The circle criterion for a class of sector-bounded dynamic nonlinearities

We present a circle criterion which is necessary and sufficient for absolute stability with respect to a natural class of sector-bounded nonlinear causal operators. This generalized circle criterion contains the classical result as a special case. Furthermore, we develop a version of the generalized criterion which guarantees input-to-state stability.


Introduction
The stability and convergence properties of Lur'e systems, a common and important class of nonlinear feedback systems, are a much researched area. Absolute stability theory seeks to conclude stability of the feedback system shown in Fig. 1  used sufficient conditions for absolute stability. It is well known that the circle criterion is not necessary for absolute stability with respect to real static nonlinearities.
Lyapunov approaches have been employed to deduce global asymptotic stability of unforced (that is, u = 0) Lur'e systems (see, for example, [2,11,13,14,16,17,19,28,29]), whilst input-output methods, pioneered by Sandberg and Zames in the 1960s, have been used to infer L 2 and L ∞ stability (see, for example, [5,28]). More recently, forced Lur'e systems have been analysed in the context of input-to-state stability (ISS) theory, with attention focussed on the extent to which results from classical absolute stability theory can be generalized to ensure certain ISS properties [1, 6-8, 10, 15, 16, 22-24]. Originating in the paper [25], ISS and its variants, such as integral inputto-state stability, are properties of general controlled nonlinear systems and, roughly, ensure a natural boundedness property of the state, in terms of initial conditions and inputs, see also the survey papers [4,26].
In the current paper, we consider the situation wherein the nonlinearity of the system shown in Fig. 1 belongs to a natural class of nonlinear causal operators which are sector bounded in an L 2 -sense and satisfy a weak local Lipschitz-type condition. In particular, the class generalizes the classical set-up and is sufficiently wide to account for operators with unbounded memory (described by nonlinear integral equations, for example) and for input-output operators of certain dynamical processes. We develop a generalized multivariable circle criterion which is necessary and sufficient for global asymptotic L 2 -stability (L 2 -GAS) of the closed-loop system shown in Fig. 1 for all operators in the class under consideration. The L 2 -GAS property implies in particular, that, for every L 2 -input u and all initial conditions, the solution of the closed-loop system converges to 0 as time goes to ∞, see Sect. 2 for details on the concept of L 2 -GAS. Furthermore, we derive an ISS version of this result, namely a circle criterion which is necessary and sufficient for ISS of the system in Fig. 1 for all nonlinear operators satisfying an exponentially weighted L 2 -sector condition. By applying the sufficiency part of the generalized circle criterion to the case wherein is the Nemytskii operator induced by a static nonlinearity, the classical circle criterion is easily recovered.
We emphasize that the nonlinearities considered are real in the sense that real input signals are mapped into real output signals. The key condition of the circle criterion is the positive realness of a certain rational matrix depending on G(s) := C(s I − A) −1 B, the transfer function of the linear system given by (A, B, C), and the (possibly dynamic) sector data. As has already been indicated above, the main contribution of the paper is the proof of the equivalence of the positive-real condition in the circle criterion and absolute stability with respect to all real nonlinear causal operators satisfying a suitable L 2 -sector condition, a result, which, in a sense, mirrors a well-known the-orem from stability radius theory: namely the identity r C (A, B, C) = r R, d (A, B, C), where r C (A, B, C) is the stability radius with respect to complex static linear perturbations and r R, d (A, B, C) is the stability radius with respect to real nonlinear causal L 2 -bounded perturbation operators, where it is assumed that A is asymptotically stable (see [13,Proposition 4.4] and [14,Theorem 5.6.20]). Furthermore, we remark that, in the complex case, the sufficiency of the circle criterion is trivial (in the sense that the proof carries over from the real case without change) and the necessity has been established in [9,Theorems 6.8 and 6.11], where it is shown, in a general infinite-dimensional systems setting, that stability with respect to all complex linear static feedbacks satisfying a sector condition determined by two matrices K 1 and K 2 implies the positive realness of (I − K 1 G)(I − K 2 G) −1 .
The layout of the paper is as follows. In Sect. 2, we discuss some preliminaries, present a number of auxiliary results and introduce the class of Lur'e systems which will be considered in the rest of the paper. Section 3 is devoted to small gain conditions for L 2 -GAS and ISS for Lur'e systems with nonlinear causal L 2 -bounded operators in the feedback loop. Contact will be made with the Aizerman conjecture and the work by Hinrichsen and Pritchard [13,14]. In Sect. 4, it is established that a natural generalization of the positive-real condition familiar from the circle criterion is sufficient for L 2 -GAS (ISS) for all nonlinear causal operators satisfying a suitable (exponentially weighted) L 2 -sector condition, whilst necessity of the positive-real condition for absolute stability is proved in Sect. 5. Finally, a proof of an auxiliary result from Sect. 2 is presented in Sect. 6.

