Low‐gain integral control for a class of discrete‐time Lur'e systems with applications to sampled‐data control

We study low‐gain (P)roportional (I)ntegral control of multivariate discrete‐time, forced Lur'e systems to solve the output‐tracking problem for constant reference signals. We formulate an incremental sector condition which is sufficient for a usual linear low‐gain PI controller to achieve exponential disturbance‐to‐state and disturbance‐to‐tracking‐error stability in closed‐loop, for all sufficiently small integrator gains. Output tracking is achieved in the absence of exogenous disturbance (noise) terms. Our line of argument invokes a recent circle criterion for exponential incremental input‐to‐state stability. The discrete‐time theory facilitates a similar result for a continuous‐time forced Lur'e system in feedback with sampled‐data low‐gain integral control. The theory is illustrated by two examples.


INTRODUCTION
We consider Proportional Integral (PI) control in the context that the plant is specified by a system of controlled nonlinear difference equations called a forced Lur'e (also Lurie or Lurye) system-a well-studied and ubiquitous class of nonlinear control systems-comprising the feedback connection of a linear control system and a static nonlinear output feedback.
Integral control is a classical control engineering technique for robustly tracking constant reference signals, and refers to the feedback connection of an integrator and a (stable) plant. Low-gain integral control is a special case wherein the integrator gain is sufficiently small, which is known to be sufficient for closed-loop stability under a sign condition on the steady-state gain. Early literature on integral control includes References 1-7. Low-gain integral control has been further generalized to, for example: discrete-time systems; sampled-data systems; input-output approaches; classes of distributed parameter systems; adaptively determined integrator gains, and; integral control in the presence of input and output nonlinearities. Literature across these areas is vast and includes  Given the importance of output regulation in applied settings, much attention has been devoted to the extension of integral control, and related notions, to nonlinear plants, with early contributions including References 18,and 19,20 are recent papers in the area. These latter works both contain a bibliographic overview of contributions to integral control in the nonlinear setting, see also Reference 21, the research monograph 22 and the references therein.
The academic study of stability and convergence properties of Lur'e systems is broadly termed absolute stability theory, and is also a much-researched area. Absolute stability theory seeks to conclude stability (referring to a number of possible notions) via the interplay of frequency-domain properties of the linear component and sector or boundedness properties of the nonlinearity. Relevant background on absolute stability theory includes the texts, 23,24 and the papers 25,26 specifically consider the discrete-time case. Recently, a line of enquiry has arisen investigating how classical absolute stability type results generalise to guarantee the so-called input-to-state stability (ISS) property; see, for instance  As is well-known, ISS is a stability concept for nonlinear control systems which accommodates the contribution of exogenous control terms. For further background on ISS, we refer the reader to the survey. 30 Here we exploit recent ISS theory for discrete-time Lur'e systems developed in Reference 31 to derive sufficient conditions for the feedback connection of a usual (linear) low-gain PI controller and a forced Lur'e system to admit an exponential disturbance-to-tracking-error estimate. This result is in the spirit of the well-known circle criterion from absolute stability theory, and is presented in Theorem 1. In the absence of forcing terms, exponential convergence of the output to the desired reference is guaranteed. The results in Reference 31 are applicable here as the studied feedback connection may itself be written as a Lur'e system with an augmented state. Our working assumption is that the nonlinear component in the Lur'e system is not known and, therefore, is not available for feedback purposes. The main technical challenge is to obtain a result which is in the spirit of low-gain integral control and absolute stability theory-namely ensuring that there is a sufficiently small positive integrator gain * such that the closed-loop feedback system has desired stability and convergence properties for all integrator gains ∈ (0, * ) and all nonlinear terms satisfying, in this case, a suitable incremental sector condition.
As an application, we consider sampled-data low-gain integral control, now of a plant specified by a system of forced and controlled Lur'e differential equations. Sampled-data control broadly refers to controlling continuous-time plants by discrete-time controllers, via the use of sample-and hold-operations. For more background on sampled data control, we refer to the texts. 32,33 Proposition 1 is analogous to Theorem 1, and provides an exponential disturbance-to-tracking-error estimate for sufficiently small sampling times and integrator gains.
The manuscript is organised as follows. Section 2 contains the discrete-time theory, and also contains comparisons with known results from the literature. The results of Section 2 are used in the context of sampled-data control in Section 3.
Two worked examples are presented in Section 4, and Section 5 is the conclusion. Proofs of our results appear in the Appendix.
We let denote the (open) right-half complex plane and the exterior of the closed unit disc, respectively. For F = R or C, we let F n and F n×m denote n-dimensional (real or complex) Euclidean space, and the space of matrices with elements in F of format n × m, respectively. We equip F n with its usual 2-norm, denoted || ⋅ ||, which is induced by the usual inner product ⟨⋅, ⋅⟩ on F n . We use the same symbol || ⋅ || to denote the induced operator norm on For F = C 0 or E, we let H ∞ (F, C p×m ) denote the Hardy space of bounded, analytic, matrix-valued functions F → C p×m , with respective norms As usual, such functions will play the role of transfer functions of stable continuous-time and discrete-time linear control systems, respectively. Given a rational, matrix-valued function H ∶ F → C p×m , we say that a matrix K ∈ C m×p is feedback admissible for H if det(I − KH) ≠ 0 on F. Furthermore, in the case that m = p, we call H positive real if Re H(s) is positive semi-definite for all s ∈ F which are not poles of H; see, for example, Reference 34. Here F = C 0 or E corresponds to positive realness in the continuous-time and discrete-time settings, respectively. It is well-known, for example from Reference 34, Proposition 3.3 or Reference 31, Lemma 3.5, that rational positive real functions have no poles in C 0 or E ∪ {∞}, respectively. Furthermore, such a function H is called strongly positive real or strictly positive real if s  → H(s) − I or s  → H(s − ) is positive real, respectively, for some > 0.
For presentational reasons, we write column vectors inline as row vectors.

