Optimization of Exact Feedback Linearization EKF Controller for Magnetic Levitation Using Cuckoo Search Algorithm

Dephney Blossom; Vidya S Rao; Kavyashree .

doi:10.12688/f1000research.165341.1

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Research Article

Optimization of Exact Feedback Linearization EKF Controller for Magnetic Levitation Using Cuckoo Search Algorithm

[version 1; peer review: awaiting peer review]

Dephney Blossom¹, Vidya S Rao ², Kavyashree .³

PUBLISHED 28 Aug 2025

Author details Author details

¹ Department of IT, Volvo group India Private Limited, Bengaluru, India
² Department Instrumentation & Control Engineering, Manipal Institute of Technology, Manipal Academy of Higher Education, Manipal, Karnataka, 576104, India
³ Manipal School of Architecture and Planning, Manipal Academy of Higher Education, Manipal, Karnataka, India

Dephney Blossom
Roles: Conceptualization, Data Curation, Formal Analysis, Funding Acquisition, Investigation, Methodology, Project Administration, Resources, Software, Validation

Vidya S Rao
Roles: Conceptualization, Methodology, Project Administration

Kavyashree .
Roles: Writing – Original Draft Preparation, Writing – Review & Editing

OPEN PEER REVIEW

REVIEWER STATUS AWAITING PEER REVIEW

This article is included in the Manipal Academy of Higher Education gateway.

Abstract

Background

Magnetic levitation is an electromechanical technique that allows objects to float in air using magnetic fields alone. This technology eliminates physical contact, thereby reducing friction, wear, and energy loss while providing high-precision and stable operation. A key challenge in maglev systems lies in their inherent instability and nonlinearity, which complicate control design. In this study, a hollow metal sphere is levitated as part of a real-time experimental setup, which requires a fine balance between gravitational and magnetic forces.

Method

To address the complexities of the Magnetic levitation system, an intelligent control strategy is employed. The core control architecture integrates an Extended Kalman Filter for accurate state estimation with Exact Feedback Linearization to manage the nonlinear dynamics of the system. Additionally, system performance is optimized through Cuckoo Search, a nature-inspired metaheuristic algorithm used for pole placement tuning within the feedback linearization framework. The Extended Kalman Filter estimates hidden states and compensates for noise and disturbances, while the Exact Feedback Linearization simplifies the system dynamics to a linear form for more straightforward controller design. The Cuckoo Search algorithm ensures optimal controller parameters by minimizing a performance index.

Result

The combined Cuckoo Search-optimized, Exact Feedback Linearization with Extended Kalman Filter demonstrates significant improvements in the robustness and stability of the Magnetic levitation system. Simulation and experimental results show enhanced tracking performance, faster settling time, and improved disturbance rejection compared to conventional methods. The system remains stable and responsive under various operating conditions, validating the effectiveness of the proposed approach.

Conclusion

The study presents a robust and intelligent control framework for magnetic levitation systems. The method effectively addresses the challenges posed by system nonlinearity and uncertainty, leading to improved control accuracy and system performance. These findings suggest that such intelligent control strategies are well-suited for practical applications of Magnetic levitation systems, will enhance reliability.

Keywords

System stability, State estimation, Cuckoo Search, Optimization, Extended Kalman Filter, Exact Feedback Linearization

Corresponding author: Vidya S Rao

Competing interests: No competing interests were disclosed.

Grant information: The author(s) declared that no grants were involved in supporting this work.

Copyright: © 2025 Blossom D et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

How to cite: Blossom D, Rao VS and . K. Optimization of Exact Feedback Linearization EKF Controller for Magnetic Levitation Using Cuckoo Search Algorithm [version 1; peer review: awaiting peer review]. F1000Research 2025, 14:832 (https://doi.org/10.12688/f1000research.165341.1) First published: 28 Aug 2025, 14:832 (https://doi.org/10.12688/f1000research.165341.1) Latest published: 28 Aug 2025, 14:832 (https://doi.org/10.12688/f1000research.165341.1)

Introduction

Magnetic levitation (maglev) has emerged as a cutting-edge electromechanical technology that enables the suspension and control of objects using magnetic fields alone, thereby eliminating the need for physical contact. Maglev systems offer numerous advantages such as precise manipulation of electromagnetic forces, frictionless motion, high positioning accuracy, mechanical wear elimination, and effective isolation from external disturbances. These benefits make maglev systems a promising solution for diverse applications such as high-speed transportation, precision manufacturing, and contactless material handling.

In experimental and practical maglev systems, such as the levitation of a hollow metal sphere, achieving a stable suspension requires finely tuned control mechanisms to counteract gravitational forces and maintain equilibrium. However, the inherently unstable and nonlinear nature of maglev dynamics, compounded by system uncertainties and external perturbations, poses significant challenges to traditional control strategies.

