# GSA-Tuning IPD Control of a Field-Sensed Magnetic Suspension System

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## Abstract

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## 1. Introduction

## 2. IPD Control of the FSMSS

_{d}and $\varphi $ can be derived easily from a previous study [6]. To eliminate the steady-state error of $\Delta \tilde{x}(k)$, the I control is added. In this case, reference input R(z) is set to null. If the function of position tracking is enabled, then reference input R(z) is set properly. The I control can achieve a zero steady-state error. The proper tuning of parameters (K

_{d}, ϕ, K

_{i}) can yield effective transient and steady-state responses. The authors of previous studies [6,7] have conducted stability analysis on the IPD control of the MSS. The stability of the overall system is guaranteed when integrator parameter K

_{i}is suitably selected.

Symbol | Explanation |
---|---|

$m$ | the mass of the controlled object |

$g$ | the gravitational acceleration |

$C$ | the force constant |

$x$ | the distance between the electromagnet and suspended object |

${x}_{0}$ | the equilibrium position of the suspended object |

$\Delta x$ | $=x-{x}_{0}$ the deviation of the distance |

$\Delta \tilde{x}$ | the measured output of the position sensor device |

$\Delta X(s)$,$\Delta X(z)$ | the Laplace transform and z transform of $\Delta x$ |

$\Delta \tilde{X}(z)$ | the z transform of the measured output $\Delta \tilde{x}(k)$ |

$i$ | the coil current |

${i}_{0}$ | the bias current of the equilibrium position |

$\Delta i$ | $=i-{i}_{0}$ the deviation of the coil current |

$\Delta I(s)$,$\Delta I(z)$ | the Laplace transform and z transform of $\Delta i$ |

$T$ | the sampling period |

$\rho $ | the linear factor of the position sensor |

$\beta $ | = ${e}^{T\sqrt{2C{i}_{o}^{2}/m{x}_{o}^{3}}}$ the derived parameter |

$\tilde{\beta}$ | = $\beta +{\beta}^{-1}$ the derived parameter |

$\mathsf{\sigma}$ | = $\sqrt{C/2m{x}_{o}}$ the derived parameter |

$\tilde{\mathsf{\sigma}}$ | = $\sigma \rho \left({\beta}^{2}-1\right)/\beta $ the derived parameter |

$G(z)$ | the z transform of $\Delta X(z)/\Delta I(z)$ |

$\tilde{G}(z)$ | the z transform of $\Delta \tilde{X}(z)/\Delta I(z)$ |

$R(z)$ | the reference input of Figure 2 |

## 3. GSA-Tuning IPD Control

_{d}is the dimension of an agent, and N

_{m}is the number of agents. The velocity of agent i can be written as follows:

_{aj}represents the active gravitational mass related to agent j, and M

_{pi}is the passive gravitational mass related to agent i. G(t) is the gravitational coefficient at time t and decreases over time for controlling the search accuracy. G(t) can be determined as follows:

_{j}is a random number in the interval [0,1]. Kbest represents the set of first agents with greater mass and the optimal fitness value. Thus, on the basis of the law of motion, the acceleration of agent i at time t in the dth dimension is written as follows:

_{ii}(t) is the inertial mass of agent i. The next search step involves identifying the values of the subsequent velocity and position of the agent. Therefore, its position and velocity can be calculated as follows:

_{i}is a random number in the interval [0, 1], and ${v}_{i}^{d}$ is the velocity of agent i in the dth dimension. This random number is employed to equip the search with a randomized feature. Gravitational and inertial masses are calculated using the fitness evaluation. A heavier mass equates to a more efficient agent. This indicates that superior agents have higher attraction and move more slowly. The gravitational and inertial masses are assumed to be equal, as displayed in the following:

Symbol | Explanation |
---|---|

${X}_{i}=\left({x}_{i}^{1},\cdots ,{x}_{i}^{d},\cdots ,{x}_{i}^{Nd}\right)$ | position of agent i |

${x}_{i}^{d}$ | dth dimension of ${X}_{i}$ |

$Nd$ | dimension of an agent |

$Nm$ | number of agents |

$t$ | index of iteration |

${t}_{\mathrm{max}}$ | total number of iterations |

${V}_{i}=\left({v}_{i}^{1},\cdots ,{v}_{i}^{d},\cdots ,{v}_{i}^{Nd}\right)$ | velocity of agent i |

${v}_{i}^{d}$ | dth dimension of ${V}_{i}$ |

${F}_{ij}^{d}(t)$ | dth dimension of gravitational force where agent j acts on agent i |

${F}_{i}^{d}(t)$ | total force that acted on agent i in dth dimension |

$G(t)$ | gravitational coefficient at time t |

$G({t}_{0})$ | initial value of $G(t)$ |

$\alpha $ | a positive constant for $G(t)$ |

$\epsilon $ | a small positive constant for Equation (10) |

${M}_{aj}$ | active gravitational mass related to agent j |

${M}_{pi}$ | passive gravitational mass related to agent i |

${R}_{ij}(t)$ | Euclidian distance between two agents i and j |

Kbest | the set of first agents with larger mass |

$rand$ | random number in the interval [0,1] |

${a}_{i}^{d}(t)$ | acceleration of agent i at time t and in dth dimension |

${M}_{ii}(t)$ | inertial mass of agent i |

${M}_{i}(t)$ | equality mass assumption for the gravitational and inertia mass for Equation (17) |

