# Comparative Study of Different Controllers for Levitating Ferromagnetic Material

Authors: T. Deepa; D. Subbulekshmi; P. Lakshmi; Marabathina Maheedhar; E. Kishore; M. Vinodharanai; T. Chockalingam
Journal: Advances in Materials Science and Engineering
 (2022)

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## Abstract

The purpose of the analysis is to levitate and stabilize a spherical ball of magnetic levitation system at the desired position using various controllers and determine the one which gives better performance. The ball which is used to levitate is a ferromagnetic material such as stainless steel. This study talks about the proportional derivative controller, proportional integral derivative (PID) controller, and linear quadratic regulator (LQR) controller in place of a physical system. The transient response of the magnetic levitation system can be modified for desirable results due to the implementation of the controller. Simulation and experimental methods are used to verify the results. Both the techniques have been made to stabilize the ball position in the desired position. PID and LQR are designed to achieve the ball position to the desired level, and it is observed that the LQR controller gives the best result.


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## Body

## 1. Introduction

The state-feedback control with the inertial delay observer (IDO) was proposed by Singru et al. [1] for precise control of a one-inch diameter steel ball in a magnetic levitation device. The levitation control differs from torque control, as demonstrated by Xue et al.’s levitation efficiency study of a type of the bearing-less switched reluctance motor (BLSRM).Ginoya et al. compared the output of a cascaded sliding mode control designed for magnetic levitation systems with electrical and electromechanical loops to that of a standard linear quadratic regulator combined with a PI controller [2]. In their paper, Azukizawa et al. [3] succeeded in enhancing the system’s levitation properties by increasing the magnetic field of the HTS magnet. A magnetic levitation control system is developed by linearizing the nonlinear system model around the operating point by Ghodsi et al. [4]. According to the results of Kim et al. [5], under cogging powers, magnetic levitation can achieve stable lavation and produce enough power for continuous operation. A prototype levitation platform with the steady-state power consumption per each HEM was created by designing a zero-power controller.The object which is used to levitate the object is stainless steel which is a ferromagnetic material. The utilization of these materials is due to their ferromagnetism, as well as their outstanding corrosion, radiation, and heat resistant qualities [6]. From an application standpoint, the magnetic properties of ferromagnetic stainless steels SUS 403, TAF, and SUS 405 can be summarised as follows. They are harder than standard soft magnetic materials but weaker than semihard materials in terms of magnetic properties, and heat treatment changes the magnetic properties. The coercive force and rectangularity are increased by quenching and tempering. Full annealing reduces saturation induction and permeability marginally.Core separation can be changed. According to Lim et al. [7], the levitation force ripples and cogging force are reduced, and the electromagnetic properties of the levitation magnet are investigated using the finite element analysis. Takao et al. [8, 9] used three models and proposed a few modifications that featured different arrangements to increase the levitation force in the system and identified that the common rail model’s levitation force was seven times greater than the maximum levitation force of the common model. According to Liu et al. [10], AC superconducting windings that are used to increase levitation forces are well suited for hydraulic turbo-generator de-load applications. The inductive eddy magnetic field interacts with the primary field, producing levitation force and rotation torque between the primary and secondary fields, as well as ac loss power analysis and estimation.Instead of incorporating both a levitation coil and a ready-made base, Bai and Lee [11] constructed an electromagnetic levitation coil with a pulse-width-modulation signal to change the magnetic levitation height. Amal et al. [12] devoted to a lumped circuit method based on a magnetic equivalent circuit for modelling and sizing of a tubular linear permanent-magnet synchronous motor (T-LPMSM). By implementing the output oversampling scheme to collect the input-output data, the subspace characteristics are used to complement the excitation of the observed data; Sun et al. [13] express the formula that can easily explore the unstable dynamics of the magnetic levitation model.Andreev’s. [14] mathematical model calculates the polarization of the sphere, and the factors affecting effective magnetic charges are determined by an axially magnetized torus. Yuming Gong analyses and concludes that, for better levitation performance of the field distribution, levitation force, and guiding force, the iron shim with 4 mm thickness is the best among 4, 6, and 8 mm iron shims due to decrease in suspension gap, which is experimented in [15]. To investigate the effect of magnetic field gradient on LFD of MP-added superconductors to the device, Abdioglu et al. [16] used the effect of magnetic flux distribution and magnetic power addition on magnetic levitation force with superconductors.Pandey et al. [17] discuss the design of a fractional sliding mode controller for a nonlinear magnetic levitation device dynamic to regulate the current through an electromagnetic coil to levitate a ferromagnetic ball. Hernandez et al. [18] build a controller as a combination of a fact and an LQR gain matrix to replace the proportional operation of fractional PID to LQR gain matrix. By using the context of LQR optimal control, Miller [19] has demonstrated that the optimally decentralized system has strong controllability and observability performance, which is the same as the optimally centralized system which has been considered. This performance of the LQR control decentralized system is enhanced by the linear periodic controller with its graphs strongly associated with the system. By minimizing the objective function of the plant system such as electronic devices and vibration reduction of structures, Teppa Garran et al. [20] have smoothened the transient response of the measured output which contains linear quadratic regulator (LQR) in the quadratic form. Norman [21] mentioned his development of a state-space approach with LQR for the modern control theory which is used to analyse the system. The developed state-space method is relatively easier for multioutput systems.Eswaran et al. [22] have derived the transfer function for the DC servo motor and controlled the motor’s speed by the PID controller. Later, they have discussed the vulnerabilities and threats that can be caused by IoT-connected devices and proposed a cybersecurity solution for safeguarding privacy. Nagarajan et al. [23] has designed a PI controller for the AC servo motor after deriving its transfer function.Furthermore, they have analysed the step response of the system obtained to the time-domain specifications. The inverted pendulum is a difficult control issue that constantly moves into an unregulated state, and the result from A.N.K. Nasir and 1M.A. Ahmad shows that LQR provided a better response than PID control strategies and is presented in the time domain [24]. Munder and Yaseen have experimented on the SIMLAB platform based on three parameters such as peak overshoot, settling time, and rise time for LQR, PID, and lead compensation controllers for magnetic levitation system. This study backs up our claim that LQR performs best in terms of peak overshoot, settling time, and rise time, with 14.6 percent, 0.199, and 0.064 for peak overshoot, settling time, and rise time, respectively [25].Novelty of the work is improving the magnetic levitation system performance by using the LQR controller than the PID controller, and comparative analysis was carried our by comparing the settling time, rise time, and peak overshoot of the magnetic levitation system when operated with PID and LQR controllers.