Gao discussed the necessity of a controller that does not require an exact mathematical model of the plant. He discussed the design of the LADRC tuning process.¹ Bingwei suggested using gray wolf optimization in order to tune. ADRC criteria to determine the best answer. Amjad described the concept and implementation of LESO and NESO. The required settings were adjusted using PSO. This shows that LESO provides a better estimation than NESO.² Amjad in his another work discussed the implementation of PSO as a search tool for tuning the parameters of ADRC. He then implemented this on a maglev system.³

Cuckoo search using Levy flights is a revolutionary metaheuristic algorithm introduced by Xin She Yang and Suash Deb. The search algorithm, which is shown in our study, was previously presented in this work.⁴ Mukul covered Levy’s flying parameters, which are applicable to many seeking and foraging algorithms, in his bachelor’s thesis.⁵ Libing discussed using CS in conjunction with levy flights as an LADRC search algorithm. The controller and observer bandwidths are the two tuning parameters for the LADRC. These parameters were adjusted using CS. In addition, maglev systems use this innovative controller-searching algorithm combination.⁶ Together with Xing Shi and Xin She Yang In their book, he covered the CS and Bat algorithms and went on to include working MATLAB code as well as pseudocode.⁷

In his book, Dan Simon covered many forms of EKF and how they were designed and implemented. He provides an example of each of the variations in EKF.⁸

In this thesis, John developed a precise feedback linearizing algorithm. The maglev system was controlled and estimated using a troller and Kalman filter. In real time, he put the same into practice.⁹ Because many laboratory setups employ Simulink in a real-time environment, Nivedita discussed how to easily create a discrete EKF in this environment.¹⁰

To address these challenges, this study explored three intelligent control methodologies that integrate advanced estimation and optimization techniques. The second method utilizes an Extended Kalman Filter (EKF) for nonlinear state estimation combined with Exact Feedback Linearization and Cuckoo Search (CS)-based pole placement tuning to achieve precise control. These controllers aim to enhance the system stability, robustness, and performance under nonlinear operating conditions. By leveraging nature-inspired optimization algorithms and modern estimation techniques, this study contributes to the development of intelligent maglev control systems capable of maintaining high performance despite inherent instability and external interference. A comparative evaluation of these methods offers insights into their practical viability and effectiveness in real-world levitation applications.

The aim of this project is to design an intelligent controller that considers both linear and nonlinear models. The designed controller was validated through a simulation and real-time implementation.

Methodology

Working principle of Maglev

Figure 1 shows a schematic of the maglev physical setup. The Maglev ball position control system consists of an electromagnetic coil, infrared light sensors, a metal ball, an analog and digital interface, and a computer controller.¹¹ Initially, when the current passes through the electromagnetic coil, it produces the necessary magnetic field. The magnetic force was controlled by controlling the current in the coil. This magnetic field forces the object, and the metallic ball in this case experiences lifting force. There are two components in the IR sensor: the transmitter and receiver of the IR light. When an object is levitated, a portion of the light is blocked. The generation of voltage depends on the amount of light falling on the receiver. The amount of voltage generated indicates the position of the object. The necessary analog and digital interfaces were also present. The sensor output is passed through an analog-to-digital converter pin. The controller calculates the control law by comparing the reference with the position command of the maglev system. The digital-to-analog converter pin receives control input from the controller in the computer. The amount of current flowing through the coil is governed by the control law. The current in the coil decreases as the object approaches the magnet and vice versa.

Figure 1. Maglev control setup.

The efficient regulation of electromagnetic force is an excellent teaching and research tool for control systems, electromagnetism, sensors, and mechatronics that aligns with Sustainable Development Goals 4 and 7. It also helps minimize energy usage. In line with Sustainable Development Goal 9, this technology immediately promotes innovations in magnetic levitation transportation, such as maglev trains and noncontact handling technologies in manufacturing.

Formulation of Maglev nonlinear modelling

Nonlinear systems can be represented by differential equation of the form,

(1)

\ddot{x} (t) = f (t, x (t), \dot{x}, w (t)) + b (t) u (t)

(2)

y (t) = C_{1} x (t) + sn (t)

where x(t) represents the position output, b(t) denotes the time-varying coefficient, w(t) represents unknown external disturbances such as vibrations, friction, and torque disturbances.

sn

represents sensor noise.

\ddot{x} (t)

denotes acceleration of the ball at position x(t).

Now, let us take $x_{1}$ = $U_{out}$ = $Position (x)$ and $x_{2}$ = $U_{in}$ = current; then, the state-space equations are given as

(3)

\dot{x_{1}} = x_{2}

(4)

\ddot{x} = (\frac{- 2 g}{i_{0}}) U_{in} + (\frac{2 g}{x_{0}}) x_{1}

(5)

y = x_{1} = Position

where

i_{0} is

Equilibrium value of the current,

x_{0}

Equilibrium value of position

x_{0}

, acceleration due to gravity

g

, and the ball’s position is taken as the output variable

y

. Equation 6 can be rewritten as

(6)

{\dot{x}}_{2} = f (i, x_{1}) + b_{0} U_{in}

(7)

b_{0} = \frac{- 2 g}{i_{0}}

(8)

f = \frac{2 g}{x_{0}} x_{1}

Cuckoo search algorithm

Brood parasitism, a unique aspect of cuckoo breeding behavior, is fascinating. When certain species of cuckoos lay their eggs in the host bird’s nest, the latter can engage in direct conflict with the intruders, discard the alien eggs, or simply depart their nest and create a new nest somewhere if it realizes that the eggs are not their own. This algorithm implements L’evy flights rather than random walks. Recent studies also indicate that CS is potentially far more efficient than PSO and genetic algorithms.¹²

