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arXiv:2303.16505v1 [eess.SY] 29 Mar 2023

On the Design of Limit Cycles of Planar Switching Afﬁne Systems

Nils Hanke1and Olaf Stursberg2

Abstract— In the context of studying periodic processes, this

paper investigates ﬁrst under which conditions switching afﬁne

systems in the plane generate stable limit cycles. Based on

these conditions, a design methodology is proposed by which

the phase portraits of the switching systems are determined

to obtain globally stable limit cycles from simple speciﬁcations,

such as given amplitudes and frequencies of desired oscillations.

As an application, the paper ﬁnally shows that an oscillator

model can be derived with a small effort from data measured

for an unknown oscillating system.

I. INTRODUCTI ON

Since oscillations occur in a large variety of domains

(technical, physical, biological, astronomical, etc.), limit cy-

cles have been a subject of research for a long period of

time, see e.g. [1], [2], [3], [4]. They have been characterized

and investigated for different nonlinear models such as

Kuramoto, Van-der-Pol, or FitzHugh-Nagumo oscillators [5],

[6], [7], [8]. The analytical characterization of limit cycles

as well as the speciﬁcation of conditions under which unique

limit cycles are observed is, however, limited to very special

cases. One thread of research has thus studied the existence

of limit cycles for systems consisting of multiple linear sys-

tems: Goncalves considered limit cycles induced by feedback

through relays with hysteresis for stable linear time-invariant

systems, and he formulated conditions as linear matrix in-

equalities (LMIs) that guarantee global asymptotic stability

of limit cycles [9]. Follow-up investigations led to necessary

conditions for regions of stability using piece-wise linear

systems (PLS) [10]. The different approach in [11] aims

at controlling the dynamics of networked piecewise-linear-

shaped Fitz-Hugh-Nagumo neurons through their nullclines,

relying on computing the oscillation period of all nodes

in a master-slave conﬁguration. Although PLS have been

widely used in studies of nonlinear dynamical systems, the

rigorous mathematical deﬁnition has been barely considered

[12], [13]. Alternative work has investigated hybrid con-

trol strategies which combine stabilizing and destabilizing

control laws to obtain the properties of limit cycles with

deﬁned amplitudes and frequencies [14]. The occurrence of a

stable limit cycle for a hybrid system in which the switching

between different modes is determined by a suitable control

strategy was described in [15].Kai and Masuda designed a

stable limit cycle as polygonal closed curve by connecting

vertices of the polygon through line segments determined by

1Nils Hanke, Control and System Theory, Dept. of Electrical

Eng. and Computer Science, University of Kassel, Germany

n.hanke@uni-kassel.de

2Olaf Stursberg, Control and System Theory, Dept. of Electri-

cal Eng. and Computer Science, University of Kassel, Germany

stursberg@uni-kassel.de

piecewise afﬁne systems [16]. Kai also provided conditions

and analytic solutions to control the piecewise afﬁne system

by state feedback such that a given polygonal closed curve

becomes a stable limit cycle [17], [18].

In contrast to the previously cited work, the present paper

follows the objective to design oscillating systems in the

plane such that globally stable limit cycles are obtained. An

important design requirement is that simple speciﬁcations

(basically only the desired frequency and amplitudes) are suf-

ﬁcient to synthesize switching piece-wise systems, which e.g.

can serve as a building block for networks of oscillators. The

class of planar switching afﬁne systems (PSAS) seems partic-

ularly suited for this purpose, as linear system theory allows

representing limit cycles very efﬁciently, as opposed to the

above-named examples of nonlinear systems with periodic

behavior. Accordingly, the paper ﬁrst formulates a theorem

for synthesizing globally stable limit cycles requiring just one

switching surface. Based on an analytic speciﬁcation of the

limit cycle including the amplitude and frequency, a design

approach is proposed that determines the phase portraits such

that the speciﬁcations for amplitude and frequency are met,

and stability is obtained. Finally the paper describes based on

an example, how the design procedure can be used to quickly

construct the periodic system to represent oscillations as

found in experimental data, but without requiring techniques

of system identiﬁcation or machine learning.