Preliminaries
Our focus is the following multivariate discrete-time controlled Lur'e system where x + (t) = x(t + 1) for all t ∈ Z + . We denote the linear data in (1) by Σ ∶= (A, B, C, D, G), with A ∈ R n×n , B ∈ R n×m 1 , C ∈ R p 2 ×n , D ∈ R n×m 2 , and G ∈ R p 1 ×n . Here m 1 , m 2 , n, p 1 , and p 2 are fixed positive integers. The term F ∶ R p 1 → R m 1 is a (nonlinear) function which shall require the properties defined in Assumption (A3) and Theorem 2.2. Roughly speaking, F is a function which will be required to satisfy a so-called incremental sector condition. As usual, the variables u, x, and y in (1) denote the input, state, and measured output, respectively, and they take values in R n , R m 2 , and R p 2 , respectively. The variables v i are exogenous input signals, which we call forcing terms. The terms v 1 , v 2 and v 3 take values in R n , R p 1 , and R p 2 , respectively.
We assume throughout that we have access to only the (noisy) output for control purposes. Thus, for output regulation of constant references, we introduce the low-gain integrator where r ∈ R p 2 is the desired reference, L I ∈ R m 2 ×p 2 is a (matrix) integrator gain, > 0 is a low-gain parameter, and w 0 ∈ R m 2 is the initial integrator state. In the case that m 2 = p 2 = 1, then we simply take L I = 1, leaving as the sole integrator gain parameter. We consider the feedback connection of (1), (2) and the PI controller u ∶= L P y + w, where L P ∈ R m 2 ×p 2 is the proportional feedback gain. Substituting the variables u and y into (1) and (2) yields the folllowing closed-loop feedback system It is clear that, for every (x 0 , w 0 ) ∈ R n × R m 2 and all , there is a unique solution of 3 which we denote by (x, w). We comment that (3) is in fact a forced Lur'e system with an augmented state. A block diagram of the closed-loop feedback system (3) is contained in Figure 1. For L P ∈ R m 2 ×p 2 , we set A L P ∶= A + DL P C, and let G CB denote the transfer function The functions G CD , G GB , G GD are defined analogously, and they capture the various input-output relationships in (3). We proceed to introduce assumptions used in our main result, Theorem 1 below, and convenient notation. To minimize disruption to the presentation, we provide commentary on the various assumptions after the statement of the theorem.
The first two assumptions pertain to the linear components of the model data Σ, L P and L I .
as the collection of forcing terms in (3) and their respective reference values (which may simply be zero). We set and define which shall play an important auxiliary and technical role in the current work. The third assumption connects the function F ∶ R p 1 → R m 1 and P(0): (A3) For all z 1 , z 2 ∈ R p 1 , there is a unique q ∈ R p 1 such that Our first lemma introduces quantities which shall appear as state-and input-limits in the feedback system (3).

Lemma 1.
Assume that Σ, L P and L I satisfy Assumptions A1-A3, and let > 0, r ∈ R p 2 and v † as in (4) be given.
Let q ∈ R p 1 be the unique solution of P(0)F(q +v 2 ) + 3 = q. Then, (x † , w † ) given by and is a constant solution of (3) with (constant) v = v † , and further satisfies Cx † = r −v 3 .

An exponential disturbance-to-state/tracking-error result
The following theorem is the main result of this section and, roughly, provides an exponential disturbance-to-state and disturbance-to-tracking-error estimate for (3) for all sufficiently small integrator gains and all nonlinear terms satisfying a given incremental sector condition. The result is in the spirit of the familiar circle criterion, but a recent version which is sufficient for exponential ISS. The derived estimates guarantee the complementary control objective of output-tracking, namely ensuring that Theorem 1. Consider (3) and assume that Σ, L P and L I satisfy A1 and A2. Let P be as in (5). Given K 1 , K 2 ∈ R m 1 ×p 1 , assume that K 1 is feedback admissible for P and G GB , and further that Then there exists * > 0 such that for every ∈ (0, * ), there exist Γ > 0 and ∈ (0, 1) such that, for every r ∈ R p 2 , every v and v † as in (4), every F ∶ R p 1 → R m 1 which satisfies A3 and and all (x 0 , w 0 ) ∈ R n × R m 2 , the solution (x, w) of (3) satisfies, for all ∈ Z + and all t ∈ N, where r † ∶= r −v 3 , and w † , x † are given by (7) and (8), respectively. The constant * depends on Σ, L P , L I , K 1 , K 2 , and the left hand side of (9), but not on F, x 0 , w 0 , or r. The constants Γ and depend on , Σ, L P , L I , K 1 , K 2 , and the left-hand side of (9), but not on F, x 0 , w 0 , or r.
Recall from the notation section that the symbols ⟨⋅ , ⋅⟩ in (9) denote the usual inner-product on (in this case) R m 1 . We provide commentary on the above theorem in terms of the result's hypotheses, conclusions and extensions.

