A Discrete-Time, Time-Delayed Lur'e Model with Biased Self-Excited Oscillations
Juan Paredes, Syed Aseem Ul Islam, Omran Kouba, Dennis S. Bernstein
AA Discrete-Time, Time-Delayed Lur’e Modelwith Biased Self-Excited Oscillations
Juan Paredes, Syed Aseem Ul Islam, Omran Kouba, and Dennis S. Bernstein
Abstract — Self-excited systems arise in many applications, suchas biochemical systems, mechanical systems with fluid-structureinteraction, and fuel-driven systems with combustion dynamics.This paper presents a Lur’e model that exhibits biased self-excited oscillations under constant inputs. The model involvesasymptotically stable linear dynamics, time delay, a washoutfilter, and a saturation nonlinearity. For all sufficiently largescalings of the loop transfer function, these components causedivergence under small signal levels and decay under largesignal amplitudes, thus producing an oscillatory response. Abias-generation mechanism is used to specify the mean of theoscillation. The main contribution of the paper is a detailedanalysis of a discrete-time version of this model.
I. I
NTRODUCTION
A self-excited system has the property that the input is con-stant but the response is oscillatory. Self-excited systems arisein numerous applications, such as biochemical systems, fluid-structure interaction, and combustion. The classical exampleof a self-excited system is the van der Pol oscillator, whichhas two states whose asymptotic response converges to a limitcycle. A self-excited system, however, may have an arbitrarynumber of states and need not possess a limit cycle. Overviewsof self-excited systems are given in [1], [2], and applicationsto chemical and biochemical systems are discussed in [3]–[5]. Self-excited thermoacoustic oscillation in combustors isdiscussed in [6]–[8]. Self-excited oscillations of a tropicalocean-atmosphere system are discussed in [9]. Fluid-structureinteraction and its role in aircraft wing flutter is discussed in[10]–[13]. Wind-induced self-excited motion and its role inthe Tacoma Bridge collapse is discussed in [14].Models of self-excited systems are typically derived in termsof the relevant physics of the application. From a systemsperspective, the main interest is in understanding the featuresof the components of the system that give rise to self-sustainedoscillations. Understanding these mechanisms can illuminatethe relevant physics in specific domains and provide unityacross various domains.A unifying model for self-excited systems is a feedbackloop involving linear and nonlinear elements; systems of thistype are called
Lur’e systems . Lur’e systems have been widelystudied in the classical literature on stability theory [15].Within the context of self-excited systems, Lur’e systemsunder various assumptions are considered in [2], [16]–[24].
Juan Paredes, Syed Aseem Ul Islam, and Dennis S. Bernstein are with theDepartment of Aerospace Engineering, University of Michigan, Ann Arbor,MI, USA. { jparedes, aseemisl, dsbaero } @umich.edu Omran Kouba is with the Department of Mathematics in the Higher Instituteof Applied Sciences and Technology, Damascus, Syria.
Application to thermoacoustic oscillation in combustors isconsidered in [25]. Self-oscillating discrete-time systems areconsidered in [26]–[29].Roughly speaking, self-excited oscillations arise from acombination of stabilizing and destabilizing effects. Destabi-lization at small signal levels causes the response to grow fromthe vicinity of an equilibrium, whereas stabilization at largesignal levels causes the response to decay from large signallevels. In particular, negative damping at low signal levels andpositive damping at high signal levels is the mechanism thatgives rise to a limit cycle in the van der Pol oscillator [30,pp. 103–107]. Note that, although systems with limit-cycleoscillations are self-excited, the converse need not be true sincethe response of a self-excited system may oscillate without thetrajectory reaching a limit cycle. Alternative mechanisms exist,however; for example, time delays are destabilizing, and Lur’emodels with time delay have been extensively considered asmodels of self-excited systems [31].The present paper considers a time-delayed Lur’e (TDL)model that exhibits self-excited oscillations. This model, whichis illustrated in Figure 1, incorporates the following compo-nents: i ) Asymptotically stable linear dynamics. ii ) Time delay. iii ) A washout (that is, highpass) filter. iv ) A continuous, bounded nonlinearity N : R → R thatsatisfies N (0) = 0 , is either nondecreasing or nonin-creasing, and changes sign (positive to negative or viceversa) at the origin. v ) A bias-generation mechanism, which produces an offsetin the oscillatory response that depends on the value ofthe constant external input.A notable feature of this model is that self-oscillations areguaranteed to exist for asymptotically stable dynamics thatare not necessarily passive as in [32]. We note that washoutfilters are used in [33] to achieve stabilization, whereas, in thepresent paper, they are used to create self-oscillations. β + G ( s ) v b W ( s ) e − T d s N× v v f yy d y f Fig. 1: Continuous-time, time-delayed Lur’e model with constant input u andbias generation. For this time-delay Lur’e model, the time-delay provides the a r X i v : . [ n li n . AO ] M a r estabilization mechanism, while, under large signal levels, thesaturation function yields a constant signal, which effectivelybreaks the loop, thus allowing the open-loop dynamics to stabi-lize the response. This stabilization occurs at large amplitude.In order to create an oscillatory response, the Lur’e modelincludes a washout filter, which removes the DC componentof the delayed signal y d and allows the saturation functionto operate in its small-signal linear region. A similar featureappears in [2], [16]–[19] in the form of the numerator s in G for the case where y represents velocity. This combination ofelements produces self-excited oscillations for all sufficientlylarge scalings of the asymptotically stable dynamics. Anadditional feature of this model is the ability to produceoscillations with a bias, that is, an offset. This is done by thebias-generation mechanism involving the scalar β. Example1.1 illustrates the response of the model in Figure 1.
Example 1.1:
Let G ( s ) = s +1 , W ( s ) = s . s +1 , and N ( y f ) = tanh( y f ) . For T d = 5 s, β = 5 , and v = 10 , the response of the TDL model is shown in Figure 2. Inparticular, the output y ( t ) converges to a periodic signal withbias vβG (0) = 50 . Next, the effect of v, β, and T d on theoscillatory response of the model will be shown. The responseof the TDL model for v = 2 . , , β = 5 , and T d = 1 , s isshown in Figure 3, while the response for v = 2 . , β = 5 , , and T d = 1 , s is shown in Figure 4. Note that, for the sametransfer function G , different values of v, β, and T d producedifferent waveforms for y ( t ) and different phase portraits of y f ( t ) versus y ( t ) . (cid:5) Fig. 2: Example 1.1 Self-excited oscillatory response of the continuous-timetime-delay Lur’e model shown in Figure 1. Note that the oscillation hasnonzero bias due to the bias-generation mechanism.