II. LIMIT CYCL E OF SWITCHING A FFI NE

SYST EMS I N THE PLANE

The underlying model class of this investigation is switch-

ing afﬁne systems deﬁned as follows: Assume a state space

X⊆Rnwith a polyhedral partition P={P1,...,Pnp}con-

sisting of fully dimensional polyhedra which are pairwise

disjoint with respect to their interior and which cover X. Let

an afﬁne autonomous dynamics:

˙x(t) = Ai·x(t) + bi(1)

be assigned to any polyhedron Pi∈P,i∈ {1,...,np}. (Note

that Aiand bimay have been obtained from designing an

afﬁne state feedback controller u(t) = Ki·x(t)+ difor a plant

model ˙x(t) = ˜

Ai·x(t)+ ˜

Bi·u(t)on Pi.) Let Tk={t0,t1,...,tk,}

denote a (possibly inﬁnite) set of switching times (extended

by the initial time t0=0). A run of (1), denoted by ¯x[0,∞[,

starting from an initialization x(t0) = x0is a sequence of

phases [tk,tk+1]in between two successive switching times,

where the instance of (1) is activated for that ifor which

x(t)∈Pifor t∈[tk,tk+1]. For any switching between two

dynamics with indices i,j∈ {1,...np}occurring at a time tk

and for a state x(tk), positioned on the shared boundary of

Piand Pj, assume the following: The left-hand limit of (1)

in time ˙x(t−

k):=lim

ε

→0Ai·x(t−

ε

) + bipoints outside of Pi,

while the right-hand limit ˙x(t+

k):=lim

ε

→0Aj·x(t+

ε

) + bj

points into Pj. This implies a unique switching time when

the run crosses the boundary of Piand Pj. Note, however,

that the destination jof switching may not be unique, if x(tk)

is contained in the boundaries of more than two polyhedra in

P. For this case, as well as for the initialization of x(t0)to a

point on the shared boundary, an additional rule for selecting

the active dynamics needs to be provided.

Given the motivation of deﬁning oscillations by a model,

which is as simple as possible (but, of course, allows for

stable limit cycles), the following considerations refer now to

states deﬁned in R2, and to switching afﬁne systems with just

P={P1,P2}. Obviously, the underlying partitioning then

reduces to two half-spaces separated by a line, referred to as

switching line below. The corresponding model is deﬁned as

follows (using Iand II to denote the two modes):

Deﬁnition 1: PSAS denoted as Σ

Given a switching line C·x=dfor x∈R2,C= [c1,1,c1,2]∈

R1×2,d∈R, the polyhedral partition results to P={PI,PII }

with PI={x|C·x≤d},PII ={x|C·x≥d}. For matrices AI∈

R2×2,bI∈R2×1,AII ∈R2×2, and bII ∈R2×1, the dynamics

assigned to this partition with t∈R≥0is:

˙x(t) = AI·x(t) + bI(2)

for x(t)∈Int(PI), and:

˙x(t) = AII ·x(t) + bII (3)

for x(t)∈Int(PII ), where I nt denotes the interior of the

respective set. For points on the switching line C·x(t) = d,

the dynamics (2) is assigned if lim

ε

→0x(t−

ε

)∈Int(PI)

applies for the predecessor in time, and the dynamics (3)

is assigned if lim

ε

→0x(t−

ε

)∈Int(PII ). For the initial time

t0,Σstarts to evolve according to (2), if x(t0)∈Int(PI),

and according to (2) if x(t0)∈Int(PI I ). For C·x(t0) = d, it is

assigned by convention that Σstarts to evolve with (3) if and

only if C·(AI I ·x(t) + bII )>0 and ( C·(AI·x(t) + bI)≥0 or

kC·(AII ·x(t) + bII)k2>kC·(AI·x(t)+ bI)k2).

While a run ¯x[0,∞[of Σfollows in general from the rules

indicated below (1) and in Def. 1, the speciﬁc instance of a

limit cycle is deﬁned next:

Deﬁnition 2: Limit Cycle of Σ

A run ¯x∗

[0,∞[of Σaccording to Def. 1 is called limit cycle,

if a ﬁnite period T∈R>0exists such that for any point

x(t)∈¯x∗

[0,∞[,t∈R≥0it applies that: x(t+T) = x(t).

