Modelling and control of periodic time-variant event graphs in dioids

Abstract

Timed Event Graphs (TEGs) can be described by time invariant (max,+) linear systems. This formalism has been studied for modelling, analysis and control synthesis for decision-free timed Discrete Event Systems (DESs), for instance specific manufacturing processes or transportation networks operating under a given logical schedule. However, many applications exhibit time-variant behaviour, which cannot be modelled in a standard TEG framework. In this paper we extend the class of TEGs in order to include certain periodic time-variant behaviours. This extended class of TEGs is called Periodic Time-variant Event Graphs (PTEGs). It is shown that the input-output behaviour of these systems can be described by means of ultimately periodic series in a dioid of formal power series. These series represent transfer functions of PTEGs and are a convenient basis for performance analysis and controller synthesis.

Appendices

Appendix A: Formula of residuation

In a complete dioid, the following formula hold for the residuation of left and right multiplication see Baccelli et al. (1992, Chap.4).

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Appendix B: Formula for floor and ceil operations (Graham et al. 1989)

For $x\in \mathbb {R}$,

$$ \begin{array}{@{}rcl@{}} \left\lfloor \lfloor x \rfloor \right\rfloor = \lfloor x \rfloor,& \qquad \lceil \lceil x \rceil \rceil = \lceil x \rceil. \end{array} $$

For $x\in \mathbb {R}$, $m\in \mathbb {Z}$ and $n \in \mathbb {N}$,

$$ \begin{array}{@{}rcl@{}} \left\lfloor\frac{x+m}{n}\right\rfloor = \left\lfloor\frac{\lfloor x\rfloor +m}{n}\right\rfloor,& \qquad \left\lceil\frac{\vphantom{\lceil x\rceil}x+m}{n}\right\rceil = \left\lceil\frac{\lceil x\rceil +m}{n}\right\rceil. \end{array} $$

For $m\in \mathbb {Z}$ and $n \in \mathbb {N}$,

$$ \begin{array}{@{}rcl@{}} \left\lfloor \frac{m}{n} \right\rfloor = \left\lceil \frac{m-n+1}{n} \right\rceil,& \qquad \left\lceil\frac{\vphantom{\lceil x\rceil}m}{n} \right\rceil = \left\lfloor \frac{m+n-1}{n} \right\rfloor. \end{array} $$

Appendix C: Proofs

1.1 C.1 Proof of Proposition 1 (relations between T-operators)

Let us recall that $y\in \mathbb {R},\ \forall n \in \mathbb {Z}, \lceil y+n \rceil = \lceil y \rceil + n$. To prove Eq. 16, because of Definition 6, ∀x ∈Σ,

$$ \begin{array}{@{}rcl@{}} \left( {\Delta}_{\omega}\delta^{\varsigma} x\right)(k) &=& \left \lceil \frac{x(k)+\varsigma}{\omega} \right\rceil \omega = \left\lceil \frac{x(k)}{\omega} + \frac{\varsigma}{\omega} +\left\lceil \frac{\varsigma}{\omega}\right\rceil - \left\lceil\frac{\varsigma}{\omega}\right\rceil \right\rceil \omega\\ &=&\left\lceil \frac{\varsigma}{\omega} \right\rceil \omega + \left\lceil \frac{x(k) + \varsigma - \omega \lceil(\varsigma\slash \omega) \rceil}{\omega} \right\rceil \omega\\ &=& \left( \delta^{\lceil\frac{\varsigma}{\omega} \rceil\omega}{\Delta}_{\omega}\delta^{\varsigma-\lceil\frac{\varsigma}{\omega}\rceil\omega}x\right)(k). \end{array} $$

Second,

$$ \begin{array}{@{}rcl@{}} \left( \delta^{\varsigma}{\Delta}_{\omega} x \right)(k) &=& \varsigma + \left \lceil \frac{x(k)}{\omega} \right\rceil \omega \\ &=& \varsigma - \left\lceil \frac{\varsigma}{\omega} \right\rceil \omega + \left\lceil \frac{\varsigma}{\omega} \right\rceil \omega + \left \lceil \frac{x(k)}{\omega} \right\rceil \omega \\ &=& \varsigma - \left\lceil \frac{\varsigma}{\omega} \right\rceil \omega + \left \lceil \frac{x(k)+\lceil\varsigma\slash\omega \rceil\omega}{\omega} \right\rceil \omega \\&=& \left( \delta^{\varsigma-\lceil\frac{\varsigma}{\omega}\rceil\omega}{\Delta}_{\omega}\delta^{\lceil\frac{\varsigma}{\omega} \rceil\omega} x\right)(k). \end{array} $$