To describe the CS algorithm, we first idealize a population of n cuckoos and n nests as a cuckoo-host system. The average host nest consists of three or four eggs, and each cuckoo can lay its eggs in a few nests in real-world cuckoo-host systems. However, restricting that one cuckoo can lay one egg at only one host nest at a time simplifies the algorithmic technique. This implies that the number of eggs nests = cuckoos. Hence, the egg’s location acts as a solution vector x to the optimization problem. Therefore, distinguishing between eggs, cuckoos, and nests is no longer necessary.⁷ Three idealized rules are given by Xin-She Yang and Suash Deb.⁴

• Nest is randomly selected by each cuckoo that can lay only one egg at a time.
• In the optimization problem, nests with a high calibre of eggs are considered the best nests and are passed to the next iteration.
• The number of host nests is fixed. An egg laid by a cuckoo is discovered by the host bird with a probability Pa of [0, 1]. This fraction Pa of n nests was replaced with new random solutions considered as new nests.

These random solutions in the interval [0,1] are compared with Pa; if r > Pa, then the position of the nest is randomly updated once. Otherwise, the position of the birds’ nests remained unchanged. A flowchart depicting the CS is shown in Figure 2.

Figure 2. Flowchart of Cuckoo search algorithm.

Different performance indicators such as ITAE, IAE, and RMSE are used by intelligent optimization algorithms as a minimizing function, which then leads to convergence of the algorithm.

(9)

\begin{matrix} IAE = \int_{0}^{t} | r (t) - y (t) | dt \\ RMSE = \sqrt{\frac{1}{T} \sum_{t = 0}^{T} {(r (t) - y (t))}^{2}} \\ ITAE = \int_{0}^{T} t | r (t) - y (t) | dt \\ ISE = \int_{0}^{T} t {(r (t) - y (t))}^{2} dt \end{matrix}

Cuckoo search based exact feedback linearization with EKF

The Maglev system is a highly nonlinear system with multiple disturbances and uncertainties acting on the system. It is crucial to design a controller that can withstand these uncertainties and disturbances, and track the reference trajectory. One such controller that considers nonlinear dynamics is the exact feedback linearization. The Nonlinear mathematical modelling of the maglev system is transformed into an equivalent linear system using a feedback linearization technique. All the states needed by the controller for implementation in real time may not be available, or the obtained states may include measurement noises from the sensor at play or the plant has process noise that tags along the actual state output. Hence, the EKF was employed to estimate the states free of process and measurement noise. Exact feedback linearization has tuning parameters; hence, the CS algorithm is deployed to intelligently find the tuning parameters of the controller. A block diagram of the proposed controller, referred to as CS-EFL-EKF, is shown in Figure 3.

Figure 3. Block diagram of CS-EFL-EKF.

Exact feedback linearization

This is one of the most common control techniques used in nonlinear systems. It applies a linear control technique to design a control law for the stabilization or tracking of nonlinear systems. The idea here is to transform a non- linear system into a fully or partially linear system by selecting a proper nonlinear transformation z = T ( x).

By applying a nonlinear state feedback variable, v = α(x)+β(x) u, a linear control approach can be used. In this study, the estimated states $\hat{x}$ of the maglev system are passed to exact feedback linearization.

Extended Kalman Filter (EKF)

High-performance modern controllers or nonlinear controllers use more than one state for their function, which may not ideally be available in real time, let alone measure the entire state vector. Hence, it necessary to develop a method to determine the states of the system. There are many limits to how well one can determine the state of the system, such as sensor signal noise, approximations in the system modelling equations, and many other external factors. Therefore, an EKF was proposed to estimate the states of the maglev system. They are widely used in nonlinear systems.

The EKF consists of a prediction step and state-correction step. In the prediction stage, the states at time k were predicted based on time k- 1. In the correction stage, the predicted states were corrected using new measurements obtained at time k. The idea of the EKF is based on the linearization of the state equation and measurement equation. The Jacobian of the state equation and measurement equation enables linearization, helping propagate the state and state co-variance in a linear format. In this study, we used a continuous EKF for the simulation and a discrete EKF to prove the real-time implementation of the EKF for the maglev system.

Consider the nonlinear system described in equation 1 and equation 2: Here, process noise w is defined as a zero-mean Gaussian white noise with covariance Q and it accounts for all plant dynamics that were neglected in the modelling. Sensor noise v is a zero-mean Gaussian white noise term with covariance R.

Continuous EKF

As per the analysis of nonlinear systems,¹³ it is known that a slight changes in the equilibrium conditions can give extremely different output. Hence, it is important to determine the initial conditions of the maglev system to be used in the EKF.

