by some further fundamental limitations. This follows from the obvious but important fact:

Fact I.1**.**

*If the system is not absolutely stable, there can be no appropriate Zames-Falb multiplier. *

However, only a few papers discuss such limitations. Megretski [46] shows that there exists a phase limitation for continuous-time Zames-Falb multipliers. Another phase limitation of Zames-Falb multipliers is given by Jönsson and Laiou [47, 48]. Such limitations are often ignored when new searches for multipliers are presented (see for example [45] and references therein). Often only $k_{N}$ is provided as an upper limit for the slope restriction.

We discuss Megretski’s phase restriction [46] with respect to a fourth-order continuous-time plant whose phase drops from $+180^{\circ}$ to $-180^{\circ}$; similarly with $k$ sufficiently big the phase of $1+kG$ drops from above $+90^{\circ}$ degrees to below $-90^{\circ}$. The limitation cannot be applied to first, second or third-order plants whose phase is in the range $(-180,90)$ degrees or $(-90,180)$ degrees; this agrees with the well-known result that the Kalman conjecture is true for such plants [49].

To the best of authors’ knowledge, no similar limitation has been developed in the discrete-time domain. Since there exist second-order discrete-time counterexamples to the Kalman conjecture whose phase is in the range $(-180,0)$ degrees [50, 51] one might expect a simpler limitation for discrete-time multipliers; this turns out to be indeed the case.

The contribution of this paper is for both continuous-time and discrete-time multipliers. We generalise Megretski’s limitation [46] for continuous-time multipliers to a wider choice of frequency intervals. Further, we show that Megrestki’s limitation [46] only applies for the class of Zames-Falb multipliers which do not require the odd condition on the nonlinearity; we provide the corresponding result when the nonliearity is odd. We discuss the limitation’s numerical calculation and demonstrate its application in the context of a classical example due to O’Shea [11, 45]. In particular we demonstrate a fourth-order counterexample of the Kalman conjecture for which the constraint is active. A further contribution of the paper is the development of a phase limitation for discrete-time Zames-Falb multipliers. This limitation is fundamentally different to Megrestki’s limitation as it only requires the phase of $(1+kG)$ to be either in the interval $(90,180)$ degrees or in the interval $(-180,-90)$ degrees. The limitation is easy to compute, and is active for a second-order discrete-time counterexample of the Kalman conjecture. This close link between the preclusion of a Zames-Falb multiplier and unstable behaviour leads us to the following conjecture as the counterpart to Fact I.1; however no proof (or counterexample) is offered in this paper:

Conjecture I.2**.**

*If there is no appropriate Zames-Falb multiplier, the system is not absolutely stable. *

One direct application of the phase limitation is to show there can be no direct discrete-time counterpart of the off-axis circle criterion. The continuous-time off-axis circle criterion is a useful graphical stability test and is shown to be a less conservative criterion compared to the circle criterion [3], [4]. The derivation is based on the phase properties of RL/RC multipliers. The direct discrete-time counterpart of the off-axis circle criterion is sometimes assumed to be true in the literature (e.g. [52], [53]). However only a highly restrictive discrete-time version is proposed in [42], without discussion as to whether the direct discrete-time counterpart off-axis circle criterion is true or false. In this paper, we show that in some cases there are no Zames-Falb multipliers with the requisite phase properties for its derivation - i.e. the direct counterpart off-axis circle criterion cannot be derived using multiplier theory. The invalidation is completed by counterexample.

Some preliminary results related with Theorem IV.3 part (i) were presented in [54].

II Notation and Preliminary results

II-A Signal spaces

For continuous-time signals let $\mathcal{L}_{2}[0,\infty)$ be the Hilbert space of square integrable and Lebesgue measurable functions $f:[0,\infty)\rightarrow\mathds{R}$ and let $\mathcal{L}_{2}$ be defined similarly for $f:\mathds{R}\rightarrow\mathds{R}$. Let $\mathcal{L}_{2e}[0,\infty)$ be the extended space of $\mathcal{L}_{2}[0,\infty)$ [43].

For discrete-time signals let $\mathds{Z}$ and $\mathds{Z^{+}}$ be the set of integer numbers and positive integer numbers including [math], respectively. Let $\ell$ be the space of all real-valued sequences, $h:\mathds{Z}^{+}\rightarrow\mathds{R}$ and let $\ell_{2}$ denote the Hilbert space of all square-summable and measurable real sequences $f:\mathds{Z}^{+}\rightarrow\mathds{R}$ ($\ell$ is the extended space of $\ell_{2}$). Similarly, we can define the Hilbert space $\ell_{2}(\mathds{Z})$ by considering real sequences $f:\mathds{Z}\rightarrow\mathds{R}$.

II-B Lur’e problem and the Kalman conjecture

The feedback interconnection system is a Lur’e system represented in Fig. 2 with both $G$ and $\phi$ mapping $\mathcal{L}_{2e}[0,\infty)\rightarrow\mathcal{L}_{2e}[0,\infty)$ (continuous time) or $\ell\rightarrow\ell$ (discrete time). The object $G$ is assumed LTI stable and the object $\phi$ memoryless and slope-restricted (see below). The interconnection relationship is

[TABLE]

The system (2) is well-posed if the map $(v,w)\mapsto(g,f)$ has a causal inverse on $\ell\times\ell$, and this feedback interconnection is $\ell_{2}$-stable if for any $f,g\in\ell_{2}$, both $w,v\in\ell_{2}$.

