On the exactness of a stability test for Lure systems with slope-restricted nonlinearities Andrey Kharitenko Carsten W. Scherer

2025-05-02 0 0 600.96KB 9 页 10玖币

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On the exactness of a stability test for Lur’e systems with

slope-restricted nonlinearities

Andrey Kharitenko, Carsten W. Scherer

October 28, 2022

Abstract

In this note it is shown that the famous multiplier abso-

lute stability test of R. O’Shea, G. Zames and P. Falb is

necessary and suﬃcient if the set of Lur’e interconnec-

tions is lifted to a Kronecker structure and an explicit

method to construct the destabilizing static nonlinear-

ity is presented.

1 Introduction

A classical problem in control theory is the stability

analysis of the so-called Lur’e systems. Studied by A.

Lur’e and V. Postnikov in 1944 [1] explicitly for the

ﬁrst time, they consist of a feedback interconnection

between a linear time-invariant system Gand a non-

linear static operator ∆.

The method of stability multipliers [2, 3] is used to

establish stability of such systems by searching for an

artiﬁcial system M∈ M∆(called multiplier) such that

−MG is strictly passive. Here M∆is a suitable set of

multipliers such that ∆M−1is a positive operator for

any M∈ M∆. Under suitable assumptions, if such a

multiplier is found, stability of the interconnection can

be deduced from the passivity- or IQC-theorem [2], [4].

Diﬀerent classes of nonlinearities ∆allow for diﬀerent

multiplier classes M∆, and a larger multiplier class

implies a less conservative stability test.

Yet the question of whether a stability criterion cre-

ated in this way is necessary is dependent on M∆and

remains open in general. Many well-known multiplier

stability criteria, such as the Popov or circle criterion,

were shown already early on to be only suﬃcient [5,

6, 7]. For the class of monotone or slope-restricted

nonlinearities, a particularly rich class of multipliers is

given by the so-called O’Shea-Zames-Falb (OZF) mul-

tipliers [8, 9, 10], which were introduced in [8] and

[9] for continuous- and discrete-time SISO intercon-

nections and extended in [11] to the MIMO case for

monotone nonlinearities. In fact, for this class of non-

linearities in discrete-time, OZF-multipliers form the

largest possible passivity multiplier set [12] and give

the least conservative results. Along with the discov-

ery of suitable search methods [13, 14, 10, 15, 16], this

has motivated its extensive use in recent times, e.g.

[17, 18, 19].

The question about the conservatism of this multi-

plier set is, therefore, of great interest [20, 21, 22] and

its exactness was conjectured in [10]. Moreover, this

question is closely related to the Kalman problem from

the 1950s, which asks for necessary and suﬃcient con-

ditions for the stability of Lur’e interconnections with

slope-restricted nonlinearities.

In this note we show that the stability test generated

by OZF-multipliers cannot diﬀerentiate between Lur’e

interconnections in diﬀerent dimensions, if the linear

part is lifted from Gto the Kronecker form G⊗Id,

and could, hence, be potentially conservative. As our

main result, we prove that, up to this lifting, the test is

indeed necessary and suﬃcient and provide an explicit

construction of the destabilizing nonlinearity. We also

point out connections to the duality bounds obtained

in [20] and to criteria for the absence of periodic oscil-

lations in nonlinear ﬁlters [23].

The paper is structured as follows. After introducing

the necessary concepts of absolute stability and inte-

gral quadratic constraints as well as stating the main

stability test in Section 2, we proceed in Section 3.1 to

show that the test cannot diﬀerentiate between inter-

connections up to a lifting. In Section 3.2 we then state

our main exactness result. To prove it, we reformulate

the infeasibility of the main stability test in Section

3.3 as a condition on a linear program and explicitly

construct a nonlinearity using duality in Section 3.4.

Finally, in Section 3.5 we prove our main result and

discuss its connections to other results in the literature

in Section 5. All technical proofs, except the one of the

main exactness result, can be found in Appendix A.1,

while Appendix A.2 contains two auxiliary facts.

