# Error-Based Switched Fractional Order Model Reference Adaptive Control for MIMO Linear Time Invariant Systems

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## Abstract

**:**

## 1. Introduction

- This paper proposes an error-based switching mechanism in the design of SFOMRAC schemes for LTI systems. The switching uses the value of the control error to decide whether to use the fractional order or the integer order in the controller parameters adaptive laws. Compared to the previous work [11], the error-based switching is more appealing in practice because it allows for making decisions based on a system signal that can be measured and used as a metric of system performance and stability.
- The SFOMRAC is proposed in this paper for multivariable systems. This is an improvement regarding previous works ([11] and references therein), where only single input, single output systems were considered.
- A complete and thorough analytical proof of stability and convergence of the resulting design is provided in this paper, where the controller will not be limited in advance to switching by a finite amount, as it was in previous works ([11] and some references therein).
- The design and analysis is also carried out for cases when system states are affected by a bounded non-parametric disturbance. This non-ideal case was not addressed in any of the previous works.
- Exhaustive simulation studies are conducted, and numerical results show that the SFOMRAC allows for obtaining a better balance among performance indicator ITAE (Integral of the Time weighted Absolute value of the Error) and control energy ISI (Integral of the Squared Control Signal) for some switching error levels, leading to an improved control strategy compared to the classical non-switched integer-order (MRAC) and fractional-order (FOMRAC) schemes.

## 2. Basic Concepts

#### 2.1. Notation and Basic Definitions

#### 2.2. Elements of Fractional Calculus

**Definition 1**

**.**

**Definition 2**

**.**

#### 2.3. Analytical Tools

**Property 1.**

**Theorem 1**

**.**Consider the system of fractional order integral equations

- i.
- There exists a unique continuous solution $y\in \mathcal{C}[0,T]$ for system (5).
- ii.
- $y\in \mathcal{C}[0,T]$ is a solution for system (5) for$${p}_{i}\left(t\right):=\sum _{k=0}^{\u2308\alpha \u2309-1}\frac{{t}_{k}}{k!}{y}_{{i}_{0}}^{\left(k\right)}$$

**Theorem 2**

**.**For a given $x\in {\mathbb{R}}^{d}$, let $u\in \left\{x\right\}+{I}_{0+}^{\alpha}\mathcal{C}\left([0,T]\right),{\mathbb{R}}^{d}$ and $V:{\mathbb{R}}^{d}\to \mathbb{R}$ satisfies the following conditions

- The function V is convex on ${\mathbb{R}}^{d}$ and $V\left(0\right)=0$.
- The function V is differentiable on ${\mathbb{R}}^{d}$.

## 3. Problem Statement and Proposed Control Scheme

#### 3.1. Control Problem

**Assumption 1.**

**Assumption 2.**

#### 3.2. Proposed Control Strategy

#### 3.3. Closed-Loop Description

#### 3.4. Main Results

**Theorem 3.**

**Proof of Theorem 3.**

**Proof of claim i.**

**Proof of claim ii.**

**Proof of claim iii.**

**Proof of claim iv.**

**Proof of claim v.**

**Remark 1.**

**Remark 2.**

**Theorem 4.**

**Proof of Theorem 4.**

## 4. Influence of Controller Parameters in the Resulting Control Energy and System Performance: Simulation Studies

#### 4.1. Performance Indices to Evaluate a Controlled System

#### 4.2. Simulation Details

- If the ${E}_{max}\le 1$, then the set of switching error levels $\u03f5$ to be tested in simulations was selected within a 0.025 difference among them in the whole interval $(0,{E}_{max})$.
- If the $0.1<{E}_{max}\le 4$, then the set of switching error levels $\u03f5$ to be tested in simulations was selected within a 0.05 difference among them in the whole interval $(0,{E}_{max})$.
- If the ${E}_{max}>4$, then the set of switching error levels $\u03f5$ to be tested in simulations was selected within a 0.125 difference among them in the whole interval $(0,{E}_{max})$.