• x(0) - Initial condition of state vector: For the maglev system, equilibrium points are considered as initial conditions
(10)
$[x_{1} (0) x_{2} (0) x_{3} (0)] - [\begin{matrix} 0.009 & 0 & 0.8 \end{matrix}]$
At static equilibrium, the time-rate derivative is equal to zero; hence, the initial velocity $x_{2} (0)$ is equal to zero. The state $x_{3} (0)$ (current) is calculated as:
(11)
$x_{3} (0) = x_{1} (0) \sqrt{\frac{grmass}{K}}$
by substituting necessary values into equation 11, $x_{3} (0)$ can be obtained.
${\hat{x}}_{1} (0)$ - Initial condition of state estimation vector: Because these are the initial state vectors passed for estimation,
(12)
$[\begin{matrix} {\hat{x}}_{1} (0) & {\hat{x}}_{2} (0) & {\hat{x}}_{3} (0) \end{matrix}] - [\begin{matrix} 0.009 * 1.05 & 0 * 1.05 & 0.8 * 1.05 \end{matrix}]$
• $P (0)$ - Initial condition of estimation error covariance: Because maglev has three states, the size of $P (0)$ matrix is 3 × 3. It is a covariance matrix; therefore, it should be a positive-definite or positive-semi-definite matrix.
Hence
(13)
$P (0) = [\begin{matrix} 0.1 & 0 & 0 \\ 0 & 0.1 & 0 \\ 0 & 0 & 0.1 \end{matrix}]$
• Q is the process noise covariance Q because maglev has three states, and the size of the Q matrix is 1 × 3 or 3 × 1 depending on the program. Because it is a covariance matrix, it should be a positive-definite or positive-semi-definite matrix.
Hence
(14)
$Q - [\begin{matrix} 0.0001 & 0.0001 & 0.0001 \end{matrix}]$
• R - Sensor noise covariance, R Because position is the only output received from the sensor in the maglev, the size of R is 1 × 1. Because it is a covariance matrix, it should be a positive-definite or positive-semi-definite matrix.
Hence
(15)
$R = 5 e - 06$

One needs to find the Jacobian matrix, which is a partial derivative evaluated at the current state estimate:

• A matrix, ${\frac{\partial f (x, w)}{\partial x} |}_{\hat{x}}$
(16)
$[\begin{matrix} 0 & 1 & 0 \\ \frac{2 K {\hat{x}}_{3}^{2}}{m {\hat{x}}_{1}^{3}} & 0 & \frac{- 2 K {\hat{x}}_{3}}{m {\hat{x}}_{1}^{2}} \\ \frac{- 4 K {\hat{x}}_{3} {\hat{x}}_{2}}{L {\hat{x}}_{1}^{3}} & \frac{2 K {\hat{x}}_{3}}{L {\hat{x}}_{1}^{2}} & \frac{- R}{L} + \frac{2 K {\hat{x}}_{2}}{L {\hat{x}}_{1}^{2}} \end{matrix}]$
It is partial derivative of f(x) given,
(17)
$f (x) = [\begin{array}{c} x_{2} \\ g - \frac{K {x_{3}}^{2}}{m {x_{1}}^{2}} \\ - \frac{xgR}{L} + \frac{2 K x_{19} x_{2}}{L {x_{1}}^{2}} \end{array}]$
• H matrix, ${\frac{\partial h (x, v)}{\partial x} |}_{\hat{x}}$
(18)
$[\begin{matrix} 1 & 0 & 0 \end{matrix}]$
This is a partial derivative of the measurement equation, y, given in equation 1 and equation 2.
• L matrix, ${\frac{∂f (x, w)}{∂w} |}_{\hat{x}}$
(19)
$\begin{matrix} [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] \end{matrix}$
This is partial derivative of f(x, w) w.r.t process noise w.
• M matrix, ${\frac{∂h (x, v)}{∂v} |}_{\hat{x}}$
(20)
$[\begin{matrix} 1 & 0 & 0 \end{matrix}]$
This is partial derivative of h(x, v) w.r.t measurement noise v.

Steps to execute continuous EKF¹⁴ is given below,

1.
(21)
$\tilde{Q} = LQ L^{T}$
2.
(22)
$\tilde{R} = MR M^{T}$
3. Kalman gain, $K$
(23)
$K = P H^{'} {\tilde{R}}^{- 1}$
Here, P is the covariance of the estimated error matrix, which is updated for every time sample in the program.
4. Derivative of state vector, $\hat{\dot{x}}$
(24)
$\hat{\dot{x}} = f (\hat{x}, w, t, u) + K [y - h (\hat{x}, v, t)]$

f ( $\hat{x}$ , w, t, u) is given in equation 18, which is added to the process noise w at the current time sample t. h ( $\hat{x}$ , v, t) is nothing but the first state in the state vector estimation because in the maglev system position is the output at the current time sample t. Integrating equation 25 will yield estimated yields that are both predicted and updated. y is the output received from the maglev.

1. Derivative of estimated error covariance, $\dot{P}$
(25)
$\dot{P} = AP + P A^{'} + \tilde{Q} - P C^{'} {\tilde{R}}^{- 1} HP$

$A$ is given by equation 26. Integrating $\dot{P}$ yields the estimated error covariance matrix for the next use in the prediction, which is the current updated value.

