Zames-Falb multipliers for absolute stability: From O'Shea's contribution to convex searches

Absolute stability attracted much attention in the sixties. Several stability conditions for loops with slope-restricted nonlinearities were developed. Results such as the Circle Criterion and the Popov Criterion form part of the core curriculum for students of control. Moreover, the equivalence of results obtained by different techniques, specifically Lyapunov and Popov's stability theories, led to one of the most important results in control engineering: the KYP Lemma.


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Absolute stability attracted much attention in the 1960s. Several stability conditions for loops with sloperestricted nonlinearities were developed. Results such as the Circle Criterion and the Popov Criterion form part of the core curriculum for students of control. Moreover, the equivalence of results obtained by different techniques, specifically Lyapunov and Popov's stability theories, led to one of the most important results in control engineering: the KYP Lemma.
For Lurye 1 systems this work culminated in the class of multipliers proposed by O'Shea in 1966 and formalized by . The superiority of this class was quickly and widely accepted. Nevertheless the result was ahead of its time as graphical techniques were preferred in the absence of readily available computer optimization. Its first systematic use as a stability criterion came 20 years after the initial proposal of the class. A further 20 years have been required to develop a proper understanding of the different techniques that can be used. In this long gestation some significant knowledge has been overlooked or forgotten. Most significantly, O'Shea's contribution and insight is no longer acknowledged; his papers are barely cited despite his original parameterization of the class.
This tutorial paper aims to provide a clear and comprehensive introduction to the topic from a user's viewpoint. We review the main results: the stability theory, the properties of the multipliers (including their phase properties, phase-equivalence results and the issues associated with causality), and convex searches. For clarity of exposition we restrict our attention to continuous time multipliers for single-input single-output results. Nevertheless we include several recent significant developments by the authors and others. We illustrate all these topics using an example proposed by O'Shea himself.
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Introduction
A feedback interconnection of a linear system and a static nonlinearity is said to be absolutely stable if the interconnection is stable (in some sense) for every nonlinearity in a given class. The theory of absolute stability has occupied an important portion of the control theory literature due to its relevance to a variety of practical control/systems engineering problems. The absolute stability problem can be studied, broadly, from either the perspective of internal stability, or from that of input-output stability. The former, and perhaps more common, approach typically involves the search for the parameters of a proposed Lyapunov function which can be used to guarantee asymptotic stability of the origin for as large a class of nonlinearities as possible. The latter approach involves the use of transfer functions called multipliers. In their classical interpretation they are used to translate one nonlinear passivity-type problem into another linear, easier to solve, passivity-type problem. The aim, again, is to choose a multiplier within a predefined class of multipliers which allows inputoutput stability to be guaranteed for as large a class of nonlinearities as possible. In this paper, attention is focused on inputoutput stability from the perspective of passivity and in particular on the properties of the so-called Zames-Falb multipliers.
The multiplier approach attracted much attention from the control community in the 1960s. One reason for this was, without the computing power of today, researchers were able to glean a great deal about the absolute stability of a system purely from the properties of the linear part. In an early paper the concept of multiplier was used by Brockett and Willems [9] and the idea developed rapidly from this (see [58]). Despite this early promise and flurry of activity, probably the most widely known absolute stability tools today are the Circle and Popov Criteria (see [79,38]) which have become standard, in part due to their simplicity and in part due to their graphical interpretations. However, when a tighter description of the nonlinearity is available, these criteria are well-known to be conservative. In such cases, the use of more general multiplier methods can be useful and, in particular, the socalled Zames-Falb multipliers can often be used to improve predictions made about stability and performance of the interconnection.
Despite their moniker, Zames-Falb multipliers were actually discovered by O'Shea (his portrait is shown in Fig. 1) in [59,60]. While the treatment of O'Shea [59] was restricted to causal multipliers, the aim of [60] was to extend this definition to noncausal multipliers: "this modification allows greater freedom in the phase variation of GðjωÞ þ 1=k outside of the 7 901 band". There were several correspondence items discussing these [88,85,23]. A rigorous and correct treatment was first given in the much-cited paper by Zames and Falb [89]. The contribution of O'Shea was fully acknowledged by all concerned at the time. As an example, Desoer and Vidyasagar [22] state that the "idea of using noncausal multipliers is due to O'Shea." However, the class of multipliers aroused little further interest for 20 years, until the proposal of Safonov and Wyetzner [65] for computer-aided search and the illustration by Megretski and Rantzer [54] of multiplier analysis embedded within the framework of IQCs. In these and subsequent papers the pioneering work of O'Shea was largely overlooked. The terminology "Zames-Falb multiplier" appears to have been coined by Chen and Wen [18,19] in their proposal for a convex search. This development, while rightly acknowledging the important work of Zames and Falb, has had an unfortunate consequence. Zames and Falb [89] focus on the relation of the nonlinearity to the monotone and bounded static nonlinearity; O'Shea's insights into the phase properties of the multipliers have been largely forgotten (with one notable exception: the discussion of Megretski [51] on phase limitation).
In this tutorial paper we re-examine Zames-Falb multipliers and, in particular, use an example of O'Shea [60] to discuss the phase properties of the Zames-Falb multipliers and how they can be used advantageously in the study of the absolute stability problem.
The remainder of the paper is structured as follows. In Section 2 we provide a brief motivating example explaining the significance of Zames-Falb multipliers, and in Section 3 we review the basics of the absolute stability problem and some approaches to its solution. In Section 4 we address at length an example previously discussed by O'Shea [60]. In particular we discuss how a number of input-output stability methods can be used for analysis. This section includes a comprehensive treatment of the application of a multiplier originally proposed by O'Shea. In Section 5 further properties of Zames-Falb multipliers are discussed and in Section 6 a brief review of start-ofthe-art computational searches is given. Further developments of Zames-Falb multipliers are discussed in Section 7 and open questions considered in Section 8. Finally in Section 9 we conclude and point to some other recent developments in the use of Zames-Falb multipliers. While we emphasise the tutorial aspect of this overview, some mathematical formalism and machinery is inevitable; this is given in the appendix.

Motivating example
Remark 1. Several concepts in this section are formally defined in Section 3 and/or the Appendix.
Since saturation is a memoryless and slope restricted nonlinearity, the Zames-Falb multipliers can be used to study the stability/robust stability of systems involving saturation [34]. We shall illustrate such analysis with an anti-windup example [39] where robust stability is to be established [73,55]. UðsÞ and YðsÞ are the Laplace transform of the plant's input and output, respectively.
Consider a plant with additive uncertainty where GðsÞ is the nominal SISO transfer function and Δ represents additive uncertainty with, for any bounded signal u, In the case where Δ is restricted to be a linear time invariant (LTI) system we may write this as the familiar H 1 norm condition Suppose the controller has the internal model control structure given by and illustrated in Fig. 2.
The robustness of such controllers is discussed at length by Morari and Zafiriou [56]. Briefly, if both G and Q are stable, then it follows from a small gain argument that the loop is stable provided Suppose now there is saturation in the loop, as in Fig. 3. Since the saturation operator is in series with Δ, a similar small gain argument [73] says that the loop remains stable provided (5) is  satisfied. In other words, the antiwindup scheme preserves the robustness (to additive uncertainty) of the loop without saturation.
But the antiwindup scheme of Fig. 3 can be notoriously sluggish. To improve matters, one suggestion in the literature [90] is the scheme of Fig. 4 with This often has much better performance, but there is no longer an a priori guarantee of stability. In our example we consider a case where the Zames-Falb multipliers can be used to establish such stability. Suppose G is first order with a delay (a standard model in the process industries): A natural choice for Q is then: In the unconstrained (saturation identity) case, robust stability is established via (5) which in this case reduces to In the constrained (saturation ≢ identity) case, establishing robust stability is much more difficult. However the constrained loop is stable [55] provided there exists a multiplier (of some form) M such that at all frequencies. It may not be possible to satisfy this inequality with a constant M (and it cannot be satisfied with constant M if J Q J 1 -γ), but for our example, it is straightforward to check that the inequality is satisfied if we choose It transpires that this belongs to the class of first order Zames-Falb multipliers provided 0 o a o 2c. In this case the robust stability of the constrained loop is established using a Zames-Falb multiplier.