To prove Eq. 17, note that ⌈(a + ς)/ω⌉ω = ⌈ς/ω⌉ω + ⌈(a + ς − ω⌈ς/ω⌉)/ω⌉ω, and therefore

$$ \begin{array}{@{}rcl@{}} \left( {\Delta}_{\omega} \delta^{\varsigma} {\Delta}_{\omega} x\right)(k) &=& \left\lceil \frac{\lceil x(k)\slash \omega \rceil \omega + \varsigma }{\omega} \right\rceil \omega \\ &=& \left\lceil \frac{\varsigma}{\omega} \right\rceil \omega + \left\lceil \left\lceil \frac{ x(k)}{ \omega} \right\rceil + \frac{ \varsigma -\omega\lceil\varsigma \slash \omega \rceil }{\omega} \right\rceil \omega \end{array} $$

since: $\lceil x(k)\slash \omega \rceil \in \mathbb {Z}$ and − 1 < (ς − ω⌈ς/ω⌉)/ω ≤ 0, finally,

$$ \begin{array}{@{}rcl@{}} \left( {\Delta}_{\omega} \delta^{\varsigma} {\Delta}_{\omega} x\right)(k) = \left\lceil \frac{\varsigma}{\omega} \right\rceil \omega + \left\lceil \frac{ x(k)}{ \omega} \right\rceil \omega = \left( \delta^{\lceil\frac{\varsigma}{\omega}\rceil\omega}{\Delta}_{\omega} x\right)(k). \end{array} $$

1.2 C.2 Proof of Proposition 2 (operator representation of a release-time-function)

First recall that release-time-functions are nondecreasing. Hence, in Eq. 9, n_ω− 1 − ω ≤ n₀ ≤ n₁ ≤⋯ ≤ n_ω− 1 ≤ n₀ + ω. Moreover, recall that the release-time-function $\mathcal {R}_{\delta ^{\varsigma }{\Delta }_{\omega } \delta ^{\varsigma ^{\prime }}}(\xi )$ of an operator $\delta ^{\varsigma }{\Delta }_{\omega } \delta ^{\varsigma ^{\prime }}$ is defined by

$$ \begin{array}{@{}rcl@{}} \mathcal{R}_{\delta^{\varsigma}{\Delta}_{\omega}\delta^{\varsigma^{\prime}}}(\xi) = \varsigma + \lceil (\xi+\varsigma^{\prime} ) \slash \omega \rceil \omega, \end{array} $$

where ξ = x(k) is a date. Thus, $\mathcal {R}_{p}$ associated with Eq. 21 is

$$ \begin{array}{@{}rcl@{}} \mathcal{R}_{p}(\xi) &=& \max(n_{0} + \lceil (\xi-(\omega-1)) \slash \omega \rceil \omega,n_{1}-\omega + \lceil \xi \slash \omega \rceil \omega, \\ &&\cdots, n_{\omega\text{-}1}-\omega + \lceil (\xi-(\omega-2)) \slash \omega \rceil \omega ). \end{array} $$

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We can evaluate the expression Eq. 35 for all dates ξ. If we choose $\xi = j\omega ,\ \ \forall j\in \mathbb {Z}_{max}$, we have

$$ \begin{array}{@{}rcl@{}} \mathcal{R}_{p}(j\omega) &=& \max(n_{0} + \lceil (j\omega-(\omega-1)) \slash \omega \rceil \omega,n_{1}-\omega + \lceil j\omega \slash \omega \rceil \omega, \\ &&\cdots, n_{\omega\text{-}1}-\omega + \lceil (j\omega-(\omega-2)) \slash \omega \rceil \omega ) \\ &=& \max(n_{0} + j\omega ,n_{1}-\omega + j\omega, \cdots, n_{\omega\text{-}1}-\omega + j\omega) \\ &=& n_{0}+j\omega. \end{array} $$

