Absolute stability analysis of discrete time feedback interconnections*

Proceedings of the 20th World Congress Proceedings of 20th The International Federation of Congress Automatic Control Proceedings of the the 20th World World The International Federation of Congress Automatic Control The International Federation of Automatic Control Toulouse, France, July 2017 Proceedings of the 20th9-14, World Congress The International Federation of Automatic Control Toulouse, France, July 9-14, 2017 Available online at www.sciencedirect.com Toulouse, France, July The International of Automatic Control Toulouse, France,Federation July 9-14, 9-14, 2017 2017 Toulouse, France, July 9-14, 2017

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IFAC PapersOnLine 50-1 (2017) 8447–8453 Absolute stability analysis of discrete Absolute Absolute stability stability analysis analysis of of discrete discrete  feedback interconnections Absolute stability analysis of discrete  feedback interconnections feedback interconnections  feedback interconnections Matthias Fetzer ∗∗ Carsten W. Scherer ∗∗

time time time time

Matthias Fetzer ∗∗ Carsten W. Scherer ∗ Matthias Matthias Fetzer Fetzer ∗ Carsten Carsten W. W. Scherer Scherer ∗∗ Matthias Fetzer Carsten W. Scherer Department of Mathematics, University of Stuttgart, Pfaffenwaldring Department of Mathematics, University of Stuttgart, Pfaffenwaldring Department of Mathematics, University of Stuttgart, Pfaffenwaldring 5a, 70569 Stuttgart, Germany (e-mail: {matthias.fetzer, Department of Mathematics, University of Stuttgart, Pfaffenwaldring 5a, 70569 Stuttgart, Germany (e-mail: {matthias.fetzer, ∗ Department of Mathematics, University of Stuttgart, Pfaffenwaldring 5a, Stuttgart, Germany (e-mail: {matthias.fetzer, carsten.scherer}@mathematik.uni-stuttgart.de). 5a, 70569 70569 Stuttgart, Germany (e-mail: {matthias.fetzer, carsten.scherer}@mathematik.uni-stuttgart.de). 5a, 70569 Stuttgart, Germany (e-mail: {matthias.fetzer, carsten.scherer}@mathematik.uni-stuttgart.de). carsten.scherer}@mathematik.uni-stuttgart.de). carsten.scherer}@mathematik.uni-stuttgart.de). Abstract: The problem of absolute stability of aa discrete time feedback interconnection is Abstract: The problem of absolute stability of time feedback interconnection is Abstract: The problem of absolute stability of aa discrete discrete time feedback interconnection is revisited. For the case of a slope-restricted nonlinearity in feedback with a linear time invariant Abstract: The problem of absolute stability of discrete time feedback interconnection is revisited. For the case of a slope-restricted nonlinearity in feedback with a linear time invariant Abstract: The problem of absolute stability of a discrete time tests feedback interconnection is revisited. For the case of a slope-restricted nonlinearity in feedback with aa linear time invariant system it is shown that several of the recently proposed stability are special cases of the revisited. For the case of a slope-restricted nonlinearity in feedback with linear time invariant system it is shown that several of the recently proposed stability tests are special cases of the revisited. For the case ofseveral a slope-restricted nonlinearity in feedback withare a test linear time invariant system it is shown that of the recently proposed stability tests special cases of the Zames-Falb criterion. Moreover a new computationally tractable stability is proposed that system it is shown that several of the recently proposed stability tests are special cases of the Zames-Falb Moreover aa new tractable stability is proposed system it is criterion. shown that several of the computationally recently proposed stability testsmultipliers. aretest special cases ofthat the Zames-Falb criterion. Moreover computationally tractable stability test is that simultaneously employs full-block circle, Yakubovich and Zames-Falb Zames-Falb criterion. Moreover a new new computationally tractable stability test is proposed proposed that simultaneously employs full-block circle, Yakubovich and Zames-Falb multipliers. Zames-Falb criterion. Moreover a new computationally tractable stability test is proposed that simultaneously employs full-block circle, Yakubovich Zames-Falb multipliers. simultaneously employs full-block circle, Yakubovich and and Zames-Falb multipliers. © 2017, IFAC (International Federationcircle, of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. simultaneously employs full-block Yakubovich and Zames-Falb multipliers. Keywords: Absolute stability, discrete-time systems, integral quadratic constraints, Keywords: Absolute stability, discrete-time systems, integral quadratic constraints, Keywords: stability, discrete-time discrete-time systems, slope-restricted nonlinearities, stability tests. Keywords: Absolute Absolute stability, systems, integral integral quadratic quadratic constraints, constraints, slope-restricted nonlinearities, stability tests. Keywords: Absolute stability, discrete-time systems, integral quadratic constraints, slope-restricted nonlinearities, stability slope-restricted nonlinearities, stability tests. tests. slope-restricted nonlinearities, stability tests. 1. INTRODUCTION related criteria (see, e.g., Ahmad et al. [2010, 2011, 2013a, 1. INTRODUCTION INTRODUCTION related criteria (see, e.g., Ahmad et al. [2010, 2011, 2013a, 1. related criteria (see, e.g., Ahmad et al. [2010, 2011, 2013a, 1. INTRODUCTION 2015], Wang et al. [2014], Boczar et al. [2015]). related criteria (see, e.g., Ahmad et al. [2010, 2011, 2013a, 2015], Wang et al. [2014], Boczar et al. [2015]). 1. INTRODUCTION related criteria (see, e.g., Ahmad et al. [2010, 2011, 2013a, 2015], Wang et al. [2014], Boczar et al. [2015]). In this paper we consider discrete time feedback intercon2015], Wang et al. [2014], Boczar et al. [2015]). In this paper we consider discrete time feedback interconIn case of just scalar nonlinearity, result obtained In this paper we consider consider discrete time feedback intercon2015], Wang etone al. [2014], Boczar et al.the [2015]). nections involving a linear time-invariant (LTI) system G In this paper we discrete time feedback interconIn case of just one scalar nonlinearity, the result obtained nections involving lineardiscrete time-invariant (LTI) system system G In case of just one scalar nonlinearity, the result Wang al. [2014], which is based on mulIn case ofet just one scalar nonlinearity, theZames-Falb result obtained obtained In this paper weoperator consider time feedback interconnections involving aaa linear time-invariant (LTI) G and a nonlinear ∆ defined via a slope-restricted or nections involving linear time-invariant (LTI) system G by by Wang et al. [2014], which is based on Zames-Falb muland a nonlinear operator ∆ defined via a slope-restricted or In case of just one scalar nonlinearity, the result tests obtained by Wang et al. [2014], which is based on Zames-Falb multipliers, is shown to subsume all earlier stability and by Wang et al. [2014], which is based on Zames-Falb mulnections involving a linear time-invariant (LTI) system G and a nonlinear operator ∆ defined via a slope-restricted or sector-bounded nonlinearity (see Fig. 1). Stability analysis and a nonlinear operator ∆ defined via a slope-restricted or tipliers, is shown to subsume all earlier stability tests and sector-bounded nonlinearity (see Fig. 1). Stability analysis by Wang et al. [2014], which is based on Zames-Falb multipliers, is shown to subsume all earlier stability tests and thus leads to the least conservative estimates for stability tipliers, is shown to subsume all earlier stability tests and andsuch a nonlinear operator ∆ defined via a slope-restricted or thus leads to the least conservative estimates for stability sector-bounded nonlinearity (see Fig. 1). Stability analysis of interconnections has a long standing history, prosector-bounded nonlinearity (see Fig. 1). Stability analysis of such interconnections has a long standing history, protipliers, is shown to subsume all earlier stability tests and thus leads to the least conservative estimates for stability margins (see also Willems and Brockett [1968]). However, thus leads to the least conservative estimates for stability sector-bounded