Preliminaries
In an early paper, Brockett and Willems [9] use the concept of a multiplier. The aim of its use is to reduce the conservatism of the open loop approach which is used to analyse the stability of the problem. The advantage of this approach is that the condition to be tested will only depend on the linear system G and the maximum slope of the nonlinearity k.

The Lurye problem
The Lurye problem consists of finding conditions on the linear system G such that the feedback interconnection between G and any nonlinearity ϕ that belong to some class of nonlinearities is stable. As stability must be ensured for the whole class, the adjective absolute is added, and this problem is also known as the absolute stability problem (see [47] for an overview).
The feedback interconnection in Fig. 5 is defined by It is usual, although not necessary, to assume that G is strictly proper. This is enough to ensure the feedback between G and any slope-restricted nonlinearity is well-posed (i.e. that u 1 and u 2 are uniquely defined given r 1 and r 2 , all on the extended spaces defined in the appendix). The system is said to be input/output stable if for any r 1 A L 2 , r 2 A L 2 we also have u 1 A L 2 and u 2 A L 2 .
In this paper we consider the class of static nonlinearities with slope less than or equal to k. With an abuse of notation we use ϕ to denote both the memoryless operator ðϕ : L 2e -L 2e Þ and its associated nonlinear function ϕ : R-R.

Definition 1.
A static nonlinearity ϕ is said to be slope restricted ϕAS½0; k if for any real number x and y we have The Linear Time Invariant (LTI) system G is given by and its transfer function is GðsÞ ¼ CðsI À AÞ À 1 B þ D. Henceforth, we will no longer distinguish between LTI operators and their transfer functions. The Rosenbrock system matrix ð17Þ will be used as shorthand. G n denotes the adjoint of G and it is given by G n ðsÞ ¼ G 0 ðÀsÞ:   We assume G is stable (i.e. A is Hurwitz); hence we may assume r 1 ¼ 0 without loss of generality. If G is unstable, the loop would be unstable with ϕ being the zero operator.

The Nyquist value and the Kalman conjecture
The class of slope restricted nonlinearities ϕAS½0; k includes the linear gains τk with τ A ½0; 1. This provides some insight to the absolute stability problem. In particular it is necessary for absolute stability that the Lurye system be stable with any such linear gain. The Nyquist value k N is the maximum value of k for which this holds: Definition 2 (Nyquist value). Let G A RH 1 with Rosenbrock system matrix (17). The Nyquist value is given by It is tautologous to say that for absolute stability we require k o k N . As a result, the inverse of the linear system ð1 þ τkGÞ needs to be bounded for all τ A ½0; 1. This fact implies that for absolute stability to hold the phase of the system ð1 þ kGÞ must be within the interval ðÀ1801; 1801Þ. Kalman [35] made the conjecture that consideration of feedback with linear gain was also sufficient for absolute stability: Kalman Conjecture [35]. Let ϕ be a memoryless nonlinearity slope-restricted on S A ½0; k. Then, the Lurye system in Fig. 5 is asymptotically stable if A À τBCkð1þτkDÞ À 1 is Hurwitz for all τ A ½0; 1.
The conjecture has played an important part in the development of the absolute stability of feedback systems containing slope-restricted nonlinearities. It is true for first, second and thirdorder continuous-time systems [8]. Thus we know a priori that a third order system is absolutely stable provided ϕAS½0; k N Þ and we can benchmark a test for stability by seeking the maximum slope value and comparing with this upper bound (e.g. [65,16]). But the conjecture is false in general and the fourth-order counterexamples proposed more than 40 years ago [26,60,82,46] can also be used as benchmarks as they can be very challenging for stability tests. We illustrate such a benchmark in this paper.

Passivity, loop transformations and multipliers
Passivity theory provides an important stability test for closed-loop systems. Conditions for stability are simplified since one element of the Lurye system (13) is LTI and stable. We can assume that r 1 ¼ 0 without loss of generality. It is sufficient for closed-loop stability that ϕ be passive and G be strictly input passive (SIP). A stable operator ϕ : L 2 -L 2 is said to be passive 2 if there exists some β r 0 such that 〈ϕu; u〉 Z β for all u A L 2 : A stable LTI system G is SIP [10] if and only if there is a δ40 such that RefGðjωÞg Z δ for all ω: If the nonlinearity ϕ is sector bounded on the interval ½0; k then the map fromũ 2 ¼ u 2 À y 2 =k to y 2 is passive. But the system shown in Fig. 6 is stable if and only if our original Lurye system is stable. Hence, via a loop transformation argument, it is sufficient for stability for G þ 1=k to be SIP. This is the Circle Criterion.
Similarly, suppose M (a "multiplier") is a biproper transfer function whose zeros and poles are all in the left half plane. Then the system shown in Fig. 7 is stable if and only if our original Lurye system is stable. If ϕM À 1 is passive, then it suffices for stability that MG be SIP.