## 2. Physical Model

### 2.1. Magnetic Levitation System

Magnetic object and magnetic field interaction is used to suspend or levitate a magnetic object. This technique is used in the magnetic levitation system to suspend a ball in its electromagnetic field. With electronic feedback control, a magnetic object can be stabilized and levitated by dynamically adjusting one or more electromagnets in the magnetic system to stabilize the magnetically levitated object at the desired position. To stabilize the levitated magnetic platform, servo control is maintained to control the field of magnetic force that levitates it.Figure1 shows the magnetic levitation system with parts labelled. The basic principle of magnetic levitation system operation is to keep ferromagnetic objects levitated using the applied voltage on the electromagnet. The object’s position is determined through a sensor.Figure 1
Magnetic levitation system.


### 2.2. Modelling of Magnetic Levitation System

The mathematical representation of the system can be categorized into 3 parts: electrical, mechanical, and sensor models. The electrical model can be viewed from the electrical equation derived using Kirchhoff’s law, where Figure2 shows the equivalent electrical circuit of the magnetic levitation system, and the system parameters are shown in Table 1.Figure 2
Electrical circuit of the Maglev system: electrical model of the system.(1)v=iR+Ldidt.Considering the inductance resistance asRL,(2)v=iRL+RS+Ldidt.Taking Laplace transform from equation (2),(3)Vs=IsRL+RS+Ls∗Is,(4)Vs=IsLs+RL+RS,(5)IsVs=1Ls+RL+RS.The mechanical model of the system is(6)F=mg−kfix2.By Newton’s second law,(7)F=md2xdt2,(8)md2xdt2=mg−kfix2.For obtaining steady-state ball positionXss which gives steady-state current iss,(9)d2xdt2=0,mg=kfissXss2,mgkfXss=iss.According to vector standard of shifted variables defined,(10)x^t≜xt−Xss,(11)i^t≜it−iss.Rearranging (10) and (11),(12)x^+Xss≜xt,i^+iss≜it.Applying the shifted variables to mechanical (8),(13)md2xdt2=mg−kfi^+issx^+Xss2.Linearizing (13) x^=0,i^=0,(14)d2xdt2=1m∂∂xmg−kfi^+issx^+Xss2x^=0,i^=0x^+∂∂xmg−kfi^+issx^+Xss2x^=0,i^=0i^,(15)d2xdt2=2kfiss2mxss3x^−2kfissmxss2i^.Laplace transform of (15) is(16)XsIs=2kfiss/mxss2s2−2kfiss2/mxss3.Equating electrical (5) and mechanical equation (16),(17)XsVs=2kfiss/mxss2s2−2kfiss2/mxss3∗1Ls+RL+RS.Table1 gives the detailed system parameters, and by substituting the above values in equation (17), the transfer function of the system is as follows:(18)xsvs=77.44s3+70.16s2−849.2s−0.5947.Table 1
The parameter value of the Maglev system.
VariablesValuesMass of ball (m)0.0571 kgInductance (L)0.017521 HResistance (R)0 .0243 ohmA0.005831x00.0225Spring constant (K)0.0001477 Nm2/A2Gravity (g)9.81 m/s2