nonlinearity (see Fig. 1). Stability analysis of such interconnections has a long standing history, probably with the works Tsypkin [1964] and Jury of suchstarting interconnections has a of long standing history, pro- margins (see also Willems and Brockett [1968]). However, bably starting with the works of Tsypkin [1964] and Jury thus leads to the least conservative estimates for stability margins (see also Willems and Brockett [1968]). However, will be revealed in the present paper, for multiple nonlimargins (see also Willems and Brockett [1968]). However, of such interconnections has a of long standing history, pro- as bably starting with the works Tsypkin [1964] and and Lee [1964]. Both approaches employed ideas developed bably starting works of Tsypkin [1964] and Jury Jury as will be revealed in the present paper, for multiple nonliand Lee [1964]. with Boththe approaches employed ideas developed developed margins (see also Willems and Brockett [1968]). However, as will be revealed in the present paper, for multiple nonlinearities it is beneficial to combine Zames-Falb multipliers as will be revealed in the present paper, for multiple nonlibably starting with the works of Tsypkin [1964] and Jury and Lee [1964]. Both approaches employed ideas by Popov for continuous-time systems (Popov [1961]). In and Lee [1964]. Both approaches employed ideas developed nearities it is beneficial to combine Zames-Falb multipliers by Popov for continuous-time systems (Popov [1961]). In as will be revealed in the present paper, for multiple nonlinearities it is beneficial to combine Zames-Falb multipliers with those corresponding to other stability criteria. nearities it is beneficial to combine Zames-Falb multipliers andPopov Lee [1964]. Both approaches employed ideas developed by for continuous-time systems (Popov [1961]). In contrast to the Popov criterion that only requires the by Popov for continuous-time systems (Popov [1961]). In with those corresponding to other stability criteria. contrast toforthe the Popov criterion criterion that (Popov only requires requires the nearities it corresponding is beneficial to to combine Zames-Falb multipliers with those other stability criteria. with those corresponding to other stability criteria. by Popov continuous-time systems [1961]). In contrast to Popov that only the existence of some sector bound on nonlinearity, it contrast to Popov thatthe only requires the one of the contributions this paper, we highlight the existence of the some sectorcriterion bound on on the nonlinearity, it As with those corresponding toof other stability criteria. As one of the contributions of this paper, we highlight the contrast to the Popov criterion thatthe only requires also the existence of some sector bound nonlinearity, it became apparent that the discrete-time counterpart As one of the contributions of this paper, we highlight the existence of some sector bound on the nonlinearity, it fundamental principles underlying all the above mentioned As one of the contributions of this paper, we highlight the became apparent that the bound discrete-time counterpart also principles underlying all the above mentioned existenceapparent of the some sector on thecounterpart nonlinearity,also it fundamental became that the discrete-time necessitated assumption of monotonicity. As one of the contributions of this paper, we highlight the fundamental principles underlying all the above mentioned became apparent that the discrete-time counterpart also stability tests and provide insights into their interrelation. fundamental principles underlying all the above mentioned necessitated the assumption assumption of monotonicity. monotonicity. tests and provide insights into their interrelation. became apparent that the discrete-time counterpart also stability necessitated the of fundamental principles underlying all the above mentioned stability tests and provide insights into their interrelation. necessitated the assumption of monotonicity. On the one this allows us to even the stability testshand, and provide insights intoshow theirhow interrelation. On the one this allows us show how even the necessitated the assumption of∆ monotonicity. stability testshand, and provide insights into their interrelation. On the one hand, thiscan allows us to to show how even the most recent versions actually be derived from On the one hand, this allows us to show how even ∆ most recent versions can actually be derived from the ∆ zz ∆ w On the one hand, thiscan allows us tobe how even and most recent versions actually derived from the ones proposed by O’Shea and Younis [1967], Willems most recent versions can actually beshow derived from the w zz ∆ ones proposed by O’Shea Younis Willems and w most recent versions canand actually be[1967], derived from the w ones proposed by O’Shea and Younis [1967], Willems and Brockett [1968] and Kulkarni and Safonov [2002]. While, ones proposed by O’Shea and Younis [1967], Willems and + G z+ Brockett [1968] and Kulkarni and Safonov [2002]. While, w G ones proposed by O’Shea and Younis [1967], Willems and Brockett [1968] and Kulkarni and Safonov [2002]. While, d + on the other we can extend the classical results Brockett [1968]hand, and Kulkarni and Safonov [2002]. While, G + G d on the other hand, we can extend the classical results d Brockett [1968] andatKulkarni and Safonov [2002].test While, on the other hand, we can extend the classical results d + in order to arrive less conservative stability for G on the other hand, we can extend the classical results Fig. 1. Feedback interconnection in order to arrive at less conservative stability test for d Fig. 1. Feedback interconnection on the other hand, we can extend the classical results in order to arrive at less conservative stability test for multiple nonlinearities. in order to arrive at less conservative stability test for Fig. 1. Feedback interconnection Fig. 1. Feedback interconnection multiple nonlinearities. Under the additional hypothesis that the derivative of in order to arrive at less conservative stability test for multiple nonlinearities. multiple nonlinearities. Fig. 1. Feedback interconnection Under the additional hypothesis that the derivative of Under the additional hypothesis that the derivative of In order to paint a clear and concise picture, we formuthe uncertainty is bounded, O’Shea and Younis [1967] Under the additional hypothesis that the derivative of multiple nonlinearities. In order to paint aa clear and concise picture, we formuthe uncertainty is bounded, O’Shea Younis [1967] order to clear and picture, we formuUnder thea additional hypothesis thatand thethe derivative of In the uncertainty is O’Shea and Younis [1967] late all criteria of integral quadratic In order to paint paintin a the clearframework and concise concise picture, we formuproposed discrete-time counterpart to celebrated the uncertainty is bounded, bounded, O’Shea and Younis [1967] late all criteria in the framework of integral quadratic proposed a discrete-time counterpart to the celebrated In order to paint a clear and concise picture, we formulate all criteria in the framework of integral quadratic the uncertainty is bounded, O’Shea and Younis [1967] constraints proposed a discrete-time discrete-time counterpart to the celebrated (IQCs) (Megretski and Rantzer [1997]). This late all criteria in the framework of integral quadratic Zames-Falb stability criterion (Zames and Falb [1968]) proposed a counterpart to the celebrated constraints (IQCs) (Megretski and Rantzer [1997]). This Zames-Falb stability criterion (Zames and Falb [1968]) late all criteria in the framework of integral quadratic constraints (IQCs) (Megretski and Rantzer [1997]). This proposed a discrete-time counterpart to the celebrated Zames-Falb stability criterion (Zames and Falb [1968]) viewpoint does indeed allow for a structural comparison constraints (IQCs) (Megretski and Rantzer [1997]). This that was later generalized in Willems and Brockett [1968]. Zames-Falb stability criterion (Zames and Falb [1968]) viewpoint does indeed allow for a structural comparison that was later generalized in Willems and Brockett [1968]. constraints (IQCs) (Megretski and Rantzer [1997]). This viewpoint does indeed allow