On Solving Controlled-Invariance Problems in Dioids Using the

PyMinMaxGD Python Scripts Library

Olivier Boutin

1 a

, Claude Martinez

2 b

and Naly Rakoto

3 c

LS2N - UMR6004, 3iL Ingénieurs, Limoges, France

LS2N - UMR6004, Nantes Université, École Centrale Nantes, F-44000 Nantes, France

LS2N - UMR6004, IMT Atlantique, Nantes, France

Keywords:

Modelling and Control of Discrete-Event Systems, Supervision Systems, Systems Synchronisation,

Controlled Invariance, Manufacturing Process Management, Dioids, Periodic Series, Algebraic Toolbox,

C++, Python, Object-Oriented Programming and Scripting Programming.

Abstract:

MinMaxGD is a C++ library developed and maintained since 2000 at the LARIS laboratory at Angers, France.

It provides ad-hoc primitives for calculations so as to handle periodic series over a dioid (namely M

Jγ, δK).

Our aim with PyMinMaxGD is to provide, for this C++ library, a Python interface using SWIG, in order to

allow scripting programming, on top of the already available programming primitives. The use of PyMin-

MaxGD is illustrated here to tackle some general problems of control using the controlled-invariance theory.

The corresponding control problems deal with the meeting of time or marking constraints or even both of

them. First, the set that corresponds to the veriﬁcation of the constraints is deﬁned, and then the supremal

controlled-invariant set included in this speciﬁcation set is calculated. In the sequel, a controller that allows

the state of the system to remain in the latter set is given.

1 INTRODUCTION

To the best of our knowledge, the main library to pro-

vide ad-hoc primitives so as to handle periodic series

over a dioid is MinMaxGD (Hardouin, 2024), a C++

library developed and maintained since 2000 (Cot-

tenceau et al., 2000) at the LARIS laboratory at

Angers, France

. A JavaScript interface, called Min-

MaxGDJS, has also been developed in 2017 (Fer-

reira Cândido et al., 2017), giving the ability to access

the periodic-series library through any Web browser

on any operating system. This makes it easier to get

insight on calculations with periodic series, without

the hassle of conﬁguring and building the original

C++ library. Another interest is to provide a platform

for giving portable demos. Nevertheless, due to its

locked Web-based interface, it lacks the power to em-

bed the calculations within some other programming

processings. Our aim with PyMinMaxGD is then to

https://orcid.org/0009-0000-7292-0672

https://orcid.org/0000-0003-1168-3284

https://orcid.org/0000-0002-5798-0359

See Section 4.1 for detailed information and a compar-

ison between such toolboxes.

provide a different interface for the original library,

meant for Python

and using SWIG (SWIG Maintain-

ers, 2024), an open-source software tool used to con-

nect libraries written in C++ with scripting languages,

in order to allow scripting programming. The pro-

duced code keeps the very same computation beneﬁts

as for the original library, extending its portability to

a scripting programming paradigm.

As a ﬁrst attempt to develop research on the con-

trol of discrete-event systems using the representa-

tion of their behaviour, within the framework of se-

ries in the M

Jγ, δK dioid focusing on the state of

a system rather than its transfer function, some as-

pects of the use of series for the controlled-invariance

theory are presented. The concept of controlled in-

variance, or (A, B)-invariance, has been introduced

independently by (Basile and Marro, 1969) and by

(Wonham and Morse, 1970). The whole geometric

approach for control, based on the then new alge-

braic geometry theory (Grothendieck and Dieudonné,

1960), has started from these works. The solution of

many classical control problems, among which the

What is actually called an "extension module" in the

Python ecosystem.

Boutin, O., Martinez, C. and Rakoto, N.

On Solving Controlled-Invariance Problems in Dioids Using the PyMinMaxGD Python Scripts Library.

DOI: 10.5220/0012936500003822

In Proceedings of the 21st International Conference on Informatics in Control, Automation and Robotics (ICINCO 2024) - Volume 1, pages 555-566

ISBN: 978-989-758-717-7; ISSN: 2184-2809

555

disturbance decoupling problem, the regulator prob-

lem, and various observer design problems, were

given using this controlled-invariance theory – see

for instance (Wonham, 1974) and (Basile and Marro,

1992) for a complete account of these results. The

controlled-invariance theory has been applied also in

the framework of systems over more general algebraic

contexts than ﬁelds, such as rings or semirings. For

the sake of extending the classical results in the latter

structures, the concept of controlled-invariant mod-

ule has been introduced; (Hautus, 1982) and (Conte

and Perdon, 1995) are pioneering references for in-

variance control over a ring.

