# Improvements on flat output characterization for fractional systems

Stéphane Victor, Pierre Melchior

► To cite this version:

Stéphane Victor, Pierre Melchior. Improvements on flat output characterization for fractional systems. Fractional Calculus and Applied Analysis, 2015, 18 (1), pp.238-260. 10.1515/fca-2015-0016. hal-01319178

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Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
IMPROVEMENTS ON FLAT OUTPUT
CHARACTERIZATION FOR FRACTIONAL SYSTEMS

Stéphane Victor and Pierre Melchior

Abstract

In trajectory planning, flatness is used to compute inputs generating
suitable trajectories, without using any integration. The flatness property
of linear controllable time-invariant fractional systems is studied. The for-
malism of polynomial matrix of the fractional differential operator is used
leading to the characterization of fractionally flat outputs. The so-called
defining matrices, which are transformations that express all system vari-
ables in function of the fractionally flat outputs and a finite number of
their time derivatives, are introduced and characterized in this fractional
context. Flatness of fractional systems is then applied to the trajectory
planning of a real thermal experiment.

MSC 2010: 26A33, 34A08 and 60G22

*Key Words and Phrases:* flatness, polynomial matrices, fractional dif-
ferentiation, fractional systems, fractional flat output, thermal systems,
trajectory planning

1. Introduction

Studies point out the importance of modeling thermal [19, 2], electro-
chemical [4], viscoelastic [18], nuclear magnetic resonance [13] and biological
systems [26, 16] as fractional (or non-integer) differential equations. Integer
models have an excessive number of state variables which poorly reproduces
dynamical aspects of this kind of systems [14, 2].

The motion planning problem, which consists of determining the in-
puts generating the reference (output) trajectories, has been handled for
fractional systems and three strategies have been developed: the first one is
based on the used of fractional prefilter [21, 32]; the second one is based
on the extension of the preshaping approach to fractional systems [23, 10]; the third one is the extension of flatness approach to fractional systems. Preliminary works have been done concerning the extension to linear fractional systems [24] and MIMO fractional systems [27, 28]. In this paper, we focus on the third strategy, namely the differential flatness, which was introduced by M. Fliess, J. Levine, Ph. Martin and P. Rouchon [6, 7, 8]. The flatness-based trajectory design is extended for linear time-invariant (LTI) fractional systems. Note that, it turns out that systems described by ordinary differential equations are differentially flat if and only if they are controllable, meaning that there exists an output of the same dimension of the control vector such that all system variables can be expressed as functions of this flat output and its successive time derivatives. This flat output may be obtained as a by-product of the Brunovský controllability canonical form [1] or by using a characterization [12], in polynomial matrix form, of the so-called *defining matrices*. Few developments have been carried out for fractional systems ([28, 29]), also the theoretical background of the fractional flatness extension is proposed in a commutative algebra.

After recalling fractional calculus framework in Section 2, Section 3 presents fractional linear flatness. In Section 4, a real thermal application modeling the transfer temperature/heat flux in an aluminum rod, shows the usefulness of the characterization to trajectory planning. A conclusion is drawn in Section 5.

## 2. Fractional Calculus

### 2.1. Definition of fractional derivatives.
The fractional derivative arose from discussions between several mathematicians such as Leibniz, Euler, Liouville, Riemann, Grünwald, Letnikov and others [25, 17, 19].

Let us introduce $\gamma \in \mathbb{R}_+$, $n = \lceil \gamma \rceil = E(\gamma) + 1 = \min\{k \in \mathbb{N} | k > \gamma\}$ and $\nu \in [0, 1[$ such that $\gamma = n - \nu$. Let $\mathbf{a}$ be a given arbitrary real number and $f$ be in $C^\infty([\mathbf{a}, +\infty[)$, the set of infinitely continuously differentiable functions from $[\mathbf{a}, +\infty[$ to $\mathbb{R}$. We denote by $f^{(k)}(t)$ the $k$-th order derivative of $f$ with respect to $t$ for every $k \in \mathbb{N}$.

The Riemann-Liouville derivative of order $\gamma$, $D_{\mathbf{a}}^{\gamma}$, is defined as the $n$-th order (integer) derivative of the Cauchy integral of order $\nu$ of $f$ at time $t$ [17]:

$$D_{\mathbf{a}}^{\gamma} f(t) = D^{n} (I_{\mathbf{a}}^{\nu} f(t)) \triangleq \left(\frac{d}{dt}\right)^{n} \left(\frac{1}{\Gamma(\nu)} \int_{\mathbf{a}}^{t} \frac{f(\tau)}{(t-\tau)^{1-\nu}} d\tau\right), \quad (2.1)$$
where $\mathbf{I}_{\mathbf{a}}^{\nu}f(t)$ is the fractional primitive of $f$ and the Euler's $\Gamma$ function, defined as

$$ \Gamma(x) = \int_{0}^{\infty} e^{-t}t^{x-1}dt, \quad \forall x \in \mathbb{R}^{*} \setminus \mathbb{N}^{-}, \qquad (2.2) $$

is the generalized factorial ($\forall n \in \mathbb{N}, \Gamma(n + 1) = n!$).

**REMARK 2.1.** Note that $\nu\Gamma(\nu) = \Gamma(\nu + 1)$.

**REMARK 2.2.** If $\gamma \in \mathbb{N}$, the fractional derivative and the integer one coincide ($\mathbf{D}_{\mathbf{a}}^{\gamma}f(t) = \mathbf{D}^{\gamma}f(t)$) and if $\gamma < 0$, $\mathbf{D}_{\mathbf{a}}^{\gamma}f(t) = \mathbf{I}_{\mathbf{a}}^{-\gamma}f(t)$.

**REMARK 2.3.** Note that $\mathbf{a}$ is not necessarily equal to 0 in the above definition, which shows that some of the past of function $f$ is required.

**REMARK 2.4.** When $n = 0$, the differentiation (2.1) boils down to an integration and when $n > 0$, it consists of the $n$-th order differentiation of the integral $\mathbf{I}_{\mathbf{a}}^{\nu}$.

**2.2. Recalls on fractional derivatives.** The main definitions and propositions are recalled in this section. We thus provide rigorous algebraic definitions and properties that will enable the extension of the flatness property to fractional systems.