The following theorem states conditions which ensure that

the run of Σfollows the limit cycle forever, if the run starts

from a point on the cycle.

Theorem 1: For a system Σas speciﬁed in Def. 1, let the

parametrization satisfy that AI,AII have only distinct negative

real eigenvalues

λ

I= [

λ

I

1,1,

λ

I

2,1]T∈R2x1,

λ

II = [

λ

II

1,2,

λ

II

2,2]T∈

R2x1, and let unique equilibrium points xI

R∈R2×1,xII

R∈R2×1

follow from the choice of Ai. Then, a unique limit cycle ¯x∗

[0,∞[

with period Taccording to Def. 2 exists with initialization

to x(ts0)with C·x(ts0) = dand for two different switching

PSfrag replacements

˙xI(ts0)

˙xI(ts1)

xII

R

˙xII (ts1)

˙xII (ts2)

xI

R

C·x(ts0) = C·x(ts2) = d

PI

PII

C·x(ts1) = d

C

¯x∗

[0,∞[

x1

x2

A∗

Fig. 1. Sufﬁcient conditions (4)-(8) for a stable limit cycle ¯x∗

[0,∞[.

points x(ts1)6=x(ts2)on the switching line, if the following

set of sufﬁcient conditions holds:

C·(AI·x(ts0) + bI)<0 (4)

C·(AI·x(ts1) + bI)>0,C·(AII ·x(ts1) + bII )>0 (5)

C·(AII ·x(ts2) + bII )<0 (6)

C·xI

R>d,C·xII

R<d(7)

x(ts2) = x(ts0),T=ts2−ts0.(8)

The meaning of (4)-(8) for a limit cycle ¯x∗

[0,∞[of Σis

illustrated in Fig. 1, using the abbreviations ˙xI(tsk):=AI·

x(tsk) + bI, and ˙xII (tsk):=AII ·x(tsk) + bII for k∈ {0,1,2}.

Proof: [Thm. 1] Given C·x(ts0) = dand C·x(ts1) = d,

any point on the limit cycle ¯x∗

[0,∞[for t∈[ts0,ts1]follows by

integrating (2) from:

x(t) = eAI(ts1−ts0)·x(ts0) +

ts1

Z

ts0

eAI(ts1−

τ

)·bId

τ

,(9)

Due to condition (4), Σis forced to activate the dynamics

(2) during t∈[ts0,ts1]with x(t)∈PI. Since (2) is stable

according to the assumptions on AI, the state is attracted

to xI

R, which is, however, positioned in PI I according to the

ﬁrst part of condition (7). This together with the condition

(5) enforces the ﬁrst switching event at a ﬁnite time ts1, by

which x(t)transitions from PIto PII , and the dynamics (3) is

activated for t∈[ts1,ts2]. With C·x(ts2) = dand by integrating

(3), any point on ¯x∗

[0,∞[for t∈[ts1,ts2]is given by:

x(t) = eAII(ts2−ts1)·x(ts1) +

ts2

Z

ts1

eAII (ts2−

τ

)·bII d

τ

.(10)

For this phase, the second part of condition (5) together with

the ﬁrst statement in (8) and (6) implies that x(t)is governed

2

PSfrag replacements

C·x=d

4

2

˜x(˜

ts1)

˜x(˜

ts3)

˜x(˜

ts5)

x(ts0) = x(ts2)

˜x(˜

ts4)

˜x(˜

ts2)1

3

x(ts1)

˜x(0)

¯x∗

[0,∞[

˜x[˜

ts1,˜

ts2]

˜x[˜

ts2,˜

ts3]

˜x(0)

˜x(˜

ts4)

˜x(˜

ts2)

˜x(˜

ts1)

˜x(˜

ts3)

˜x(˜

ts5)

PI

PII

x1

x2

Fig. 2. Convergence for an initialization ˜x(0)in the interior (green) and

exterior (red) of ¯x∗

[0,∞[(black) with switching line C·x=d(blue).

