Fractional Order, Interval Parameter, State Space Model of the One-Dimensional Heat Transfer Process

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Fractional Order, Interval Parameter, State Space Model of the One-Dimensional Heat Transfer Process

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Krzysztof Oprzędkiewicz^*

This version is not peer-reviewed

Submitted:

11 June 2024

Posted:

12 June 2024

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Abstract

In the paper the new, fractional order, state space model of a heat transfer in a one- 1 dimensional metallic rod is addressed. The fractional orders of derivative along space and time 2 are not exactly known and they are described by intervals. The proposed model is the interval 3 expanding of the state space fractional model of the one dimensional heat transfer. It is expected to 4 better describe the reality because interval order of each real process is hard to estimate. Elementary 5 properties of the proposed model are analysed. Theoretical results are experimentally validated using 6 data from real experimental laboratory system.

Keywords:

Subject: Engineering - Control and Systems Engineering

1. Introduction

The modeling of processes and plants hard to describe using another mathematical tools is one of the main areas of application of the Fractional Order (FO) calculus.

Fractional order models of various physical phenomena have been presented by many Authors for years. Fundamental results are presented e.g. by [1,2] (the heating of an one dimensional beam), [3] (FO models of chaotic systems and Ionic Polymer Metal Composites), [4]. FO models of diffusion processes are considered u.a. by [5] [6], and [7]. A collection of results using new Atangana-Baleanu operator can be found in [8]. This book presents i.e. the FO blood alcohol model, the Christov diffusion equation and fractional advection-dispersion equation for groundwater transport processes.

Recently FO models are used u.a. to describe a spread of diseases. As examples can be given the modeling of the dynamics of COVID using Caputo-Fabrizio operator presented by [9] or the model of transmission of Zika virus using Atangana-Baleanu operator is proposed by [10].

Models of temperature fields are considered by many Authors, e.g. [11,12]. Analytical solution of the two-dimensional, IO heat transfer equation is presented in the paper [13]. Numerical methods of solution of PDE-s can be found, e.g., in [14]. Fractional Fourier integral operators are analyzed u.a. by [15]. It is important to note that a significant part of known investigations deals only with a steady-state temperature fields with omitting their dynamics.

The state space FO models of the one dimensional heat transfer have been proposed in many previous papers of author from years [16,17,18,19,20,21,22,23]. These models used different FO operators: Grünwald-Letnikov, Caputo, Caputo-Fabrizio and Atangana-Baleanu as well as discrete operators CFE and FOBD. Each proposed model has been thoroughly theoretically and experimentally verified. In addition, each of them assures better accuracy in the sense of square cost function than its IO analogue.

The sensitivity of the FO model to uncertainty of coefficients describing the heat conduction and heat exchange was analysed in the paper [24]. In this paper the uncertainty of fractional orders has not been considered. Such an analysis is given in this paper.

The papers [25] and [26] present the generalization of FO models mentioned above to the two dimensional case.

This article presents the generalization of the model given by [20] to the case when both fractional orders are described by intervals. Such an approach should better describe reality, because really the fractional orders are hard to estimate in contrast to parameters describing the heat transfer and heat conduction. From knowledge of author it turns out that such a model has not been proposed yet. The interval fractional order system was considered e.g. by [27], but this paper deals with the interval state operator and known interval order of derivative along the time.

The paper is organized as follows. Preliminaries recall some basic ideas from fractional calculus, necessary to present results. Next the experimental system and its FO model with interval orders is presented and analyzed. Finally theoretical results are validated using experiments.

2. Preliminaries

At the beginning the non integer order, integro-differential operator is presented (see e.g. [1,4,28,29]).

Definition 1.

(The elementary non integer order operator) The non integer order integro-differential operator is defined as follows:

_{a} D_{t}^{α} g (t) = \{\begin{matrix} \frac{d^{α} g (t)}{d t^{α}} α > 0 \\ g (t) α = 0 \\ \int_{a}^{t} g (τ) {(d τ)}^{- α} α < 0 \end{matrix} .

(1)

where a and t denote time limits for operator calculation,

α \in R

denotes the non integer order of the operation.