The analysis and examples in the paper focus on a discrete-time version of the time-delayed Lur’e model with the standardsaturation function. This setting simplifies the analysis ofsolutions as well as the numerical simulations.The contents of the paper are as follows. Section II considersa discrete-time linear feedback model and analyzes the rangeof values of α for which the closed-loop model is asymp-totically stable. Section III extends the problem in SectionII by including a saturation nonlinearity. This discrete-timeLur’e model is shown to have an asymptotically oscillatoryresponse for sufficiently large values of the loop gain. SectionIV extends the Lur’e model to include a bias-generationmechanism.Preliminary results relating to the present paper appear in[34]. Key differences between [34] and the present paperinclude the following: 1) Lemma 2.1 and v ) of Theorem 2.2 are Fig. 3: Example 1.1: For v = 2 . , , β = 5 , and T d = 1 , s, (a) shows theresponse y ( t ) , and (b) shows y ( t ) versus y f ( t ) for t > s. Note that theshapes of the oscillations are distinct for different parameters under the samelinear dynamics given by G. Fig. 4: Example 1.1: For v = 2 . , β = 5 , , and T d = 1 , s, (a) shows theresponse y ( t ) , and (b) shows y ( t ) versus y f ( t ) for t > s. Note the effectof the value of β on the output offset and how the shape of the oscillation iskept for different values of β. not given in [34]; 2) due to limited space, no proofs are givenin [34]; and 3) the present paper includes several examplesthat do not appear in [34].Define Z (cid:52) = { . . . , − , , , . . . } , N (cid:52) = { , , , . . . } , and P (cid:52) = { , , . . . } . For all polynomials p, spr( p ) denotes the max-imum magnitude of all elements of roots( p ) . For all nonzero z = x + y ∈ C , where x and y are real, arg z = atan2( y, x ) ∈ ( − π, π ] denotes the principal angle of z . Let P (cid:52) = N/D be atransfer function with no zeros on the unit circle, where N and D are coprime, m (cid:52) = deg N and n (cid:52) = deg D. For all θ ∈ [0 , π ] , writing P ( z ) = N ( z ) D ( z ) = K (cid:81) mi =1 ( z − z i ) (cid:81) nj =1 ( z − p i ) , where K is a nonzeroreal number, ∠ P ( e θ ) ∈ R denotes the unwrapped phase anglef P evaluated at θ ∈ ( − π, π ] , such that ∠ P ( e θ ) (cid:52) = m (cid:88) i =1 arg( e θ − z i ) − n (cid:88) j =1 arg( e θ − p i ) . Unlike θ (cid:55)→ arg P ( e θ ) , which may be discontinuous on [0 , π ] ,the function θ (cid:55)→ ∠ P ( e θ ) is C on [0 , π ] . In addition, forall θ ∈ [0 , π ] , there exists r θ ∈ Z such that ∠ P ( e θ ) =arg P ( e θ ) + 2 πr θ . II. T
IME -D ELAYED L INEAR F EEDBACK MODEL
In this section we consider the discrete-time, time-delayedLur’e model shown in Figure 5, where α ∈ R , G is a strictlyproper asymptotically stable SISO transfer function with nozeros on the unit circle, G d ( z ) (cid:52) = 1 /z d is a d -step delay, where d ∈ N , and W ( z ) (cid:52) = ( z − /z is a washout (that is, highpass)filter. Let G = N/D , where the polynomials N and D arecoprime, D is monic, n (cid:52) = deg D , and m (cid:52) = deg N. αG ( z ) W ( z ) G d ( z ) y f yy d Fig. 5: Discrete-time time-delayed linear feedback system.
Let ( A, B, C, be a minimal realization of G whoseinternal state at step k is x k ∈ R n . Furthermore, considerthe realization ( N d , e d,d , e T1 ,d , of G d with internal state x d ,k ∈ R d , where N d is the standard d × d nilpotent matrixand e i,d is the i th column of the d × d identity matrix I d . Finally, let (0 , , − , be a realization of W with internalstate x f ,k ∈ R , and let α be a real number that scales G. Then, the discrete-time, time-delayed linear feedback modelshown in Figure 5 has the closed-loop dynamics x k +1 x d ,k +1 x f ,k +1 = A αBe T1 ,d − αBe d,d C N d e T1 ,d x k x d ,k x f ,k , (1)with output y k = (cid:2) C (cid:3) x k x d ,k x f ,k (2)and internal signals y d ,k = e T1 ,d x d ,k , (3) y f ,k = − x f ,k + y d ,k . (4)For all d ∈ N and α ∈ R , define L d,α ( z ) (cid:52) = αG ( z ) W ( z ) G d ( z ) = α ( z − N ( z ) z d +1 D ( z ) . (5)Furthermore, for all d ∈ N , define L d (cid:52) = L d, = GW G d . Finally, for all d ∈ N and α ∈ R , note that the closed-loop transfer function of the time-delayed linear feedback model isgiven by L d,α − L d,α = α ( z − N ( z ) p d,α ( z ) , (6)where p d,α ( z ) (cid:52) = z d +1 D ( z ) − α ( z − N ( z ) . (7)Note that, for all α ∈ R , p d,α .The following lemma is needed for the proof of Theorem2.2. Lemma 2.1:
Let p and q be monic polynomials with realcoefficients, assume that deg q < deg p , assume that all ofthe roots of p are in the open unit disk, and, for all α ∈ R , define p α (cid:52) = p + αq. Then, there exist α n < , α p > , and δ > such that spr( p α n ) = spr( p α p ) = 1 , for all α ∈ ( α n , α n + δ ) ∪ ( α p − δ, α p ) , spr( p α ) < , and, for all α ∈ ( α n − δ, α n ) ∪ ( α p , α p + δ ) , spr( p α ) > . Proof.