Similarly ∀i = {1,⋯ , (ω − 1)},

$$ \begin{array}{@{}rcl@{}} \mathcal{R}_{p}(i+j\omega) &=& \max(n_{0} + \lceil (i+j\omega-(\omega-1)) \slash \omega \rceil \omega,\\&& n_{1}-\omega + \lceil(i+j\omega) \slash \omega \rceil \omega, \\ && \cdots, n_{\omega\text{-}1}-\omega + \lceil (i+j\omega-(\omega-2)) \slash \omega \rceil \omega ) \\ &=& n_{i}+\left\lceil (i+j\omega-(\omega-1))/\omega\right\rceil\omega = n_{i}+j\omega. \end{array} $$

Hence we have shown that,

$$ \begin{array}{@{}rcl@{}} \mathcal{R}_{p}(\xi) = \left\{\begin{array}{llll} n_{0}+\omega j &\text{if } \xi = 0+\omega j, \\ n_{1}+\omega j &\text{if }\xi = 1+\omega j, \\ \qquad {\vdots} \\ n_{\omega-1}+\omega j &\text{if }\xi = (\omega-1)+\omega j. \end{array}\right. \end{array} $$

1.3 C.3 Proof of Proposition 5 (product of polynomials)

Due to Eq. 23$p_{1}={\bigoplus }_{i=1}^{I}v_i\gamma ^{n_i}$ and $p_{2}={\bigoplus }_{l=1}^{L}\bar {v}_l\gamma ^{\nu _l}$ can be expressed with a common period ω = lcm(ω₁,ω₂):

$$ \begin{array}{@{}rcl@{}} p_{1} = {\bigoplus}_{i=1}^{I}\left( {\bigoplus}_{j=1}^{J_{i}}\delta^{\varsigma_{i_{j}}}{\Delta}_{\omega}\delta^{\varsigma_{i_{j}}^{\prime}} \right)\gamma^{n_{i}}, \quad p_{2} ={\bigoplus}_{l=1}^{L}\left( {\bigoplus}_{k=1}^{K_{l}}\delta^{\tau_{l_{k}}}{\Delta}_{\omega}\delta^{\tau_{l_{k}}^{\prime}} \right)\gamma^{\nu_{l}}. \end{array} $$

Then the product is obtained by

$$ \begin{array}{@{}rcl@{}} p_{1}\otimes p_{2} &=& \left( {\bigoplus}_{i=1}^{I}\left( {\bigoplus}_{j=1}^{J_{i}}\delta^{\varsigma_{i_{j}}}{\Delta}_{\omega}\delta^{\varsigma_{i_{j}}^{\prime}} \right)\gamma^{n_{i}} \right)\left( {\bigoplus}_{l=1}^{L}\left( {\bigoplus}_{k=1}^{K_{l}}\delta^{\tau_{l_{k}}}{\Delta}_{\omega}\delta^{\tau_{l_{k}}^{\prime}} \right)\gamma^{\nu_{l}} \right) \\ &=& {\bigoplus}_{i=1}^{I}{\bigoplus}_{l=1}^{L}\left( \left( {\bigoplus}_{j=1}^{J_{i}}\delta^{\varsigma_{i_{j}}}{\Delta}_{\omega}\delta^{\varsigma_{i_{j}}^{\prime}} \right)\left( {\bigoplus}_{k=1}^{K_{l}}\delta^{\tau_{l_{k}}}{\Delta}_{\omega}\delta^{\tau_{l_{k}}^{\prime}} \right)\right) \gamma^{n_{i}+\nu_{l}}\\ &=& {\bigoplus}_{i=1}^{I}{\bigoplus}_{l=1}^{L}\left( {\bigoplus}_{j=1}^{J_{i}}{\bigoplus}_{k=1}^{K_{l}}\delta^{\varsigma_{i_{j}}}{\Delta}_{\omega}\delta^{\varsigma_{i_{j}}^{\prime}} \delta^{\tau_{l_{k}}}{\Delta}_{\omega}\delta^{\tau_{l_{k}}^{\prime}} \right) \gamma^{n_{i}+\nu_{l}} \\ &=& {\bigoplus}_{i=1}^{I}{\bigoplus}_{l=1}^{L}\left( {\bigoplus}_{j=1}^{J_{i}}{\bigoplus}_{k=1}^{K_{l}}\delta^{\varsigma_{i_{j}}+\lceil (\varsigma_{i_{j}}^{\prime}+\tau_{l_{k}})/\omega \rceil \omega }{\Delta}_{\omega}\delta^{\tau_{l_{k}}^{\prime}} \right) \gamma^{n_{i}+\nu_{l}}, \end{array} $$

with J_i ≤ ω, K_l ≤ ω and complexity $\mathcal {O}(2\omega I L)$.