Zames-Falb theorem
O'Shea [59,60] proposed a set of multipliers appropriate for sloperestricted nonlinearities. This included an extension to noncausal multipliers. The machinery was formalized by Zames and Falb [89] in their seminal paper.
Theorem 1 (Zames et al. [89]). Consider the feedback system in Fig. 5 with G A RH 1 , and a slope-restricted nonlinearity ϕAS½0; k. Assume that the feedback interconnection is well-posed. Then suppose that there exists a convolution operator M : L 2 ðÀ1; 1Þ-L 2 ðÀ1; 1Þ whose impulse response is of the form mðtÞ ¼ δðtÞÀ where δ is the Dirac delta function and Assume that: Loop transformation. If the mapφ fromũ 2 ¼ u 2 À y 2 =k to y 2 is passive, then it suffices for stability to test whetherG ¼ G þ 1=k is strictly input passive, wherẽ y 1 ¼ y 1 þ u 1 =k. Fig. 7. Multiplier theory. If the map S 1 ¼ ϕM À 1 is passive then it suffices for stability to test whether S 2 ¼ MG is strictly input passive.
(ii) either hðtÞ Z 0 for all t A R and h i Z 0 for all i A N, or ϕ is odd; and (iii) there exists δ40 such that Then the feedback interconnection (13) is L 2 -stable. □ The corresponding class of multipliers is known as the class of Zames-Falb multipliers.
Definition 3. The class of Zames-Falb multipliers M is given by all transfer functions M A L 1 whose inverse Laplace transform 3 is given by where This class class will be used for slope-restricted and odd nonlinearities. If the nonlinearity is non-odd, only a subclass of multiplier can be used. Although such definitions may appear formidable at first sight, it is usual to consider only subclasses. Most searches are restricted to rational Zames-Falb multipliers (the class RM), where h i ¼ 0 for all i and M A RL 1 . An exception is the search of [65] where instead hðtÞ ¼ 0 for all t, so the multiplier is a sum of delayed impulses.
In addition, if M is a Zames-Falb multiplier we can always find a factorization M ¼ M a M c where M c ; M À 1 c A H 1 and M a ; M À 1 a A H À 1 , i.e. M n a ; ðM n a Þ À 1 A H 1 . This is the cornerstone of [89] to formalize the use of the class of multiplier proposed by [60]. In the jargon, this factorization is referred to as a canonical factorization (see [13], and the references therein).
It is emphasized that the causality assumption of the real systems G and ϕ is not required on the multiplier, since it is just a mathematical "device". Hence M is not required to be causal. It is required to be bounded in the sense that its impulse response has finite L 1 -norm (24). In particular this ensures that M can be factorized into a causal and bounded operator M c and an anticausal and bounded operator M a . For LTI systems, the use of the bilateral Laplace transform leads to duality properties. 4 Loosely speaking, if a system is assumed to be causal, M A H À 1 means that the impulse response of the system is unbounded; if a system is assumed to be bounded, M A H À 1 means that the impulse response mðtÞ is zero for all t 4 0.
It can be shown that the L 1 norm condition (21) on M and the slope-restriction on ϕ ensures This guarantees the block S 2 in Fig. 7 to be positive. Similarly, the phase property on MðG þ 1=kÞ (22) ensures Nevertheless, stability cannot be ensured since M is not causal. Zames and Falb used the canonical factorization M ¼ M a M c to show stability. The properties of the inner product in Eqs. (25) and (26) mean we can write 〈M n a y 2 ; M cũ 〉Z 0; and 〈M cỹ1 ; M n a u 1 Þ〉 Z 0: ð27Þ As a result, both the blocks S 1 and S 2 in Fig. 8 are stable and positive, and hence passive. Therefore, the feedback interconnection between S 1 and S 2 is stable (by passivity) and equivalent to our original Lurye problem.

O'Shea's example
Our standard problem is the Lurye problem (13) depicted in Fig. 5. The nonlinearity ϕ from u 2 to y 2 is static and sloperestricted to the interval S½0; k. Brockett and Willems [9] suggested plants with the structure would be challenging to analyse. O'Shea [60] chose a subclass of the form where the symmetry aids both intuitive understanding and the ability to find solutions by hand. The symmetry is given by GðjωÞ ¼ G n ðjω À 1 Þ. This turns out to be a challenging feature for several classes of multipliers.
If the nonlinearity ϕ is replaced with a linear gain k, then the loop is stable for all k and for all 0 o ζ r1; i.e. the Nyquist value is infinite when ζ is in this range (the phase never drops below À 1801, see Fig. 12). But if ϕ is a saturation block in series with a gain k, then it is possible to find values of k and ζ that are apparently unstable. For example, Fig. 9 shows the result generated in Simulink when ζ ¼ 0:1 and k ¼ 2000. Such phenomena were first observed by Fitts [26], and have attracted much attention as counterexamples to the Kalman conjecture. Barabanov [8] questioned the validity of Fitts' original counterexample; this has led to considerable discussion [45,43,46]. O'Shea [61] showed that such loops could be guaranteed stable for all k 4 0 provided 1=2 o ζ r1. For most of our discussion we will fix ζ ¼ 0:6. Fig. 10 shows such stable behavior generated in Simulink when ζ ¼ 0:6 and k ¼ 2000.
In the following subsections, we will consider how various standard criteria can be used to judge stability. In particular we will be able to associate a particular range of k for each criterion where stability can be guaranteed.

Passivity
In our problem the nonlinearity ϕ from u 2 to y 2 is passive. It would therefore suffice for the phase of G to lie on the interval ðÀ901; þ901Þ. However G is not passive; its phase approaches þ 1801 at low frequency and À1801 at high frequency (Fig. 12). Hence the passivity theorem cannot be used directly to establish stability for any k 4 0.

Small gain theorem
The L 2 gain of the nonlinearity from u 2 to y 2 is k. That is to say, for any u 2 A L 2 , we must have J y 2 J 2 r k J u 2 J 2 . It follows by the small gain theorem that the loop is guaranteed stable provided The H 1 norm of G is J G J 1 ¼ 0:6944 (see Fig. 11). Hence we may conclude the loop is stable for

Circle Criterion
Although the passivity theorem cannot be invoked directly, it can be used indirectly. The nonlinearity ϕ is sector bounded; it follows we can use a loop transformation and apply the Circle Criterion ( Fig. 6). It is thus sufficient for G þ1=k to be SIP for stability. For our example we find ( Fig. 12) Re GðjωÞ È É 4 À0:0868 for all ω: It follows that the loop is stable provided k o 1 0:0868 ¼ 11:52:

Popov Criterion
For the Popov Criterion we test whether MðG þ 1=kÞ is SIP where M is a Popov multiplier of the form This is a standard and well-known result, although the case with ηo0 is often ignored [38,80]. In fact it can be derived as a corollary of the Zames-Falb theorem [9,14].     12. Nyquist plot of G when ζ ¼ 0:6. The plot never encircles the À 1 point, so the Nyquist value is infinite. The phase drops from þ 1801 to À 1801, so G is not passive. The Nyquist plot of G is always to the right of À 0:0868, so the Circle Criterion confirms stability for k o 1=0:0868.
One might naïvely expect the Popov Criterion to offer an improvement over the Circle Criterion. However for this example the symmetry of G ensures this is not the case. The Popov plot in Fig. 13 provides a result no better than the Circle Criterion and shows that the implicit Popov multiplier is 1þ 0s, since the dashed line is vertical. The reason is simple: suppose k 411:52 (the maximum k for which the Circle Criterion guarantees stability). There is a frequency interval where the phase of 1 þ kGðjωÞ is greater than þ 901 and a frequency interval where it is less than À901. See Fig. 14 for the case k ¼ 15. Any Popov multiplier that raises the phase at high frequency (i.e. with positive coefficient) cannot reduce the phase at low frequency. Conversely, any Popov multiplier that reduces the phase at low frequency cannot raise the phase at high frequency. In brief, if k 411:52 and given some ηAR, there must exist some frequency ω where A typical result with η positive is shown in Fig. 15; a negative η would result in a similar but opposite effect.
In the 1960s and 1970s, several other frequency domain conditions based on stability multipliers were tested using graphical interpretations; see [58] as a classical textbook on this topic and Section 3.4.1 in [7] for a recent overview. In particular, Table 3.1 in [7] can be used to show that the symmetry of the phase prevents other classes of multipliers, such as the Yakubovich multipliers [87], the RL multipliers [9] or the RC multipliers [9], from improving on the Circle Criterion for this example. Similarly the Off-axis Circle Criterion [20] uses either RL or RC multipliers. Note that Park [62] provides a convex search for the Yakubovich multipliers.