#### 4.3. Analysis of the Results Obtained from Simulation Studies

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Evolution of plant output and control signal for Plant 1 when controlled using the SFOMRAC, FOMRAC and IOMRAC, with the best results for three different scenarios, using Reference Model 3.

**Figure 2.**Evolution of plant output and control signal for Plant 3 when controlled using the SFOMRAC, FOMRAC and IOMRAC, with the best results for three different scenarios, using Reference Model 2.

**Figure 3.**Evolution of the plant output and control signal for Plant 3 when controlled using the SFOMRAC, FOMRAC and IOMRAC, with the best results for three different scenarios, using Reference Model 3.

${\mathit{A}}_{\mathit{p}}$ | ${\mathit{B}}_{\mathit{p}}$ | ${\mathit{A}}_{\mathit{m}}$ | ${\mathit{B}}_{\mathit{m}}$ | ||
---|---|---|---|---|---|

Plant 1 | −1 | 1 | Reference model 1 | −0.5 | 0.5 |

Plant 2 | −10 | 10 | Reference model 2 | −1 | 1 |

Plant 3 | 1 | 1 | Reference model 3 | −5 | 5 |

Plant 4 | 10 | 10 | Reference model 4 | −10 | 10 |

**Table 2.**Details of the schemes obtaining the lowest values for functional J for stable plants, using different weighting factors ${w}_{1},{w}_{2}$.

Plant 1: Stable with pole in $s=-1$ | |||||||||||

${\mathit{B}}_{\mathit{m}}$ | ${\mathit{A}}_{\mathit{m}}$ | ${\mathit{w}}_{\mathit{1}}$ | ${\mathit{w}}_{\mathit{2}}$ | min
$\mathit{J}$ | Controller | $\mathit{\gamma}$ | ${\mathit{\alpha}}_{\mathbf{0}}$$\mathit{\alpha}$ | ${\mathit{E}}_{\mathit{max}}$ | $\mathit{\u03f5}$ | $\mathit{ISI}$ | $\mathit{ITAE}$ |