## 2.1. Magnetic Levitation System

Magnetic object and magnetic field interaction is used to suspend or levitate a magnetic object. This technique is used in the magnetic levitation system to suspend a ball in its electromagnetic field. With electronic feedback control, a magnetic object can be stabilized and levitated by dynamically adjusting one or more electromagnets in the magnetic system to stabilize the magnetically levitated object at the desired position. To stabilize the levitated magnetic platform, servo control is maintained to control the field of magnetic force that levitates it.Figure1 shows the magnetic levitation system with parts labelled. The basic principle of magnetic levitation system operation is to keep ferromagnetic objects levitated using the applied voltage on the electromagnet. The object’s position is determined through a sensor.Figure 1
Magnetic levitation system.


## 2.2. Modelling of Magnetic Levitation System

The mathematical representation of the system can be categorized into 3 parts: electrical, mechanical, and sensor models. The electrical model can be viewed from the electrical equation derived using Kirchhoff’s law, where Figure2 shows the equivalent electrical circuit of the magnetic levitation system, and the system parameters are shown in Table 1.Figure 2
Electrical circuit of the Maglev system: electrical model of the system.(1)v=iR+Ldidt.Considering the inductance resistance asRL,(2)v=iRL+RS+Ldidt.Taking Laplace transform from equation (2),(3)Vs=IsRL+RS+Ls∗Is,(4)Vs=IsLs+RL+RS,(5)IsVs=1Ls+RL+RS.The mechanical model of the system is(6)F=mg−kfix2.By Newton’s second law,(7)F=md2xdt2,(8)md2xdt2=mg−kfix2.For obtaining steady-state ball positionXss which gives steady-state current iss,(9)d2xdt2=0,mg=kfissXss2,mgkfXss=iss.According to vector standard of shifted variables defined,(10)x^t≜xt−Xss,(11)i^t≜it−iss.Rearranging (10) and (11),(12)x^+Xss≜xt,i^+iss≜it.Applying the shifted variables to mechanical (8),(13)md2xdt2=mg−kfi^+issx^+Xss2.Linearizing (13) x^=0,i^=0,(14)d2xdt2=1m∂∂xmg−kfi^+issx^+Xss2x^=0,i^=0x^+∂∂xmg−kfi^+issx^+Xss2x^=0,i^=0i^,(15)d2xdt2=2kfiss2mxss3x^−2kfissmxss2i^.Laplace transform of (15) is(16)XsIs=2kfiss/mxss2s2−2kfiss2/mxss3.Equating electrical (5) and mechanical equation (16),(17)XsVs=2kfiss/mxss2s2−2kfiss2/mxss3∗1Ls+RL+RS.Table1 gives the detailed system parameters, and by substituting the above values in equation (17), the transfer function of the system is as follows:(18)xsvs=77.44s3+70.16s2−849.2s−0.5947.Table 1
The parameter value of the Maglev system.
VariablesValuesMass of ball (m)0.0571 kgInductance (L)0.017521 HResistance (R)0 .0243 ohmA0.005831x00.0225Spring constant (K)0.0001477 Nm2/A2Gravity (g)9.81 m/s2


