# Comparison between the Second and Third Generations of the CRONE Controller: Application to a Thermal Diffusive Interface Medium

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## Abstract

**:**

## 1. Introduction

- A crossover frequency ω
_{cg}= 1 rad/s; - A phase margin M
_{φ}= 3 dB;

## 2. Plant Modeling

#### 2.1. Partial Differential Equations (PDE)

_{d}represents the thermal diffusivity of the material, and T(x,t) represents the temperature at the position x for time t.

#### 2.2. Plant Transfer Function

_{d}is the thermal effusivity, λ represents the thermal conductivity, ρ is the medium density, and C

_{p}is the medium heat.

_{Lx}≈ ω

_{L}. Thus, Equation (3) can be written as follows, where this latter defines the validation model of the system:

_{x}= α

_{d}/x

^{2}.

#### 2.3. Material Characteristics

_{L}and ω

_{x}for three values of the length, L, and the temperature sensor position, x.

## 3. CRONE Controllers Presentation

_{cg}. Thus, the phase margin variation only results from the plant variation. This strategy has to be used when frequency ω

_{cg}is within a frequency range where the plant phase is constant. In this range, the plant variations are only gain-like. This first generation uses the a priori calculation where the controller transfer function is calculated directly based on the user specifications.

_{cg}, and the plant phase variation is canceled by those of the controller. Thus, there is no phase margin variation when the frequency of ω

_{cg}varies. Such a controller produces a constant open-loop phase whose Nichols locus is a vertical straight line named the frequency template. This controller is synthesized a posteriori, where its transfer function is deduced from the open-loop transfer function.

## 4. Second CRONE Generation

#### 4.1. Introduction

_{A}, ω

_{B}]):

_{cg}is the frequency for which the uncertainties do not lead to any phase variation, n ∈

**ℝ**and n ∈ [1,2].

_{cg}, the Black–Nichols plot of the open-loop transfer function β(s) is a vertical asymptote with a constant phase equal to n, as shown in Figure 2. This asymptote allows having [48]:

- a robust phase margin M
_{φ}equal to (2−n)×π/2; - a robust resonance factor Q
_{T}, defined as follows:$${Q}_{T}=\frac{\underset{\omega}{\mathrm{sup}}\left|T\left(j\omega \right)\right|}{\left|T\left(j0\right)\right|}=\frac{1}{\mathrm{sin}\left(n\pi /2\right)};$$ - a robust gain module M
_{m}, defined as follows:$${M}_{m}=\underset{\omega}{\mathrm{inf}}\left|\beta \left(j\omega \right)+1\right|={\left(\underset{\omega}{\mathrm{sup}}\left|S\left(j\omega \right)\right|\right)}^{-1}=\mathrm{sin}\left(n\pi /2\right);$$

_{l}and ω

_{h}represent the low and high transitional frequencies, n is the fractional order (varying between 1 and 2) around the frequency ω

_{cg}, n

_{l}and n

_{h}are the system behavior at low and high frequencies, and β

_{0}is a constant value that assures a crossover frequency ω

_{cg}. It is expressed in Equation (10):

_{A}, ω

_{B}], and it belongs to the nominal crossover frequency ω

_{cgnom}. In order to respect the robustness of the stability degree, it is necessary to define the margins for ω

_{cg}, such as:

_{l}and ω

_{h}), which help getting the fractional order behavior between ω

_{A}and ω

_{B}. Previous studies have shown that it is sufficient for ω

_{l}to be one decade less ω

_{B}, whereas for ω

_{l}, it must be one decade above ω

_{h}[49].

_{cgnom}as being the geometric median of ω

_{l}and ω

_{h}. Added to that, a new parameter r, being the ratio of ω

_{B}and ω

_{A}, is introduced:

_{l}and ω

_{h}could be written with respect to ω

_{cgnom}and r as follows:

_{0}(jω):

_{F}(jω) to the rational form C

_{R}(jω). Several methods could be applied in this case; however, one simple method is based on the representation of the function using a recursive distribution of poles and zeros. Each pole and zero form a cell. The higher the number of cells, the most accurate the results are, but the more complex the transfer function would be. However, four to eight cells would be enough, as the fractional frequency range is below three decades [50]. Another option exists in using the CRONE toolbox, which can give the rational representation of the fractional form since it knows the frequency response of C

_{F}(jω) [51,52].