The maglev system can be controlled using the estimated state vector obtained from the continuous EKF in the exact feedback linearization.

Discrete EKF

All real-time systems are discrete. A discrete EKF is easy to implement in real time because it requires fewer samples compared to a continuous EKF.

1. F matrix s
(26)
${\frac{\partial f (x_{k - 1}, w_{k - 1})}{\partial x} |}_{{\hat{x}}_{k - 1}}$
This is the same as the A matrix in equation 16. Here, instead of continuous time, the F matrix is calculated for discrete-time samples k.
2. L matrix,
(27)
${\frac{\partial f (x_{k - 1}, w_{k - 1})}{\partial w} |}_{{\hat{x}}_{k - 1}}$
This is the same as equation 19, except that it is calculated at discrete time sample k-1.
3. M matrix,
(28)
${\frac{\partial h (x_{k}, v_{k})}{\partial v} |}_{{\hat{x}}_{k}}$
It is same as equation 21 except it is calculated at discrete time sample k.
4. H matrix,
(29)
${\frac{\partial h (x_{k,}, v_{k})}{\partial x} |}_{{\hat{x}}_{k}}$
It is same as equation 19 except it is calculated at discrete time sample k.
5. Calculate the estimated error vector $P_{k}^{-}$ :
(30)
$P_{k}^{-} = A {P_{k - 1}}^{+} A^{'} + LQ$
6. Calculate the estimated state vector ${\hat{x}}_{k}^{-}$ :
(31)
${\hat{x}}_{k}^{-} = Af ({\hat{x}}_{k - 1}^{+})$
7. Kalman gain, $K$
(32)
$K = {P_{k}}^{-} H^{'} (H {P_{k}}^{-} H^{'} + MR M^{'})$
8. ${\hat{x}}_{k}^{+}$ :
(33)
${\hat{x}}_{k}^{+} = {\hat{x}}_{k}^{-} + K (y - h_{k} ({\hat{x}}_{k}^{-}))$
where y is the actual output obtained from the maglev system.
9. $P_{k}^{+}$ :
(34)
$P_{k}^{+} = (I - KH) P_{k}^{-}$

The maglev system can be controlled using the estimated state vector obtained from the discrete EKF in the exact feedback linearization.

Results

Simulation of CS based exact feedback linearization with EKF

Simulations were performed using MATLAB. This simulation introduced PID- and CS-based exact feedback linearization with the EKF for comparison and analysis. Process and measurement noises were added to the system to determine the effectiveness of the controllers. As seen in the methodology, the exact feedback linearizing controller requires feedback gains k₁, k₂ and k₃. This can be obtained from a single controller bandwidth p. Using an intelligent searching algorithm, the CS can obtain the optimal value of p. The sampling time for the simulation is 0.001 s. The fitness function for the CS algorithm is a performance indicator, namely, the RMSE.

Figure 4 shows the fitness value (RMSE) for each iteration. It converges to zero and determines the optimal tuning parameter, p.

Figure 4. Individual optimal fitness values obtained from CS for EFL.

Table 1 shows p-values obtained from CS. Using this value, the tuning parameters of EFL are calculated, yielding k₁, k₂, k₃ and are tabulated in the same manner.

Table 1. Values obtained from CS for EFL.

Iteration	RMSE	p	k₁	k₂	k₃
8	0.003122	1.9917293e+01	7.9012e+03	1.1901e+03	59.7519

Here, the reference is passed to the maglev system, along with its derivatives. Because the relative degree r = 3, the reference is differentiated three times. As seen from equation 15, the Q value and equation 16 R value are used to compute the Gaussian noise. MATLAB has a function called wgn which generates white Gaussian noise based on the Q and R values.

Figure 5 shows the step-tracking signal, and it can be seen that the output tracks the given reference effectively in the presence of process noise (w) and measurement noise (v).

Figure 5. Step position tracking for CS-EFL-EKF.

Figure 6 shows the response of the continuous EKF. It can be seen that it accurately estimates the actual states, even in the presence of w and v.

Figure 6. EKF output-with step input.

Figure 7 shows the magnitude of the position-tracking error and the control signal used to control the maglev system.

Figure 7. Step position tracking error and control signal for CS-EFL-EKF.

A sine reference signal is also passed to check the efficiency and performance of the proposed controller, and the same conclusions are drawn, as seen from the step reference signal. The responses are presented in Figure 8, Figure 9, and Figure 10.

Figure 8. Sine position tracking for CS-EFL-EKF.

Figure 9. Step signal EKF output.

Figure 10. Sine position tracking error and control signal for CS-EFL-EKF.

Performance indicators, such as RMSE, IAE, ITAE, and ISE, should be calculated for the proposed controller to analyze its effectiveness. The results are shown in Table 2.

Table 2. Performance indicators of CS-EFL-EKF.

Signal	ISE	RMSE	ITAE	IAE
Step	0.000874	0.000540	0.010259	0.018877
Sine	0.000023	0.000108	0.004024	0.004976

Now, one can be compared with the proposed controller. Because PID is a linear controller that uses a transfer function to describe the maglev system, a comparison is made only by adding measurement noise. However, the proposed controller has both process and measurement noises. Figure 11 shows the response of the step reference signal.

