# Fractional Order PID Control of Rotor Suspension by Active Magnetic Bearings

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## Abstract

**:**

## 1. Introduction

## 2. Fractional Order Calculus and Control

#### 2.1. Fractional Order Calculus Definition and Applications

#### 2.2. Fractional Order PID Control

#### 2.2.1. Frequency Domain Characteristics

#### 2.2.2. Time Domain Characteristics

#### 2.3. Fractional Order PID Tuning Methods

#### 2.4. Formulation of the Objective Function

#### 2.4.1. Time Domain Objectives

#### 2.4.2. Frequency Domain Objectives

- Sensitivity function:$$S(s)=\frac{1}{1+L(s)};$$
- Complementary sensitivity function:$$T(s)=\frac{L(s)}{1+L(s)};$$
- Disturbance sensitivity:$${S}_{\mathrm{d}}(s)=\frac{G(s)}{1+L(s)};$$
- Control sensitivity:$${S}_{\mathrm{u}}(s)=\frac{C(s)}{1+L(s)},$$

- Disturbance rejection objective function:$${J}_{\mathrm{d}}={\u2225\frac{1}{s}{S}_{\mathrm{d}}(s)\u2225}_{\infty};$$
- Control output objective function:$${J}_{\mathrm{c}}={\u2225{S}_{\mathrm{u}}(s)\u2225}_{\infty};$$
- Robust stability objective function:$${J}_{\mathrm{S}}={\u2225{W}_{\mathrm{S}}(S)S(s)\u2225}_{\infty};$$
- Noise rejection objective function:$${J}_{\mathrm{T}}={\u2225{W}_{\mathrm{T}}(s)T(s)\u2225}_{\infty}.$$

- Set-point tracking objective function:$${J}_{\mathrm{t}}={\u2225\frac{1}{s}S(s)\u2225}_{2}.$$

#### 2.4.3. Optimization Algorithms

- Genetic Algorithm (GA)The GA algorithm is the heuristic optimization algorithm influenced by the concept of the population genetics. The following are the steps involved in the algorithm:
- Initialize population: The population of the search is set by converting the controller parameters to binary strings known as chromosomes, where each chromosome represents a possible solution of the problem. Note that the size of the population for each generation is set by the user at the beginning.
- Objective evaluation: Each generated chromosome is evaluated based on the specified objective function.
- Selection: Chromosomes will be selected based on the level of their fitness. The higher fitness level of an individual chromosome, the better chance it will be selected.
- Crossover: The selected chromosomes will randomly exchange some bit(s) to generate the offspring for an evaluation in the next iteration. This process helps expand the possibility of the search space.
- Mutation: The mutation operator will make some small random change to the surviving chromosomes. This process prevents the solutions from being trapped in local minima. Typically, a low mutation rate is chosen, otherwise the search will become totally random.
- Elitism: The best found solution in each generation may be lost in the subsequent generation due to the crossover and mutation processes. Therefore, elitism is introduced to simply copy the best found chromosomes to the next generation. Normally, the number of elites is chosen to be a small fraction of the overall population so that the optimization process is not biased on these solutions.

- Differential Evolution (DE)The DE algorithm is introduced by Storn and Prince in 1995 [37] and is essentially the refined version of the GA with some changes that overcome the disadvantage of the GA. The DE algorithm has an optimization process similar to the GA, except that the DE algorithm uses floating point number instead of bit representation for solution vectors [38]. Therefore, instead of using logical operators used in the GA, the DE algorithm uses arithmetic operators for mutation and crossover processes, which lower the computational complexity and facilitate greater flexibility in the design of the mutation distribution [39]. The optimization step of the DE algorithm is slightly different from the GA and can be summarized as follows:
- Initialize population: The population is initialized randomly and uniformly distributed in the range of the specified lower and upper bound of the variables. The size of the population generated is prescribed by the designer.
- Mutation: First, three solution vectors from the initial population are chosen randomly. Then the donor vector ${d}_{i}$ is generated by adding the weighted difference of the first two vectors, ${x}_{r1}$ and ${x}_{r2}$, to the base solution vector ${x}_{r0}$, as follows,$${d}_{i}={x}_{r0}+F({x}_{r1}-{x}_{r2}).$$There are some modifications to let ${x}_{r0}$ be the best solution from the initial population. Generally, the factor F can vary between 0 and 2, which makes the DE algorithm sensitive to the choice of F. As suggested in [38], a good initial value of F is between 0.5 and 1.
- Crossover: To increase the diversity of the population, the donor vector ${d}_{i}$ exchanges its components with the base vector ${x}_{r0}$ to form the trial vector ${u}_{i}$ with the crossover probability ${C}_{r}$. The range of ${C}_{r}$ is between 0 and 1, where the value of ${C}_{r}$ equals to 1 means that all components of the donor vector ${d}_{i}$ are replaced by the base vector ${x}_{r0}$. The initial good guess of ${C}_{r}$ value is 0.5 in order to maintain the diversity of the population.
- Selection: At this stage, the trial vector ${u}_{i}$ is evaluated. If the trial vector yields the lower value of the objective function, then it replaces the corresponding base vector ${x}_{r0}$ in the next generation. Otherwise, the base vector ${x}_{r0}$ is retained in the population. Hence, the population either gets better or remains the same (with respect to the minimization of the objective function), but never deteriorates.