## 3. Controller Design for the Magnetic Levitation System

### 3.1. PID Controller

PID (proportional integral derivative) controllers are the most accurate and stable controllers, which controls process variables using the control loop feedback mechanism. To keep the actual output from a process as close to the set point or target output as possible the PID controller uses a closed-loop control feedback mechanism. PID controller’s main purpose is to force feedback to match a set point. The proportional, integral, and derivative are individually adjusted or turned in a PID controller.General PID algorithm form is given as(19)ut=Kp∗et+Ki∫etdt+Kddetdt.The PID controller is designed, and the proportional, integral, and derivative values were derived using the Ziglar Nicholas method.The mechanism of PID controller gain is proportional term generates a result that is proportionate to the current error value. By multiplying the error by a constant Kp, also known as the proportional gain constant, the proportional response can be changed. The integral term’s impact is proportional to the magnitude of the error as well as the duration of the error. In a PID controller, the integral is the total of the instantaneous error over time, which represents the accumulated offset that should have been corrected earlier. The integral gain (Ki) is then multiplied by the accumulated error and added to the controller output. Determine the slope of the error over time and multiply this rate of change by the derivative gain Kd to get the derivative of the process error. The derivative gain, Kd, is the magnitude of the derivative term’s contribution to the total control action.For the magnetic levitation system,(i)
Kp = 0.000547(ii)
Ki = 5.94e-09(iii)
Kd = 12.6This value is fed to the PID controller of the closed-loop system.


### 3.2. LQR Controller

The linear quadratic regulator (LQR) enables the high-performance design and the closed-loop stable state of the system using a method that provides optimally controlled feedback gains. It uses the state-space method to analyse a system which is a method in modern control theory. A multioutput system is relatively simple when working with the state-space method. Full-state feedback can be used to stabilize the system.Algebraic Riccati equation is(20)ATP+PA−PBR−1BT+Q=0,(21)K=R−1BTP.Table4 shows the LQR method decreases the amount of work that the control systems’ engineer has to undertake in order to optimise the controller. However, the engineer must still define the cost function parameters and compare the results to the design objectives. This usually means that controller development will be an iterative process in which the engineer evaluates the “best” controllers generated by simulation and then tweaks the parameters to produce a controller that is more in line with the design goals. The linear-quadratic regulator (LQR), a feedback controller whose equations are given below, provides one of the theory’s fundamental results.To find LQR gain values, Ricatti equation (20)is(22)−ATP+PA−PBR−1BT+Q=0,where the LQR gain values were found using the Ricatti equation (20):(23)ATP+PA−PBR−1BT+Q=0,where(24)A=A−BR−1BT,Q=Q−BR−1BT,K=R−1BTP.Substituting the values of A, B, Q, and R in equation (20), the gain values obtained are 0.0211, 1.6985, and 0.0018 as derived.


## 3.1. PID Controller

PID (proportional integral derivative) controllers are the most accurate and stable controllers, which controls process variables using the control loop feedback mechanism. To keep the actual output from a process as close to the set point or target output as possible the PID controller uses a closed-loop control feedback mechanism. PID controller’s main purpose is to force feedback to match a set point. The proportional, integral, and derivative are individually adjusted or turned in a PID controller.General PID algorithm form is given as(19)ut=Kp∗et+Ki∫etdt+Kddetdt.The PID controller is designed, and the proportional, integral, and derivative values were derived using the Ziglar Nicholas method.The mechanism of PID controller gain is proportional term generates a result that is proportionate to the current error value. By multiplying the error by a constant Kp, also known as the proportional gain constant, the proportional response can be changed. The integral term’s impact is proportional to the magnitude of the error as well as the duration of the error. In a PID controller, the integral is the total of the instantaneous error over time, which represents the accumulated offset that should have been corrected earlier. The integral gain (Ki) is then multiplied by the accumulated error and added to the controller output. Determine the slope of the error over time and multiply this rate of change by the derivative gain Kd to get the derivative of the process error. The derivative gain, Kd, is the magnitude of the derivative term’s contribution to the total control action.For the magnetic levitation system,(i)
Kp = 0.000547(ii)
Ki = 5.94e-09(iii)
Kd = 12.6This value is fed to the PID controller of the closed-loop system.