Notation, terminology and auxiliary results
In this section, we present and discuss a number of preliminaries required for the development of the main results of the paper.

Notation
The fields of real and complex numbers are denoted by R and C, respectively. We set R + := [0, ∞) and, for α ∈ R, C α := {s ∈ C : Re s > α}. Let C α denote the closure of C α , that is, C α = {s ∈ C : Re s ≥ α}. Throughout, let F be the field of real or complex numbers, R or C, respectively. The matrix space F m× p is endowed with the operator norm induced by the 2-norm. For M ∈ C m× p , let M T and M * denote the transposition and Hermitian transposition of M, respectively, and, if m = p, we set We say that the matrix M ∈ C m×m is Hurwitz if all its eigenvalues have negative real parts. For M ∈ C m× p and N ∈ C q× p , we write We say that ϕ : R + × R p → R m is a (real) locally Lipschitz Carathéodory function if the function t → ϕ(t, z) is Lebesgue measurable for every z ∈ R p and z → ϕ(t, z) is locally Lipschitz, uniformly in t on compact intervals. The set of all locally Lipschitz Carathéodory functions R + × R p → R m is denoted by C(R + × R p , R m ).
We will make use of the Hardy spaces H ∞ p×m and H 2 p×m of holomorphic functions C 0 → C p×m with respective norms given by where the norm on the RHS is the operator norm induced by the 2-norm. We recall that a holomorphic function is in H 2 p×m if, and only if, it is the Laplace transform of a square-integrable function.

Stability and stability radii in the frequency domain
Let H be a rational matrix (the coefficients of the entries are not required to be real) of format p × m, frequently interpreted as the transfer function matrix of a finitedimensional linear time- For the special case wherein K(s) is constant, K(s) ≡ K , we introduce some convenient notation: If the rational matrix H is stable, we define the F-stability radius of H by We shall refer to the F-stability radius as the real or complex stability radius, depending on whether F = R or F = C, respectively. If H is stable and (A, B, C, D) is a stabilizable and detectable realization of H, then r F (H) coincides with the stability radius r F (A; B, C, D; C 0 ) of A with respect to the weighting (B, C, D) and the stability region C 0 as defined in [14,Sect. 5.3]. It is clear that r R (H) ≥ r C (H) and if H(s) ≡ 0, then there exists a destabilizing feedback K ∈ F m× p with K = r F (H). We note that r F (H) can also be expressed in the following form Furthermore, it is well known from [12,Proposition 2.1] or [14,Theorem 5.3.9] that The above identity implies that, in the context of linear output feedback with complex gains, the small-gain condition is sharp in the sense that there exists a destabilizing output feedback K ∈ C m× p such that H H ∞ K = 1 (provided that H(s) ≡ 0).
The following result shows that the complex stability radius plays a key role in the context of stabilization and destabilization by real dynamic linear feedback.
The key part of the proof of the above result relies on a construction from [ . We recall that a square rational matrix H is positive real if Re H(s) is positive semidefinite for all s ∈ C 0 which are not poles of H. It is well known that if H is positive real, then H is holomorphic in C 0 . Furthermore, H is said to be strictly positive real (strongly positive real ) if there exists ε > 0 such that s → H(s − ε) is positive real (H − ε I is positive real). If H is strongly positive real and does not have any poles on the imaginary axis, then H is strictly positive real.
The following characterization of positive-real properties in terms of norm conditions will be used later on.  We describe a number of scenarios in which the real supremum-value property holds. The positive realness of the function s → s −1 H 0 implies that I + 2kH is also positive real for all k ∈ (0, κ), and thus, it follows from [20, Lemma 3.10] that H −k I H ∞ = 1/k for all k ∈ (0, κ). As H −k I (0) = (1/k)I , we see that H −k I has the real supremumvalue property for every k ∈ (0, κ).
(e) In the previous examples, the supremum is achieved at s = 0. But there are many examples of transfer functions having the real supremum-value property for which the supremum is achieved at s = iω 0 for some ω 0 ∈ (0, ∞) and not at s = 0. Here, we provide one such example. Let a, b > 0 and consider the strictly proper stable rational function H given by .