Definition II.1**.**

(Memoryless slope-restricted nonlinearity) The nonlinearity $\phi:\mathcal{L}_{2e}[0,\infty)\rightarrow\mathcal{L}_{2e}[0,\infty)$ or $\phi:\ell\rightarrow\ell$ is said to be memoryless and slope-restricted in $S[0,k]$, if there is a function $N:\mathds{R}\rightarrow\mathds{R}$ such that $(\phi u)(t)=N(u(t))$ or $(\phi u)(k)=N(u(k))$, $N(0)=0$, and

[TABLE]

*In addition, $\phi$ is said to be odd if $N$ is odd, i.e. $N(x)=-N(-x)$, for all $x\in\mathds{R}$. *

We define the Nyquist value and state the Kalman conjecture for both continuous-time and discrete-time systems.

Definition II.2** (Nyquist value).**

Given a stable LTI system $G$, the Nyquist value $k_{N}$ is the supremum of all the positive real numbers $k$ such that $\tau kG$ satisfies the Nyquist Criterion for all $\tau\in[0,1]$. It can also be expressed as:

[TABLE]

*with $G$ evaluated over all frequencies (i.e. $\omega\in\mathbb{R}$ for continuous-time systems and $\omega\in[0,2\pi]$ for discrete-time systems). *

Conjecture II.3** (Kalman Conjecture, [55]).**

Let $\phi$ be a memoryless slope-restricted nonlinearity such that there exists a continuously differentiable $N:\mathds{R}\rightarrow\mathds{R}$ and $S>0$ such that $\phi(v)(t)=N(v(t))$ (or $\phi(v)(k)=N(v(k))$) and

[TABLE]

*Then the negative feedback interconnection of the continuous-time (or discrete-time) LTI systems $G\sim[A,B,C,0]$ and $\phi$ (Fig 2) is globally asymptotically stable if $A-BCk$ is Hurwitz (Schur) for all $k\in[0,S]$. *

There exist fourth-order continuous-time counterexamples to the Kalman conjecture [56, 49, 57] and second-order discrete-time counterexamples [50, 51].

II-C Zames-Falb multipliers

The characteristics of continuous-time Zames-Falb multipliers is given in the following theorem that defines two different classes of multipliers.

Theorem II.4**.**

(Continuous-time Zames-Falb multipliers, [12].) Consider the continuous-time feedback system in Fig. 2 with $G$ a stable LTI system and $\phi$ memoryless and slope-restricted in $S[0,k]$. Suppose that there exists an LTI multiplier $M:\mathcal{L}_{2}\rightarrow\mathcal{L}_{2}$ whose transfer function has the form

[TABLE]

such that the impulse response $h$ of $H$ satisfies

[TABLE]

Moreover, let us assume that either $\phi$ is odd or $h(t)>0$. Suppose further there is some $\delta>0$ such that

[TABLE]

Then the feedback interconnection (2) is $\mathcal{L}_{2}$-stable.  ∎

Remark II.5**.**

With some abuse of notation, we denote $h(t)$ as the addition of a real-valued function $h_{a}(t)$ and impulses at different instants, i.e.

[TABLE]

Definition II.6**.**

The class of continuous-time Zames-Falb multipliers $\mathcal{M}^{c}$ is defined as the LTI systems $M:\mathcal{L}_{2}\rightarrow\mathcal{L}_{2}$ whose transfer function has the form

[TABLE]

such that the impulse response $h$ of $H$ satisfies that $h(t)\geq 0$ for all $t$ and

[TABLE]

Definition II.7**.**

The class of continuous-time “odd” Zames-Falb multipliers $\mathcal{M}^{c}_{\text{odd}}$ is defined as the LTI systems $M:\mathcal{L}_{2}\rightarrow\mathcal{L}_{2}$ whose transfer function has the form

[TABLE]

such that the impulse response $h$ of $H$ satisfies

[TABLE]

By definition, $\mathcal{M}^{c}\subset\mathcal{M}^{c}_{\text{odd}}$.

The counterpart result in discrete time is given in the following theorem and it also defines two different classes of multipliers:

Theorem II.8**.**

(Discrete-time Zames-Falb multipliers, [14], [43]) Consider the discrete-time feedback system in Fig. 2 with $G$ a stable LTI system and $\phi$ memoryless and slope-restricted in $S[0,k]$. Suppose that there exists an LTI multiplier $M:\ell_{2}\rightarrow\ell_{2}$ whose transfer function has the form

[TABLE]

such that the impulse response $h$ of $H$ satisfies that $h_{0}=0$ and

[TABLE]

Moreover, let us assume that either the nonlinearity is odd or $h_{i}\geq 0$. Suppose further

[TABLE]

Then the feedback interconnection (2) is $\ell_{2}$-stable.  ∎

Remark II.9**.**

*Inequality (8) is evaluated over $\omega\in\mathbb{R}$ whereas inequality (16) is evaluated over the frequency interval $\omega\in[0,2\pi]$. Hence, by the Extreme Value Theorem [58], it is unnecessary to define any $\delta>0$ for the discrete case corresponding to that used in the continuous case. *

Similarly to the previous definitions, we can define the classes of multipliers $\mathcal{M}^{d}$ and $\mathcal{M}^{d}_{\text{odd}}$.