#### 4.2. First Case Study

#### 4.2.1. Plant Parameters

- -
- Aluminum, L = 1 m and x = 0.5 cm → ω
_{L}= 0.97 10^{−4}rad/s and ω_{x}= 3.88 rad/s; - -
- Copper, L = 1.1 m and x = 0.55 cm → ω
_{L}= 0.97 10^{−4}rad/s and ω_{x}= 3.87 rad/s; - -
- Iron, L = 0.49 m and x = 0.243 cm → ω
_{L}= 0.96 10^{−4}rad/s and ω_{x}= 3.89 rad/s.

#### 4.2.2. Synthesis Model

_{2}(s) in Equation (16) will be considered as the synthesis model that will be easier to use in order to calculate the controller transfer function. Interested readers can refer to a previous work of the authors for more details about the calculations [53].

_{2}(jω) (in blue) and of H(x,jω,L) (in green) obtained with aluminum for L = 1 m and x = 0.5 cm. It is well noted the coherence of both plots (for the gain and the phase) in the frequency range around ω

_{cg}.

#### 4.2.3. Controller Transfer Function

- -
- n
_{l}= 2, in order to assure a null training error; - -
- n
_{h}= 1.5, in order to limit the input sensitivity; - -
- Q
_{T}= 3 dB or M_{Φ}= 45° → n = (180°−M_{Φ})/90° = 1.5; - -
- ω
_{cgnom}= 1 rad/s ;

^{z}when z tends towards zero.

_{x})

^{0.5}, one can obtain:

_{x}= 3.88 rad/s.

_{cg}. For the low frequencies, the controller has an integrator behavior, whereas for high frequencies, it has a proportional behavior.

#### 4.2.4. Performance Analysis

#### 4.3. Second Case Study

#### 4.3.1. Plant Parameters

- -
- Aluminum, L = 1 m and x = 0.5 cm → ω
_{L}= 0.97 10^{−4}rad/s and ω_{x}= 3.88 rad/s; - -
- Copper, L = 1.1 m and x = 1 cm → ω
_{L}= 0.97 10^{−4}rad/s and ω_{x}= 1.17 rad/s; - -
- Iron, L = 0.49 m and x = 0.1 cm → ω
_{L}= 0.96 10^{−4}rad/s and ω_{x}= 23 rad/s.

#### 4.3.2. Synthesis Model

_{L}and ω

_{x}remain unchanged as they were presented for the first case study (refer to Section 4.2.2 and Section 4.2.3).

#### 4.3.3. Controller Transfer Function

#### 4.3.4. Performance Analysis

_{R}(s) controller.

## 5. Third CRONE Generation

#### 5.1. Introduction

_{i}and s = +jω ∈ C

_{j}. The real order a determines the phase placement in the Nichols chart, whereas the imaginary part b shows its angle with respect to the vertical axis, as shown in Figure 10.

_{m}(s) is the set of models defined within a frequency range, which allows us to write:

#### 5.2. CRONE Toolbox

#### 5.3. Case Study

#### 5.3.1. Plant Parameters

#### 5.3.2. Synthesis Model

#### 5.3.3. Performance Analysis

_{R}(s) (a), the Nichols plot of the open loop (b), the Bode gain diagrams for the sensitivity functions S(s) (c) and T(s) (d) (as presented in system (6)), for aluminum (in blue), copper (in green) and iron (in red).

## 6. Conclusions and Future Works

- -
- Implement this system on a real test bench;
- -
- Study the accuracy of this system when varying the position of the temperature sensors; this deviation is due involuntarily when implementing the test bench;
- -
- Apply other regulators to control this fractional order plant as the sliding mode control (with its multiple types), H
_{inf}robust control, and much more; - -
- Introduce some estimators to evaluate the temperature value at some location where the temperature sensor can’t be placed.