Routine calculations yield
showing that H has the real supremum-value property with the supremum achieved at precisely two points in C 0 , namely ±i √ ab. ♦

Nonlinear operators
For q ∈ [1, ∞] and J ⊂ R an interval, let L q (J , R n ) denote the usual Lebesgue space of functions defined on J with values in R n . The local version of and equal to 0 on (t, ∞). We recall that an operator defined on L 2 (R + , R m ) or L 2 loc (R + , R m ) and mapping into L 2 loc (R + , R p ) is said to be causal if π t = π t π t for all t ≥ 0. If an operator defined on L 2 (R + , R m ) is causal, then it naturally extends to a causal operator on The causality of guarantees that (w) is a well-defined function in L p loc (R + , R p ). Furthermore, we note that a causal operator : L 2 loc (R + , R m ) → L 2 loc (R + , R p ) can be "localized" as follows: for every τ ∈ (0, ∞) and every w ∈ L 2 loc ([0, τ ), R m ), we define (w) ∈ L 2 loc ([0, τ ), R p ) by setting Again, this definition is meaningful by the causality of . For τ ≥ 0, the shift (or delay) operator S τ : has a transfer function H ∈ H ∞ p×m in the sense that, for every w ∈ L 2 (R + , R m ), the Laplace transform of H w is given by Hw, where w denotes the Laplace transform of w. Conversely, every H ∈ H ∞ p×m such that H(s) is real for all s ∈ (0, ∞) is the transfer function of a bounded linear shift-invariant operator H : We recall that the L 2 -induced operator norm of a bounded linear shift-invariant operator equals the H ∞norm of its transfer function.
We say that a causal operator : We set The operators in F R L p×m are precisely the input-output operators of asymptotically stable finite-dimensional linear time-invariant real state-space systems with m inputs and p outputs. Trivially, every operator in F R L m× p is causal and weakly Lipschitz.
In the following, let N (R + × R m , R p ) be the set of static nonlinearities ϕ ∈ C(R + × R m , R p ) which are uniformly linearly bounded, that is, The symbol ϕ should not be confused with the function (t, z) → ϕ(t, z) . We provide two classes of examples of operators which are causal and weakly Lipschitz.
Simple examples show that, in the absence of this lower semi-continuity condition, it is possible that N ϕ < ϕ . Finally, it is well known that N ϕ is continuous as a map from L 2 (R + , R m ) to L 2 (R + , R p ), see, for example, [18, Theorem 2.14].
Both of these operators are causal, weakly Lipschitz and linearly bounded. Operators of the form K • N ϕ and N ϕ • H, where K ∈ F R L p× p and H ∈ F R L m×m , are special instances of these types of operators. ♦