Hypotheses
The hypotheses of Theorem 1 are the Assumptions A1-A3, the positive real assumptions (i) and (ii), and the incremental sector condition (9) on F. In the context of low-gain PI control of discrete-time linear systems, Assumptions A1 and A2 are known together to be sufficient for low-gain output regulation; see, for example Reference 14, theorem 2.5, remark 2.7. Observe that the invertibility and spectrum requirement in Assumption A2 necessitates that m 2 = p 2 and that L I is also invertible.
To discuss Assumption A3 requires more information on P in (5). In overview, P plays an important auxiliary and technical role in the current work by capturing essential input-output features of the linear components in the closed-loop Lur'e system (3)-particularly for small > 0. For which purpose, for > 0 we introduce and  ∶= and the associated transfer function K (z) = (I −  ) −1 . It is straightforward to see that  ,  and  comprise the linear components of the closed-loop Lur'e system (3) with combined state (x, w). A calculation shows that P(0) = K (1) for all > 0-the steady-state gain of K . Assumption A3 plays a key role in the proof of Lemma 1 which, essentially, entails that, for all reference terms r and persistent forcing terms v † , there is a unique constant solution (x † , w † ) of the feedback system (3) Assumptions of this type appear in other nonlinear integral control works, such as Reference 18, assumptions N.2 and N.3. That a steady-state gain P(0) = K (1) should appear in a condition for equilibria is natural.
Assumption A3 is always satisfied if P(0) = 0. For nonzero P(0), a sufficient condition for A3 to hold is that F is globally Lipschitz with Lipschitz constant less than 1∕||P(0)||. In this case the continuous function is a contraction (for all z 1 , z 2 ∈ R p 1 ), the upshot of which is that there is a unique solution of (6) by the Contraction Mapping Principle. The assumptions (i) and (ii) are various (strengthened) positive-real hypotheses. Note that the former is in a "continuous-time" sense-meaning positive real on the open right-half complex plane C 0 , and the latter is in a "discrete-time" sense-meaning positive real on the exterior of the closed complex unit disc E 0 .
That a positive-real condition on C 0 appears in a discrete-time result is as follows. We prove Theorem 1 by applying a recent circle criterion for exponential ISS (Reference 31, corollary 3.7) to the closed-loop Lur'e system (3) which, note, depends on > 0. Thus, we seek hypotheses on the model data that are both independent of and guarantee that (I − K 2 K )(I − K 1 K ) −1 is positive real on E 0 for all sufficiently small > 0.
Very roughly, a careful argument shows that K (z) approaches P(s), for some s ∈ iR ∪ {∞}, as → 0 and z → 1. In other words, there is no single limit of K (z) as (z, ) → (1, 0), rather the "limit" depends on the behavior of (z − 1)∕ as z → 1 and → 0. The upshot is that the conjunction of the positive-real assumptions in (i) and (ii) are sufficient for (12) (The precise result is statement (c) of Lemma 3). The condition (9) is an incremental sector condition for F. To motivate this assumption note that, by Reference 31, corollary 3.7, positive-realness of (I − K 2 G GB )(I − K 1 G GB ) −1 and the (usual) sector condition are together sufficient (up to some minor technical assumptions) for exponential ISS of x given by the first equation in (3), with forcing term Dw + v 1 + DL P v 3 . The comparable hypotheses in Theorem 1 are the (stronger) positive-realness assumptions (i) and (ii), and the sector condition (9), the latter of which is simply an incremental version of (13). That an incremental condition should appear as a sufficient condition in Theorem 1 is unsurprising as, roughly, output regulation to a desired set point r introduces a new equilibrium (x † , w † ) into (3) when unforced, (see Lemma 1), which varies as r varies. Roughly, the conclusions of Theorem 1 follow once the shifted state (x − x † , w − w † ) is shown to be exponentially ISS. This leads to the value of (and difficulty in establishing) Theorem 1. Chiefly, the hypotheses imposed are on the to-be-controlled Lur'e system (1) in terms of Assumptions A1-A3, which are primarily input-output/steady-state gain conditions, the positive-real conditions, and the already-mentioned incremental sector condition (9). Indeed, the small integrator parameter > 0 does not appear in the hypotheses of Theorem 1. However, in the spirit of low-gain integral control, the exponential disturbance-to-state and disturbance-to-tracking-error stability conclusions obtained are valid for all sufficiently small and, in the spirit of absolute stability theory, for all nonlinear terms F satisfying (9). The dependence of the constants on the various terms is carefully stated.
Put differently, a slight strengthening of sufficient conditions for exponential ISS of the nonlinear plant become, in conjunction with Assumptions A1-A3, sufficient conditions for the stability of the closed-loop feedback system (3), for all sufficiently small integrator gains.
Conclusions A consequence of the estimate (10) In general, the rate of the above convergence will depend on the rate of convergence of v(t) to v † . However, if v = 0 and v † = 0, then the estimate (10) ensures that the convergence is exponentially fast. Note that Cx is the "true" output, whilst y = Cx + v 3 is the measured output, which is subject to the forcing (noise, measurement error) term v 3 . Furthermore, we see that constant or convergent plant state forcing terms v 1 and v 2 are rejected, and constant or convergent output forcing terms v 3 lead to an asymptotic tracking-offset ofv 3 for the true output. These features are consistent with low-gain PI control of linear control systems.

Extensions
It is straightforward to show that, under the hypotheses of Theorem 1, there exist M 1 > 0 and ∈ (0, 1) such that, for every r ∈ R p 2 , and for all x 0 ∈ R n , the solution x of (1) with u = L P y + w † , with y given by (1), satisfies, for all t ∈ N and ∈ Z + , ) .
In other words, a P-control plus a suitable constant provides a method for the output to track any prescribed reference.
Our proof of Theorem 1 shows that, if v = 0 and v † = 0, then the incremental sector condition (9) on F in Theorem 1 can weakened to sup z∈R p 1 z≠0 for all q = q(r) as in Lemma 1. However, verifying (14) requires additional knowledge of F and the linear data to determine q, and may only be suitable if, in practice, output-tracking of only a few references r is required. Finally, by way of extensions and variations of Theorem 1, we comment that: • Theorem 1 extends to the situation wherein the integrator state w is subject to a forcing term v 4 , that is, (2) is replaced by . Roughly, the hypotheses of of Lemma 1 and Theorem 1 do not change, and the stability conclusions of Theorem 1 remain valid, but the additional terms v 4 andv 4 are introduced into v and v † in (4) and then the estimate (10), and the resulting equilibria x † , w † , and r † change accordingly. For brevity, we do not give a formal statement.
• The conclusions of Theorem 1 remain true if the control variable u is replaced by u = L P (r − y) + w (noting sign change on L P here), although the resulting equilibria x † , w † , and r † change accordingly.
• Theorem 1 remains true if the state space for (1), which is currently R n , is replaced by a (possibly infinite-dimensional) Hilbert space X. We have presented the finite-dimensional case for simplicity, but the proof for more general X basically remains unchanged. Indeed, the key results from the literature used in the proof are those in References 31 and 14, which both treat the infinite-dimensional case. The various transfer functions, such as G CB , are no longer necessarily rational, but essentially the same positive realness arguments apply as in the rational case.
In the single-input, single-output setting (m 1 = m 2 = p 1 = p 2 = 1), the positive real condition (i) in Theorem 1 essentially follows from the positive realness condition (ii) and an additional single point condition, detailed in the next lemma. (3) with m 1 = m 2 = p 1 = p 2 = L I = 1 and assume that Σ and L P satisfy Assumptions A1 and A2. Let P be as in (5). Assume that K 1 ,

