# A Model-Based Prognostic Framework for Electromechanical Actuators Based on Metaheuristic Algorithms

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Related Work

## 3. Materials and Methods

- ${k}_{1}$: Dry friction. When ${k}_{1}=1$, the resulting friction is the nominal value multiplied by three.
- ${k}_{2}$: Backlash. When ${k}_{2}=1$, the backlash magnitude is the nominal value multiplied by one hundred.
- ${k}_{3}$, ${k}_{4}$, ${k}_{5}$: Short circuit (SC). Being a three-phase motor, each coefficient is linked to a short circuit in one phase.
- ${k}_{6}$, ${k}_{7}$: Static eccentricity. These coefficients are linked to the modulus and phase of the eccentricity in the rotor. Under nominal conditions, the phase corresponds to 0 rad, so ${k}_{7}=0.5$.
- ${k}_{8}$: Proportional gain drift (PGD). ${k}_{8}=1$ is linked to an increase of 50 per cent in the proportional gain, while ${k}_{8}=0$ determines a 50 per cent decrease. The nominal value is ${k}_{8}=0.5$.

#### 3.1. Employed Algorithms

- PSO, since it resulted as one of the most used ones;
- DE, to represent the evolutionary algorithm category;
- GWO, which we selected among new algorithms.

#### 3.1.1. Evolutionary Algorithms

- Population: The solution “pool”, which is initialized at the start of the process;
- Variety: The population must be varied enough to explore the solution space effectively;
- Heredity: This values is linked to the capability of passing a characteristic to the offspring;
- Selection: For artificial algorithms, selection must only occur in the desired direction, which is a key parameter to ensure that only the best solutions will be reproduced.

**Differential Evolution.**One of the most famous EA is the differential evolution strategy [43,44], which is also one of the tested algorithms in this work. The main concept of this algorithm follows the genetic principles.

#### 3.1.2. Swarm Intelligence Methods

**Particle Swarm Optimization.**Being a SI algorithm, PSO [46] draws its inspiration from the movement of bird flocks or fish schools. In fact, starting from a population of potential solutions (i.e., particles) and moving them throughout the search space, this methodology can solve optimization problems by following rigid mathematical formulas. As said before, the optimization is guaranteed by the fact that there is a capillary and diffused intelligence: the movement of each particle, and hence, its path, is affected by each particle’s local best known position and the best known locations in the search space (these are known because the knowledge is shared by particles). That is precisely why the swarm is able to iteratively identify the optimum solutions by information sharing. Some initial parameters shall be defined, such as the population size, particle initial placements and speed, and particle inertia. After the initialization set up, each particle is given a random neighborhood, and by travelling, the best overall position is discovered. The position associated with the optimal global location are updated, so that each particle knows it. A detailed examination of the solution space is possible, thanks to the velocities’ inherent stochastic component [47].The pseudo-code for this algorithm is reported in Figure 4. More information on the algorithm implementation can be found in [41]. The employed code routine iteratively runs until 200 iterations are reached or until the error between two successive runs is less than ${10}^{-9}$.

**Grey Wolf Optimization.**If PSO was generically inspired by birds’ and fishes’ movement, GWO [48] has a more precise inspiring animal: wolves. This optimization technique follows the idea of the rigid hierarchical scales among grey wolf population’s members. After choosing the size of the wolf pack and the initial positions of each “animal”, an initial population hierarchy is defined by looking at the fitness function values. The decision of the number of wolves is crucial, as both the accuracy of the algorithm and the execution time are affected by this parameter. The higher the fitness value, the higher the hierarchical position in the scale. In this way, the individuals with lower scores will be less influential in the optimization process, while the better-positioned animals will lead the process. In other words, each wolf represents a distinct solution to the problem.