Lur'e systems with nonlinear causal operators in the feedback loop
Throughout the paper, let (A, B, C) ∈ R n×n × R n×m × R p×n and let G be the transfer function of the linear system given by and consider the following initial-value probleṁ Note that the differential equation in (2.3) is a forced Lur'e system given by the linear system (A, B, C), the nonlinearity and the forcing (or input) function u, see Fig. 1.
, we obtain the standard ODE initial-value probleṁ as a special case of (2.3). Obviously, as ϕ is memoryless, specification of the initial segment v is redundant. An absolutely continuous function x : By a suitable modification of the arguments used in [13] (where the uncontrolled case u = 0 is treated), it can be shown that, for every (t 0 , 3) has a solution on [t 0 , τ ) and that, for given t 0 < τ ≤ ∞, there exists at most one solution

Remark 2.6
The case wherein t 0 = 0 has a special feature which we wish to highlight. In this case, the initial segment v is simply a point in coincide on the open interval (0, τ ), and so, they are equal almost everywhere in [0, τ ).
Consequently, if t 0 = 0, then the initial segment v is irrelevant and, without loss of generality, we may assume that 3) defined on [t 0 , ∞) and the following two conditions are satisfied: When the system (2.3) is considered without forcing (u = 0), then the origin is said to be globally asymptotically stable (GAS) if there exists κ ≥ 0 such that, for and (ii) holds with u = 0. If property (ii) is satisfied with u = 0 and for some (fixed) t 0 ≥ 0, then the origin is said to be globally attractive at time t 0 .
Not surprisingly, if, in (2.3), ∈ F R L m× p , then the above stability and attractivity concepts are closely related to well-known frequency-domain properties. (

3) If (A, B, C) is stabilizable and detectable and the origin of (2.3) is globally
We remark that if F is not stable, then statement (2) is in general not true.

Proof of Lemma 2.7. Let
can be written as We also note that there exists c ≥ 0 (not depending on t 0 , x 0 , v or u) such that Consequently, on [t 0 , ∞), Using Laplace transformation, a routine calculation shows that where x * and u * denote the Laplace transforms of x * and u * , respectively.
The components of Cx * are strictly proper rational functions, and so since C x * (t) → 0 as t → ∞ (because, by hypothesis, x(t) → 0 as t → ∞), we conclude that C x * (t) converges to 0 exponentially fast as t → ∞. This in turn implies via (2.8) that, for every x 0 ∈ R n , the function (I − G(s)F(s)) −1 C(s I − A) −1 x 0 does not have any We proceed in two steps.
Step 1. Define a rational matrix H by H(s) : To this end, set T(s) := A + BF(s)C and note that Step 2. Using that G F is stable, the Hurwitz property of A F and Step 1, it follows from (2.6) and (2.9) that there exists a constant b ≥ 0 (not depending on t 0 , x 0 , v or u) such that By detectability, there exists H ∈ R n× p such that A + HC is Hurwitz. As x satisfieṡ for suitable a ≥ 0 (not depending t 0 , x 0 , v or u), we conclude that (2.3) with = F is L 2 -GAS.

