Convex searches for discrete-time Zames-Falb multipliers

1 Con v e x searches for discrete-time Zames–F alb multipliers Joaquin Carrasco, Member , IEEE, W illiam P . Heath, Member , IEEE, Nur Syazreen Ahmad, Member , IEEE, Shuai W ang, Member , IEEE and Jingfan Zhang, Student Member , IEEE Abstract —In this paper we dev elop and analyse con vex searches for Zames–Falb multipliers. W e present two different approaches: Infinite Impulse Response (IIR) and Finite Impulse Response (FIR) multipliers. The set of FIR multipliers is complete in that any IIR multipliers can be phase-substituted by an arbitrarily large order FIR multiplier . W e show that searches in discrete-time for FIR multipliers are effecti ve ev en f or large orders. As expected, the numerical results pr ovide the best ` 2 - stability r esults in the literatur e f or slope-r estricted nonlinearities. Finally , we demonstrate that the discrete-time search can provide an effective method to find suitable continuous-time multipliers. Index T erms —Zames–Falb multipliers, absolute stability , Lur’e problem. I . I N T RO D U C T I O N The stability of a feedback interconnection between a linear time-in variant system G and any nonlinearity φ within the class of nonlinearities Φ is referred to as the Lur’e problem (see Section 1.3 in [1] for a history of this problem). As the stability is obtained for the whole class of nonlinearities, the adjective “absolute” or “rob ust” is added. In the classical solution of this problem frequenc y-domain conditions on the linear system are determined by the class of nonlinearites. The inclusion of a multiplier reduces the conservati veness of the approach. The stability problem is translated into the search for a multiplier M which belongs to the class of multipliers associated with the class of nonlinearities Φ , where G and M satisfy some frequency conditions. The class of Zames–Falb multipliers is defined both for the continuous-time domain [2] and for the discrete-time domain [3] (see [4] for a tutorial on Zames–Falb multipliers for the continuous-time domain). Loosely speaking, a Zames–Falb multiplier preserv es the positi vity of a monotone and bounded nonlinearity . Hence if an L TI plant G is in negati ve feedback with a monotone and bounded nonlinearity then stability is guaranteed if there is a multiplier M such that Re { M G } > 0 , (1) J. Carrasco, W . P . Heath, and J. Zhang are with the School of Electrical and Electronic Engineering, Univ ersity of Manchester , Sackville St. Building, Manchester M143 9PL, UK. e-mail: joaquin.carrascogomez@manchester .ac.uk, william.heath@manchester .ac.uk, jingfan.zhang@postgrad.manchester .ac.uk. N. S. Ahmad is with the School Of Electrical & Electronic Engineering, USM, Engineering Campus, Seberang Perai Selatan, Nibong T ebal, Penang, 14300, Malaysia. e-mail: syazreen@usm.my S. W ang is a Senior Researcher with Robotics X in T encent, High T ech- nology Park, Nanshan District, Shenzhen, 518057, China. e-mail: shawnsh- wang@tencent.com with M and G e valuated over all frequencies (i.e. at j ω , ω ∈ R for continuous-time systems and at e j ω , ω ∈ [ 0 , 2 π ] for discrete-time systems). Similarly (and by loop tranformation) if an L TI plant G is in negati ve feedback with an S [ 0 , k ] slope- restricted nonlinearity , then stability is guaranteed if there is a multiplier M such that Re { M ( 1 + kG ) } > 0 , (2) with M and G e valuated over all frequencies. In addition a wider class of multipliers is av ailable if the nonlinearity is odd; multipliers for quasi-odd multipliers can also be derived [5]. A. Ovevie w of sear ches of Zames–F alb multipliers in Continuous-time T o date, most of the literature on search methods for Zames- Falb multipliers has been focused on continuous-time systems, where three types of method hav e been developed: a) F inite Impulse Response (FIR): Searches over sums of Dirac delta functions are proposed and developed in [6], [7] and [8]. The main adv antage of this method is the simplicity and versatility of using impulse responses for the multiplier . Howe ver the searches require a sweep over all frequencies, which can lead to unreliable results in some cases [9]. More- ov er , the choice of times for the Dirac delta functions is heuristic. b) Basis functions: In [10] and [11] it is proposed to parameterise the multiplier in terms of causal basis functions e + i ( t ) = t i e − t u ( t ) where u ( t ) is the unit (or Hea viside) step function, and anticausal basis functions e − i ( t ) = t i e t u ( − t ) , with i = 1 , . . . , N for some N . As an advantage o ver the FIR method, the positivity of M ( 1 + k G ) can be tested through the KYP lemma. Moreov er the search provides a complete search ov er the class of rational multipliers as N approaches infinity [12]. The method provided significant adv antages, such as the combination with other nonlinearities [13]. Nonetheless if N is required to be large then the search becomes numerically ill-conditioned. W ith small N there is conservatism for odd nonlinearities, since the impulse of the multiplier is allowed to change sign. In fact the results reported in [10] for SISO examples are not significantly better for odd nonlinarities than for non-odd. c) Restricted structure rational multipliers: In [16] an LMI method is proposed where the L 1 norm of a low- order causal multiplier is bounded in a con vex manner (see also [17]). Se veral extensions hav e been proposed: adding 2 a Popov multiplier [18], de veloping an anticausal counter - part [9], and increasing the order of the multiplier [19]. The method is quasi-conv ex and effecti ve but does not provide a complete search. It has two further drawbacks: the bound of the L 1 -norm may be conserv ativ e and it can only be applied if the nonlinearity is odd. In [21], [4], it has been sho wn that their relativ e perfor- mances vary with different examples. It must be highlighted that results in the basis functions can be significantly improved by manually selecting the parameters of the basis [14], [15]. Similarly , manual tuning of delta functions can be useful for time-delay systems [22]. In addition, there are se veral other stability tests in the literature, where either the Zames–Falb multipliers are not explicitly inv oked or extensions to the Zames–Falb multipliers are proposed. These can all be viewed as searches ov er subclasses of Zames–Falb multipliers [20], [21]. In particular , the Off-Axis Circle Criterion is a powerful technique that uses graphical tools to ensure the existence of a possibly high-order multiplier by using graphical methods [23], hence avoiding the use of an optimization tool. It can be used to establish a large set of plants that satisfy the Kalman conjecture [24], [25]. B. Zames–F alb multipliers in Discr ete-time domain In [3], [26], the discrete-time counterparts of the Zames– Falb multipliers [2] are given. The conditions are the natural counterparts to the continuous-time case, where the L 1 -norm is replaced by the ` 1 -norm and the frequency-domain inequal- ity must be satisfied on the unit circle. In the continuous- time case, the use of improper multipliers has generated “extensions” of the original that hav e been analysed in [20], [21]. In the discrete-time case, the conditions for the Zames– Falb multipliers are necessary and sufficient to preserve the positivity of the nonlinearity [26]; it follo ws that the class of Zames–Falb multipliers is the widest class of multipliers that can be used. The result has been extended to MIMO systems [27], repeated nonlinearities in [28] and MIMO re- peated nonlinearities in [29]. These works are focused on the description of the av ailable multipliers, but no explicit search method is discussed. Modern digital control implementation requires a complete study in the discrete-time domain. In addition the possibility of using the Zames–Falb multipliers for studying the stability and robustness properties of input-constrained model predic- tiv e control (MPC) [30] provides an inherent moti vation for discrete-time analysis, since MPC is naturally formulated in discrete time. Recently , Zames–Falb multipliers in discrete- time hav e been attracting attention in their use to ensure con vergence rates of optimization algorithms [31], [33]. More generally , the absolute stability problem of discrete- time Lur’e systems with slope–restricted nonlinearities con- tinues to attract attention. Recent studies include [34], [35], [36], [55] which all take a L yapunov function approach; as an adv antage they generate easy-to-check Linear Matrix Inequality (LMI) conditions. Ho wev er one might expect that improv ed results could be obtained via a multiplier approach, since this provides a more general condition. In f act some of these approaches can be interpreted as a search over a small subclass of Zames–Falb multipliers; see [36] for further details. Although this paper deals with SISO systems, it must be highlighted that a tractable stability test using Zames–Falb multipliers for MIMO nonlinearities has been proposed in [37]. The differences between continuous-time and discrete-time Lur’e systems are non-trivial. As an example, second-order counterexamples to the discrete-time Kalman conjecture have been found [38], [39]. For continuous-time systems the Kalman conjecture holds for first, second, and third order plants [40]. This is reflected by phase restrictions that can be placed on discrete-time Zames–Falb multipliers that are different in kind to their continuous-time counterparts [41]. In this paper we propose se veral searches for SISO L TI discrete-time Zames–Falb multipliers. The search of multipli- ers can be carried out with two different approaches: a) Infinite impulse response (IIR) multiplier: The search is the counterpart of the method proposed by T urner et al. [16], [9], presented in [42] and included for the sake of completeness. The multipliers are parametrised in terms of their state-space representation, and classical multiobjectiv e techniques are used to produce an LMI search. b) F inite impulse r esponse (FIR) multiplier: This search can be considered as the counterpart of both Safonov’ s and Chen and W en’ s methods ([6], [11]). Initial results were presented in [43]. Here, two alternati ve v ersions are provided: firstly we propose an ad hoc factorization which leads to a hard-factorization of the multiplier; secondly we use standard lifting techniques, e.g. [44], whose factorization need not be hard but can provide other advantages. Numerical results and some computational consideration are discussed in Section V . In Section VI we consider how the discrete-time FIR search may be used effecti vely to find continuous-time multipliers. W e show by numerical examples that tailoring the method can match or beat searches proposed in the literature for rational transfer functions. W e must highlight that discrete-time Zames–Falb multi- pleirs hav e been defined as L TV operators [3]. Ho wev er , we reduce our attention to L TI Zames–Falb multiplier . In the spirit of [20], it remains open whether the restriction to L TI Zames– Falb multiplier can be made without loss of generality when G is an L TI system. Moreov er we ha ve conjectured that if there is no suitable Zames–Falb multiplier for a plant G and gain k smaller than its Nyquist gain (see Section II for a definition), then there exists a slope-restricted nonlinearity in [ 0 , k ] such that the feedback interconnection between G and the nonlinearity is unstable [41]. Howe ver , further work is required to prove or disprove these conjectures. I I . N OTA T I O N A N D P R E L I M I NA RY R E S U L T S Let Z and Z + be the set of integer numbers and positi ve integer numbers including 0, respectively . Let ` be the space of all real-v alued sequences, h : Z + → R . Let ` 1 ( Z ) be the space of all absolute summable sequences, so gi ven a sequence h : Z → R such that h ∈ ` 1 , then its ` 1 -norm is k h k 1 = ∞ ∑ k = − ∞ | h k | , (3) 3 where h k means the k th element of h . In addition, let ` 2 denote the Hilbert