Estimating the maximal integrator gain and exponential decay rate
Theorem 1 guarantees the existence of two quantities which play an essential role-the so-called maximal integrator gain * > 0 and closed-loop exponential decay rate ∈ (0, 1) in (10). Since the conclusions of Theorem 1 are only guaranteed to hold for integrator gains ∈ (0, * ), determining suitable * is of great practical interest. Unfortunately, to the best of our knowledge, both * and are rather difficult to calculate exactly, and we proceed to discuss how they may be estimated, essentially by carefully inspecting the proof of Theorem 1.
In both cases, the key objects are the discrete-time triple ( , , ) in (11), and the corresponding transfer function K . The proof of Theorem 1 relies on the properties that and that (12) holds, for all ∈ (0, * ), for some * > 0. These claims are established in the technical result Lemma 3. The term * in (12) is that which appears in Theorem 1.
Restricting attention to the single-input single-output case in the control loop (meaning m 2 = p 2 = 1 = L I ), a consequence of the research of Coughlan 8 and his PhD thesis 35 is that (15) holds with 0 ∶= 1∕|f (G CD )| by Reference 35, ) .
It is shown in Reference 35, proposition 12.1.3, that . The quantity f (G CD ) can be difficult to compute, and a more conservative option for * is In particular, our line of argument gives that * is bounded from above by 0 . The condition (12) for * does not appear to produce a constructive method for determining * . Heuristically, the condition (12) can be tested graphically for candidate * > 0. We now turn attention to . By way of context, for usual low-gain PI control of linear systems, the exponential decay rate equals ( ) < 1. Presently, the key stability result invoked in the proof of Theorem 1 is Reference 31, corollary 3.7, which itself draws upon Reference 31, theorem 3.2. An inspection of that proof (Reference 31, column 1, p. 3031) shows that any such that will satisfy (10). Here M ∶= (K 1 + K 2 )∕2 and L ∶= (K 1 − K 2 )∕2. This latter condition does not seem to constructively determine , but the above H ∞ -norm may be computed for candidate . In particular, our argument gives that is bounded from below by ( + M).

2.6
Comparisons with existing literature We provide some comparisons between our results and others available in the literature. Much attention has been devoted to output regulation of linear control systems subject to input saturation, including specifically in the discrete-time case the works References 36 and 37. Output regulation is a more general problem than integral control, in that the desired reference signal is typically generated by a so-called exosystem, such as periodic signals, and naturally need not be constant. At the heart of these works are stabilisation of discrete-time linear control systems by bounded (specifically saturated) feedback, including References 38 and 39.
The overlap between References 36,37 and the present work is minimal, however, owing essentially to the fact that the nonlinear term irreconcilably appears in different places in the models under consideration. We consider linear PI control applied to a nonlinear system, while References 36 and 37 consider nonlinear (saturated) controls of a linear system. Our work is closer in spirit to Reference 18, which considers the same problem we do, but for rather general continuous-time nonlinear control systems. The key assumptions in Reference 18 are that the nonlinear plant is globally uniformly exponentially stable, for all constant input signals, and that the steady-state gain map is monotone. The conclusions pertain to global asymptotic (exponential) stability, rather than input-to-state stability considered here. The approach taken in Reference 18 is somewhat different to that here, as there the low-gain integral parameter is viewed as "sufficiently small" so that the closed-loop feedback system may be rewritten as a standard singular perturbation model, and singular perturbation techniques applied. The recent paper 20 effectively generalises and strengthens, 18 by weakening various assumptions and by including an anti-windup component in the integrator. The conclusions of Reference 20 are local, but only local assumptions are imposed, and also does not consider ISS properties.
Finally, we comment that the paper 40 considers the (exponential) synchronization problem of two continuous time Lur'e systems using PID control, but that work does not consider output tracking and the overlap is otherwise minimal.

SAMPLED-DATA LOW-GAIN INTEGRAL CONTROL OF LUR'E SYSTEMS
As an application we show that the theory developed in Section 2 facilitates the sampled-data low-gain integral control of the following multivariate continuous-time controlled Lur'e systeṁ Here u is a control input, and v 1 is an exogenous input, x is the state variable and y is the (true) output. We assume that F is locally Lipschitz, and we let H CB (s) ∶= C(sI − A) −1 B denote the transfer function associated with the continuous-time triple (A, B, C), and similarly for H CD and so on. The linear data (A, B, C, D, G) in (16) is as in Section 2.
For fixed sampling-period > 0, we define the sampling operator  by which is defined on all continuous functions R + → R p 2 and returns a sequence in  (Z + , R p 2 ). The zero-order hold operator  is defined as which maps  (Z + , R m 2 ) into the set of step-functions mapping R + → R m 2 . We assume that the (noisy) sampled output is another forcing term, is available for feedback purposes. We consider the feedback connection of (16) and the discrete-time low-gain integrator where, as before, r ∈ R p 2 is the desired reference, L I ∈ R m 2 ×p 2 is a (matrix) integrator gain, > 0 is a low-gain parameter, and w 0 ∈ R m 2 is the initial integrator state. If m 2 = p 2 = 1, then we simply take L I = 1, so that is the only integrator gain parameter.
The plant (16) and controller (17) are connected in feedback via the held integral control u = (w), which yields the closed-loop feedback systeṁx , where x is an absolutely continuous function R + → R n and w ∈  (Z + , R m 2 ), is a solution of (18) if, for all k ∈ Z + and all t ∈ [0, k ], and (18b) and (18c) are satisfied. Existence of a unique solution of (18) is a consequence of the above equality and (18).
Our main result of this section is the following proposition which, roughly, shows that, for sufficiently small sampling-times and sufficiently small integrator gains, the feedback system (18) admits an exponential disturbance-to-state and disturbance-to-output estimate. Output tracking is guaranteed in the absence of forcing. For simplicity, we take the nominal forcing value v † as zero.
it follows that, for all z 1 , z 2 ∈ R p 1 , there is a unique solution q ∈ R p 1 to Let r ∈ R p 2 . The following statements hold.
We provide commentary on the above result. First, the Assumptions B1-B3 are continuous-time analogues of Assumptions A1-A3. Here we are assuming that the to-be-controlled system (16) already has stable linear part A, and so no proportional feedback is included (i.e., there is no L P term in u). Including a sampled-and-hold proportional feedback term in (18) would complicate the analysis even further, and is beyond the scope of the present contribution.
Second, Assumption B2 requires that p 2 = m 2 and L I is invertible, and reduces to the usual low-gain integral control requirement that H CD (0) > 0 when k 2 = p 2 = 1.
Third, in light of combined assumptions on H GB and Q, it follows from Reference 34, corollary 4.5, that the hypothesis of strict positive realness of (I − K 2 H GB )(I − K 1 H GB ) −1 and (I − K 2 Q)(I − K 1 Q) −1 is equivalent to that of strong positive realness of these functions.
Fourth, our proof shows that in fact, there is some * > 0, such that for each ∈ (0, * ), there is some * > 0 such that the conclusions of the above proposition hold. In other words, all sufficiently small sampling periods "work," but the permitted small integrator gains will depend in general on the sampling period.
Unfortunately, estimating the maximal integrator gain * in the sampled-data setting seems even more challenging than in the wholly discrete-time situation considered in Section 2. Indeed, the proof of Proposition 1 extracts a discrete-time forced Lur'e system from (18)  and G ∶= where A ∶= e A , B ∶= ∫ 0 e As B ds and D ∶= ∫ 0 e As D ds.
The guidelines in Section 2.5 after Theorem 1 now apply for determining * , replacing (11) by (25). For example, * ∈ (0, 0 ( )), where 0 is such that (A ) < 1. The additional difficulty comes from the fact that in (25) must satisfy some additional "sufficiently small" properties, which are not constructive, although these may be artifacts of our proof.