II-D Off-axis circle criterion

The continuous-time off-axis circle criterion is given.

Lemma II.10**.**

**(Off-axis circle criterion for continuous-time

systems, [3])*** Consider the feedback system in Fig. 2 with $G$ LTI stable and $\phi$ is slope-restricted in $S[0,k]$. Suppose that the Nyquist plot of the linear part of the system $G(j\omega)$ lies entirely to the right of a straight line passing through the point $(-\frac{1}{k}+\delta,0)$ where $\delta>0$ and $\phi$ is monotonically increasing. Then the feedback interconnection (2) is $\mathcal{L}_{2}$-stable. *

For discrete time, only a highly restrictive version is proposed.

Lemma II.11**.**

(Reduced off-axis circle criterion for discrete-time systems, [42])* Let the Nyquist plot of $G(e^{j\omega})$ for all $0\leq\omega\leq\pi$ lie entirely to the right of a straight line, whose slope $k$ is nonnegative passing through $(-\frac{1}{K_{2}},0)$. Let $\omega_{0}$ be such that $\mbox{Re }G(e^{j\omega_{0}})=-\frac{1}{K_{2}}$ and $\mbox{Re }G(e^{j\omega})\geq-\frac{1}{K_{2}}$ for $\omega\geq\omega_{0}$ and $\mbox{Im }G(e^{j\omega})\leq 0$ for $\omega_{0}\geq\omega\geq 0$. Then the system is asymptotically stable for all monotone $\phi$ with slope restriction $K_{2}$ in the feedback path if*

[TABLE]

where $\theta$ is the angle made by the straight line and the imaginary axis, i.e., $\theta=\cot^{-1}k$. If $\mbox{Im }G(e^{j\omega})\geq 0$ for $\omega_{0}\geq\omega\geq 0$, the same argument can be used to prove the asymptotic stability of the system with nonpositive $k$ and

[TABLE]

II-E Further mathematical notation

For the convenience of solving potential numerical issues, the notation of $O(\cdot)$ is given.

Definition II.12**.**

The condition

[TABLE]

means that there exist $M$ and $t_{0}$ such that

[TABLE]

The floor function, denoted by $\lfloor\nu\rfloor$, is defined by

[TABLE]

III Continuous phase limitations and the Kalman conjecture

Megretski presents in [46] a phase limitation for continuous-time Zames-Falb multipliers. In this section we generalise the result to a wider set of frequency intervals, and derive separate results for both $\mathcal{M}^{c}$ and $\mathcal{M}^{c}_{odd}$. Although it is stated in [46] that the result there is valid for $\mathcal{M}^{c}_{odd}$ (in the terminology of this paper) we show by counterexample that it is in fact valid for $\mathcal{M}^{c}$ only. Finally, we bridge the limitation of [46] with the Kalman conjecture; this is the key motivation to develop a different set of phase limitations for the discrete-time Zames-Falb multipliers.

III-A Phase limitations

Definition III.1**.**

Let $0<a<b<c<d$, $\kappa>0$, $\lambda>0$ and $\mu>0$. Define

[TABLE]

and

[TABLE]

where

[TABLE]

and

[TABLE]

with

[TABLE]

Lemma III.2**.**

If $\lambda$ and $\mu$ are chosen such that

[TABLE]

*then $\rho^{c}$ and $\rho^{c}_{odd}$ in Definition III.1 are well-defined; that is to say $\rho^{c}<\infty$ and $\rho^{c}_{odd}<\infty$. *

Proof:

See Appendix.

Remark III.3**.**

The direct calculation of the ratios $\psi(t)/\phi(t)$ and $\psi(t)/\tilde{\phi}(t)$ is numerically ill-conditioned for small $t$ since, with the choice (28), we have $\psi(t)=0+O(t^{3})$, $\phi(t)=0+O(t^{2})$ and $\tilde{\phi}(t)=0+O(t^{2})$, all as $t\rightarrow 0$. Nevertheless, the same construction ensures we can write

[TABLE]

with

[TABLE]

*We use this relation for small $t$ in the numerical examples below. *

Theorem III.4** (Continuous-time phase limitations).**

Let $M$ be a continuous-time Zames-Falb multiplier. Suppose

[TABLE]

and

[TABLE]

for some $\rho>0$. Then under the conditions of Lemma III.2.

1. (i)

$\rho<\rho^{c}$* if $M\in\mathcal{M}^{c}$,* 2. (ii)

$\rho<\rho^{c}_{odd}$* if $M\in\mathcal{M}^{c}_{odd}$.*

Proof:

See Appendix.