space of all square-summable real sequences f : Z + → R with the inner product defined as h f , g i = ∞ ∑ k = 0 f k g k , (4) for f , g ∈ ` 2 , k ∈ Z + . Similarly , we can define the Hilbert space ` 2 ( Z ) by considering real sequences f : Z → R . W e use 0 i to denote a row vector with i entries, all equal to zero. Similarly 0 denotes a matrix with zero entries where the dimension is obvious from the context. W e use I i to denote the i × i identity matrix. The standard notation RL ∞ is used for the space of all real rational transfer functions with no poles on the unit circle. If G ∈ RL ∞ , its norm is defined as k G k ∞ = sup | z | = 1 | G ( z ) | . Furthermore RH ∞ is used for the space of all real rational transfer functions with all poles strictly inside the unit circle. Similarly , RH − ∞ is used for the space of all real rational transfer functions with all poles strictly outside the unit circle. With some reasonable abuse of the notation, gi ven a rational transfer function H ( z ) analytic on the unit circle, k H k 1 means the ` 1 - norm of impulse response of H ( z ) . Let ¯ M denote a linear time in variant operator mapping a time domain input signal to a time domain output signal and let M denote the corresponding transfer function. W e consider that the domain of conv ergence includes the unit circle, so that the ` 1 -norm of the in verse z-transform of M is bounded if M ∈ RL ∞ . W e say the multiplier ¯ M is causal if M ∈ RH ∞ , ¯ M is anticausal if M ∈ RH − ∞ , and ¯ M is noncausal otherwise. See [45] for further discussion on causality and stability . Henceforth, we will use M for both the operator and its transfer function. A discrete L TI causal system G has the state space real- ization of ( A , B , C , D ). That is to say , assuming the input and output of G at sample k are u k and y k , respecti vely , and the inner state is denoted as x k , the following relationship is satisfied G : ( x k + 1 = Ax k + Bu k , y k = C x k + Du k , (5) in short G ∼  A B C D  . (6) Its transfer function is gi ven by G ( z ) = C ( zI − A ) − 1 B + D , where z is the z-transform of the forward (or left) shift operator . In fact, this notation is not always adopted in the literature since the definition of the z-transform is not uniform in the use of z or z − 1 . See [45], [47]. The discrete-time v ersion of the KYP lemma will be used to transfer frequency domain inequalities into LMIs: Lemma II.1. (Discr ete KYP lemma, [48]) Given A, B, M , with det ( e j ω I − A ) 6 = 0 for ω ∈ R and the pair ( A , B ) contr ollable, the following two statements ar e equivalent: (i)  ( e j ω I − A ) − 1 B I  ∗ M  ( e j ω I − A ) − 1 B I  ≤ 0 . (7) - 6 - ?  m  f v w φ G m − g Fig. 1. Lur’e problem (ii) Ther e is a matrix X ∈ R n × n such that X = X > and M +  A > X A − X A > X B B > X A B > X B  ≤ 0 . (8) The corresponding equivalence for strict inequalities holds even if the pair ( A , B ) is not contr ollable. Throughout this paper , the superscript ∗ stands for conjugate transpose. Remark II.2. State space r epr esentations such as (5) are appr opriate for causal systems, but not for anticausal and noncausal systems. These can be repr esented in state space as descriptor systems. The KYP lemma has been e xtended to descriptor systems in [49] for continuous-time LTI systems. In [50] an appr oach to the analysis of discrete singular systems is pr esented; however it is restricted to causal systems. In this work we e xploit the structur e of our multipliers to find causal systems that have the same fr equency response on the unit cir cle. Hence the classical KYP lemma suffices. The discrete-time Lur’e system is represented in Fig. 1. The interconnection relationship is ( v k = f k + ( Gw ) k , w k = − φ ( v k ) + g k . (9) The system (9) is well-posed if the map ( v , w ) 7→ ( g , f ) has a causal in verse on ` × ` , and this feedback interconnection is ` 2 -stable if for an y f , g ∈ ` 2 , both w , v ∈ ` 2 . The memoryless nonlinearity φ : R 7→ R with φ ( 0 ) = 0 is said to be bounded if there exists C such that | φ ( x ) | < C | x | for all x ∈ R and φ is said to be monotone if for any two real numbers x 1 and x 2 then 0 ≤ φ ( x 1 ) − φ ( x 2 ) x 1 − x 2 . (10) Moreov er , φ is slope-restricted in the interv al S [ 0 , K ] , hence- forth φ K , if 0 ≤ φ K ( x 1 ) − φ K ( x 2 ) x 1 − x 2 ≤ K , (11) for all x 1 6 = x 2 . Finally , the nonlinearity φ is said to be odd if φ ( x ) = − φ ( − x ) for all x ∈ R . Zames–Falb multipliers preserve the positi vity of the class of monotone nonlinearities [2], [3]. Then a loop transformation allows us to obtain the follo wing result for slope restricted nonlinearities: Theorem II.3 ([26], [3]) . Consider the feedback system in F ig. 1 with G ∈ RH ∞ , and φ is a slope-restricted in S [ 0 , K ] . Suppose that ther e exists a multiplier M : ` 2 ( Z ) 7→ 4 ` 2 ( Z ) whose impulse response is m : Z 7→ R and satisfies ∑ ∞ k = − ∞ | m k | < 2 m 0 , Re { M ( z )( 1 + K G ( z )) } > 0 ∀| z | = 1 , (12) and either m k ≤ 0 for all k 6 = 0 or φ is also odd. Then the feedback interconnection (9) is ` 2 -stable. The above theorem leads to the definition of the class of Zames–Falb multipliers: Definition II.4. (DT LTI Zames–F alb multipliers [3]) The class of discr ete-time SISO LTI Zames–F alb multipliers con- tains all LTI convolution operators M : ` 2 ( Z ) 7→ ` 2 ( Z ) whose impulse r esponse is m : Z 7→ R satisfies ∑ ∞ k = − ∞ | m k | < 2 m 0 . W ithout loss of gener ality , the value of m 0 can be chosen to be 1. Remark II.5. An important subclass of Zames–F alb multipli- ers is obtained by adding the limitation m k ≤ 0 , which must be used if we only have information about slope-r estriction of the nonlinearity . Remark II.6. It is also standard to write Definition II.4 using the ` 1 -norm by stating the condition as k M k 1 < 2 . Definition II.7. (Nyquist value) Given G ∈ RH ∞ , the Nyquist value k N is the supremum of all the positive r eal numbers K such that τ K G ( z ) satisfies the Nyquist Criterion for all τ ∈ [ 0 , 1 ] . It can also be expr