Small gain conditions for L 2 -GAS and ISS
The first result in this section is a small-gain theorem for L 2 -GAS.
We note that (3.1) is equivalent to the small-gain condition G K H ∞ − K < 1, where we adopt the common engineering jargon wherein the norms of operators or transfer functions are referred to as gains of the corresponding systems.
Proof of Theorem 3.1 Let satisfy (3.1). We proceed in two steps.
Step 1. Assume that G ∈ H ∞ p×m and K = 0. Note that, by stabilizability and detectability, the matrix A is Hurwitz.
As in [13, proof of Theorem 3.12], it can be shown that there exists a constant c ≥ 0 (not depending on (t 0 , x 0 , v, u)) such that and routine arguments using the variation-of-parameters formula, the Hurwitz property of A and the linear boundedness of show that (2.3) is L 2 -GAS. We remark that [13,Theorem 3.12] is about GAS in the uncontrolled case u = 0 and not about L 2 -GAS, but an inspection of the proof shows that obvious modifications of the arguments in [13] establishes L 2 -GAS.
Step 2. Let (A K , B K , C K , D K ) be a minimal realization of K (with state dimension n K ) and setÂ , it now follows from Step 1 thaṫ is L 2 -GAS. Let x be the solution of (2.3). Obviously, Defining z as in (2.5) (with F replaced by K), we have thaṫ and there exists c ≥ 0 (not depending on (t 0 , x 0 , v, u)) such that ] v , and we conclude from (3.4) and (3.6) that ζ satisfies (3.2) with ζ 0 = col(x 0 , z(t 0 )),v = v andû = col (u, 0). The claim now follows from (3.5) and the L 2 -GAS property of (3.2).
Corollary 3.2 Let K ∈ F R L m× p with transfer function K, let ρ > 0 and assume that (A, B, C) is stabilizable and detectable. If 3) is L 2 -GAS for all causal and weakly Lipschitz nonlinearities : Proof By Proposition 2.1, ρ < r C (G K ), and thus, the claim follows from Theorem 3.1.
Roughly speaking, the above corollary says that if we consider all real feedback operators in the ball of radius ρ centred at K, then stability for all real linear compensators implies stability for all real nonlinear operators. As such, the corollary is reminiscent of the Aizerman conjecture.
The next result is a straightforward consequence of Proposition 2.1, Lemma 2.7 and Theorem 3.1.

Corollary 3.3 Let K ∈ F R L m× p with transfer function K, let ρ > 0, and assume that (A, B, C) is stabilizable and detectable. System (2.3) is L 2 -GAS for all causal and weakly Lipschitz nonlinearities
: Let : L 2 loc (R + , R p ) → L 2 loc (R + , R m ) be causal and weakly Lipschitz, and let ν ≥ 0. It is a routine exercise to show that the operator ν : L 2 loc (R + , R p ) → L 2 loc (R + , R m ) defined by ν (w) := e ν · (e −ν · w) is also causal and weakly Lipschitz. The operator ν is called the ν-exponential weighting of and will play a key role in the context of exponential input-to-state stability (ISS), a concept which we will now define. We say that (2.3) is exponentially ISS if there exist positive constants and γ such that, for all (t 0 , (3.7) A sufficient condition for exponential ISS of (2.3) is provided by the next result.

Corollary 3.4 Let
: L 2 loc (R + , R p ) → L 2 loc (R + , R m ) be causal and weakly Lipschitz and K ∈ F R L m× p with transfer function K. If (A, B, C) is stabilizable and detectable, G K ∈ H ∞ p×m and there exists μ > 0 such that

then (2.3) is exponentially ISS.
We identify a number of scenarios for which (3.8) is satisfied for sufficiently small μ > 0.
(b) Let L be a shift-invariant bounded linear operator from L 2 (R + , R p ) into itself such that the transfer function L of L is bounded and holomorphic on C −ε for some ε > 0. Then, for every ν ∈ (0, ε), L ν is a shift-invariant bounded linear operator from L 2 (R + , R p ) into itself with transfer function L ν (s) = L(s − ν) and thus L ν = sup Re s>−ν L(s) . It follows from [3, Theorem 3.7] that L ν −L H ∞ → 0 as ν → 0, and so, L ν − L → 0 as ν → 0. Let ϕ ∈ N (R + × R p , R m ) and consider the operator := N ϕ • L. It is clear that ν = N ϕ ν • L ν and thus (3.9) Consequently, assuming that G ∈ H ∞ p×m , the condition ϕ L < r C (G) is sufficient for (3.8) to hold with K = 0 and for sufficiently small μ > 0.
(c) In most instances, the inequality in (3.9) will of course be strict. Here, we consider a special case of part (b) for which equality holds. Let ϕ be time-independent and L = S τ , where τ > 0 and S τ is the delay (or shift) operator defined in (2.2). It is clear that S ν τ = e τ ν for all ν ≥ 0, and it can be proved that Hence, assuming that G ∈ H ∞ p×m , the condition ϕ < r C (G) is necessary and sufficient for (3.8) to hold with = N ϕ • S τ and K = 0 and for sufficiently small μ>0. ♦
The following result is an immediate consequence of Corollary 3.4 and part (a) of Example 3.5.