Finally, systems over a semiring, often called

“Max-Plus

systems”, have been thoroughly studied

in the founding reference (Baccelli et al., 1992). They

have also received attention from the perspective of

controlled-invariance theory, the ﬁrst results being

due to (Katz, 2007). It is stated there that there ex-

ists a maximal controlled-invariant module in every

given module in R

max

and many other semirings. It

is pointed out that the computation of this maximal

set is an open problem in general and important par-

ticular cases where the problem is solvable are identi-

ﬁed. Sufﬁcient conditions that allow this computa-

tion are often met in practice. In (Cárdenas et al.,

2015), it was shown that a feedforward control law

is suitable to maintain the state of a system within the

controlled-invariant set. In (Cárdenas et al., 2017), a

procedure to ﬁnd the greatest controlled-invariant set

included in the admissible set deﬁned by some con-

straints has been presented. There, it has also been

shown that a static non linear state feedback control

law is suitable to maintain the state of a system within

the controlled-invariant set. Some other problems of

control have been tackled in (Martinez et al., 2022)

and (Animobono et al., 2022). All the aforemen-

tioned work on the use of controlled-invariance theory

have been done within the canonical Z

max

and Z

min

dioid frameworks. Based on our information, there

is no research that has tackled the use of controlled-

invariance theory in the context of a dioid of series

such as M

Jγ, δK. The purpose of the control prob-

lems given in the present article is to illustrate, for the

ﬁrst time, that controlled invariance can also be useful

for the class of systems that are modelled with series,

and particularly using the C++ MinMaxGD library.

The remainder of this article is organised as fol-

lows: the following section will brieﬂy deﬁne and

describe the theoretical elements based on which our

modelling framework works; then, Section 3 will in-

troduce how control theory can be applied in this

Often denoted thanks to Z

max

or R

max

for example,

depending on the exact set of studied values.

mathematical context; afterwards, in Section 4, more

insight will be given on other toolboxes for such work

and why none of them suits our needs, as well as more

details on the features offered by our developments;

subsequently, Section 5 will provide two use cases of

our library, based on a typical problem of a physical

system having to follow a speciﬁcation, reinterpreted

as a synchronisation problem and a time constraint

meeting, and within the speciﬁc M

Jγ, δK dioid, in-

cluding detailed Python scripts; ﬁnally, a last section

will give some conclusions and openings of this work.

2 BACKGROUND

2.1 Dioids and Systems Representation

A dioid is a set, denoted D, endowed with two bi-

nary operations, a sum (⊕) and a product (⊗). The

complete deﬁnition of a dioid (or idempotent semir-

ing) can be found, for example, in (Brunsch et al.,

2013, Section 1.3). The identity element of ⊕, ε, is

also called the zero element of D, while the identity

element of ⊗, e, is also called the unit element of D.

Due to the idempotency property, any dioid can

also be endowed with a natural (partial) order deﬁned

by a ⪯ b ⇔ a ⊕ b = b.

As in a classical ﬁeld, the binary operations can

be extended to the matrix case in dioids (Cohen et al.,

1989, p.42). For the matrices A, B ∈ D

n×m

and C ∈

m×p

one gets

(sum) [A ⊕ B]

i j

= [A]

i j

⊕ [B]

i j

;

(product) [A ⊗C]

i j

k=1



[A]

⊗ [C]

k j



Remark 2.1. It is worth noticing that the product of

n×m

is not always commutative, i.e. A ⊗ B may be

different from B ⊗ A.

Let us now introduce a 2-dimensional dioid de-

noted B Jγ, δK (Cohen et al., 1989). It is the set of

formal power series in two variables γ and δ with

Boolean coefﬁcients, i.e B = {ε, e} and the exponents

of variables γ and δ are in Z.

The interpretation

of a monomial γ

is that,

considering a previous monomial γ

′

, the epoch be-

ing γ

, k − k

′

events occur after t −t

′

units of time,

which leads to a compact modelling of information.

Graphically, a series can be seen as a set of dots of a

point cloud in the event-time plane (see Figure 1)

Following the discussion in (Baccelli et al., 1992, Sec-

tion 5.4.4).

Bearing in mind that the time part of the modelling is

now an ordinate.

ICINCO 2024 - 21st International Conference on Informatics in Control, Automation and Robotics

556

Figure 1: Graphical representation of series s = γ

⊕

⊕ γ

, s ∈ B Jγ, δK.

Actually, the “South-East cone” of each point in

the plane matches a corresponding monomial of a

series. This establishes an equivalence relation and

implies that series may have a “minimal” representa-

tion (Cohen et al., 1989, Comment ii) p.51), based on

the “absorption” of some monomials by others. The

resulting dioid of equivalence classes in B Jγ, δK is de-

noted M

Jγ, δK.

Remark 2.2. In order to fully deﬁne M

Jγ, δK, its

zero and unit elements are deﬁned as ε = γ

+∞

−∞

and

e = γ

respectively

(Cohen et al., 1986, p. 991).

In the next section, complex systems described

by M

Jγ, δK series are shown to have their graphi-

cal counterparts in the form of Timed Event Graphs,

which also consists in compact modelling.

2.2 Timed Event Graphs

Timed Event Graphs (TEGs) deﬁne a class of timed

Petri nets where there are no conﬂicts (Ramchandani,

1974, Chapter 4).

Figure 2 shows two examples of TEGs. Please

note that transition t

is downstream of t

, for reasons

out of the scope of this study.

2.3 Modules in M

Jγ, δK

The concept of modules on a semiring is analogous to

the concept of vector spaces on a ﬁeld. Modules over

a semiring are often called semimodules, and modules

over a dioid are sometimes called moduloid, but the

term module shall rather be used in this article, fol-

lowing the track of (Wagneur, 1996). Here, modules

+∞ and −∞ being respectively the zero element of

min

and Z

max

. 0 is their common unit element.