**2.2.1. Fractional integrability:** Recall $\nu \in [0, 1[$, and $t \in [a, +\infty[$. Setting

$$ g_{t}^{\nu}(\tau) = \frac{1}{\Gamma(\nu + 1)} (t^{\nu} - (t - \tau)^{\nu}), \qquad (2.3) $$

the Stieltjes measure $dg_{t}^{\nu}(\tau) \stackrel{\Delta}{=} \frac{\partial g_{t}^{\nu}}{\partial \tau}(\tau)d\tau = \frac{1}{\Gamma(\nu)}(t - \tau)^{\nu-1}d\tau$ is indeed absolutely continuous with respect to the Lebesgue measure $d\tau$, and its density $\frac{\partial g_{t}^{\nu}}{\partial \tau}$ is in $L^{1}(\mathbf{a}, t)$ since

$$ \|\frac{\partial g_t^\nu}{\partial \tau}\|_{L^1(\mathbf{a}, t)} = \int_{\mathbf{a}}^{t} \left| \frac{\partial g_t^\nu}{\partial \tau}(\tau) \right| d\tau = g_t^\nu(t) - g_t^\nu(\mathbf{a}) = \frac{|t - \mathbf{a}|^\nu}{\Gamma(\nu + 1)} < +\infty, $$

for all $t \ge \mathbf{a}$. Moreover, with these notations, the fractional primitive $\mathbf{I}_{\mathbf{a}}^{\nu}f(t)$ reads

$$ \mathbf{I}_{\mathbf{a}}^{\nu}f(t) = \int_{\mathbf{a}}^{t} f(\tau)dg_{t}^{\nu}(\tau). \qquad (2.4) $$

Since

$$ |\mathbf{I}_{\mathbf{a}}^{\nu}f(t)| \le \left\|\frac{\partial g_{t}^{\nu}}{\partial \tau}\right\|_{L^{1}(\mathbf{a}, t)} \cdot \|f\|_{L^{\infty}(\mathbf{a}, t)} < +\infty, \qquad (2.5) $$
we immediately deduce that the fractional primitive is well-defined for all $f \in L^\infty(\mathbf{a}, t)$, and a fortiori $f \in C^\infty([\mathbf{a}, +\infty[)$, all $\nu \in [0, 1[$ and all finite $t \ge \mathbf{a}$.

The definition (2.4) may be extended to any $\nu + k$ for an arbitrary integer $k$:

$$ \mathbf{I}_{\mathbf{a}}^{\nu+k} f(t) = \int_{\mathbf{a}}^{t} f(\tau) dg_{t}^{\nu+k}(\tau), \qquad (2.6) $$

since $dg_t^{\nu+k} = \frac{1}{\Gamma(\nu+k)} (t-\tau)^{\nu+k-1} d\tau$ properly defines a Stieltjes measure.

**THEOREM 2.1.** For all $\nu + k - n \ge 0$:

$$ \mathbf{D}^n \mathbf{I}_{\mathbf{a}}^{\nu+k} f(t) = \mathbf{I}_{\mathbf{a}}^{\nu+k-n} f(t). \qquad (2.7) $$

*Proof.* The proof is by induction. After sketching the first step, the remainder follows exactly the same lines and is left to the reader:

$$ \begin{aligned} \mathbf{D} \left( \mathbf{I}_{\mathbf{a}}^{\nu+k+1} f(t) \right) &= \frac{d}{dt} \left( \int_{\mathbf{a}}^{t} f(\tau) dg_{t}^{\nu+k+1}(\tau) \right) \\ &= \frac{d}{dt} \left( \int_{\mathbf{a}}^{t} \frac{f(\tau)}{\Gamma(\nu + k + 1)} (t - \tau)^{\nu+k} d\tau \right) \\ &= \int_{\mathbf{a}}^{t} \frac{f(\tau)}{\Gamma(\nu + k)} (t - \tau)^{\nu+k-1} d\tau \\ &= \mathbf{I}_{\mathbf{a}}^{\nu+k} f(t). \end{aligned} $$

□

2.2.2. **Fractional differentiability:** $n$ is assumed positive ($n > 0$). To prove that definition (2.1) is well-defined for all $n$ and $f \in C^\infty([\mathbf{a}, +\infty[)$, note that $\nu - 1 < 0$ implies that $g_t^{\nu-1}$ and its derivative with respect to $\tau$ are unbounded around $\tau = t$, and therefore it is not possible to differentiate directly $\mathbf{I}_{\mathbf{a}}^{\nu} f(t)$ with respect to $t$. We first prove the following:

**LEMMA 2.1.**

$$ \mathbf{I}_{\mathbf{a}}^{\nu} f(t) = \mathbf{I}_{\mathbf{a}}^{\nu+k} f^{(k)}(t) + \sum_{j=0}^{k-1} \frac{(-1)^j (t-\mathbf{a})^{\nu+j}}{\Gamma(\nu+j+1)} \delta_{\mathbf{a}}^{(j)} f \qquad (2.8) $$

for an arbitrary $k \in \mathbb{N}$, all $f \in C^\infty([\mathbf{a}, t])$ and all $t \ge \mathbf{a}$, where the distribution $\delta_{\mathbf{a}}^{(j)}$ is the $j$th derivative of the Dirac measure at $\mathbf{a}$, namely $\delta_{\mathbf{a}}^{(j)} f = (-1)^j f^{(j)}(\mathbf{a})$.
*Proof.* The proof of this formula is only done for $k = 1$, the generalization for $k > 1$ follows exactly the same lines and therefore is left to the reader.

Using (2.4) and the integration by parts formula, we get

$$ 
\begin{aligned}
\mathbf{I}_{\mathbf{a}}^{\nu} f(t) &= f(t) g_{t}^{\nu}(t) - f(\mathbf{a}) g_{t}^{\nu}(\mathbf{a}) - \int_{\mathbf{a}}^{t} f'(\tau) g_{t}^{\nu}(\tau) d\tau \\
&= \left( f(t) - f(\mathbf{a}) - \int_{\mathbf{a}}^{t} f'(\tau) d\tau \right) \frac{t^{\nu}}{\Gamma(\nu + 1)} \\
&\quad + f(\mathbf{a}) \frac{(t - \mathbf{a})^{\nu}}{\Gamma(\nu + 1)} + \int_{\mathbf{a}}^{t} f'(\tau) \frac{(t - \tau)^{\nu}}{\Gamma(\nu + 1)} d\tau \\
&= \frac{(t - \mathbf{a})^{\nu}}{\Gamma(\nu + 1)} \delta_{\mathbf{a}} f + \mathbf{I}_{\mathbf{a}}^{\nu+1} f'(t),
\end{aligned}
$$

after noticing that $f(t) - f(\mathbf{a}) - \int_{\mathbf{a}}^{t} f'(\tau) d\tau = 0$ and that $dg_{t}^{\nu+1}(\tau) = \frac{(t-\tau)^{\nu}}{\Gamma(\nu+1)} d\tau$, which achieves the proof of (2.8). $\square$

**REMARK 2.5.** Since $\nu + k > 0$ for all $k$, (2.8) is valid even if $t = \mathbf{a}$.

**REMARK 2.6.** Relation (2.8) implies that the integral operator $\mathbf{I}_{\mathbf{a}}^{\nu}$ maps $\mathbb{C}^{\infty}([\mathbf{a}, t])$ into $\mathbb{C}^{\infty}([\mathbf{a}, t])$, which proves that (2.1) is well-defined.