for a structural comparison Zames-Falb stability criterion (Zames and Falb [1968]) that was later generalized in Willems and Brockett [1968]. of stability multipliers and enables us to identify, among viewpoint does indeed allow for a structural comparison O’Shea and Younis [1967] already claim that their criterion that was later generalized in Willems and Brockett [1968]. of stability multipliers and enables us to identify, among O’Shea and Younis [1967] already claim that their criterion viewpoint does indeed allowenables for aleast structural comparison of stability multipliers and us to identify, among that was later generalized in Willems and Brockett [1968]. O’Shea and Younis [1967] already claim that their criterion the multitude of possibilities, the conservative of the of stability multipliers and enables us to identify, among is less restrictive than the one proposed by Jury and Lee O’Shea and Younis [1967] already claim that their criterion the multitude of possibilities, the least of the is less restrictive than the one proposed by Jury and Lee of stability multipliers and enables us conservative to identify, among the multitude of possibilities, the least conservative of the O’Shea and Younis [1967] already claim that their criterion is less restrictive than the one proposed by Jury and Lee corresponding stability tests. Moreover, it is then easy the multitude of possibilities, the least conservative of the [1964], which was the most effective test at that time. is less restrictive than the one proposed by Jury and Lee corresponding stability tests. Moreover, it is then easy [1964], which was the most effective test at that time. the multitude of possibilities, the least conservative of the corresponding stability tests. Moreover, it is then easy is less restrictive than the one proposed by Jury and Lee [1964], which was the most effective test at that time. to reveal that (at least for scalar nonlinearities) both corresponding stability tests. Moreover, it is then easy [1964], which was the most effective test at that time. to reveal that (at least for scalar nonlinearities) both Following these early results, many researchers have concorresponding stability tests. Moreover, it is then easy to reveal that (at least for scalar nonlinearities) both [1964], which was the most effective test at that time. discrete time counterparts of the Popov and the Yakuboto reveal that (at least for scalar nonlinearities) both Following early results, many researchers have contime of the Popov and the YakuboFollowingtothese these early results, many researchers haveextencon- discrete tributed this field of study and, in particular, Following these early results, many researchers have conto reveal thatcounterparts (at[1965], least Dewey for scalar nonlinearities) both discrete time counterparts of Popov and the vich (Yakubovich and Jury [1966]) stability discrete time counterparts of the the Popov and the YakuboYakubotributed to this field of study and, in particular, extenvich (Yakubovich [1965], Dewey and Jury [1966]) stability Following these early results, many researchers have contributed to this field of study and, in particular, extended the above described stability tests to multi-variable tributed to this field of study and, in particular, extendiscrete time counterparts of the Popov and the Yakubovich (Yakubovich [1965], Dewey and Jury [1966]) stability criteria are already included in the on Zamesvich (Yakubovich [1965], Dewey and one Jurybased [1966]) stability ded the above described stability tests to multi-variable criteria are already included in the one on Zamestributed to this field ofPark study and, in[1998], particular, extended the described stability tests to nonlinearities (see, e.g, and Kim Haddad and ded the above above described stability tests to multi-variable multi-variable vich (Yakubovich [1965], Dewey and Jurybased [1966]) stability criteria are already included in the one based on ZamesFalb multipliers. This should be contrasted with the sicriteria are already included in the one based on Zamesnonlinearities (see, e.g, Park and Kim [1998], Haddad and Falb multipliers. This should be contrasted with the sided the above described stability tests to multi-variable nonlinearities (see, e.g, Park and Kim [1998], Haddad and Bernstein [1994], Kulkarni and Safonov [2002]). Recently nonlinearities (see, e.g, Park and Kim [1998], Haddad and criteria are already included in the one based on ZamesFalb multipliers. This should be contrasted with the situation in continuous time (see, e.g., Fetzer and Scherer Falb multipliers. This should be contrasted with the siBernstein [1994], Kulkarni and [2002]). Recently tuation in continuous time (see, e.g., Fetzer and Scherer nonlinearities (see, e.g, Parkinterest andSafonov Kim [1998], Haddad and Bernstein [1994], Kulkarni and Safonov [2002]). Recently there seems to be renewed in the subject with seBernstein [1994], Kulkarni and Safonov [2002]). Recently Falb multipliers. This should be contrasted with the situation in continuous time (see, e.g., Fetzer and Scherer [2017, 2016b]) where both tests may only be approximately tuation in continuous time (see, e.g., Fetzer and Scherer there seems to be renewed interest in the subject with se[2017, 2016b]) where both tests may only be approximately Bernstein [1994], Kulkarni and Safonov [2002]). Recently there seems to be renewed interest in the subject with several publications proposing seemingly different yet closely there seems to be renewed interest in the subject with setuation in continuous time (see, e.g., Fetzer and Scherer [2017, 2016b]) where both tests may only be approximately handled by using multipliers (Safonov and [2017, 2016b]) whereZames-Falb both tests may only be approximately veral publications seemingly different yet closely by using Zames-Falb multipliers (Safonov and there seems to be proposing renewed interest in the subject se- handled veral proposing seemingly different yetwith closely veral publications closely  The publications [2017, 2016b]) where both tests may only2014]). be approximately handled by using Zames-Falb multipliers (Safonov and Wyetzner [1987], Carrasco et al. [2013, handled by using Zames-Falb multipliers (Safonov and authors would proposing like to thankseemingly the Germandifferent Researchyet Foundation  Wyetzner [1987], Carrasco et al. [2013, 2014]). veral publications proposing seemingly different yet closely The authors would like to thank the German Research Foundation  handled by usingCarrasco Zames-Falb multipliers (Safonov and  The Wyetzner [1987], et al. [2013, 2014]). authors would like to thank the German Research (DFG) for financial support of the project within theFoundation Cluster of Wyetzner [1987], Carrasco et al. [2013, 2014]). The authors would like to thank the German Research Foundation (DFG) for financial support of the project within the Cluster of Apart from the work of Kulkarni and Safonov [2002],  The authors (DFG) for financial of the Cluster of Wyetzner [1987], Carrasco et al. [2013, 2014]). Excellence Simulation Technology (Grant number: 310/2) would support like to thank theproject Germanwithin Research Foundation (DFG) for in financial support of the the project theEXC Cluster of Apart from the work Kulkarni and Safonov [2002], Excellence in Simulation Technology (Grant within number: EXC 310/2) Apart from the work of of Kulkarni and Safonov [2002], all the above discussed stability tests employ diagonally Apart from the work of Kulkarni and Safonov [2002], Excellence Simulation Technology (Grant number: 310/2) at the University of Stuttgart. (DFG) for in financial support of the project within theEXC Cluster of Excellence in Simulation Technology (Grant number: EXC 310/2) all the above discussed stability tests employ diagonally at the University of Stuttgart. Apart from the work of Kulkarni and Safonov [2002], all the above discussed stability tests employ diagonally all the above discussed stability tests employ diagonally at the University of Stuttgart. Excellence in Simulation Technology (Grant number: EXC 310/2) at the University of Stuttgart. all the above discussed stability tests employ diagonally at the University of Stuttgart. ∗ ∗ ∗ ∗