The process is iterated and terminated when a specified number of iterations is exceeded or the value of the objective function falls below a prescribed value. For more variations of the mutation and crossover scheme, see [38]. - Particle Swarm Optimization (PSO)The PSO algorithm is a population-based stochastic optimization technique developed by Eberhart and Kennedy in 1995 [40]. The method is inspired by the social behavior of bird flocking, fish schooling, etc. Unlike the GA and DE algorithms that use the concept of the fittest to survive, the PSO algorithm uses the particles that constitute a swarm to move around the prescribed space in order to find the best solution. Each particle adjusts its position ${x}_{j}$ based on its experience from previous iterations as well as the experience of other particles. The two important values used for adjusting the moving direction in the concept of PSO are the best solution of each particle (pbest) and the best solution of the entire swarm (gbest). The searching algorithm to calculate the new position of each particle based on pbest and gbest is described in the following equations:$$\begin{array}{}(13\mathrm{a})& \hfill {v}_{j,N}^{(t+1)}& =w\xb7{v}_{j,N}^{(t)}+{c}_{1}{r}_{1}(pbes{t}_{j,N}-{x}_{j,N}^{(t)})+{c}_{2}{r}_{2}(gbest-{x}_{j,N}^{(t)}),\hfill (13\mathrm{b})& \hfill {x}_{j,N}^{(t+1)}& ={x}_{j,N}^{(t)}+{v}_{j,N}^{(t+1)},\hfill \end{array}$$
- -
- n is the number of particles (population size);
- -
- m is the dimension of the problem (number of variables);
- -
- ${c}_{1}$, ${c}_{2}$ are the acceleration factors;
- -
- ${r}_{1}$, ${r}_{2}$ are the uniformly distributed numbers between 0 and 1.

$$w={w}_{\text{max}}-\frac{{w}_{\text{max}}-{w}_{\text{min}}}{ite{r}_{\text{max}}}\times iter$$

#### 2.5. Fractional Order PID Controller Implementation

## 3. System Description and Modelling

#### 3.1. Overview of the Test Rig

#### 3.2. Rotor Lateral Dynamics

#### 3.3. Rotor Axial Dynamics

## 4. Fractional Order Control of Rotor Suspension

#### 4.1. Control Design Specifications

- Zone A: newly commissioned machines normally fall into this zone
- Zone B: acceptable for unrestricted long-term operation
- Zone C: unsatisfactory for long-term continuous operation
- Zone D: sufficiently severe to cause damage to the machine

#### 4.2. Design and Experimental Test of FOPID Controller for Rotor Lateral Dynamics

#### 4.2.1. Design of FOPID Controller

- Stability of closed-loop system (${J}_{1}$): Closed-loop stability will be determined by the number of poles that have a positive real part. The optimization goal of this objective is zero.
- Stability margin (${J}_{2}$): A peak magnitude of the sensitivity function as described in Section 2.4 will be used to determine a stability margin.
- Vibration level (${J}_{3}$): A maximum magnitude of forced responses among three cases (translate mode, conical mode, and overhung cantilevered) as illustrated in Figure 17. This objective will be used to determine the maximum vibration.
- Integral square error (ISE) of a unit step response (${J}_{4}$): Instead of specifying transient response performance separately, the performance index ISE will be used in order to reduce conflict between different performances.

#### 4.2.2. Experimental Test of Lateral Rotor Suspension

#### 4.3. Design and Experimental Test of FOPID Controller for Rotor Axial Dynamics

#### 4.3.1. Design of FOPID Controller

- Stability of the closed-loop system (${J}_{1}$): Closed-loop stability will be determined by the number of poles that have positive real part. The optimization goal of this objective is zero.
- Tracking error performance: ${J}_{2}={\u2225{W}_{3}S{W}_{2}\u2225}_{\infty}.$
- Control effort according to the reference input signal: ${J}_{3}={\u2225{W}_{4}SK{W}_{2}\u2225}_{\infty}.$
- Transmission of the input disturbance to the control output: ${J}_{4}={\u2225-{W}_{3}SG{W}_{1}\u2225}_{\infty}.$
- Closed-loop dynamics from the reference input to the rotor position: ${J}_{5}={\u2225-{W}_{4}T{W}_{1}\u2225}_{\infty}$.

#### 4.3.2. Experimental Test of Axial Rotor Suspension

## 5. Discussion

## Author Contributions

## Conflicts of Interest

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**Figure 17.**Unbalance values and locations as specified in API 617 [22].