Lemma III.2 and Theorem III.4, with the choice $\kappa=1$, $c=ra$, $d=rb$ and hence $\lambda/\mu=r^{2}$, are in [46]. An interpretation of Theorem III.4 with these values is illustrated in Fig. 3 (see also [45]). According to the constraints on the coefficients of continuous Zames-Falb multipliers, if the phase is simultaneously greater than $\tan^{-1}\rho$ on $\omega\in[a,b]$ (in Region I) and smaller than $-\tan^{-1}\rho$ on $\omega\in[ra,rb]$ (in Region II), then $\rho<\rho^{c}$ if $M\in\mathcal{M}^{c}$ and $\rho<\rho^{c}_{odd}$ if $M\in\mathcal{M}^{c}_{odd}$.

Remark III.5**.**

*It is straightforward to produce the counterpart of Theorem III.4 with (31) and (32) replaced by $\mbox{Im}(M(j\omega))<-\rho\mbox{Re}(M(j\omega))$ for all $\omega\in[a,b]$ and $\mbox{Im}(M(j\omega))>\kappa\rho\mbox{Re}(M(j\omega))$ for all $\omega\in[c,d]$ respectively. *

Remark III.6**.**

In [46], Megretski uses a positive sign in the exponential of the Laplace transform:

[TABLE]

*This is the standard convention in the Physics literature (see for example [59]) but opposite to that used in [12]. The apparent discrepancy has no significant consequence for the analysis of phase limitations since if $M(s)$, with impulse response $m(t)$, is a Zames-Falb multiplier then $M(-s)$, with impulse response $m(-t)$, is also a Zames-Falb multiplier. *

It is natural to ask whether a phase limitation over a single frequency range can be constructed in a similar manner. This is not possible in continuous time, as any corresponding definition of $\rho^{c}$ or $\rho^{c}_{odd}$ would be unbounded as $t$ approaches [math]. Loosely speaking, we can generate a multiplier in $\mathcal{M}^{c}$ with phase arbitrarially close to $\pm 90^{\circ}$ over an arbitrararily large frequency inteterval by selecting $h(t)=(1-\epsilon)\delta(t-t^{*})$ with $\epsilon>0$ arbitrarially close to [math] and $t^{*}$ arbitrarially close to 0. But we construct such phase limitations for discrete-time multipliers below, in Section IV.

III-B *Numerical example *

Here we illustrate Theorem III.4 with a numerical example. Let $a=1.6$ and $b=2.25$. Let $\kappa=1$, $c=ra$ and $d=rb$ with $r=2.1$. Then a sweep over time intervals followed by local numerical search gives

[TABLE]

and

[TABLE]

Now consider the multiplier

[TABLE]

with $h(t)=-0.9\delta(t+1)$. Figure 4 shows that the relations (31) and (32), or equivalently

[TABLE]

and

[TABLE]

are satisfied simultaneously for $\rho=\rho^{c}$ but not for $\rho=\rho^{c}_{odd}$. This is consistent with Theorem III.4 as $M\in\mathcal{M}^{c}_{odd}$ but $M\notin\mathcal{M}^{c}$. It is a counterexample to the false claim in [46] that the phase limitation of Theorem III.4 part (i) is applicable to the wider class $M\in\mathcal{M}^{c}_{odd}$.

Remark III.7**.**

Both in this numerical example and at the end of Section III-A we consider multipliers where $h$ takes the form $h(t)=(1-\varepsilon)\delta(t-\tau)$ for some $\tau$. There is a close link with Theorem III.4. Specifically if $|\psi(\tau)|/\phi(\tau)=\rho^{c}$ and $\varepsilon\rightarrow 0$ then

[TABLE]

Compare (80) in the proof of Theorem III.4. Similarly if $|\psi(\tau)|/\tilde{\phi}(\tau)=\rho^{c}_{odd}$ then

[TABLE]

*We discuss the corresponding relations at greater length in Section IV-B for the discrete-time case. *

III-C Counterexamples to Kalman conjecture via phase limitations

It is instructive to interpret the phase limitations of Theorem III.4 in a manner consistent with known results about the Kalman conjecture.

On the one hand, it is well-known that first, second and third order plants hold the Kalman conjecture [49]. The phase of such plants cannot reach both Regions I and II in Fig. 3. So the phase limitations cannot apply to these plants. First-order plants do not require a dynamic multiplier, second-order plants require a dynamics multiplier with a tunable zero and a pole at infinity, i.e. a Popov multiplier, and-third order plant requires both a tunable pole and zero, i.e. first order RL/RC multipliers. In all these cases, only a first-order multiplier is required, and we know that there is no phase limitation in the selection of such multipliers [39].

Remark III.8**.**

Although the off-axis circle criterion is also based on RL/RC multipliers, it is not sufficient to show all third-order plants hold the Kalman conjecture. For example, the off-axis circle criterion with the plant

[TABLE]

*guarantees stability with $k<3.928$, whereas the multiplier $M=(s+1)/(s+\epsilon)$ guarantees stability for any positive $k$ with a sufficiently small value $\epsilon>0$. *

On the other hand the phase limitations may be applied to fourth-order plants, and these in turn may be counterexamples to the Kalman conjecture by: a) showing numerically that a phase limitation can be applied to a well-known plant, and b) showing that the Lur’e system with this plant and a slope-restricted nonlinearity may be unstable.