**COROLLARY 2.1.** The operator $\mathbf{D}_{\mathbf{a}}^{\gamma}$ maps $\mathbb{C}^{\infty}([\mathbf{a}, +\infty])$ to $\mathbb{C}^{\infty}([\mathbf{a}, +\infty])$ and

$$ \mathbf{D}_{\mathbf{a}}^{\gamma} f(t) = \mathbf{I}_{\mathbf{a}}^{\nu} f^{(n)}(t) + \sum_{j=0}^{n-1} \frac{(t-\mathbf{a})^{\nu-n+j}}{\Gamma(\nu-n+j+1)} \delta_{\mathbf{a}}^{(j)} f. \quad (2.9) $$

*Proof.* By differentiating equation (2.8) $n$ times, and applying theorem 2.1, we get:

$$ \mathbf{D}_{\mathbf{a}}^{\gamma} f(t) = \mathbf{I}_{\mathbf{a}}^{\nu+k-n} f^{(k)}(t) + \sum_{j=0}^{k-1} \mathbf{D}^{n} \left( \frac{(-1)^{j}(t-\mathbf{a})^{\nu+j}}{\Gamma(\nu+j+1)} \delta_{\mathbf{a}}^{(j)} f \right). $$

We thus have the relation (2.9) with $k = n$. $\square$

**REMARK 2.7.** In systems theory, the natural signal space generally considered is the space of causal signals, namely the space of functions $f: \mathbb{R} \to \mathbb{R}$, which are infinitely differentiable and such that $f(t) = 0$ for all $t \in ]-\infty, \mathbf{a}]$. For such signals, $f(\mathbf{a}) = 0$ and $f^{(k)}(\mathbf{a}) = 0$, for all $k \ge 1$. We restrict our attention to this class of functions.
Therefore, the signal space is defined by

$$ \mathfrak{H}_a \triangleq \{f : \mathbb{R} \mapsto \mathbb{R} | f \in \mathbb{C}^{\infty}([a, +\infty]), f(t) = 0, \forall t \le a\}. $$

Consequently, we have that $f^{(k)}(a) = 0$ for all $f \in \mathfrak{H}_a$ and all $k \in \mathbb{N}$. As a result, we have:

**THEOREM 2.2.** For all $f \in \mathfrak{H}_a$ and $\gamma = n - \nu$, we have

$$ \mathbf{D}_a^{\gamma} f(t) = \mathbf{I}_a^{\nu} \mathbf{D}^n f(t). \qquad (2.10) $$

## 2.3. Commutativity.

### 2.3.1. Commutativity of fractional integrals:

**THEOREM 2.3.** For all $\gamma, \kappa > 0$, and all $f \in \mathbb{C}^{\infty}([a, \infty])$, we have:

$$ \mathbf{I}_a^{\gamma} \mathbf{I}_a^{\kappa} f(t) = \mathbf{I}_a^{\kappa} \mathbf{I}_a^{\gamma} f(t) = \mathbf{I}_a^{\gamma+\kappa} f(t). \qquad (2.11) $$

*Proof.* Knowing that

$$ \mathbf{I}_a^{\gamma} \mathbf{I}_a^{\kappa} f(t) = \frac{1}{\Gamma(\gamma)\Gamma(\kappa)} \int_a^t \frac{d\tau}{(t-\tau)^{1-\gamma}} \int_a^{\tau} \frac{f(u)du}{(\tau-u)^{1-\kappa}}, $$

and by interchanging the order of integration by Fubini's theorem, we get

$$ \mathbf{I}_a^{\gamma} \mathbf{I}_a^{\kappa} f(t) = \frac{1}{\Gamma(\gamma)\Gamma(\kappa)} \int_a^{\tau} f(u)du \int_a^t \frac{d\tau}{(t-\tau)^{1-\gamma} (\tau-u)^{1-\kappa}}. $$

Let us first compute $\int_a^t (t-\tau)^{\gamma-1} (\tau-u)^{\kappa-1} d\tau$. With the change of variable $\tau = u + s(t-u)$, $d\tau = ds(t-u)$, it comes:

$$ \begin{aligned} & \int_a^t (t-\tau)^{\gamma-1} (\tau-u)^{\kappa-1} d\tau \\ = & \int_0^1 (t-u-s(t-u))^{\gamma-1} (u+s(t-u)-u)^{\kappa-1} d\tau \\ = & (t-u)^{\gamma+\kappa-1} \int_0^1 s^{\kappa-1} (1-s)^{\gamma-1} ds \qquad (2.12) \\ = & (t-u)^{\gamma+\mu-1} B(\gamma, \kappa). \end{aligned} $$

In (2.12), $\int_0^1 s^{\kappa-1} (1-s)^{\gamma-1} ds$ corresponds to the definition of the Euler Beta function $B(\gamma, \kappa)$, [20]. We have the well known formula relating Beta and Gamma functions:

$$ B(\gamma, \kappa) = \frac{\Gamma(\gamma)\Gamma(\kappa)}{\Gamma(\gamma + \kappa)}. $$
Finally,

$$ \mathbf{I}_{\mathbf{a}}^{\gamma} \mathbf{I}_{\mathbf{a}}^{\kappa} f(t) = \frac{1}{\Gamma(\gamma + \kappa)} \int_{\mathbf{a}}^{t} \frac{f(\tau)}{(t - \tau)^{1-\gamma-\kappa}} d\tau = \mathbf{I}_{\mathbf{a}}^{\gamma+\kappa} f(t). \quad (2.13) $$

The other equality in (2.13) is proven in the same way.

□

### 2.3.2. Commutativity of fractional derivatives:

**THEOREM 2.4.** If $f \in \mathfrak{H}_a$ and $\gamma = n - \nu$, with $n \in \mathbb{N}$ and $\nu \in [0, 1[$, we have:

$$ \mathbf{D}_{\mathbf{a}}^{\gamma} (\mathbf{I}_{\mathbf{a}}^{\gamma} f(t)) = \mathbf{I}_{\mathbf{a}}^{\gamma} (\mathbf{D}_{\mathbf{a}}^{\gamma} f(t)) = f(t). \quad (2.14) $$

*Proof.* See [22] adapted to our signal space.

$$ \begin{aligned} \mathbf{D}_{\mathbf{a}}^{\gamma} (\mathbf{I}_{\mathbf{a}}^{\gamma} f(t)) &= \mathbf{D}^{n} \mathbf{I}_{\mathbf{a}}^{\nu} \mathbf{I}_{\mathbf{a}}^{\gamma} f(t) \quad (\text{by the definition of } \mathbf{D}_{\mathbf{a}}^{\gamma}) \\ &= \mathbf{D}^{n} \mathbf{I}_{\mathbf{a}}^{\nu+\gamma} f(t) \quad (\text{by Theorem 2.3}) \\ &= \mathbf{D}^{n} \mathbf{I}^{\gamma} f(t) = f(t) \quad (\text{since } \nu + \gamma = n), \end{aligned} $$

where we have written the usual rational differential $\mathbf{D}^n$ and integral $\mathbf{I}^n$ in place of $\mathbf{D}_a^n$ and $\mathbf{I}_a^n$ respectively.