#### 3.2. Models

- Controller: This block is essentially composed of a PID controller. In fact, even if there are much more advanced and sophisticated control logics (e.g., [53]), PID controllers are still the way to go and they are still chosen even in complex systems, as they are easy to implement and tune. PID controllers are composed of three separated branches, where the proportional, differential, and integral action are calculated. The controller aim is comparing the command signals with the actual signal obtained from the motor transmission dynamics block, hence closing the control loop. In this particular case, both position and speed can be monitored. This block outputs the reference current ${I}_{REF}$, obtained from the motor torque thanks to the torque constant, which is finally passed to the inverter.
- Inverter: This block contains Clarke-Park equations, and it provides the motor block with the three voltages (one for each phase) for the PMSM motor by performing the corresponding pulse width modulation (PWM). A very complicated physics-based process is handled by Simscape, a specific Simulink library, capable of providing electrical simulation packages. The main actions inside this block are the calculation of the electrical angle starting from the motor position, the splitting of ${I}_{REF}$ into the three phase currents (with Clarke-Park equations), the PWM process, and the calculation of the three phase voltages, using the fed-back currents.
- Sinusoidal BLDC motor: This block is able to simulate the electrical and magnetic interactions inside a PMSM. It contains Simscape elements, and it manages three main processes:
- The calculation of the counter-electromotive force coefficient ${c}_{j}$ for each phase. This is achieved with the multiplication of the back EMF coefficients (obtained with experimental test campaigns) with three sine waves ${120}^{\circ}$ out of phase from each other.
- The implementation of the motor resitive-inductive circuit. A set of mathematical equations (Equation (5)) that model the three star connected LR branches is solved and phase currents (${i}_{j}$) are, hence, calculated. The resistance ${R}_{j}$ and inductance ${L}_{j}$ of the motor are taken from equipment data sheets.$$\begin{array}{c}\sum _{j=1}^{3}{i}_{j}=0\\ {V}_{j}-{c}_{j}\omega ={R}_{j}{i}_{j}+{L}_{j}\frac{\mathrm{d}{i}_{j}}{\text{}\mathrm{d}t},\end{array}$$
- The calculation of the motor available torque. Three different electromotive coefficients are used to calculate the motor torque along with the relative phase currents:$${T}_{m}=\sum _{j=1,2,3}{i}_{j}{c}_{j},$$

- Motor transmission dynamics: this final block compares the available torque with the external requested torque and solves a second-order dynamical system (Equation (7) comprehensive of multiple non linearities, such as dry friction and backlash [52]). The outputs of this block are the motor position and speed, which are looped back to the controller.$${T}_{m}-{T}_{l}={J}_{m}\frac{{\mathrm{d}}^{2}{\theta}_{m}}{\mathrm{d}{t}^{2}}+{C}_{m}\frac{\mathrm{d}{\theta}_{m}}{\mathrm{d}t}$$

## 4. Results

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

MEA | More Electric Aircraft |

EMA | Electro-Mechanical Actuator |

PHM | Prognostic and Health Management |

PMSM | Permanent Magnet Syncrhonous Motor |

DE | Differential Evolution |

PSO | Particle Swarm Optimization |

GWO | Grey Wolf Optimization |

NTB | Numerical Test Bench |

LCC | Life Cycle Costs |

EHA | Electro-Hydraulic Actuators |

CBM | Condition-Based Maintenance |

RAMS | Reliability, Availability, Maintenability, and Safety |

TLBO | Teaching–Learning-Based Optimization |

HGS | Hunger Game Search |

VNS | Variable Neighbourhood Search |

ACO | Ant Colony Optimization |

CSO | Cuckoo Search Optimization |

MM | Monitoring Model |

RM | Reference Model |

SC | Short Circuit |

FDI | Failure Detection and Identification |

ConOps | Concept of Operation |

TLP | Top Level Parameter |

MSA | Metaheuristic Search Algorithm |

EA | Evolutionary Algorithm |

SI | Swarm Intelligence |

GA | Genetic Algorithm |

BLDC | BrushLess Direct Current |

PID | Proportional Integral Derivative |

PWM | Pulse Width Modulation |

FMECA | Failure Mode Effect and Criticality Analysis |

PC | Performance Coefficient |

PHMC | Prognostic and Health Management Computer |

EMF | Electro-Motive Force |

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**Figure 1.**FDI methodology overview [10].

**Figure 3.**Pseudo-code for DE algorithm, as taken from [41].

**Figure 4.**Pseudo-code for PSO algorithm, as taken from [41].

**Figure 5.**Pseudo-code for GWO algorithm, as taken from [41].

**Figure 6.**RM model structure as taken from [41].

**Figure 7.**Comparison between different algorithms: mean percentage error [%]. For each failure mode and for each algorithm, the failure magnitudes are selected as follows: high magnitude (${k}_{i}=0.75$) and low one (${k}_{i}=0.25$).

**Figure 8.**Comparison between different algorithms: computational cost [s]. For each failure mode and for each algorithm, the failure magnitudes are selected as follows: high magnitude (${k}_{i}=0.75$) and low one (${k}_{i}=0.25$).