Let k be the smallest positive integer such that h ( x ) (cid:52) = x k [ p ( x ) q (1 /x ) − q ( x ) p (1 /x )] is a polynomial, and define Z (cid:52) = { z ∈ C : | z | = 1 , h ( z ) = 0 , and q ( z ) (cid:54) = 0 } . Note that Z has at most deg h elements. Furthermore, since h ( z ) = 0 for all z ∈ Z and p and q have real coefficients, itfollows that, for all z ∈ Z , (cid:18) p ( z ) q ( z ) (cid:19) = p (¯ z ) q (¯ z ) = p (1 /z ) q (1 /z ) = p ( z ) q ( z ) , which implies that, for all z ∈ Z , p ( z ) /q ( z ) ∈ R .Next, define A (cid:52) = {− p ( z ) /q ( z ) : z ∈ Z} , and, in the casewhere A is not empty, let A = { α , . . . , α m } , where m ≤ deg h and α < · · · < α m . Note that, since, for all z ∈ Z ,p ( z ) (cid:54) = 0 , it follows that / ∈ A . Now, let α be a real numberthat is not contained in A , and suppose that spr( p α ) = 1 . Then, there exists z α ∈ C such that p α ( z α ) = 0 and | z α | = 1 . To show that q ( z α ) (cid:54) = 0 , suppose that q ( z α ) = 0 . Then,since p α ( z α ) = 0 , it follows that p ( z α ) = 0 , which, sinceall of the roots of p are in the open unit disk, implies that spr( p α ) < , which is a contradiction. Hence, q ( z α ) (cid:54) = 0 . Next, to show that spr( p α ) (cid:54) = 1 , note that p α ( z α ) = 0 implies p α ( z α )= p ( z α ) + αq ( z α )= p ( z α ) + αq ( z α )= p ( z α ) + αq ( z α )= p (1 /z α ) + αq (1 /z α ) . (8)Since, in addition, α = − p ( z α ) /q ( z α ) , it follows from (8) that p (1 /z α ) − ( p ( z α ) /q ( z α )) q (1 /z α ) = 0 . (9)Now, multiplying both sides of (9) by q ( z α ) implies q ( z α ) p (1 /z α ) − p ( z α ) q (1 /z α ) = − h ( z α ) /z kα = 0 , nd thus h ( z α ) = 0 . Hence, z α ∈ Z , and thus α = − p ( z α ) /q ( z α ) ∈ A , which is a contradiction. Therefore, forall ˆ α / ∈ A , spr( p ˆ α ) (cid:54) = 1 . Next, let j ∈ { , , . . . , m } , and define I j (cid:52) = ( α j , α j +1 ) , where α (cid:52) = −∞ and α m +1 (cid:52) = ∞ . For all α ∈ I j , it followsfrom the continuity of α (cid:55)→ spr( p α ) that either spr( p α ) < or spr( p α ) > . Next, write p ( x ) = a n x n + · · · + a , q ( x ) = b d x d + · · · + b such that b d (cid:54) = 0 , a n (cid:54) = 0 , and d < n , and let z ,α , . . . , z n,α be the roots of p α . Then, the coefficient of x d in p α is relatedto the roots of p α by a d + αb d = a n ( − n − d (cid:88) (cid:89) j ∈ B z j,α , (10)where the sum is taken over all (cid:0) nn − d (cid:1) subsets B of { , . . . , n } with n − d elements. It thus follows from (10) that | a d + αb d | ≤ | a n | (cid:18) nn − d (cid:19) spr( p α ) n − d , which implies that lim α →−∞ spr( p α ) = lim α →∞ spr( p α ) = ∞ . Hence, for all α ∈ I ∪ I m , spr( p α ) > . Next, since spr( p ) = spr( p ) < , / ∈ A , and, for all α ∈ I ∪ I m , spr( p α ) > , it follows that there exists a unique j ∈ { , . . . , m − } such that ∈ I j . Hence, for all α ∈ I j , spr( p α ) < . Now, define j n (cid:52) = min { j ∈ { , . . . , m − } : spr( p α ) < for all α ∈ I j } ,j p (cid:52) = max { j ∈ { , . . . , m − } : spr( p α ) < for all α ∈ I j } . Then, for all α ∈ I j n ∪ I j p , spr( p α ) < and, for all α ∈ I j n − ∪ I j p +1 , spr( p α ) > , and thus it follows fromthe continuity of spr and the intermediate value theorem that spr( p α j n ) = spr( p α j p+1 ) = 1 . Furthermore, since j n ≤ j ≤ j p , it follows that α j n < and α j p +1 > . Hence, defining α n (cid:52) = α j n and α p (cid:52) = α j p +1 , which, as an aside, shows that A has at least two elements, it follows that α n < , α p > , and spr( p α n ) = spr( p α p ) = 1 , and, furthermore, there exists δ > such that, for all α ∈ ( α n , α n + δ ) ∪ ( α p − δ, α p ) , spr( p α ) < , and, for all α ∈ ( α n − δ, α n ) ∪ ( α p , α p + δ ) , spr( p α ) > , which completes the proof. (cid:3) The following result shows that, for sufficiently large valuesof the delay d, the linear closed-loop system is not asymptoti-cally stable outside of a bounded interval of values of α. Thisresult also shows that, for asymptotically large d, this rangeof values of α is finite and symmetric. Theorem 2.2:
The following statements hold: i ) For all d ∈ N , there exist α d, , α d , α d, > suchthat α d, < α d < α d, , spr( p d,α d ) = 1 , for all α ∈ ( α d, , α d ) , spr( p d,α ) < , and, for all α ∈ ( α d , α d, ) , spr( p d,α ) > . ii ) For all d ∈ N , there exist α d, , α d , α d, < suchthat α d, < α d < α d, , spr( p d,α d ) = 1 , for all α ∈ ( α d , α d, ) , spr( p d,α ) < , and, for all α ∈ ( α d, , α d ) , spr( p d,α ) > . Furthermore, there exists ¯ d ∈ N such that the followingstatements hold: iii ) For all d > ¯ d and θ ∈ (0 , π ] , L d ( e θ ) (cid:54) = 0 and dd θ ∠ L d ( e θ ) < . (11) iv ) For all d > ¯ d, there exist α d, l < and α d, u > suchthat p d,α is asymptotically stable if and only if α ∈ ( α d, l , α d, u ) , and p d,α is not asymptotically stable if andonly if α ∈ ( −∞ , α d, l ] ∪ [ α d, u , ∞ ) . v ) Define α ∞ (cid:52) = min θ ∈ (0 ,π ] (cid:12)(cid:12)(cid:12)(cid:12) D ( e θ )( e θ − N ( e θ ) (cid:12)(cid:12)(cid:12)(cid:12) . (12)Then, α ∞ > , for all d > ¯ d, α ∞ ≤ min {− α d, l , α d, u } , and lim d →∞ − α d, l = lim d →∞ α d, u = α ∞ . (13) Proof. i ) and ii ) follow from Lemma 2.1. To prove iii ),note that, for all θ ∈ (0 , π ] , G ( e θ ) (cid:54) = 0 , W ( e θ ) (cid:54) = 0 , and G d ( e θ ) (cid:54) = 0 , and thus L d ( e θ ) (cid:54) = 0 . Next, let θ ∈ (0 , π ] , andnote that sin θ cos θ − θ cos θ cos θ − sin θ − (sin θ + cos θ )= cos θ − sin θ = sin π + θ cos π + θ , which implies arg(cos θ − sin θ ) = arg(cos π + θ + sin π + θ ) . Hence ∠ W ( e θ ) = arg( e θ − − arg( e θ )= arg(cos θ − sin θ ) − θ = arg(cos π + θ + sin π + θ ) − θ = π + θ − θ = π − θ . (14)Next, letting d ∈ N , it follows from (14) that ∠ L d ( e θ ) = ∠ G ( e θ ) + ∠ W ( e θ ) + ∠ G d ( e θ )= ∠ G ( e θ ) + π − θ − dθ = ∠ G ( e θ ) + π − ( d + ) θ. (15)Now, let ¯ d ∈ N satisfy max θ ∈ [0 ,π ] dd θ ∠ G ( e θ ) ≤ ¯ d + . herefore, for all θ ∈ (0 , π ] and d > ¯ d, dd θ ∠ L d ( e θ ) = dd θ ∠ G ( e θ ) − d − ≤ max θ ∈ (0 ,π ] dd θ ∠ G ( e θ ) − d − ≤ ¯ d + − d − < . To prove iv ), note that iii ) implies that, for all d > ¯ d , ∠ L d ( e θ ) is a decreasing function of θ on (0 , π ] . Hence,for all α > , all crossings of the positive real axis by theNyquist plot of L d,α ( e θ ) = αL d ( e θ ) as θ increases over theinterval ( − π, π ] occur from the first quadrant to the fourthquadrant. Next, note that, for all d > ¯ d and θ ∈ (0 , π ] , | L d,α ( e θ ) | = α | L d ( e θ ) | is a increasing function of α on (0 , ∞ ) , and that, for all d > ¯ d and α > , since all of the polesof L d,α are in the open unit disk, it follows that spr( p d,α ) > if and only if the number of clockwise encirclements of of the Nyquist plot of L d,α ( e θ ) over θ ∈ ( − π, π ] is at leastone. Therefore, for all d > ¯ d and α , α > such that spr( p d,α ) > and α > α , the Nyquist plot of L d,α ( e θ ) over θ ∈ ( − π, π ] has at least one clockwise encirclement of . Furthermore, for all d > ¯ d and α , α > such that spr( p d,α ) < and α < α , the Nyquist plot of L d,α ( e θ ) over θ ∈ ( − π, π ] has zero encirclements of . Hence, i ) implies that there exists a unique α d, u > such that spr( p d,α d, u ) = 1 , for all α ∈ [0 , α d, u ) , spr( p d,α ) < , and, forall α ∈ [ α d, u , ∞ ) , spr( p d,α ) ≥ . Similarly, ii ) implies thatthere exists a unique α d, l < such that spr( p d,α d, l ) = 1 , forall α ∈ ( α d, l , , spr( p d,α ) < and, for all α ∈ ( −∞ , α d, l ) , spr( p d,α ) > . Hence, iv ) holds.To prove v ), let α ∈ R and d ≥ ¯ d. Note that roots( p d,α ) has at most n + d +1 elements and that λ ∈ roots( p d,α ) if andonly if L d,α ( λ ) = 1 . Now, let λ = ρe θ ∈ roots( p d,α ) , where ρ ∈ [0 , ∞ ) and θ ∈ ( − π, π ] . Writing G ( z ) = K (cid:81) mk =1 ( z − z k ) (cid:81) nk =1 ( z − p k ) ,it follows from L d,α ( λ ) = 1 that | α | = | λ d +1 | (cid:81) nk =1 | λ − p k || K || λ − | (cid:81) mk =1 | λ − z k | = ρ d + n − m (cid:81) nk =1 | e θ − p k ρ || K || e θ − ρ | (cid:81) mk =1 | e θ − z k ρ | . (16)The case where α > is considered henceforth. Let A d, + denote the set of all α > such that roots( p d,α ) has at leastone element with magnitude 1. It follows from i ) that A d, + is not empty. Furthermore, iv ) implies that A d, + ⊆ [ α d, u , ∞ ) . Finally, i ) and iv ) imply that spr( p d,α d, u ) = 1 , and thus α d, u =min A d, + . For all α ∈ A d, + , let θ α ∈ (0 , π ] satisfy p d,α ( e θ α ) = 0 . Itthus follows from (16) with ρ = 1 that, for all α ∈ A d, + ,α = g ( θ α ) , (17)where g : (0 , π ] → (0 , ∞ ) is defined by g ( θ ) (cid:52) = (cid:81) nk =1 | e θ − p k || K || e θ − | (cid:81) mk =1 | e θ − z k | . Since g is continuous and lim θ ↓ g ( θ ) = ∞ , it follows that g has a global minimizer. Hence, define the set of minimizersof g by M (cid:52) = { θ ∈ (0 , π ] : g ( θ ) = α ∞ } , where α ∞ (cid:52) = min θ ∈ (0 ,π ] g ( θ )= min θ ∈ (0 ,π ] (cid:81) nk =1 | e θ − p k || K || e θ − | (cid:81) mk =1 | e θ − z k | = min θ ∈ (0 ,π ] (cid:12)(cid:12)(cid:12)(cid:12) D ( e θ )( e θ − N ( e θ ) (cid:12)(cid:12)(cid:12)(cid:12) . Hence, the minimum in (12) exists, is positive, and is indepen-dent of d. Furthermore, for all α ∈ A d, + , α ∞ ≤ g ( θ α ) = α, and thus α ∞ ≤ min A d, + . Next, we show that there exists r ∈ Z such that ∠ L d (1) +2 r π ∈ (0 , π ) and that, for all d ≥ ¯ d, there exists r d ∈ Z suchthat ∠ L d ( e π ) + 2 r d π ∈ [ − π, .We now consider the case where G (1) > ; the case G (1) < is addressed by the case where α < . Let r ∈ Z satisfy ∠ G (1) + 2 r π = 0 , and note that (15) implies that, for all d ≥ ¯ d, ∠ L d (1) (cid:52) = lim θ ↓ ∠ L d ( e θ ) = π + ∠ G (1) . It thusfollows that ∠ L d (1) + 2 r π = π ∈ (0 , π ) . (18)Next, (15) with θ = π implies that, for all r ∈ Z , ∠ L d ( −
1) + 2 rπ = ∠ G ( − − dπ + 2 rπ. (19)Let r m ∈ Z satisfy ∠ G ( −
1) + 2 r m π ∈ {− π, } . Consider thecase where G ( − > , and thus ∠ G ( − r m π = 0 . Define r d, p (cid:52) = r m + (cid:98) d (cid:99) . In the case where d is even, it follows from(19) with r = r d, p that ∠ L d ( −
1) + 2 r d, p π = 0 . Likewise, inthe case where d is odd, ∠ L d ( −
1) + 2 r d, p π = − π . Hence, ∠ L d ( −
1) + 2 r d, p π ∈ {− π, } . (20)Similarly, in the case where G ( − < , define r d, n (cid:52) = r m + (cid:98) d +12 (cid:99) so that ∠ L d ( −
1) + 2 r d, n π ∈ {− π, } . (21)Note that, in the case where d is even, r d, p = r d, n = r m + d , whereas, in the case where d is odd, r m + d − = r d, p
1) + 2 r d π ∈ {− π, } ⊂ [ − π, . (23)Next, let d ≥ ¯ d. Then, iii ) implies that, θ (cid:55)→ ∠ L d ( e θ ) iscontinuous and decreasing on (0 , π ] . It thus follows from (18)that r = 14 − π ∠ L d (1) (24)nd from (23) that r d ∈ { r d, min , r d, max } , where r d, min (cid:52) = − − π ∠ L d ( − ,r d, max (cid:52) = − π ∠ L d ( − . Since ∠ L d ( −
1) = ∠ L d ( e π ) < ∠ L d (1) , it follows from (24)that r < − π ∠ L d ( −
1) = 14 + r d, max = 34 + r d, min < r d, min < r d, max , and thus r < r d + 1 , which implies that r ≤ r d . Then, let d ≥ ¯ d. Since (18) implies that ∠ L d (1)+2 r π > and (23) implies that ∠ L d ( −
1) + 2 r d π ≤ , it follows that,for all r ∈ { r , . . . , r d } , ∠ L d (1) + 2 rπ > , (25)and ∠ L d ( −