Figure 11. Step Reference tracking of PID with v vs CS-EFL-EKF with w and v.

Figure 12 shows the plot for the sine reference signal, and the same conclusions can be drawn as for the step reference signals.

Figure 12. Sine Reference tracking of PID with v vs CS-EFL-EKF with w and v.

When measured noise v and processed noise w hit the maglev system, the PID that works fine in the article¹⁵ will fail, as clearly shown in Figure 11. In contrast, the CS-EFL-EKF tracks the reference position signal exactly, which proves the efficacy of the system.

Cuckoo search based exact feedback linearization with EKF

In the Real-time implementation of the CS-EFL-EKF, a discrete EKF is used to estimate the states. The discrete EKF accurately estimated the states. However, exact feedback linearization cannot be used to control the maglev system. This can be attributed to many reasons, one of which is the current age of the maglev laboratory setup. It has been in use for more than 10 years. The exact feedback linearization controller depends heavily on system parameters, such as resistance, inductance, and magnetic constants. These values may have changed over time in the hardware implementation. Because maglev is inherently unstable and non-linear, with incorrect parameter values, the controller effects can be drastically affected. Figure 13 shows the control algorithm designed in the Simulink environment. Figure 14 shows the discrete EKF algorithm in the Simulink environment.

Figure 13. CS-EFL-EKF setup in Simulink.

Figure 14. Discrete EKF setup in Simulink.

Figure 15 shows the response obtained using the discrete EKF. The discrete EKF accurately estimated the output position.

Figure 15. Actual position v/s estimated position.

In Figure 15, when the pulse-width modulated reference signal is input, the actual maglev ball output position is shown as a red signal. The estimated position of the ball (maglev) is shown in blue and coincides exactly with the actual output position. This demonstrates the efficacy of the CS-EFL-EKF design. In addition, the position tracking error dies asymptotically, as highlighted in the Figure 6 and Figure 7.

Table 3 compares the performance indicators for PID with v and CS-EFL-EKF controllers. Based on the percentage change of improvements mentioned in Table 3, it can be said that the performance of CS-EFL-EKF is relatively better than that of the traditional PID controller. This improves the stability of the controller even when process and measurement noise act on the system.

Table 3. Comparison of the performance indicators.

Signal	Controller	ISE	RMSE	ITAE	IAE
Step	PID¹⁵	6.708425	0.047288	6.485768	3.255486
	CS-EFL-EKF	0.000853	0.000533	0.010027	0.018463
	Improvement	99.987%	98.8728%	99.8453%	99.4328%
Sine	PID¹⁵	3.966528	0.044534	2.628576	2.384513
	CS-EFL-EKF	0.000044	0.000149	0.007575	0.007488
	Improvement	99.9988%	99.6654%	99.718%	99.6859%

Conclusion

Detailed nonlinear modelling of the Maglev system was performed. Finally, a cuckoo search-based exact feedback linearizing controller with an EKF is designed. The CS was used to determine the tuning parameters of the EFL controller. Owing to the presence of the EKF, which can estimate the states even in the presence of process noise and measurement noise, the proposed controller can enable the maglev system to track the given reference and achieve stability. According to the performance indicators, the proposed controller yielded better results than the traditional PID controller in the presence of noise. The same was validated by simulation. Real-time implementation of the controller was not achieved because of the extreme dependence of the physical parameters in the play. However, the EKF accurately estimated real-time states.

Hence, it can be seen that the introduction of intelligence into the conventional controller can effectively improve the transient and steady-state responses, dynamic response, and interference rejection capability. However, the control signal usage will be higher in the case of intelligent controllers.

Ethical declarations

No animal or human subjects were used in this work. This manuscript is an original paper and has not been published in other journals. The corresponding author, on behalf of all the authors declare that we have followed all the accepted ethical standards in this research work

Data availability statement

The underlying dataset for this study has been deposited in the public GitHub repository: https://github.com/drkavyashree/Cuckoo-Search-Algorithm.git

Data License: Apache License 2.0, January 2004

This study does not involve human participants or animal subjects. The research is based entirely on a real-time experimental setup and MATLAB simulations. Therefore, ethical approval from an Institutional Review Board (IRB) is not required, and no human or animal ethical considerations apply.

There are no ethical or legal barriers to sharing the data on a controlled basis. The IRB approval is not applicable in this case, data sharing is permitted upon request for academic and non-commercial research purposes.

Source code

Open Science Framework: Extended data for ‘Optimization of Exact Feedback Linearization EKF Controller for Magnetic Levitation Using Cuckoo Search Algorithm’ is archived at https://doi.org/10.17605/OSF.IO/B2UD9.¹⁶

This project contains the following underlying data:

• ekf_pid.mat – Input data for EKF Controller
• ekf_pid_sine.mat - Input data (Sine input) for EKF Controller
• r1.mat – Hardware data for the system
• r2.mat - Hardware data for the system
• r3.mat - Hardware data for the system
• r.mat - Hardware data for the system
• sin2.mat - Input data (Sine input) for EKF Controller
• sine3.mat - Input data (Sine input) for EKF Controller
• sine.mat - Input data (Sine input) for EKF Controller
• step2.mat - Input data (Step input) for EKF Controller

Data are available under the terms of the Data are available under the terms of the Creative Commons Zero “No rights reserved” data waiver (CC0 1.0 Universal).