by the stable dynamics (3) and is attracted to the stable

equilibrium point xII

R, which is contained in PIaccording to

the second part of condition (7). Thus, the second switching

event takes place at a ﬁnite time ts2, enforcing that PII is

left, and the same situation as for the beginning of the ﬁrst

phase holds again. Due to the ﬁrst part of condition (8), the

point of initialization xs0is reached, and the limit cycle is

closed by concatenation of the two pieces of ¯x∗

[0,∞[for [ts0,ts1]

and [ts1,ts2], where for both phases the assumption on the

parameters of (2) and (3) as stated in the theorem ensure

unique solutions by (9) and (10). Obviously, the period Tof

one cycle is equal to the sum of the two phases, as implied

by the second part of (8). By repeated concatenation of the

two alternating phases the complete and unique limit cycle

¯x∗

[0,∞[is obtained.

The principle of the conditions stated in Theorem 1 for

realizing the limit cycle is that in any phase, the dynamics

gears towards an equilibrium point that is unattainable, since

a different subsystem is activated before the equilibrium

point is reached. At the same time, the gradient conditions

formulated for the switching line are determined such that

the line deﬁnitely crossed before the turn of the trajectory

towards the currently relevant equilibrium point occurs.

Note that the formulation in Theorem 1 with choosing the

initial state x(ts0)being positioned on the switching line is

used only for notational convenience, i.e., the extension to

assigning x(0)to any arbitrary point on the limit cycle is

straightforward.

Next global stability of the limit cycle is considered and

discussed based on the Fig. 2 which is oriented to the

structure of Fig. 1. In Fig. 2, an arbitrary initialization of

the state to an ˜x(0)outside of the limit cycle (black curve)

is chosen, and the run from this point is denoted by ˜x[0,∞[

(shown by a red dashed line). The alternating activation of

the two dynamics (2) and (3) follows the same pattern as

explained in the proof of Theorem 1 for the motion on

¯x∗

[0,∞[: After the ﬁrst phase of ˜x[0,∞[in PIwith (2) for the

time interval t∈[0,˜

ts1], a switching to (3) in PI I takes place

before the ﬁrst dynamics is again activated at ˜

ts2, and so

on. It will be shown in a subsequent theorem that, due to

the construction of Σaccording to Theorem 1, the run ˜x[0,∞[

approaches ¯x∗

[0,∞[in a spiral from the outside. In contrast,

Fig. 2 (green dashed line) shows for the initialization to an

˜x(0)inside of the limit cycle that the run ˜x[0,∞[converges

to ¯x∗

[0,∞[from the inside. To formalize this argumentation, a

deﬁnition of stability and a corresponding theorem are stated

next.

Deﬁnition 3: Stability of a Limit Cycle of Σ

A limit cycle ¯x∗

[0,∞[is called globally stable, if independent

of the initialization x(0) = x0∈R2every trajectory converges

towards ¯x∗

[0,∞[.

Theorem 2: For the system Σaccording to Def. 1, let

a unique limit cycle ¯x∗

[0,∞[be obtained by enforcing the

conditions stated in Theorem 1. Then, ¯x∗

[0,∞[is globally stable

according to Def. 3.

Proof: For ˜x(0)outside of ¯x∗

[0,∞[, convergence of ˜x[0,∞[

to ¯x∗

[0,∞[requires that the sequence of switching points

˜x(˜

ts1),˜x(˜

ts3),˜x(˜

ts5),... converges to x(ts1), and likewise that

the sequence ˜x(˜

ts2),˜x(˜

ts4),˜x(˜

ts5),... converges to x(ts0):

lim

i→∞k˜x(˜

tsi)−x(ts1)k=0 for odd i,(11)

lim

i→∞k˜x(˜

tsi)−x(ts2)k=0 for even i(12)

with x(ts2) = x(ts0); see also Fig. 2 (marked in red). The

existence of the switching points ˜x(tsi)for ﬁnite ˜

tsias

reached by continuous evolution from ˜x(tsi−1)follows from

the stability of AI(or AII respectively) and the associated

equilibrium points xI

R(or xII

R) on the reverse side of the

switching line. The same reasoning leads to the existence

of ˜x(ts1)as reached from ˜x(0).