1) + 2 rπ ≤ . (26)Thus, since θ → ∠ L d ( e θ ) is decreasing and continuous on (0 , π ] , it follows from (25), (26), and the intermediate valuetheorem that, for all r ∈ { r , . . . , r d } , there exists a unique θ r,d ∈ (0 , π ] such that ∠ L d ( e θ r,d ) + 2 rπ = 0 . (27)Furthermore, let r l , r h ∈ { r , . . . , r d } such that r l ≤ r h , andlet θ r l ,d , θ r h ,d ∈ (0 , π ] satisfy ∠ L d ( e θ r l ,d ) + 2 r l π = ∠ L d ( e θ r h ,d ) + 2 r h π = 0 . (28)In the case where r l = r h , (28) implies that θ r l ,d = θ r h ,d .In the case where r l < r h , (28) implies that ∠ L d ( e θ r h ,d ) < ∠ L d ( e θ r l ,d ) , and, since θ → ∠ L d ( e θ ) is decreasing on (0 , π ] ,θ r l ,d < θ r h ,d . Hence, in the case where r = r d , it followsthat θ r ,d = θ r d ,d , and, in the case where r < r d , it followsthat θ r ,d < θ r +1 ,d < · · · < θ r d ,d . Next, let d ≥ ¯ d, and let r ∈ { r , . . . , r d } and θ r,d ∈ (0 , π ] satisfy (27), so that ∠ L d ( e θ r,d ) is an integer multipleof π. Therefore, L d ( e θ r,d ) is a positive number, and thus L d,α r ( e θ r,d ) = 1 , where α r (cid:52) = 1 /L d ( e θ r,d ) > . Therefore, p d,α r ( e θ r,d ) = 0 , and thus α r ∈ A d, + , which implies that θ r,d ∈ Θ d, + , where Θ d, + (cid:52) = { θ ∈ (0 , π ] : there exists α ∈ A d, + such that L d,α ( e θ ) = 1 } . Now suppose that, for all d ≥ ¯ d, there exists θ ∈ Θ d, + \{ θ r ,d , θ r +1 ,d , . . . , θ r d ,d } . Hence there exists r ∈ Z \{ r , . . . , r d } such that ∠ L d ( e θ ) + 2 rπ = 0 . In the casewhere r < r , it follows from (18) that ∠ L d (1) + 2 rπ ≤ ∠ L d (1) + 2( r − π = − π. (29) Since θ → ∠ L d ( e θ ) is decreasing on (0 , π ] , (29) implies that,for all θ ∈ (0 , π ] , ∠ L d ( e θ ) + 2 rπ < − π < L d ( e θ ) + 2 rπ, (30)which is a contradiction. Hence, r > r d . Similarly, supposingthat r > r d also leads to a contradiction. Therefore, for all d ≥ ¯ d, Θ d, + = { θ r ,d , θ r +1 ,d , . . . , θ r d ,d } . Next, for all d ≥ ¯ d and r ∈ { r , . . . , r d } , adding πr toboth sides of (15) with θ = θ r,d , it follows from (27) that θ r,d = 4 rπ + 2 ∠ G ( e θ r,d ) + π d + 1 . (31)Then, (31) implies that, for all d ≥ ¯ d and r ∈ { r +1 , . . . , r d } ,θ r,d − θ r − ,d = 22 d + 1 [2 π + ∠ G ( e θ r,d ) − ∠ G ( e θ r − ,d )] . (32)In the case where r = r , it follows from (31) with r = r that, for all d ≥ ¯ d,θ r ,d = 4 r π + 2 ∠ G ( e θ r ,d ) + π d + 1 . (33)It thus follows from (33) that, for all θ ∈ (0 , π ] , there exists d θ, l ≥ ¯ d such that, for all d ≥ d θ, l ,θ r ,d < θ. (34)In the case where r = r d , it follows from (22) and (31) with r = r d that, for all d ≥ ¯ d,θ r d ,d = 4 r d π + 2 ∠ G ( e θ rd,d ) + π d + 1 ≥ r m + d − π + 2 ∠ G ( e θ rd,d ) + π d + 1= π − π − ∠ G ( e θ rd,d ) + 2 πr m )2 d + 1 . (35)Hence, for all θ ∈ (0 , π ) , − π − ∠ G ( e θ rd,d ) + 2 πr m )2 d + 1 + π − θ ≤ θ r d ,d − θ, which implies that there exists d θ, r ≥ ¯ d such that, for all d ≥ d θ, r , θ r d ,d > θ. (36)Furthermore, (35) implies that π ≥ lim d →∞ θ r d ,d ≥ π − lim d →∞ π − ∠ G ( e θrd,d )+2 πr m )2 d +1 = π. Hence, lim d →∞ θ r d ,d = π. (37)Next, let θ ∞ ∈ M. We first consider the case where θ ∞ ∈ (0 , π ) . It follows from (34) and (36) with θ = θ ∞ that, forall d ≥ max { d θ ∞ , l , d θ ∞ , r } , there exists r ∈ { r + 1 , . . . , r d } such that θ r − ,d ≤ θ ∞ ≤ θ r,d . (38)t follows from (32) and (38) that, for all ε > , there exists d θ ∞ , m ≥ max { d θ ∞ , l , d θ ∞ , r } such that, for all d ≥ d θ ∞ , m , there exists r ∈ { r + 1 , . . . , r d } such that ≤ θ ∞ − θ r − ,d ≤ θ r,d − θ r − ,d < ε (39)and ≤ θ r,d − θ ∞ ≤ θ r,d − θ r − ,d < ε. (40)Now, for all ˜ d ≥ , define ψ θ ∞ , ˜ d (cid:52) = argmin θ ∈ Θ ˜ d, + | θ ∞ − θ | ∈ (0 , π ] . It follows from (38) that, for all ˜ d ≥ max { d θ ∞ , l , d θ ∞ , r } , thereexists r ∈ { r + 1 , . . . , r ˜ d } such that ψ θ ∞ , ˜ d ∈ { θ r − , ˜ d , θ r, ˜ d } , and thus, for all ε > , (39) and (40) imply | ψ θ ∞ , ˜ d − θ ∞ | ∈ { θ ∞ − θ r − , ˜ d , θ r, ˜ d − θ ∞ } < ε. Hence, lim ˜ d →∞ ψ θ ∞ , ˜ d = θ ∞ ∈ (0 , π ) . (41)In the case where θ ∞ = π, (37) implies lim ˜ d →∞ ψ π, ˜ d = π. (42)Hence, (41) and (42) imply lim ˜ d →∞ ψ θ ∞ , ˜ d = θ ∞ ∈ (0 , π ] . (43)Since (17) implies that, for all d ≥ ¯ d, α d, u = min θ ∈ Θ d, + g ( θ ) , and, for all θ ∞ ∈ M, α ∞ = g ( θ ∞ ) , it follows from (43) that lim d →∞ α d, u = α ∞ . (44)Similarly, in the case where α < , lim d →∞ − α d, l = α ∞ . (45)Finally, (44) and (45) imply lim d →∞ − α d, l = lim d →∞ α d, u = min θ ∈ (0 ,π ] (cid:12)(cid:12)(cid:12)(cid:12) D ( e θ )( e θ − N ( e θ ) (cid:12)(cid:12)(cid:12)(cid:12) . (cid:3) Proposition 2.3:
Let α ∈ R , d ≥ , and θ ∈ (0 , π ] , andassume that p d,α ( e θ ) = 0 . Then, α = e ( d +1) θ ( e θ − G ( e θ ) . (46)Furthermore, writing G − ( e θ ) = a + b, where a, b ∈ R , itfollows that b = − a sin dθ − sin ( d + 1) θ cos dθ − cos ( d + 1) θ (47)and α = a cos dθ − cos ( d + 1) θ . (48) Proof. (46) follows from p d,α ( e θ ) = 0 . Furthermore, (46)implies that α = [(cos dθ − cos ( d + 1) θ ) + (sin dθ − sin ( d + 1) θ )] G − ( e θ )2 − θ ) , (49) and thus α = f + g − θ , (50)where f (cid:52) = a [cos dθ − cos ( d + 1) θ ] − b [sin dθ − sin ( d + 1) θ ] , (51) g (cid:52) = b [cos dθ − cos ( d + 1) θ ] + a [sin dθ − sin ( d + 1) θ ] . (52) Since α is real, (50) implies that g = 0 , and thus (52) implies(47). Next, combining (47) with (51) yields f = a [cos dθ − cos ( d + 1) θ ] + [sin dθ − sin ( d + 1) θ ] cos dθ − cos ( d + 1) θ = a − dθ cos ( d + 1) θ − dθ sin ( d + 1) θ cos dθ − cos ( d + 1) θ = a − θ cos dθ − cos ( d + 1) θ . (53) Finally, combining g = 0 and (53) with (50) yields (48). (cid:3) Example 2.4:
Let G ( z ) = z + p , where p ∈ ( − , , and let e θ (cid:54) = 1 , where θ ∈ (0 , π ] , be a root of p d,α on the unit circle.Writing G − ( e θ ) = a + b, it follows that a = cos θ + p and b = sin θ, and (46) and (48) have the form α ( θ ) = e ( d +2) θ + pe ( d +1) θ e θ − θ + p cos dθ − cos ( d + 1) θ , (54)which implies | α ( θ ) | = (cid:114) p + 2 p cos θ + 12 − θ . (55)Furthermore, it follows from (47) that sin ( d + 2) θ = (1 − p ) sin ( d + 1) θ + p sin dθ. (56)Since L d has d + 2 poles in the open unit disk and one zeroat 1, it follows that there exist exactly d + 1 distinct values θ , . . . , θ d +1 of θ ∈ [0 , π ] that satisfy (56). The correspondingvalues of α ( θ i ) are given by α ( θ i ) = cos θ i + p cos dθ i − cos ( d + 1) θ i = − cos ( d + 2) θ i + (1 − p ) cos ( d + 1) θ i + p cos dθ i − θ i . (57)Next, v ) in Theorem 2.2 and (55) imply that α ∞ = min θ ∈ (0 ,π ] (cid:12)(cid:12)(cid:12)(cid:12) e θ + pe θ − (cid:12)(cid:12)(cid:12)(cid:12) = min θ ∈ (0 ,π ] (cid:114) p + 2 p cos θ + 12 − θ . (58)Hence, it follows from (55) and (58) that α ∞ = min θ ∈ (0 ,π ] | α ( θ ) | . (59)etting θ ∗ ∈ (0 , π ] be a minimizer of (55), it follows that d | α | dθ (cid:12)(cid:12)(cid:12)(cid:12) θ = θ ∗ = − | α ( θ ∗ ) | sin θ ∗ (2 p + 4 p + 2)(2 − θ ∗ ) = 0 , (60)which implies that θ ∗ = π. Hence, (58) implies α ∞ = 1 − p ∈ (0 , . (61)For p = , d = 6 , and d = 7 , Figure 6 shows α ( θ i ) and | α ( θ i ) | versus θ i . Note that, for both values of d, the minimumvalue of | α ( θ i ) | is α ∞ = , as stated by (61), which occurs at θ = π . Finally, Figure 7 shows α d, l and α d, u versus d for p =0 . , which indicates that lim d →∞ − α d, l = lim d →∞ α d, u = α ∞ , as stated in (13). Fig. 6: Example 2.4: For p = , d = 6 , and d = 7 , (a) and (b) show α ( θ i ) versus θ i , and (c) and (d) show | α ( θ i ) | versus θ i . Note that the sign of α ( θ i ) alternates. The dashed lines indicate ± α ∞ = ± . Fig. 7: Example 2.4: For p = , α d, l and α d, u versus d . As d → ∞ , α d, l and α d, u converge to − α ∞ and α ∞ , respectively, where α ∞ = . Special case:
For p = 0 , (55) becomes | α ( θ ) | = 1 √ − θ , (62)and (56) becomes sin ( d + 1) θ = sin ( d + 2) θ. (63)Note that, for all i ∈ Z , sin( d +1) θ = sin[(2 i +1) π − ( d +1) θ ] . Therefore, (63) holds if and only if θ = k +12 d +3 π . Hence, θ ∈ [0 , π ] satisfies (63) if and only if there exists i ∈ { , . . . , d +1 } such that θ k (cid:52) = (cid:16) i +12 d +3 (cid:17) π. For these d + 2 values of θ, (57)implies that the corresponding values of α ( θ ) are given by α ( θ i ) = cos θ i cos dθ i − cos( d + 1) θ i = cos( d + 1) θ i − cos( d + 2) θ i − θ i . (64)Next, it can be shown that, for all i ∈ { , . . . , d } ,α ( θ i ) α ( θ i +1 ) < . Note that θ d +1 = π and α ( θ d +1 ) =( − d +1 12 . Hence, | α ( θ d +1 ) | = . Furthermore, in the casewhere d is even, α d, l = α ( θ d +1 ) = − < and α d, u = α ( θ d ) > > , whereas, in the case where d is odd, α d, l = α ( θ d ) < − < and α d, u = α ( θ d +1 ) = > . In addition,although lim d →∞ α ( θ d ) does not exist, it follows from (62)that lim d →∞ | α ( θ d ) | = lim d →∞ (cid:113) − ( d +12 d +3 ) π = , whichconfirms (13) and (61). For d = 10 and d = 11 , Figure 8shows α ( θ i ) and | α ( θ i ) | versus θ i . Note that, for both valuesof d, the minimum value of | α ( θ i ) | is , which occurs at θ = π . Finally, Figure 9 shows α d, l and α d, u versus d, whichindicates that lim d →∞ α d, l = − and lim d →∞ α d, u = . (cid:5) Fig. 8: Example 2.4: For p = 0 , d = 10 , and d = 11 , (a) and (b) show α ( θ i ) versus θ i , and (c) and (d) show | α ( θ i ) | versus θ i . The dashed linesindicate ± α ∞ = ± . Fig. 9: Example 2.4: For p = 0 , α d, l and α d, u versus d . As d → ∞ , α d, l and α d, u converge to − α ∞ and α ∞ , respectively, where α ∞ = . Example 2.5:
Let G ( z ) = N ( z ) D ( z ) = z + 0 . ± . z ( z − . ± . . igure 10 shows that, for all d ≥ , there exists α d, l < such that p d,α if and only if α ∈ ( α d, l , as stated in iv ) fromTheorem 2.2. Furthermore, define α uc ( θ ) (cid:52) = (cid:12)(cid:12)(cid:12)(cid:12) D ( e θ )( e θ − N ( e θ ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) ( e θ − . ± . e θ − e θ + 0 . ± . (cid:12)(cid:12)(cid:12)(cid:12) , such that α ∞ = min θ ∈ (0 ,π ] α uc ( θ ) . Figure 11 shows that α uc has a minimum at θ ≈ . π, which implies that α ∞ ≈ α uc (0 . π ) = 0 . . Finally, Figure 12 shows α d, l and α d, u versus d ≥ , which shows that lim d →∞ α d, l = − α ∞ and lim d →∞ α d, u = α ∞ , as stated in v ) from Theorem 2.2. (cid:5) Fig. 10: Example 2.5: spr( p d,α ) versus − α, for α ≤ . (a) shows that, for d = 0 , α , l as defined in iv ) from Theorem 2.2 doesn’t exist. (b) shows that,for d ≥ , α , l as defined in iv ) from Theorem 2.2 exists.Fig. 11: Example 2.5: α uc ( θ ) versus θ for θ ∈ (0 , π ] . Note that lim θ ↓ α uc ( θ ) = ∞ and that min θ ∈ (0 ,π ] α uc ( θ ) = α uc ( θ ∞ ) ≈ . , where θ ∞ ≈ . π. Fig. 12: Example 2.5: α d, l and α d, u versus d ≥ . As d → ∞ , α d, l and α d, u converge to − α ∞ and α ∞ , respectively, where α ∞ ≈ . . III. T
IME -D ELAYED L UR ’ E M ODEL
Inserting the saturation nonlinearity following the washoutfilter W in Figure 5 yields the TDL model shown in Figure13, which has the closed-loop dynamics x k +1 x d ,k +1 x f ,k +1 = A e d,d C N d e T1 ,d x k x d ,k x f ,k + αB v f ,k , (65) with y k , y d ,k , and y f ,k given by (2), (3), and (4), respectively,where v f ,k = sat δ ( y f ,k ) is the output of the saturation function sat δ : R → R , where δ > , defined by sat δ ( u ) = (cid:40) u, | u | ≤ δ, sign( u ) , | u | > δ. (66) αG ( z ) W ( z ) G d ( z )sat δ v f yy d y f Fig. 13: Discrete-time time-delayed Lur’e model.