1 | 0 | 0 | FOMRAC | 1 | 0.1 | - | - | 436.156 | 3435.24 | ||

0 | 1 | 0 | SFOMRAC | 10 | 0.5 | 0.0897 | 0.05 | 498.243 | 0.235 | ||

0.5 | −0.5 | 0.5 | 0.5 | 0.297 | SFOMRAC | 1 | 0.1 | 0.381 | 0.15 | 459.518 | 918.165 |

0.3 | 0.7 | 0.228 | SFOMRAC | 1 | 0.1 | 0.381 | 0.30 | 481.697 | 177.248 | ||

0.7 | 0.3 | 0.263 | SFOMRAC | 1 | 0.1 | 0.381 | 0.05 | 444.048 | 2129.20 | ||

1 | 0 | 0 | FOMRAC | 1 | 0.1 | - | - | 437.463 | 3403.41 | ||

0 | 1 | 0 | SFOMRAC | 10 | 0.5 | 0.161 | 0.05 | 500.033 | 0.346 | ||

1 | −1 | 0.5 | 0.5 | 0.297 | SFOMRAC | 1 | 0.1 | 0.461 | 0.15 | 460.725 | 896.975 |

0.3 | 0.7 | 0.226 | SFOMRAC | 1 | 0.1 | 0.461 | 0.30 | 480.037 | 219.197 | ||

0.7 | 0.3 | 0.264 | SFOMRAC | 1 | 0.1 | 0.461 | 0.10 | 445.614 | 2083.60 | ||

1 | 0 | 0 | FOMRAC | 1 | 0.1 | - | - | 438.530 | 3370.86 | ||

0 | 1 | 0 | SFOMRAC | 10 | 0.3 | 0.387 | 0.05 | 502.547 | 0.301 | ||

5 | −5 | 0.5 | 0.5 | 0.274 | SFOMRAC | 1 | 0.1 | 0.721 | 0.20 | 470.487 | 511.406 |

0.3 | 0.7 | 0.201 | SFOMRAC | 1 | 0.1 | 0.721 | 0.30 | 481.916 | 192.159 | ||

0.7 | 0.3 | 0.258 | SFOMRAC | 1 | 0.1 | 0.721 | 0.05 | 447.679 | 2008.66 | ||

1 | 0 | 0 | FOMRAC | 1 | 0.1 | - | - | 438.661 | 3336.77 | ||

0 | 1 | 0 | SFOMRAC | 10 | 0.3 | 0.541 | 0.05 | 503.534 | 0.298 | ||

10 | −10 | 0.5 | 0.5 | 0.276 | SFOMRAC | 1 | 0.1 | 0.815 | 0.15 | 465.228 | 765.521 |

0.3 | 0.7 | 0.201 | SFOMRAC | 1 | 0.1 | 0.815 | 0.30 | 484.020 | 157.073 | ||

0.7 | 0.3 | 0.259 | SFOMRAC | 1 | 0.1 | 0.815 | 0.05 | 448.148 | 1990.14 | ||

Plant 2: Stable with pole in $s=-10$ | |||||||||||

1 | 0 | 0 | FOMRAC | 1 | 0.1 | - | - | 435.101 | 3456.47 | ||

0 | 1 | 0 | SFOMRAC | 10 | 0.9 | 0.5185 | 0.05 | 497.045 | 0.104 | ||

0.5 | −0.5 | 0.5 | 0.5 | 0.317 | SFOMRAC | 1 | 0.1 | 0.352 | 0.15 | 458.488 | 895.718 |

0.3 | 0.7 | 0.249 | SFOMRAC | 1 | 0.1 | 0.352 | 0.25 | 475.929 | 262.677 | ||

0.7 | 0.3 | 0.273 | SFOMRAC | 1 | 0.1 | 0.352 | 0.05 | 442.999 | 2123.50 | ||

1 | 0 | 0 | FOMRAC | 1 | 0.1 | - | - | 436.386 | 3418.62 | ||

0 | 1 | 0 | SFOMRAC | 10 | 0.6 | 0.588 | 0.05 | 498.551 | 0.049 | ||

1 | −1 | 0.5 | 0.5 | 0.314 | SFOMRAC | 1 | 0.1 | 0.377 | 0.15 | 459.609 | 879.322 |

0.3 | 0.7 | 0.244 | SFOMRAC | 1 | 0.1 | 0.377 | 0.30 | 479.012 | 196.760 | ||

0.7 | 0.3 | 0.273 | SFOMRAC | 1 | 0.1 | 0.377 | 0.05 | 444.543 | 2077.013 | ||

1 | 0 | 0 | FOMRAC | 1 | 0.1 | - | - | 431.396 | 3390.67 | ||

0 | 1 | 0 | SFOMRAC | 10 | 0.4 | 0.157 | 0.15 | 499.800 | 0.008 | ||

5 | −5 | 0.5 | 0.5 | 0.317 | SFOMRAC | 1 | 0.1 | 0.447 | 0.15 | 462.557 | 788.290 |

0.3 | 0.7 | 0.243 | SFOMRAC | 1 | 0.1 | 0.447 | 0.25 | 475.490 | 294.819 | ||

0.7 | 0.3 | 0.279 | SFOMRAC | 1 | 0.1 | 0.447 | 0.05 | 446.567 | 2002.01 | ||

1 | 0 | 0 | FOMRAC | 1 | 0.1 | - | - | 437.528 | 3387.29 | ||

0 | 1 | 0 | SFOMRAC | 10 | 0.5 | 0.311 | 0.05 | 499.963 | 0.009 | ||

10 | −10 | 0.5 | 0.5 | 0.321 | SFOMRAC | 1 | 0.1 | 0.524 | 0.15 | 464.131 | 735.770 |

0.3 | 0.7 | 0.243 | SFOMRAC | 1 | 0.1 | 0.524 | 0.25 | 477.619 | 250.128 | ||

0.7 | 0.3 | 0.281 | SFOMRAC | 1 | 0.1 | 0.524 | 0.05 | 447.013 | 1982.203 |

**Table 3.**Details of the schemes obtaining the lowest values for functional J for unstable plants, using different weighting factors ${w}_{1},{w}_{2}$.