2 2

Figure 2: Two TEGs. The TEG above is meant to be syn-

chronised with a takt time (Haghsheno et al., 2016) given

by the TEG below, thanks to the study to come in Section 5.

and submodules of M

Jγ, δK, and more speciﬁcally

ﬁnitely generated modules, i.e. those that are gener-

ated with a ﬁnite family of vectors of series deﬁned in

Jγ, δK, are of high interest.

Let us consider a matrix M of size n × m with its

entries in M

Jγ, δK, n and m being positive integers.

Let us denote ImM as the submodule of M

Jγ, δK

that is generated by the columns of M, hence ImM =

{x ∈ M

Jγ, δK

| ∃v ∈ M

Jγ, δK

, x = Mv}. By

deﬁnition, a submodule M of M

Jγ, δK

is ﬁnitely

generated if there exists a positive integer q and a ma-

trix M ∈ M

Jγ, δK

n×q

such that M = ImM. Finitely

generated modules over M

Jγ, δK

may be repre-

sented in several ways. One can refer to the origi-

nal results of (Butkovi

c and Hegedüs, 1984) where

it is established that the family of ﬁnitely gener-

ated submodules of R

max

coincides with the family

of ﬁnitely generated cones that are sets of the form

Cone(R, Q) = {x ∈ R

max

| R⊗x = Q⊗x} where R and

Q are matrices of size p×n, p being a positive integer.

This result is applied on submodules of M

Jγ, δK

Theorem 2.1. Given a module M ⊂ M

Jγ, δK

, the

two following assertions are equivalent.

(i) There exist a positive integer q and a matrix M ∈

Jγ, δK

n×q

such that M = ImM.

(ii) There exist a positive integer p and matrices

R, Q ∈ M

Jγ, δK

p×n

such that M = Cone(R, Q).

Algorithms that permit to switch from a cone rep-

resentation to an image representation have been pre-

sented ﬁrst in (Butkovi

c and Hegedüs, 1984), and

reﬁned by Allamigeon et al. in (Allamigeon et al.,

2010).

For any set S ⊂ M

Jγ, δK

, its image by M is

denoted by MS. The preimage by M of any set T ⊂

Jγ, δK

is denoted M

−1

T . The difference of two

sets S , S

′

⊂ M

Jγ, δK

, denoted S ⊖ S

′

is deﬁned as

{x” ∈ M

Jγ, δK

|∃x ∈ S , x

′

∈ S

′

, x” ⊕ x

′

= x}.

On Solving Controlled-Invariance Problems in Dioids Using the PyMinMaxGD Python Scripts Library

557

3 CONTROL PROBLEMS AND

ALGORITHMS

3.1 Controlled Invariance

Let us consider here a controlled system of the form:

x = Ax ⊕ Bu, (1)

where A ∈ M

Jγ, δK

n×n

, B ∈ M

Jγ, δK

n×m

, n and

m are positive integers, and x ∈ M

Jγ, δK

and u ∈

Jγ, δK

. The variable x refers to the state trajec-

tory of the system, and u is its control input.

Note that, in opposition to systems over R

max

there is no speciﬁc mention of the initial state x

be-

cause each entry of the state vector is a series that

represents all the trajectory, including the initial state.

The following deﬁnition of controlled invariance is a

transcription of the deﬁnition stated for systems over

max

, see for instance (Martinez et al., 2022).

Deﬁnition 3.1. A set M is said to be controlled-

invariant if, for every vector x ∈ M , there exists a

control u such that the solution of system (1) is en-

tirely in M .

Research about controlled-invariant sets proper-

ties are still in progress. Nevertheless, some proper-

ties are already well known; for instance, the case of

the submodules of R

max

, or over various other semir-

ings, has particularly been studied. See (Katz, 2007)

and (Di Loreto et al., 2010), for instance.

Proposition 3.1. A set M ⊂ M

Jγ, δK

controlled-invariant if and only if the following

inclusion is satisﬁed: AM ⊂ M ⊖ ImB. Equiva-

lently, M is controlled-invariant if and only if the

following inclusion is satisﬁed:

M ⊂ A

−1

(M ⊖ ImB).

Proof By deﬁnition, the set M is controlled-

invariant if each of its elements x satisﬁes Ax ⊕ Bu ∈

M for some vector u ∈ M

Jγ, δK

. The result fol-

lows since Bu remains in ImB when u varies, and by

deﬁnition of ⊖ (Katz, 2007, Section 2).

Theorem 3.1. The following properties are veri-

ﬁed (Cárdenas et al., 2017).

(i) A module M ⊂ M

Jγ, δK

is controlled-invariant

if and only if:

AM ⊂ M ⊖ ImB .

(ii) A module M ⊂ M

Jγ, δK

generated by the

columns of matrix M ∈ M

Jγ, δK

n×q

is controlled-

invariant if and only if there exist matrices U ∈

Jγ, δK

m×q

and V ∈ M

Jγ, δK

q×q

such that the

following identity is veriﬁed:

AM ⊕BU = MV .

(iii) A module M ⊂ M

Jγ, δK

such that M =

ImM = Cone(R, Q), for matrices M ∈ M

Jγ, δK

n×q

and R, Q ∈ M ∈ M

Jγ, δK

p×n

is controlled-invariant

if and only if there exists a matrix U ∈ M

Jγ, δK

m×q

such that the following equality is veriﬁed:

R(AM ⊕BU) = Q(AM ⊕ BU) .