${m}_{i}(t)$ | calculated variable for ${M}_{i}(t)$ |

$fi{t}_{i}(t)$ | fitting function (or fitness) |

$best(t)$ | strongest agent in the population |

$worst(t)$ | weakest agent in the population |

Procedure | Operation Details |
---|---|

Step 1: | Randomized initial controller parameters (K_{d}, ϕ, K_{i}) in the stable range Equations (6)–(7) of all agents. Set following parameters: $Nd$, $Nm$, ${t}_{\mathrm{max}}$, $G({t}_{0})$, $\alpha $, $\epsilon $, and $Kbest$. |

Step 2: | Execute the control system simulation (or experiment) for all agents of t-iteration. |

Step 3: | Calculate fitness $fi{t}_{i}(t)$. |

Step 4: | Calculate formulae sequentially for Equations (20), (21), (18), (19), (17), (11), (12), (10), (13), (14), (15), and (16). |

Step 5: | Update controller parameter position ${X}_{i}(t+1)$. Specify the stable range for Equations (6)–(7) of three controller parameters (K_{d}, ϕ, K_{i}). |

Step 6: | Check the stopping criteria. If they are satisfied, then stop. Otherwise, proceed to Step 2. |

_{d}, ϕ and K

_{i}to enable the output response $\Delta \tilde{X}(z)$ to track reference input $R(z)$. The search procedure of the proposed IPD–GSA control (IPD control by GSA tuning) is listed in Table 3. For explaining the proposed IPD–GSA control, three flowcharts were plotted, as displayed in Figure 4, Figure 5 and Figure 6. Figure 4 displays the system operation process. This process includes three main subprocesses: initialization, a control system simulation, and the GSA. A flowchart of the control system simulation is provided in detail in Figure 5. The program of the control system simulation is executed N

_{m}times for every iteration. The GSA flowchart is displayed in detail in Figure 6. First, the fitting functions are calculated for N

_{m}agents. Thereafter, the formulas of the GSA are calculated sequentially. Finally, the positions of agents are updated. The stable range of Equations (6)–(7) of the three controller parameters (K

_{d}, ϕ, and K

_{i}) is specified for every updated position.

## 4. Simulation

_{d}is obtained as follows:

_{m}= 4. Additionally, let the dimension of each agent $Nd$ = 3, with the position of ith agent as ${X}_{i}=\left({x}_{i}^{1},{x}_{i}^{2},{x}_{i}^{3}\right)$. Each agent contains three controller parameters (i.e., K

_{d}, ϕ, and K

_{i}); in other words, ${K}_{d}={x}_{i}^{1}$, $\varphi ={x}_{i}^{2}$, and ${K}_{i}={x}_{i}^{3}$. The design constraints of each agent are defined in Equations (28)–(29). The total number of iterations ${t}_{\mathrm{max}}$ is 100. The initial value of the gravitational coefficient $G({t}_{0})$ is set as 100, and $\alpha $ is set to 20. A small positive constant $\epsilon $ is set to 10. The set of first agents with greater mass $Kbest$ is set to the total number of agents ($Nm$). The fitness $fi{t}_{i}(t)$ for the i’s iteration is assigned as defined in the following function:

_{p}is the peak overshoot, ${\gamma}_{1}=10$ is the weighting factor of M

_{p}, and ${\gamma}_{2}=10$ is the weighting factor of the integral squared error.

_{p}and $\sum _{k=1}^{100}{e}_{1}^{2}\left(k\right)$ of Equation (30) are calculated from system responses of Equations (22)–(26) for the i’s iteration. The IPD controller (K

_{d}, ϕ, K

_{i}) of i’s iteration is ${X}_{i}=\left({x}_{i}^{1},{x}_{i}^{2},{x}_{i}^{3}\right)$. The simulations were performed using MATLAB software. The (K

_{d}, ϕ, K

_{i}) iterative curve is provided in Figure 7. After 100 iterations, the position of four agents approximated $X=[16.2138,-0.8646,0.4972]$. Optimal fitness (best(t)) approximated 0.1230. Therefore, the optimal IPD controller could be obtained as follows: ${K}_{d}=16.2138$, $\varphi =-0.8646$, and ${K}_{i}=0.4972$. The output response is displayed in Figure 8; the horizontal axis represents the sampling time, the unit is 1 mini-s, and the vertical axis is the measured output ($\Delta \tilde{x}$) of the MSS. The red line represents the step input ($r$), the amplitude of which is 0.1 units. The blue curve indicates the measured output ($\Delta \tilde{x}$) of the MSS. As displayed in Figure 8, the steady-state error and overshoot to a step input were absent. For the MSS, this indicated strong performance.

## 5. Experiments and Results

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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Li, J.-H.; Chiou, J.-S. GSA-Tuning IPD Control of a Field-Sensed Magnetic Suspension System. *Sensors* **2015**, *15*, 31781-31793.
https://doi.org/10.3390/s151229879

**AMA Style**

Li J-H, Chiou J-S. GSA-Tuning IPD Control of a Field-Sensed Magnetic Suspension System. *Sensors*. 2015; 15(12):31781-31793.
https://doi.org/10.3390/s151229879

**Chicago/Turabian Style**

Li, Jen-Hsing, and Juing-Shian Chiou. 2015. "GSA-Tuning IPD Control of a Field-Sensed Magnetic Suspension System" *Sensors* 15, no. 12: 31781-31793.
https://doi.org/10.3390/s151229879