The circle criterion
In this section, we formulate a number of circle criteria for Lur'e systems of the form (2.3), including an ISS version. The textbook form of the circle criterion is contained in our considerations as a special case.
Throughout, let K 1 , K 2 ∈ F R L m× p be given and let : then is said to satisfy a sector condition (determined by the sector data K 1 and K 2 ). Similarly, we say that satisfies a strict sector condition if sup w∈L 2 , π t w =0, t≥0 The next lemma relates the above sector conditions to L 2 norm conditions and facilitates the proofs of the main results of this section In particular, if (4.1) holds, then is linearly bounded.
It is convenient to define the following sets of sector-bounded operators: The following corollary is a straightforward consequence of Lemma 4.1.

Corollary 4.2
The following statements hold.
Let ϕ ∈ C(R + × R p , R m ) and K 1 , K 2 ∈ R m× p and note that Throughout the rest of the section, let K 1 and K 2 be the transfer functions of the operators K 1 ∈ F R L m× p and K 2 ∈ F R L m× p , respectively. To formulate the main result of this section, we introduce the following assumption. (A) There exists an operator K # ∈ F R L p×m such that K # (K 2 − K 1 ) = I and ( If assumption (A) holds, then, for all y ∈ im (K 2 −K 1 ), we have that (K 2 −K 1 )K # y = y, showing that (K 2 −K 1 )K # = 1, and consequently, (K 2 − K 1 )K # H ∞ = 1. Below we describe some situations in which assumption (A) is satisfied.