Not all submodules of M

Jγ, δK

are controlled-

invariant. But one can show that any module M ⊂

Jγ, δK

contains a maximal controlled-invariant

submodule, denoted V

⋆

(A, B). It is shown in

(Di Loreto et al., 2010) that the supremal controlled-

invariant submodule included in a ﬁnitely generated

module is the limit of the sequence {V

} deﬁned as

follows:

= M ; V

i+1

= M ∩ A

−1

⊖ ImB) , for i ≥ 0 .

(2)

The successive terms of the sequence are ﬁnitely gen-

erated modules.

As it is mentioned in (Cárdenas et al., 2017), the

matrices M

and (R

, Q

) can be computed iteratively

such that V

= ImM

= Cone(R

, Q

) seeking for a so-

lution of the following equation:



i−1

A R

i−1

R ε







i−1

A Q

i−1

Q ε





. (3)

At each step, one computes M

and U

and, then,

one computes R

et Q

using theorem 2.1. The limit

of the sequence is the intersection of modules V

Finally, if for some integer i, the equality V

i+1

is true, then the sequence stabilises, which leads

to V

= V

, for every k ≥ i, and therefore V

= V

Following (2), V

= M ∩A

−1

⊖ ImB) can be de-

duced. Thus, using Proposition 3.1, V

can be con-

sidered as controlled-invariant set, which is included

in M and, therefore, it is included into the maximal

controlled-invariant set V

⋆

(A, B).

3.2 Some Control Problems

The ﬁrst problem tackled here and illustrated in the

example in subsection 5.1 is a problem of synchro-

nisation of two systems, say (Σ

) and (Σ

), that are

deﬁned as follows:

(Σ

)

(

x = A ⊗ x ⊕ B⊗ u

y = C ⊗ x

(Σ

)

(

w = D ⊗ w

v = E ⊗w

Several variants of this problem have been studied

and presented in (Martinez et al., 2022) in the R

max

dioid. Systems (Σ

) and (Σ

) are said to have syn-

chronous outputs if their output solutions coincide,

ICINCO 2024 - 21st International Conference on Informatics in Control, Automation and Robotics

558

i.e. y = v. When no input is applied (say u = ε),

systems are said synchronous if y = v. In practical

situations in industry, the notion of synchronisation

is related to the takt time that somehow deﬁnes the

tempo of the factory (Haghsheno et al., 2016). The

problem that is addressed here is that of the forced

synchronisation, which is formulated as follows.

Problem 3.1. (Forced synchronisation)

System (Σ

) and (Σ

) are synchronised if there ex-

ists a control law u that forces synchronisation be-

tween them, i.e. the outputs of both systems (Σ

) and

(Σ

) are equal: y = v. Such a control u is said to be

admissible for the synchronisation of (Σ

) and (Σ

In (Martinez et al., 2022), the forced synchroni-

sation problem was slightly different, seeking for a

given k

, after which synchronisation would be possi-

ble; then the system was called k

-synchronisable. In-

deed, state trajectories that verify the solution of prob-

lem 3.1 can be considered as the target to be reached,

seeking for a set of coreachable x

and u

This extension needs more research work in the

Jγ, δK dioid.

The second problem tackled here and illustrated

in the second example in subsection 5.2 concerns the

veriﬁcation of time constraints related to bounded du-

ration for some operations. This problem has been

addressed in (Kim and Lee, 2016) and continued in

(Jacob and Amari, 2017). Then, it has been formu-

lated as a controlled-invariant problem in (Cárdenas

et al., 2017). In this problem, the constraints limit the

maximal duration of certain tasks; that is to say, the

minimal duration is given by the delay τ associated to

a place of a TEG and the constraint to be veriﬁed is

that the sojourn time of tokens is lower than a certain

max

, considering the following system (Σ):

(Σ)

(

x = A ⊗ x ⊕ B⊗ u ,

y = C ⊗ x .

Problem 3.2. (Constraints veriﬁcation) The series x

that models the state behaviour should verify some

additional constraints of the form:

(duration)

−τ

max

⊗ x

≤ x

(4)

To take into account several constraints, then, for

each of them, one has to write a single inequality as

in (4). These inequalities are gathered in a matrix in-

equality as follows:

E ⊗ x ≤ x.

The set of solutions of this inequality is Im E

⋆

4 TOOLBOXES FOR

CALCULATIONS OVER DIOIDS

4.1 Existing Toolboxes

Almost a dozen of initiatives have been encountered

while looking for a toolbox that could suit our needs.

Table 1 below summarises all these initiatives and

projects, for the ease of comparison

As a matter of facts, none of them actually in-

clude the four following characteristics together, that

are deﬁnitely needed for our research:

1. including both dating and counting dynamics at

the same time;

2. possibility to integrate self-deﬁned routines, like

plug-ins, without having the burden of compiling

the core components every time, for ﬂexibility;

3. possibility to automatise simulations benchmarks,

and re-run parts of them interactively, as with

scripts;

4. interaction with other pieces of software.

For all these reasons, the development of our own

toolbox, described in the following subsection, has

been decided.