Plant 3: Unstable with pole in $s=1$ | |||||||||||

${\mathit{B}}_{\mathit{m}}$ | ${\mathit{A}}_{\mathit{m}}$ | ${\mathit{w}}_{\mathbf{1}}$ | ${\mathit{w}}_{\mathbf{2}}$ | min
$\mathit{J}$ | Controller | $\mathit{\gamma}$ | ${\mathit{\alpha}}_{\mathbf{0}}$$\mathit{\alpha}$ | ${\mathit{E}}_{\mathit{max}}$ | $\mathit{\u03f5}$ | $\mathit{ISI}$ | $\mathit{ITAE}$ |

1 | 0 | 0 | FOMRAC | 10 | 0.7 | - | - | 496.808 | 5.841 | ||

0 | 1 | 0 | FOMRAC | 10 | 0.8 | - | - | 496.872 | 2.882 | ||

0.5 | −0.5 | 0.5 | 0.5 | 0.0001 | FOMRAC | 10 | 0.7 | - | - | 496.808 | 5.841 |

0.3 | 0.7 | 0.00014 | FOMRAC | 10 | 0.7 | - | - | 496.808 | 5.841 | ||

0.7 | 0.3 | 0.00006 | FOMRAC | 10 | 0.7 | - | - | 496.808 | 5.841 | ||

1 | 0 | 0 | FOMRAC | 10 | 0.7 | - | - | 498.921 | 5.706 | ||

0 | 1 | 0 | FOMRAC | 10 | 0.9 | - | - | 500.300 | 2.529 | ||

1 | −1 | 0.5 | 0.5 | 0.00011 | FOMRAC | 10 | 0.7 | - | - | 498.921 | 5.706 |

0.3 | 0.7 | 0.00015 | FOMRAC | 10 | 0.7 | - | - | 498.921 | 5.706 | ||

0.7 | 0.3 | 0.00006 | FOMRAC | 10 | 0.7 | - | - | 498.921 | 5.706 | ||

1 | 0 | 0 | SFOMRAC | 10 | 0.5 | 0.447 | 0.05 | 502.681 | 7.637 | ||

0 | 1 | 0 | SFOMRAC | 10 | 0.8 | 0.553 | 0.05 | 505.226 | 1.105 | ||

5 | −5 | 0.5 | 0.5 | 0.00023 | SFOMRAC | 10 | 0.5 | 0.447 | 0.05 | 502.681 | 7.637 |

0.3 | 0.7 | 0.00033 | SFOMRAC | 10 | 0.5 | 0.447 | 0.05 | 502.681 | 7.637 | ||

0.7 | 0.3 | 0.00014 | SFOMRAC | 10 | 0.5 | 0.447 | 0.05 | 502.681 | 7.637 | ||

1 | 0 | 0 | SFOMRAC | 4 | 0.5 | 0.736 | 0.05 | 503.248 | 69.481 | ||

0 | 1 | 0 | SFOMRAC | 10 | 0.8 | 0.719 | 0.05 | 506.839 | 0.921 | ||

10 | −10 | 0.5 | 0.5 | 0.0017 | SFOMRAC | 6 | 0.5 | 0.684 | 0.05 | 503.395 | 14.353 |

0.3 | 0.7 | 0.0014 | SFOMRAC | 6 | 0.5 | 0.684 | 0.05 | 503.395 | 14.353 | ||

0.7 | 0.3 | 0.0015 | SFOMRAC | 4 | 0.5 | 0.736 | 0.05 | 503.248 | 69.481 | ||

Plant 4: Unstable with pole in $s=10$ | |||||||||||

1 | 0 | 0 | SFOMRAC | 7 | 0.3 | 0.053 | 0.05 | 497.173 | 52.706 | ||

0 | 1 | 0 | SFOMRAC | 10 | 0.8 | 0.491 | 0.35 | 497.227 | 0.346 | ||

0.5 | −0.5 | 0.5 | 0.5 | 0.00012 | SFOMRAC | 10 | 0.7 | 0.231 | 0.2 | 497.198 | 0.554 |