EXAMPLES
We illustrate our results through two examples.

Example 1.
We consider the low-gain PI control of a scalar difference equation, a so-called Ricker model, namely for the Gold-spotted grenadier anchovy (Coilia dussumieri), see Reference 41. Here the state x(t) describes the biomass of mature individuals in a population at time-step t ∈ Z + , and and are positive parameters denoting the natural mortality and fishing mortality, respectively. The positive parameter > 0 is the maximum per-capita reproduction rate and > 0 affects the density-dependent mortality near equilibrium abundance (Reference 41, supporting information). Although the model (27) is simple, its inclusion is intended to illustrate the key ideas behind our results, without being obscured by numerous technical details. Note that (27) is a (scalar) unforced discrete-time Lur'e system with A = a ∶= e −( + ) , B = G = 1 and F(z) = f (z) ∶= ze − z . (Strictly speaking, f is only defined for nonnegative arguments and, to fit the framework of the current paper, we define f on all of R by extending by zero. This extension, although artifactual, is not seen in physically motivated examples.) We assume that y = x + v 3 , where v 3 is a measurement error term, so that C = 1. For simplicity, we assume that the other exogenous forcing terms v 1 and v 2 in (1) are zero. The closed-loop feedback system (3) simplifies to where L P ∶= k ∈ R is a proportional feedback parameter, r is the desired reference and is the small integrator parameter. We set L I = 1. We shall assume that k ∈ (−a, 0), so that a + k ∈ (0, 1). The zero equilibrium of the uncontrolled model (27) is globally asymptotically stable if ≤ 1 − a, and if > 1 − a, then is a nonzero equilibrium of the uncontrolled model (27), corresponding to a persistent population. We shall consider the latter situation and seek to apply Theorem 1 to (28) to raise x to a limiting population r > x ♯ . For this purpose, we verify the hypotheses of this result. Assumption A1 is satisfied as + > 0, and by our choice of k. Since G = C = B = D, the four transfer functions G CB , G CD etc. are all equal, and are all equal to In light of the above, Assumption A2 is trivially satisfied. Note that G is a linear fractional transformation, and hence so is J ∶= (1 − k 2 G)∕(1 − k 1 G) for all real k 1 and k 2 , as J is the composition of linear fractional transformations. Hence, the image of the complex unit circle under J is a circle whenever 1 − k 1 G(±1) ≠ 0, which is symmetric with respect to the real axis and crosses the real axis at J(−1) and J(1). Thus, J is strongly positive real whenever k 1 and k 2 are such that Here P in (5) is given by Therefore, Assumption A3 is also trivially satisfied, independently of f . Furthermore, (1 − k 2 P)∕(1 − k 1 P) is strictly positive real by Lemma 2 whenever (29) holds.
Since v † = 0, for each desired reference r ≥ 0, the terms 1 , 2 and 3 in Lemma 1 simplify somewhat, and the resulting unique q ∈ R in Lemma 1 is simply given by q = 3 = r.
To summarise, to track a particular (single) reference r, the hypotheses of Theorem 1 reduce to finding real k 1 and k 2 such that (29) holds and f satisfies the incremental sector condition (14) with q = r. The former condition is algebraic and the latter may be verified graphically in this scalar case. Note that the these hypotheses are in fact independent of the functional form of f . Common to all the following numerical simulations we take which yield x ♯ = 1.3946 < r. These parameter values have been chosen somewhat arbitrarily. We further take k 1 ∶= 0.0444 > 0.0403 = f (r)∕r and k 2 ∶= −0.125, which have been chosen so that the strong positive-real condition (29) and the sector condition (14) hold -the latter is seen graphically in Figure 2A.
Numerical simulations of the closed-loop feedback system (28) with model data as in (30) and v 3 = 0 are contained in Figure 2B. The integrator gains are varied with values contained in Table 1. In each case, convergence x(t) → r as t → ∞ is observed. The solution to the uncontrolled model (27) is also plotted for comparison.
We proceed to discuss various aspects of Theorem 1, starting with the choice of small integrator gain. Recall that estimating the maximal permitted integrator gain * , the existence of which is guaranteed by Theorem 1, is difficult in general, but becomes more tractable in this simple example. We follow the approach discussed in Section 2.5. Here Routine stability analysis (such as the Jury criterion) gives ( ) < 1 whenever 0 < < 0 ∶= 1 − (a + k) (= 0.9 with the numerical values in (30)). Further straightforward calculations give The technical result, Lemma 3, ensures that (1 − k 2 K )∕(1 − k 1 K ) is positive real as ↘ 0. Global exponential stability of the closed-loop feedback system is guaranteed for all ∈ (0, 0 ) such that (1 − k 2 K )∕(1 − k 1 K ) is positive real. Both conditions are satisfied by the gains used in Figure 2.
Next, regarding robustness with respect to the nonlinear term f , the maximal integrator gain * is independent of f . Figure 3B contains simulations of (28) with model data as in (30), = 0.3, v 3 = 0, and with nonlinear terms given by for positive constants c 1 , c 2 , and c 3 . The constants have been chosen so that f i (r) = f (r) for each i. The above functions have been chosen somewhat arbitrarily, and all satisfy the sector condition (14) with q = r. In each case, convergence of x(t) → r as t → ∞ is observed. The difference in transient behavior appears negligible. Finally, Figure 4 contains simulations of (28) with model data as in (30) As expected by the estimate (10), we see that the deviation |x(t) − r| is bounded, and grows as ||v 3 || ∞ = i grows.