1.4 C.4 Proof of Lemma 1 (ultimate domination)

Recall that $(\gamma ^{\nu } \delta ^{\tau })^{*}\delta ^{\varsigma }{\Delta }_{\omega } \delta ^{\varsigma ^{\prime }}= \delta ^{\varsigma }{\Delta }_{\omega } \delta ^{\varsigma ^{\prime }}(\gamma ^{\nu } \delta ^{\tau })^{*}$ (Proposition 4, therefore $\tau _{1} = k_{1}\omega , \ k_{1}\in \mathbb {N}$ (resp. $\tau _{2} = k_{2}\omega , k_{2}\in \mathbb {N}$) and inequality Eq. 26 can be expressed by

$$ \begin{array}{@{}rcl@{}} \underset{j\geq K}{\bigoplus} \delta^{\varsigma_{2}+j\tau_{2}}{\Delta}_{\omega}\delta^{\varsigma_{2}^{\prime}}\gamma^{n_{2}+j\nu_{2}}\preceq \underset{i\geq 0}{\bigoplus} \delta^{\varsigma_{1}+i\tau_{1}}{\Delta}_{\omega}\delta^{\varsigma_{1}^{\prime}}\gamma^{n_{1}+i\nu_{1}}. \end{array} $$

It exists a positive integer K such that inequality Eq. 26 holds, if and only if $x \in \mathbb {N},\forall x \geq K,\ \exists y \in \mathbb {N}$ such that

$$ \begin{array}{@{}rcl@{}} \delta^{x\tau_{2}}\delta^{\varsigma_{2}}{\Delta}_{\omega}\delta^{\varsigma_{2}^{\prime}} \preceq \delta^{y\tau_{1}}\delta^{\varsigma_{1}}{\Delta}_{\omega}\delta^{\varsigma_{1}^{\prime}};\ \ n_{2}+x\nu_{2} \geq n_{1}+y\nu_{1}. \end{array} $$

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Since $\delta ^{\varsigma _{1}}{\Delta }_{\omega } \delta ^{{\varsigma }_{1}^{\prime }}$ and $\delta ^{\varsigma _{2}}{\Delta }_{\omega } \delta ^{{\varsigma }_{2}^{\prime }}$ are assumed to be canonical monomials then $\varsigma _{1}^{\prime }<\omega$ and $\varsigma _{2}^{\prime }<\omega$. Furthermore, since s₁ is in the commute form τ₁ is a multiple of ω and therefore $\tau _{1}+\varsigma _{1}^{\prime }>\varsigma _{2}^{\prime }$. We can now rewrite Eq. 36,

$$ \begin{array}{@{}rcl@{}} &&\delta^{x\tau_{2}}\delta^{\varsigma_{2}}{\Delta}_{\omega}\delta^{\varsigma_{2}^{\prime}} \preceq \delta^{(y-1)\tau_{1}}\delta^{\varsigma_{1}}{\Delta}_{\omega}\delta^{\varsigma_{1}^{\prime}+\tau_{1}};\ n_{2}+x\nu_{2} \geq n_{1}+y\nu_{1} \\ &\Leftrightarrow\ &\varsigma_{2}+x\tau_{2} \leq \varsigma_{1}+(y-1)\tau_{1}; \ \ n_{2}+x\nu_{2} \geq n_{1}+y\nu_{1} \\ &\Leftrightarrow\ &\frac{\varsigma_{2}+x\tau_{2} -\varsigma_{1}+\tau_{1}}{\tau_{1}}\leq y \leq \frac{n_{2}+x\nu_{2} - n_{1}}{\nu_{1}}. \end{array} $$

Such an integer $y \in \mathbb {Z}$ exists, if

$$ \begin{array}{@{}rcl@{}} 1 \leq \frac{n_{2}+x\nu_{2} - n_{1}}{\nu_{1}}-\frac{\varsigma_{2}+x\tau_{2} - \varsigma_{1}+\tau_{1}}{\tau_{1}}. \end{array} $$

This holds for a sufficiently large x, given by

$$ \begin{array}{@{}rcl@{}} x \geq K_{1} = \left\lceil\frac{2\nu_{1}\tau_{1}+\nu_{1}(\varsigma_{2}-\varsigma_{1})+\tau_{1}(n_{1}-n_{2})}{\tau_{1}\nu_{2}-\tau_{2}\nu_{1}} \right\rceil. \end{array} $$

In addition y has to be positive, which is guaranteed, if $x \geq K_{2}= \left \lceil (n_{1}-n_{2})\slash v_{2} \right \rceil$. Hence, we can give an upper bound for K in Eq. 26, i.e., $K = \max \limits \left (0,K_{1},K_{2}\right )$.