Copyright © 2017 IFAC 8781 Copyright © 2017, 2017 IFAC 8781Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright 8781 Copyright © © 2017 2017 IFAC IFAC 8781 Peer review under responsibility of International Federation of Automatic Copyright © 2017 IFAC 8781Control. 10.1016/j.ifacol.2017.08.757

Proceedings of the 20th IFAC World Congress 8448 Matthias Fetzer et al. / IFAC PapersOnLine 50-1 (2017) 8447–8453 Toulouse, France, July 9-14, 2017

structured multipliers even if considering repeated multivariable nonlinearities. As another contribution of this paper we demonstrate how unstructured full-block multipliers may be combined with diagonal (full-block) ZamesFalb multipliers for non-repeated (repeated) nonlinearities in order to generate more powerful novel tests. The paper is structured as follows. We set the stage in Section 2 by defining the class of nonlinearities and the interconnection as well as by briefly sketching the framework of IQCs. In Sections 3 and 4 we derive our full-block stability multipliers and compare them to previous ones. After discussing the implementation of these multipliers in Section 5, we close with numerical examples in Section 6. Notation: We denote by e the all ones vector in Rn and for A ∈ Rn×m we use A = (Aij ) ≥ 0 if Aij ≥ 0 for all i, j. If A, B are square matrices, we write diag(A, B) for the block diagonal matrix with blocks A, B. The subset of symmetric matrices in Rn×n is denoted by Sn . If α = diag(αi ), β = diag(βi ) ∈ Rk×k are diagonal, we express by [α, β] the set of diagonal matrices {diag(δ1 , . . . , δk ) : αi ≤ δi ≤ βi for all i ∈ {1, . . . , k}}. D⊂ C is the open unit disc and T its boundary. k2 (k2e ) denotes the space of (locally) square summable functions mapping N0 into Rk ; ∞ k 2 2 is equipped with the norm u = j=0 u(j)2 . 2. PRELIMINARIES

We consider the following class of nonlinearities. Definition 1. Let µ ≤ 0 ≤ ν. Then ϕ : R → R is sloperestricted, in short ϕ ∈ slope(µ, ν), if ϕ(0) = 0 and ϕ(x) − ϕ(y) ϕ(x) − ϕ(y) ≤ sup <ν (1) µ≤ x−y x−y x=y

for all x, y ∈ R, x = y. If µ = 0 and the bound on the right is absent, ϕ is just monotone and we write ϕ ∈ slope(0, ∞). The nonlinearity ϕ is said to be sector-bounded if (ϕ(x) − αx)(βx − ϕ(x)) ≥ 0 for all x ∈ R (2) and some α ≤ 0 ≤ β; this is expressed as ϕ ∈ sec[α, β]. With such nonlinearities ϕ1 , . . . , ϕk , let Φ : Rk → Rk be T given as Φ(x1 , . . . , xk ) = (ϕ1 (x1 ) . . . ϕk (xk )) and let the operator ∆Φ be defined as (∆Φ (z))(t) := Φ(z(t)) for all t ∈ N0 , z ∈ k2e . (3) We write ∆Φ ∈ slope(µ, ν) or ∆Φ ∈ sec[α, β] if ϕj ∈ slope(µj , νj ) or ϕj ∈ sec[αj , βj ] for all j ∈ {1, . . . , k} and with µ = diag(µj ), ν = diag(νj ), α = diag(αj ), β = diag(βj ), respectively. For the special case when all nonlinearities coincide, i.e., ϕj = ϕ for all j, we say that Φ is a repeated nonlinearity and indicate this by ∆Φ ∈ slope(µI, νI) or ∆Φ ∈ sec[αI, βI] for the operator.

Given such a nonlinearity ∆Φ , we consider its feedback interconnection (see Fig. 1) with a stable LTI system G described through a state-space realization as follows: x(t + 1) = Ax(t) + Bw(t), x(0) = 0, w = ∆Φ (z) z(t) = Cx(t) + Dw(t) + d(t). (4)

Here we assume that A ∈ Rn×n is Schur stable, i.e., eig(A) ⊂ D, and d ∈ k2 .

The interconnection (4) is said to be well-posed if for each d ∈ k2 and each τ ∈ [0, 1] there exists a unique response

z ∈ k2e of (4) with ∆Φ replaced by τ ∆Φ which depends causally on d. Moreover, (4) is stable if ∃γ > 0 :

z ≤ γd

for all

d ∈ k2 .

(5)

We capture the action of ∆Φ by means of IQCs: Two signals z, w ∈ k2 with z-transforms zˆ, w ˆ are said to satisfy the IQC defined by a multiplier Π that is measurable, bounded and Hermitian valued on T if  iω   2π  iω ∗ zˆ(e ) zˆ(e ) iω Π(e ) dω ≥ 0. ΣΠ (z, w) = w(e ˆ iω ) w(e ˆ iω ) 0 A causal operator ∆ : k2 → k2 satisfies the IQC imposed by Π in case that (6) ΣΠ (z, ∆(z)) ≥ 0 for all z ∈ k2 .

Let us finish this section by stating a particular version of the IQC stability result (see, e.g., Megretski and Rantzer [1997], Kao [2012] and, for a more general setting, Fetzer and Scherer [2016a]) adapted to our special configuration. Theorem 2. Assume that the interconnection (4) with ∆Φ as in (3) is well-posed. Then (4) is stable if (1) τ ∆Φ satisfies the IQC defined by Π for all τ ∈ [0, 1]; (2) the following FDI holds:   ∗ G(eiω ) () Π(eiω ) ≺ 0 for all ω ∈ [0, 2π]. (7) I The verification of well-posedness for ∆Φ ∈ sec[α, β] or ∆Φ ∈ sec(µ, ν) in Fetzer and Scherer [2017] literally carries over to discrete-time interconnections; in the sequel we tacitly assume that (4) is well-posed. 3. PRINCIPLES OF STABILITY MULTIPLIERS In order to highlight the key underlying concepts for defining the subsequently appearing stability multipliers, we divide this section according to the generating principles. As will become apparent, all the multipliers employed in the papers cited in the introduction either rely on a subgradient argument or on polytopic bounding for the creation of inequalities. Another focus of this section is the formulation of stability test using full-block multipliers. 3.1 Methods based on polytopic bounding Let ∆Φ ∈ sec[α, β]. Conceptually, the circle criterion exploits the simple fact that w(t) = ∆Φ (z)(t) = Φ(z(t)) for z ∈ k2 can be expressed, due to (2), as (8) w(t) = ∆(t)z(t) for all t ∈ N0 with ∆(t) ∈ [α, β]; indeed we can take ∆(t) = diag(δj (t)) and δj (t) = ϕj (zj (t))/zj (t) if zj (t) = 0 or δj (t) = 0 if zj (t) = 0 for all j ∈ {1, . . . , k}. If we now choose any element Π in the class of full-block multipliers       T I Π[α, β] = Π ∈ S2k () Π  0, ∀∆ ∈ [α, β] (9) ∆ (see Iwasaki and Hara [1998] and Scherer [2001]), we obviously infer     z(t) I = ()T Π z(t) ≥ 0 ∀t ∈ N0 . ()T Π w(t) ∆(t)

By summation we conclude that ∆Φ satisfies the IQC (10) ΣΠ (z, ∆Φ (z)) ≥ 0 for all z ∈ k2 .