In order to establish (11) and (12), the mapping from ˜x(˜

tsi)

to the next intersection with the switching line in the same

direction is:

˜x(˜

tsi+2) = eAI(˜

tsi+2−˜

tsi+1)eAII (˜

tsi+1−˜

tsi)˜x(˜

tsi)

+eAI(˜

tsi+2−˜

tsi+1)

˜

tsi+1

Z

˜

tsi

eAII (˜

tsi+1−

τ

)bII d

τ

+

˜

tsi+2

Z

˜

tsi+1

eAI(˜

tsi+2−

τ

)bId

τ

(13)

The sequence of distances of the switching points ˜x(˜

tsi)for

odd ito x(ts1)(and of ˜x(˜

tsi)for even ito x(ts2)respectively)

is measured by a discrete Lyapunov function, deﬁned only

on the switching line (C·˜x(˜

tsi) = d) with ∆xd=˜x(˜

tsi)−x(ts1)

for odd i, and ∆xd=˜x(˜

tsi)−x(ts2)for even i.

Vd(˜x(˜

tsi)) = k∆xdk2=∆xT

d∆xd,d∈ {1,2}.(14)

Similar to the procedure in [15], [19], decrease of Vdaccord-

ing to:

Vd(˜x(˜

tsi+2)) −Vd(˜x(˜

tsi)) <0 (15)

3

results as follows:

Deﬁne the convergence ratio

α

˜

tsi+2,˜

tsifor a 2-periodic limit

cycle gives a recursive deﬁnition of the distances marked by

1 and 3, respectively 2 and 4 in Fig. 2:

k˜x(˜

tsi+2)−x(tsd)k2

k˜x(˜

tsi)−x(tsd)k2≤

α

˜

tsi+2,˜

tsi(16)

Thus, (16) and the deﬁnition of the discrete Lyapunov

functions in (15) give a relation for any switch:

Vd(˜x(˜

tsi+2)) ≤

α

˜

tsi+2,˜

tsiVd(˜x(˜

tsi)) (17)

Given a stable limit cycle with switching sequence ˜

ts1,˜

ts2...

then for any initial switching point the solution will asymp-

totically converge to the limit cycle if:

α

˜

tsi+2,˜

tsi<1 (18)

The recursive representation in (13) can be used for odd

and even ito represent the inequality in (16). Since both

subsystems are Hurwitz and the time differences are positive

every exponential function can become upper bounded lower

than one. By comparing the time differences to an equal

representation of the limit cycle (ﬁt (10) into (9)) the

afﬁne parts are ensured to be lower than one additionally.

Thus (16) holds with respect to Theorem 1. The discrete

Lyapunov functions decrease and therefore the initialisation

˜x(˜

tsi)converges to x(ts1)or x(ts2)on the limit cycle. Finally,

the argumentation in the proof of Theorem 1 shows that

any initialization to ˜x(0)on ¯x∗

[0,∞[means that ¯x∗

[0,∞[is never

left, thus completing the proof. The same reasoning can

be applied for an initialization ˜x(0)inside of ¯x∗

[0,∞[and the

situation shown in Fig. 2 (green), leading to the result that

the ˜x(˜

tsi)converge from the interior of ¯x∗

[0,∞[to x(ts0), or

x(ts1)respectively.

The initialisation excludes cases in which ˙x=0 shows up.

III. LIM IT CYCLE D ESI GN

Based on the conditions for the existence of limit cycles

presented before, this section shows how a speciﬁc stable

limit cycle can be constructed starting from speciﬁcations

for the amplitude and frequency of the oscillation of Σ.

Deﬁnition 4: Amplitude of the Limit cycle

Let x(t) = [x1(t),x2(t)]Tdescribe a point moving along the

limit cycle ¯x∗

[0,∞[with period Tfor t∈[i·T,(i+1)·T],

i∈N∪ {0}. In addition, let A∗= [A∗

1,A∗

2]Tdenote the

center point in between the two switching points on the

switching line C·x(ts0) = dand C·x(ts1) = d, see Fig. 3.