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Representation of diffusive interface medium along with the sensors and the heating element.

**Figure 4.**Bode plots of H(x,jω,L) for aluminum (in blue), copper (in green), and iron (in red) for the first case study.

**Figure 5.**Bode plots of P

_{2}(jω) (in blue) and of H(x,jω,L) (in green) obtained with aluminum for L = 1 m and x = 0.5 cm.

**Figure 6.**Bode diagrams for the fractional controller ${\tilde{C}}_{F}\left(s\right)$ (in blue) and the rational form ${C}_{R}\left(s\right)$ (in green).

**Figure 7.**Open-loop Black–Nichols plots (

**a**), closed-loop step responses (

**b**), control inputs for a step input of 1 °C (

**c**) for aluminum (in blue), copper (in green), and iron (in red).

**Figure 8.**Bode plots of H(x,jω,L) for aluminum (in blue), copper (in green), and iron (in red) for the second case study.

**Figure 9.**Open-loop Black–Nichols plots (

**a**) and closed-loop step responses (

**b**) for aluminum (in blue), copper (in green) and iron (in red).

**Figure 11.**Frequency responses for: Bode diagrams of the controller C

_{R}(s) (

**a**), the Nichols plot of the open-loop function (

**b**), the sensitivity function S(s), and (

**c**) complementary sensitivity function T(s) (

**d**) for aluminum (in blue), copper (in green), and iron (in red).

**Figure 12.**Time domain responses for: closed-loop step input regarding the temperature T(t,x) (

**a**) and the input control signal (

**b**) for a step input of 1 °C for aluminum (in blue), copper (in green), and iron (in red).

Material | ${\mathit{\alpha}}_{\mathit{d}}$ | ${\mathit{\eta}}_{\mathit{d}}$ | ${\mathit{H}}_{0}$ | ${\mathit{\varpi}}_{\mathit{L}}$ (rad/s) | ${\mathit{\varpi}}_{\mathit{x}}$ (rad/s) | ||||
---|---|---|---|---|---|---|---|---|---|

m^{2}/s | W·K^{−1}·m^{−2}·s^{0.5} | K·s^{0.5}·W^{−1} | L = 0.25 m | L = 0.5 m | L = 1 m | x = 0 cm | x = 0.5 cm | x = 1 cm | |

Cop. | 117 × 10^{−6} | 3.72 × 10^{4} | 0.269 | 19 × 10^{−4} | 4.68 × 10^{−4} | 1.17 × 10^{−4} | Infinite | 4.68 | 1.17 |

Alu. | 97 × 10^{−6} | 2.41 × 10^{4} | 0.416 | 16 × 10^{−4} | 3.88 × 10^{−4} | 0.97 × 10^{−4} | Infinite | 3.88 | 0.97 |

Iro. | 23 × 10^{−6} | 1.67 × 10^{4} | 0.596 | 3.68 × 10^{−4} | 0.92 × 10^{−4} | 0.23 × 10^{−4} | Infinite | 0.92 | 0.23 |

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**MDPI and ACS Style**

Moreau, X.; Abi Zeid Daou, R.; Christophy, F.
Comparison between the Second and Third Generations of the CRONE Controller: Application to a Thermal Diffusive Interface Medium. *Fractal Fract.* **2018**, *2*, 5.
https://doi.org/10.3390/fractalfract2010005

**AMA Style**

Moreau X, Abi Zeid Daou R, Christophy F.
Comparison between the Second and Third Generations of the CRONE Controller: Application to a Thermal Diffusive Interface Medium. *Fractal and Fractional*. 2018; 2(1):5.
https://doi.org/10.3390/fractalfract2010005

**Chicago/Turabian Style**

Moreau, Xavier, Roy Abi Zeid Daou, and Fady Christophy.
2018. "Comparison between the Second and Third Generations of the CRONE Controller: Application to a Thermal Diffusive Interface Medium" *Fractal and Fractional* 2, no. 1: 5.
https://doi.org/10.3390/fractalfract2010005