Conversely,

$$ \begin{aligned} \mathbf{I}_{\mathbf{a}}^{\gamma} (\mathbf{D}_{\mathbf{a}}^{\gamma} f(t)) &= \mathbf{I}_{\mathbf{a}}^{\gamma} \mathbf{D}^{n} \mathbf{I}_{\mathbf{a}}^{\nu} f(t) \quad (\text{by the definition of } \mathbf{D}_{\mathbf{a}}^{\gamma}) \\ &= \mathbf{I}_{\mathbf{a}}^{\gamma} \mathbf{I}_{\mathbf{a}}^{\nu} \mathbf{D}^{n} f(t) \quad (\text{by Theorem 2.2}) \\ &= \mathbf{I}_{\mathbf{a}}^{\gamma+\nu} \mathbf{D}^{n} f(t) = \mathbf{I}^{n} \mathbf{D}^{n} f(t) = f(t). \end{aligned} $$

□

Theorem 2.4 allows us to define negative order fractional integrals as positive order fractional derivatives and conversely:

**DEFINITION 2.1.** For all $f \in \mathfrak{H}_a$ and $\gamma > 0$, we set:

$$ \mathbf{I}_{\mathbf{a}}^{-\gamma} f(t) = \mathbf{D}_{\mathbf{a}}^{\gamma} f(t), \quad \mathbf{D}_{\mathbf{a}}^{-\gamma} f(t) = \mathbf{I}_{\mathbf{a}}^{\gamma} f(t). \quad (2.15) $$

Let $\gamma, \kappa > 0$ be non integer numbers, where $\gamma = n - \nu$ and $\kappa = m - \mu$ with $\nu, \mu \in [0, 1[$ and $n, m \in \mathbb{N}$. The composition of the Riemann-Liouville fractional derivative operators $\mathbf{D}_a^\gamma$ and $\mathbf{D}_a^\kappa$ does not make sense in general outside the space $\mathfrak{H}_a$. Nevertheless, this composition, restricted to $\mathfrak{H}_a$, is commutative, as in the followings:

**THEOREM 2.5.** If $f \in \mathfrak{H}_a$, the following commutativity result holds:

$$ \mathbf{D}_{\mathbf{a}}^{\kappa} (\mathbf{D}_{\mathbf{a}}^{\gamma} f(t)) = \mathbf{D}_{\mathbf{a}}^{\gamma} (\mathbf{D}_{\mathbf{a}}^{\kappa} f(t)) = \mathbf{D}_{\mathbf{a}}^{\gamma+\kappa} f(t). \quad (2.16) $$
*Proof.* Following the same lines as in the proof of theorem 2.4, we have:

$$ 
\begin{aligned}
\mathbf{D}_{\mathfrak{a}}^{\gamma} \mathbf{D}_{\mathfrak{a}}^{\kappa} f(t) &= \mathbf{D}^{n} \mathbf{I}_{\mathfrak{a}}^{\nu} (\mathbf{I}_{\mathfrak{a}}^{\mu} \mathbf{D}^{m}) f(t) = \mathbf{D}^{n} \mathbf{I}_{\mathfrak{a}}^{\nu+\mu} \mathbf{D}^{m} f(t) \\
&= \mathbf{I}_{\mathfrak{a}}^{\nu+\mu-n-m} f(t) = \mathbf{D}_{\mathfrak{a}}^{\gamma+\kappa} f(t).
\end{aligned}
$$

The converse is proven in the same way.

□

**2.4. D<sub>a</sub><sup>γ</sup>-polynomial Algebra.** Given a positive fractional order γ, we now restrict the domain of the fractional operator D<sub>a</sub><sup>γ</sup> to ℏ<sub>a</sub>. For the sake of simplicity, we will keep the notation D<sub>a</sub><sup>γ</sup> in place of D<sub>a</sub><sup>γ</sup>|<sub>ℏ<sub>a</sub></sub>.

We consider arbitrary polynomials of the indeterminate D<sub>a</sub><sup>γ</sup> with real coefficients, of the form Σ<sub>k=0</sub><sup>K</sup> a<sub>k</sub>D<sub>a</sub><sup>kγ</sup> and call them D<sub>a</sub><sup>γ</sup>-polynomials. Their degree is defined as usually.

ℝ [D<sub>a</sub><sup>γ</sup>] denotes the set of such D<sub>a</sub><sup>γ</sup>-polynomials endowed with the usual addition + and multiplication × of polynomials. It is immediately verified that (ℝ [D<sub>a</sub><sup>γ</sup>], +, ×) is a (commutative) principal ideal domain (see Theorem 2.5).

**DEFINITION 2.2.** If γ is a positive real number such that there exist μ > 0 and r ∈ ℝ satisfying γ = rμ, we say that γ is commensurate of order μ.

Also, any D<sub>a</sub><sup>γ</sup>-polynomial can be expressed as a D<sub>a</sub><sup>μ</sup>-polynomial. Consequently, if γ is commensurate of order μ, the ring ℝ [D<sub>a</sub><sup>γ</sup>] is homomorphic to ℝ [D<sub>a</sub><sup>μ</sup>].

**THEOREM 2.6 (Equivalence).** If P ∈ ℝ [D<sub>a</sub><sup>γ</sup>], Q ∈ ℝ [D<sub>a</sub><sup>κ</sup>] and if there exists μ such that γ and κ are commensurate of order μ, then P = Q is equivalent to:

(i) P ∈ ℝ [D<sub>a</sub><sup>μ</sup>], Q ∈ ℝ [D<sub>a</sub><sup>μ</sup>],

(ii) the coefficients of P and Q expressed as D<sub>a</sub><sup>μ</sup>-polynomials are equal.

**REMARK 2.8.** In ℝ [D<sub>a</sub><sup>γ</sup>], the properties of polynomial division and Bézout identity, as well as the definitions of G.C.D, L.C.M., and prime polynomials, are the same as in the principal ideal domain of formal real polynomials.

If p, q ∈ ℝ, ℝ [D<sub>a</sub><sup>γ</sup>]<sup>p×q</sup> is called the set of D<sub>a</sub><sup>γ</sup>-polynomial matrices of size p × q, whose entries are D<sub>a</sub><sup>γ</sup>-polynomials.
When $p = q$, we denote by $GL_p(\mathbb{R}[\mathbf{D}_a^\gamma])$ the group of unimodular $\mathbf{D}_a^\gamma$-polynomial matrices, i.e. the set of invertible (square) $\mathbf{D}_a^\gamma$-polynomial matrices whose inverse is also a $\mathbf{D}_a^\gamma$-polynomial matrix.

$\mathbf{D}_a^\gamma$-polynomial matrices enjoy the following very important property [3, Chap. 8], [31]:

**THEOREM 2.7 (Smith diagonal decomposition).** Let $A$ be a $(p \times q)$ $\mathbf{D}_a^\gamma$-polynomial matrix, with $p \le q$ (resp. $p \ge q$), there exist two matrices $V \in GL_p(\mathbb{R}[\mathbf{D}_a^\gamma])$ and $U \in GL_q(\mathbb{R}[\mathbf{D}_a^\gamma])$ such that:

$$VAU = [\Delta \quad 0_{p,q-p}] \quad (\text{resp.} \quad \begin{bmatrix} \Delta \\ 0_{p-q,q} \end{bmatrix}), \qquad (2.17)$$

where $\Delta = \text{diag}\{\delta_1, \dots, \delta_\sigma, 0, \dots, 0\} \in \mathbb{R}[\mathbf{D}_a^\gamma]^{p \times p}$ (resp. $\mathbb{R}[\mathbf{D}_a^\gamma]^{q \times q}$). In $\Delta$, the integer $\sigma \le \min(p, q)$ is the rank of $A$ and every non zero $\mathbf{D}_a^\gamma$-polynomial $\delta_i$, for $i = 1, \dots, \sigma$, is a divisor of $\delta_j$ for all $\sigma \ge j \ge i$.