O'Shea's multiplier
O'Shea [60] proposed the multiplier with p 4 0 sufficiently small. This is sufficient to ensure the phase of MðG þ 1=kÞ lies above À901 and below þ 901 as in see Fig. 16. This in turn is sufficient for stability even though the multiplier has a pole in the right half plane, i.e. the multiplier is noncausal.   In particular, the existence of O'Shea's multiplier M with the property that MðG þ 1=kÞ has phase on the interval ðÀ901; 901Þ guarantees the existence of a Zames-Falb multiplier (Definition 3) satisfying the conditions of the Zames-Falb theorem (Theorem 1). Hence the existence of O'Shea's multiplier is sufficient for stability.
The multiplier M suggested by O'Shea (31) itself is not within the class of Zames-Falb multipliers M. We can write with lows. Put q ¼ ð2 À 2pÞ=ð2 À pÞ and choose ρ40 such that q o 1 À ρ, for example ρ ¼ p=ð4 À 2pÞ. We can then write Then we can write for ε40 sufficiently small. The phases of MðG þ 1=kÞ and M PE ðG þ 1=kÞ are compared in Fig. 17.
In short, the noncausal multiplier (31) can be used to guarantee the absolute stability of our example for any positive k provided p 4 0 is chosen sufficiently small. A similar analysis guarantees stability for any positive k provided the damping ratio ζ 4 0:5. A formal proof requires the concept of phase-equivalence [14,15] which we discuss further in the next section.

Further properties
The class of Zames-Falb multiplier is one of the possible classes that have been proposed for analyzing the stability of the Lurye system (13). The aims of this section are to discuss the phase properties of the Zames-Falb multipliers and the equivalence between different classes of multipliers, where it will be shown that phase is a key factor.

Positivity
One trivial property of the multiplier is that it must be a positive system. By definition, a multiplier is required to preserve the positivity of the class of nonlinearities. As mentioned by Carrasco et al. [13], when a scaled identity is within the class of nonlinearities, then the multiplier itself needs to be positive. Referring to Fig. 8, this can easily be seen because Thus if we consider the particular caseφũ 2 ¼ kũ 2 with k 4 0, then and hence The phase of the multiplier is required to be within the interval ½À901; 901. However, we cannot consider this as a limitation of the multiplier as the phase of ð1 þ kGÞ must belong to the interval ðÀ1801; 1801Þ to satisfy the necessity of the Kalman Conjecture.
In the Nyquist diagram, we can find further restrictions on the Nyquist plot of the multiplier M A M. In particular, we can use L 1 norm properties to ensure that the Nyquist plot belongs to a circle with center ð1; 0Þ and radius 1 [66]; see Fig. 18. Loosely speaking, it is due to the fact that the L 1 norm of a system is always greater than its H 1 norm. See [79] for further details.

Noncausal multipliers
Undoubtedly, O'Shea's main contribution to multiplier theory was the introduction of noncausal multipliers (see [22, p. 227]). The motivation was to increase the flexibility of the phase of the multiplier. In this section, we demonstrate this concept. An analytic result can be obtained for rational first order Zames-Falb multipliers.  Lemma 1. Given ϵ40, there exists a first-order causal Zames-Falb multiplier such that its phase is 901 À ϵ at some frequency. However, if M c is a causal rational first-order Zames-Falb multiplier, then ∠M c ðjωÞ 4 À arcsinð1=3Þ for all ωAR. Lemma 2. Given ϵ40, there exists an anticausal first-order Zames-Falb multiplier such that its phase is À 90 ○ þ ϵ at some frequency.
However, if M ac is an anticausal rational first-order Zames-Falb multiplier, then ∠M ac ðjωÞ o arcsinð1=3Þ for all ωAR.
The proofs of these results are straightforward, but they show the significant reduction on the selection of the phase of the multiplier if we limit ourselves solely to causal or anticausal multipliers for a fixed order (see Fig. 19). However, it can be easily shown that a causal multiplier can reach any phase by considering an infinite dimensional multiplier. Once again the result is trivial by using the multiplier MðsÞ ¼ 1 þ z 1 e 7 s . Hence one could be tempted to think that there is no phase limitation if the order is infinite.

Phase limitations of Zames-Falb multipliers
It has been established that the conditions on the Zames-Falb multiplier require some limitation in the selection of the phase. 5 The lack of such limitation would imply that we can make any biproper plant appear passive if its phase is within the interval ðÀ180 ○ ; 180 ○ Þ; hence the Kalman conjecture would be true. As the Kalman conjecture is known to be false, the L 1 condition on the multiplier must have an interpretation as a phase limitation. One such characterization of a limitation is given in [52] (see Fig. 20).
Lemma 4 (Megretski [52]). Let b 4 a 4 0, r4 b=a be real numbers. Let If it has been shown that there is no limitation in the phase of a causal or anticausal multiplier, Megretski's result shows that there is a limitation based on the rate of change of the phase. It can be easily shown with O'Shea's example. Let us consider k ¼ 1; then the required phase properties of the multiplier are presented in Table 1.
As O'Shea [60] mentions, the proposed multiplier ensures stability for any gain for ζ 4 0:5. For ζ o 0:5, the problem remains unsolved, and searches can be tested with this example. However, Fig. 9 shows that no multiplier can ensure stability for large k when ζ ¼ 0:1. So the limiting factor for the phase of multiplier is not the phase itself, but how fast it changes. So, for ζ ¼ 0:5 there exists a multiplier able to change its phase from À 791 up to 791 in two decades; but for ζ ¼ 0:1, there is no Zames-Falb multiplier which can change its phase form À 881 up to 881 in two decades, preserving the properties for the rest of frequencies.
The analysis of Megretski [52] is not definitive. It is clear that restrictions to subclasses of Zames-Falb multipliers, such as the causal multipliers, impose further limitations in phase. However no analytic result has yet been provided. Fig. 21 shows the phase of two causal but irrational multipliers. Their phase spans the interval ðÀ901; þ901Þ with very fast transitions from À901 up to þ 901 but slow transitions from þ 901 down to À 901.  If the phase of a multiplier is above tan À 1 ρ for a range of frequencies ða; bÞ, then the multiplier cannot reach a phase below À tan À 1 ρ over the range of frequencies ðra; rbÞ.