1.5 C.5 Proof of Proposition 6 (sum of ultimately periodic series)

We distinguish two cases first: σ(s₁) = σ(s₂). By defining N = lcm(ν₁,ν₂) = k₁ν₁ = k₂ν₂ and T = k₁τ₁ = k₂τ₂, then $(\gamma ^{\nu _{1}}\delta ^{\tau _{1}})^{*}$ and $(\gamma ^{\nu _{2}}\delta ^{\tau _{2}})^{*}$ can be written as

$$ \begin{array}{@{}rcl@{}} q_{1}^{\prime}(\gamma^{N}\delta^{T})^{*}&=&(e \oplus \gamma^{\nu_{1}}\delta^{\tau_{1}} \oplus {\cdots} \oplus \gamma^{(k_{1}-1)\nu_{1}}\delta^{(k_{1}-1)\tau_{1}})(\gamma^{k_{1}\nu_{1}}\delta^{k_{1}\tau_{1}})^{*} ,\\ q_{2}^{\prime}(\gamma^{N}\delta^{T})^{*}&=&(e \oplus \gamma^{\nu_{2}}\delta^{\tau_{2}} \oplus {\cdots} \oplus \gamma^{(k_{2}-1)\nu_{2}}\delta^{(k_{2}-1)\tau_{2}})(\gamma^{k_{2}\nu_{2}}\delta^{k_{2}\tau_{2}})^{*} . \end{array} $$

Thus the sum can be written as: $s_{1} \oplus s_{2} = p_{1} \oplus p_{2} \oplus (q_{1}q_{1}^{\prime }\oplus q_{2}q_{2}^{\prime })(\gamma ^N\delta ^T)^{*}$.

Second, σ(s₁) > σ(s₂). Note that series s₁,s₂ can be expressed with a common period thus one can write,

$$ \begin{array}{@{}rcl@{}} s_{1} \oplus s_{2} &=&\tilde{p}_{1} \oplus\tilde{p}_{2} \oplus {\bigoplus}_{i = 1}^{I}\delta^{\varsigma_{1i}}{\Delta}_{\omega}\delta^{\varsigma_{1i}^{\prime}}\gamma^{n_{1i}}(\gamma^{k_{1}\nu_{1}}\delta^{\bar{\tau_{1}}})^{*} \\ &&\oplus {\bigoplus}_{j = 1}^{J}\delta^{\varsigma_{2j}}{\Delta}_{\omega}\delta^{\varsigma_{2j}^{\prime}}\gamma^{n_{2j}}(\gamma^{k_{2}\nu_{2}}\delta^{\bar{\tau_{2}}})^{*}. \end{array} $$

Due to Lemma 1, we can show that s₁ ⊕ s₂ is ultimately dominated by s₁.

1.6 C.6 Proof of Proposition 7 (product of ultimately periodic series)

Recall that s₁ and s₂ can be expressed in the commute form, Proposition 4. Then product of two series $s_{1} = p_{1}\oplus q_{1}(\gamma ^{\nu _{1}}\delta ^{\tau _{1}})^{*}$ and $s_{2}= p_{2}\oplus (\gamma ^{\nu _{2}}\delta ^{\tau _{2}})^{*}q_{2}$ can be written as

$$ \begin{array}{@{}rcl@{}} s_{1}\otimes s_{2} =& p_{1}p_{2} \oplus p_{1}q_{2}(\gamma^{\nu_{2}}\delta^{\tau_{2}})^{*} \oplus p_{2}q_{1}(\gamma^{\nu_{1}}\delta^{\tau_{1}})^{*} \oplus q_{1}(\gamma^{\nu_{1}}\delta^{\tau_{1}})^{*}(\gamma^{\nu_{2}}\delta^{\tau_{2}})^{*}q_{2}. \end{array} $$