Next an idea of complete Gamma Euler function is recalled (see for example [29]):

Definition 2.

(The complete Gamma function)

Γ (x) = \int_{0}^{\infty} t^{x - 1} e^{- t} d t .

(2)

An idea of Mittag-Leffler function needs to be given next. It is a non integer order generalization of exponential function

e^{λ t}

and it plays crucial role in the solution of a FO state equation. The two parameter Mittag-Leffler function is defined as follows:

Definition 3.

(The two parameter Mittag-Leffler function)

E_{α, β} (x) = \sum_{k = 0}^{\infty} \frac{x^{k}}{Γ (k α + β)} .

(3)

For

β = 1

we obtain the one parameter Mittag-Leffler function:

Definition 4.

(The one parameter Mittag-Leffler function)

E_{α} (x) = \sum_{k = 0}^{\infty} \frac{x^{k}}{Γ (k α + 1)} .

(4)

The fractional order, integro-differential operator (1) can be expressed by three fundamental definitions, given by Grünwald and Letnikov (GL definition), Riemann and Liouville (RL definition) and Caputo (C definition). Relations between Caputo and Riemann-Liouville, between Riemann-Liouville and Grünwald-Letnikov operators are given e.g. in [4,30]. Discrete versions of these operators are deeply analysed by [31]. The C definition has a simple interpretation of an initial condition (it is analogical as in integer order case) and intuitive Laplace transform. Additionally its value from a constant equals to zero, in contrast to e.g. RL definition. That’s why in the further consideration the C definition will be applied. It is recalled beneath.

Definition 5.

(The Caputo definition of the FO operator)

_{0}^{C} D_{t}^{α} f (t) = \frac{1}{Γ (V - α)} \int_{0}^{\infty} \frac{f^{(V)} (τ)}{{(t - τ)}^{α + 1 - V}} d τ .

(5)

In (5) V is a limiter of the non integer order:

V - 1 \leq α < V

. If

V = 1

then consequently

0 \leq α < 1

is considered and the definition (5) takes the form:

_{0}^{C} D_{t}^{α} f (t) = \frac{1}{Γ (1 - α)} \int_{0}^{\infty} \frac{\dot{f} (τ)}{{(t - τ)}^{α}} d τ .

(6)

Finally a fractional linear state equation using Caputo definition needs to be recalled. It takes the following form:

\begin{matrix} _{0}^{C} D_{t}^{α} x (t) = A x (t) + B u (t) \\ y (t) = C x (t) \end{matrix} .

(7)

where

α \in (0, 1)

is the fractional order of the state equation,

x (t) \in R^{R}

u (t) \in R^{L}

y (t) \in R^{P}

are the state, control and output vectors respectively,

A, B, C

are the state, control, and output matrices.

3. The Experimental System and Its Model with Fractional, Interval Orders

The experimental heat plant is shown in Figure 1. It has the form of a thin copper rod

260 [m m]

long. To simplify assume that its length equals to 1.0. This allows to express the localization and length of heater and RTD sensors with respect to 1.0. The rod is heated by the electric heater

Δ x_{0} = 0.14

long located at the end of the rod. An output temperature is measured with the use of miniature RTD sensors (Pt100) long

Δ x

attached in points: 0.29, 0.50 and 0.73 of rod length. The input signal of the system is the standard current from range

0 - 20 [m A]

. It is amplified to the range

0 - 1.5 [A]

and it is the input signal for the heater. Signals from the RTD-s are read directly by analog input module in the PLC. Data from PLC are collected by SCADA application. The whole system is connected via PROFINET. The temeprature distribution with respect to time and length is shown in the Figure 2.

The fundamental, time continuous model describing the heat conduction and dissipation in the rod is the partial differential equation of the parabolic type with the homogeneous Neumann boundary conditions at the ends, the homogeneous initial condition, the heat exchange along the length of rod and distributed control and observation. This equation with integer orders of both differentiations has been discussed in many papers, for example in [32,33,34].