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Since the class Π[α, β] was originally defined to handle time-varying parametric uncertainties in polytopes, these multipliers are said to be generated by polytopic bounding. The same conclusion can be drawn for the following subset of so-called diagonally structured multipliers:    −αβλ α+β λ  2 Πdr [α, β] := λ = diag(λ1 , ..., λk ) > 0 . α+β λ −λ  2

This class is obtained by exploiting the sector bounds (2) in an standard S-procedure argument. We emphasize that, even in all recent papers on this subject, only Πdr [α, β] is considered, which causes unnecessary conservatism. Note that the definition of Π[α, β] involves infinitely many constraints. For a thorough discussion of various relaxation schemes that enable the use of this class in computations, we refer the reader to Fetzer and Scherer [2017]. It is now straightforward to derive a full-block circle and a full-block Yakubovich stability criterion in our setting. Circle criterion Let ∆Φ ∈ sec[α, β]. Since (10) holds for all Π ∈ Π[α, β], Theorem 2 implies the following result. Corollary 3. The interconnection (4) with ∆Φ ∈ sec(α, β) is stable if there exists some Π ∈ Π[α, β] with   ∗  G(z) G(z) Π ≺ 0 for all z ∈ T. (11) I I Note that we use the symbol z for the frequency variable in (11) in order to distinguish it from the signal z. As emphasized above, all discrete-time circle criteria for stability in the literature restrict the search of Π to the subclass Πdr [α, β] ⊂ Π[α, β] of diagonally structured multipliers. Therefore, full-block multipliers will not be worse than the conventional ones, and it can be concluded from numerical examples that they typically reduce conservatism significantly (Fetzer and Scherer [2017]). Yakubovich criterion Let us now turn to the discrete time analogue of the Yakubovich criterion for ∆Φ ∈ slope(µ, ν). For convenience we assume that the nonlinearities ϕj are continuously differentiable for j ∈ {1, . . . , k}.

Choose z ∈ k2 and let w := ∆Φ (z). By the mean value theorem there exist ξjt (depending on t ∈ N0 ) such that wj (t + 1) − wj (t) = ϕj (zj (t + 1)) − ϕj (zj (t)) = ϕj (ξjt )(zj (t + 1) − zj (t)). Since the slope restriction Φ ∈ slope(µ, ν) translates into µj ≤ ϕj (ξ) ≤ νj for all j ∈ {1, . . . , k}, ξ ∈ R, we infer, in complete analogy to the circle criterion, that there exist ∆(t) ∈ [µ, ν] with w(t + 1) − w(t) = ∆(t)(z(t + 1) − z(t)) for all t ∈ N0 . Thus for Πy ∈ Π[µ, ν] we obtain   z(t + 1) − z(t) T ≥ 0 for all t ∈ N0 . (12) ( ) Πy w(t + 1) − w(t)

As the time shift in the outer factors of (12) gives rise to a multiplication with z −1 in the frequency domain, we obtain, by summation and with Parseval’s theorem, the IQC ΣΠ (z, w) ≥ 0 for the dynamic multiplier   (z −1)I 0 ∗ Π(z) := ( ) Πy = | z −1|2 Πy . (13) 0 (z −1)I

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Again with Theorem 2 we arrive at the following result. Corollary 4. The interconnection (4) with ∆Φ ∈ slope(µ, ν) is stable if there exists some Πy ∈ Π[µ, ν] with   (z −1)G(z) ( )∗ Πy ≺ 0 for all z ∈ T. (z −1)I Remark 5. One can show (Fetzer [2017]) that this remains correct even if we just assume (1) (implying that the ϕj are only differentiable almost everywhere).

Combined polytopic criterion Let us now discuss how we may combine the circle and Yakubovich multipliers and embed the combination into a more general class of completely unstructured multipliers. Assume that ∆Φ ∈ sec[α, β] ∩ slope(µ, ν). Of course, the most simple way of exploiting both constraints simultaneously is to just add up the according multipliers. The corresponding FDI then reads as   G(z)   Πc 0  I  ≺ 0 for all z ∈ T (14) ( )∗ 0 Πy  (z −1)G(z)  (z −1)I with some Πc ∈ Π[α, β] and Πy ∈ Π[µ, ν]. Yet, this obviously results in a potentially conservative block diagonal structure. Based on the above described generating principle, the generalization to unstructured multipliers is simple. Indeed, for w = ∆Φ (z) and z ∈ k2 we have      w(t) ∆c (t) 0 z(t) = w(t + 1) − w(t) 0 ∆y (t) z(t + 1) − z(t)

with suitable ∆c (t) ∈ [α, β], ∆y (t) ∈ [µ, ν] and for all t ∈ N0 . Since diag(∆c (t), ∆y (t)) ∈ [diag(α, µ), diag(β, ν)], we infer for any Πcy ∈ Π[diag(α, µ), diag(β, ν)] that   z(t) T  z(t + 1) − z(t)  ( ) Πcy   ≥ 0 for all t ∈ N0 . w(t) w(t + 1) − w(t) In exactly the same way as described above this leads to a stability test that is formulated with the FDI   G(z)  (z −1)G(z)  (15) ( )∗ Πcy   ≺ 0 for all z ∈ T. I (z −1)I We arrive at the following general full-block stability test. Corollary 6. The interconnection (4) with ∆Φ ∈ sec[α, β]∩ slope(µ, ν) is stable if there exists some matrix Πcy ∈ Π[diag(α, µ), diag(β, ν)] with (15). 3.2 Subgradient based arguments

We start this subsection by giving a direct convexity proof for full-block FIR Zames-Falb IQCs as originally proposed by Willems and Brockett [1968]. The derivation will serve as a foundation for the subsequent comparison of multiplier classes in the literature. Full-block FIR Zames-Falb multipliers Let us recall some definitions introduced in Willems and Brockett [1968]. Definition 7. Let M = (Mij ) ∈ Rk×k . Then M is a Zmatrix if Mij ≤ 0 for i = j. Moreover, M is doubly hyperdominant if it is a Z-matrix and if, in addition, M e ≥ 0 and eT M ≥ 0.