Clearly, p₁p₂ is a polynomial (Proposition 5). $(\gamma ^{\nu _{1}}\delta ^{\tau _{1}})^{*}(\gamma ^{\nu _{2}}\delta ^{\tau _{2}})^{*} = (\gamma ^{\nu _{1}}\delta ^{\tau _{1}} \oplus \gamma ^{\nu _{2}}\delta ^{\tau _{2}})^{*} = s_3$ is an ultimately periodic series in ${\mathcal{M}}_{in}^{ax} [\![ \gamma ,\delta ]\!]$, therefore it is also a series in $\mathcal {T}^{*} [\![ \gamma ]\!]$ and $q_{1}s_{3}q_{2} = \tilde {s}_{3}$ as well. $p_{1}q_{2}(\gamma ^{\nu _{2}}\delta ^{\tau _{2}})^{*} = \tilde {s}_{2}$ (resp. $p_{2}q_{1}(\gamma ^{\nu _{1}}\delta ^{\tau _{1}})^{*}= \tilde {s}_{1}$) are two series in $\mathcal {T}^{*} [\![ \gamma ]\!]$. Finally we have a sum $p_{1}p_{2}\oplus \tilde {s}_{1}\oplus \tilde {s}_{2}\oplus \tilde {s}_{3}$ of ultimately periodic series in $\mathcal {T}^{*} [\![ \gamma ]\!]$, Proposition 6 Appendix C5.

1.7 C.7 Proof of Proposition 8 (Kleene star of a polynomial)

We first investigate a particular case, in which the star of a series in $\mathcal {T}^{*} [\![ \gamma ]\!]$ can be calculated similarly to the star of a simple monomial in $\mathcal {T}^{*} [\![ \gamma ]\!]$, see Eq. 24. Consider the following series $s\in \mathcal {T}^{*} [\![ \gamma ]\!]$ where w.l.o.g. τ is a multiple of ω, see Proposition 4 commute form,

$$ \begin{array}{@{}rcl@{}} s = \tilde{S}{\Delta}_{\omega}\delta^{\varsigma^{\prime}} = \left( {\bigoplus}_{i =1}^{I}\gamma^{n_{1i}}\delta^{\varsigma_{1i}} \oplus {\bigoplus}_{j = 1}^{J} \gamma^{n_{2j}}\delta^{\varsigma_{2j}} (\gamma^{\nu}\delta^{\tau})^{*}\right){\Delta}_{\omega}\delta^{\varsigma^{\prime}}, \end{array} $$

where $\tilde {S}= P\oplus Q(\gamma ^{\nu }\delta ^{\tau })^{*}\in {\mathcal{M}}_{in}^{ax} [\![ \gamma ,\delta ]\!]$. The product ss can be written as

$$ \begin{array}{@{}rcl@{}} ss &=& (P\oplus Q(\gamma^{\nu}\delta^{\tau})^{*}){\Delta}_{\omega}\delta^{\varsigma^{\prime}}(P\oplus Q(\gamma^{\nu}\delta^{\tau})^{*}){\Delta}_{\omega}\delta^{\varsigma^{\prime}}\\ &=&\tilde{S}{\Delta}_{\omega}\delta^{\varsigma^{\prime}}P{\Delta}_{\omega}\delta^{\varsigma^{\prime}} \oplus \tilde{S}{\Delta}_{\omega}\delta^{\varsigma^{\prime}} Q(\gamma^{\nu}\delta^{\tau})^{*}){\Delta}_{\omega}\delta^{\varsigma^{\prime}}\\ && \text{since}, {\Delta}_{\omega}(\gamma^{\nu}\delta^{\tau})^{*} = (\gamma^{\nu}\delta^{\tau})^{*}{\Delta}_{\omega}\\ &=&\tilde{S}{\Delta}_{\omega}\delta^{\varsigma^{\prime}}P{\Delta}_{\omega}\delta^{\varsigma^{\prime}} \oplus \tilde{S}(\gamma^{\nu}\delta^{\tau})^{*}{\Delta}_{\omega}\delta^{\varsigma^{\prime}}Q{\Delta}_{\omega}\delta^{\varsigma^{\prime}}\\ &&\text{due to (17)}, {\Delta}_{\omega}\delta^{\varsigma^{\prime}}P{\Delta}_{\omega} = P^{\prime}{\Delta}_{\omega},\ \ {\Delta}_{\omega}\delta^{\varsigma^{\prime}}Q{\Delta}_{\omega} = Q^{\prime}{\Delta}_{\omega}\\ &=&\tilde{S}(P^{\prime}\oplus Q^{\prime}(\gamma^{\nu}\delta^{\tau})^{*}) {\Delta}_{\omega}\delta^{\varsigma^{\prime}}= \tilde{S}\hat{S}{\Delta}_{\omega}\delta^{\varsigma^{\prime}} \end{array} $$