References

1. Yoo D, Yau SS-T, Gao Z: Optimal fast tracking observer bandwidth of linear extended state observer. International Journal of Control. 2007; 80(1): 102–111. Publisher Full Text
2. Humaidi AJ: Experimental design and Verification of Extended State Observers for Magnetic Levitation System Based on PSO. The Open Electrical Electronic Engineering Journal. 2018; 12: 110–120. Publisher Full Text
3. Humaidi AJ, Badr HM, Hameed AH: PSO Based Active Disturbance Rejection Control for Position Control of Mag netic Levitation System. 5th International Conference on Control, Decision and Information Technologies. 2018; 8. Publisher Full Text
4. Yang X-S, Deb S: Cuckoo Search via Levy Flights. World Congress on Nature Biologically Inspired Computing. 2009.
5. Singh MK: Evaluating levy flight parameters for Random Searches in a 2D Space, Bachelor of Science in Mechanical Engineering. Massachusetts Institute of Technology; 2013.
6. Wei L, Fan K, Zhou X, et al.: Linear Active Disturbance Rejection Control of Magnetic Levitation System Based on Cuckoo Search. IEEE 11th Data Driven Control and Learning Systems Conference. 2022. Publisher Full Text
7. Yang X-S, He X-S: Bat Algorithm and Cuckoo Search Al gorithm, Chapter 2, Nature-Inspired Computation and Swarm Intelligence.2020. Publisher Full Text
8. Simon D: Chapter 13, Optimal State Estimation: Kalman, H, and Non-linear Approaches. John Wiley Sons, Inc; 2006. Publisher Full Text
9. Henley JA: Design And Implementation Of A Feedback Lin earizing Controller And Kalman Filter For A Magnetic Levitation System. 2007. Thesis for Master of Science in mechanical engineering, University of Texas at Arlington, Texas. Reference Source
10. Nivedita PC: Object Tracking in Simulink using Extended Kalman Filter. International Journal of Advanced Research in Computer and Communication Engineering. 2015 July 2015; 4(7). Publisher Full Text
11. Feedback Instrument Magnetic Levitation manua.
12. Ahmad I, Javaid MA: Nonlinear Model and Con trol design for Magnetic Levitation System. 9th WSEAS International Conference on Signal processing, Robotics and Automation. 2010.
13. Slotine J-J-E, Li W: Applied Nonlinear Control. Prentice Hall; 1991. D-13-040890-5.
14. Wei Z, Huang Z, Zhu J: Position Control of Magnetic Levitation Ball Based on an Improved Adagrad Algorithm and Deep Neural Network Feed forward Compensation Control. Hindawi Mathematical Problems in Engineering. 2020; 2020: 1–13. Publisher Full Text
15. Blossom D, Rao VS: Design of Metaheuristically Supervised Linear ADRC for a Magnetic Levitation System Control.Dassan P, Thirumaaran S, Subramani N, editors. Intelligent Computing, Smart Communication and Network Technologies. ICICSCNT 2023. Communications in Computer and Information Science. Vol. 1970. . Cham: Springer; 2024. Publisher Full Text
16. Kavyashree, Rao VS, Blossom D: Optimization of Exact Feedback Linearization EKF Controller for Magnetic Levitation Using Cuckoo Search Algorithm.2025. Publisher Full Text

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Version 1

VERSION 1 PUBLISHED 28 Aug 2025

Author details Author details

¹ Department of IT, Volvo group India Private Limited, Bengaluru, India
² Department Instrumentation & Control Engineering, Manipal Institute of Technology, Manipal Academy of Higher Education, Manipal, Karnataka, 576104, India
³ Manipal School of Architecture and Planning, Manipal Academy of Higher Education, Manipal, Karnataka, India

Dephney Blossom
Roles: Conceptualization, Data Curation, Formal Analysis, Funding Acquisition, Investigation, Methodology, Project Administration, Resources, Software, Validation

Vidya S Rao
Roles: Conceptualization, Methodology, Project Administration

Kavyashree .
Roles: Writing – Original Draft Preparation, Writing – Review & Editing

Competing interests

No competing interests were disclosed.

Grant information

The author(s) declared that no grants were involved in supporting this work.

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Published: 28 Aug 2025, 14:832

https://doi.org/10.12688/f1000research.165341.1

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Blossom D, Rao VS and . K. Optimization of Exact Feedback Linearization EKF Controller for Magnetic Levitation Using Cuckoo Search Algorithm [version 1; peer review: awaiting peer review]. F1000Research 2025, 14:832 (https://doi.org/10.12688/f1000research.165341.1)

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[1] 1. Yoo D, Yau SS-T, Gao Z: Optimal fast tracking observer bandwidth of linear extended state observer. International Journal of Control. 2007; 80(1): 102–111. Publisher Full Text

[2] 2. Humaidi AJ: Experimental design and Verification of Extended State Observers for Magnetic Levitation System Based on PSO. The Open Electrical Electronic Engineering Journal. 2018; 12: 110–120. Publisher Full Text

[3] 3. Humaidi AJ, Badr HM, Hameed AH: PSO Based Active Disturbance Rejection Control for Position Control of Mag netic Levitation System. 5th International Conference on Control, Decision and Information Technologies. 2018; 8. Publisher Full Text

[4] 4. Yang X-S, Deb S: Cuckoo Search via Levy Flights. World Congress on Nature Biologically Inspired Computing. 2009.