Example 4.3 (a)
Assume that the sector data K 1 and K 2 are static, that is, K 1 = N K 1 and K 2 = N K 2 for some K 1 , K 2 ∈ R m× p . If ker(K 2 − K 1 ) = {0}, then, setting K := K 2 − K 1 , the matrix K T K is invertible and K # := (K T K ) −1 K T is a left-inverse of K . Moreover, K K # is the orthogonal projection onto im K along ker K # = ker K T = (im K ) ⊥ , and thus, K K # = 1. It follows that assumption (A) holds for N K 1 and N K 2 , provided that ker( Consequently, assumption (A) is satisfied if, and only if, (4.11) holds. In other words, assumption (A) is equivalent to det(K 2 − K 1 ) being a unit in the ring R R H ∞ .
(c) Assume that there exist K 1 , K 2 ∈ R m× p and K ∈ F R L p× p such that K 1 = N K 1 • K and K 2 = N K 2 • K. Assumption (A) holds if ker(K 2 − K 1 ) = {0} and inf s∈C 0 | det K(s)| > 0, where K is the transfer function of K.
(d) Assume that there exist K ∈ R m× p and L 1 , L 2 ∈ F R L p× p such that K 1 = N K • L 1   (1) Proof Let L and M be as in (4.3). By assumption (A), the operator L # := 2K # ∈ F R L p×m is a left-inverse of L and such that LL # ≤ 1. (1) Let ∈ S(K 1 , K 2 ). An application of statement (2) of Lemma 4.1 shows that there exists θ ∈ (0, 1) such that Defining a causal weakly Lipschitz operator : we obtain that where we have used that LL # ≤ 1. By hypothesis, H is positive real and thus it follows from (4.12) that LG M H ∞ ≤ 1 (see statement (1) of Lemma 2.2). We may now conclude that We now proceed in two steps.
Step 1: (4.16) is L 2 -GAS. As LG M is the transfer function of (Ã,B,C), it follows from (4.14) and Theorem 3.1 that (4.16) is L 2 -GAS (where the roles of G and K in Theorem 3.1 are played by LG M and 0, respectively), provided that (Ã,B,C) is stabilizable and detectable. We will show thatÃ is Hurwitz, which trivially implies stabilizability and detectability of (Ã,B,C). To this end, definê 0) and note that G M is the transfer function of (Â,B,Ĉ). Now, LG M is stable and so G M is stable because the transfer function of L # is a stable left-inverse of L. It now follows as in Step 2 of the proof of Theorem 3.1 thatÂ is Hurwitz which in turn implies thatÃ is also Hurwitz.
(2) Set denote the transfer function of K by K and, for ρ ≥ 0, define The idea is to prove that, for sufficiently small ρ > 0, the conditions of statement (1) hold with K 1 and K 2 replaced by K 1 − ρK and K 2 + ρK, respectively. A routine calculation shows that By hypothesis, H 0 = H is strictly positive real and hence does not have any poles in C 0 . As H 0 (∞) = I , H 0 is also holomorphic at ∞ and we conclude that H 0 is stable. It follows from [9, Corollary 4.5] that H 0 is strongly positive real, and an application of statement (2) of Lemma 2.2 shows that Consequently, and thus, H ρ is positive real for every ρ ∈ [0, ρ * ] by statement (1) of Lemma 2.2.
(2) Defining K as in (4.20), denoting the transfer function of K by K and using the arguments from the proof of statement (2) of Theorem 4.4, it can be shown that, for sufficiently small ρ > 0, the rational matrix (I − (K 1 − ρK)G)(I − (K 2 + ρK)G) −1 is positive real and ∈ S e (K 1 − ρK, K 2 + ρK). It now follows from statement (1) that (2.3) is exponentially ISS.
The textbook version of the circle criterion given in [2,11,17,28] is essentially statement (2) of Corollary 4.6 in the absence of forcing (u = 0); that is, the stability conclusion in the textbook version is global exponential stability for the unforced version of (2.4). The proofs in [2,11,17,28] are based on the positive-real lemma and Lyapunov theory, whilst Corollaries 4.5 and 4.6 have been derived by a small-gain argument.
If, in statement (1), the sector condition (4.10) is replaced by the weaker sector condition where α : R + → R + is an arbitrary K ∞ function (that is, α is continuous, strictly increasing and surjective), then, in general, system (2.4) is not exponentially ISS, but it is still ISS in the usual sense (see, for example, [4,25,26]

