A rolled-off passivity theorem

Given two nonlinear systems which only violate incremental passivity when their incremental gains are sufficiently small, we give a condition for their negative feedback interconnection to have finite incremental gain, which generalizes the incremental small gain and incremental passivity theorems. The property may be determined graphically by plotting the Scaled Relative Graphs (SRGs) of the systems, which provides engineering significance to the mathematical result.


Introduction
The small gain and passivity theorems are two of the fundamental pillars of nonlinear input/output systems theory, originating in a seminal paper of Zames [1].The former theorem states that the negative feedback of two systems is stable if the product of their gains is less than one; the latter guarantees stability if one system is passive and the other is strongly passive with finite gain.
Zames provides two versions of each theorem.The first version requires gain and phase properties to be satisfied with respect to the reference input/output pair u 0 (t) = 0, y 0 (t) = 0, and guarantees boundedness, or continuity of the operator at u 0 .The second version requires properties to be satisfied incrementally, or with respect to every input/output trajectory, and guarantees the stronger property of continuity everywhere.These stronger theorems are known as the incremental small gain and incremental passivity theorems.
The conditions of the small gain and passivity theorems are restrictive, and there are many feedback systems which are stable, but do not meet the assumptions of either theorem.A common issue in practice is that a system would satisfy the conditions of the passivity theorem, were it not for high frequency dynamics destroying passivity.The input/output gain, however, is small at these high frequencies.This issue played an important part in the development of adaptive control -see [2] and references therein.The prevalence of such systems has motivated several specialized stability results.The LTI mixed small gain/passivity condition of Griggs et al. [3] divides the frequency spectrum into frequencies at which two systems are passive, frequencies at which they have small gain, and frequencies at which they satisfy both criteria.This is generalized directly to nonlinear systems by Forbes and Damaren [4], using the terminology hybrid small gain/passivity, and is connected to the Generalized KYP lemma of Iwasaki and Hara [5] in reference [6].The nonlinear generalization of Griggs et al. [7] uses a pair of linear operators to define a "blended" supply rate which represents mixed small gain and passivity.The roll-off IQC was introduced by Summers, Arcak and Packard [8] to capture the roll-off of input/output gain at high frequency.All these mixed small gain/passivity results are non-incremental, guaranteeing boundedness of the feedback system, but not continuity.The secant condition also mixes small gain and passivity, guaranteeing the stability of a cascade of output-strictly passive systems with a condition on the product of their secant gains, which capture both gain and passivity information [9,10].The secant condition readily applies to the incremental case [11].Recent work has also generalized the passivity theorem to infinite dimensional LTI systems [12,13].
The subject of this paper is an incremental rolled-off passivity theorem, which applies to incrementally stable systems which only violate incremental passivity when their incremental input/output gain is small.Rather than requiring gain to roll-off over any particular frequency range, we simply require the gain to roll off when the phase shift, measured as an angle in signal space, exceeds π 2. The idea takes inspiration from the blended supply rate of Griggs et al. [7], however, rather than using a smoothly blended supply, we simply split the space of input signals into those pairs of signals where the two systems have incremental small gain, are incrementally passive, or both.We call this property incremental (µ, γ)-dissipativity.Unlike the results of references [4,7,14], we do not require systems to be incrementally (µ, γ)-dissipative for the same partition of signals.This maintains the "worst case" nature of the small gain and passivity theorems, but simplifies the verification of the property.The resulting condition for finite incremental gain bears a strong resemblance to the classical incremental small gain condition.
Methods exist for determining when LTI systems satisfy a mixed or hybrid small gain/passivity property [6,15].Determining whether such properties hold for a pair of nonlinear systems, however, is often difficult.In contrast, incremental (µ, γ)-dissipativity can be read directly from the Scaled Relative Graph (SRG) of a nonlinear system.SRGs have recently been introduced by Ryu, Hannah and Yin [16] for the study of monotone operator methods in optimization.The SRG allows incremental properties of operators, such as Lipschitz continuity and monotonicity, to be determined graphically, and leads to intuitive and rigorous proofs of convergence using geometric transformations in the complex plane.The SRG is particularly suited to proving tightness of bounds, and novel tightness results have been obtained [16,17].In reference [18], the author and his colleagues showed that the SRG generalizes the Nyquist diagram of an LTI transfer function (generalized by Pates [19]), and applied SRG techniques to the study of feedback systems.Theorem 2 of reference [18] shows that if SRGs corresponding to two systems are separated (and remain separated as one SRG is scaled), the negative feedback of the two systems has finite incremental gain.The distance between the two SRGs is a nonlinear stability margin.This result generalizes the Nyquist criterion [20] to stable nonlinear operators.The rolledoff passivity condition that we give in this paper guarantees that the relevant SRGs are separated, and is a special case of [18,Thm. 2].In order to keep this paper self-contained, we give a direct proof of the result.
There is, of course, a large body of work on more general stability theorems for feedback systems.For systems of the Lur'e form, that is, an LTI dynamic component in negative feedback with a static nonlinearity, dynamic multipliers, such as those proposed by Popov [21] and Zames, Falb and O'Shea [22,23,24], can be used to make the LTI component passive, without affecting the passivity of the static nonlinearity.The passivity theorem can then be used to conclude stability.Methods involving the gap and related metrics prove stability by showing the input/output graphs of two systems are separated [25,26,27,28,29].Absolute stability and multiplier methods are unified in the IQC framework introduced by Megretski and Rantzer [30].These methods guarantee boundedness of the input/output operator, but in general give no guarantee of continuity.Indeed, it has been shown that large classes of multipliers, while preserving passivity of static nonlinearities, destroy their incremental passivity, making it difficult to conclude continuity from multiplier methods [31].Continuity at a particular point in signal space, which isn't necessarily u 0 (t) = 0, y 0 (t) = 0, can be guaranteed by the use of differential techniques [32,33].The primary benefits of the result we present here are the guarantee of continuity everywhere, the ability of incremental (µ, γ)-dissipativity to capture common real world effects, and the ability to verify the required properties graphically.
After introducing some necessary preliminaries and giving a brief review of the theory of SRGs in Section 2, we formally introduce the property of incremental (µ, γ)-dissipativity in Section 3, and give it a graphical interpretation.We then state the main result of this paper in Section 4, Theorem 1, and give a direct proof of the result.An example is given in Section 5.