The FO expanding of this model is presented with details in the papers [16,17]. In this paper its version with interval fractional order along the length is proposed. It takes the following form:

\{\begin{matrix} ^{C} D_{t}^{α} Q (x, t) = a_{w} \frac{\partial^{β} Q (x, t)}{\partial x^{β}} - R_{a} Q (x, t) + b (x) u (t) \\ \frac{\partial Q (0, t)}{d x} = 0, t \geq 0 \\ \frac{\partial Q (1, t)}{d x} = 0, t \geq 0 \\ Q (x, 0) = Q_{0}, 0 \leq x \leq 1 \\ y (t) = k_{0} \int_{0}^{1} Q (x, t) c (x) d x . \end{matrix}

(8)

In (8)

a_{w}

R_{a}

are known coefficients of heat conduction and heat exchange,

k_{0}

is a steady-state gain.

The non integer orders

α

and

β

are not exactly known and they are described by the following intervals:

α = [\underset{̲}{α}; \bar{α}] \subset R^{+} .

(9)

where:

\begin{matrix} \underset{̲}{α} = α_{o} (1 - δ), \\ \bar{α} = α_{o} (1 + δ) . \end{matrix}

(10)

where

0.0 < δ < 1.0

. This deviation can be also expressed in percents:

0 < δ < 100 %

β = [\underset{̲}{β}; \bar{β}] \subset R^{+} .

(11)

where:

\begin{matrix} \underset{̲}{β} = β_{o} (1 - Δ), \\ \bar{β} = β_{o} (1 + Δ) . \end{matrix}

(12)

where

0.0 < Δ < 1.0

or equivalently

0 < Δ < 100 %

The intervals

α

and

β

build the vector of uncertain orders q:

q = [α; β] \subset I (R^{2}) .

(13)

All the vectors q build the space of uncertain orders Q:

Q = {q = q = [α; β] : \underset{̲}{α} \leq α \leq \bar{α}, \underset{̲}{β} \leq β \leq \bar{β}} .

(14)

The set Q can be interpreted as the rectangle in the

R^{2}

plane stretched on the following vertices:

\begin{matrix} q_{l l} = [\underset{̲}{α}, \underset{̲}{β}], \\ q_{l h} = [\underset{̲}{α}, \bar{β}], \\ q_{h l} = [\bar{α}, \underset{̲}{β}], \\ q_{h h} = [\bar{α}, \bar{β}] . \end{matrix}

(15)

The center of the rectangle Q is defined by the nominal values of orders:

q_{o} = [α_{o}, β_{o}] .

(16)

The interval form of orders

α

and

β

expands the heat transfer equation (8) to the infinite set of equations, limited by vertices (15).

The differential equation (8) can be expressed as beneath:

\{\begin{matrix} ^{C} D_{t}^{α} Q (t) = A Q (t) + B u (t) \\ Q (0) = Q_{0} \\ y (t) = k_{0} C Q (t) \end{matrix}

(17)

where:

\{\begin{matrix} A Q = a_{w} \frac{\partial^{β} Q (x)}{\partial x^{β}} - R_{a} Q, \\ D (A) = \{Q \in H^{2} (0, 1) : Q^{'} (0) = 0, Q^{'} (1) = 0\}, \\ a_{w}, R_{a} > 0, \\ H^{2} (0, 1) = \{u \in L^{2} (0, 1) : u^{'}, u^{″} \in L^{2} (0, 1)\}, \\ C Q (t) = 〈 c, Q (t) 〉, B u (t) = 〈 b u (t) 〉 \\ Q (t) = {[q_{1} (t), q_{2} (t) . .]}^{T} . \end{matrix}

(18)

In (18)

D (A)

denotes the field of the state operator A,

^{'}

^{″}

denotes the first and second derivative along the length,

< . . >

is the standard scalar product.

The basis of the state space is spanned by eigenvectors of the state operator A:

h_{n} = \{\begin{matrix} 1, n = 0 \\ \sqrt{2} c o s (n π x), n = 1, 2, . . . \end{matrix} .

(19)

3.1. The Decomposition of the Spectrum

The eigenvalues of the state operator A depend only on interval order

β

λ_{n} (β) = - a_{w} {(n π)}^{β} - R_{a}, n = 0, 1, 2, . . .

