Discrete-time Flatness and Linearization along Trajectories Bernd KolarJohannes DiwoldConrad Gst ottner

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Discrete-time Flatness and Linearization

along Trajectories

Bernd Kolar ∗Johannes Diwold ∗∗ Conrad Gst¨ottner ∗∗

Markus Sch¨oberl ∗∗

∗Magna Powertrain Engineering Center Steyr GmbH & Co KG,

Steyrer Str. 32, 4300 St. Valentin, Austria

(e-mail: bernd kolar@ifac-mail.org)

∗∗ Institute of Automatic Control and Control Systems Technology,

Johannes Kepler University Linz, Altenbergerstraße 66, 4040 Linz,

Austria (e-mail: johannes.diwold@jku.at, conrad.gstoettner@jku.at,

markus.schoeberl@jku.at)

Abstract: The paper studies the relation between a nonlinear time-varying ﬂat discrete-time

system and the corresponding linear time-varying systems which are obtained by a linearization

along trajectories. It is motivated by the continuous-time case, where it is well-known that the

linearization of ﬂat systems along trajectories results in linear time-varying systems which are

again ﬂat. Since ﬂatness implies controllability, this constitutes an important veriﬁable necessary

condition for ﬂatness. In the present contribution, it is shown that this is also true in the discrete-

time case: We prove that the linearized system is again ﬂat, and that a possible ﬂat output is

given by the linearization of a ﬂat output of the nonlinear system. Analogously, the map that

describes the parameterization of the system variables of the linear system by this ﬂat output

coincides with the linearization of the corresponding map of the nonlinear system. The results

are illustrated by two examples.

Keywords: discrete-time systems; ﬂatness; linearization; controllability; time-varying systems

1. INTRODUCTION

The concept of ﬂatness has been introduced in the 1990s

by Fliess, Levine, Martin and Rouchon for nonlinear

continuous-time systems, see e.g. Fliess et al. (1992), Fliess

et al. (1995), or Fliess et al. (1999). Since ﬂatness allows

an elegant solution for motion planning problems and

a systematic design of tracking controllers, it is of high

practical relevance and belongs to the most popular non-

linear control concepts. Nevertheless, checking the ﬂatness

of a nonlinear multi-input system is known as a highly

nontrivial problem, for which still no complete systematic

solution in the form of veriﬁable necessary and suﬃcient

conditions exists (see e.g. Nicolau and Respondek (2016),

Nicolau and Respondek (2017), or Gst¨ottner et al. (2021)

for recent contributions in this ﬁeld). For this reason,

also necessary conditions for ﬂatness are of interest to

be able to prove at least that a given system is not ﬂat.

One such necessary condition is based on the fact that

the linearization of a ﬂat continuous-time system along

a trajectory yields a linear time-varying system which is

again ﬂat and hence controllable (see e.g. Rudolph (2021)).

Since the latter property can be checked easily for linear

systems, this connection between a nonlinear system and

its linearization constitutes an important necessary condi-

tion for the ﬂatness of continuous-time systems.

?This work has been supported by the Austrian Science Fund

(FWF) under grant number P 32151.

The purpose of the present contribution is to investigate

the relation between a ﬂat system and its linearization

along a trajectory in the discrete-time case. Since the

linearization of a nonlinear system along a trajectory

leads in general to a linear time-varying system but the

literature has addressed so far only the time-invariant

case, we ﬁrst need to discuss the concept of discrete-time

ﬂatness for time-varying systems. As proposed in Diwold

et al. (2022b), we consider discrete-time ﬂatness as the

existence of a one-to-one correspondence of the system

trajectories to the trajectories of a trivial system. This

leads naturally to a formulation which takes into account

both forward- and backward-shifts of the system variables

as it is also proposed in Guillot and Mill´erioux (2020). The

point of view adopted e.g. in Sira-Ramirez and Agrawal

(2004), Kaldm¨ae and Kotta (2013), or Kolar et al. (2016),

where discrete-time ﬂatness is deﬁned by replacing the

time derivatives of the continuous-time case by forward-

shifts, is included as a special case and denoted within the

present paper as forward-ﬂatness.

As our main result, we prove that the linearization of a ﬂat

discrete-time system along a trajectory is again ﬂat, and

that a possible ﬂat output is given by the linearization of a

ﬂat output of the nonlinear system. Furthermore, we show

that the corresponding parameterization of the system

variables by the ﬂat output and its shifts coincides with the

linearization of the parameterization of the nonlinear sys-

tem. Like in the continuous-time case, this connection be-

tween nonlinear system and linearized system establishes

arXiv:2210.09247v1 [math.OC] 17 Oct 2022

an important necessary condition for ﬂatness. Even though

for discrete-time systems the property of forward-ﬂatness

can be checked eﬃciently by a generalization of the test

for static feedback linearizability (see Kolar et al. (2022))

which is based on a certain decomposition property derived

in Kolar et al. (2021), for the more general case including

both forward- and backward-shifts of the system variables

a computationally feasible test does not yet exist. 1Hence,

as we shall illustrate by our second example, the derived

necessary condition is a useful possibility to prove that a

given discrete-time system is not ﬂat.