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It is said to be doubly dominant if, for all i ∈ {1, . . . , n}, n n   Mii ≥ |Mij | and Mii ≥ |Mji | . j=1, j=i

j=1, j=i

Remark 8. Any doubly dominant M can be decomposed as M = Md +Mod where Md and Mod contain the diagonal and off-diagonal elements; then Md − |Mod | is doubly hyperdominant if we take the absolute value element-wise. The following lemma provides the foundation for discrete time Zames-Falb multipliers. We formulate it for repeated nonlinearities that comprise scalar ones as a special case. Lemma 9. Let Φ ∈ slope(0I, ∞I). If M ∈ Rk×k is doubly hyperdominant then Φ(x)T M x ≥ 0 for all x ∈ Rk . In case that, in addition, ϕ is odd, this holds for any doubly dominant matrix M . This is a matrix version of a result in Willems and Brockett [1968]; our direct proof highlights the role of the underlying principles, namely convexity and permutation invariance. Proof. Suppose that ϕ is not necessarily odd and choose the convex primitive Iϕ satisfying Iϕ (0) = 0. Define the convex function Ψ(x) := Iϕ (x1 ) + · · · + Iϕ (xk ). Since ∇Ψ(x) = Φ(x) we infer by convexity that Φ(x)T (x − y) ≥ Ψ(x) − Ψ(y) for all x, y ∈ Rk . (16) We now exploit that Φ is repeated by observing Ψ(P x) = Ψ(x) for any permutation matrix P . Thus (16) implies

Φ(x)T (x − P x) ≥ 0 for all x ∈ Rk . By the Birkhoff-von Neumann theorem [Horn and Johnson 1985] we infer for all doubly stochastic matrices S that Φ(x)T (I − S)x ≥ 0 for all x ∈ Rk . For the given Z-matrix M with M e ≥ 0 and eT M ≥ 0 it is now clearly possible to choose r > 0 small enough such that I − rM ≥ 0 and 1 − reT M e ≥ 0. Thus   I − rM rM e ≥0 S := reT M 1 − reT M e and S is obviously doubly stochastic. As just seen, we can conclude that  T   1 Φ(x) x Φ(x)T M x = (I − S) ≥ 0 for x ∈ Rk . 0 0 r If ϕ is also odd, the result follows from |Φ(x)| = Φ(|x|) for all x ∈ R and with M = Md + Mod in Remark 8. Indeed, since Md − |Mod | is doubly hyperdominant, we get Φ(x)T M x = Φ(x)T Md x + Φ(x)T Mod x

≥ |Φ(x)|T Md |x| − |Φ(x)|T |Mod ||x|

= Φ(|x|)T (Md − |Mod |)|x| ≥ 0 for x ∈ Rk .

 It is now standard to extend Lemma 9 from monotone to slope-restricted nonlinearities (see, e.g., Zames and Falb [1968], D’Amato et al. [2001]). For later reference, we state the result in terms of a quadratic form as follows. Corollary 10. Let Φ ∈ slope(µI, νI) with µ ≤ 0 ≤ ν and assume that M is doubly hyperdominant or that ϕ is odd and M is doubly dominant. Then     νI −I x T 0 MT () ≥ 0 for all x ∈ Rk . −µI I Φ(x) M 0

The extension to infinite block matrices defining operators on k2 follows Willems and Brockett [1968]. Suppose that M = (Mij )i,j∈Z is an infinite block matrix with Mij ∈ Rk×k such that there exists some b ≥ 0 with   Mji  ≤ b and Mij  ≤ b for all j ∈ Z. (17) i∈Z

i∈Z

k k It  is then well-known that M : 2 → 2 , (M x)i := j∈Z Mij xj defines a bounded linear operator. Now suppose that M is a Z-matrix. Due to (17) and if e∞ ∈ k2e is the sequence of all-ones vectors then M e∞ and eT∞ M are well-defined sequences in k2e . Let us assume that, in addition, M e∞ ≥ 0 and eT∞ M ≥ 0 element-wise. Then we obtain the following result as a consequence of Corollary 10. Corollary 11. (Zames-Falb IQC). With µ ≤ 0 ≤ ν let Φ ∈ slope(µI, νI) and assume that M with (17) is either an (infinite) doubly hyperdominant matrix or that ϕ is odd and M is doubly dominant. Then     ν id − id z T 0 MT () ≥ 0 ∀z ∈ k2 . −µ id id ∆Φ (z) M 0 (18)

For the subsequent discussion it suffices to restrict the attention to block Toeplitz matrices with the structure   .. .. .. .. .. .. .. . . . . . . .    . . . 0 Ml+ . . . M0 . . . M−l− 0 . . .   . . . 0 Ml+ . . . M0 . . . M−l− 0 . . . M =   . . . 0 Ml+ . . . M0 . . . M−l− 0 . . .    .. .. .. .. .. .. .. . . . . . . .

(19) for some chosen l± ∈ N0 ; it is then required that M0 is a Z-matrix, M−j ≤ 0 for j ∈ {1, . . . , l− }, and Mj ≤ 0 for j ∈ {1, . . . , l+ } as well as     T Mj ≥ 0 and Mj e ≥ 0 e j∈J

j∈J

with J = {l− , . . . , l+ }. For y ∈ k2 we then infer   j  Mj z yˆ(z) =: HM (z)ˆ y (z) M y(z) = j∈J

and, due to the structure of M ,   1  T y(z) = MjT j yˆ(z) = HM (1/ z)T yˆ(z). M z j∈J

Based on (19) let us now define the class of FIR ZamesFalb multipliers as the set      ∗  0 HM νI −I T ΠM (µI, νI) = Π  Π = () HM 0 −µI I (20) where HM (z)∗ = HM (1/ z)T and  z j Mj . (21) HM (z) = j∈J

If ∆Φ ∈ slope(µI, νI) then (18) implies (via Parseval’s theorem) for all Π ∈ ΠM (µI, νI) that

(22) ΣΠ (z, ∆Φ (z)) ≥ 0 for all z ∈ k2 The multiplier classes corresponding to (20) for ν = ∞ and µ = −∞ are derived analogously and take the form

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    ∗ ∗  −µ(HM + H M ) HM ΠM (µI, ∞I) = Π  Π = HM 0    (23) ∗  ν(HM + HM ) −HM ΠM (µI, ∞I) = Π  Π = ∗ 0 −HM (24) respectively. This leads us to the following stability result, as a consequence of (22) and Theorem 2. Corollary 12. (FIR Zames-Falb criterion). Let µ ≤ 0 ≤ ν, ∆Φ ∈ slope(µI, νI) and assume that M in (19) is either a doubly hyperdominant matrix or that ϕ is odd and M is doubly dominant. Then the interconnection (4) is stable if there exists Π ∈ ΠM (µI, νI) such that  ∗   G(z) G(z) Π(z) ≺ 0 for all z ∈ T. (25) I I Multipliers for some non-repeated uncertainty ∆Φ ∈ slope(µ, ν) may be obtained by choosing scalar functions HM,j for each ϕj and combining them diagonally, which just amounts to the restriction that all Mj are diagonal; we denote the respective multiplier class by ΠM (µ, ν). 4. RELATION TO MULTIPLIERS IN THE LITERATURE First note that the complete class of full-block Zames-Falb multipliers was already described in Kulkarni and Safonov [2002]. The multipliers in (20), (21) are the full-block versions of the FIR Zames-Falb multipliers as suggested for the scalar case in Wang et al. [2014] and can be easily implemented numerically; this renders the results in Kulkarni and Safonov [2002] computational. 4.1 Zames-Falb multipliers of order one In this section we prove that both the criteria proposed in Tsypkin [1964] and in Jury and Lee [1964] as well as all later derivatives thereof (see, e.g., Haddad and Bernstein [1994], Ahmad et al. [2010, 2011, 2013b, 2015]) are special cases of (20), (21) for l− = l+ = 1. This reveals that all variants of the discrete-time counterpart to the Popov criterion are rendered obsolete by a Zames-Falb stability test using (20), (21) with l± ≥ 1.