This decomposition over $\mathbb{R}[\mathbf{D}_a^\gamma]$ is done in practice by the same algorithm as the one over $\mathbb{R}[s]$, $s$ being a complex variable. It consists in computing a diagonal form by repeatedly using greatest common divisors (see e.g. [9, 31]).

**DEFINITION 2.3.** Given $A$, a $p \times q$ $\mathbf{D}_a^\gamma$-polynomial matrix, we say that $U$ is a right-Smith factor, and we note $U \in \mathbb{R} - \text{Smith}(A)$, if there exists $V \in GL_p(\mathbb{R}[\mathbf{D}_a^\gamma])$ such that (2.17) holds true. Symmetrically, we say that $V$ is a left-Smith factor, and we note $V \in \mathbb{L} - \text{Smith}(A)$, if there exists $U \in GL_q(\mathbb{R}[\mathbf{D}_a^\gamma])$ such that (2.17) holds true.

**DEFINITION 2.4 (Hyper-regularity).** Given a $p \times q$ $\mathbf{D}_a^\gamma$-polynomial matrix $A$, we say that $A$ is hyper-regular if, and only if, in (2.17), we have $\Delta = I$.

Note that a $p \times p$ square matrix $A$ is hyper-regular if, and only if, it is unimodular.

### 3. Fractional linear flatness

All considered signals will be defined in $\mathfrak{H}_a$, so that the commutativity property holds. For notation simplicity, a polynomial $A(\mathbf{D}_a^\gamma)$ will shortly be denoted as $A$, a $n \times n$ identity matrix by $I_n$ and a $n \times m$ zero matrix by $0_{n,m}$.
**3.1. Linear fractional systems.** Let a linear fractional system be given by the following state space representation

$$A_{\gamma}(\mathbf{D}_{\mathbf{a}}^{\gamma})x = B(\mathbf{D}_{\mathbf{a}}^{\gamma})u \quad (3.1)$$

with $A_{\gamma}$ a $n \times n$ matrix whose entries are $\mathbf{D}_{\mathbf{a}}^{\gamma}$-polynomials and $B$ is a $n \times m$ matrix of rank $m$, with $1 \le m \le n$.

Following [5] (see also [12]), we introduce the system module $\mathfrak{M}_{A_{\gamma},B}$ corresponding to (3.1) as follows: $\mathfrak{M}_{A_{\gamma},B}$ is the finitely generated quotient module

$$\mathfrak{M}_{A_{\gamma},B} = [x_0, u_0]/[A_{\gamma}x_0 - Bu_0], \quad (3.2)$$

where the components of $x_0 = (x_{0,1}, \dots, x_{0,n})$ and $u_0 = (u_{0,1}, \dots, u_{0,m})$ are not all zeros but otherwise arbitrary elements of $\mathfrak{H}_{\mathbf{a}}$ and where we have denoted by $[x_0, u_0]$ the $\mathbb{R}[\mathbf{D}_{\mathbf{a}}^{\gamma}]$-module generated by the components of $[x_0, u_0]$, and by $[A_{\gamma}x_0 - Bu_0]$ the submodule generated by the rows of $A_{\gamma}x_0 - Bu_0$.

According to [5] (see also e.g. [3]), $\mathfrak{M}_{A_{\gamma},B}$ can be decomposed into the following direct sum:

$$\mathfrak{M}_{A_{\gamma},B} = \mathcal{T} \oplus \mathcal{F},$$

where the uniquely defined module $\mathcal{T}$ is torsion and $\mathcal{F}$ is free. $\mathcal{F}$ is unique up to isomorphism.

The system is said free controllable if and only if $\mathcal{T} = \{0\}$.

**3.2. Flat output of a linear non integer system.**

**THEOREM 3.1.** The pair $(A_{\gamma}, B)$ defined in (3.1) is controllable iff one of the equivalent statements hold true ([11, 15]):

* A and B are left coprime;

* the $n \times (n+m)$ matrix $[A_{\gamma}, -B]$ has rank $n \forall \mathbf{D}_{\mathbf{a}}^{\gamma}$.

The system (3.1), supposed controllable, is said differentially flat [7, 8] if there exist a set of independent variables, called flat outputs, such that all system variables are function of the flat outputs and a finite number of their time derivatives:

$$\begin{align}
x_i &= \sum_{j=1}^{m} \sum_{k=0}^{\alpha_j} p_{i,j,k} \mathbf{D}_{\mathbf{a}}^{k\gamma} z_j, \quad i = 1, \dots, n, \tag{3.3} \\
u_l &= \sum_{j=1}^{m} \sum_{k=0}^{\alpha_j+1} q_{l,j,k} \mathbf{D}_{\mathbf{a}}^{k\gamma} z_j, \quad l = 1, \dots, m, \nonumber
\end{align}$$

where $z$ is defined as a linear combination of $x$, $u$ and a finite number of fractional $\gamma$-derivatives of $u$.
Using $\mathbf{D}_a^\gamma$-polynomial matrices, (3.3) reads

$$x = Pz, \quad u = Qz, \tag{3.4}$$

where $P$ (resp. $Q$) is $n \times m$ (resp. $m \times n$), of components $P_{i,j} = \sum_{k=0}^{\alpha_j} p_{i,j,k} \mathbf{D}_a^{k\gamma}$ (resp. $Q_{l,j} = \sum_{k=0}^{\alpha_{j+1}} q_{l,j,k} \mathbf{D}_a^{k\gamma}$).

The matrices $P$ and $Q$ satisfying (3.4) are called *defining matrices* of the fractional flat output $z$.

**THEOREM 3.2.** If all the diagonal terms of $A_\gamma$ are non null and if the matrices $A_\gamma$ and $B$ are left coprime then there exist a variable $z = (z_1, ..., z_m)$, flat output of (3.1) where the defining matrices $P$ and $Q$ are given by:

$$RA_\gamma P = 0_{n-m,n}, \tag{3.5}$$

$$A_\gamma P = BQ, \tag{3.6}$$

with $R$ an arbitrary matrix of rank $n-m$ orthogonal to $B$ (i.e. $RB = 0_{n-m,m}$), and with $P$ and $Q$ of rank $m$ for all $\mathbf{D}_a^\gamma$ and right coprime.

Moreover, a flat output $z$ of the controllable system (3.1) always exists (and so do $P$ and $Q$). The proof is an easy adaptation of [11] for $\mathbf{D}_a^\gamma$-polynomial matrices.