The time-varying amplitude A(t)is then deﬁned as the two

Euclidean norms AI(t) = kx(t)−A∗k2for x(t)∈PIand

AII (t) = kx(t)−A∗k2for x(t)∈PII . Furthermore, let AI

max

denote the maximum of AI(t) = kx(t)−A∗k2over x(t)∈PI,

and AII

max the maximum of AI I (t) = kx(t)−A∗k2over

x(t)∈PII . Finally, AI

min and AII

min denote the corresponding

minimum amplitudes.

Deﬁnition 5: Limit cycle frequency

Given the limit cycle ¯x∗

[0,∞[of Σwith period Tas deﬁned

in (8), the frequency of the corresponding oscillation of x(t)

is

ω

=2

π

T.

To fully determine Σ(while considering the conditions

provided in Theorem 1), the equations (9) and (10) are solved

by use of the decomposition:

eAi·t=Wi·eAiD·t·W−1

i=Wi·"e

λ

i

1,2·t0

0e

λ

i

2,2·t#·W−1

i,(19)

where AiD∈R2x2denotes the diagonalised matrix to Ai∈

R2x2with eigenvalues

λ

i= [

λ

i

1,2,

λ

i

2,2]T∈R, and the eigen-

vector matrix Wi∈R2x2.

Solving equation (9) with (19) leads to (20), while solving

(10) likewise leads to (21):

x(t)=

e

λ

I

1,1td1(S1+S5

λ

I

1,1

)+e

λ

I

2,1td1(S2+S6

λ

I

2,1

)+H1

e

λ

I

1,1td1(S3+S7

λ

I

1,1

)+e

λ

I

2,1td1(S4+S8

λ

I

2,1

)+H2

(20)

x(t)=

e

λ

II

1,2td2(S9+S13

λ

II

1,2

)+e

λ

II

2,2td2(S10 +S14

λ

II

2,2

)+H3

e

λ

II

1,2td2(S11 +S15

λ

II

1,2

)+e

λ

II

2,2td2(S12 +S16

λ

II

2,2

)+H4

(21)

H1=−S5

λ

I

1,1

−S6

λ

I

2,1

,H2=−S7

λ

I

1,1

−S8

λ

I

2,1

,td1=ts1−ts0(22)

H3=−S13

λ

II

1,2

−S14

λ

II

2,2

,H4=−S15

λ

II

1,2

−S16

λ

II

2,2

,td2=ts2−ts1(23)

The substitutions S1to S16 used in these equations are listed

in the appendix.

To determine a limit cycle complying with the speciﬁca-

tions, the entries of AI,bI,AII and bII are computed. Since

each subsystem has six degrees of freedom, a total of twelve

equations is needed to obtain a stable limit cycle by two

switching planar afﬁne subsystems. Selected points are used

to ﬁx the degrees of freedom. Two of these points are the

switching points satisfying C·x(ts1) = dand C·x(ts2) = d,

used twice for both subsystems. (Inserting the coordinates

of these points into (20) and (21) leads to eight equations.)

Furthermore, for each of the two polytopes PIand PII , an

arbitrarily chosen additional point can be selected. Using

a maximum or minimum amplitude appears as reasonable

choice to determine these points. If these are inserted into

(20) and (21), the required set of twelve equations is obtained

to ﬁx all degrees of freedom. These equations together with

the conditions of Theorem 1 determine Σas well as the limit

cycle.