essed as: k N = sup { K ∈ R + : inf ω {| 1 + τ k G ( e j ω ) |} > 0 ) , ∀ τ ∈ [ 0 , 1 ] } . (13) In terms of its state space r ealization (5) , k N is the supremum of K such that all eig en values of (A − BKC ) ar e located in the open unit disk, with K in the interval [ 0 , k N ] . Remark II.8. The Kalman conjecture is not valid for discrete- time systems even for plants of or der 2 [38], [39]. Ther e is no a priori guarantee (e xcept for first or der systems) that if k is less than the Nyquist value for the plant then the negative feedback inter connection of the plant and a nonlinearity slope- r estricted in S [ 0 , k ] is stable . I I I . S E A R C H E S F O R I I R M U LT I P L I E R S In III-A we present a search for discrete-time causal mul- tipliers that is the counterpart to the search for continuous- time causal multipliers presented in [16] (see also [17]). In Section III-B we present the anticausal counterpart, similar in spirit to the continuous-time anticausal search of [9]. The results in this section were fully presented in [42], so proofs are omitted. When the multiplier is parameterised in terms of its state- space representation as in [16], [17], we require the following bound [51] for all the searches. Lemma III.1 ([51]) . Consider a dynamical system G r ep- r esented by (5) and x 0 = 0 . Suppose that ther e exist µ > 0 , 0 < λ < 1 and P = P > such that  A > P A − λ P A > PB ? B > PB − µ I  < 0 , (14)  ( λ − 1 ) P + C > C C > D ? ( µ − γ 2 ) I + D > D  < 0 . (15) Then k G k 1 < γ . Furthermor e, A has all its eigen values in the open unit disk. The use of this result is a fundamental limitation of this method as the parameterisation of the multipliers requires their causality to be established before carrying out the search. Another important feature of this method is that it requires the nonlinearity to be odd as it is not possible to ensure the positivity of the impulse response of the multiplier . A. Causal searc h In the spirit of [16], a search ov er the class of causal discrete-time Zames–Falb multipliers is presented as follo ws: Proposition III.2. Let G ( z ) ∼  A g B g C g D g  wher e A g ∈ R n × n , B g ∈ R n × 1 , C g ∈ R 1 × n and D g ∈ R 1 × 1 . Let φ k be an odd nonlinearity slope-r estricted in S [ 0 , K ] . W ithout loss of g enerality , assume that the feedback inter connection of G and a linear gain K is stable. Define A p , B p , C p and D p as follows: A p = A g ; (16) B p = B g ; (17) C p = k C g ; (18) D p = 1 + k D g . (19) Assume that ther e exist positive definite symmetric matrices S 11 > 0 , P 11 > 0 , unstructur ed matrices ˆ A , ˆ B and ˆ C with the same dimension as A u , B u , and C u , respectively , and positive constants 0 < µ < 1 and 0 < λ < 1 such that the LMIs (20) , (21) , and (22) (given on the following page) ar e satisfied. Then the feedback inter connection (1) is ` 2 -stable. Remark III.3. Similar to the continuous case, the inequal- ities (20) , (21) , and (22) are not LMIs if λ is defined as variable. Hence , the use of this r esult r equir es a linear sear ch of λ over the interval between 0 and 1 . Remark III.4. The c hange of variable is the same as in the continuous case. Ther efor e the multiplier can be r ecover ed following [17] using A u = − ( P 11 − S 11 ) − 1 ˆ A , (23) B u = − ( P 11 − S 11 ) − 1 ˆ B , (24) C u = ˆ C . (25) Remark III.5. Under further conditions, e.g. D p = 0 , it is possible to extend this method with a first or der anticausal component in the multiplier , i.e. M ( z ) = ( 1 + m − 1 z ) + M c ( z ) , under the constraint | m − 1 | < 1 . The development of the result is similar with the use of the state-space r epr esentation of zG ( z ) . 5 LMIs in Proposition III.2:       − S 11 ? ? ? ? − S 11 − P 11 ? ? ? − C p − ˆ C − C p − D > p − D p ? ? S 11 A p S 11 A p S 11 B p − S 11 ? P 11 A p + ˆ B C p + ˆ A P 11 A p + ˆ B C p P 11 B p + ˆ B D p − S 11 − P 11       < 0 , (20)   λ ( S 11 − P 11 ) ? ? 0 − µ I ? − ˆ A − ˆ B S 11 − P 11   < 0 , (21)   ( λ − 1 )( P 11 − S 11 ) ? ? 0 ( µ − 1 ) I ? ˆ C 0 − I   < 0 . (22) B. Anticausal multiplier The anticausal counterpart of the abov e search can be stated as follows: Proposition III.6. Let G ∈ RH ∞ be repr esented in the state space by A g , B g , C g and D g wher e A g ∈ R n × n , B g ∈ R n × 1 , C g ∈ R 1 × n and D g ∈ R 1 × 1 . Let φ K an odd nonlinearity slope- r estricted in S [ 0 , K ] . W ithout loss of generality , assume that the feedback inter connection of G and a linear gain k is well- posed and stable. Define A p , B p , C p and D p as follows: A p = A g − B g ( kD g + 1 ) − 1 KC g ; (26) B p = − B g ( K D g + 1 ) − 1 ; (27) C p = ( K D g + 1 ) − 1 k C g ; (28) D p = ( K D g + 1 ) − 1 . (29) Assume that ther e exist positive definite symmetric matrices S 11 > 0 , P 11 > 0 , unstructur ed matrices ˆ A u , ˆ B u and ˆ C u , and positive constants 0 < µ < 1 and 0 < λ < 1 such that the LMIs (20) , (21) , and (22) are satisfied, then the feedbac k inter connection (1) is ` 2 -stable. Remark III.7. Once the sear ch has pr ovided the matrices A u , B u , and C u , then the multiplier is given by: M ac ( z ) = C u  z − 1 I − A u  − 1 B u + 1 , (30) which can be written as M ac ( z ) ∼  A −> u A −> u C > u B > u A −> u 1 − B > u A −> u C > u  , (31) if A u is non-singular . If A u is singular , then the result is still valid b ut the multiplier does not have a forwar d r epresen- tation. Note that the re gion of con ver gence of this tr ansfer function does not include z = ∞ and the term m 0 in the in verse z-tr ansform of M ac ( z ) corresponds with M ac ( 0 ) , i.e. ( Z − 1 ( M ac ))( 0 ) = M ac ( 0 ) . I V . S E A R C H E S F O R F I R M U L T I P L I E R S In this section, we restrict our attention to FIR multipliers, i.e. M ( z ) = n b ∑ i = − n f m i z − i , (32) where n b ≥ 0 and n f ≥ 0. Without loss of generality we set m 0 = 1. If the nonlinearity is not odd we consider only the subclass of Zames–Falb multipliers with m i ≤ 0 for all i ∈ Z \{ 0 } . The multiplier M is said to be causal if n b ≥ 0 and n f = 0, it is said to