(20)

and consequently for the interval

β_{I}

each eigenvalue expands to the interval too:

λ_{I n} = [λ_{n} (\underset{̲}{β}); λ_{n} (\bar{β})] .

(21)

Consequently the diagonal state operator has interval, diagonal form:

A_{I} = d i a g {λ_{I 0}, λ_{I 1}, λ_{I 2}, . . .} .

(22)

The spectrum

σ

of the state operator A is a set of all its eigenvalues:

σ (A_{I}) = {λ_{I 0}, λ_{I 1}, λ_{I 2}, . . .} .

(23)

For exactly known

β

the spectrum contains negative, real, single, separated eigenvalues ordered by indices n:

λ_{0} > λ_{1} > λ_{2} . . .

. The most poorly damped eigenvalue

λ_{0}

is called the dominant (leading) eigenvalue. This property allows to easily decompose a system into single, scalar subsystems associated to particular eigenvalues. This is presented with details by [16,17].

However for interval

β_{I}

situation gets more complicated due to particular eigenvalues expand to intervals. This implies that eigenvalues associated to different modes can partially overlap. In such a situation the spectrum decomposition is impossible because there exist two or more different, indistinguishable eigenvalues. This situation is illustrated by the Figure 3.

The above situation can occur if the interval

β_{I}

is too wide. Simultaneously, narrow range of uncertainty guarantees the distinguishability of the spectrum. The maximum values of

Δ_{d}

and

Δ_{a}

assuring the distinguishability are described by the following Propositions.

Proposition 1.

(The maximum size of

Δ_{d}

assuring the distinguishability of any two eigenvalues) Consider the interval spectrum (23) of the system (17). Assume that the relative uncertainty of the order β is equal Δ.

The size

Δ_{d}

assuring the distinguishability of two eigenvalues

n_{1}

and

n_{2}

is expressed by the following inequality:

Δ_{d} \leq \frac{L - 1}{L + 1},

(24)

where:

L = \frac{ln (n_{2} π)}{ln (n_{1} π)} .

(25)

Proposition 2.

(The maximum size of

Δ_{a}

assuring the distinguishability of two adjacent eigenvalues)

Consider the interval spectrum (23) of the system (17). Assume that the relative uncertainty of the order β is equal Δ.

The size

Δ_{a}

assuring the distinguishability of the two adjacent eigenvalues n and

n + 1

is expressed by the following inequality:

Δ_{a} \leq \frac{L_{a} - 1}{L_{a} + 1},

(26)

where:

L_{a} = \frac{ln (n π)}{ln ((n + 1) π)} .

(27)

The proof of both propositions is given beneath.

Proof.

Consider two different interval eigenvalues and denote they by

n_{1}

and

n_{2}

. Assume that they are partially overlapped. This is expressed as follows:

\begin{matrix} λ_{n_{1}} (\bar{β}) \geq λ_{n_{2}} (\underset{̲}{β}) \Leftrightarrow \\ - a_{w} {(n_{1} π)}^{β + Δ β} - R_{a} \geq - a_{w} {(n_{2} π)}^{β - Δ β} - R_{a} \Leftrightarrow \\ {(n_{1} π)}^{β (1 + Δ)} \geq {(n_{2} π)}^{β (1 - Δ)} . \end{matrix}

(28)

Using logarithm function we obtain:

β (1 + Δ) ln (n_{1} π \geq β (1 - Δ) ln (n_{2} π) .

(29)

After some elementary transformations we obtain (24) and the proof is completed. Assuming

n_{1} = n

and

n_{2} = n + 1

gives directly condition (26). □

It is important to note that the maximum uncertainty permitted from point of view of spectrum decomposition does not depend on the value of

β

and it is determined only by location of eigenvalues, described by their indices.

Using the condition (24) the maximum order N assuring the distinguishability of all eigenvalues for given uncertainty

Δ

can be given too. It is described by the following proposition.

Proposition 3.

(The maximum dimension of the model N guaranteeing the distinguishability of the spectrum )

Consider the interval spectrum (23) of the system (17). Assume that the relative uncertainty of the order β is equal Δ.

