Total stability and integral action for discrete-time nonlinear systems

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S. Zobolia, D. Astolﬁa, V. Andrieua

aUniversit´e Lyon 1, Villeurbanne, France – CNRS, UMR 5007, LAGEPP, France.

Abstract

Robustness guarantees are important properties to be looked for during control design. They ensure stability of closed-

loop systems in face of uncertainties, unmodeled eﬀects and bounded disturbances. While the theory on robust stability is

well established in the continuous-time nonlinear framework, the same cannot be stated for its discrete-time counterpart.

In this paper, we propose the discrete-time parallel of total stability results for continuous-time nonlinear system. This

enables the analysis of robustness properties via simple model diﬀerence in the discrete-time context. First, we study

how existence of equilibria for a nominal model transfers to suﬃciently similar ones. Then, we provide results on the

propagation of stability guarantees to perturbed systems. Finally, we relate such properties to the constant output

regulation problem by motivating the use of discrete-time integral action in discrete-time nonlinear systems.

1. Introduction

Stability of an equilibrium is of utter importance in the

design of feedback controllers. Stability properties are typ-

ically inferred through the analysis of the system model in

closed-loop. However, it is well known that uncertainties

inherently exist in control applications, due to the pres-

ence of unmodeled eﬀects and parameters mismatches. To

tackle the issue, robust control and robust stability anal-

ysis have become fundamental tools for control design [1].

In practice, due to the nature of digital controllers, these

laws are implemented in a discrete-time form. The prob-

lem of synthesizing such robust controllers can be cast in

an optimization framework, both in the linear and the non-

linear scenario, e.g. [2–6]. Nevertheless, a strong theo-

retical foundation turns out to be a fundamental design

tool. While the theory on robust stability and regulation

is well-developed and mature for continuous-time systems,

see e.g., [7, 8], the discrete-time nonlinear scenario still

misses some important results. Although it is known that,

for suﬃciently small sampling times, continuous-time re-

sults are valid in the discrete framework, such an approach

may be restrictive or inapplicable for some control appli-

cations.

In the context of discrete-time nonlinear systems, some

necessary local conditions linked to robustness appeared in

[9]. Here, the authors state that a necessary condition for

local stability of nonlinear discrete-time autonomous sys-

tems comes from the solvability of a nonlinear equation.

Also, results on robust stability appeared in [10, Chapter

5]. Therein, it is shown that if the origin of the nominal

?Research partially funded by the ANR Delicio project.

Email addresses: samuele.zoboli@univ-lyon1.fr (S. Zoboli),

daniele.astolfi@univ-lyon1.fr (D. Astolﬁ),

vincent.andrieu@univ-lyon1.fr (V. Andrieu)

system is locally stable and Lipschitz, then it is also locally

robustly stable for bounded disturbances. More recently,

in the context of converse Lyapunov theorems for discrete-

time systems, [11, 12] proved that a necessary condition

for robust stability is the existence of a smooth Lyapunov

function. Yet, to the best of authors’ knowledge, there

are no results mimicking well-established general results

of “total stability” for continuous-time systems. The con-

cept of total stability was ﬁrstly introduced in the works of

Dubosin [13], Gorsin [14], Malkin [15], and more recently

studied in [8, 16, 17]. In this context, robustness proper-

ties are analyzed directly via nonlinear unstructured mod-

els diﬀerences. This allows inferring the preservation of a

stable equilibrium point for suﬃciently similar plants by

means of simple model comparison [8, Lemma 4, 5].

The goal of this paper is therefore to translate such re-

sults to the discrete-time scenario. We draw conclusions

similar to the continuous case, yet under some fundamen-

tal diﬀerences, given by the discrete nature of the system.

In particular, we show that stability properties of the equi-

librium of a nominal model imply the existence and sta-

bility of an equilibrium (possibly diﬀerent from the for-

mer) for any perturbed system suﬃciently “close” to the

nominal one. The result is proved under some regularity

assumptions and bounded mismatches. Moreover, we pro-

vide a counterexample highlighting that some results from

the continuous-time scenario may not apply in the dis-

cretized framework. This disproves some arguments of [9].

Finally, we link the obtained results on robust stability to

the robust output regulation problem, building on recent

forwarding techniques [18, 19]. We justify the addition of

an integral action for rejecting constant disturbances or

tracking constant references. More speciﬁcally, we show

that if the true model of the plant to be controlled is suﬃ-

ciently close to one used for controller design, then output

Preprint submitted to Automatica October 10, 2022

arXiv:2210.03450v1 [eess.SY] 7 Oct 2022

regulation is still achieved.

The rest of the paper is organized as follows: Section 2

presents the main result of the paper on total stability;

Section 3 applies the result to the problem of constant

output regulation; Section 4 comments and concludes the

paper.