be anticausal if n b = 0 and n f ≥ 0, and it is said to be noncausal if n b > 0 and n f > 0. T wo different searches are included as they provide alter- nativ e insights on the design of the multiplier . T o conclude the section, we show that any Zames–F alb multiplier can be phase-substituted by an appropriate FIR multiplier . A. Har d-F actorizations of Zames–F alb multiplier s In this section we de velop an LMI search for FIR Zames– Falb multipliers. In Lemma IV .1 we sho w that the ` 1 condition can be expressed with linear constraints. In Lemma IV .3 we show that although our multiplier is noncausal, the positivity condition can be expressed in terms of a nonsingular state- space representation, leading to an LMI formulation. Our main stability result is stated in Theorem IV .4. It is possible to show that the LMI requires a positiv e definite matrix, so it is a hard- factorization. W e seek a Zames–F alb multiplier M ( z ) with structure of (32) and m 0 = 1 such that Re { M ( z )( 1 + K G ( z )) } > 0 for all | z | = 1 . (33) Lemma IV .1. If M ( z ) has the structur e of (32) with m 0 = 1 , then M ( z ) is a Zames–F alb multiplier pro vided m i ≤ 0 for i = − n f , . . . , − 1 and i = 1 , . . . , n b , (34) and n b ∑ i = − n f m i > 0 . (35) If the nonlinearity is odd then we can write m i = m + i − m − i for i = − n f , . . . , n b (we define m + 0 = 1 and m − 0 = 0 ) and M ( z ) is a Zames–F alb multiplier pr ovided: m + i ≥ 0 and m − i ≥ 0 for i = − n f , . . . , n b , (36) and n b ∑ i = − n f m + i + n b ∑ i = − n f m − i < 2 . (37) 6 Pr oof. This follows immediately from Theorem II.3. The decomposition for odd nonlinearities is the Jordan measure decomposition (e.g. [52]). Remark IV .2. If the nonlinearity is not odd this leads to n f + n b + 1 linear constraints while if the nonlinearity is odd this leads to 2 n f + 2 n b + 1 linear constraints. Giv en P ( z ) = 1 + kG ( z ) , condition (33) can be written: M ( z ) P ( z ) + [ M ( z ) P ( z )] ∗ > 0 for all | z | = 1 . (38) Howe ver , since M is noncausal and P ∈ RH ∞ , it follows that M P does not hav e a nonsingular state-space description. This is addressed in Lemma IV .3 below . First we define some quantities. Let P ( z ) hav e state-space description P ∼  A p B p C p D p  , , (39) where A ∈ R n p × n p . Let n = max ( n f , n b ) and define ˜ A =   A p B p 0 0 0 I n − 1 0 0 0   and ˜ B =   0 0 1   . (40) where ˜ A ∈ R ( n p + n ) × ( n p + n ) . Also let C n =  C p D p 0 n − 1  , (41) and C d , i =  0 n p + n − i 1 0 i − 1  for i = 1 , . . . , n f , (42) where n p is the dimension of A p . Define C i as C i = C n ˜ A n − i + − i ∑ j = 1  C n ˜ A n − i − j − 1 ˜ B  C d , j for i = − n f , . . . − 1 , (43) C 0 = C n ˜ A n , (44) C i = C n ˜ A n − i for i = 1 , . . . , n b , (45) and D i as D i = C n ˜ A n − i − 1 ˜ B for i = − n f , . . . , − 1 , (46) D 0 = C n ˜ A n − 1 ˜ B , (47) D i = 0 for i = 1 , . . . , n b . (48) Then we can say: Lemma IV .3. Suppose P ( z ) is a causal and stable discrete- time transfer function with state-space description (39) and suppose M ( z ) is a noncausal FIR transfer function given by (32) with m 0 = 1 . Ther e exist P i ( z ) for i = − n f , . . . , n b with nonsingular state-space repr esentation such that M ( z ) P ( z ) + [ M ( z ) P ( z )] ∗ = n b ∑ i = − n f m i ( P i ( z ) + [ P i ( z )] ∗ ) for all | z | = 1 . (49) Furthermor e the statement M ( z ) P ( z ) + [ M ( z ) P ( z )] ∗ > 0 for all | z | = 1 , (50) is equivalent to the statement that ther e exists a matrix X ∈ R ( n p + n ) × ( n p + n ) such that X = X > and  ˜ A > X ˜ A − X ˜ A > X ˜ B ˜ B > X ˜ A ˜ B > X ˜ B  − M > f Π M f < 0 , (51) with Π =  0 m m > 0  , (52) m > =  m − n f , . . . , m − 1 , 1 , m 1 , . . . m n b  , (53) and M f =      C − n f D − n f . . . . . . C n b D n b 0 1      , (54) with ˜ A, ˜ B, C i and D i given by (40), (43-45) and (46-48). Pr oof. W e can write M ( z ) P ( z ) = n b ∑ i = − n f m i z − i P ( z ) . (55) Hence we must choose causal P i ( z ) for i = − n f , . . . , n b such that P i ( z ) + [ P i ( z )] ∗ = z − i P ( z ) +  z − i P ( z )  ∗ for all | z | = 1 . (56) It follows immediately that for i = 0 , . . . , n b we can choose P i ( z ) = z − i P ( z ) . (57) When i is negativ e, z − i P ( z ) is not causal (beware: if i is negati ve then z − i is anticausal). W e can partition z − i P ( z ) into causal and anticausal parts z − i P ( z ) = P A C i ( z ) + P C i ( z ) . (58) The partition is standard since P A C i is FIR (e.g. [45]). If we write P as P ( z ) = ∞ ∑ k = 0 p k z − k , (59) then, for i = − n f , . . . , − 1, we ha ve P A C i ( z ) = − i − 1 ∑ k = 0 p k z − i − k = D p z − i + − i − 1 ∑ k = 1 C p A k − 1 p B p z − i − k , (60) and P C i ( z ) = z − i P ( z ) − P A C i ( z ) = C p A − i p ( zI − A p ) − 1 B p + C p A − i − 1 p B p . (61) Then we can choose P i ( z ) = P C i ( z ) + P A C i ( z − 1 ) . (62) W e parameterize each P i ( z ) as follo ws. Let n = max ( n f , n b ) . Define ˜ A and ˜ B as (40) and C n as (41). Then z − n P ( z ) = C n ( zI − ˜ A ) − 1 ˜ B . (63) 7 When i is positi ve we can write P i ( z ) = z − i P ( z ) = C n ˜ A n − i ( zI − ˜ A ) − 1 ˜ B = C i ( zI − ˜ A ) − 1 ˜ B + D i for i = 1 , . . . , n b , (64) where C i and D i are giv en by (45) and (48) respectiv ely . Similarly P 0 ( z ) = P ( z ) = C n ˜ A n ( zI − ˜ A ) − 1 ˜ B + C n ˜ A n − 1 ˜ B = C 0 ( zI − ˜ A ) − 1 ˜ B + D 0 , (65) where C 0 and D 0 are giv en by (44) and (47) respecti vely . When i is ne gativ e, we write P i ( z ) = C p A − i p ( zI − A p ) − 1 B p + C p A − i − 1 p B p + D p z − i + − i − 1 ∑ k = 1 C p A k − 1 p B p z − i − k . (66) The state space realization of the delay operator z − j is formu- lated as z − j = C d , j ( zI − ˜ A ) − 1 ˜ B , (67) with C d , i giv en by (42). So we can write this P i ( z ) = C n ˜ A n − i ( zI − ˜ A ) − 1 ˜ B + C n ˜ A n − i − 1 ˜ B + C n ˜ A n − 1 ˜ Bz − i + − i − 1 ∑ k = 1 C n ˜ A n + k − 1 ˜ Bz − i − k = C i ( zI − ˜ A ) − 1 ˜ B + D i for i = − n f , . . . , − 1 , (68) where C i and D i are giv en by (43) and (46) respecti vely . Finally we can write M ( z ) P ( z ) + [ M ( z ) P ( z )] ∗ =      P − n f ( z ) . . . P n b ( z ) 1      ∗  0 m m > 0       P − n f ( z ) . . . P n b ( z ) 1      =  ( zI − ˜ A ) − 1 ˜ B 1  ∗ M > f Π M f  ( zI − ˜ A ) − 1 ˜ B 1  . (69) The result then follo ws immediately from the KYP Lemma for discrete-time systems (Lemma II.1). W e can now state our main result. Theorem IV .4. Consider the feedback system in Fig .1 with G ∈ RH ∞ , and φ is a nonlinearity slope-r estricted in S [ 0 , k ] . Suppose we can find m and X such that the LMI (51) is sat- isfied under the conditions of Lemma IV .3 with the additional constraints either (34) and (35) or φ is also odd and (36) and (37). Then the feedbac k inter connection (9) is ` 2 -stable. Pr oof. This follows immediately from Lemma IV .1, Lemma IV .3 and Theorem II.3. Remark IV .5. Theor em IV .4 gives an LMI condition for stability . The symmetric matrix X has ( n + n p )( n + n p + 1 ) / 2 independent parameters while the parameter vector m has n f + n b fr ee variables when the nonlinearity is not odd and 2 n b + 2 n f fr ee variables when the nonlinearity is odd. When the nonlinearity is not odd ther e are n f + n b + 1 linear constraints on m and when the nonlinearity is odd ther e ar e 2 n f + 2 n b + 1 linear constraints. Proposition IV .6. If ther e exists X = X T satisfies (51) in Lemma IV .3, then X > 0 . Pr oof. It follows since the diagonal matrix block M T f Π M f with the ( n + n p ) first rows and columns is zero, hence condition (51) requires ˜ A T X ˜ A − X < 0 with all eigen values of ˜ A in the open unit disk, hence X > 0. B. Alternative implementation of FIR sear ch In this section we provide a causal-factorization ap- proach which is widely discrete-time for general robust tech- niques [44], b ut here we focus on Zames–Falb multipliers. One can think of this technique as the discrete-time counterpart of factorization approach in [13] for general continuous-time multipliers. By the IQC theorem, we seek a Zames–Falb multiplier such that  − G ( z ) 1  ∗  0 K M ∗ ( z ) K M ( z ) − ( M ( z ) + M ∗ ( z ))   − G ( z ) I  > 0 ∀| z | = 1 . Substituting the Zames–Falb multiplier M ( z ) by its FIR form (32) with n b = n f = n , then the IQC multiplier can be factorized via lifting as follo ws  0 K M ∗ ( z ) K M ( z ) − ( M ( z ) + M ∗ ( z ))  = Ψ ( z ) ∗ κ ( k , m ) Ψ ( z ) , where Ψ ( z ) =                   1 0 z − 1 0 z − 2 0 . . . . . . z − n 0 0 1 0 z − 1 0 z − 2 . . . . . . 0 z − n                   , and κ ( k , m ) is gi ven in (70) in ne xt page. Theorem IV .7. Consider the feedback system in Fig .1 with P ∈ RH ∞ , and φ is a nonlinearity slope-restricted in S [ 0 , K ] . Let Ψ ( z )  − G ( z ) 1  ∼  ˆ A ˆ B ˆ C ˆ D  , and m > =  m − n , . . . , m − 1 , 1 , m 1 , . . . m n  . If ther e exist X = X T and m such that  ˆ A > X ˆ A − X ˜ A > X ˜ B ˜ B > X ˜ A ˜ B > X ˜ B  +  ˆ C ˆ D  T κ ( k , m )  ˆ C ˆ D  < 0 , (71) 8 κ ( k , m ) =                   0 0 0 · · · 0 km 0 km 1 km 2 · · · k m n 0 0 0 · · · 0 km − 1 0 0 · · · 0 0 0 0 · · · 0 km − 2 0 0 · · · 0 . . . . . . . . . · · · . . . . . . . . . . . . · · · . . . 0 0 0 · · · 0 km − n 0 0 · · · 0 km 0 km − 1 km − 2 · · · km − n − 2 m 0 − m 1 − m − 1 − m 2 − m − 2 · · · − m n − m − n km 1 0 0 · · · 0 − m 1 − m − 1 0 0 · · · 0 km 2 0 0 · · · 0 − m 2 − m − 2 0 0 · · · 0 . . . . . . . . . · · · . . . . . . . . . . . . · · · . . . km n 0 0 · · · 0 − m n − m − n 0 0 · · · 0                   (70) n ∑ i = − n | m i | < 2 , (72) and either m i ≤ 0 for all i 6 = 0 or φ is odd, then the feedback inter connection (9) is ` 2 -stable. Pr oof. The proof follows by the application of the KYP lemma, as (71) is equiv alent to (12); hence the conditions of Theorem II.3 hold, and stability is then guaranteed. Remark IV .8. Conditions for quasi-odd, quasi-monotone non- linearities [5] can be straightforwar dly implemented. Remark IV .9. In this factorization, it is not possible to ensur e X > 0 . The introduction of the condition X > 0 would r educe the class of available multipliers. C. Phase-Equivalence In the spirit of [20], [21], we can state the phase-equi valence between the full class of L TI Zames–Falb multipliers and FIR Zames–Falb multipliers as follows: Lemma IV .10. Given P ∈ RH ∞ , if there exists a Zames–F alb multiplier M such that Re { M ( z ) P ( z ) } > 0 ∀| z | = 1 , (73) then there exists an FIR Zames–F alb multiplier M FIR such that Re { M FIR ( z ) P ( z ) } > 0 ∀| z | = 1 . (74) Pr oof. Given an L TI Zames–F alb multiplier M ( z ) = ∞ ∑ i = − ∞ m i z − i , and ∞ ∑ i = − ∞ | m i | < 2 m 0 , (75) for any ε > 0, there exists N such that − N − 1 ∑ i = − ∞ | m i | + ∞ ∑ i = N + 1 | m i | < ε . (76) W e can write M ( z ) = N ∑ i = − N m i z − i + M t ( z ) = M FIR ( z ) + M t ( z ) , (77) with k M t k ∞ ≤ k M t k 1 < ε . Meanwhile, as P ( z ) and M ( z ) are continuous functions in the unit circle, by the e xtreme v alue theorem [46], there exists δ 1 > 0 such that Re { M ( z ) P ( z ) } ≥ δ 1 for all | z | = 1 . (78) Let us choose N such that (76) is satisfied with ε = δ 1 2 k P k ∞ . Then for all z satisfying | z | = 1 we find Re { M ( z ) P ( z ) } = Re { M FIR ( z ) P ( z ) } + Re { M t ( z ) P ( z ) } ≤ Re { M FIR ( z ) P ( z ) } + | M t ( z ) P ( z ) | ≤ Re { M FIR ( z ) P ( z ) } + | M t ( z ) || P ( z ) | ≤ Re { M FIR ( z ) P ( z ) } + k M t k ∞ k P k ∞ ≤ Re { M FIR ( z ) P ( z ) } + δ 1 2 , (79) Finally , rearranging using (79) and using (73), it follo ws that Re { M FIR ( z ) P ( z ) } ≥ Re { M ( z ) P ( z ) } − δ 1 2 ≥ δ 1 2 > 0 for all | z | = 1 . (80) V . N U M E R I C A L R E S U L T S A. Comparison with other r esults T able I presents the numerical examples that we analyse. All six plants are taken from previous papers [36], [39]. Results are shown in T able II. W e have run results in Theorem IV .4 for values of n = n b = n f between 1 and 100, and optimal results are presented in T able II indicating n ∗ the optimal value of n. The FIR search is significantly better than all competiti ve results in the literature, it beats