The maximum size N of the model assuring the distinguishability of all eigenvalues meets the following inequality:

\frac{N}{{(N - 1)}^{D}} \geq π^{D - 1},

(30)

where:

D = \frac{1 + Δ}{1 - Δ} .

(31)

Proof.

Computing L from (24) we obtain:

L \geq D

Assuming

n_{2} = N

and

n_{1} = N - 1

in (25) yields:

\begin{matrix} ln (n) + ln (π) \geq D ln (N - 1) + D ln (π) \Leftrightarrow \\ ln (N) - D (ln (N - 1)) \geq (D - 1) ln (π) \Leftrightarrow \\ ln (\frac{N}{{(N - 1)}^{D}}) \geq ln (π^{(D - 1)}) . \end{matrix}

(32)

From (32) we obtain (30) and the proof is completed. □

The dependence (30) gives the implicit condition to estimate N. However it can be easily applied numerically. This will be shown in the section “Simulations”.

3.2. The Input and Output Operators

Next recall the form of input and output operators, presented e.g. in [35]. They do not depend on interval orders of the system.

The input operator B describes the location and construction of heater. It is as follows:

B = {[b_{0}, b_{1}, b_{2}, . . .]}^{T} .

(33)

where

b_{n} = 〈 b, h_{n} 〉

b (x)

is the shaping function of the heater:

b (x) = \{\begin{matrix} 1, x \in [0, x_{u}] \\ 0, x \notin [0, x_{u}] \end{matrix} .

(34)

According to (19) and (34) each element

b_{n}

takes the following form:

b_{n} = \{\begin{matrix} x_{u}, n = 0, \\ \frac{\sqrt{2} s i n (n π x_{u})}{n π}, n = 1, 2, . . . \end{matrix} .

(35)

The output operator C describes the location and size of RTD-s. It is as beneath:

C = [c_{0}, c_{1}, c_{2}, . . .] .

(36)

In (36)

c_{n} = 〈 c, h_{n} 〉

c (x)

is the output sensor function:

c (x) = \{\begin{matrix} 1, x \in [x_{1}, x_{2}] \\ 0, x \notin [x_{1}, x_{2}] \end{matrix} .

(37)

Using (19) and (37) yields:

c_{n} = \{\begin{matrix} x_{2} - x_{1}, n = 0, \\ \frac{\sqrt{2} (s i n (n π x_{2}) - s i n (n π x_{1}))}{n π}, n = 1, 2, . . . \end{matrix}

(38)

It is important to note that the operators B and C do not depend on the uncertain orders

α

and

β

The shaping function

b (x)

and sensor function

c (x)

are piecevise constant functions.

3.3. The Step and Impulse Responses of the System with Interval Orders

If the control is the Heaviside function:

u (t) = 1 (t)

then the analytical formula of step response is as beneath:

y_{\infty} (t) = \sum_{n = 0}^{\infty} \frac{(E_{α} (λ_{n} t^{α}) - 1 (t))}{λ_{n}} b_{n} c_{n} .

(39)

where

E_{α}

is the one parameter Mittag-Leffler function,

λ_{n}

b_{n}

and

c_{n}

are expressed by (20), (35) and (38) respectively. Analogically we can compute the impulse response:

g_{\infty} (t) = \sum_{n = 0}^{\infty} c_{n} b_{n} t^{α - 1} E_{α, α} (λ_{n} t^{α}) .

(40)

In (40)

E_{α, α}

is two parameter Mittag-Leffler function.

The non integer order model described by (17) - (40) is infinite dimensional. It is obvious that its practical use requires its reduction to finite dimensional form. This can be done by truncating further modes in state equation (17) and computing solution (39) or (40) as a finite sum. Consequently operators A, B and C reduce to matrices as well as the both time responses take the form of finite sums:

y (t) = y_{0} \sum_{n = 1}^{N} \frac{(E_{α} (λ_{I n} t^{α}) - 1 (t))}{λ_{I n}} b_{n} c_{n} .

(41)

g (t) = \sum_{n = 0}^{N} c_{n} b_{n} t^{α - 1} E_{α, α} (λ_{I n} t^{α}) .

(42)

In (41) and (42) N is the order of finite approximation. Its value determines the accuracy of the model. It can be estimated using numerical or analytical methods (see [17]).