1.1. Notation

R, resp. N, denotes the set of real numbers, resp. non-

negative integers. R≥0denotes the set of nonnegative real

numbers. In this work, we deﬁne a time-invariant nonlin-

ear dynamical discrete-time system as x+=f(x), where

x∈Rnis the state solution evaluated at timestep k∈N

with initial condition x0, and x+is the state solution at

step k+ 1. Sets are denoted by calligraphic letters and,

for a given set X, we identify its boundary by ∂X. The

notation X \ G identiﬁes the intersection between Xand

the complement of G. When a set Xis strictly included

in a set G, we use the notation X$G. We use |·|as

the norm operator for matrices and vectors. Moreover, we

denote as d(x, X) a generic distance function between any

point x∈Rnand a closed set X ⊂ Rn. For instance, one

may choose d(x, X) = infz∈X |x−z|. Given a function

α:R≥0→R≥0we say α∈ K if it is continuous, zero

at zero and strictly increasing. Similarly, we say α∈ K∞

if α∈ K and lims→∞ α(s) = ∞. Moreover, for a square

matrix Awe denote by λmax(A) its maximum eigenvalue.

As a ﬁnal note, let P∈Rn×nbe a symmetric positive deﬁ-

nite matrix. For any A∈Rn×nand for an arbitrary scalar

r:= r1r2with r1, r2>0, the generalized Schur’s comple-

ment implies that the following inequalities are equivalent

A>P A −rP 0⇐⇒ −r1P A>P

P A −r2P0.

2. Total stability results

In this section, we study how the stability properties of

the origin of a given discrete-time autonomous nonlinear

system

x+=f(x),(1)

transfer to systems described by a suﬃciently similar dif-

ference equation

x+=ˆ

f(x),(2)

where f:Rn→Rn,ˆ

f:Rn→Rnare continuous func-

tions. We propose two diﬀerent results. The ﬁrst one links

the existence of an equilibrium for system (1) to the exis-

tence of an equilibrium for the perturbed system (2). We

show that an equilibrium for (2) exists, provided that the

two models are locally close enough. More precisely, the

result holds if the functions fand ˆ

fare not too diﬀerent

in the C0norm, and system (1) presents an attractive for-

ward invariant set containing its equilibrium and which is

homeomorphic to the unit ball. The second result consid-

ers the case where both the dynamics and the Jacobians

of the two systems (1), (2) are suﬃciently similar. Under

such conditions, we show that the existence of a locally ex-

ponentially stable equilibrium for (1) implies the existence

of a locally exponentially stable equilibrium for (2) close

to it. Moreover, we present a lower bound on the size of

the domain of attraction of the equilibrium for (2).

2.1. Existence of equilibrium

We now present the minimal assumption required to

show the existence of an equilibrium for (2). To this end,

we introduce the following notation. Given a positive func-

tion V:A ⊆ Rn→R≥0and a positive real number c > 0,

we denote the sublevel set of such a function as

Vc(V) := {x∈ A :V(x)≤c}.(3)

Assumption 1. Let Abe an open subset of Rn. There

exists a C0function V:A → R≥0satisfying V(0) = 0

and such that the following holds:

1. there exists a positive real number ¯csuch that the set

V¯c(V)is homeomorphic to the unit ball;

2. there exists ρ∈(0,1) such that

V(f(x)) ≤ρV (x),∀x∈ V¯c(V).(4)

Remark 1. Assumption 1 is not requiring the nominal

system (1) to be asymptotically stable. It solely assumes

the existence of an attractive forward invariant compact set

which is homeomorphic to the unit ball. However, since V

is not strictly positive outside of the origin, Vmay have

local minima and does not allow to conclude asymptotic

stability of the origin.

The ﬁrst result is formalized by the following proposi-

tion. More comments on the assumption are postponed

after the proof.

Proposition 1. Let Assumption 1 hold. Then, for any

positive c≤¯cthere exists a positive real number δsuch

that, for any continuous function ˆ

f:Rn→Rnsatisfying

|ˆ

f(x)−f(x)|< δ, ∀x∈ V¯c(V)(5)

the corresponding system (2) admits an equilibrium point

xe∈ Vc(V). Moreover, such systems has no other equilib-

rium in the set V¯c(V)\ Vc(V).

Proof. Consider c≤¯cand let ˜ρbe any positive real

number satisfying

ρ < ˜ρ < 1.

Since the set V¯c(V) is homeomorphic to the unit ball, it

is bounded. Moreover, the function Vbeing continuous,

Vc(V) is a compact subset. Next, we deﬁne the function

p:R≥0→Ras

p(s) = max

x∈V¯c(V)

v∈Rn:|v|=1 V(f(x) + sv)−¯c,(6)

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摘要：

Totalstabilityandintegralactionfordiscrete-timenonlinearsystemsS.Zobolia,D.Astola,V.AndrieuaaUniversiteLyon1,Villeurbanne,France{CNRS,UMR5007,LAGEPP,France.AbstractRobustnessguaranteesareimportantpropertiestobelookedforduringcontroldesign.Theyensurestabilityofclosed-loopsystemsinfaceofuncertaintie...

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Total stability and integral action for discrete-time nonlinear systems

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