Signals and systems
We model systems as operators on a Hilbert space, which is a vector space of signals, equipped with an inner product ⟨⋅ ⋅⟩ and induced norm ⋅ ∶= ⟨⋅ ⋅⟩.
where ū(t) denotes the conjugate transpose of u(t).The inner product on We write L 2 for L 2 (R n ), where the dimension n is immaterial.
By an operator (on a Hilbert space H), we mean a possibly multi-valued map H ∶ H → H.The graph, or relation, of H, is the set {(u, y) u ∈ H, y ∈ H(u)}.We will use the notions of an operator and its relation interchangeably, and denote them the same way.The usual operations on functions can be extended to relations: Note that H −1 always exists, but is not an inverse in the usual sense.In particular, it is in general not the case that H −1 H = I.If, however, H is an invertible function, its functional inverse coincides with its relational inverse, so the notation H −1 can be used without ambiguity.
An operator H ∶ H → H is said to have finite incremental gain, or be Lipschitz continuous, if there exists some nonnegative µ < ∞ such that, for all u 1 , u 2 ∈ H, We say that an operator We say that H is ε-input strictly incrementally positive if, for all u 1 , u 2 ∈ H, Incremental positivity is, in general, a weaker property than incremental passivity (as defined by Zames [1]), however, the two are equivalent for causal operators on L 2 [34, p. 174].Incremental positivity is otherwise known as (operator) monotonicity, as introduced by Minty in 1961 [35], and popularized by Rockafellar [36].Monotonicity has since become a fundamental property in the field of mathematical optimization [37].We use the term incremental positivity, adopted by Zames [1], Desoer and Vidyasagar [34], and others, partly because the results of this paper form a natural extension of their work, and partly to avoid confusion with the unrelated notion of monotonicity introduced by Hirsch and Smith [38].

Scaled relative graphs
In this section, we briefly introduce the theory of SRGs.We give only the theory required for the proof of Theorem 1, and refer the interested reader to reference [16] for the complete theory of SRG interconnections and their use in the theory of optimization, reference [18] for the use of SRGs in systems theory and their relation to the Nyquist diagram, and references [16,17,18,19] for the computation of SRGs for particular systems.