Necessity of the circle criterion
Here, we will investigate scenarios in which the (strict) positive realness of (I − K 1 G)(I − K 2 G) −1 is necessary for absolute stability. In particular, we will show that the positive real conditions in Theorem 4.4 are necessary for absolute stability with respect to all real nonlinear causal operators satisfying the relevant L 2 -sector condition. Throughout this section, let K 1 , K 2 ∈ F R L m× p and let K 1 and K 2 denote the transfer functions of K 1 and K 2 , respectively Proposition 5.1 Let (A, B, C) be stabilizable and detectable.
(1) Assume that < 1. Then, and statement (3) of Lemma 4.1 yields that ∈ S(K 1 , K 2 ), whence ∈ F R L m× p ∩ S(K 1 , K 2 ). By hypothesis, the origin of (2.3) is globally attractive at time 0, and hence, (2.3) is also L 2 -GAS by Lemma 2.7. Since x satisfies (5.2), we conclude that x(t) → 0 as t → ∞, which in turn implies that ζ(t) → 0 as t → ∞. This shows that the origin of (5.1) is globally attractive at time 0 for every ∈ F R L m×m such that < 1, and so, in particular,Ã is Hurwitz andG ∈ H ∞ m×m . An application of statement (1) It follows from statement (2) of Lemma 2.2 that H is strongly positive real and H ∈ H ∞ p×m and therefore H is also strictly positive real.
The following corollary shows that stability for all (real) nonlinear causal operators in a sector and the corresponding positive realness property are equivalent.    K 2 ). Then, trivially, the origin of (2.3) is globally attractive for all ∈ F R L m× p ∩ S e (K 1 , K 2 ). Since F ν −F → 0 as ν → 0 for every F ∈ F R L m× p , it follows from Lemma 4.1 that F R L m× p ∩ S e (K 1 , Invoking Proposition 5.1 shows that H is positive real. We remark that statement (1) of Corollary 5.2 has some overlap with [28,Theorem 126,Sect. 6.6] where, in a single-input single-output context, it is shown that stability (in the sense of L 2 -input-output theory) for all causal nonlinearities in a L 2 -sector (defined by static sector data) implies the positive-real condition of the circle criterion. The proof given in [28] relies on a result on minimal norm destabilization by linear delayed feedback of the form ρS τ , where ρ ∈ R (see [28,Lemma 112, Sect. 6.6]) and does not generalize to the general multivariable set-up considered in this paper.
Finally, we look at a scenario wherein, in the context of (real) static nonlinearities, positive realness is necessary for absolute stability. (2.4) and assume that (A, B, C) is stabilizable and detectable. Let K 1 , K 2 ∈ R m× p , define L := 1 2 (K 2 − K 1 ) and M :=
(1) If the origin of (2.4) is globally attractive for all ϕ given by ϕ(t, z) = K z with K ∈ R m× p satisfying sup z∈R p , z =1 K z − K 1 z, K z − K 2 z < 0, (5.3) and LG M has the real supremum-value property, then (I − K 1 G)(I − K 2 G) −1 is positive real.
(2) If the origin of (2.4) is globally attractive for all ϕ given by ϕ(t, z) = K z with K ∈ R m× p satisfying K z − K 1 z, K z − K 2 z ≤ 0 ∀ z ∈ R p , (5.4) and LG M has the real supremum-value property, then (I − K 1 G)(I − K 2 G) −1 is strictly and strongly positive real.
We remark that in general, on its own, global attractivity for all K ∈ R m× p satisfying ( (1) Let F be an arbitrary element in R m×m such that F < 1 and set K := F L + M ∈ R m× p . Then, sup z∈R p , z =0 K z − Mz Lz < 1, and since K z − K 1 z, K z − K 2 z = K z − Mz 2 − Lz 2 for all z ∈ R p and ker L = {0}, we see that K satisfies (5.3). Now, the argument from the proof of statement (1) of Proposition 5.1 can be used to conclude that all F ∈ R m×m such that F < 1 are in S R (G), whence 1 ≤ r R (G). AsG as the real supremum-value property, Lemma 2.3 shows that r R (G) = r C (G), and so, 1 ≤ r C (G) = 1/ G H ∞ . As in the proof of statement (1) of Proposition 5.1, it now follows that (I − K 1 G)(I − K 2 G) −1 is positive real.
(2) Arguing as in the prove of statement (1), we see that all F ∈ R m×m such that F ≤ 1 are in S R (G), implying that 1 < r C (G) = 1/ G H ∞ . As in the proof of statement (2) of Proposition 5.1, it can be shown that (I − K 1 G)(I − K 2 G) −1 is strictly and strongly positive real.
In the remark below, we describe how Proposition 5.3 can be used in the context of positive systems.

Remark 5.4
Let the matrices B and C be nonnegative and assume that there exists F ∈ R m× p such that A + B FC is Hurwitz and Metzler. Let P ∈ R m×m be nonnegative and set K 1 := F − P and K 2 := F + P. Then, defining L and M as in (4.3), we see that LG M = PG F is the transfer function of the stable positive system (A + B FC, B, PC), and so, by part (b) of Example 2.4, has the real supremum-value property. Proposition 5.3 shows that if the origin of (2.4) is globally attractive for all ϕ given by ϕ(t, z) = K z with K ∈ R m× p satisfying (5.3), then (I − K 1 G)(I − K 2 G) −1 is positive real (strictly and strongly positive real if global attractivity holds for all K satisfying (5.4)). ♦ Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Appendix
For completeness, we give a proof of Proposition 2.1.

Proof of
showing that H F = H K (I − (F − K)H K ) −1 has a pole at iω † , completing the proof.