Equivalences
The first equivalence result between classes of multiplier was given by Falb and Zames [24]. In this paper, they show that given any RL and RC multiplier (for definitions, see [24,9]), a Zames-Falb multiplier with the same phase can be found. Then it is no longer important that the class of Zames-Falb multipliers does not include some RL and RC multipliers, since we can always find a "substitute" in the class.
A formal definition is required. However, we need to limit our set of interest. It is a key step in order to be able to establish formal substitution and equivalence results. As we have mentioned, the necessity of the Kalman conjecture is required in order to ensure absolute stability; hence we will restrict our attention to plants where this property is required.
Definition 5. The set SR is given by the plantsG ¼ 1 þ kG with the following properties: 1 þkG is stable. ð1 þ τkGÞ À 1 is stable for any τ A ½0; 1.
Loosely speaking, given some plant G and maximum slope k, then ð1 þ kGÞ∉SR, the existence of a Zames-Falb multiplier such that ð1 þ kGÞM is positive can be dismissed. Once we have introduced this class of LTI systems, then formal definitions can be given. This property has already been used in Section 4.5, and the phase-substitution of multipliers with the addition of the Popov term has been explained. Another simple but insightful equivalence is described as follows: Definition 7 (Park's multipliers [62]). The class of Park's multipliers is defined as follows: for any two scalars a and b.
It is straightforward to show that Park's multipliers can be linked with the class developed by Yakubovich [87]. However, a more detailed analysis will show that we can find a Zames-Falb multiplier with the same phase properties.
Firstly, let us find the zeros of a Park multiplier. If b 4 0, then the zeros will be labelled as follows: whereas if b o0 we will use ; This ensures that j z 1 j o j z 2 j . Note that z 1 z 2 o 0. Secondly, the phase of M P ðjωÞ is given by the phase of its zeros, since the phase of its poles cancels out, i.e.
Finally, the zero with larger absolute value, z 2 can be transformed into a pole reflected in the imaginary axis since ∠ðjω À z 2 Þ ¼ À∠ðjω þ z 2 Þ. As a result, the phase of any Park multiplier is given by So we have found a phase-substitute multiplier that can be rewritten as follows: Then, we can check that M A M since where m is the inverse Laplace transform of and we have used that z 1 z 2 o0 and j z 1 j o j z 2 j . This example and the previous example illustrate the concept of phase-substitution. Using this concept, it can be shown that a search over the whole class of Zames-Falb multipliers would be enough to obtain the best possible result compared with any other class of multipliers in the literature [14]. Nevertheless the significant difficulties in obtaining a convex search over the whole class of Zames-Falb multipliers mean that the parameterizations of other classes of multipliers may still be useful. We discuss searches in the following section.

Convex searches
Section 4 has illustrated how various stability tests can be used to find the maximum slope of the nonlinearity ϕ for which stability is guaranteed. This, and other sections, have also illustrated how the selection of an appropriate Zames-Falb multiplier can enable more accurate statements regarding the absolute stability of a Lurye system to be made. Using an example from O'Shea [60] it has been shown that how the selection of such a multiplier may be achieved for a relatively simple system. For more complex systems, it is somewhat more difficult to choose the most "appropriate" Zames-Falb multiplier because the set of such multipliers is extremely large and, in fact, infinite dimensional. Typically, we would like to choose a multiplier which allows us to make the least conservative statements about, for example: 1. the size of slope for which stability is guaranteed; 2. the L 2 gain from a given input to a given output [71,53].
Choosing such a multiplier which either maximizes the slope size or minimizes the L 2 gain is not trivial. In this section, several automated searches for rational ðM A RMÞ multipliers are introduced. The searches described are based on Chen and Wen [18,19], which is similar to that implemented in the IQC toolbox [53,36,21], and also more recent approaches of the authors e.g., [74,16]. The technique of Safonov and Wyetzner [65] and Gapski and Geromel [28], which has recently been updated in [17], is briefly discussed in Section 6.7.

Linear search in k
Modern searches for multipliers are somewhat different to classical graphical criteria (see [58]). In graphical criteria with simple multipliers (e.g. Circle, Popov), k max is found directly via a plot of GðjωÞ, even though an auxiliary multiplier is implicit (see Fig. 13).
The search for more sophisticated multipliers requires a different approach. In this case, a linear search over k is carried out. Given a value k f , then a search over M is carried out to find a suitable multiplier forG ¼ G þ 1=k f , i.e. we search for a multiplier for all ω40. If the search is successful, the multiplier found is suitable for any k ok f , i.e.
where we have used that RefMðjωÞg 4 0; hence we can increase k f . If the search is unsuccessful, we reduce k f until a successful search is obtained. Before computational methods were available attempts were made to interpret the Zames-Falb multipliers graphically e.g., [27,48]. It must be highlighted that these geometrical methods also require a linear search. In this tutorial, we focus on the development of convex searches, but Section 3.4.1 of Altshuller [6] provides an overview of such graphical methods (see [5] for an example).

Time and frequency domain conditions
Recall that the system in Fig. 5  A central issue, which will recur throughout this section, is that of combining, in an efficient and tractable manner, these two conditions. In particular, it is generally difficult to provide a frequency domain characterization of the L 1 norm, although, as discussed in Section 5, the L 1 norm requirement appears to place a limit on the rate-of-change of phase of the multiplier [52].
A result which we shall invoke several times in this section is the so-called Positive Real Lemma given below. This can be interpreted as a special case of the KYP lemma [63].

1.
GðjωÞ ⋆ þ GðjωÞ Z 0 8 ωAR; if and only if ðA; BÞ is controllable and there exists a P ¼ P 0 such that GðjωÞ ⋆ þ GðjωÞ 4 0 8 ωAR; if and only if there exists a P ¼ P 0 such that The Positive Real Lemma provides a connection between positive realness in the frequency domain and a matrix inequality. It is of central importance in casting conditions involving Zames-Falb multipliers as LMI's.

Preliminary manipulations
The main goal of this section is to translate the positive real condition (40) and the L 1 condition (41) into tractable, automated searches. Two approaches to this will be described, but both ways share some initial manipulation which will be covered here. Assume first that HðsÞ has state-space realization ð42Þ where the matrices are to be determined. 7 Here and elsewhere G has the following state-space realization: ð43Þ Given these two state-space realizations, it then follows that ð44Þ where ð45Þ Our objective then becomes a search, over the multiplier statespace matrices , in order to maximize the scalar k which represents the slope restriction of our nonlinearity. Invoking Lemma 5 now shows that the positive real condition (40) can, equivalently, be expressed as a search over real symmetric matrices P such that the following matrix inequality is satisfied Using the realization (45), this inequality can be written as Note that because the matrices ðA I ; B I ; C I ; D I Þ are affine functions of multiplier matrices this matrix inequality is nonlinear, due to products of and P, and therefore not amenable to efficient solution. While this section does not seem to have eased the difficulty of the search for Zames-Falb multipliers, it transpires that the inequality (47) is a useful stepping stone towards a convex search. The following two sections will show two different approaches for simplifying this inequality and imposing the L 1 bounds (41).

Structured multipliers
The problems with searches for multipliers as they stand are two-fold: the troublesome L 1 condition (41) and the nonlinear matrix inequality (47) that arises as a result of the positive real condition (40). The approach advocated by [18,19] (see also [32,53]) is to structure the multipliers in such a way that (i) L 1 norm bounds may easily be obtained and (ii) the nonlinear matrix inequality (47) becomes a linear matrix inequality. The approach described here follows [18]. The first observation to make is that if a transfer function, HðsÞ, is given a first order structure, it is easy to calculate its L 1 norm as illustrated below.

Example 1 (A first order multiplier). Let
Thus the L 1 bound is simply κ=ao1. For fixed a this is simply a linear inequality in κ and the associated state-space matrices are Notice that, under the assumption that a is constant, three of the four state-space matrices are constant, which means that inequality (47) is actually linear. Therefore, with this structure of multiplier we have obtained a linear inequality for the L 1 norm and a linear matrix inequality for the positive real condition: a tractable search.
The basic approach by [18] is to extrapolate from the above example. By restricting attention to a sub-class of Zames-Falb multipliers, the L 1 norm conditions become simple linear inequalities. Also, because this ensures that matrices A H and B H are constant, the nonlinear matrix inequality (47) becomes linear.