where $\hat {S}= P^{\prime }\oplus Q^{\prime }(\gamma ^{\nu }\delta ^{\tau })^{*}\in {\mathcal{M}}_{in}^{ax} [\![ \gamma ,\delta ]\!]$ is a series given by

$$ \begin{array}{@{}rcl@{}} \hat{S} = {\bigoplus}_{i =1}^{I}\gamma^{n_{1i}}\delta^{\lceil(\varsigma_{1i}+\varsigma^{\prime})\slash\omega\rceil\omega} \oplus {\bigoplus}_{j = 1}^{J} \gamma^{n_{2j}}\delta^{\lceil(\varsigma_{2j}+\varsigma^{\prime})\slash\omega\rceil\omega} (\gamma^{\nu}\delta^{\tau})^{*}. \end{array} $$

The star s^∗ is an ultimately periodic series in $\mathcal {T}^{*} [\![ \gamma ]\!]$, which can be obtained by

$$ \begin{array}{@{}rcl@{}} s^{*} &=& \mathrm{e}\oplus\tilde{S}{\Delta}_{\omega}\delta^{\varsigma^{\prime}}\!\! \oplus\underbrace{ \tilde{S}{\Delta}_{\omega}\delta^{\varsigma^{\prime}}\tilde{S}{\Delta}_{\omega}\delta^{\varsigma^{\prime}}}_{\tilde{S}\hat{S}{\Delta}_{\omega}\delta^{\varsigma^{\prime}}} \!\oplus \underbrace{\tilde{S}{\Delta}_{\omega}\delta^{\varsigma^{\prime}}\tilde{S}{\Delta}_{\omega}\delta^{\varsigma^{\prime}}\tilde{S}{\Delta}_{\omega}\delta^{\varsigma^{\prime}}}_{\tilde{S}\hat{S}^{2}{\Delta}_{\omega}\delta^{\varsigma^{\prime}}} \!\oplus{\cdots} \\ &=&\mathrm{e} \oplus\hat{S}^{*} \tilde{S}{\Delta}_{\omega}\delta^{\varsigma^{\prime}}=\mathrm{e} \oplus\hat{S}^{*}s. \end{array} $$

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Second, a polynomial in $\mathcal {T}^{*} [\![ \gamma ]\!]$ can be partitioned into a sum of sub-polynomials in the following form

$$ \begin{array}{@{}rcl@{}} p &=& \left( {\bigoplus}_{i =1}^{I}\gamma^{\nu_{1i}}\delta^{\varsigma_{1i}}\right) {\Delta}_{\omega} \oplus \left( {\bigoplus}_{j = 1}^{J} \gamma^{\nu_{2j}}\delta^{\varsigma_{2j}}\right){\Delta}_{\omega}\delta^{-1} {\cdots} \\ &&\oplus \left( {\bigoplus}_{k = 1}^{K} \gamma^{\nu_{\omega k}}\delta^{\varsigma_{\omega k}}\right){\Delta}_{\omega}\delta^{1-\omega}, \\ &=&{\bigoplus}_{l=0}^{\omega-1}p_{l}= p_{0} \oplus p_{1} \oplus {\cdots} \oplus p_{\omega-1}. \end{array} $$

where, $p_l = \bigoplus _{i}\gamma ^{\nu _i}\delta ^{\varsigma _i}{\Delta }_{\omega } \delta ^{-l}$. Since (a ⊕ b)^∗ = (a^∗b)^∗a^∗,

$$ \begin{array}{@{}rcl@{}} p^{*} = \left( (\underbrace{p_{0} \oplus {\cdots} \oplus p_{\omega-2}}_{\bar{p}_{\omega-2}})^{*} p_{\omega-1}\right)^{*}(\underbrace{p_{0} \oplus {\cdots} \oplus p_{\omega-2}}_{\bar{p}_{\omega-2}} )^{*}. \end{array} $$