4.2 The Proposed Toolbox

The distribution of our piece of software is available

online (Boutin and Martinez, 2024), under GPLv3

license (Free Software Foundation, Inc., 2007). It

mainly consists of a single SWIG interface ﬁle of

about 900 lines of code so far, on top of the Min-

MaxGD library, based on roughly 5 000 lines of

code spread over 11 ﬁles deﬁning 4 classes and

a few high-level functions and constants. It has

been compiled and installed

with the following soft-

ware requirements: 6.1 64-bit (SLTS

) Linux®

kernel; g++ 11.3.0; SWIG 4.0.2; Python 3.11.0; the

libpython3.11-dev and python-dev-is-python3

Linux packages and the matplotlib, numpy, pillow

and kiwisolver Python libraries.

Apart from the implementation of the scripts de-

signed to support the algorithms mentioned in Sec-

Note that symbol B there stands for the Boolean set

endowed with two binary operations “and” and “or” and

only composed of the elements e and ε.

Only once for the inclusion of the core toolbox, as

opposed to the need to compile C++-only code for each

new calculation design or update; then only the user-deﬁned

scripts need to be updated and loaded interactively.

Meaning Super Long Time Support (Civil Infrastruc-

ture Platform™, 2023).

On Solving Controlled-Invariance Problems in Dioids Using the PyMinMaxGD Python Scripts Library

559

Table 1: Comparison of available dioid toolboxes.

Name Implementation Evolution from Available dioids Last version

MinMaxGD (Hardouin, 2024) C++ M

γ, δ

03 Jul. 2024

ContainerMinMaxGD (Corronc, 2013) C++ Fork of MinMaxGD Intervals in Z

min

21 Apr. 2011

MinMaxGDJS (Ferreira Cândido et al.,

2017)

JavaScript wrapping over

C++

Fork of MinMaxGD M

γ, δ

03 May 2019

ETVO (Cottenceau et al., 2022) C++ / WebAssembly

wrapping

Fork of MinMaxGD M

γ, δ

and ET (see (Trunk,

2019, Chap. 5))

20 May 2023

Maxpluspy (Lahaye, 2019) Python Z

max

and “Z

max

-automata” 27 Nov. 2019

PetriTUB (Bednar et al., 2024) Python Z

max

and Z

min

, from Petri nets 14 Mar. 2024

MaxPlus Arithmetic Toolbox (Chance-

lier et al., 2015)

Already included in Sci-

coslab Toolbox

Fork of Scilab 4 Z

max

03 Oct. 2015

MaxPlus.jl (Quadrat, 2024) Julia Port from MaxPlus

Arithmetic Toolbox

max

and Z

min

30 Apr. 2024

Max-Plus Algebra Toolbox (Sta

nczyk,

2016)

Matlab/Octave plug-in Z

max

and Z

min

14 Jun. 2016

TropicalNumbers (Liu, 2023) Julia Z

max

, Z

min

, (R

, max, ×) and B 26 Sep. 2023

tion 3, our toolbox offers the following new function-

alities compared to the original MinMaxGD toolbox:

• possibility to type user-friendly symbols (+, *, /,

) for matrix calculations and the + and * sym-

bols for the sum and the product between either

monomials or polynomials, instead of using ver-

bose functions names with parameters;

• comparison via <, <=, > or >= for elements belong-

ing either to series or series matrices;

• series matrices concatenation, if compatible

(when they share the same number of rows);

• matrix import/export from raw text ﬁles based on

a dedicated syntax;

• use of actual γ and δ symbols when displaying

monomials, as well as the ε symbol for the zero el-

ement with a precision of the corresponding scope

(monomial, polynomial or series);

• ﬁgure generation for the graphical representation

of polynomials in the form of “point clouds” and

series in the form of “step functions” (with the op-

tion of showing several series in the same ﬁgure).

All the ﬁgures shown in this article were actually

produced thanks to our toolbox;

As of 6 September 2024.

Used as “left division”, for commutativity issues

exposed in Remark 2.1. The \ character cannot not be

overloaded, as it is reserved in Python for line joining

or escape sequence speciﬁcations. See docs.python.org/

3/reference/lexical_analysis.html#explicit-line-joining

and docs.python.org/3/reference/lexical_analysis.html#

escape-sequences

• user-friendly deep copy of matrices instead of

standard Python shallow copies

, when needed.

All these features are embedded in current version

1.2 of our toolbox, together with a SWIG typemap

between the internal C++ 2-element array, for the

cells of series matrices, and a Python 2-tuple to en-

able user-friendly assignment syntax (of the kind

M[row,col] = given_series), as in the original

C++ code. For instance, the point cloud of Figure 1

can be obtained thanks to the following simple script:

# Toolbox functionalities loading first

from pyminmaxgd import *

# Then a call to the high-level routine

point_cloud(mono(1,1) + mono(3,2) + mono(4,5))

Two examples of use cases of our toolbox are pre-

sented in the following section.

5 EXAMPLES

5.1 Synchronising Production with a

Takt Time Clock

Some problems of synchronisation have been pre-

sented in (Martinez et al., 2022) and (Animobono

et al., 2022). An example of synchronisation of a

simple production plant that has to produce items at

a given pace is presented here. The “physical” sys-

tem Σ

and the takt time clock Σ

(its speciﬁcation)

are modelled by TEGs, represented in Figure 2. The

following systems of equation in dioid M

Jγ, δK is

used to describe their behaviour:

See docs.python.org/3/library/copy.html.