## 3.2. LQR Controller

The linear quadratic regulator (LQR) enables the high-performance design and the closed-loop stable state of the system using a method that provides optimally controlled feedback gains. It uses the state-space method to analyse a system which is a method in modern control theory. A multioutput system is relatively simple when working with the state-space method. Full-state feedback can be used to stabilize the system.Algebraic Riccati equation is(20)ATP+PA−PBR−1BT+Q=0,(21)K=R−1BTP.Table4 shows the LQR method decreases the amount of work that the control systems’ engineer has to undertake in order to optimise the controller. However, the engineer must still define the cost function parameters and compare the results to the design objectives. This usually means that controller development will be an iterative process in which the engineer evaluates the “best” controllers generated by simulation and then tweaks the parameters to produce a controller that is more in line with the design goals. The linear-quadratic regulator (LQR), a feedback controller whose equations are given below, provides one of the theory’s fundamental results.To find LQR gain values, Ricatti equation (20)is(22)−ATP+PA−PBR−1BT+Q=0,where the LQR gain values were found using the Ricatti equation (20):(23)ATP+PA−PBR−1BT+Q=0,where(24)A=A−BR−1BT,Q=Q−BR−1BT,K=R−1BTP.Substituting the values of A, B, Q, and R in equation (20), the gain values obtained are 0.0211, 1.6985, and 0.0018 as derived.


## 4. Results and Discussion

### 4.1. PID Controller Output

Experimental output of the control signal and ball position of a magnetic levitation system using the PID controller for various input signals are shown in Figure3. The values are(i)
Kp: 32(ii)
Ki: 0.05(iii)
Kd: 0.17Figure 3
STEP output and control signal for the PID controller.
(a)(b)Step, sinusoidal, and square response of the magnetic levitation system is shown in Figures3–5, respectively, by using the PID controller. When we observe the response of actual ball position, it is unable to reach desired ball position. Table 2 shows the PID output inference.Figure 4
SINE output and control signal for the PID controller.
(a)(b)Figure 5
SQUARE output and control signal for the PID controller.
(a)(b)Table 2
PID output inference.
PID controllerStepSquareSinePeak overshoot3.081 ×10−23.13 ×10−21.943 ×10−2Settling time0.5079 ×10−20.7765 ×10−21.102 ×10−2Rise time1.43 ×10−21.0067 ×10−21.26 ×10−2


### 4.2. Inference from the PID Graph

### 4.3. LQR Controller Output

The experimental output of the ball position and control signal of a magnetic levitation system using an LQR controller are shown in Figure4.Figures6–8 show the step response, sinusoidal, and square response of the magnetic levitation system by using the LQR controller. By using the PID controller, when we observe the response of actual ball position, compared to the PID controller, it is able to reach desired ball position. Table 3 shows the LQR output inference.Figure 6
STEP output and control signal for the LQR controller.
(a)(b)Figure 7
SINE output and control signal for the LQR controller.
(a)(b)Figure 8
SQUARE output and control signal for the LQR controller.
(a)(b)Table 3
LQR output inference.
LQRStepSquareSinePeak overshoot1.56 ×10−21.441 ×10−23.1 ×10−2Settling time0.588 ×10−21.075 ×10−20.8802 ×10−2Rise time1.154 ×10−20.8357 ×10−20.546 ×10−2


### 4.4. Inference from the LQR Graph

Table4 shows the response of the magnetic levitation system for various response using PID and LQR controllers. For the PID controller, peak overshoot for step, square, and sine is 3.081 × 10−2, 3.13 × 10−2, and 1.943 × 10−2, respectively. Settling time is 0.5079 × 10−2, 0.7765 × 10−2, and 1.102 × 10−2 for step, square, and sine response, respectively. Rise time for step, square, and sine is 1.43 × 10−2, 1.0067 × 10−2, and 1.26 × 10−2. For the LQR controller, peak overshoot for step, square, and sine is 1.56 × 10−2, 1.441 × 10−2, and 3.1 × 10−2, respectively. Settling time is 0.588 × 10−2, 1.075 × 10−2, and 0.8802 × 10−2 for step, square, and sine response, respectively. Rise time for step, square, and sine is 1.154 × 10−2, 0.8357 × 10−2, and 0.546 × 10−2.Table 4
Response of the magnetic levitation system with PID and LQR controllers.
ControllersResponseStepSquareSinePIDPeak overshoot3.081 ×10−23.13 ×10−21.943 ×10−2Settling time0.5079 ×10−20.7765 ×10−21.102 ×10−2Rise time1.43 ×10−21.0067 ×10−21.26 ×10−2LQRPeak overshoot1.56 ×10−21.441 ×10−23.1 ×10−2Settling time0.588 ×10−21.075 ×10−20.8802 ×10−2Rise time1.154 ×10−20.8357 ×10−20.546 ×10−2