TA B L E 1 Low-gain integrator parameters
In closing we comment that, for the models (27) and (28) to be biologically meaningful, x(t) ≥ 0 for every t ∈ Z + is required, which is not guaranteed by Theorem 1. To ensure that x(t) ≥ 0 is not violated, some constraint on w(t) in the difference equation for x in (28) is required, which is beyond the scope of the present contribution. Roughly, when trying to make x larger than its initial value, as is the case here, our simulations show that x does indeed remain nonnegative.

Example 2.
We consider low-gain sampled-data integral control of the following mass-spring-damper system with forcing, namely Here z(t) denotes the displacement of the mass from rest at time t, u is a control signal and v ∈ L ∞ loc (R + ) is a forcing term. The displacement is assumed to be measured, giving the observed variable y = z. Moreover, m s , k s , d s > 0 are constants, and f ∶ R → R is locally Lipschitz with f (0) = 0. The above model has linear damping, and the restoring force depends nonlinearly on z(t).
Writing (34) in first-order form, and connecting via sample-and-hold with a low-gain integrator gives (18) with and F ∶= f .
Furthermore, v 1 ∶= Dv and, for simplicity, v 2 is assumed equal to zero. We seek to apply Proposition 1. For this purpose, we proceed to verify the hypotheses. Assumption (B1) holds as m s , k s , d s are all positive. Furthermore, H CD (0) = 1∕k m > 0 and a straightforward calculation shows that Q(0), so that Assumptions (B2) and (B3) are satisfied.  (30), v 3 = 0 and functions f i .

F I G U R E 4
Simulation results from Example 1. State x(t) given by (28) plotted against t with model data as in (30) and v 3 as in (33).
For illustrative numerical simulations of the sampled-data low-gain integral control feedback system (18), we take With these choices, it follows that k 1 ∶= −0.2 and k 2 ∶= 0.5 renders both (I − K 2 H GB )(I − K 1 H GB ) −1 and (I − K 2 Q)(I − K 1 Q) −1 strictly positive real. Thus, the conclusions of Proposition 1 apply for every r ∈ R such that the function f satisfies the incremental sector condition (14) with q = r -in particular, for the present example we choose a saturated deadzone type function where sign(0) ∶= 0 and sign(z) ∶= z∕|z| otherwise. We explore varying , , x 0 and v. We simulated (18) numerically in Matlab R2020a. Specifically, the ode113 command was used to solve the differential equation (18a) over the interval [k , (k + 1) ], and then the difference equation (18c) was iterated once via (18a) and (18b). This provides the initial data to solve the differential equation (18a) on the next interval [(k + 1) , (k + 2) ], and the process repeats. Figure 5A contains a contour plot of (A ), where A is as in (25), against varying sampling-period and integrator gain . Choosing , > 0 such that (A ) < 1 is a requirement for the conclusions of Proposition 1 to hold, but is not sufficient by itself; see the commentary after the statement of the proposition. However, we use the contour plot to provide at least a guide for choosing and . Indeed, Figure 5B plots (y)(k) against k for varying , with additional model data As expected by the estimate (23), convergence over time of the sampled-output y to r is observed. The convergence appears slower as increases. Figure 6A plots (y)(k) against k for varying initial states x 0 i , with additional model data As expected by the estimate (23), convergence over time of the sampled-output y to r is observed. Finally, we illustrate the exponential disturbance-to-tracking error estimate by varying the forcing term v = v j . Figure 6B contains three simulations with additional model data Again, as expected by the estimate (23), we see that the deviation |y(k ) − r| is bounded, and grows as ||v j || L ∞ = 2j grows.

SUMMARY
Low-gain PI control for a class of multivariate discrete-time Lur'e systems has been considered. Specifically, exponential disturbance-to-state and disturbance-to-tracking-error stability properties of the feedback connection of a Lur'e system and the usual linear low-gain integral controller have been investigated. Our main result is Theorem 1, which provides a sufficient condition in the spirit of an incremental circle criterion for these stability properties to hold for all sufficiently small integrator gains and, in particular, ensures that the usual control objective y(t) → r as t → ∞ is achieved in the absence of persistent forcing or measurement error. The other key hypotheses for Theorem 1 are properties of the linear system, namely a stabilizability condition Assumption A1, a familiar sign condition on the steady-state gain Assumption A2, and a condition ensuring suitable state limits exist in closed loop Assumption A3. The rationale for our choice of controller is that the nonlinear term F in the to-be-controlled system (1) is not known and cannot be used in the design of the integrator. As such, we have sought to understand the performance (at least theoretically and qualitatively) of this controller when connected to Lur'e systems. A moral of our work is that, broadly, under certain assumptions, a linear PI controller qualitatively performs as expected when connected in feedback to a Lur'e system. As an application, in Section 3 we considered sampled-data low-gain integral control of continuous-time controlled Lur'e systems, wherein a continuous-time Lur'e system is connected in feedback via sampleand hold-operations to the discrete-time low-gain integrator considered in Section 3. Proposition 1 is the main result of this section.
On the one hand, our results are in the spirit of low-gain integral control in that we conclude closed-loop stability for all sufficiently small integrator gains, based on assumptions which are independent of the integrator gain. On the other hand, our result is in the spirit of the circle criterion-a classical absolute stability result-both in terms of assumptions (briefly, positive realness on the linear data and an incremental sector condition on the nonlinear term) and conclusions which ensure stability for all such nonlinear terms.
In closing, we comment that the previous works 42, 43 by the current authors have considered the utility of low-gain integral control for population management of linear models arising in theoretical ecology, as a potential tool for population conservation. However, linear models allow for unbounded, exponential growth which is not ecologically realistic and, in fact, nonlinear systems of Lur'e type are known to arise naturally in this setting; see, for example Reference 44. Therefore, the current paper in part paves the way for low-gain integral control in much more realistic ecological settings.

ACKNOWLEDGMENTS
The authors' contribution to an early draft of this work was financially supported by the EPSRC grant EP/I019456/1. Richard Rebarber's contribution to this work was partially supported by NSF Grant DMS-1412598. The authors are grateful to Hartmut Logemann, who commented on a draft of this work, and several anonymous reviewers whose comments helped them to improve the work.

CONFLICT OF INTEREST
The authors have declared no conflict of interest.