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560

(Σ

)

x = A ⊗ x ⊕ B⊗ u ,

(Σ

)

w = A

⊗ w,

with

A =

, B =

and







ε ε γ

ε ε

ε γ







The behaviour of some internal transitions of the

TEGs of Figure 2 are represented in Figure 3 (x

is the

state representation of transition t

Figure 3: State behaviour of systems internal transitions.

In order to solve the problem of synchronisation,

let us ﬁrst build an extended system Σ with matrices

and B

as follows:







ε ε ε

ε ε ε ε γ

ε ε γ

ε ε

ε ε ε γ



















The state X of Σ has to remain in a module K =

ImK that veriﬁes the following constraint:

S ⊗ X ≤ T ⊗ X,

with S and T as follows:

S =

ε ε e ε ε

e e ε ε ε

T =

e e ε ε ε

ε ε e ε ε

Therefore K can be obtained:

K =







ε ε e ε

ε ε ε e

ε ε e e

e ε ε ε

ε e ε ε







of which module K is not controlled-invariant.

Hence, the sequence described in equation (3) is ap-

plied and the supremal controlled-invariant module

included in the module K is M

= ImM

, with M

described below.







ε e ε e ε e e

e ε e ε e ε ε

e e e e e e e







It also permits to ﬁnd matrix U





And ﬁnally, a control law u = FX can be found.

The corresponding feedback is obtained with U

F ⊗ M

F =



ε γ



. (5)

See in Figure 4 the controller that has been added to

the original TEG of Figure 2.

After applying the controller, the new behaviour

of the internal transitions of the system are repre-

sented in Figure 5. One can easily see that the dy-

namics of the transitions of the upper TEG of Figure 2

match as closely as possible

the cyclic behaviour of

the lower TEG.

5.2 Verifying Time Constraints

Problems of duration control are relatively frequent

in industry, one of them concerns the control of semi-

conductor wafer fabrication with cluster tools (Kim

Dynamics of t

and t

actually coincide.

On Solving Controlled-Invariance Problems in Dioids Using the PyMinMaxGD Python Scripts Library

561

2 2

Figure 4: The upper TEG of Figure 2 is synchronised with

a takt time thanks to a feedback (in red).

and Lee, 2016; Jacob and Amari, 2017; Cárdenas

et al., 2017). Some operations that occur in process

modules, or chambers, have a limitation in terms of

duration variation or a maximal duration that should

not be exceeded. This kind of problems has al-

ready been tackled as a controlled-invariant problem

in (Cárdenas et al., 2017) in dioid R

max

. The exam-

ple presented here have also been solved in (Kim and

Lee, 2016; Jacob and Amari, 2017). The example

given in (Cárdenas et al., 2017) is transposed here

into the M

Jγ, δK dioid and solved with scripts writ-

ten thanks to our toolbox. The system given hereafter

has to verify time constraints given as an inequality.

The goal is to seek the supremal controlled-invariant

set included in the set of solution of that constraint

inequality. The systems behaviour follows (1) with:

A =







ε γ

100

ε ε ε γ

280

ε ε

ε γ

115

ε ε ε γ

295

ε ε

ε γ

ε ε ε γ

240

ε ε

ε γ

ε ε ε γ

255

ε ε

ε ε ε ε e ε ε ε

ε e ε ε ε ε ε ε

ε ε ε e ε ε ε ε

ε ε ε ε ε ε e ε







, B =













The state trajectories have to verify the constraints

expressed by the following inequality:

E ⊗ x ≤ x,

with matrix E:

E =







ε ε ε ε ε ε ε ε

−110

ε ε ε ε ε ε ε

ε ε ε ε ε ε ε ε

ε ε γ

−250

ε ε ε ε ε







the set of solution is ImE

⋆

with

⋆







e ε ε ε ε ε ε ε

ε e ε ε ε ε ε ε

ε ε e ε ε ε ε ε

ε ε ε e ε ε ε ε

ε ε ε ε e ε ε ε

−110

ε ε ε ε e ε ε

ε ε ε ε ε ε e ε

ε ε γ

−250

ε ε ε ε e







By calculation, the supremal controlled-invariant

module is M

= ImM

, M

being described in (6).

is also feedback controlled-invariant, so U

F ⊗ M

F =



ε γ

115

ε γ

180

295

ε ε ε



(7)

Two simulations are presented in Figure 6. The

initial state do not belong to M

and there is no feed-

back applied. Let us observe that there is a variation

of the sojourn time between x

and x

and its duration

may exceed 110 time units. In Figure 7, the initial

state belongs to M

and feedback F is applied. Ob-

serve the regularity of the sojourn time between x

and x

and its duration never exceeds 110 time units.

5.3 Scripting with PyMinMaxGD

All the calculations and ﬁgures presented in this sec-

tion have been realised using Python scripts within

our toolbox and are available in our distribution.