A. Example

The aforementioned procedure is illustrated by an exam-

ple: Assume that the frequency

ω

=1.824Hz of a limit cycle,

the minimum amplitude of AI

min(tI

min) = 0.25, and the max-

imum amplitude AII

max(tII

max) = 1.3778 (for tI

min,tII

max ∈R≥0)

4

are given as speciﬁcations for system design. The switching

line is deﬁned by C= [1,0]and d=0. Since Tis equal to the

sum of the two phases, ts1=0.801s and ts2=2.644s are cho-

sen, leading to switching points xs p (ts1) = [0,−2.394]Tand

xsp (ts0) = xsp (ts2) = [0,−5.160]T. Using A∗= [0,−3.778]T

and Def. 4 results in xI

min(tI

min) = [−0.237,−3.6959]T,tI

min =

0.2825s, tI

min ≤ts1and xII

max(tII

max) = [0.1142,−2.395]T, as

well as tII

max =0.0482s, tII

max ≤ts2. Relevant points of con-

struction for this example are illustrated in Fig.3. The four

PSfrag replacements

x1

x2

-2.5

-4

-5.5

-0.25 00.25

C·x=dxI

min(tI

min)

AI

min(tI

min)

xII

max(tII

max)

AII

max(tII

max)

xsp (ts0),xsp (ts1)

A∗

Fig. 3. Characterizing points of the design example.

points and their coordinates determine twelve equations as

previously explained. Solving the equation system leads to

an oscillator system of type Σwith two afﬁne dynamics

parameterized by:

AI=−3 1

3−2,bI=3

−3(24)

AII =−4 1

−3 0.25,bII =5

0.75(25)

The Figs. 4 and 5 show the resulting stable limit cycle and the

course of the amplitude over time, starting from an arbitrarily

chosen initial state x(0) = [−1.5,−4.5]T. The characterizing

points used for design are, of course, located on the limit

cycle.

PSfrag replacements

−2

−2

−2.5

−3

−3.5

−4

−4.5

−5

−1.5−1−0.500.511.52

x1

x2

x(0)

Fig. 4. Limit cycle with characterizing points of the design example.

IV. LI MIT C YCLE IDENTIFICATIO N

By employing the design principles introduced before, this

section proposes an approach to identify stable limit cycles

PSfrag replacements

0510

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

t

A(t)

AII

max(tII

max)

AI

min(tI

min)

Fig. 5. Amplitude of the limit cycle for the design example.

from measured (possibly noisy) data of nonlinear systems.

Consider an oscillating chemical reaction, a so-called relax-

ation oscillator, which is characterized by alternating phases

of fast and slow reaction. The oscillator shows the basic

mechanism of cyclical oxidation and reduction of palladium.

Let y1describe the state of oxidation of a palladium catalyst

and y2the CO concentration in the reactor. Additionally,

α

denotes the ﬂow rate through the reactor, and Qindicates

the division into active and passive regions. For the purpose

of this section, the following model is subsequently used for

data generation – in practice, however, the model would be

supposed to be unknown, and only periodic data would be

available:

˙y1=[Θ(y1,y2,Q)−y1]·

β

(26)

˙y2=−Θ(y1,y2,Q)·y2+

α

·y0−

α

·[1−Θ(y1,y2,Q)]·y2(27)

Θ(y1,y2,Q)=Θ0·(f(y1,Q)−y2),f(y1,Q)=e

−y2

1

Q(28)

Θ0is the Heaviside step function, and the frequency

β

of

the oscillation is:

β

=Θ(y1,y2,Q)·

β

+(1−Θ(y1,y2,Q))·

β

0(29)

with

β

0describing the speed of reduction, and

β

the speed of

oxidation. The y1-y2phase space is divided by the boundary

line f(y1,Q)(see Fig. 6 magenta) into an active and a passive

area with different dynamic behaviour. In the active area,

equation (28) is used in (26), (27) and (29) being 1 and

0 in the passive area. White noise is added to the right-

hand sides of (26) and (27) to represent unknown inﬂuences

of the chemical process and generate appropriated data for

the identiﬁcation process. More detailed information on the

model can be found in [20].

Given data from the simulation of the model (for a ﬂow

rate of

α

=0.83), the objective is to identify a PSAS

that represents the oscillating behavior. When detecting the

switching points as well as two more characteristic points,

the aforementioned procedure can be used to obtain the

5

model Σwith the following parameterization of the afﬁne

dynamics:

AI=−0.01 ·

α

0

0−1,bI=0.01 ·

α

0.9·

α

(30)

AII =0.1 0

0−

α

,bII =0

0.9·

α

(31)

The nonlinear switching line (Fig. 6 magenta) was approxi-

mated by C= [0.4115,1]and d=1.132 (Fig. 6 blue). This

line was determined based on the two switching points which

divide the active and passive regions. For the identiﬁcation,

the maxima of the amplitudes on the two regions were used

to generate the corresponding equations. Thus, the scheme as

in Fig. 3 can be applied and enables successful identiﬁcation.