Let us hence assume Φ ∈ slope(µ, ν), choose l− = l+ = 1, and select M with the zeroth block row (. . . 0 M1 M0 M−1 0 . . .) where M0 ≥ 0, M−1 ≤ 0, M1 ≤ 0 are diagonal and satisfy eT (M−1 + M0 + M1 ) ≥ 0. (M−1 + M0 + M1 )e ≥ 0, These requirements are obviously fulfilled for the more special (and potentially restrictive; see Example 15) choices  M0 = Λ + Λ,  M1 = −Λ. M−1 = −Λ,

 ≥ 0. We denote the resulting multiplier with diagonal Λ, Λ classes corresponding to (20)–(24), respectively, by Π(Λ, (µ, ν), Π(Λ, (µ, ∞) and Π(Λ, (−∞, ν). (26) Λ) Λ) Λ)

As all multiplier classes in the present paper are convex cones, their combination is just obtained by summing them (µ, ∞) + Π(Λ , (−∞, ν) and up. Therefore, Π(Λ , 1 Λ1 ) 2 Λ2 )  i and Λ, Λ  as described (µ, ν) (with varying Λi , Λ Π(Λ, Λ) above) define valid multiplier classes for ∆Φ ∈ slope(µ, ν).

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Table 1 illustrates how various stability tests proposed in the literature relate to Corollary 12 with (26). In the second column we state the uncertainty class under consideration in the respective paper as listed in the first column, while the third one gives the employed multiplier combination. Even if considering uncertainties in slope(µ, ν), several papers just only exploit the fact that they are contained in either slope(µ, ∞) or slope(−∞, ν). Yet some use, e.g., the information ∆Φ ∈ slope(0, ν) = slope(−∞, ν) ∩ slope(0, ∞) by additively combining multipliers for slope(−∞, ν) and slope(0, ∞), respectively. We devote the subsequent section to the proof that this is not beneficial if compared to using the dedicated multipliers for the class slope(0, ν) directly. Several more aspects are worth pointing out in Table 1. All cited papers employ a combination of diagonally structured circle criterion multipliers (from the class Πdr [α, β]) and first order Zames-Falb multipliers. Ahmad et al. [2013b] include one of Yakubovich type (Πy,dr [µ, ν], see (28)), but this is also covered by Zames-Falb multipliers as shown later. Also note that several approaches either take  = 0 which, of course, increases conservativeness Λ = 0 or Λ if computing stability margins. Thus, a combination of the multipliers proposed in the present paper is guaranteed to lead to the same or improved stability estimates. We can as well conclude that both the multipliers proposed by Tsypkin [1964] and Jury and Lee [1964] (as well as the later proposed derivatives thereof) are special cases of Zames-Falb multipliers of order one. Hence, ΠM (µI, νI) with M as in (19) and l± = 1 could be seen as a full-block generalization of Tsypkin multipliers that, to the best of the authors knowledge, have not been described anywhere in the literature. As a side-remark, we emphasize that our approach does not require the LTI system in the loop to be strictly proper, as is typically encountered in the literature. 4.2 Redundant multiplier combinations We have seen that a large number of papers handle ϕ ∈ slope(µ, ν) by combining Zames-Falb multipliers for slope(µ, ∞) and slope(−∞, ν); let us now settle that it is more beneficial to work with the single class of dedicated multipliers ΠM (µ, ν). Lemma 13. Let Π ∈ ΠM (µ, ν), Π1 ∈ ΠM (µ, ∞), Π2 ∈ ΠM (−∞, ν) be three given Zames-Falb multipliers. Then  ∈ ΠM (µ, ν) there exists another Zames-Falb multiplier Π  such that Π  Π + Π1 + Π2 on T. Proof. Omitted(see Fetzer [2017]).



In summary, we can just work with the tightest slope restriction in Corollary 12, i.e., ϕ ∈ slope(µ, ν), since the validity of (25) for a combination of multipliers as in Lemma 13 implies the existence of some multiplier in ΠM (µ, ν) also satisfying (25). In case of a single nonlinearity, let us finally stress that Yakubovich multipliers for ∆Φ ∈ slope(µ, ν) are also covered by first order Zames-Falb multipliers. Indeed if k = 1, Π[µ, ν] can be parameterized as       −2µν µ + ν  Π[µ, ν] = Π  Π = λ , λ>0 (27) µ + ν −2

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Table 1. Overview of some multiplier classes employed in the literature Reference Tsypkin [1964] Jury and Lee [1964] Haddad and Bernstein [1994] Ahmad et al. [2010, 2011] Ahmad et al. [2013b] Ahmad et al. [2015]

Uncertainty class sec[0, β] ∩ slope(0, ∞)

Multiplier class combination Πdr [0, β] + Π (0, ∞) Πdr [0, β] + Π

sec[0, β] ∩ slope(0, ν) sec[0, β] ∩ slope(0, ν)

sec[0, β] ∩ slope(0, ν) sec[0, ν] ∩ slope(0, ν)

(Fetzer and Scherer [2017]). The claim then follows by 2 the simple observation that, with (27), |z −1| Π[µ, ν] = Π(λ,λ) (µ, ν). Also for k > 1 this shows that diagonally structured Yakubovich multipliers 2 Πy,dr [µ, ν] := |z −1| Πdr [µ, ν] (28) offer no benefit if combined with Zames-Falb multipliers. Yet, this no longer holds true for the full-block versions (see Example 16). 5. IMPLEMENTATION OF MULTIPLIERS In order to keep the derivation and comparison of multipliers as insightful as possible, we relied on the formulation of our tests in terms of FDIs in the frequency domain. Still, the translation to LMIs with an insightful structure is routine. Indeed, via multiplication with 1 = 1/(z z) > 0 on T, observe that (15) holds on T iff we have  1  0 z   1 − 1 0  G(z) ∗ z  ≺ 0 ∀ z ∈ T; (29) () Πcy  1   0 I z 0 1 − 1z    Ψcy (z)

clearly Ψcy is a proper and stable transfer function.

In order to render Corollary 12 computational for some pair l = (l− , l+ ) let us, for brevity of display, consider the case l+ ≥ l− with the definitions T  ψl+ = I 1z I . . . zl1+ I , Ψl+ = diag(ψl+ , ψl+ )

and the square matrix Pl ∈ R(l+ +1)k×(l+ +1)k given by   M0 M−1 . . . M−l− 0 . . . 0  M1 0 . . . 0 0 . . . 0 Pl =  . (30) .. . . .. .. .. ..   ... . . . . . . Ml+ 0 . . . 0 0 . . . 0 Then the multiplier (20), (21) may be expressed as    νI −I ∗ 0 PlT ∗ Ψ l+ ; (31) () Πl T Ψl+ := () −µI I Pl 0 again Ψzf = T Ψl+ is proper and stable. The case of l− > l+ is treated analogously.