Specifically we will consider the phase limitation of Theorem III.4 part (i) with $\kappa=1$, $c=ra$, and $d=rb$; that is to say the original result of [46] applied to $\mathcal{M}^{c}$. A particularly suitable example to show this limitation is O’Shea example [11, 45]:

[TABLE]

since the symmetry of the problem simplifies the selection of the parameters. In this example, O’Shea showed that there is a Zames-Falb multiplier for any $k$ if $\xi>0.5$. The following result shows that it is not possible to reach an arbitrary large $k$ for any $\xi\leq 0.25$. For the case $\xi=0.25$, the phase of $G(s)$ is above $177.98^{\circ}$ over the interval $[a,b]$ where $a=0.02249$ and $b=0.03511$; hence it is below $-177.98^{\circ}$ over the interval $[1/b,1/a]$ by using the symmetry of the plant. Then a suitable Zames-Falb multiplier for this plant would require a phase below $-87.98^{\circ}$ over the interval $[a,b]$ and above $87.98^{\circ}$ over the interval $[1/b,1/a]$. The phase limitation ensures that there is no Zames-Falb multiplier with such a phase characteristic, since $\tan^{-1}\rho^{c}\approx 87.79^{\circ}$. Strictly speaking, we have used the counterpart of Theorem III.4 mentioned in Remark III.5.

Although numerical reliability can be problematic in the discussion of the Kalman conjecture [57], simulations of the plant with asymmetrical saturation show a time evolution that does not appear to settle to zero, supporting the validity of Conjecture I.2. The simulation shown in Figure 5 has been run in MATLAB R2013, using the solver ode45, with maximum step size of 0.0001 s, and relative tolerance of $10^{-3}$. The nonlinearity $\phi$ is described by the nonlinear function

[TABLE]

the input $g$ is given by

[TABLE]

and $f(t)=0$. The relevance of this counterexample to the Kalman conjecture is that we can show that there is no Zames-Falb multiplier with $h(t)\geq 0$ for the system. The asymmetry of the nonlinearity seems to be a key factor as simulations with symmetric saturations show stable behaviour. The importance of asymmetry in the stability of Lur’e systems with saturation has been discussed recently [36, 60].

Remark III.9**.**

*The magnitude of the response in Fig. 5 is bounded. Since the plant $G$ is stable and $\phi$ is sector-bounded it follows that if $g\in\mathcal{L}_{\infty}$ then all signals must be in $\mathcal{L}_{\infty}$. *

IV Discrete-time phase limitation

In this section we develop phase limitations for discrete-time Zames-Falb multipliers. Their derivation is in the spirit of Megretski’s limitation [46] and Theorem III.4 for continuous-time multipliers. However their properties are simpler and consistent with the existence of second-order discrete-time counterexamples to the Kalman conjecture [50, 51]. In particular, and by contrast with their continuous-time counterparts, they are concerned with properties over a single interval $\omega\in[a,b]$.

It is worth highlighting that if a discrete-time multiplier preserves the positivity of all monotone and bounded nonlinearities then either it is a Zames-Falb multiplier or there exists a Zames-Falb multiplier with the same phase [14], [43]. Hence any phase limitation on the discrete-time Zames-Falb multipliers is also a limitation for any discrete-time multiplier.

IV-A Phase limitations

Definition IV.1**.**

Let $0\leq a<b\leq\pi$. Define

[TABLE]

and

[TABLE]

where

[TABLE]

and

[TABLE]

with

[TABLE]

Lemma IV.2**.**

*Both $\rho^{d}$ and $\rho^{d}_{odd}$ in Definition IV.1 are well-defined; that is to say $\rho^{d}<\infty$ and $\rho^{d}_{odd}<\infty$. *

Proof:

See Appendix.

Theorem IV.3** (Discrete-time phase limitations).**

Let $M$ be a discrete-time Zames-Falb multiplier. Suppose

[TABLE]

for some $\rho>0$. Then

1. (i)

$\rho<\rho^{d}$* if $M\in\mathcal{M}^{d}$,* 2. (ii)

$\rho<\rho^{d}_{odd}$* if $M\in\mathcal{M}^{d}_{odd}$.*

Proof:

See Appendix.

An interpretation of Theorem IV.3 is illustrated in Fig. 6. According to the constraints on the coefficients of discrete-time Zames-Falb multipliers, if the phase is greater than $\tan^{-1}\rho$ on $\omega\in[a,b]$ (in Region A), then $\rho<\rho^{d}$ if $M\in\mathcal{M}^{d}$ and $\rho<\rho^{d}_{odd}$ if $M\in\mathcal{M}^{d}_{odd}$.