The time responses (41) and (42) are functions of interval orders collected in the vectors (13):

y (t) = y (t, q)

g (t) = g (t, q)

. This expands they to sectors limited by the borders of intervals

α

and

β

\begin{matrix} y_{l l} (t) = y (t, q_{l l}), \\ y_{l h} (t) = y (t, q_{l h}), \\ y_{h l} (t) = y (t, q_{h l}), \\ y_{h h} (t) = y (t, q_{h h}) . \end{matrix}

(43)

\begin{matrix} g_{l l} (t) = g (t, q_{l l}), \\ g_{l h} (t) = g (t, q_{l h}), \\ g_{h l} (t) = g (t, q_{h l}), \\ g_{h h} (t) = g (t, q_{h h}) . \end{matrix}

(44)

In (43) and (44)

q_{. .}

are defined by (15). Analogically the nominal time responses are expressed as follows:

y_{o} (t) = y (t, q_{o}),

(45)

g_{o} (t) = g (t, q_{o}) .

(46)

where

q_{o}

is the vector of nominal parameters (16).

The most obvious way to describe the sensitivity of the step response

y (t)

to uncertainty of the orders

α

is to propose the sensitivity functions in the form

\frac{\partial y (t)}{\partial δ}

and

\frac{\partial y (t)}{\partial Δ}

. However such explicit functions are hard to compute.

The sensitivity can be also tested using the following relative dynamic deviations

E_{. .} (t)

E_{. .} (t) = \frac{e_{. .} (t)}{y_{o} (t)} \cdot 100 % .

(47)

where

_{. .} =_{l l},_{l h},_{h l},_{h h}

and:

\begin{matrix} e_{l l} (t) = | y_{o} (t) - y_{l l} (t) |, \\ e_{l h} (t) = | y_{o} (t) - y_{l h} (t) |, \\ e_{h l} (t) = | y_{o} (t) - y_{h l} (t) |, \\ e_{h h} (t) = | y_{o} (t) - y_{h h} (t) | . \end{matrix}

(48)

3.4. The Steady-State Response

The steady-state response of the system is as follows:

y_{N}^{s s} (β) = \sum_{n = 0}^{N} y_{n}^{s s} (β),

(49)

where

y_{n}^{s s}

are expressed as follows:

y_{n}^{s s} (β) = \frac{b_{n} c_{n}}{a_{w} {(n π)}^{β} + R_{a}},

(50)

in (50)

b_{n}

and

c_{n}

are described by (35) and (38) respectively.

The responses

y_{n}^{s s}

do not depend on the order

α

, but they are affected by the interval order

β

only. From the point of view of the geometric interpretation the rectangle (14), (15) reduces to the section limited by the limits of the order

β

. This is expressed as beneath:

\begin{matrix} lim_{t \to \infty} y_{n l l} (t) = lim_{t \to \infty} y_{n h l} (t) = y_{n l}^{s s}, \\ lim_{t \to \infty} y_{n l h} (t) = lim_{t \to \infty} y_{n h h} (t) = y_{n h}^{s s} . \end{matrix}

(51)

where nominal and limit values re described a s follows:

\begin{matrix} y_{n l}^{s s} = y_{n}^{s s} (\bar{β}), \\ y_{n o}^{s s} = y^{s s} (β_{o}), \\ y_{n h}^{s s} = y_{n}^{s s} (\underset{̲}{β}) . \end{matrix}

(52)

and consequently the whole response (49) takes the form of the following interval:

y_{N h, o, l}^{s s} = \sum_{n = 0}^{N} y_{n h, o, l}^{s s} .

(53)

The relative deviation

E^{s s}

of the steady-state response from its nominal value is the limit value of deviation

E_{. .} (t)

expressed by (47):

E_{l, h}^{s s} = lim_{t \to \infty} E_{l l, h l, l h, h h} (t) .

(54)

Using (53) the deviation (54) is expressed as:

E_{l, h}^{s s} = \frac{| y_{N o}^{s s} - y_{N l, h}^{s s} |}{y_{N o}^{s s}} \cdot 100 % .