DATA AVAILABILITY STATEMENT
Data sharing not applicable to this article as no datasets were generated or analyzed during the current study. The MAT-LAB routines used to generate the numerical examples and figures in Section 4 are available from the corresponding author on request.

APPENDIX A. PRELIMINARIES
The Appendix is divided into three sections. The second and third contain the proofs of results in Sections 2 and 3, respectively. The first gathers some common preliminaries.
For K 1 , K 2 as in the statements of Theorem 1 or Proposition 1, define Routine calculations give that the sector condition (9) can be rewritten as for some > 0 (independent of z 1 , z 2 ). In particular, the above inequality gives that F is globally Lipschitz. Moreover, the inequality (A2) entails that L is bounded from below, and so L * L is invertible. Therefore, L ♯ = (L * L) −1 L * is a left inverse for L. Further calculations from (A2) yield that for some ∈ (0, 1) (e.g. Reference 31, proof of Corollary 3.7).

APPENDIX B. PROOFS FOR SECTION 2
Recall the notation A L P ∶= A + DL P C, which satisfies (A L P ) < 1 if 1 holds, and that P is defined in (5).
Proof. (Proof of Lemma 1) Let r, and v † be as described. We claim that where q is as the statement of the result. The existence of such a q is guaranteed by hypothesis 3. Indeed, we compute Gx ♯ using the definitions (7) and (8) to yield that where the final equality follows from (6). From the assumed invertibility of G CD (1) (and consequently of L I ), it follows that Therefore, again using (7) and (8), we compute that establishing the second equality in (B1). Next, rearranging (8) and invoking (B1) gives and, evidently, (B1) also yields that completing the proof.  K 1 P). With a slight abuse of notation, we view P as a map from the extended complex plane to itself. Note that trivially every complex s belongs to a set of the form + iR with real -these are lines parallel to the imaginary axis. If G CB (1)G GD (1) = 0, then P is obviously constant, and hence so is M. Thus, by the strong positive realness hypothesis on G GB . If G CB (1)G GD (1) ≠ 0, then P is a linear fractional transformation. By the assumption of same signs of 1 − K 1 G GB (1) and 1 − K 1 P(0), it follows by continuity that there exists sufficiently small * ∈ (−G CD (1), 0) such that 1 − K 1 P( * ) and 1 − K 1 G GB (1) have the same sign. Recall that G CD (1) > 0 by 2. Let ≥ * . By construction of * , it follows that P( + iR) is bounded, and hence P( + iR) is a circle. Moreover, P( + iR) is symmetric with respect to the real axis as The circle P( + iR) crosses the real axis at P(∞) = G GB (1) and P( ). The assumption on the signs of being equal implies that 1 − K 1 P( ) ≠ 0, as t  → P(t) is monotone. In particular, M has no poles with real part greater than or equal to * . Now M is the composition of linear fractional transformations, and so is itself a linear fractional transformation. In particular, M( + iR) is a circle (as M( + iR) is bounded), which is also symmetric with respect to the real axis as P is. Therefore, M( + iR) crosses the real axis at M( ) and M(∞). Since t  → M(t) is a real-valued, continuous function, which is positive at zero and infinity by hypothesis, it follows that We conclude that M is strictly positive real, as required. ▪ The next lemma collects properties of the linear components of the feedback Lur'e system (3), and is an essential ingredient in the proof of Theorem 1.

Lemma 3.
Assume that Σ, L P and L I satisfy Assumptions (A1) and (A2) and let > 0. Define  ,  and  as in (11). The following statements hold.
Proof. (a) The claim is well-known and is a key ingredient for low-gain integral control of linear systems; see, for example, Reference 14, theorem 2.5, remark 2.7.
(c) The proof is essentially a careful continuity argument. For symmetric matrices Q 1 and Q 2 , the notation Q 2 ≼ Q 1 or Q 1 ≽ Q 2 means that Q 1 − Q 2 is positive semi-definite. We shall frequently use the routinely-established claim that and, as a corollary, if Q 1 is positive definite, and ||Q 1 − Q 2 || is sufficiently small, then Q 2 is positive definite as well.
For notational convenience, define  by which is a continuous function of a matrix variable Γ (whenever K 1 is feedback admissible).
Observe that K 1 is feedback admissible for K , as K is strictly proper for all > 0. In particular, (K ) is well defined for all > 0. Moreover, (K ) is rational, and so holomorphic on E with the possible exception of (necessarily isolated) poles. For > 0, let Λ ⊆ E denote the (possibly empty) set of poles of (K ). We shall prove that there exists * > 0 such that, for all ∈ (0, * ), from which the claimed positive-realness follows Reference 31, lemma 3.5. For this purpose, observe that (P) is rational and strictly positive real by hypothesis, with P ∈ H ∞ (C 0 , C p 1 ×m 1 ) by 2. Hence, (P) does not have any poles in some open right-half complex plane containing C 0 . Since (P(s)) → (G GB (1)) as |s| → ∞, with positive-definite real part by hypothesis, the hypotheses of Reference 34, theorem 4.4, are satisfied and, by that result, there exist 1 , > 0 such that Fix ∈ (0, * ) and let (x, w) denote the solution of (3). Letting q, w † and x † be as in (6), (7), and (8), respectively, we introduce the shifted variablesx ∶= x − x † ,w ∶= w − w † andṽ ∶= v − v † . It is routine to verify that whereF ∶ R p 1 → R m 1 is given byF Similarly, since Cx † = r −v 3 , we see thatw where 2 ∶= −ṽ 3 . Combining the equalities from (B12) to the definition of 2 gives Clearly, (B14) is a forced Lur'e system with nonlinear termF and transfer function K . Since F is globally Lipschitz (see Appendix A), estimating || 1 || and || 2 || yields a positive constant c 1 such that The incremental sector condition (9) entails that for some > 0. In light of the above inequality, the positive-realness of K from statement (c) of Lemma 3, and the bounds (B15), an application of Circle Criterion for exponential ISS (Reference 31, corollary 3.7) to (B14) gives an estimate of the form (10) for the variablesx andw. The result of Reference 31, corollary 3.7 invokes theorem 3.2 of Reference 31. For ease of verification, Table B1 relates the relevant notation of these results to the notation used here. We note that Reference, 31 corollary 3.7, requires that Σ (notation there) is (exponentially) stabilizable and detectable-which is trivially satisfied in the present setting as ( ) < 1 by statement (a) of Lemma 3. Furthermore, K 1 TA B L E B1 Relationship between notation used in the derivation of (10)

Notation in Reference 31
Notation here Σ = (A, B, B e , C, D, D e ) ( , , I, , 0, 0) -see (11) fF K 0 is required to be an admissible feedback operator, meaning K 1 ∈ A(D) in notation of Reference 31. However, this holds since D = 0 and so trivially I − DK 1 = I is invertible. The estimate for Cx − r † follows from the estimate (10) and the fact that .