## 4.1. PID Controller Output

Experimental output of the control signal and ball position of a magnetic levitation system using the PID controller for various input signals are shown in Figure3. The values are(i)
Kp: 32(ii)
Ki: 0.05(iii)
Kd: 0.17Figure 3
STEP output and control signal for the PID controller.
(a)(b)Step, sinusoidal, and square response of the magnetic levitation system is shown in Figures3–5, respectively, by using the PID controller. When we observe the response of actual ball position, it is unable to reach desired ball position. Table 2 shows the PID output inference.Figure 4
SINE output and control signal for the PID controller.
(a)(b)Figure 5
SQUARE output and control signal for the PID controller.
(a)(b)Table 2
PID output inference.
PID controllerStepSquareSinePeak overshoot3.081 ×10−23.13 ×10−21.943 ×10−2Settling time0.5079 ×10−20.7765 ×10−21.102 ×10−2Rise time1.43 ×10−21.0067 ×10−21.26 ×10−2


## 4.2. Inference from the PID Graph

## 4.3. LQR Controller Output

The experimental output of the ball position and control signal of a magnetic levitation system using an LQR controller are shown in Figure4.Figures6–8 show the step response, sinusoidal, and square response of the magnetic levitation system by using the LQR controller. By using the PID controller, when we observe the response of actual ball position, compared to the PID controller, it is able to reach desired ball position. Table 3 shows the LQR output inference.Figure 6
STEP output and control signal for the LQR controller.
(a)(b)Figure 7
SINE output and control signal for the LQR controller.
(a)(b)Figure 8
SQUARE output and control signal for the LQR controller.
(a)(b)Table 3
LQR output inference.
LQRStepSquareSinePeak overshoot1.56 ×10−21.441 ×10−23.1 ×10−2Settling time0.588 ×10−21.075 ×10−20.8802 ×10−2Rise time1.154 ×10−20.8357 ×10−20.546 ×10−2


## 4.4. Inference from the LQR Graph

Table4 shows the response of the magnetic levitation system for various response using PID and LQR controllers. For the PID controller, peak overshoot for step, square, and sine is 3.081 × 10−2, 3.13 × 10−2, and 1.943 × 10−2, respectively. Settling time is 0.5079 × 10−2, 0.7765 × 10−2, and 1.102 × 10−2 for step, square, and sine response, respectively. Rise time for step, square, and sine is 1.43 × 10−2, 1.0067 × 10−2, and 1.26 × 10−2. For the LQR controller, peak overshoot for step, square, and sine is 1.56 × 10−2, 1.441 × 10−2, and 3.1 × 10−2, respectively. Settling time is 0.588 × 10−2, 1.075 × 10−2, and 0.8802 × 10−2 for step, square, and sine response, respectively. Rise time for step, square, and sine is 1.154 × 10−2, 0.8357 × 10−2, and 0.546 × 10−2.Table 4
Response of the magnetic levitation system with PID and LQR controllers.
ControllersResponseStepSquareSinePIDPeak overshoot3.081 ×10−23.13 ×10−21.943 ×10−2Settling time0.5079 ×10−20.7765 ×10−21.102 ×10−2Rise time1.43 ×10−21.0067 ×10−21.26 ×10−2LQRPeak overshoot1.56 ×10−21.441 ×10−23.1 ×10−2Settling time0.588 ×10−21.075 ×10−20.8802 ×10−2Rise time1.154 ×10−20.8357 ×10−20.546 ×10−2


## 5. Conclusion

From the implementation of the PID and LQR controller, we have analysed the transient response of the physical magnetic levitation system. For step response, peak overshoot is 1.56 × 10−2 by using the LQR controller. Similarly, by using the LQR controller, peak overshoot of the square response is 1.441 × 10−2, and for sine response, peak overshoot is 3.1 × 10−2. For step response, peak overshoot is 3.081 × 10−2 by using the PID controller. Similarly, by using the PID controller, peak overshoot of the square response is 3.13 × 10−2, and for sine response, peak overshoot is 1.943 × 10−2. It is observed from the obtained results that the LQR controller gives us more satisfactory results among the PID controllers tested. This is verified using the difference between the graphs and the obtained peak overshoot, settling point, and oscillation of the output signal response.



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