To analyze the self-oscillating behavior of the time-delayedLur’e model, we replace the saturation nonlinearity by its de-scribing function. Describing functions are used to characterizeself-excited oscillations in [2, Section 5.4] and [15, pp. 293–294]. The describing function Ψ δ ( ε ) for sat δ for a sinusoidalinput with amplitude ε > is given by Ψ δ ( ε ) = π (cid:20) sin − (cid:0) δε (cid:1) + (cid:0) δε (cid:1) (cid:113) − (cid:0) δε (cid:1) (cid:21) , if ε > δ , otherwise . (67)Note that, for ε > δ , the function Ψ δ confined to ( δ, ∞ ) withcodomain (0 , is decreasing, one-to-one, and onto. Let p d,α,ε be the characteristic polynomial of the linearized time-delayLur’e model, such that p d,α,ε ( z ) (cid:52) = z d +1 D ( z ) − α Ψ δ ( ε )( z − N ( z ) . (68)For all ε l > , ε u > , θ l ∈ R , and θ u ∈ R such that ε l < ε u and θ l < θ u , define the rectangle Γ θ l ,θ u ,ε l ,ε u (cid:52) = { ( θ, ε ) : θ l < θ < θ u and ε l < ε < ε u } Lemma 3.1:
Let α ∈ R , and let θ ∈ Θ be such that sign α = sign α and | α | < | α | , where α (cid:52) = α ( θ ) , andlet d > ¯ d . Then, the following statements hold: i ) There exist ε > , θ l > , θ u > , ε l > , and ε u > such that ε l < ε u , θ l < θ u , ( θ , ε ) ∈ Γ θ l ,θ u ,ε l ,ε u , and, inthe rectangle Γ θ l ,θ u ,ε l ,ε u , ( θ, ε ) = ( θ , ε ) is the uniquesolution of p d,α,ε ( e θ ) = 0 . ii ) dd ε Ψ δ ( ε ) (cid:12)(cid:12)(cid:12)(cid:12) ε = ε (cid:54) = 0 . (69) iii ) dd θ Im[ L d ( e θ )] (cid:12)(cid:12)(cid:12)(cid:12) θ = θ (cid:54) = 0 . (70) Proof.
To prove i ) , note that, for sign α = sign α and | α | > | α | , there exists ε > δ such that α = Ψ δ ( ε ) α .Therefore, p d,α ( e θ ) = p d,α,ε ( e θ ) = 0 . Furthermore, thereexists a rectangle Γ θ l ,θ u ,ε l ,ε u , where θ l > , θ u > , ε l > ,ε u > , ε l < ε u and θ l < θ u , such that ( θ , ε ) ∈ Γ θ l ,θ u ,ε l ,ε u nd Θ ∩ ( θ l , θ u ) = θ . Hence, in the rectangle Γ θ l ,θ u ,ε l ,ε u , ( θ, ε ) = ( θ , ε ) is the unique solution of p d,α,ε ( e θ ) = 0 .To prove ii ) , note that dd ε Ψ δ ( ε ) (cid:12)(cid:12)(cid:12)(cid:12) ε = ε = − δ (cid:112) ε − δ πε < . (71)To prove iii ) , writing G ( e θ ) = a n − b n , where a n = aa + b and b n = ba + b . Then, L d ( e θ ) = G ( e θ ) G d ( e θ ) W ( e θ )= ( a n − b n )( e − dθ − e − ( d +1) θ )= ( a n (cos dθ − cos ( d + 1) θ ) − b n (sin dθ − sin ( d + 1) θ ))+ ( − a n (sin dθ − sin ( d + 1) θ ) − b n (cos dθ − cos ( d + 1) θ ))= fa + b − ga + b . (72) Since θ ∈ Θ and α ( θ ) (cid:54) = 0 , it follows from (47) and (72)with θ = θ that g = 0 and thus Re[ L d ( e θ )] (cid:54) = 0 , Im[ L d ( e θ )] = 0 . (73)Furthermore, differentiating ∠ L d ( e θ ) with respect to θ yields dd θ ∠ L d ( e θ ) = dd θ atan (cid:18) Im[ L d ( e θ )]Re[ L d ( e θ )] (cid:19) = Re[ L d ( e θ )] dd θ Im[ L d ( e θ )] − Im[ L d ( e θ )] dd θ Re[ L d ( e θ )] | L d ( e θ ) | . (74)It follows from (73) and (74) that dd θ ∠ L d ( e θ ) (cid:12)(cid:12)(cid:12)(cid:12) θ = θ = Re[ L d ( e θ )] dd θ Im[ L d ( e θ )] (cid:12)(cid:12) θ = θ | L d ( e θ ) | . (75)It follows from (11) that, for all d > ¯ d, dd θ ∠ L d ( e θ ) (cid:12)(cid:12) θ = θ < . Hence, it follows from (75) that dd θ Im[ L d ( e θ )] (cid:12)(cid:12) θ = θ (cid:54) = 0 . (cid:3) Theorem 3.2:
Consider the discrete-time time-delayedLur’e model in Figure 13, assume that x (cid:54) = 0 , and let α ∈ ( −∞ , α d, l ) ∪ ( α d, u , ∞ ) . Then, there exists a nonconstantperiodic function τ : N → R such that lim k →∞ | y k − τ k | = 0 . Proof.
Lemma 3.1 implies that the assumptions of Theorem7.4 in [15, pp. 293, 294] are satisfied. It thus follows that theresponse is asymptotically periodic. (cid:3)
It can be seen that Theorem 3.2 holds in the case where thesaturation function is replaced by an odd sigmoidal nonlinear-ity such as atan or tanh.