For instance, the high-level script that gives the

results shown in Section 5.1 is the following one:

from pyminmaxgd import *

# CIlib: controlled-invariance library

from CIlib import eye, mpsolve

from CIlib import computeModule, normalize

from CIlib import computeVM, fb_calcul

# Model of the production system

A = smatrix(2, 2)

# File definition of the matrix cells

load(A, "A.sm")

# Only entry for the production system

B = smatrix(2, 1)

B[1, 0] = mono(0, 0)

# Model of the specifications to follow

Ar = smatrix(3, 3)

load(Ar, "Ar.sm")

# Extended matrices,

# including both systems

Ab = smatrix(A, smatrix(2, 3))

Arr = smatrix(smatrix(3, 2), Ar)

Abt = smatrix(transpose(Ab), transpose(Arr))

Ab = transpose(Abt)

Bbt = smatrix(transpose(B), smatrix(1, 3))

Bb = transpose(Bbt)

# x has to remain in the module

# defined by S x <= T x

S = smatrix(2, 5)

load(S, "S.sm")

T = smatrix(2, 5)

load(T, "T.sm")

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562







ε ε ε ε ε ε ε ε ε ε ε e ε

ε ε ε e e e e e e e γ

−35

ε ε

ε ε ε ε ε ε ε ε ε ε ε ε e

ε ε ε γ

−130

−120

−95

−30

−55

−30

−20

e ε ε

ε ε ε ε γ

−170

ε γ

−170

ε ε γ

−225

−205

ε ε

e ε ε ε ε ε ε ε ε ε ε γ

−110

ε e ε γ

−245

−180

−245

−235

−215

ε ε

ε ε e ε ε ε ε ε ε ε ε ε γ

−250







(6)

Figure 5: State behaviour of systems internal transitions,

after synchronisation.

Figure 6: State behaviour of systems internal transitions x

and x

# Computation of the admissible domain,

# i.e. where x should remain in ImS0,

# solution of S x <= T x,

S0 = mpsolve(S, T, eye(5, 5))

# Computation of the Module,

# thanks to Theorem 3.1

module = computeModule(S0, Ab, Bb)

# Normalization of the module

# in order to reduce its size

normalModule = normalize(module)

# Computation of V_M^

(A,B)

# This function is not detailed here.

# See the reference [Cárdenas et al.

# '17] for more details

# alpha is a kind of starting point

Figure 7: State behaviour of systems internal transitions x

and x

. Controller F is applied to the system, initial state

starts in M

alpha = mono(0, 29)

M, V = computeVM(normalModule, Ab, Bb, alpha)

# The corresponding static feedback

F = fb_calcul(M, V, Ab, Bb)

print("F=\n", F, sep="")

save(F, "F.sm")

The possibility to load matrices from external ﬁles

has been used in the previous script, in order to sim-

plify it. In the end, we store the feedback result in a

new ﬁle, for the sake of the persistence of the data.

The ﬁle contents of the four previously loaded

series-matrix are displayed below.

Series matrix A :

2 x 2

| 0 | 1 |

0 | (1,2)[e]* | (1,2)[e]* |

1 | (1,3)[e]* | (1,2)[e]* |

Series matrix Ar :

3 x 3

| 0 | 1 | 2 |

0 | eps | eps | (1,1)[e]* |

1 | (1,7)[e]* | eps | eps |

2 | eps | (1,2)[e]* | eps |

Series matrix S :

2 x 5

| 0 | 1 | 2 | 3 | 4 |

0 | eps | eps | (e)[e]* | eps | eps |

1 | (e)[e]* | (e)[e]* | eps | eps | eps |

Series matrix T :

On Solving Controlled-Invariance Problems in Dioids Using the PyMinMaxGD Python Scripts Library

563

2 x 5

| 0 | 1 | 2 | 3 | 4 |

0 | (e)[e]* | (e)[e]* | eps | eps | eps |

1 | eps | eps | (e)[e]* | eps | eps |

The last export of the ﬁrst script indeed yields the

same result as the one shown in (5):

1 x 5

| 0 | 1 | 2 | 3 | 4 |

0 | (1,3)[e]* | (1,2)[e]* | (1,2)[e]* | eps | (1,1)[e]* |

As for the high-level script that gives the results

shown in Section 5.2, its content is as follows:

# In this file we tackle the control

# of DESs that has to meet time

# constraints: example of Jae Kim and

# Tae-Eog Lee, published in 2003

from pyminmaxgd import *

# CIlib: controlled-invariance library

from CIlib import computeModule, normalize

from CIlib import computeVM, fb_calcul

from CIlib import ones, Includespan

from CIlib import getfullcol

A = smatrix(8, 8)

load(A, "A2.sm")

B = smatrix(8, 1)

B[1, 0] = mono(0, 0)

E = smatrix(8, 8)

load(E, "E2.sm")

print("E=", E)

Es = star(E)

print("E*=", Es)

# Computation of the Module

module = computeModule(Es, A, B)

normalModule = normalize(module)

alpha = mono(-30, 300)

M, V = computeVM(module, A, B, alpha,

memorySafe=True)

F = fb_calcul(M, V, A, B)

print("F=\n", F, sep="")

save(F, "F2.sm")

# Now some simulations using F

answer = Includespan(M, A * M + B * F * M)

print("Is M (A+B*F) invariant ?", answer)

x0 = ones(A.getcol(), 1)

print("with x0=\n", x0)

print("Is x0 in ImM ?", Includespan(M, x0))

xk1 = A * x0

xk2 = A * xk1

xk3 = A * xk2

xk4 = A * xk3

xk5 = A * xk4

xk6 = A * xk5

xk7 = A * xk6

xk8 = A * xk7

xk9 = A * xk8

s = x0 + xk1 + xk2 + xk3 + xk4 + xk5

s = s + xk6 + xk7 + xk8 + xk9

print("Sequence s from x0 out of ImM:", s)

answer = Includespan(M, s)

print("x remains in ImM w/o fb?", answer)