classical searched as the Tsypkin Criterion [53], [54] as well as the most recent result in the L yapunov literature [36], [55]. It is worth highlighting that these L yapunov methods correspond with particular cases of FIR Zames–Falb multipliers, besides small numerical discrep- ancies. Results [36] corresponds with the case n b = n f = 1, whereas results in [55] correspond with the case n b = n f = 2, besides small numerical discrepancies. Results ha ve been ob- tained by using CVX [56], [57] with the SeDuMi solver [58]. Roughly speaking, the higher the order of the multiplier , the better the results. Howe ver , there is a small deteriora- tion due to numerical issues as n increase. W e show that the maximum slope suf fers also a small deterioration as n increases by including the values of the maximum slope with n b = n f = 100. Figure 2 shows this deterioration as n increases for Example 1. W e associate this deterioration to the numerical error associated with an increment in the size of the matrices in the LMIs. 9 0 1 02 03 04 05 06 07 08 09 0 1 0 0 n 12.9 13 13.1 13.2 13.3 13.4 13.5 Maximum slope Fig. 2. Maximum slope for Example 1 as n = n f = n b increases. T ABLE I E X AM P L E S Ex. Plant 1 [36] G 1 ( z ) = 0 . 1 z z 2 − 1 . 8 z + 0 . 81 2 [36] G 2 ( z ) = z 3 − 1 . 95 z 2 + 0 . 9 z + 0 . 05 z 4 − 2 . 8 z 3 + 3 . 5 z 2 − 2 . 412 z + 0 . 7209 3 [36] G 3 ( z ) = − z 3 − 1 . 95 z 2 + 0 . 9 z + 0 . 05 z 4 − 2 . 8 z 3 + 3 . 5 z 2 − 2 . 412 z + 0 . 7209 4 [36] G 4 ( z ) = z 4 − 1 . 5 z 3 + 0 . 5 z 2 − 0 . 5 z + 0 . 5 4 . 4 z 5 − 8 . 957 z 4 + 9 . 893 z 3 − 5 . 671 z 2 + 2 . 207 z − 0 . 5 5 [36] G 5 ( z ) = − 0 . 5 z + 0 . 1 z 3 − 0 . 9 z 2 + 0 . 79 z + 0 . 089 6 [39] G 6 ( z ) = 2 z + 0 . 92 z 2 − 0 . 5 z There are small numerical differences between results with both factorizations. In general, there is a slightly better per - formance of the hard factorization presented in Section IV .A. For instance, maximum slope in Example 1 is 13.5215 with n f = n b = 28, whereas the soft factorization in Section IV .B reaches 13.5162 with n = 11. Similar deterioration is observed, maximum slope is reduced to 13.5001 when n = 100. As e xpected, results for odd nonlinearities are always better than results for non-odd nonlinearities. Although it is natural as the set of multiplier increase and phase retrictitions are reduced, this contrasts with the SISO results reported in [10] for the continuous case. In Examples 1 to 4 the FIR results beat all others in the literature. In Example 5 both the FIR results and others in the literature achiev e the Nyquist value. Example 6 is used in [39] to sho w that stability is deteriorated by the lack of symmetry . From [39], we expect that a maximum slope above 1 for odd nonlinearities and below 1 for non-odd nonlinearities. B. Structur e of Multipliers It is worth highlighting the sparsity in the structure of the multiplier . In Figure 3, we show the terms above 10 − 5 . The structure of the multiplier can be explained as it reaches it maximum allowed phase over some particular range of frequencies when it has an sparse structure [41], therefore the optimization use only the positions in the multiplier which are useful to correct the phase of the ( 1 + k G ) in the region when it is not positi ve. 0 5 10 15 20 25 30 35 40 n -40 -30 -20 -10 0 10 20 30 40 Terms above 10 -5 in the multiplier Fig. 3. Pattern of the significant terms of the multipliers for Example 1 as n = n f = n b increases. C. Computational time It is interesting to analyse the performance of the search as n increases. As expected, the computational time increases in a polynomial fashion. Ho wev er , it is worth highlighting that the use of the Jordan measure decomposition in (37) increases the computational time as the number of v ariables in the multiplier is doubled. The code is run in MA TLAB R2017a with Mac Book Pro 2.3 GHz Intel Core i5 and 8GB 2133 MHz LPDDR3. 0 1 02 03 04 05 06 07 08 09 0 1 0 0 n 0 50 100 150 200 250 300 Time (s) Fig. 4. Computational time require to find the maximum slope in Example 1 with a precision of 10 − 5 in the bisection algorithm. The bisection method is started with k min = 0 and k max = 1 . 1 k N . The case m i ≤ 0 in red (slope- restricted nonlinearities), and the in blue the most general class of multipliers (slope-restricted and odd nonlinearities). V I . A P P L I C A T I O N T O S A F O N OV ’ S M E T H O D Safonov proposed the first numerical method to search for Zames–Falb multipliers [6]. Different modifications hav e been proposed [7], [8] to produce numerical optimization of the multiplier . In this section, we pro vide a dif ferent approach, which require manual tuning from the user, b ut may be used to test the conservatism of fully-autonomous numerical searches. 10 T ABLE II S L OP E - R ES T R IC T E D R E S ULT S B Y U S I NG D I FFE R E N T S TA BI L I T Y C R IT E R IA . Criterion Odd nonlinearity? Ex. 1 Ex. 2 Ex. 3 Ex. 4 Ex. 5 Ex. 6 Circle Criterion [53] N 0.7934 0.1984 0.1379 1.5312 1.0273 0.6510 Tsypkin Criterion [54] N 3.8000 0.2427 0.1379 1.6911 1.0273 0.6510 Ahmad et. al. (2015), Thm 1 [36] N 12.4178 0.72614 0.30267 2.5911 2.4475 0.9067 Park et al. (2018) N 12.9960 0.7396 0.3054 2.5904 2.4475 0.9108 Causal DT Zames–Falb Y 12.4355 0.7687 0.2341 3.3606 2.3328 0.9222 Anticausal DT Zames–Falb Y 1.4994 0.4816 0.3058 3.2365 2.4474 1.0869 FIR Zames–Falb ( n f = 1, n b = 1) N 12.9957 0.7397 0.3054 2.5904 2.4475 0.9108 FIR Zames–Falb ( n f = 1, n b = 1) Y 12.9957 0.7783 0.3076 3.1350 2.4475 1.0869 FIR Zames–Falb ( n f = 2, n b = 2) N 12.9957 0.7397 0.3054 2.5904 2.4475 0.9115 FIR Zames–Falb ( n f = 2, n b = 2) Y 12.9957 0.7783 0.3076 3.1350 2.4475 1.0869 FIR Zames–Falb ( n f = 100, n b = 100) N 13.0280 0.7948 0.3113 3.8234 2.4475 0.9115 FIR Zames–Falb ( n f = 100, n b = 100) Y 13.5124 1.1047 0.3115 3.8196 2.4469 1.0849 FIR Zames–Falb ( n f = n b = n ∗ ) N 13.0284 (6) 0.8015 (12) 0.3120 (12) 3.8240 (24) 2.4475 (1) 0.9115 (2) FIR Zames–Falb ( n f = n b = n ∗ ) Y 13.5251 (28) 1.1073 (7) 0.3126 (4) 3.8304 (7) 2.4475 (1) 1.0869 (1) Nyquist V alue N/A 36.1000 2.7455 0.3126 7.9070 2.4475 1.0870 Note that other manual tunings of rational multipliers ha ve been suggested in the literature [13], [15], which also lead to improv ements over fully-autonomous searches. The idea is straightforward. Given a continuous plant G ( s ) we find the maximum slope as follows: 1) Choose a sampling time T s and find the discrete-time counterpart G d ( z ) . 