2 Notation and Preliminary Results

2.1 Notation

The standard inner product in Rdis denoted as hu, vi=

u>v, while kMk= ¯σ(M) is the spectral norm of a

real matrix. The identity matrix in Rd×dis denoted

by Id. Moreover, `2e

ddenotes the linear space of all se-

quences x:N0→Rd, while `2

dis the subspace of square

summable sequences equipped with the inner product

hx, yi=P∞

k=0 hxk, yki. The power of a signal x∈`2e

is deﬁned by pow(x)2= lim supN→∞ 1

NPN

k=0kxkk2∈

[0,∞]. The space of linear bounded operators on `2

and the (induced) norm thereon are denoted by L(`2

and k.k, respectively. The Kronecker product of two

matrices Aand Bis denoted by A⊗Band the direct

sum of two vector spaces Xand Yby X⊕Y. If N∈N

and x∈`2e

d, then PNxdenotes the cutoﬀ projection

deﬁned as (PNx)k=xkif k6Nand (PNx)k= 0 for

k > N. A relation R⊆`2e

m×`2e

lis said to be bounded

if there exist γand βsuch that for all (x, y)∈Rand

N∈N0we have kPNyk6γkPNxk+β. The inﬁmum

arXiv:2210.14992v1 [math.OC] 26 Oct 2022

of all such values γ, called the gain of R, is denoted

by kRk. The relation Ris said to be total if for each

x∈`2e

mthere exists some y∈`2e

lwith (x, y)∈R

and causal if for all N∈N0and all (x, y),(u, v)∈R

such that PNx=PNuit follows that there exists some

w∈`2e

lwith (u, w)∈Rand PNy=PNw. Operators

G:`2e

m→`2e

lare identiﬁed with their graph relation

R={(x, G(x)) |x∈`2e

m}. The transfer function of

a stable ﬁnite-dimensional linear time-invariant (LTI)

system Gis denoted by G(z). Finally we deﬁne the

unit circle by T={z∈C| |z|= 1}.

2.2 Well-posedness and stability

Let G:`2e

m→`2e

lbe linear and bounded and ∆ ⊆

`2e

l×`2e

ma relation. Consider the standard feedback

interconnection (Fig. 1) between Gand ∆ deﬁned by

e2=Ge1+u2and (e2, e1−u1)∈∆,(1)

where the input u= (u1, u2) belongs to `2e

m×`2e

`2e

m+l. The interconnection (1) is said to be well-posed

if the interconnection relation

R(∆) = {(u, e)∈`2e

l+m×`2e

l+m|(1) is satisﬁed}(2)

with e= (e1, e2) is total and causal. It is stable if,

additionally, R(∆) is bounded. If ∆is a set of rela-

tions, we say that (1) is robustly stable against ∆if

it is stable for each ∆ ∈∆and sup∆∈∆kR(∆)k<∞

holds. Note that this is a uniform notion of stability

for the class ∆.

∆

Figure 1: Standard feedback interconnection

2.3 Slope-restricted relations

In the following we consider any relation f⊆Rd×Rd

as a multifunction with domain dom(f) = {x∈Rd|

∃y∈Rd: (x, y)∈f}and the value f(x) = {y∈Rd|

(x, y)∈f}at x∈Rd; we write f:Rd⇒Rd.

A multifunction f:Rd⇒Rdis said to be cycli-

cally monotone if for any cyclic sequence x1, . . . , xn∈

dom(f), xn+1 =x1, and yj∈f(xj), j= 1, . . . , n it

holds that Pn

j=1 hyj, xj+1 −xji60.

Following [17] and with λ∈(−∞,0], κ∈(0,∞],

we deﬁne for every f:Rd⇒Rdthe multifunction

Tλ,κ,f = (f−λI)◦(I−1

κf)−1so that y∈Tλ,κ,f (x)

iﬀ there exists some z∈dom(f) and u∈f(z) with

x=z−1

κuand y=u−λz.