Let us define by $\bar {p}_{l} := p_0\oplus {\cdots } \oplus p_l$, thus we can write the star $\bar {p}_{l}^{*}$ in a recursive form

$$ \begin{array}{@{}rcl@{}} \bar{p}_{l}^{*} = \left( \bar{p}_{l-1}^{*}p_{l}\right)^{*}\bar{p}_{l-1}^{*}. \end{array} $$

(38)

When we choose l = 1 we obtain $\bar {p}_{1}^{*} = \left (p_0^{*}p_{1} \right )^{*}p_0^{*}$, since $\bar {p}_0=p_0= {\bigoplus }_{i =1}^{I}\gamma ^{\nu _{1i}}\delta ^{\varsigma _{1i}}{\Delta }_{\omega }$. Due to Eq. 37, $p_0^{*}$ is given by

$$ \begin{array}{@{}rcl@{}} p_{0}^{*} &= \mathrm{e} \oplus \underbrace{\left( {\bigoplus}_{i=1}^{I}\gamma^{\nu_{i}}\delta^{\lceil \varsigma_{i}\slash \omega \rceil\omega} \right)^{*}{\bigoplus}_{i =1}^{I}\gamma^{\nu_{1i}}\delta^{\varsigma_{1i}}}_{\tilde{S}_{0}}{\Delta}_{\omega}. \end{array} $$

This star can be rewritten as $p_0^{*} = \mathrm {e} \oplus (\tilde {S}_0){\Delta }_{\omega }$ where $\tilde {S}_0$ is a series in ${\mathcal{M}}_{in}^{ax} [\![ \gamma ,\delta ]\!]$. The product $p_0^{*}p_{1}$ is ultimately periodic series in $\mathcal {T}^{*} [\![ \gamma ]\!]$, since

$$ \begin{array}{@{}rcl@{}} p_{0}^{*}p_{1} &=& (\mathrm{e} \oplus (\tilde{S}_{0}){\Delta}_{\omega})\left( {\bigoplus}_{j = 1}^{J} \gamma^{\nu_{2j}}\delta^{\varsigma_{2j}}{\Delta}_{\omega}\delta^{-1}\right),\\ &=& {\bigoplus}_{j = 1}^{J} \gamma^{\nu_{2j}}\delta^{\varsigma_{2j}}{\Delta}_{\omega}\delta^{-1} \oplus \tilde{S}_{0} {\Delta}_{\omega}\left( {\bigoplus}_{j = 1}^{J} \gamma^{\nu_{2j}}\delta^{\varsigma_{2j}}{\Delta}_{\omega}\delta^{-1}\right),\\ &=& \left( {\bigoplus}_{j = 1}^{J} \gamma^{\nu_{2j}}\delta^{\varsigma_{2j}} \oplus \tilde{S}_{0} {\bigoplus}_{j = 1}^{J} \gamma^{\nu_{2j}}\delta^{\lceil\varsigma_{2j}\slash \omega\rceil\omega}\right){\Delta}_{\omega}\delta^{-1},\\ &=& \tilde{S}_{01} {\Delta}_{\omega}\delta^{-1}, \end{array} $$

where $\tilde {S}_{01}$ is again a series in ${\mathcal{M}}_{in}^{ax} [\![ \gamma ,\delta ]\!]$. Therefore, the star $(p_0^{*}p_{1})^{*}$ can be calculated by using Eq. 37. It is an ultimately periodic series $\mathcal {T}^{*} [\![ \gamma ]\!]$. Then $\bar {p}_{1}^{*} = (p_0^{*}p_{1})^{*}p_0^{*}$ is the product of two ultimately periodic series in $\mathcal {T}^{*} [\![ \gamma ]\!]$, see Proposition 7 Appendix C6. In a similar way with $\bar {p}_{1}^{*}$ we can solve successively the recursive Eq. 38 ∀i ∈{1,⋯,ω − 1}.

1.8 C.8 Proof of Proposition 9 (Kleene star of an ultimately periodic series)

Recall that for r = (γ^νδ^τ), qr^∗ = r^∗q, Proposition 4. The star of ultimately periodic series can be rewritten as a star of polynomials s^∗ = (p ⊕ qr^∗)^∗ = p^∗(qr^∗p^∗)^∗ = p^∗(q(r ⊕ p)^∗)^∗ = p^∗(e ⊕ q(q ⊕ r ⊕ p)^∗), Baccelli et al. (1992).