**COROLLARY 3.1.** Let $B$ be hyper-regular. There exists a $(n-m) \times n$ matrix $R$ of rank $n-m$ such that $R$ is orthogonal to $B$. From the Smith decomposition of $B$, it comes straightforwardly that:

$$R = [0_{n-m,m}, I_{n-m}] V_B. \tag{3.7}$$

*Proof.* $B$ being an $n \times m$ hyper-regular matrix, there exist a $n \times n$ matrix $V_B \in \mathbf{L} - \mathbf{Smith}(B)$ and a $m \times m$ matrix $U_B \in \mathbf{R} - \mathbf{Smith}(B)$ such that:

$$V_B B U_B = \begin{bmatrix} I_m \\ 0_{n-m,m} \end{bmatrix}, \tag{3.8}$$

Knowing that $V_B$ is $n \times n$, let us define a $m \times n$ matrix $V_1$ and a $(n-m) \times n$ matrix $V_2$ such that:

$$V_B = \begin{bmatrix} V_1 \\ V_2 \end{bmatrix}. \tag{3.9}$$

From (3.8) and (3.9),

$$\left\{ \begin{array}{l} V_1 B U_B = I_m \\ V_2 B U_B = 0_{n-m,m} \end{array} \right. ,$$
$U_B$ being invertible, $V_2B = 0_{n-m,m}$ and as $rank(V_2) = n-m$, $R = V_2$. We
can then express $V_B$ as:

$$
V_B = \begin{bmatrix} V_1 \\ R \end{bmatrix}. \tag{3.10}
$$

Furthermore, from (3.8), $B = V_B^{-1} \begin{bmatrix} I_m \\ 0_{n-m,m} \end{bmatrix} U_B^{-1}$. R is orthogonal to $B$ ($RB = 0$), also R can be defined as:

$$
R = [0_{n-m,m}, I_{n-m}] V_B.
\qquad \square
$$

LEMMA 3.1. Let B be hyper-regular. $RA_\gamma$ is hyper-regular if and only if $R[A_\gamma, -B]$ is hyper-regular.

Proof.

* Assume that $RA_\gamma$ is hyper-regular, then there exist a $(n-m) \times (n-m)$ matrix $V_{RA_\gamma} \in L - \text{Smith}(RA_\gamma)$ and a $n \times n$ matrix $U_{RA_\gamma} \in R - \text{Smith}(RA_\gamma)$ such that:

$$
V_{RA_\gamma} RA_\gamma U_{RA_\gamma} = [I_{n-m}, 0_{n-m,m}].
$$

By defining the invertible matrices $V_{R[A_\gamma, -B]}(\mathbf{D}_a^\gamma) = V_{RA_\gamma}$ and:

$$
U_{R[A_\gamma, B]} = \begin{bmatrix} U_{RA_\gamma} & 0_{n,m} \\ 0_{m,n} & I_m \end{bmatrix},
$$

it comes that

$$
\begin{align*}
& V_{R[A_\gamma, -B]} R[A_\gamma, -B] U_{R[A_\gamma, -B]} (\mathbf{D}_a^\gamma) \\
&= V_{RA_\gamma} (\mathbf{D}_a^\gamma) [RA_\gamma, 0_{n-m,m}] \begin{bmatrix} U_{RA_\gamma} (\mathbf{D}_a^\gamma) & 0_{n,m} \\ 0_{m,n} & I_m \end{bmatrix}, \\
&= V_{RA_\gamma} [RA_\gamma U_{RA_\gamma}, 0_{n-m,m}] = [I_{n-m}, 0_{n-m,m}],
\end{align*}
$$

which proves that $R[A_\gamma, -B]$ is hyper-regular.

* Conversely, let us now assume that $R[A_\gamma, -B]$ is hyper-regular, then there exist a $(n-m) \times (n-m)$ matrix

$$
V_{R[A_\gamma, -B]} \in L - \text{Smith}(R[A_\gamma, -B])
$$

and a $(n+m) \times (n+m)$ matrix $U_{R[A_\gamma, -B]} \in \mathbb{R} - \text{Smith}(R[A_\gamma, -B])$
such that:

$$
V_{R[A_\gamma, -B]} R[A_\gamma, -B] U_{R[A_\gamma, -B]} = [I_{n-m}, 0_{n-m,2m}]. \quad (3.11)
$$
As $rank(U_{R[A_γ, -B]}) = n + m$, let us define the $n \times (n+m)$ submatrix $U_{n,n+m}$ and the $m \times (n+m)$ submatrix $U_{m,n+m}$ such that:

$$ U_{R[A_γ, -B]} = \begin{bmatrix} U_{n,n+m} \\ U_{m,n+m} \end{bmatrix}, \qquad (3.12) $$

where $rank(U_{n,n+m}) = n$ and $rank(U_{m,n+m}) = m$.

When detailing equation (3.11) with the expression (3.12), we read:

$$ V_{R[A_γ, -B]} R[A_γ, -B] U_{R[A_γ, -B]} \qquad (3.13) $$

$$ = [V_{R[A_γ, -B]} R A_γ, -V_{R[A_γ, -B]} R B] U_{R[A_γ, -B]}, $$

$$ = [V_{R[A_γ, -B]} R A_γ, 0_{n-m,m}] \begin{bmatrix} U_{n,n+m} \\ U_{m,n+m} \end{bmatrix}, \qquad (3.13) $$

$$ = V_{R[A_γ, -B]} R A_γ U_{n,n+m}. $$

Knowing that $rank(U_{n,n+m}) = n$, let us define the $n \times n$ submatrix $U_n$ such that $rank(U_n) = n$, and such that

$$ U_{n,n+m} = [U_n, 0_{n,m}]. $$

Thus, (3.13) becomes:

$$ \begin{aligned} V_{R[A_γ, B]} R A_γ U_{n,n+m} &= V_{R[A_γ, -B]} R A_γ U_n \\ &= [I_{n-m}, 0_{n-m,2m}], \end{aligned} $$

which proves that $RA_γ$ is hyper-regular.

□

**THEOREM 3.3.** If $B$ is hyper-regular then the pair $(A_γ, B)$ is controllable if and only if $F = RA_γ$ is hyper-regular.

*Proof.*

* Assume that $RA_γ$ is hyper-regular, then there exist a $(n-m) \times (n-m)$ matrix $V_{RA_γ} \in L - \text{Smith}(RA_γ)$ and a $n \times n$ matrix $U_{RA_γ} \in R - \text{Smith}(RA_γ)$ such that:

$$ V_{RA_γ} R A_γ U_{RA_γ} = [I_{n-m}, 0_{n-m,m}]. \qquad (3.14) $$

We define the $n \times (n-m)$ submatrix $U_{n,n-m}$ and the $n \times m$ submatrix $U_{m,n}$ such that:

$$ U_{RA_γ} = [U_{n,n-m}, U_{n,m}]. $$
Equation (3.14) then becomes:

$$ \begin{cases} V_{RA_{\gamma}} R A_{\gamma} U_{n,n-m} = I_{n-m} \\ V_{RA_{\gamma}} R A_{\gamma} U_{n,m} = 0_{n-m,m} \end{cases}, $$

The $n \times n$ Left-Smith factor $V_B$ of $B$, defined in (3.10), admits an inverse $V_B^{-1}$.

We introduce a $n \times m$ submatrix $V'_1$ and a $n \times (n-m)$ submatrix $V'_2$ such that

$$ V_B^{-1} = [V'_1, V'_2]. $$

Note that:

$$ \begin{aligned} V_B^{-1} V_B &= [V'_1, V'_2] \begin{bmatrix} V_1 \\ R \end{bmatrix} \\ &= V'_1 V_1 + V'_2 R = I_n. \end{aligned} \quad (3.15) $$

By introducing

$$ X_n = \begin{bmatrix} V_{RA_{\gamma}} R \\ V_1 \end{bmatrix} $$

and

$$ X'_n = [V'_2 V_{RA_{\gamma}}^{-1}, V'_1], $$

and knowing (3.15), we compute

$$ \begin{aligned} X'_n X_n &= [V'_2 V_{RA_{\gamma}}^{-1}, V'_1] \begin{bmatrix} V_{RA_{\gamma}} R \\ V_1 \end{bmatrix} \\ &= V'_1 V_1 + V'_2 R = I_n, \end{aligned} \quad (3.16) $$

which proves that $X_n$ is invertible.