A class of causal positive multipliers
In order to describe the work of [18] concisely, we first consider the following sub-class of Zames-Falb multipliers: where h þ c ðtÞ ¼ L À 1 fH þ c ðsÞg is such that h þ c ðtÞ ¼ 0 8 t o0 and h þ c ðtÞ Z 0 8 t Z0. In other words, the multiplier is assumed causal and the impulse response h þ c ðtÞ is positive. An example of such a function is shown in Fig. 22. These assumptions will be relaxed in subsequent sections. This representation of the multiplier has several advantages: firstly it significantly reduces the complexity of the L 1 inequality, viz: In this case, the following theorem can be used for approximating such functions: Theorem 2 (On approximation in L 1 ½0; 1Þ, Szegö [69]). For any hðtÞ Z 0 A L 1 ½0; 1Þ and any ϵ40 there exists κ þ c i and N such that The implications of this are the following: by choosing N large enough, h þ c ðtÞ can be arbitrarily well approximated by a sum of orthogonal functions. This means that, for h þ c ðtÞ, we can always find a phase equivalent multiplier represented by a sum of first order functions. In addition, with this approximation, the L 1 norm can be calculated explicitly and the L 1 constraints are linear The above follows from considering the Laplace Transform of e À t t i for i A 0; 1; The main issue in the application of this result is the requirement that h þ c ðtÞ Z 0 for all t Z 0. To guarantee this, note that h þ c ðtÞ is given by for some N and for some κ þ c i to be chosen. Then h þ c ðtÞ Z 0 for all t Z 0 if Chen and Wen [19] have shown that this time-domain condition is equivalent to A state-space realization of S þ c ðsÞcan then be given as where the structure of the above state-space matrices is given by [19]. In particular C þ c and D þ c are structured affine functions of the κ þ c i . Then applying Lemma 5 to inequality (54) we obtain the matrix inequality  For a fixed k, inequality equations (54), (56) and (51) form a system of linear matrix inequalities. Thus, in this case, the search for a multiplier becomes an LMI problem.

Noncausal multipliers
A restriction in the results derived so far is that we have assumed the multiplier, MðsÞ is causal, that is hðtÞ ¼ 0 8 t o 0. As explained earlier in the paper, this may cause some conservatism. However, this assumption can be removed relatively easily by structuring hðtÞ as the sum of a causal part and an anticausal part, viz, where h þ c A L 1 ½0; 1Þ and h þ a A L 1 ðÀ1; 0. An example of such an hðtÞ is shown in Fig. 23. A similar approximation to that given in Theorem 2 can then be used to approximate the anti-causal part of hðtÞ as which is again a sum of orthogonal terms. Recalling the approximation (52) for the causal part of h þ ðtÞ, the L 1 constraint (41) can again be replaced by the linear inequality which is a linear inequality in κ þ c i and κ þ a i . In the same way as before if we assign ð60Þ where A þ a and B þ a are constant, and C þ a ðκ þ a i Þ and D þ a ðκ þ a i Þ are affine functions of κ þ a i .Then following a similar line of reasoning to that in Eqs. (53)- (54), it follows that h þ a ðtÞ Z 0 providing there exists a symmetric matrix X þ a satisfying the LMI: In this case, a state-space realization for HðsÞ is given by This realization can then be used to obtain an LMI from (47).

Non-positive multipliers
Previously it was assumed that hðtÞ ¼ h þ ðtÞ Z 0 8 t. This assumption can be relaxed by assuming hðtÞ is the difference between two positive impulse responses From this it follows that a bound on the L 1 norm is given by Because both h þ ðtÞ and h À ðtÞ are assumed positive for all t, arguments mirroring those in the two above subsections can be used to derive LMI conditions for both the strict positive real condition (47) and the guarantees of positivity. The L 1 constraint can again be simplified to the inequality which is linear in κ þ c i , κ þ a i , κ À c i and κ À a i : we have a system of LMI's as before.

Remarks on the structured approach
There are two main criticisms which could be levelled at the fixed structure approach.
Complexity/conservatism of approximation: In common with all Zames-Falb multiplier searches, the approach of [18] searches over only a subset of these multipliers, namely over the set The order of the multiplier is proportional to 4N in this case, where N is a free parameter indicating the accuracy of the approximation: large N will imply that RM N is in some sense a denser approximation of RM, but large N implies a large computational burden as the larger number of states imply numerous LMI variables. In short: there is a clear trade-off between conservatism and computational efficiency, and the choice of the integer N may be problem-dependent. In particular, as N increases, the problem becomes ill-conditioned.
The decomposition hðtÞ ¼ h þ ðtÞÀh À ðtÞ: Although this decomposition is not conservative when N approaches infinity (see [78]), for finite values of N it may introduce some conservatism, since a triangular inequality is used to bound the L 1 -norm. This is confirmed by numerical results [15]. In particular, if hðtÞ changes sign (see Fig. 24) a nonsmooth function must be approximated with smooth functions, which is only possible if N approaches infinity. As previously mentioned, the problem becomes ill-conditioned as N increases; hence this decomposition may be conservative in practice.

Plant-order multipliers
An issue with the structured multipliers introduced in the foregoing section is that the conservatism is heavily dependent on the choice of N which determines the order of the multiplier. Indeed, for certain choices of multiplier a very large N would need to be chosen in order to reduce the conservatism to acceptable levels. In addition, although there is a clear trade-off between computational requirements and conservatism, the choice of N for a particular problem is by no means obvious.
For this reason, in a series of papers [74,75,71,12,76,16] the authors have developed an alternative method for searching for multipliers based on a change of variables similar to that used in H 1 controller design [67]. The main idea is that, if the multiplier is unstructured, but its order is the same as the plant, then a change of variables may be used to "linearize" some of the resulting matrix inequalities. The approach also uses a pseudo-LMI approach in order to bound the L 1 norm of the transfer function HðsÞ; the approach is conservative but it can be easily accommodated in the plant-order approach. The restriction of attention to plant order multipliers, RM PO & RM obviously introduces some conservatism a priori but the resulting search procedure is, with a slight caveat, entirely systematic.