In this way we obtain the following combined stability test. Corollary 14. Let ∆Φ ∈ sec[αI, βI] ∩ slope(µI, νI) and fix l− , l+ ≥ 0. Suppose there exists some Πcy ∈ Π[diag(αI, µI), diag(βI, νI)] and some M as in (19) that is either doubly hyperdominant or doubly dominant (if ϕ is odd) with     Ψcy G ∗ Πcy 0 ≺ 0 on T. (32) () I 0 Πl Ψzf Then the interconnection (4) is stable.

  (−∞, ν) Πdr [0, β] + Π (−∞, ν) (0, Λ) Πdr [0, β] + Π (−∞, ν) + Π(Λ,0) (0, ∞) (0, Λ) Πdr [0, β] + Πy,dr [0, ν] + Π (−∞, ν) + Π (0, ∞) (0, Λ1 ) (Λ2 , Λ2 ) Πdr [0, ν] + Π (0, ν) (Λ, Λ) (0,Λ)

sec[0, β] ∩ slope(−ν, ν)

(0,Λ)

This also holds for ∆Φ ∈ sec[α, β] ∩ slope(µ, ν) if just restricting all matrices Mj in (30) to be diagonal and assuming that Πcy ∈ Π[diag(α, µ), diag(β, ν)].

It is now routine to turn the verification of (32) (for some Πcy , Πl satisfying the respective constraints) into an LMI by means of the KYP lemma (Rantzer [1996]). 6. EXAMPLES Let us finally provide some numerical illustrations that have all been generated with Matlab’s LMI toolbox. Example 15. Let us first adopt an example from Ahmad et al. [2015], where G is given by  0.2 −0.2  z −0.92 . G(z) = − z −0.98 0.3 0.1 z −0.97 z −0.91

Our goal is to estimate the largest r > 0 such that the feedback interconnection (4) remains stable for all ∆Φ ∈ slope(0, ν) with ν = diag(r, r). We first assume that the nonlinearities are non-identical. As can be inferred from Table 1, the stability test proposed in Ahmad et al. [2015] is the least conservative of all listed approaches. The maximal r estimated therein is r = 3.556. Using diagonally structured circle and Zames-Falb multipliers (with l± = 1), we can still improve on that and obtain r = 3.808 which is already very close to the Nyquist value of rN = 3.85. This supports the fact that diagonally structured first order Zames-Falb multipliers may already lead to improved estimates if compared to the Popov tests in the literature. If we further assume that the nonlinearities are repeated, stability can be guaranteed up to rN by means of full-block multipliers. Example 16. Let the LTI system G in (4) be defined by     0.74 −0.3 0 0 −0.1 2.2 0.2  0 0.98 0  0.3 0  0 0      0 0.97 0 0  , B =  0.5 0  , A= 0  0   0 0 0.72 0 −0.5 0.5  0 0.1 0.31 0 0.9 0 0.05   −0.21 −0.4 −0.01 0.40 0 C= , D = 0, 0.3 0.3 −0.3 0 −0.36 and assume for simplicity that ∆Φ ∈ slope(µ, ν) with µ = 0 and ν = I, yet with non-identical functions ϕj . Let us now compare the standard approach from the literature, namely a combination of diagonally structured Zames-Falb and circle multipliers, with Corollary 14. In order to contrast both approaches, we compute 2 -gain estimates, i.e., the smallest γ > 0 such that (5) holds. This is easily achieved by only minor alterations to the FDI (32) (see, e.g., Veenman et al. [2016]). Table 2 nicely illustrates that, unlike the diagonally structured circle and Yakubovich ones, the combined polytopic multipliers also

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Table 2. 2 -gain estimates for Example 16 multiplier Πdr [µ, ν] + ΠM (µ, ν) Corollary 14

l± = 1 148.43 95.23

l± = 2 104.35 76.47

l± = 3 89.39 69.99

l± = 4 82.08 66.58

provide additional benefit if employed together with those for the Zames-Falb criterion. 7. CONCLUSION In this paper we propose a general framework that allows for the modular and combined application of different stability multipliers. It is important to note that the proof of stability relies only on the standard IQC theorem and does not require any additional arguments, as is typical for Lyapunov approaches. Moreover, we reveal that the formulation in terms of multipliers improves the insight into the generating principles for stability criteria substantially and allows to categorize the approaches in the literature. Finally, we derive new and completely unstructured fullblock multipliers that are, in numerical examples, shown to lead to less conservative stability estimates if compared to existing criteria. REFERENCES Ahmad, N.S., Carrasco, J., and Heath, W.P. (2013a). LMI searches for discrete-time Zames-Falb multipliers. In 52nd IEEE Conf. Decision and Control, 5258–5263. Ahmad, N.S., Carrasco, J., and Heath, W.P. (2015). A less conservative LMI condition for stability of discretetime systems with slope-restricted nonlinearities. IEEE T. Automat. Contr., 60(6), 1692–1697. Ahmad, N.S., Heath, W.P., and Li, G. (2010). Lyapunov functions for discrete-time multivariable Popov criterion with indefinite multipliers. In 49th IEEE Conf. Decision and Control, 1559–1564. Ahmad, N.S., Heath, W.P., and Li, G. (2013b). LMI-based stability criteria for discrete-time Lur’e systems with monotonic, sector- and slope-restricted nonlinearities. IEEE T. Automat. Contr., 58(2), 459–465. Ahmad, N.S., Heath, W., and Li, G. (2011). Lyapunov functions for generalized discrete-time multivariable Popov criterion. IFAC Proceedings Volumes, 44(1), 3392 – 3397. 18th IFAC World Congress. Boczar, R., Lessard, L., and Recht, B. (2015). Exponential convergence bounds using integral quadratic constraints. In 54th IEEE Conf. Decision and Control. Carrasco, J., Heath, W.P., and Lanzon, A. (2013). Equivalence between classes of multipliers for slope-restricted nonlinearities. Automatica, 49(6), 1732 – 1740. Carrasco, J., Heath, W.P., and Lanzon, A. (2014). On multipliers for bounded and monotone nonlinearities. Systems & Control Letters, 66, 65 – 71. D’Amato, F.J., Rotea, M.A., Megretski, A.V., and J¨ onsson, U.T. (2001). New Results for Analysis of Systems with Repeated Nonlinearities. Automatica, 37(5). Dewey, A. and Jury, E. (1966). A stability inequality for a class of nonlinear feedback systems. IEEE T. Automat. Contr., 11(1), 54–62. Fetzer, M. (2017). From classical absolute stability tests towards a comprehensive robustness analysis. Ph.D. thesis, University of Stuttgart. Fetzer, M. and Scherer, C.W. (2016a). A General Integral Quadratic Constraints Theorem with Applications to

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