Here, we use the convention 1/∞= 0 and note

that T0,∞,f =fby deﬁnition. The multifunction

f:Rd⇒Rdis said to be [λ, κ]-slope-restricted (writ-

ten f∈sloped[λ, κ]) if Tλ,κ,f is cyclically monotone.

Every relation f⊆Rd×Rddeﬁnes a static relation

∆ = ∆f⊆`2e

d×`2e

don signals by (w, z)∈∆ iﬀ zk∈

f(wk) for all k∈N0. In the sequel we work with the

class

∆d(λ, κ) := {∆f|f∈sloped[λ, κ] total and 0 ∈f(0)}.

2.4 Stability multipliers

Any shift-invariant operator M∈ L(`2

d) has an inﬁnite

block-Toeplitz representation M= (Mi−j)∞

i,j=0 with

Mj∈Rd×d. Conversely, any block-Toeplitz matrix

(Mi−j)∞

i,j=0 that satisﬁes P∞

j=−∞kMjk<∞deﬁnes

an operator M∈ L(`2

d) by M x = (P∞

j=0 Mij xj)∞

i=0.

A scalar matrix M= (mi−j)∞

i,j=0 ∈RN0×N0is said to

be doubly hyperdominant, if mj60 for j6= 0 and

P∞

j=−∞ mj>0 where the convergence of the series is

assumed by deﬁnition. We denote the set of all dou-

bly hyperdominant Toeplitz matrices by H. The above

terminology is also used for ﬁnite matrices M∈RN×N

whenever Nis clear from the context. A block-Toeplitz

matrix M= (Mi−j)∞

i,j=0 is said to be doubly hyper-

dominant, if it is doubly-hyperdominant as a scalar

matrix.

The IQC theorem [24, 25, 26, 27] permits to verify

the stability of the interconnection of Gand ∆fas

follows.

Theorem 1. Let Gbe linear, bounded and causal on

d. Assume that the feedback interconnection of Gand

∆is well-posed for every ∆∈∆d(λ, κ). Then the

interconnection is robustly stable against ∆d(λ, κ)if

there exists some  > 0and M∈ H ⊗ Idsuch that

Gw

w,ΠM(λ, κ)Gw

w6−kwk2

2for all w∈`2

(3)

where

ΠM(λ, κ) := −λ(M+M∗)M∗+λ

κM

M+λ

κM∗−1

κ(M+M∗).

This is the Zames-Falb stability test that is the subject

of study in this note.

2.5 Positive deﬁnite functions and se-

quences

A function f:T→Cis said to be positive deﬁnite

(p.d.) if f(z−1) = f(z)∗for all z∈Tand for any ﬁnite

subset {z1, . . . , zn} ⊆ Tthe matrix (f(zkz−1

j))n

k,j=1 is

hermitian and positive semi-deﬁnite. Any p.d. func-

tion fsatisﬁes f(1) >0 as well as f(z)∗=f(z∗) and

|f(z)|6f(1) for any z∈T. A well-known theorem

from harmonic analysis reads as follows.

Theorem 2 (Bochner).For a function f:T→Cthe

following statements are equivalent:

1. fis continuous and positive deﬁnite

2. there exists some ﬁnite, nonnegative measure µon

Zsuch that f(z) = RZztdµ(t)for all z∈T.

A ﬁnite set {cj}N−1

j=0 of complex numbers is said

to be positive deﬁnite if the circulant matrix C=

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OntheexactnessofastabilitytestforLur'esystemswithslope-restrictednonlinearitiesAndreyKharitenko,CarstenW.SchererOctober28,2022AbstractInthisnoteitisshownthatthefamousmultiplierabso-lutestabilitytestofR.O'Shea,G.ZamesandP.FalbisnecessaryandsucientifthesetofLur'einterconnec-tionsisliftedtoaKroneckers...

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On the exactness of a stability test for Lure systems with slope-restricted nonlinearities Andrey Kharitenko Carsten W. Scherer

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