(55)

4. Experiments and Simulations

The experiments were done using the system presented in the section I. The experimental step response was realized by change of input signal from

40 %

60 %

of its maximum range. The time-spatial temperature distribution is shown in the Figure 2. During experiments the sample time was equal:

h = 1 [s]

, the number of samples was equal:

K_{s} = 300

and the final time was equal:

t_{f} = 300 [s]

. The size and location of heater and sensors are given in the Table 1.

The analysis of sensitivity was done using parameters of the model estimated numerically via minimization of the MSE cost function for all sensors together (see [17]- caption of the Fig. 6). These parameters are recalled in the Table 2. The values of

α

and

β

are set as the nominal values

α_{o}

and

β_{o}

The impact of uncertainties

δ

and

Δ

to model was examined numerically. Results are presented in the next subsections.

4.1. The Analysis of the Spectrum Decomposition

At the beginning the impact of the uncertainty

Δ

to spectrum decomposition was tested using conditions (24), (26) and (30).

The general condition (24) was examined for indices

n_{1}

and

n_{2}

describing rather distant eigenvalues. Results are completed in the Table 3.

Next the distinguishability of adjacent eigenvalues was tested using condition (26). Results are illustrated by the Figure 4.

The Table 3 and Figure 4 show that the distinguishability is harder to keep for “further” eigenvalues with higher indices n.

Finally the maximum order N assuring the distinguishability of the whole spectrum was computed using condition (30). To make it easier denote the left and right side of inequality (30) by

f_{1}

and

f_{2}

\begin{matrix} f_{1} (N, Δ) = \frac{N}{{(N - 1)}^{D}}, \\ f_{2} (N, Δ) = π^{D - 1} . \end{matrix}

(56)

Using this notation the condition (30) turns to the following form:

f_{1} (N, Δ) \leq f_{2} (N, Δ) .

(57)

The condition (57) was tested numerically for

Δ = 0.01

and

Δ = 0.005

. Numerical solution of the inequality (57) is shown in the Figure 5.

The diagrams in the Figure 5 show that for

Δ = 0.01

the maximum size of model is equal 13 and for

Δ = 0.005

this maximum size is equal 23. In general, the increasing of the uncertainty

Δ

causes the decreasing of the order of the model N assuring the distinguishability of the spectrum.

4.2. The Sensitivity of the Dynamics

The sensitivity of the step response to uncertainties of both orders was tested using dynamic relative deviations (47) computed using data from the Table 2. Both uncertainties was set to:

δ = 0.01 = 1 %

and

Δ = 0.01 = 1 %

. All vertex step responses for all outputs are compared to nominal one and experimental in the Figure 6. Time trends of all deviations for all outputs of the system

E_{. .} (t)

are shown in the Figure 7.

In diagrams Figure 6 and Figure 7 it can be noted that the shape and maximum of deviation (47) strongly depends on the observed output. It is smallest for sensor no 2 and biggest for sensor no 3. Next, the impact of the uncertainty

δ

goes to zero for

t \to \infty

4.3. The Sensitivity of the Steady-State Response

The relative steady-state error (55) was examined for small uncertainty included in the range

0 % < Δ < 2 %

and big uncertainty from range

0 % < Δ < 80 %

separately. Results are illustrated by Figure 8 and Figure 9.

The analysis of diagrams Figure 8 and Figure 9 allows to conclude that:

The sensitivity of the steady-state response to uncertainty $Δ$ strongly depends on the localisation of the RTD. Its is highest for the most distant sensor 3 and smaller for sensors 1 and 2.
For small uncertainty the deviation $E^{s s}$ is approximately a linear function of $Δ$ and both deviations $E_{h, l}^{s s}$ are approximately the same.
For bigger uncertainty the deviation $E^{s s}$ is a nonlinear function of $Δ$ and nonlinearity is variuos for different outputs. Additionally, the upper deviation $E_{h}^{s s}$ is much bigger than lower $E_{l}^{s s}$ .

5. Final Conclusions

The results presented in the paper can be summarized that the uncertainty of both fractional orders impacts to crucial properties of the model: the distinguishability of the spectrum, the step and steady-state responses. The disturbation of order

α

affects only to dynamics, but the disturbation of the order

β

expressed by

Δ

disturbs all considered properties of the model.