TLP | Physical Failure | Effect at $({\mathit{k}}_{\mathit{i}}=0)$ | Effect at $({\mathit{k}}_{\mathit{i}}=1)$ |
---|---|---|---|

${k}_{1}$ | Dry friction | No effect (Nominal friction) | $300\%$ of nominal friction |

${k}_{2}$ | Backlash | No effect (Nominal backlash) | 100 times nominal backlash |

${k}_{3}$ | Short circuit (Phase A) | No effect (No SC on Phase A) | Complete SC on phase A |

${k}_{4}$ | Short circuit (Phase B) | No effect (No SC on phase B) | Complete SC on phase B |

${k}_{5}$ | Short circuit (Phase C) | No effect (No SC on phase C) | Complete SC on phase C |

${k}_{6}$ | Eccentricity modulus | No effect (No eccentricity) | Maximum Eccentricity |

${k}_{7}$ | Eccentricity phase | $-{180}^{\circ}$ | ${180}^{\circ}$ |

${k}_{8}$ | PGD | $50\%$ of nominal proportional gain | $150\%$ of nominal proportional gain |

**Table 2.**PMSM motor parameters. The motor part number is S 1FK7060- 2AC71-1CA0 provided by Siemens. The numbers 60 K and 100 K refer to overtemperature values of 60 K and 100 K.

Characteristic | Value |
---|---|

Rated speed (100 K) | 2000 rpm |

Number of poles | 8 |

Rated torque (100 K) | 5.3 Nm |

Rated current | 3.0 A |

Static torque (60 K) | 5.00 Nm |

Static torque (100 K) | 6.0 Nm |

Stall current (60 K) | 2.55 A |

Stall current (100 K) | 3.15 A |

Efficiency | 90.00 |

**Table 3.**Different optimization algorithms outcomes with single failures [10]. The values related to best performing algorithm (PSO) are highlighted in bold, along with the relative PCs.

Failures | DE | PSO | GWO | ||||||
---|---|---|---|---|---|---|---|---|---|

Time (s) | Err. (%) | PC (%) | Time (s) | Err. (%) | PC (%) | Time (s) | Err. (%) | PC (%) | |

Friction | 3015 | 1.30 | 56.97 | 1342.5 | 1.30 | 80.76 | 2865.5 | 1.20 | 62.26 |

Backlash | 2895 | 1.45 | 50.21 | 980.5 | 1.18 | 86.28 | 2378 | 1.28 | 63.49 |

Short Circuit | 2925.5 | 3.08 | 52.32 | 1566 | 2.64 | 78.12 | 2709 | 2.13 | 69.54 |

Eccentricity | 2995.5 | 2.46 | 62.30 | 2202 | 2.45 | 72.45 | 2479.5 | 2.75 | 65.24 |

Prop. Gain | 2853 | 6.54 | 58.61 | 1403 | 6.38 | 80.14 | 2721.5 | 6.42 | 61.24 |

Total | 2936.8 | 2.96 | 56.75 | 1498.8 | 2.79 | 79.24 | 2530.7 | 2.75 | 64.00 |

**Table 4.**Different optimization algorithms outcomes with multiple failures [10]. The values related to best performing algorithm (PSO) are highlighted in bold, along with the relative PCs.

Time (s) | Err. (%) | PC (%) | |
---|---|---|---|

DE | 1777.0 | 4.21 | 60.95 |

PSO | 1131.4 | 3.37 | 80.09 |

GWO | 1816.6 | 4.33 | 58.94 |

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## Share and Cite

**MDPI and ACS Style**

Baldo, L.; Querques, I.; Dalla Vedova, M.D.L.; Maggiore, P.
A Model-Based Prognostic Framework for Electromechanical Actuators Based on Metaheuristic Algorithms. *Aerospace* **2023**, *10*, 293.
https://doi.org/10.3390/aerospace10030293

**AMA Style**

Baldo L, Querques I, Dalla Vedova MDL, Maggiore P.
A Model-Based Prognostic Framework for Electromechanical Actuators Based on Metaheuristic Algorithms. *Aerospace*. 2023; 10(3):293.
https://doi.org/10.3390/aerospace10030293

**Chicago/Turabian Style**

Baldo, Leonardo, Ivana Querques, Matteo Davide Lorenzo Dalla Vedova, and Paolo Maggiore.
2023. "A Model-Based Prognostic Framework for Electromechanical Actuators Based on Metaheuristic Algorithms" *Aerospace* 10, no. 3: 293.
https://doi.org/10.3390/aerospace10030293