By defining the invertible matrix $Z_{n+m}$:

$$ Z_{n+m} = \begin{bmatrix} U_{RA_{\gamma}} & 0_{n,m} \\ 0_{m,n} & -U_B \end{bmatrix}, $$

we first compute

$$ \begin{aligned} X_n [A_{\gamma}, -B] Z_{n+m} &= \begin{bmatrix} V_{RA_{\gamma}} R A_{\gamma} U_{RA_{\gamma}} & 0_{n-m,m} \\ V_1 A_{\gamma} U_{RA_{\gamma}} & V_1 B U_B \end{bmatrix} \\ &= \begin{bmatrix} [I_{n-m}, 0_{n-m,n}] & 0_{n-m,m} \\ V_1 A_{\gamma} U_{RA_{\gamma}} & V_1 B U_B \end{bmatrix} \\ &= \begin{bmatrix} I_{n-m} & 0_{n-m,n} & 0_{n-m,m} \\ K_{m,n-m} & K_m & I_m \end{bmatrix}, \end{aligned} $$
where the $m \times (n-m)$ submatrix $K_{m,n-m}$ and the $m \times m$ submatrix $K_m$ are defined such that:

$$V_1 A_\gamma U_{RA_\gamma} = [K_{m,n-m}, K_m].$$

If we introduce the invertible matrices

$$U_{[A_\gamma, -B]} = Z_{n+m} \begin{bmatrix} I_{n-m} & 0_{n-m,n} & 0_{n-m,m} \\ 0_{m,n-m} & 0_m & I_m \\ 0_{m,n-m} & I_m & -K_m \end{bmatrix}$$

and

$$V_{[A_\gamma, -B]} = Z_{n+m} \begin{bmatrix} I_{n-m} & 0_{n-m,n} \\ -K_{m,n-m} & I_m \end{bmatrix} X_n,$$

it turns out that we define the Smith decomposition of $[A_\gamma, -B]$ as

$$\begin{align*} V_{[A_\gamma, -B]} [A_\gamma, -B] U_{[A_\gamma, -B]} \\ &= \begin{bmatrix} I_{n-m} & 0_{n-m,m} \\ -K_{m,n-m} & I_m \end{bmatrix} \begin{bmatrix} I_{n-m} & 0_{n-m,n} & 0_{n-m,m} \\ K_{m,n-m} & I_m & 0_m \end{bmatrix} \\ &= \begin{bmatrix} I_{n-m} & 0_{n-m,n} & 0_{n-m,m} \\ 0_{m,n-m} & I_m & 0_m \end{bmatrix}, \\ &= [I_n, 0_{n,m}]. \end{align*}$$

Finally, $[A_\gamma, -B]$ is hyper-regular, and particularly, $rank([A_\gamma, -B]) = n$, which proves, according to Theorem 3.1, that the pair $(A_\gamma, B)$ is controllable.

* Conversely, if the pair $(A_\gamma, B)$ is controllable, the matrix $[A_\gamma, -B]$ is hyper-regular, also there exist invertible matrices $V_{[A_\gamma, -B]}$ and $U_{[A_\gamma, -B]}$ such that

$$V_{[A_\gamma, -B]} [A_\gamma, -B] U_{[A_\gamma, -B]} = [I_n, 0_{n,m}]. \quad (3.17)$$

Moreover, $rank(RA_\gamma) = \min(rank(R), rank(A_\gamma)) = n-m$, and there exist a $(n-m) \times (n-m)$ matrix $V_{RA_\gamma} \in L - \text{Smith}(RA_\gamma)$ and a $n \times n$ matrix $U_{RA_\gamma} \in R - \text{Smith}(RA_\gamma)$ such that:

$$V_{RA_\gamma} RA_\gamma U_{RA_\gamma} = [\Delta_{n-m}, 0_{n-m,m}].$$

Knowing (3.10), we define the invertible matrices

$$T_1 = \begin{bmatrix} I_m & 0_{m,n-m} \\ 0_{n-m,m} & V_{RA_\gamma} \end{bmatrix} V_B = \begin{bmatrix} V_1 \\ V_{RA_\gamma} R \end{bmatrix}$$

and

$$T_2 = \begin{bmatrix} 0_{n,m} & U_{RA_\gamma} \\ -U_B & -U_B V_1 A_\gamma U_{RA_\gamma} \end{bmatrix}.$$
The product

$$ T_1[A_\gamma, -B] T_2 = \begin{bmatrix} V_1 A_\gamma & -V_1 B \\ V_{RA_\gamma} R A_\gamma & -V_{RA_\gamma} R B \end{bmatrix} T_2 \\ = \begin{bmatrix} I_m & 0_{m,n} \\ 0_{n-m,m} & [\Delta_{n-m}, 0_{n-m,m}] \end{bmatrix}, $$

which also corresponds to a Smith decomposition of $[A_\gamma, -B]$.

Finally, $\Delta_{n-m} = I_{n-m}$, which proves that $RA_\gamma$ is hyper-regular.

□

### 3.3. Defining matrix characterization.

**THEOREM 3.4.** Following the assumptions of Theorem 3.2, the defining matrices $P$ and $Q$ are given by:

$$ P = U_F \begin{bmatrix} 0 \\ I \end{bmatrix}, \qquad (3.18) $$

$$ Q = U_B [I, 0] V_B A_\gamma U_F \begin{bmatrix} 0 \\ I \end{bmatrix}, \qquad (3.19) $$

where $U_B \in \mathbb{R} - \text{Smith}(B)$, $V_B \in \mathbb{L} - \text{Smith}(B)$, and $U_F \in \mathbb{R} - \text{Smith}(F)$ with $F = RA_\gamma$.

*Proof.* Let us first define $P$. By introducing the hyper-regular matrix $F = RA_\gamma$ which admits a Smith decomposition with $V_F \in \mathbb{L} - \text{Smith}(F)$ and $U_F \in \mathbb{R} - \text{Smith}(F)$, it then comes that:

$$ F = V_F^{-1} [I, 0] U_F^{-1}. $$

From (3.6), $RA_\gamma P = RBQ = 0$ as $RB = 0$. Also,

$$ FP = V_F^{-1} [I, 0] U_F^{-1} P = 0. $$

The first defining matrix $P$ can now be defined as:

$$ P = U_F \begin{bmatrix} 0 \\ I \end{bmatrix}. $$

Finally, to define $Q$, let us first introduce a matrix $T$, left inverse of $B$:

$$ TB = I. $$

Thanks to $B$'s Smith decomposition (3.8), $T$ can be easily defined as:

$$ T = U_B [I, 0] V_B. $$
<figure><img src="image_2.png" /><figcaption>FIGURE 1. Insulated aluminium rod heated by a resistor</figcaption></figure>

Now, by using (3.6), Q is expressed as:

$$ 
\begin{aligned}
Q &= TA_{\gamma}P \\
&= U_{B}[I, 0]V_{B}A_{\gamma}U_{F} \begin{bmatrix} 0 \\ I \end{bmatrix}.
\end{aligned}
$$

□

3.4. **Dynamic inversion of a fractional system.** If a trajectory has been defined for the system output y, the fractional flat output z can be obtained from this variable.