ColA = A.getcol()

x1 = getfullcol(M, 0)

x2 = getfullcol(M, 1)

V4 = smatrix(ColA, 1)

V4[1, 0] = mono(0, 180)

V4[3, 0] = mono(0, 60)

V4[4, 0] = mono(0, 10)

V4[6, 0] = mono(0, 0)

V5 = smatrix(ColA, 1)

V5[0, 0] = mono(0, 110)

V5[5, 0] = mono(0, 0)

V6 = smatrix(ColA, 1)

V6[2, 0] = mono(0, 250)

V6[7, 0] = mono(0, 0)

x0 = x1 + x2 + V4 + V5 + V6

print("with x0=\n", x0)

print("Is x0 in ImM ?", Includespan(M, x0))

Abf = A + B * F

xl1 = Abf * x0

xl2 = Abf * xl1

xl3 = Abf * xl2

xl4 = Abf * xl3

xl5 = Abf * xl4

xl6 = Abf * xl5

xl7 = Abf * xl6

xl8 = Abf * xl7

xl9 = Abf * xl8

l = x0 + xl1 + xl2 + xl3 + xl4 + xl5

l = l + xl6 + xl7 + xl8 + xl9

print("Sequence l starting from x0 in ImM", l)

answer = Includespan(M, l)

print("x remains in ImM w/ fb ?", answer)

# Should yield True

blist = [s[0, 0], s[5, 0]]

glist = [l[0, 0], l[5, 0]]

legend = ["x1", "x6"]

title1 = "Without controller"

title2 = "With controller F"

multidraw(blist, legend, mono(5, 1000), title1)

multidraw(glist, legend, mono(5, 1000), title2)

A special Boolean parameter memorySafe has

been used in order to call a memory-safe internal al-

gorithm for the RAM size. Please note the use of the

multidraw() function, in order to draw several series

trajectories at the same time in the very same ﬁgure.

The last export of the second main script indeed

yields the same result as the one shown in (7):

ICINCO 2024 - 21st International Conference on Informatics in Control, Automation and Robotics

564

1 x 8

| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |

0 | eps | (1,115)[e]* | eps | (1,180)[e]* | (1,295)[e]* | eps | eps | eps |

The ﬁle contents of the two previously loaded

series-matrix are displayed below.

Series matrix A :

8 x 8

| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |

0 | eps | (1,100)[e]* | eps | eps | (1,280)[e]* | eps | eps | eps |

1 | eps | (1,115)[e]* | eps | eps | (1,295)[e]* | eps | eps | eps |

2 | eps | (1,5)[e]* | eps | eps | (1,240)[e]* | eps | eps | eps |

3 | eps | (1,20)[e]* | eps | eps | (1,255)[e]* | eps | eps | eps |

4 | eps | eps | eps | (1,0)[e]* | eps | eps | eps | eps |

5 | eps | (1,0)[e]* | eps | eps | eps | eps | eps | eps |

6 | eps | eps | eps | (1,0)[e]* | eps | eps | eps | eps |

7 | eps | eps | eps | eps | eps | eps | (1,0)[e]* | eps |

Series matrix E :

8 x 8

| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |

0 | eps | eps | eps | eps | eps | eps | eps | eps |

1 | eps | eps | eps | eps | eps | eps | eps | eps |

2 | eps | eps | eps | eps | eps | eps | eps | eps |

3 | eps | eps | eps | eps | eps | eps | eps | eps |

4 | eps | eps | eps | eps | eps | eps | eps | eps |

5 | (0,-110)[e]* | eps | eps | eps | eps | eps | eps | eps |

6 | eps | eps | eps | eps | eps | eps | eps | eps |

7 | eps | eps | (0,-250)[e]* | eps | eps | eps | eps | eps |

6 CONCLUSION

To the best of our knowledge, nobody had applied the

controlled-invariance theory on systems described in

Jγ, δK ever before.

Thanks to PyMinMaxGD, the Python toolbox de-

veloped on top of the C++ MinMaxGD library, it is

now possible to easily compute a control law that en-

forces a takt time for a given production system. With

the same library a solution of a problem of time con-

straints control expressed in the M

Jγ, δK dioid has

been given. More examples and problems will be in-

vestigated using the same framework. Some other fu-

ture work will also deal with the compatibility and

compilation of our toolbox on main other operating

systems (OS’s), namely macOS® and Windows®. In

the meantime, a ﬁrst workaround for users of these

OS’s could be to install a Linux virtual machine on

their computers, which should work ﬁne with nowa-

days technology.

ACKNOWLEDGEMENTS

Linux is the registered trademark of Linus Torvalds in

the U.S. and other countries.

For the purpose of Open Access, a CC-

BY public copyright licence (available at https://

creativecommons.org/licenses/by/4.0/) has been ap-

plied by the authors to the present document and will

be applied to all subsequent versions up to the Author

Accepted Manuscript arising from this submission.

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