The paper is organized as follows: First, Section 2 deals

with the concept of discrete-time ﬂatness for time-varying

systems. The core of the paper is then contained in Section

3, which studies the relation between a ﬂat system and

the linear time-varying system obtained by a linearization

along a trajectory. The presented results are illustrated by

two examples in Section 4.

Notation Since we apply diﬀerential-geometric concepts,

we use index notation and the Einstein summation conven-

tion to keep formulas short and readable. However, to high-

light the summation range especially for double sums, we

also frequently indicate the summation explicitly. For coor-

dinates that represent forward- or backward-shifts of sys-

tem variables, we use a notation with subscripts in brack-

ets. For instance, the α-th forward- or backward-shift of a

component yj,j∈ {1, . . . , m}of a ﬂat output ywith α∈Z

is denoted by yj

[α], and y[α]= (y1

[α], . . . , ym

[α]). Furthermore,

to facilitate the handling of expressions which depend on

diﬀerent numbers of shifts of diﬀerent components of a ﬂat

output, we use multi-indices. If A= (a1, . . . , am) is some

multi-index, then y[A]= (y1

[a1], . . . , ym

[am]).

2. FLATNESS OF TIME-VARYING DISCRETE-TIME

SYSTEMS

In this contribution, we consider nonlinear time-varying

discrete-time systems

xi,+=fi(k, x, u), i = 1, . . . , n (1)

with dim(x) = n, dim(u) = m, and smooth functions

fi(k, x, u). In addition, we assume that the system (1)

meets the submersivity condition

rank(∂(x,u)f) = n , (2)

which is quite common in the discrete-time literature, for

all time-steps k.

As proposed in Diwold et al. (2022b), where only time-

invariant systems are considered, we call a time-varying

discrete-time system (1) ﬂat if there exists a one-to-one

correspondence between its trajectories (x(k), u(k)) and

the trajectories y(k) of a trivial system with dim(y) =

dim(u). The trajectories of a trivial system are not re-

stricted by any diﬀerence equation and hence completely

free. By one-to-one correspondence, we mean that the

values of x(k) and u(k) at a time-step kare determined by

an arbitrary but ﬁnite number of future and past values of

y(k), i.e., by the trajectory y(k) in an arbitrarily large but

1An interesting approach can be found in Kaldm¨ae (2022) but

requires the solution of partial diﬀerential equations.

Fig. 1. One-to-one correspondence of the trajectories.

ﬁnite time window. Conversely, the value of y(k) at a time-

step kis determined by an arbitrary but ﬁnite number of

future and past values of x(k) and u(k). Consequently,

the one-to-one correspondence of the trajectories can be

expressed by maps of the form

(x(k), u(k)) = F(k, y(k−r1), . . . , y(k+r2)) (3)

and

y(k) = Φ(k, x(k−q1), u(k−q1), . . . , x(k+q2), u(k+q2))

(4)

with suitable integers r1, r2, q1, q2that describe the length

of the corresponding ﬁnite time windows, cf. Fig. 1. Since

the number of forward- and backward-shifts in (3) and (4)

can of course be diﬀerent for the individual components

of y,x, and u, we will later use appropriate multi-indices

where it is important.

In the remainder of this section, the framework used in

Diwold et al. (2022b) for the analysis of ﬂat time-invariant

discrete-time systems is adapted to the time-varying case.

First, it is important to note that the representation of a

trajectory of the system (1) by both sequences x(k) and

u(k) contains redundancy, as these sequences are coupled

by the system equations (1). By a repeated application of

(1), all forward-shifts x(k+α), α≥1 of the state variables

are determined by x(k) and the input trajectory u(k+α)

for α≥0:

x(k+ 1) = f(k, x(k), u(k))

x(k+ 2) = f(k+ 1, x(k+ 1), u(k+ 1))

In the case rank(∂xf) = n, the same is also true for the

backward-direction. However, even if the system meets

only the weaker submersivity condition (2), there exist

mfunctions g(k, x, u) such that the (n+m)×(n+m)

Jacobian matrix ∂xf ∂uf

∂xg ∂ug(5)

is regular for all k. With such functions, the map

x+=f(k, x, u)

ζ=g(k, x, u)(6)

is locally invertible for all k, and by a repeated application

of its inverse

(x, u) = ψ(k, x+, ζ) (7)

all backward-shifts x(k−β), u(k−β), β≥1 of the state-

and input variables are determined by x(k) and backward-

shifts ζ(k−β), β≥1 of the system variables ζdeﬁned by

(6):

(x(k−1), u(k−1)) = ψ(k−1, x(k), ζ(k−1))

(x(k−2), u(k−2)) = ψ(k−2, x(k−1), ζ(k−2))

Hence, every trajectory of the system (1) is uniquely

determined both in forward- and backward-direction by

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Discrete-timeFlatnessandLinearizationalongTrajectoriesBerndKolarJohannesDiwoldConradGstottnerMarkusSchoberlMagnaPowertrainEngineeringCenterSteyrGmbH&CoKG,SteyrerStr.32,4300St.Valentin,Austria(e-mail:berndkolar@ifac-mail.org)InstituteofAutomaticControlandControlSystemsTechnology,JohannesK...

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Discrete-time Flatness and Linearization along Trajectories Bernd KolarJohannes DiwoldConrad Gst ottner

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