Remark IV.4**.**

It is straightforward to produce the counterpart of Theorem IV.3 with (51) replaced by

[TABLE]

An algorithm for finding the phase limitation in Theorem IV.3 part (i) for a second order plant is given in [54]. For a given stable plant $G$ and a value of $k$ such that $0<k<k_{N}$ the phase of an ideal multiplier is obtained as

[TABLE]

Then the algorithm increases $k$ until the existence of such a multiplier can be discarded by using the limitation presented in Theorem IV.3.

IV-B Integral bound and sparsely parametrized multipliers

Theorem IV.3 gives relative bounds on the real and imaginary parts of a Zames-Falb multiplier’s frequency response over an interval $[a,b]$. It is straightforward to derive a closely related result in terms of the integrals over the same interval.

Theorem IV.5**.**

Let $M$ be a discrete-time Zames-Falb multiplier. Suppose

[TABLE]

for some $\rho>0$. Then

1. (i)

$\rho<\rho^{d}$* if $M\in\mathcal{M}^{d}$,* 2. (ii)

$\rho<\rho^{d}_{odd}$* if $M\in\mathcal{M}^{d}_{odd}$.*

Proof:

See Appendix.

Remark IV.6**.**

*Theorem IV.5 is stronger than Theorem IV.3 in the sense that condition (51) is sufficient for condition (54) but not necessary. Theorem IV.3 may be derived as a Corollary of Theorem IV.5 by applying the Mean Value Theorem [58]. *

Theorem IV.5 gives a tight phase limitation in the sense that we can associate a set of sparsely parameterized multipliers with Theorem IV.5 as follows.

Proposition IV.7**.**

---

1. (i)

For a given $a$ and $b$, define the set $\mathcal{N}^{d}\subset\mathbb{Z}$ as the set of integers $n$ such that $\psi_{d}(n)/\phi_{d}(n)=\rho^{d}$. Then multipliers of the form

[TABLE]

with

[TABLE]

satisfy (54) with $\rho$ arbitrarily close to $\rho^{d}$ in the limit as $\varepsilon\rightarrow 0$. 2. (ii)

For a given $a$ and $b$, define the set $\mathcal{N}^{d}_{odd}\subset\mathbb{Z}$ as the set of integers $n$ such that $\psi_{d}(n)/\tilde{\phi}_{d}(n)=\rho^{d}_{odd}$. Then multipliers of the form

[TABLE]

with

[TABLE]

satisfy (54) with $\rho$ arbitrarily close to $\rho^{d}_{odd}$ in the limit as $\varepsilon\rightarrow 0$.

Proof:

See Appendix.

Remark IV.8**.**

It is, once again, straightforward to produce the counterpart of Theorem IV.5 with (54) replaced by

[TABLE]

*Similarly for Theorem IV.7 with (i) $\psi_{d}(n)/\phi_{d}(n)=-\rho^{d}$ and (ii) $\psi_{d}(n)/\tilde{\phi}_{d}(n)=-\rho^{d}_{odd}$. *

As an illustrative example, suppose $a=0.7$ and $b=0.77501$ (approx.). Then

[TABLE]

and

[TABLE]

Fig 7 shows the phases of the limiting cases $M(z)=1-z^{8}$, $M(z)=1-z^{-9}$ and linear combinations of the form $M(z)=1-\lambda z^{8}-(1-\lambda)z^{-9}$ with $0<\lambda<1$. It can be seen that the phases are near to $\tan^{-1}\rho^{d}$ over the interval $[a,b]$. However they always have values both above and below, indicating that Theorem IV.3 is not tight in the same sense as Theorem IV.5.

Remark IV.9**.**

*A similar analysis is possible for continuous-time multipliers. Compare Remark III.7. *

IV-C Discrete-time counterparts of the off-axis circle criterion

The off-axis circle criterion [3] (Theorem II.10) is a useful frequency-based graphical stability test for continuous-time systems. It is sometimes assumed (e.g. in [52]) that its discrete-time counterpart is true. We state this as a conjecture:

Conjecture IV.10**.**

*Consider the feedback system in Fig. 2 with $G\in\mathbf{RH}_{\infty}$, and $\phi$ is slope-restricted in $S[0,k]$. Suppose that the Nyquist plot of the linear part of the system $G(e^{j\omega})$ lies entirely to the right of a straight line passing through the point $(-\frac{1}{k}+\delta,0)$ where $\delta>0$ and $\phi$ is monotonically increasing. Then the feedback interconnection (2) is $\ell_{2}$-stable. *

A geometrical interpretation of both Theorem II.10 for continuous-time systems and Conjecture IV.10 for discrete-time systems is given in Fig 8.

The phase-limitation on discrete-time Zames-Falb multipliers carries the implication that there can be no multiplier construction corresponding to that for RL/RC multipliers of [42] used to prove Theorem II.10. We summarise the argument as follows:

Under the conditions of Conjecture IV.10 there is some $\theta$ in $(-90,90)$ degrees such that the phase of $1+kG$ always lies in the interval $(-90-\theta,90-\theta)$ degrees. Hence an ideal LTI multiplier with constant phase $\theta$ would render the real part of $M(1+kG)$ positive over all frequencies. 2. 2.

