Dynamical analysis and controllers performance evaluation for single degree-of-freedom system

Ivan Isho Gorial

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Dynamical analysis and controllers performance evaluation for single degree-of-freedom system

Ivan Isho Gorial

| 12 ago 2020

International Journal on Smart Sensing and Intelligent Systems

Volume 13 (2020): Numero 1 (January 2020)

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Article Category: Research-Article

Pubblicato online: 12 ago 2020

Pagine: 1 - 12

Ricevuto: 02 lug 2020

DOI: https://doi.org/10.21307/ijssis-2020-018

Parole chiave
FLC, LPID, NPID, MSDS, SDOF, Mamdani system, Defuzzification

© 2020 Ivan Isho Gorial published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Nomenclature

m

mass

k

spring stiffness coefficient

b

viscous damping coefficient

F

force applied to the system externally

x

displacement

\dot{x}

velocity

\ddot{x}

acceleration ( $a$ )

zero

positive medium

positive small

negative medium

negative small

x_{i}

point in the conclusion universe (i = 1, 2, 3,…)

μ_{c} (x_{i})

membership value of the result of the conclusion set

γ

design parameter

ω

natural frequency of the exciting force

Vibration is a big problem in many engineering systems. Active control for vibrations is very important topic with a lot of applications and fiscal benefits. An SDOF spring mass damper system represents many engineering structures. For example, in facing road bump, the vibration in a car can be modeled using the SDOF system. Also, the SDOF system is easy to analyze. It is easier to develop complicated systems after starting from a benchmark problem like SDOF problem. The comfort and maneuverability of a vehicle depend quite a lot on the vibration of the vehicle. Most of the vehicles implement a semi-active suspension system. Although they perform adequately, demand for smooth and cost-effective active-vibration controlled suspension is high (Sharma and Singh, 2014).

The momentous contributions of MSDS are notably reflected in the fields of automation and mechatronics engineering. It is extensively used as a compliant actuator in robot manipulator (Li and Yin, 2017), serves as a shock absorber in the vehicle suspension system (Akpakpavi, 2017; Allamraju, 2016; El-Nasser et al., 2015; Katal and Singh, 2012; Srinivasan et al., 2016), piezoelectric vibration energy harvester (Bahiah et al., 2017; Caliò et al., 2014; Kundu and Nemade, 2016), used as anti-vibration devices for sky-scraping building to improve serviceability and suitable for teaching dynamic modeling in control systems. In recent times, the deployment of the MSD system is significantly on high demand in health services to aid the control of rehabilitation robots in reinstating the incapacitated and aged people and in a hybrid vehicle suspension system to enhance passenger trip comfort and vehicle stability.

Furthermore, several control methods had been suggested and tested and new control schemes are still emerging: Vargas and Bruneau (2004) defined metallic dampers were to be structural fuses. Sánchez and Pugnaloni (2011) studied the response of an SDOF mechanical system composed of a primary mass and a particle damper.

Also, Shen et al. (2015) studied analytically the SDOF system under the control of three semi-active methods. The nonlinear model of the rotary axial eddy current damper and the energy dissipation was investigated by Liang et al. (2019). Bhutta et al. (2016) studied mathematical modeling of a double forced double MSDS.

Moreover, Orhorhoro et al. (2016) reported the research work carried on the MSD model in a phase variable form. Sharma and Singh (2014) controlled the vibrations of MSDS utilizing fuzzy logic.

However, vibration technology grew and took on a more interdisciplinary character. This was caused by more demanding performance criteria and specifications for the design of all types of machines and structures. So, considerable attention has been paid to research, development, and implementation of structural control devices. The essential aim of this project is to dynamical analysis for MSDS with SDOF and to compare performance evaluation of three different controllers including constructing a rule base that works effectively.

The rest of this paper is ordered as follows: methodology of mechanical system model with SDOF and controller design is given in the second section. The third section clarifies simulation results and discussion. Conclusion is given in the fourth section.

Methodology of mechanical system model with SDOF and controller design

Many times, the SDOF framework can be used for modeling the machine. In addition, visualization and analysis are very simple. When designing a control approach, an SDOF problem is recommended, which is a benchmark problem (Sharma and Singh, 2014; Frankovský et al., 2011). The SDOF mechanical system is shown in Figure 1 with MSD. A harmonically variable force: $F (t) = F_{0} \sin (ω t)$ F(t) = F0 sin (ωt) is used for the model excitation.

Therefore, trying to establish a mathematical model that depicts the dynamics of the MSDS and manipulates certain system parameters to restrict the force to have a desired response and performance of the system.

Dynamic motion equation that applies to the mechanical system (Fig. 1) in the form of a vector: (1) $m a = F_{r} + F_{d} + F (t) .$

Substituting the forces equation of the motion: (2) $m \ddot{x} = - F_{r} - F_{d} + F (t),$ (2)where F_r is the reaction force in the spring; F_d the damping force; and F(t) the excitation force.