APPENDIX C. PROOFS FOR SECTION 3
The proof of Proposition 1 draws upon the material in Section 2. It is reasonably lengthy, but none of the steps are too involved. Some notation and routine consequences are needed. The outline is as follows: • By considering the evolution of x at the sampling points k for k ∈ Z + , we extract a discrete-time Lur'e system with states (x(k ), w(k)), to which Theorem 1 applies.
• However, to apply Theorem 1 at the sampling points, an error term is introduced.
• A small-gain feedback connection argument then enables us to eliminate this error term and obtain an exponential ISS estimate for x at the sampling points.
• As is standard for sampled-data arguments, it then remains to estimate x(t) for t between the sampling points.
Proof. (Proof of Proposition 1) The proof is divided into steps. ▪ Step 1: Gathering notation and consequences For > 0, we define A , B , and D as in (26). Since A is assumed Hurwitz, it is clear that (A ) < 1 for all > 0. We let G GB denote the transfer function of the discrete-time triple (A , B , G), and analogously for G CB , G CD , and G GD . It is straightforward to show that and similarly for the other transfer functions. Note that −A is invertible, as A is assumed Hurwitz.
Lemma 4. Given the notation and assumptions in Appendix C so far, it follows that there exists 0 > 0 such that, for all ∈ (0, 0 ), the function (I − K 2 G GB )(I − K 1 G GB ) −1 is strongly positive real.
Proof. The proof is essentially another careful continuity argument, and is somewhat similar to the proof of statement (c) of Lemma 3, mutatis mutandis, with the role of small > 0 there played by small > 0 here. We give a brief outline. The function (H GB ) satisfies the hypotheses of Reference 34, theorem 4.4 and, by that result, there exist 1 , > 0 such that where, recall,  is defined in (B2). Note, from their respective definitions in (26), that and, trivially, for > 0 that G GB (z) may be expressed as In light of (C4), for fixed 1 > 0, we can fix sufficiently large R > 0 such that, for all ∈ (0, 1 ) for some small 2 > 0, and so, for these z and Re (G GB (z)) ≽ I − 3 I = (1 − 3 )I, for some 3 ∈ (0, 1). Here we have used that (0) = I is trivially symmetric positive definite. By choosing 2 ∈ (0, 1 ) sufficiently small, it follows that if z ∈ E and |z − 1|∕ ≤ R, then − < Re ( z − 1 ) .
In light of (C3) and (C4), we can choose 0 < min{ 1 , 2 } such that, for all ∈ (0, 0 ), if z ∈ E and |z − 1|∕ ≤ R, then ||G GB (z) − H GB ((z − 1)∕ )|| < 4 , for some small 4 > 0 which is independent of ∈ (0, 0 ). By the continuity of  and (C2), provided that 4 is sufficiently small, the above estimate entails if z ∈ E and |z − 1|∕ ≤ R, then Re (G GB (z)) ≽ 1 I, completing the proof. ▪ The conjunction of the hypotheses (B1)-(B3) and the equalities (C1) yields that the discrete-time model data (A , B , C, D , G) satisfies Assumptions A1-A3 (there with L P = 0), for all > 0. An application of Lemma 1, gives w † and x † as in (7) and (8), respectively, with the terms G CB (1) replaced by G CB (1), and likewise for the other steady-state gains. In light of (C1), it is clear that w † and x † are in fact independent of > 0, and that Cx † = r, since all thev i terms are here equal to zero.
By construction, it follows that By rewriting the above as taking the limit as t ↘ 0, and invoking (C3), we conclude that (21) holds. This proves statement (1).
Step 2: Extracting a discrete-time Lur'e system at the sampling points In what follows, to make the exposition clearer, and unless stated otherwise, k ∈ N and m ∈ Z + are arbitrary. For , > 0, let (x, w) denote the solution of (18). We define whereF ∶ R p 1 → R m 1 is defined as in (B13) withv 2 = 0. We observe that (C7) is a special case of (B14), although the term 2 depends on x, and so is not an exogenous forcing term.
Step 3: Applying Theorem 1 to (C7) The remaining hypotheses of Theorem 1 are the positive real hypotheses, which we proceed to verify. In light of the equalities (C1), it follows that (I − K 2 P)(I − K 1 P) −1 is strongly positive real, since (I − K 2 Q)(I − K 1 Q) −1 is (see commentary after statement of the proposition). Recall that P is as in (5), but with linear data (A , B , C, D , G). An application of Lemma 4 yields the existence of 0 > 0 such that (I − K 2 G GB )(I − K 1 G GB ) −1 is strongly positive real for all ∈ (0, 0 ).
All the hypotheses of Theorem 1 are satisfied for each ∈ (0, 0 ), and an application of that result yields the existence of 0 = 0 ( ) > 0 such that, for all ∈ (0, 0 ), there exist c 1 , c 2 > 0 and 1 ∈ (0, 1) such that Here, and in what follows, c i > 0 will be multiplicative constants which appear in estimates, i ∈ (0, 1) shall be discrete-time exponential decay rates, and i > 0 are continuous-time exponential decay rates, that is, they shall appear in terms of the form e − i t . It is clear that there exist c 3 , c 4 > 0 such that where, recall, is as in (24). The conjunction of (C8) and (C9) nearly yields the estimate (22) for ||w − w † || -but there is still an additive term 2 involving the approximation error. We investigate this term next.
Step 4: A small-gain feedback connection argument We consider the term 2 . The positive-real condition on (I − K 2 H GB )(I − K 1 H GB ) −1 entails that (see, e.g., Reference 31,equations (32) to (34), for a discrete-time argument-the continuous-time case is the same). Recall that L and M are defined in (A1), and L ♯ is a particular left inverse of L.
Next, observe that x in (18a) may be expressed aṡ so that