Example 3.3:
Let G ( z )=1 /z. Figure 14 shows the transientresponse and asymptotic oscillatory response for α = 1 . , d = 0 , and δ = 1 along with plot of v f ,k and y f ,k .Figure 14(a) shows that, for k > , y k is a nonconstantperiodic function. Furthermore, Figure 14(b) shows how thesaturation nonlinearity acts upon y f ,k , which results in thesaturated signal v f ,k ∈ [ − δ, δ ] . Note that v f ,k and y f ,k arealso nonconstant periodic functions for k > . Figure 15 shows α ( θ i ) versus θ i for d = 0 and d = 1 .For α = 0 . , only in the case d = 1 has α ( θ i ) such that sign( α ( θ i )) = sign( α ) and | α ( θ i ) | < | α | . For α = 1 . , bothmodels meet the conditions for α .Figure 16 shows the response of y k for δ = 1 and allpossible pairs of d = 0 , and α = 0 . , . . For α = 0 . , onlythe model with d = 1 yields a limit cycle. For α = 1 . , bothmodels yield oscillations. This follows from the conditions for α stated in the previous paragraph and in Lemma 3.1.Finally, Figure 17 shows the magnitude of the frequencyresponse for models with α = 1 . , δ = 1 , and d = 0 , . Notethat the frequencies corresponding to the magnitude peaksare similar to the values of θ i shown in Figure 15 such that sign( α ( θ i )) = sign( α ) and | α ( θ i ) | < | α | . (cid:5) Fig. 14: Example 3.3: For d = 0 , δ = 1 , and α = 1 . , (a) shows y k , and(b) shows v f ,k and y f ,k . The saturation nonlinearity, with δ = 1 , saturatesthe values of y f ,k , resulting in v f ,k ∈ [ − δ, δ ] . Fig. 15: Example 3.3: For d = 0 and d = 1 , these plots show the values α ( θ i ) of α for which the closed-loop dynamics have a pole on the unit circleat the angle θ i for i = 1 , . . . , d + 2 . For the case d = 0 , where θ = 1 . and θ = π , the time-delayed Lur’e model has self-excited oscillations ifand only if either α > or α < − , while, for the case d = 1 , where θ = 0 . , θ = 1 . and θ = π , the TDL model has self-excitedoscillations if and only if either α > or α < − . . For all valuesof α corresponding to the shaded regions, the response of the TDL modeloscillates. IV. T
IME -D ELAYED L UR ’ E M ODEL WITH B IAS G ENERATION
We now modify the discrete-time time-delay Lur’e modelby including the bias-generation mechanism shown in Figure ig. 16: Example 3.3: Response y k of the TDL model for d = 0 , and α = 0 . , . with δ = 1 .Fig. 17: Example 3.3: Frequency response of y k for d = 0 , with α = 1 . and δ = 1 . Note that, for d = 0 , the peak is located at θ , whereas, for d = 1 , the peak is located at θ .
1. The corresponding closed-loop dynamics are thus given by x k +1 x d ,k +1 x f ,k +1 = A e d,d C N d e T1 ,d x k x d ,k x f ,k + B v b ,k , (76)with y k , y d ,k , and y f ,k given by (2), (3), and (4), respectively,where β is a constant, v b ,k = ( β + v f ,k ) v k , (77)and v f ,k = sat δ ( y f ,k ) . Note that the constant α is now omitted.Instead, the constant input v is injected multiplicatively insidethe loop, thus playing the role of α . This feature allows theoffset of the oscillation to depend on the external input. Theresulting bias ¯ y of the periodic response is thus given by ¯ y = vβG (1) . (78) Example 4.1:
Let G ( z ) = 1 /z, d = 0 , β = 2 . , v = 1 . , and δ = 1 . Figure 19(a) shows that the output y k is oscillatorywith offset ¯ y = vβG (1) = 2 . . Figure 19(b) shows v f ,k and y f ,k . Note that, as in Example 3.3, despite the offset ¯ y of y k , the signals y f ,k and v f ,k oscillate without an offset. β + G ( z ) v b W ( z ) G d ( z )sat δ × v v f yy d y f Fig. 18: Discrete-time time-delayed Lur’e model with constant input v andbias generation. Finally, Figure 19(c) shows the magnitude of the frequencyresponse for y k − ¯ y . Note that the peak is located near thesame frequency as in Example 3.3, and thus the oscillationfrequency remains the same with the addition of the bias-generation mechanism. (cid:5) Fig. 19: Example 4.1: For v = 1 . , β = 2 . , d = 0 , and δ = 1 , (a) shows y k and the offset ¯ y , (b) shows v f ,k and y f ,k , and (c) shows the frequencyresponse of y k − ¯ y. Example 4.2:
Let G ( z ) = z − . e ± π/ ( z − . e ± π/ )( z − . e ± π/ ) ,d = 4 , β = 15 v = 1 , and δ = 1 . Figure 20(a) shows that theoutput y k is oscillatory with offset ¯ y = vβG (1) = 5 . . Figure 20(b) shows that v f ,k and y f ,k have an oscillatoryresponse without an offset, as in previous cases. Finally, Figure20(c) shows the magnitude of the frequency response for y k − ¯ y . (cid:5) ig. 20: Example 4.2: For v = 1 , β = 15 , d = 4 , and δ = 1 , (a) shows y k and the offset ¯ y , (b) shows v f ,k and y f ,k , and (c) shows the frequencyresponse of y k − ¯ y. V. C
ONCLUSIONS AND F UTURE E XTENSIONS
This paper presented and analyzed a discrete-time Lur’emodel that exhibits self-excited oscillations. This model in-volves an asymptotically stable linear system, a time de-lay, a washout filter, and a saturation nonlinearity. It wasshown that, for sufficiently large loop gains, the responseconverges to a periodic signal, and thus the system has self-excited oscillations. A bias-generation mechanism provides aninput-dependent oscillation offset. The amplitude and spectralcontent of the oscillation were analyzed in terms of thecomponents of the model.An immediate extension of this work is to consider the casewhere G has zeros on the unit circle. The main results of thispaper appear to be valid for this case, although the proofs aremore intricate. Extension to sigmoidal nonlinearities such asatan and tanh as well as relay nonlinearities is of interest. Inaddition, continuous-time, time-delay Lur’e models describedby iv ) in Section I are of interest. Finally, future work will usethis discrete-time self-excited model for system identificationand adaptive stabilization.VI. A CKNOWLEDGMENTS
This research was supported by NSF grant CMMI 1634709,“A Diagnostic Modeling Methodology for Dual RetrospectiveCost Adaptive Control of Complex Systems.” R
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Proc. Amer. Contr. Conf. ,Denver, July 2020. A UTHOR B IOGRAPHIES
Juan Paredes received the B.Sc. degree in mechatronics engi-neering from the Pontifical Catholic University of Peru and aM.Sc. degree in aerospace engineering from the University ofMichigan in Ann Arbor, MI. He is currently a PhD candidatein the Aerospace Engineering Department at the University ofMichigan. His interests are in autonomous flight control andcontrol of combustion.
Syed Aseem Ul Islam received the B.Sc. degree in aerospaceengineering from the Institute of Space Technology, Islamabadand is currently pursuing the Ph.D. degree in flight dynamicsand control from the University of Michigan in Ann Arbor.His interests are in data-driven adaptive control for aerospaceapplications.
Omran Kouba received the Sc.B. degree in Pure Mathematicsfrom the University of Paris XI and the Ph.D. degree inFunctional Analysis from Pierre and Marie Curie University inParis, France. Currently he is a professor in the Department ofMathematics in the Higher Institute of Applied Sciences andTechnology, Damascus (Syria). His interests are in real andcomplex analysis, inequalities, and problem solving.
Dennis S. Bernstein received the Sc.B. degree from BrownUniversity and the Ph.D. degree from the University of Michi-gan in Ann Arbor, Michigan, where he is currently profes-sor in the Aerospace Engineering Department. His interestsare in identification, estimation, and control for aerospaceapplications. He is the author of