Definition
We define the SRG on a general, possibly complex, Hilbert space H.The angle between u, y ∈ H is defined as If u 1 = u 2 and there exist corresponding outputs The SRG of an operator is a region in the extended complex plane, symmetric about the real axis, from which properties of the operator can be easily read.Each point on the SRG gives the relative gain and phase shift of the operator for one or more particular pairs of inputs.The SRG can be thought of as a nonlinear generalization of the Nyquist diagram.This is elaborated on in the following example.
Example 1.The upper half of the SRG of a stable LTI transfer function G is the hyperbolic-convex hull of the upper half of its Nyquist diagram [18,Thm. 4].Mathematically, this is the set where Co is the convex hull, Nyquist Intuitively, the hyperbolic-convex hull is obtained by taking the convex hull with arcs centred on the real axis, rather than straight lines.For example, the SRG of 1 (s + 1) is its Nyquist diagram, the circle with centre at 0.5 and radius 0.5.
The SRG of e −s (s + 1) is illustrated in Figure 1.⌟ We also give a simple example of the SRG of a nonlinear operator.
Example 2. Define the unit saturation sat(⋅) by The SRG of the unit saturation is the closed disc with centre at 0.5 and radius 0.5: {z z − 0.5 ≤ 0.5}.This is proved in [18,Prop. 12].⌟

Determining system properties from the SRG
The properties which can be read from the SRG are those that define SRGfull classes.If A is a class of operators, we define the SRG of A by Note that a class of operators can be a single operator, and the operators in a class need not act on the same Hilbert space.
A class A, or its SRG, is called SRG-full if The value of SRG-fullness is in the implication SRG (H) ⊆ SRG (A) ⇒ H ∈ A. This allows class membership to be determined graphically.Two examples of SRG-full classes are the class of operators which share a finite incremental gain bound, and the class of incrementally positive operators.
Proposition 1.Let H denote an arbitrary Hilbert space.An operator H ∶ H → H obeys for every u 1 , u 2 ∈ H, y 1 ∈ H(u 1 ) and y 2 ∈ H(u 2 ) if, and only if, its SRG belongs to the closed disc of radius µ: H obeys ) and y 2 ∈ H(u 2 ) if, and only if, its SRG belongs to the closed right half complex plane: Proof.The proof may be found in the proof of [16,Prop. 3.3], or the proof of Lemma 1 in Section 3.

System interconnection
The SRGs of interconnected systems can be determined or approximated from the SRGs of the components.For a full treatment of SRG interconnections, we refer the reader to [16], Theorems 4.1-4.5.Here, we describe what happens to the SRG under input and output scaling, summation, inversion and composition.
If C, D ⊆ C, we define the operation C + D to be the Minkowski sum of C and D, that is, We define inversion in the complex plane by re jω ↦ (1 r)e jω .Throughout this paper, any inversion of a complex number refers to this operation, which maps points outside the unit circle to the inside, and vice versa.The points 0 and ∞ are exchanged under inversion.This operation only differs from the usual complex inversion in that the complex conjugate is not taken; this is left out for convenience, as it allows us to work in the upper half complex plane.This is possible as the SRG is symmetric about the real axis.
Given an operator A, the operator αA is defined by u ↦ αA(u), and the operator Aα is defined by u ↦ A(αu).These definitions extend to classes of operators in the natural way.Given two classes of operators A and B, their sum A + B is defined to be {A Define the line segment between Define the right-hand arc, Arc + (z, z), between z and z to be the arc between z and z with centre on the origin and real part greater than or equal to Re(z).The left-hand arc, Arc − (z, z), is defined the same way, but with real part less than or equal to Re(z) (see [16] for a more formal definition).A class of operators A is said to satisfy the right hand (resp.left hand) arc property if, for all z ∈ SRG (A), Arc + (z, z) ∈ SRG (A) (resp.Arc − (z, z) ∈ SRG (A)).
Furthermore, if A is SRG-full, then αA and Aα are SRG-full.Proposition 3. If A is a class of operators, then Proposition 4. Let A and B be classes of operators, such that ∞ ∉ SRG (A) and ∞ ∉ SRG (B).Then:

if either A or B satisfies the chord property, then SRG (A + B) ⊆ SRG (A)+ SRG (B).
Infinity can be allowed by setting SRG (A + B) = {∞} if SRG (A) = ∅ and ∞ ∈ SRG (B).Proposition 5. Let A and B be classes of operators, such that SRG (A) and SRG (B) are nonempty and bounded.Then: 1. if A and B are SRG-full, then SRG (AB) ⊇ SRG (A) SRG (B).