0.3 | 0.7 | 0.00008 | SFOMRAC | 10 | 0.7 | 0.231 | 0.2 | 497.198 | 0.554 | ||

0.7 | 0.3 | 0.00016 | SFOMRAC | 10 | 0.7 | 0.231 | 0.2 | 497.198 | 0.554 | ||

1 | 0 | 0 | SFOMRAC | 9 | 0.3 | 0.05 | 0.05 | 498.704 | 28.036 | ||

0 | 1 | 0 | SFOMRAC | 10 | 0.8 | 0.51 | 0.5 | 498.761 | 0.286 | ||

1 | −1 | 0.5 | 0.5 | 0.00011 | SFOMRAC | 10 | 0.7 | 0.234 | 0.2 | 498.726 | 0.601 |

0.3 | 0.7 | 0.00008 | SFOMRAC | 10 | 0.7 | 0.234 | 0.2 | 498.726 | 0.601 | ||

0.7 | 0.3 | 0.00014 | SFOMRAC | 10 | 0.7 | 0.234 | 0.2 | 498.726 | 0.601 | ||

1 | 0 | 0 | SFOMRAC | 10 | 0.2 | 0.089 | 0.075 | 499.862 | 6.280 | ||

0 | 1 | 0 | SFOMRAC | 10 | 0.9 | 0.791 | 0.75 | 500.154 | 0.216 | ||

5 | −5 | 0.5 | 0.5 | 0.00026 | SFOMRAC | 10 | 0.4 | 0.157 | 0.15 | 499.900 | 1.609 |

0.3 | 0.7 | 0.00023 | SFOMRAC | 10 | 0.4 | 0.157 | 0.15 | 499.900 | 1.609 | ||

0.7 | 0.3 | 0.00025 | SFOMRAC | 10 | 0.2 | 0.089 | 0.075 | 499.862 | 6.280 | ||

1 | 0 | 0 | SFOMRAC | 10 | 0.2 | 0.162 | 0.15 | 500.00 | 1.399 | ||

0 | 1 | 0 | SFOMRAC | 10 | 0.8 | 0.761 | 0.75 | 500.364 | 0.168 | ||

10 | −10 | 0.5 | 0.5 | 0.00008 | SFOMRAC | 10 | 0.2 | 0.162 | 0.15 | 500.00 | 1.399 |

0.3 | 0.7 | 0.00012 | SFOMRAC | 10 | 0.2 | 0.162 | 0.15 | 500.00 | 1.399 | ||

0.7 | 0.3 | 0.00005 | SFOMRAC | 10 | 0.2 | 0.162 | 0.15 | 500.00 | 1.399 |

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**MDPI and ACS Style**

Aguila-Camacho, N.; Gallegos, J.A.
Error-Based Switched Fractional Order Model Reference Adaptive Control for MIMO Linear Time Invariant Systems. *Fractal Fract.* **2024**, *8*, 109.
https://doi.org/10.3390/fractalfract8020109

**AMA Style**

Aguila-Camacho N, Gallegos JA.
Error-Based Switched Fractional Order Model Reference Adaptive Control for MIMO Linear Time Invariant Systems. *Fractal and Fractional*. 2024; 8(2):109.
https://doi.org/10.3390/fractalfract8020109

**Chicago/Turabian Style**

Aguila-Camacho, Norelys, and Javier A. Gallegos.
2024. "Error-Based Switched Fractional Order Model Reference Adaptive Control for MIMO Linear Time Invariant Systems" *Fractal and Fractional* 8, no. 2: 109.
https://doi.org/10.3390/fractalfract8020109