Next, the impact of the disturbation

Δ

is determined by its size and the place of measurement. Small disturbation (circa 1%) strongly disturbs the output 3 furthest from the heater and slightly interferes with the measurement at output 2, placed halfway along the length of the rod. In turn, a large disturbance (greater than 50%) disturbs output 1 the least.

Further research of the presented issues will include among others theoretical justification of numerical results and investigation of discrete, approximated models using CFE and FOBD approximations.

Acknowledgment

This paper was supported by the AGH project no 16.16.120.773.

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Figure 1. The experimental system

Figure 2. The spatial-time temperature distribution in the plant

Figure 3. The part of specrum with eigenvalues

λ_{n 1}

and

λ_{n 2}

: overlapped in the top and non overlapped in the bottom.

Figure 3. The part of specrum with eigenvalues

λ_{n 1}

and

λ_{n 2}

: overlapped in the top and non overlapped in the bottom.

Figure 4. The maximum uncertainty

Δ_{a}

assuring the distinguishability of two adjacent eigenvalues.

Figure 4. The maximum uncertainty

Δ_{a}

assuring the distinguishability of two adjacent eigenvalues.

Figure 5. The maximum N assuring the distinguishability of all eigenvalues for

Δ = 0.01

(left) and

Δ = 0.005

(right).

Figure 5. The maximum N assuring the distinguishability of all eigenvalues for

Δ = 0.01

(left) and

Δ = 0.005

(right).

Figure 6. The vertex step responses (43) of the model for

δ = 0.01

and

Δ = 0.01

for outputs 1-3 (1-top, 3-bottom)

Figure 6. The vertex step responses (43) of the model for

δ = 0.01

and

Δ = 0.01

for outputs 1-3 (1-top, 3-bottom)

Figure 7. The relative deviations (47) for

δ = 1 %

and

Δ = 1 %

for outputs 1-3 (1-top, 3-bottom).

Figure 7. The relative deviations (47) for

δ = 1 %

and

Δ = 1 %

for outputs 1-3 (1-top, 3-bottom).

Figure 8. The relative steady-state deviation (55) as a function of the small uncertainty

0 % < Δ < 2 %

for all outputs of the plant.

Figure 8. The relative steady-state deviation (55) as a function of the small uncertainty

0 % < Δ < 2 %

for all outputs of the plant.

Figure 9. The relative steady-state deviation (55) as a function of the big uncertainty

0 % < Δ < 80 %

for all outputs of the plant.

Figure 9. The relative steady-state deviation (55) as a function of the big uncertainty

0 % < Δ < 80 %

for all outputs of the plant.

Table 1. The size and the location of the heater and the sensors

$Δ x_{u}$	$Δ x$	$x_{1}$	$x_{2}$	$x_{3}$
0.14	0.06	0.26	0.47	0.70

Table 2. The parameters of the model

$α_{o}$	$β_{o}$	$a_{w}$	$R_{a}$	N
0.930289	1.998808	0.000410	0.0677066	22

Table 3. The maximum size

Δ_{d}

allowing to distinguish two different eigenvalues

Table 3. The maximum size

Δ_{d}

allowing to distinguish two different eigenvalues

$n_{1}$	$n_{2}$	$Δ_{d}$
2	10	0.3045
2	30	0.4242
5	20	0.2011
10	25	0.1173
15	25	0.0622
20	25	0.0262
25	30	0.0205

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Fractional Order, Interval Parameter, State Space Model of the One-Dimensional Heat Transfer Process

Abstract

1. Introduction

2. Preliminaries

3. The Experimental System and Its Model with Fractional, Interval Orders

3.1. The Decomposition of the Spectrum

3.2. The Input and Output Operators

3.3. The Step and Impulse Responses of the System with Interval Orders

3.4. The Steady-State Response

4. Experiments and Simulations

4.1. The Analysis of the Spectrum Decomposition

4.2. The Sensitivity of the Dynamics

4.3. The Sensitivity of the Steady-State Response

5. Final Conclusions

Acknowledgment

References

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