2) Choose n f and n b . Find the discrete-time Zames–Falb multiplier M d ( z ) = n b ∑ i = − n f m i z − i , corresponding to the maximum K d such that Re { M d ( z )( 1 + K d G d ( z )) } > 0 for all | z | = 1 . 3) (Optional) Choose ε > 0. For − n f ≤ i ≤ n b , if | m i | < ε , set m i = 0 for tractability . 4) Define M ( s ) = n b ∑ i = − n f m i e − iT s s . It follows immediately that M ( s ) belongs to the appro- priate class of Zames–F alb multipliers. 5) Find the maximum K such that Re { M ( s )( 1 + K G ( s )) } > 0 for all Re { s } = 0 . Numerical r esults All the follo wing results are taken from [21] and gi ven in T able III. Here we just provide details of the suitable multi- plier obtained by the above method. W e have used standard command in MA TLAB c2d to perform the discretization. W e use ε = 10 − 3 in Step 3. A summary of the results is giv en in T able IV, but we provide detailed information for each example. a) Example 1: Choose T s = 0 . 05, N f = 1, N b = 1. The discrete search leads then to the continuous-time multiplier giv en by M ( s ) = − 0 . 5436 e 0 . 05 s + 1 − 0 . 4561 e − 0 . 05 s . The multiplier reaches the Nyquist value in this example (K=4.5984) which matches the best results reported in [9]. Ex. G ( s ) 1 G 1 ( s ) = s 2 − 0 . 2 s − 0 . 1 s 3 + 2 s 2 + s + 1 2 G 2 ( s ) = − G 1 ( s ) 3 G 3 ( s ) = s 2 s 4 + 0 . 2 s 3 + 6 s 2 + 0 . 1 s + 1 4 G 4 ( s ) = − G 3 ( s ) 5 G 5 ( s ) = s 2 s 4 + 0 . 0003 s 3 + 10 s 2 + 0 . 0021 s + 9 6 G 6 ( s ) = − G 5 ( s ) 7 G 7 ( s ) = s 2 s 3 + 2 s 2 + 2 s + 1 8 G 8 ( s ) = 9 . 432 ( s 2 + 15 . 6 s + 147 . 8 )( s 2 + 2 . 356 s + 56 . 21 )( s 2 − 0 . 332 s + 26 . 15 ) ( s 2 + 2 . 588 s + 90 . 9 )( s 2 + 11 . 79 s + 113 . 7 )( s 2 + 14 . 84 s + 84 . 05 )( s + 8 . 83 ) 9 G 9 ( s ) = s 2 s 4 + 5 . 001 s 3 + 7 . 005 s 2 + 5 . 006 s + 6 T ABLE III C O NT I N U OU S - TI M E E X A M PL E S F RO M [ 2 1 ] b) Example 2: Choose T s = 0 . 05, N f = 0, N b = 1. The discrete search leads then to the continuous-time multiplier giv en by M ( s ) = 1 − 0 . 9551 e − 0 . 05 s . The multiplier reaches the Nyquist value in this example (K=1.0894) which matches the best results reported in [9]. c) Example 3: Choose T s = 0 . 1, N f = 20, N b = 0. The discrete search leads then to the continuous-time multiplier giv en by M ( s ) = 1 − 0 . 6507 e 1 . 9 s − 0 . 3493 e 2 s . The multiplier reaches K = 1 . 945, a 21% improvement over the best results reported in [9]. d) Example 4: Choose T s = 0 . 02, N f = 1, N b = 80. The discrete search leads then to the continuous-time multiplier giv en by M ( s ) = − 0 . 9186 e 0 . 02 s + 1 − 0 . 0809 e − 1 . 6 s . The multiplier reaches K = 1 . 29, a 2% improv ement ov er the best results reported in [9]. e) Example 5: Choose T s = 0 . 02, N f = 0, N b = 50. The discrete search leads then to the continuous-time multiplier giv en by M ( s ) = 1 − 0 . 8902 e − 0 . 02 s + 0 . 1087 e − s . The multiplier reaches K = 0 . 0055, a 65% improvement over the best results reported in [9]. 11 Ex.1 Ex. 2 Ex. 3 Ex. 4 Ex. 5 Ex. 6 Ex. 7 Ex. 8 Ex. 9 Best results in [21] 4.5949 1.0894 1.6122 1.2652 0.00333 0.00333 10,000+ 87.3854 91.0858 Algorithm in Section VI 4.5949 1.0894 1.945 1.29 0.0055 0.0039 Unreliable Unreliable 91.0858 Nyquist value 4.5894 1.0894 ∞ 3.5000 ∞ 1.7142 ∞ 87.3854 ∞ T ABLE IV C O MPA R IS O N B E T WE E N B E S T R E SU LT S R E PO RT E D I N [ 9 ] A N D C O NT I N U OU S T IM E M ET H O D I N S E C TI O N V I . f) Example 6: Choose T s = 0 . 02, N f = 50, N b = 0. The discrete search leads then to the continuous-time multiplier giv en by M ( s ) = 1 − 0 . 7909 e 0 . 02 s + 0 . 2090 e s . The multiplier reaches K = 0 . 0039, a 20% improvement over the best results reported in [9]. g) Example 7: For this example the method is poor . W e must sample at T s < 0 . 0002 to achie ve a Nyquist value of ov er 10,000. But with T s so small, we require N f and N b intractably large to obtain good multipliers. For e xample, choosing T s = 0 . 0001, and N f = N b = 50 gi ves a maximum k = 28 . 6. By contrast, setting T s = 0 . 001 gi ves a maximum k = 768. Setting T s = 0 . 01 sets it back to k = 147. h) Example 8: Again for this example the method is poor . Extreme care must be taken when discretizing the model. Setting T s = 0 . 001 and N b = N f = 40 yields a maximum k = 64. Other methods yield the Nyquist value, which is circa 87. i) Example 9: Choose T s = 0 . 01, N f = 70, N b = 1. The discrete search leads then to the continuous-time multiplier giv en by M ( s ) = 1 − 0 . 976 e − 0 . 01 s − 0 . 0013 e 0 . 48 s − 0 . 0227 e 0 . 7 s . (81) The multiplier reaches K = 360, a 395% improvement over the best results reported in [9]. Figure 5 sho ws that the phase of M ( s )( 1 + 360 G 9 ( s )) is in the interval ( − 90 , 90 ) . 10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 2 10 3 10 4 Frequency (rad/unit of time) -100 -80 -60 -40 -20 0 20 40 60 80 100 Phase (degrees) Fig. 5. Phase of M ( s )( 1 + 360 G 9 ( s )) where M ( s ) is given by (81). V I I . C O N C L U S I O N S The results in this paper provide the best results in the literature for absolute stability of discrete-time L TI systems in feedback interconnection with slope-restricted nonlinearities. W e hav e dev eloped two search methodologies for discrete- time Zames–Falb multiplier: IIR and FIR. In contrast with continuous-time domain, one of the av ailable searches is better for all examples. W e show the superiority of these searches with respect to the recent method based on L yapunov func- tions, whose results are similar to our search with n b = n f = 2. 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