In their proof of the continuous off-axis circle criterion Cho and Narendra [3] show that it is possible to construct RL/RC multipliers whose phase is arbitrarily close to some constant $\theta$ degrees over an arbitrarily large interval. We show that for some values of $\theta$ this may not be possible for any discrete-time LTI multiplier. 3. 3.

If a discrete-time LTI multiplier preserves the positivity of a slope-restricted nonlinearity then there is a Zames-Falb multiplier with the same phase [14], [43], so we can limit our set of multipliers to the class of LTI Zames-Falb multipliers. 4. 4.

If $\theta>\tan^{-1}(2/\pi)\approx 32.48^{\circ}$ then Theorem IV.3 precludes any such construction of a Zames-Falb multiplier since if $a\rightarrow 0^{+}$ and $b\rightarrow\pi^{-}$ then $\rho^{d}\rightarrow\tan^{-1}(-2/\pi)$.

Hence the phase limitation can be used to invalidate Conjecture IV.10 when $\theta>\tan^{-1}(2/\pi)$. Smaller values can be obtained by using different values of $a$ and $b$. It follows that any counterpart of the off-axis circle criterion in discrete-time must take into account specific information about frequency intervals. This is true of the more limited result originally derived by Narendra and Cho [42] (Theorem II.11). In fact it can be shown that the counterexamples to the Kalman conjecture of [50] and [51] are also counterexamples to Conjecture IV.10.

IV-D Finite search in discrete-time domain

Here we provide a result which simplifies the numerical implementation. Although the definitions of $\rho^{d}$ and $\rho^{d}_{odd}$ are given with an infinite number of terms, they can be calculated using a finite number $n=n_{N}$ given in Lemma IV.11.

Lemma IV.11**.**

Let $0\leq a<b\leq\pi$, then

[TABLE]

and

[TABLE]

with

[TABLE]

where

[TABLE]

Proof:

See Appendix.

Suppose we wish to find a phase limitation over the interval $\omega\in[0.7,0.75]$. Applying Lemma IV.11 we find $\nu=55.2$ and hence $|\psi_{d}(n)|/\phi_{d}(n)<|\psi_{d}(1)|/\phi_{d}(1)$ for all $n>n_{N}$, with

[TABLE]

Hence it is sufficient to search over the integers $1\leq n\leq 55$ for $\rho^{d}$. The numerical results shown in Fig. 9 demonstrate that $|\psi_{d}(n)|/\phi_{d}(n)<|\psi_{d}(1)|/\phi_{d}(1)$ for all $n>18$. In fact the maximum occurs at $n=-9$.

IV-E Numerical example

Let us consider the negative feedback interconnection between the plant

[TABLE]

and a slope-restricted nonlinearity. This second-order plant is a counterexample to the discrete-time Kalman conjecture as there is a periodic solution when $k=2.1$ [50], and the Nyquist value is $k_{N}=3.61$. Using the algorithm of [24] we find there exists a Zames-Falb multiplier for non-odd nonlinearities when $\hat{k}_{ZF}=1.3028$.

Using the phase limitation result given in Theorem IV.3 part (i), it is possible to show that there is no Zames-Falb multiplier for any $k>k_{PL}=1.4603$. Fig. 10 illustrates that the phase limitation results indicate there can be no appropriate Zames-Falb multiplier when $k=1.5$. The phase limitation is given by $\tan^{-1}\rho^{d}=66.7137^{\circ}$, where $\rho^{d}$ is obtained using Definition IV.1 with $a=0.7198$ and $b=0.8996$. By contrast, Fig. 11 shows that this limitation is not active when $k=\hat{k}_{ZF}$; this is expected since we have been able to find a suitable Zames-Falb multiplier for this value of the gain. A complete list of slope restriction results of $G(z)$ in (67) is given in Table II. The result of the reduced off-axis circle criterion $k_{R}$ shows conservativeness compared to all the other results in the Table. The (false) result from the direct discrete-time counterpart of the off-axis circle criterion is greater than the slope obtained by phase limitation, i.e. $k_{O}>k_{PL}$; this demonstrates that Conjecture IV.10 is false.

Finally, using combination of deadzone and saturation as nonlinearity, we are able to find periodic solution with $\hat{k}_{C}=1.3666$. These results are consistent with Conjecture I.2, i.e. $\hat{k}_{ZF}<\hat{k}_{C}<k_{PL}$.

V Conclusions

In this paper we have demonstrated the connection between phase limitations of Zames-Falb multipliers and the Kalman conjecture.

In continuous time, we have generalised a limitation proposed by Megretski, clarified its remit and illustrated its effect with a numerical example. In particular we show it can be applied to a fourth-order plant where the resulting numerical implementation shows instability. It remains open which choice of intervals $[a,b]$ and $[c,d]$ and scaling parameter $\kappa$ in Theorem III.4 provides most insight.

Motivated by this connection and recent results on the Kalman conjecture in discrete time, we have derived a more simple phase limitation for discrete-time Zames-Falb multipliers. Numerical results in discrete time are easier to obtain and we show that the slope restriction obtained by using phase limitation theorems can be about 40% of the Nyquist value even for some second-order examples. Thus the phase limitation can be directly useful when forming benchmarks for searches over Zames-Falb multipliers. Further, the phase limitation can be used to show there can be no direct counterpart in discrete time (Conjecture IV.10) to the off-axis circle criterion for continuous-time systems (Theorem II.10).