[5] 5. Singh MK: Evaluating levy flight parameters for Random Searches in a 2D Space, Bachelor of Science in Mechanical Engineering. Massachusetts Institute of Technology; 2013.

[6] 6. Wei L, Fan K, Zhou X, et al.: Linear Active Disturbance Rejection Control of Magnetic Levitation System Based on Cuckoo Search. IEEE 11th Data Driven Control and Learning Systems Conference. 2022. Publisher Full Text

[7] 7. Yang X-S, He X-S: Bat Algorithm and Cuckoo Search Al gorithm, Chapter 2, Nature-Inspired Computation and Swarm Intelligence.2020. Publisher Full Text

[8] 8. Simon D: Chapter 13, Optimal State Estimation: Kalman, H, and Non-linear Approaches. John Wiley Sons, Inc; 2006. Publisher Full Text

[9] 9. Henley JA: Design And Implementation Of A Feedback Lin earizing Controller And Kalman Filter For A Magnetic Levitation System. 2007. Thesis for Master of Science in mechanical engineering, University of Texas at Arlington, Texas. Reference Source

[10] 10. Nivedita PC: Object Tracking in Simulink using Extended Kalman Filter. International Journal of Advanced Research in Computer and Communication Engineering. 2015 July 2015; 4(7). Publisher Full Text

[11] 11. Feedback Instrument Magnetic Levitation manua.

[12] 12. Ahmad I, Javaid MA: Nonlinear Model and Con trol design for Magnetic Levitation System. 9th WSEAS International Conference on Signal processing, Robotics and Automation. 2010.

[13] 13. Slotine J-J-E, Li W: Applied Nonlinear Control. Prentice Hall; 1991. D-13-040890-5.

[14] 14. Wei Z, Huang Z, Zhu J: Position Control of Magnetic Levitation Ball Based on an Improved Adagrad Algorithm and Deep Neural Network Feed forward Compensation Control. Hindawi Mathematical Problems in Engineering. 2020; 2020: 1–13. Publisher Full Text

[15] 15. Blossom D, Rao VS: Design of Metaheuristically Supervised Linear ADRC for a Magnetic Levitation System Control.Dassan P, Thirumaaran S, Subramani N, editors. Intelligent Computing, Smart Communication and Network Technologies. ICICSCNT 2023. Communications in Computer and Information Science. Vol. 1970. . Cham: Springer; 2024. Publisher Full Text

[16] 16. Kavyashree, Rao VS, Blossom D: Optimization of Exact Feedback Linearization EKF Controller for Magnetic Levitation Using Cuckoo Search Algorithm.2025. Publisher Full Text

Optimization of Exact Feedback Linearization EKF Controller for Magnetic Levitation Using Cuckoo Search Algorithm

Abstract

Background

Method

Result

Conclusion

Keywords

Introduction

Methodology

Working principle of Maglev

Figure 1. Maglev control setup.

Formulation of Maglev nonlinear modelling

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

Cuckoo search algorithm

Figure 2. Flowchart of Cuckoo search algorithm.

(9)

Cuckoo search based exact feedback linearization with EKF

Figure 3. Block diagram of CS-EFL-EKF.

Exact feedback linearization

Extended Kalman Filter (EKF)

Continuous EKF

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

(18)

(19)

(20)

(21)

(22)

(23)

(24)

(25)

Discrete EKF

(26)

(27)

(28)

(29)

(30)

(31)

(32)

(33)

(34)

Results

Simulation of CS based exact feedback linearization with EKF

Figure 4. Individual optimal fitness values obtained from CS for EFL.

Table 1. Values obtained from CS for EFL.

Figure 5. Step position tracking for CS-EFL-EKF.

Figure 6. EKF output-with step input.

Figure 7. Step position tracking error and control signal for CS-EFL-EKF.

Figure 8. Sine position tracking for CS-EFL-EKF.

Figure 9. Step signal EKF output.

Figure 10. Sine position tracking error and control signal for CS-EFL-EKF.

Table 2. Performance indicators of CS-EFL-EKF.

Figure 11. Step Reference tracking of PID with v vs CS-EFL-EKF with w and v.

Figure 12. Sine Reference tracking of PID with v vs CS-EFL-EKF with w and v.

Cuckoo search based exact feedback linearization with EKF

Figure 13. CS-EFL-EKF setup in Simulink.

Figure 14. Discrete EKF setup in Simulink.

Figure 15. Actual position v/s estimated position.

Table 3. Comparison of the performance indicators.

Conclusion

Ethical declarations

Data availability statement

Source code

References

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