Fig. 6 compares the limit cycle from the model for data

generation (referred to by OPd ) for

α

=0.83 in red (without

noise), and the limit cycle of the model identiﬁed as PSAS

(referred to by ΣPd) in black. The difference is negligible,

i.e., the model ΣPd (whose structure is signiﬁcantly simple

than that of OPd ) can be used for the analysis of the system.

PSfrag replacements

0.30.4

0.4

0.5

0.5

0.6

0.6

0.7

0.7

0.8

0.8

0.9

0.9

1

1

1.1

1.1

1.2

1.2

C,d

(28)

ΣPd

OPd

x(0)

y2,x2

y1,x1

Fig. 6. Identiﬁcation of the limit cycle for the reaction system.

V. CONCLUS IONS

The paper has proposed a new method for synthesizing

models of type planar switching afﬁne systems to represent

oscillating behavior. The design rules guarantee the global

stability and uniqueness of the resulting limit cycle. The

advantages of the model are manifold: (i) linear system

theory is sufﬁcient to analyze the oscillating behavior; (ii)

the design of afﬁne controllers for the two afﬁne dynamics

is suited to instantiate the embedded subsystems; (iii) the

oscillations are robust in the sense that (due to the property of

global stability) deviations from the limit cycles do not lead

to divergence from ¯x∗

[0,∞[; (iv) the design rules build on very

few speciﬁcations for amplitudes and the frequency to obtain

the desired oscillations – in system identiﬁcation, the low

effort for design and the few parameters in the model have

to be contrasted to numeric procedures (such as machine

learning) to obtain a model with typical many parameters

from a large set of data.

Future work includes investigation in higher dimensions

and extensions to more subsystems and switching surfaces. In

addition, the coupling of several oscillators of the proposed

type will be investigated, as well as the application for bio-

logical systems with respect to modeling periodic rhythms.

APPE NDI X

A. Substitutions used for computing the limit cycle analyti-

cally: If the index of a Substitution is lower or equal than

eight choose R :=I, e :=0otherwise choose R :=II, e :=1.

S1,9=x1(tse)vR

1,1vR

2,2−x2(tse)vR

1,1vR

1,2

vR

1,1vR

2,2−vR

2,1vR

1,2

,S2,10 =−x1(tse)vR

1,2vR

2,1+x2(tse)vR

1,1vR

1,2

vR

1,1vR

2,2−vR

2,1vR

1,2

(32)

S3,11 =x1(tse)vR

2,1vR

2,2−x2(tse)vR

2,1vR

1,2

vR

1,1vR

2,2−vR

2,1vR

1,2

,S4,12 =−x1(tse)vR

2,2vR

2,1+x2(tse)vR

2,2vR

1,1

vR

1,1vR

2,2−vR

2,1vR

1,2

(33)

S5,13 =bR

1vR

1,1vR

2,2−bR

2vR

1,1vR

1,2

vR

1,1vR

2,2−vR

2,1vR

1,2

,S6,14 =−bR

1vR

1,2vR

2,1+bR

2vR

1,1vR

1,2

vR

1,1vR

2,2−vR

2,1vR

1,2

(34)

S7,15 =bR

1vR

2,1vR

2,2−bR

2vR

2,1vR

1,2

vR

1,1vR

2,2−vR

2,1vR

1,2

,S8,16 =−bR

1vR

2,2vR

2,1+bR

2vR

2,2vR

1,1

vR

1,1vR

2,2−vR

2,1vR

1,2

(35)

ACKNOWLEDGMENT

The authors gratefully acknowledge partial ﬁnancial sup-

port from the German Research Foundation (DFG) through

the Research Training Group ”Biological Clocks on Multiple

Time Scales”.

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