Causal plant order multipliers
In this section we restrict ourselves to causal plant order multipliers and where degðHðsÞÞ ¼ degðGðsÞÞ ¼ n p . The assumption that H A RH 1 implies that the matrix A H is Hurwitz, but no structure is imposed. This absence of structure in the multiplier is required for the change of variables proposed later, but it also means that calculating J hj 1 accurately is generally difficult. Instead, a convenient upper bound from the literature will be used.
Theorem 3 (Scherer et al. [67] and Abedor et al. [1]). Let H A RH 1 . Then ð67Þ It is emphasized that Theorem 3 gives only an upper bound on the L 1 norm, J h J 1 r ξ; it may be extremely conservative. Another issue with Theorem 3 is that the two matrix inequalities are "not quite" LMIs due to the presence of the free scalar λ40. The consequence of this is that the plant order searches proposed here will take the form of LMI's plus a line search, which is computationally cumbersome compared to an LMI, but relatively easyand entirely systematicto implement. The first step in obtaining convenient plant-order multiplier searches is the partitioning of the matrix P ¼ P 0 in the positive real condition (47). It is assumed that P 4 0 and therefore that its inverse Q À 1 ¼ P exists. This allows one to write where each of the sub-matrices, P ij ; Q ij A R nÂn . Based on this partitioning, the following matrices are defined: Using the congruence transformation diagðΠ 1 ; IÞ and noting that Π 1 P ¼ Π 2 , the positive real condition (47) is equivalent to After some algebra, it can be deduced that this inequality is equivalent to  (66) and (67) is made. With this choice, the congruence transformation diagðQ 12 ; IÞ is applied to inequality (66) and the congruence transformation diag ðQ À 1 11 Q 12 ; I; IÞ is made to inequality (67). Under these congruence transformations, inequality equations (66) and (67) then become equivalent to Together, for fixed k, inequalities equations (71), (76) and (77) form a system of linear matrix inequalities plus a line search over λ40. This problem can be solved relatively easily using modern software and the multiplier can be recovered by using Eqs. (72)-(75).

Anticausal multipliers
A key restriction so far is that MðsÞðHðsÞÞ is assumed causal. Similar results but with MðsÞ assumed anticausal can be obtained with the aid of the following result.
Theorem 4 (Carrasco et al. [16]). Let H A RH À 1 . Then ð79Þ The consequence of Y o 0 is that when applying the KYP (Lemma 5), for nonsingularity of P to be guaranteed (as we need to use Q ¼ P À 1 ), instead of stipulating P 4 0, instead it is stipulated that P o0. Using a similar reasoning to before, it then follows that the positive real condition is satisfied if the matrix inequality (71) is satisfied. Similarly, invoking Theorem 4 and applying similar reasoning to the causal case, the L 1 inequalities become the following: Together inequality equations (71), (80) and (81) form a system of LMI's plus a line search over λ40. This set of LMI's is similar to the causal result given earlier but, since it results in the return of anticausal multipliers can sometimes yield much less conservative results.

Including Popov multipliers
Popov multipliers are not bounded on the imaginary axis and so, strictly speaking, do not belong to the class of Zames-Falb multipliers. However, following arguments given in Section 4 (see [14]), they can be considered as anticausal relaxations in the case of causal Zames-Falb multipliers; or, as causal relaxations in the case of anticausal multipliers. Space prohibits a full discussion, but it suffices to say that they are useful in the plant-order searches proposed earlier (see [71,76]).

Remarks on the plant-order approach
The plant order approach is a systematic search for Zames-Falb multipliers, but the nature of the approach is inherently restrictive: the order is fixed a priori, the manner of including the L 1 constraint has several sources of conservatism and, excluding the Popov terms, the multiplier returned is either causal or anticausal. A common criticism of multiplier techniques is the poor scaling of dynamic multiplier searches with problem complexity. The plant order approach described here also suffers from that due to the inclusion of the line search over λ, and, to a lesser extent, due to the full-block nature of the matrix variables in the LMIs. Another subtle issue with the plant order approach is that, as described here, it is only applicable to the case of ϕ being odd. This is not the case with the structured approach of [19] or the approach using irrational multipliers [17].

Application of search techniques to O'Shea's example
This section illustrates the application of the search techniques described to O'Shea's example. The Zames-Falb searches are compared to the well-known Circle Criterion, Park's Criterion [62] and also the non-rational Zames-Falb searches of [17], which have not been described in detail in the paper. The Zames-Falb searches used are the causal [74] and anticausal plant-order searches [16] described in Section 6.5, these same searches with the addition of Popov multipliers [71,76,16], and the structured searches of [18] from Section 6.4. The plant-order searches were performed by solving the LMI's given earlier together with a 100 element line search over logarithmically spaced λ. The Chen and Wen search is performed with a 18th order multiplier, comprising a 9th order causal and a 9th order anticausal part. Results are tested using the IQC toolbox [36].
The results of the various searches are shown in Table 2  The structured search of Chen and Wen deserves some explanation: the performance of this technique is highly dependent on multiplier order. For ζ A ½0:2; 0:6 with a 18th order multiplier, Chen and Wen's method took a similar computation time compared with other methods but provided significantly greater slope values. For higher order multipliers, the results deteriorate. This is likely to be due to numerical issues associated with the factorials in the basis representation. For ζ o0:05, Chen and Wen's method provided more conservative estimates of slope size, irrespective of multiplier order, than the plant order search. For low order multipliers this is due to the restricted bases ð1=ðs 7 1Þ N Þ used, while for high order multipliers numerical issues again become significant.
This suggests that results can be improved by a better selection of poles and this is indeed the case. Note that the IQC-toolbox can be used as a manual search tool by choosing an adequate location of the poles, hence the multiplier is given by for some selection p c 4 0 and p a 4 0. For a discussion in the selection of these values, see [32]. An inexperienced user may perform the search using a swap over these two values or a Monte Carlo approach. However performance then depends on the user's ability. For example, when ζ 4 0:5, experienced users will understand that the phase of this plant requires a selection of the poles that concurs with the solution proposed by O'Shea: a very fast causal pole and an anticausal pole at À 1. It does not seem surprising as O'Shea proposed such solution by hand 50 years ago. O'Shea's example illustrates clearly the complexities in finding the "best" multiplier, and the current lack of a complete and tractable method.

Safonov's search
It is the first tractable search proposed in the literature, Safonov and Wyetzner [65] proposed a search where the parametrization of the multiplier contained irrational terms, i.e. mðtÞ ¼ δðtÞÀ The main advantage of this search is the simplicity to test the time domain condition. However, it is not possible to check the frequency domain condition in a convex manner. The lack of an LMI implementation reduces the usefulness of this search since it cannot straightforwardly be combined with other classes of multiplier.
Originally, the impulses where equally distributed over a range of times resulting in a large optimization problem. To reduce the computational burden, Gapski and Geromel [28] reduced the size of the optimization by proposing an iterative method where the position of a new impulse δðt À t N þ 1 Þ is obtained if the search with N impulses fails. Recently, a new sub-algorithm has been proposed to improve this selection of the new impulse [17]. In Table 2, we have used the code developed by Chang and Safonov. 8 The results for ζ 4 0:55 are very good, but this search is not able to improve Circle criterion results for ζ o 0:5. Once again, anti-symmetry of the phase of ð1 þ kGÞ is a possible explanation. The search for the new t N þ 1 is designed to correct only one region of frequencies where there is a lack of positivity. So it is possible this the selected exponential for the multiplier is only able to "fix" one of both regions where there is lack of positivity. Note that as ζ approaches zero, non-positive regions are closer each other.