if either A or B satisfies an arc property, then SRG (AB) ⊆ SRG (A) SRG (B).
Under additional assumptions, unbounded and empty SRGs can be allowed -see the discussion following [16,Thm. 4.5].
We conclude this preliminary section with a simple example of SRG composition.

Rolled-off passivity
In this section, we define a property which captures systems which have a roll-off in gain as their phase shift increases.Given a pair of input signals, the system must either be incrementally passive, with an incremental gain which is finite but in general large, or have small incremental gain.This is formalized as follows.
Definition 1.Let H ∶ L 2 → L 2 and µ, γ, ε ≥ 0. We say that H is ε-strongly incrementally (µ, γ)-dissipative if, for all u 1 , u 2 ∈ L 2 and all y 1 ∈ H(u 1 ), y 2 ∈ H(u 2 ), either: or both: and or all of (1), (2a) and (2b) hold.If ε = 0, we simply say that H is incrementally (µ, γ)-dissipative.⌟ Incremental (µ, γ)-dissipativity is defined independently of the frequency spectra of signals, however it captures those systems which are incrementally passive except for high frequency dynamics, when the system has low gain.Incremental (µ, γ)-dissipativity is easily verified for systems with low-pass dynamics whose passivity is destroyed by effects such as input saturation and small delays, as explored further in the example of Section 5.
Incremental (µ, γ)-dissipativity has an appealing graphical interpretation, developed in the following lemma.This lemma is especially useful as it allows the property of incremental (µ, γ)-dissipativity to be easily determined from the SRG of a system.Lemma 1.Let µ, γ > 0, ε ≥ 0, and let S ε µ,γ be the class of operators which are ε-strongly incrementally (µ, γ)-dissipative.Then where as shown in Figure 3. Furthermore, S ε µ,γ is SRG-full.Proof.We begin by showing SRG (S ε µ,γ ) ⊆ D 1 ∪D 2 .Let H ∈ S ε µ,γ and u 1 , u 2 ∈ L 2 be arbitrary inputs.Then, by assumption, for all y 1 ∈ H(u 1 ), y 2 ∈ H(u 2 ), either (1) is true, or (2a) and (2b) are true, or all three inequalities are true.Suppose first that (1) is true.Then We now treat the second case.Suppose that (2a) and (2b) are true.Note that, for z ∈ z H (u 1 , u 2 ) corresponding to outputs y 1 , y 2 , It then follows from Equation (2a Since u 1 and u 2 were arbitrary, it follows that SRG (S ε µ,γ ) ⊆ D 1 ∪ D 2 .To show the opposite inclusion, let z ∈ D 1 ∪D 2 be arbitrary, and consider The fact that A z ∈ S ε µ,γ is shown using the following argument, which also proves SRG-fullness of S ε µ,γ .
We conclude this section with two examples of incrementally (µ, γ)-dissipative systems.
Example 4. We begin by revisiting the LTI transfer function of Example 1, G = e −s (s + 1).Its SRG is shown again in the left of Figure 4. We can read directly from the SRG that this system is (µ 1 , γ 1 )-dissipative, with µ 1 = 0.7581 and γ 1 = 1.The circles with centres at the origin and radii of µ Example 5. We now revisit the system of Example 3 -the cascade of the transfer function 1 (s + 1) with a unit saturation.In Example 3, a bounding approximation of the SRG of this system was obtained -this is repeated on the right of Figure 4.It can be read directly from the figure that this system is incrementally (µ 2 , γ 2 )-dissipative, with µ 2 = 0.5 and γ 2 = 1.Again, the relevant circles are shown in the figure as dashed lines.For any μ > µ 2 , it can be verified graphically that there exists an ε > 0 such that this system is ε-strongly incrementally (μ, γ 2 )-dissipative.⌟ Then the feedback interconnection of H 1 and H 2 shown in Figure 5 maps L 2 to L 2 and has finite incremental gain from u to y if Note that setting µ 1 = µ 2 = 0 and letting γ 1 , γ 2 → ∞ recovers the incremental passivity theorem, and setting γ 1 < µ 1 , γ 2 < µ 2 recovers the incremental small gain theorem, for operators on L 2 [34].The focus for the remainder of this paper will be cases where neither the incremental passivity theorem nor the incremental small gain theorem apply.We do not make any assumptions about causality of operators, and neither do we give any guarantees -causality must be treated separately.Theorem 1 follows as a corollary of [18,Thm. 2].In the remainder of this section, we give a direct proof.The proof proceeds by taking bounding approximations of the SRGs of H 1 and H 2 , and using them to construct a bounding approximation of the SRG of the closed loop operator.This approximation is a bounded region in the complex plane, from which we conclude finite incremental gain, via Proposition 1.
Given two arbitrary systems H 1 and H 2 , it is not necessarily true that their feedback interconnection is admissible, that is, a well-defined operator on L 2 .If the systems satisfy the conditions of Theorem 1, however, admissibility is guaranteed.This is proven using a homotopy argument similar to Megretski and Rantzer [30].We scale the feedback operator by a gain τ ∈ (0, 1], and show that the mapping from τ to the closed loop incremental gain is continuous.This shows the finite incremental gain of H 1 is preserved as the feedback is introduced, and in particular, the closed loop system continues to map L 2 into L 2 .Note that the standard form of the incremental passivity theorem [34,Thm. 30,p. 184] requires admissibility as an assumption.The extra strength of Theorem 1 comes, loosely speaking, from the additional assumption that both operators H 1 and H 2 have finite incremental gain. Proof of Theorem 1.Let µ 1 , µ 2 , γ 1 , γ 2 and ε satisfy the conditions of the theorem.Let S µ1,γ1 be the class of operators which are incrementally (µ 1 , γ 1 )dissipative, and S ε µ2,γ2 be the class of operators which are ε-strongly incrementally (µ 2 , γ 2 )-dissipative.