As all system variables can be expressed by the flat output z, so is the desired output $y_{des}$:

$$ y_{des} = CPz, \qquad (3.20) $$

with $x = Pz$.

THEOREM 3.5. With $W = CP$, we can define $V_W \in L - \text{Smith}(W)$ and $U_W \in \mathbb{R} - \text{Smith}(W)$ such that:

1) if $r \le m$, $U_W W V_W = [I, 0]$, then

$$ W_{inv} = U_W \begin{bmatrix} I \\ 0 \end{bmatrix} V_W; \qquad (3.21) $$

2) if $r > m$, $U_W W V_W = \begin{bmatrix} I \\ 0 \end{bmatrix}$, then

$$ W_{inv} = U_W [I, 0] V_W. \qquad (3.22) $$

4. Flatness applied to an aluminium rod

4.1. **Plant description.** A long aluminium rod heated by a resistor is considered. To ensure unidirectional heat transfer, the entire surface of the rod is insulated by a foam as shown in Figure 1. The input signal is a thermal flux generated by a resistor glued at one end and the output signal
is the temperature of the rod measured at a distance $x^* = 5$mm from the heated end. The injected heat flux is controlled by a computer through an on-off transistor with a controlled amplitude of the input voltage.

### 4.2. Physical modeling.
To demonstrate the fractional behavior of this thermal system, the rod is modeled under specific assumptions,

(1) the rod is perfectly isolated,

(2) the rod is considered as a semi-infinite homogeneous plane medium with conductivity $\lambda$ and diffusivity $\alpha$,

(3) at rest, the rod is at ambient temperature, so that there is no thermal exchange with the environment,

(4) losses on the surface where the thermal flux is applied are neglected. The electrical energy consumed in the resistor is assumed to be totally transformed into thermal energy and diffused by conduction unidirectionally.

the model linking the temperature to the injected flux density is fractional [2]. Care is taken so that assumption (3) is fulfilled. A time period of 24 hours is observed between two consecutive experiments so that the temperature of the aluminium rod cools down to the ambient temperature and hence initial conditions are zero at the beginning of each experiment, since there is initially no thermal exchange with the environment.

The *oosrivcf* algorithm, *srivcf* algorithm developed in [30] and combined with gradient-based differentiation order estimation, is applied (see e.g. [28] for the details) resulting to the following model

$$G(s) = \frac{1.394.10^{-5}}{s^{3\nu} - 0.069s^{2\nu} + 0.058s^{\nu}} \times e^{-3s}. \quad (4.1)$$

As assumptions (1) and (2) cannot be satisfied in practice, there is heat leakage. Moreover limit conditions must be taken into account on the true geometry of the aluminium rod. As a result, the commensurate order, which is an additional degree of freedom, converges to $\nu = 0.26$ and not 0.5 as expressed theoretically in [2].

### 4.3. Linear fractional flatness.
The aim here is to control the temperature at a specific point of the rod $x^* = 5$mm. From model (4.1), the pseudo-state-space representation is deduced as follows:

$$A = \begin{bmatrix} 0.069 & -0.058 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix}, C = \begin{bmatrix} 0 & 0 & 1.394.10^{-5} \end{bmatrix} \quad (4.2)$$

$$B = \begin{bmatrix} 1 & 0 & 0 \end{bmatrix}^T, D = 0.$$
Let us apply fractional flatness to path planning. It is assumed that the first and second derivatives of the temperature are equal to zero at initial and final temperatures. The metallic rod will have to rise 30°C above the initial temperature in 1250 seconds and will stabilize for the same duration, according to the following chosen path, a Polynomial Interpolation of degree (PI5):

$$y(t) = q_i + 80(q_f - q_i) \frac{t^3}{t_f^3} - 240(q_f - q_i) \frac{t^4}{t_f^4} + 192(q_f - q_i) \frac{t^5}{t_f^5}, \quad (4.3)$$

with $q_i = 0°C$, $q_f = 30°C$, and $t_f = 2500s$. The desired output is illustrated in full line in the third plot of Figure 2.

Equation (4.2) has the form of (3.1) with

$$A_{\gamma} = \begin{bmatrix} s^{\nu} - 0.069 & 0.058 & 0 \\ -1 & s^{\nu} & 0 \\ 0 & -1 & s^{\nu} \end{bmatrix}. \quad (4.4)$$

Then, compute a matrix R orthogonal to B:

$$R^T = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \quad (4.5)$$

$$R^T A_{\gamma} = \begin{bmatrix} -1 & s^{\nu} & 0 \\ 0 & -1 & s^{\nu} \end{bmatrix}. \quad (4.6)$$

Thus, according to Theorem 3.2, a flat output is obtained by computing $P(s)$ solution of $R^T A_{\gamma} P = \begin{bmatrix} -P_1 + s^{\nu}P_2 \\ -P_2 + s^{\nu}P_3 \end{bmatrix} = 0$, which yields $P(s) = \begin{bmatrix} s^{2\nu} & s^{\nu} & 1 \end{bmatrix}^T$. $P(s)$ is clearly of rank 1 for all $s^{\nu}$. Finally, the input command $u$ (to be applied to the system in order to get the desired output), and the state $x$ are computed according to (3.4). Thus, the effective output is also generated with (3.20). Figure 2 illustrates the input command $u$, the fractional flat output (which has no physical meaning) and the effective output which is compared to the desired path.

## 5. Conclusion

The mathematical background on fractional calculus has been thoroughly studied so that flatness properties may be extended to controllable fractional systems. Also, integrability and differentiability properties of the fractional derivative have enabled us to well define the signal space and therefore to work in a commutative algebra on $D_a^{\gamma}$-polynomials and $D_a^{\gamma}$-polynomial matrices. These tools accompanied with the Smith diagonal decomposition were necessary to extend the flatness principles to fractional systems. Consequently, an improved characterization of fractional linear
<figure><img src="image_3.png" /><figcaption>FIGURE 2. Simulation of the metallic rod in open-loop: input commands ($u = \Phi$), flat output, desired and effective (dashed) outputs</figcaption></figure>

flat outputs for linear fractional systems has been proposed in polynomial matrix form. The algebraic formulation of flatness enables to determine the fractional flat outputs that define completely the trajectories of a fractional linear system. Finally, flatness properties have been applied to a real heat application: the model linking the measured temperature and the injected flux density is non integer. Simulations have been presented, where flatness principles enable to define the density flux input in order to follow a desired temperature path. For future works, the closed loop control of the heat system will be studied. A robust and efficient control could be proposed, such as a CRONE controller, in order to guarantee robust path tracking.

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Univ. Bordeaux, CNRS, IMS UMR 5218, Bordeaux INP/
ENSEIRB-MATMECA, Talence, 33405, France

e-mails: { stephane.victor, pierre.melchior } @ims-bordeaux.fr