The IQC framework
We have framed our discussion in terms of passivity rather than IQCs (integral quadratic constraints) [54] which provided a new framework for multiplier theory. Whereas passivity, dissipativity, and Lyapunov theories can be used in any nonlinear interconnection, the IQC framework restricts its attention to the Lurye system. It therefore provides a natural framework in which to work with Lurye problems in general, and Zames-Falb multipliers in particular. Although Megretski and Rantzer [54] provide a self-contained frequency domain result, the IQC framework was developed in a combination of frequency and time domains by Yakubovich [87]. A natural extension of the original IQC framework to obtain stability conditions including Zames-Falb multipliers using convolution results in [23] has been developed in Altshuller et al. [4] and Altshuller [7].
Furthermore IQC theory provides not only self-contained stability results, but also computational tools to test stability conditions. For example, the search of Chen and Wen [18,19] is encapsulated within the IQC-β toolbox [36,53]. The IQC framework is especially useful if there is more than one nonlinearity or uncertainty in a feedback loop (e.g. [34]); the motivating example of Section 2 is most naturally expressed in the IQC framework.
Zames-Falb multipliers were used as an illustrative example by Megretski and Rantzer [54]. This sparked renewed interest in their properties and applications. It is therefore worth asking what advantages the IQC framework offers over and above the userfriendliness noted above. IQC theory dispenses with the requirement that multipliers can be factorized, and this is often claimed as an advantage over classical methods [54]. But the requirement that the Zames-Falb multipliers must be factorizable is no restriction on their generality [89]. It turns out that this is also the case whenever multipliers are used for a class of nonlinearities that includes a finite gain [13]. On the other hand the IQC framework allows additional properties of the nonlinearities to be included in the analysis (for example [71] include a tighter sector condition); to the best of the authors' knowledge the results of such analysis cannot be obtained via classical techniques. The relation between passivity theory and the IQC theorem is explored by Carrasco et al. [13]. The relation between dissipativity (and hence Lyapunov methods) and the IQC theorem is beginning to be understood [86,77,68,11].

MIMO nonlinearities
Similarly, our treatment has been restricted to SISO systems with slope restricted nonlinearities. The generalization of the Zames-Falb theorem to MIMO nonlinearities is discussed by Safonov and Kulkarni [66]. In particular, it is necessary that the nonlinearity can be expressed as the derivative of a convex potential function. This condition is natural for SISO nonlinearities, but may be restrictive in the MIMO case. In fact the condition was recognized in the classical literature [83]. It can be shown that the quadratic program used in input-constrained model predictive control satisfies the conditions for Zames-Falb multipliers [31]. Often attention is limited to diagonal nonlinearities; if the nonlinearities are repeated then the symmetry may be usefully exploited [21,40,72]. More generally, it is possible to construct specific multipliers appropriate for nonlinearities with repeated blocks [49].

Discrete-time multipliers
O'Shea also pioneered the equivalent multipliers for use with discrete-time systems [61]. Formal treatments can be found by [84] and [82]; their MIMO extension and necessity properties are given by Willems and Brockett [40] and Willems [49]. The discrete-time counterpart to the search for causal multipliers of Turner et al. [74] (and the anti-causal search of [16]) is developed by Ahmad et al. [2]. A search for a first-order FIR Zames-Falb multiplier is presented by Ahmad et al. [3]. A generalization to FIR Zames-Falb multipliers is presented by Wang et al. [81]. This approach appears highly promising, as it seems to combine the best aspects of the searches of Chen and Wen [18] and Safonov and Wyetzner [65]. To analyse the effectiveness of this method, the discrete counterpart of Lemma 4 by Megretski has been developed by Wang et al. [80]. An interesting application has been given in [44], where analysis and design of optimization algorithms have been carried out using Zames-Falb multipliers.

Different nonlinearities
Both Rantzer [64] and Materassi and Salapaka [51] allow relaxations on the condition of the nonlinearity. The nonlinearity considered in [64] is a perturbation of a nominal odd saturation function, while that considered by Materassi and Salapaka [51] is some perturbation of a more general nominal odd nonlinearity. In both cases positivity is preserved by further limiting the L 1 norm condition. The multipliers of Rantzer [64] can be shown to be applicable to a wider class of nonlinearity [42,41].

Open questions
Although there have been many recent advances in Zames-Falb multiplier theory, some questions are still open even for SISO systems in continuous-time. Here we provide some of these open questions.
Complete search: Different searches have been proposed in the literature. Results in this paper and comparisons by Carrasco et al. [15] show that no complete tractable search can be found in the literature. Advantages and drawbacks of each technique have been mentioned in Section 6. The development of a complete and tractable search remains an open challenge.
Instability criteria: Currently, searches are tested using only the Nyquist value as an upper bound. However, it is well known that this is not a tight bound for fourth-order systems or higher order.
For O'Shea's example small values of ζ produce unstable behavior. The maximum slope for searches decreases significantly as ζ approaches zero, though the Nyquist value remains infinity. In discrete-time, second-order counterexamples to the Kalman conjecture are given by Heath et al. [30], so the Nyquist values is only tight in general for first order systems. Tractable instability criteria must be developed to be able to understand the real conservativeness of the classes of multipliers and searches over these classes. Existence of limit cycles for this particular problem is discussed by Leonov and Kuznetsov [46], whereas general results are also available in the literature (e.g. [50]).
Dual problem: The dual problem in robustness analysis has been proposed by Jönsson [32] and Jönsson and Rantzer [33]. The limitations given by Megretski [52] provide a method to find when a Zames-Falb multiplier cannot be found. However, it is not understood how to use these limitations to discard the existence of a Zames-Falb multiplier suitable for a plant G and slope k.
Stability conjecture: Related with the two previous questions, we can state the following conjecture: Conjecture 1. Suppose we have a stable plant G and a constant k ok N such that there is no Zames-Falb multiplier M satisfying RefMðjωÞð1 þ kGðjωÞÞg Z δ for any δ40. Then there exists a slope-restricted nonlinearity ϕAS½ 0; k such that the feedback interconnection between G and ϕ is not Completeness: In discrete-time, the Zames-Falb multipliers are the only multipliers that can preserve the positivity of monotone and bounded nonlinearities [83,40,49]. In continuous time other multipliers, such as Popov multipliers, can also preserve the positivity of the nonlinearity yet are not themselves Zames-Falb multipliers. Nevertheless, all other classes of multipliers in the literature have been shown to be phase-equivalent to Zames-Falb multipliers [14,15]. It is an open question whether any possible class of multipliers preserving the positivity of monotone and bounded nonlinearities is phase-contained in the class of Zames-Falb multipliers.
Synthesis: The use of multipliers for synthesis has already been proposed by Veenman and Scherer [78]. Moreover, Zames-Falb multipliers have been already used in synthesis techniques [57,37]. The development of a convex synthesis technique using the Zames-Falb multipliers for anti-windup design is still open.
Local stability: The use of input-output stability criteria provides many advantages. However, open-loop unstable systems only can be locally stable in closed-loop. The use of IQCs for local stability has been considered by Fang et al. [25]. However, it is unknown whether multiplier theory has any role in the analysis of local stability.

Conclusion
This tutorial has attempted to provide a coherent introduction to the topic of Zames-Falb multipliers. We have shown a motivating example for using Zames-Falb multipliers in the robustness analysis of antiwindup. Their definitions, phase properties, and searches have been presented.
We have devoted a significant part of the paper to describing O'Shea's contribution, most notably O'Shea's observation that noncausal multipliers provide significant advantages over causal multipliers, in particular with respect to their phase properties. We have also shown, using O'Shea's original set of examples, the complexity in multiplier searches: for some ðζ 40:5Þ, a manual search remains best; while for others ðζ o0:5Þ the best achievable slope remains unknown and for different values of ζ different automated searches appear better.