Im
ε The shortest distance between the two SRGs, that is, min{ ) and all z 2 ∈ SRG (−τ S ε µ2,γ2 ), is given by This is determined from the figure above, allowing for the fact that one or both SRGs may have µ j > γ j .By the assumption of the theorem, all of these values are positive for τ ∈ (0, 1].Applying Propositions 2 (with α = −1) and 4, it follows that SRG (S ) is bounded away from zero by a distance r τ .Applying Propositions 1 and 3 allows us to conclude a finite incremental gain bound of 1 r τ for the class of operators (4), as illustrated below.

Feedback example
Example 6.Consider the feedback system shown in Figure 6.The forward path H 1 consists of a delayed first order lag.The feedback path H 2 consists of a unit saturation cascaded with a first order lag and static gain k.
The Note that neither H 1 nor H 2 is incrementally positive (their SRGs are not contained in the right half plane), nor do they obey the incremental small gain condition for k > 1 (the product of the maximum moduli of their SRGs exceeds 1).However, it follows from Theorem 1 that the feedback system has finite incremental L 2 gain for all 0 < k < 1.3191.⌟

Conclusions
Theorem 1 guarantees finite incremental gain of the negative feedback interconnection of two systems, where, for every pair of input/output trajectories, the systems satisfy either an incremental small gain condition or an incremental passivity condition.This property, which we call incremental (µ, γ)-dissipativity, captures systems which have either small gain or small phase shift (or both), and includes systems with low-pass dynamics whose passivity is destroyed at high frequencies by effects such as saturation and delay.
A primary advantage of incremental (µ, γ)-dissipativity is that it can be verified graphically from the SRG of a system, as shown in Lemma 1.This makes the property both intuitive, and simple to verify.
The set CD is defined to be the Minkowski product of C and D,CD ∶= {cd c ∈ C, d ∈ D}.The set αC is defined by αC ∶= {αc c ∈ C}.

Example 3 .Figure 2 :
Figure 2: Illustration of the composition used in Example 3. The SRG on the right, a cardioid, is a bound on the SRG of a unit saturation cascaded with the transfer function 1 (s + 1).