Based on the results of this paper, we propose Conjecture I.2, which seems to be compatible with current state-of-the-art knowledge and results for both continuous and discrete-time domains.

There is plenty of scope for future work. It seems possible that the phase limitations might be used to provide a more computationally efficient search fo appropriate multipliers. Phase limitations for the class of Zames-Falb multipliers available when the nonlinearity is quasi-odd [36] require further research.

VI Acknowledgements

We would like to thank the anonymous reviewers for their helpful suggestions.

VII Appendix

VII-A Proof of Lemma III.2

Both the functions

[TABLE]

and

[TABLE]

are monotone non-decreasing in $\omega$ when $t>0$. It follows that $\phi(t)>0$ and $\tilde{\phi}(t)>0$ when $t>0$. In addition

[TABLE]

and

[TABLE]

Finally

[TABLE]

and

[TABLE]

so the choice (28) ensures

[TABLE]

and

[TABLE]

VII-B Proof of Theorem III.4

Suppose (31) and (32) hold for some multiplier $M(j\omega)=1-H(j\omega)$. Then

[TABLE]

and

[TABLE]

where $h$ in the impulse response of $H$. Hence integrating (31) and (32) over their respective intervals gives

[TABLE]

and

[TABLE]

Summing the two inequalities, multiplied by $\lambda$ and $-\mu$ respectively, gives

[TABLE]

1. (i)

If $M\in\mathcal{M}^{c}$ then $\|h\|_{1}<1$ and $h(t)\geq 0$ for all $t$. So

[TABLE]

and hence we can write (80) as

[TABLE]

But, since $\phi$ is an even function and $\psi$ is an odd function,

[TABLE]

Further, since $\phi$ is non-negative,

[TABLE]

Hence $\rho<\rho^{c}$. 2. (ii)

If $M\in\mathcal{M}^{c}_{odd}$ then we can only say $\|h\|_{1}<1$. Nevertheless,

[TABLE]

and hence (80) leads to

[TABLE]

But, since $\tilde{\phi}$ is also an even function and (as before) $\psi$ is an odd function,

[TABLE]

Further, since $\tilde{\phi}$ is non-negative,

[TABLE]

Hence $\rho<\rho^{c}_{odd}$.

VII-C Proof of Lemma IV.2

The result is immediate following a similar argument to the proof of Lemma III.2. In particular, as $\phi_{d}$ and $\tilde{\phi}_{d}$ are evaluated for discrete values of $n\geq 1$, their limiting behaviour as $n\rightarrow 0$ need not be considered.

VII-D Proof of Theorem IV.3

Suppose (51) holds for some multiplier $M(e^{j\omega})=1-H(e^{j\omega})$. Then

[TABLE]

and

[TABLE]

where $h$ is the impulse response of $H$. Hence integrating (51) over the interval $[a,b]$ gives

[TABLE]

1. (i)

If $M\in\mathcal{M}^{d}$ then $h_{0}=0$, $\|h\|_{1}<1$ and $h_{n}\geq 0$ for all $n$. So

[TABLE]

and hence we can write (91) as

[TABLE]

But, since $\phi_{d}$ is an even function and $\psi_{d}$ is an odd function,

[TABLE]

Further, since $\phi_{d}$ is non-negative,

[TABLE]

Hence $\rho<\rho^{d}$. 2. (ii)

If $M\in\mathcal{M}^{d}_{odd}$ then we can only say $h_{0}=0$ and $\|h\|_{1}<1$. Nevertheless,

[TABLE]

and hence (91) leads to

[TABLE]

But, since $\tilde{\phi}_{d}$ is also an even function and (as before) $\psi_{d}$ is an odd function,

[TABLE]

Further, since $\tilde{\phi}_{d}$ is non-negative,

[TABLE]

Hence $\rho<\rho^{d}_{odd}$.

VII-E Proof of Theorem IV.5

Substituting

[TABLE]

and

[TABLE]

into (54) leads to (91). The proof is then identical to that of Theorem IV.3.

VII-F Proof of Proposition IV.7

1. (i)

Let $M_{n}(z)=1-z^{-n}$ with $n\in\mathcal{N}([a,b])$. Then

[TABLE]

Integrating over the interval yields

[TABLE]

Furthermore, if

[TABLE]

with

[TABLE]

then we may write

[TABLE]

The proof follows straightforwardly. 2. (ii)

Similar.

VII-G Proof of Lemma IV.11

Let

[TABLE]

where we have used that $(x-\sin x)$ is a monotonically increasing function; and

[TABLE]

For $n>\nu$, we know $(b-a)n-2>0$. In addition,

[TABLE]

so

[TABLE]

As a result,

[TABLE]

Finally, it is easy to check that

[TABLE]

and hence the same relation holds for $|\psi_{d}(n)|/\tilde{\phi}_{d}(n)$.