**Figure 19.**Magnitude plots of the sensitivity functions of the FOPID controllers with different tuning algorithms.

**Figure 20.**Forced response with unbalance mass placing for three excitation cases as specified in API 617 [22].

**Figure 21.**Magnitude plots of the sensitivity functions at zero and maximum continuous speed under the FOPID controllers tuned by the DE algorithm.

**Figure 22.**Bode plots of the lateral AMB sensitivity function at the motor side and compressor side under three different controllers.

**Figure 29.**Bode plots of the thrust AMB sensitivity and complementary sensitivity functions under the FOPID controller tuned by the DE algorithm.

Radial AMB | I_{b} (A) | K_{x} (N/m) | K_{i} (N/A) |
---|---|---|---|

Motor side | 3 | 1.27 × 10^{6} | 199.34 |

Compressor side | 4 | 2.26 ×10^{6} | 265.86 |

Thrust AMB | 3 | 4.23 × 10^{6} | 664.12 |

AMB | Motor Side | Compressor Side | Thrust |
---|---|---|---|

Amplifier gain (A/V) | 1.5 | 1.5 | 1.5 |

Amplifier bandwidth (rad/s) | 5026.5 | 5026.5 | 5026.5 |

Sensor gain (V/m) | 3.937 × 10^{4} | 3.937 × 10^{4} | 3.937 × 10^{4} |

Sensor bandwidth (rad/s) | 1.26 × 10^{4} | 1.26 × 10^{4} | 1.26 × 10^{4} |

Maximum slew rate (N/s) | 2.2 × 10^{6} | 2.2 × 10^{6} | 1.9 × 10^{6} |

Zone Limit | Maximum Displacement | Peak Sensitivity |
---|---|---|

A/B | <0.3 ${C}_{min}$ | < 3 (9.5 dB) |

B/C | <0.4 ${C}_{min}$ | <4 (12 dB) |

C/D | <0.5 ${C}_{min}$ | <5 (14 dB) |

**Table 4.**Comparison of performances of the FOPID controllers tuned by different evolutionary algorithms.

Specifications | GA | PSO | DE |
---|---|---|---|

Sensitivity function peak | 2.4414 | 2.3947 | 2.2727 |

Peak unbalance vibration (mm) | 0.0029 | 0.0027 | 0.0024 |

Controller output peak (V) | 0.3909 | 0.3639 | 0.4176 |

Overshoot (%) | 6.1020 | 5.7632 | 2.7300 |

Rise time (s) | 0.0077 | 0.0077 | 0.0071 |

Settling time (s) | 0.0205 | 0.0161 | 0.0151 |

Bearing | K_{P} | K_{I} | K_{D} | λ | μ |
---|---|---|---|---|---|

Motor side (MS) | 0.1752 | 0.120 | 0.0011 | 0.752 | 0.942 |

Compressor side (CS) | 0.1795 | 0.112 | 0.0010 | 0.834 | 0.902 |

Specifications | PID | FOPID | LQG |
---|---|---|---|

Sensitivity function peak | 2.6742 | 2.2727 | 2.4794 |

Peak unbalance vibration (mm) | 0.0037 | 0.0024 | 0.0026 |

Controller output peak (V) | 0.4228 | 0.3340 | 0.1810 |

Overshoot (%) | 0.172 | 0.033 | 0.178 |

Rise time (s) | 0.003 | 0.003 | 0.005 |

Settling time (s) | 0.023 | 0.042 | 0.016 |

Bandwidth (rad/s) | 12,757 | 13,759 | 14,377 |

Controller dimension as implemented | 6 | 7 | 11 |

Controller Characteristics | GA | DE | PSO |
---|---|---|---|

Infinity norm of objective functions | 1.061 | 1.034 | 1.035 |

Bearing | K_{P} | K_{I} | K_{D} | λ | μ |
---|---|---|---|---|---|

Thrust AMB | 0.0743 | 11.3933 | 0.00468 | 0.6249 | 0.9068 |

Specifications | IOPID | FOPID | H_{∞} |
---|---|---|---|

Sensitivity function peak | 3.312 | 2.482 | 2.433 |

Infinity norm of objectives | 1.380 | 1.034 | 0.907 |

Controller bandwidth (rad/s) | 2045 | 2841 | 2754 |

Controller order as implemented | 4 | 6 | 8 |

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**MDPI and ACS Style**

Anantachaisilp, P.; Lin, Z. Fractional Order PID Control of Rotor Suspension by Active Magnetic Bearings. *Actuators* **2017**, *6*, 4.
https://doi.org/10.3390/act6010004

**AMA Style**

Anantachaisilp P, Lin Z. Fractional Order PID Control of Rotor Suspension by Active Magnetic Bearings. *Actuators*. 2017; 6(1):4.
https://doi.org/10.3390/act6010004

**Chicago/Turabian Style**

Anantachaisilp, Parinya, and Zongli Lin. 2017. "Fractional Order PID Control of Rotor Suspension by Active Magnetic Bearings" *Actuators* 6, no. 1: 4.
https://doi.org/10.3390/act6010004