The reaction force in the spring is: (3) $F_{r} = k x .$

The force of damping is as follows: (4) $F_{d} = b \dot{x} .$

The force of excitation is as follows: (5) $F (t) = F_{0} \sin (ω t) .$

After substitution of the relation (3), (4), (5) to Equation (2), we obtain the mechanical system acceleration with SDOF as follows: (6) $\ddot{x} = - \frac{b}{m} \dot{x} - \frac{k}{m} x + \frac{1}{m} F (t) .$

As a consequence, we obtain the mechanical system’s kinematic values (displacement, velocity, and acceleration) in relation to time t (Table 1).

Table 1.

Values of system parameters.

Symbol	Value	Unit
m	2	kg
k	250	N/m
b	2.236	–

Figure 2 shows the system of MSDS without any controller.

In addressing the control of position and velocity of MSDS, several control algorithms have been proposed and tested. Three different controllers are designed and a comparative analysis is performed on SDOF of MSDS for model derived through Newton’s law. These suggested controllers are designed independently from each other as follows.

LPID controller

PID (proportional integral derivative) control is a control loop feedback mechanism that is mostly used in industrial control systems. PID control calculates an error value and applies a correction based on proportional, integral, and derivative terms (Samin et al., 2011; Nafea et al., 2018; Sartika et al., 2019).

For the same system in Figure 2, we insert PID controller before the plant. In continuous time, the ideal PID controller is given as: (7) $u_{P I D} = k_{p} e + k_{i} \int e d t + k_{d} \dot{e} .$

Ziegler and Nichols proposed rules for determining values of k_p, k_i, and k_d based on Katsuhiko (2010) and Ang et al. (2005) and clarified in Table 2.

Table 2.

Values of gains.

Proportional gain (k_p)	Integral gain (k_i)	Derivative gain (k_d)
0.42	0.08	0.28

NPID controller

The nonlinear PID control law is proposed based on Al-Samarraie et al. (2016) by replacing the integral of the error function with the integral of the error saturation function, by adjusting the parameters of the saturation function: (8) $u_{N P I D} = k_{p} e + k_{i} \int_{0}^{t} s a t_{γ} e d t + k_{d} \dot{e},$ (8)where Sat_γ is the saturation function given by: (9) $S a t_{γ} (e) = γ \times s i g n (e) .$

Moreover, the design parameter γ for the NPID controller, γ = 1.5.

FLC

Velocity and displacement of the SDOF system were taken as input to the fuzzy controller and the actuation force was output, whereas fuzzy logic-based control system was used in the nonlinear active vibration control system. Now, as shown in Figure 3 that clarifies the system of MSD with FLC, five membership function in the first input and five membership function in the second input so that (25 rules) was designed as FLC in addition to, five membership function in the output. These all by utilizing MATLAB fuzzy logic toolbox.

FLC utilizes the Mamdani system which in the consequent part uses fuzzy sets. To make an FLC (Mamdani controller), the first step is to specify the use of variables of linguistic and also define the statement of the problem for this function, as shown in Figures 4–7.

1st input displacement (error) variable.

2nd input velocity (change in error) variable.

Velocity and displacement of the sprung mass were fed as input to the fuzzy controller, which was based on 25 rules developed by heuristics. The controller was found to work satisfactorily in perturbed conditions also. Illustration of the rule base of MSDS control is as follows.

For discrete sets, center of gravity for singletons (COGS) is determined as follows (Alan, 2010): (10) $u_{C O G S} = \frac{\sum_{i} μ_{c} (x_{i}) x_{i}}{\sum_{i} μ_{c} (x_{i})} .$

Simulation results and discussion

The simulation results are obtained in two ways: without a controller and with controller, i.e. FLC, LPID, and NPID. The three cases for the force are considered here as follows.

Case 1

F = 0.5 N .

F = 0.5N.

Figure 8 illustrates rule viewer, and Figures 8 and 9 clarify a fuzzy inference system (FIS) rule viewer and three-dimensional plot of the output surface viewer and simulation results controller for MSDS, respectively.

Surface viewer for error vs. change of error.

Moreover, the displacement and velocity with three independent controllers and without controller are displayed in Figures 10 and 11, respectively.

Displacement without controlled and with controlled.

Velocity without controlled and with controlled.

Table 3 shows the response of SDOF system and percentage reduction in response output for case 1.

Table 3.

Response of SDOF system and percentage reduction in response output.

Case 1
Type	Absolute maximum displacement (m)	Reduction (%)	Absolute maximum velocity (m/s)	Reduction (%)
Without controller	1.213 × 10⁻²	–	1.544 × 10⁻¹	–
With controller
FLC	2.280 × 10⁻³	81.203%	4.493 × 10⁻²	70.900%
LPID	1.564 × 10⁻²	28.936%	1.501 × 10⁻¹	2.784%
NPID	9.680 × 10⁻³	20.197%	7.245 × 10⁻²	53.076%

Case 2

F = F_{0} \times \sin (w t),

where $A$ A is the amplitude, $α = \frac{2 π}{T}$ .

Take T = 2πs, F = 0.5N, so that, α = 1 rad/s, and F = sin(t)N.

However, the displacement and velocity with three independent controllers and without controller are displayed in Figures 12 and 13, respectively.

Table 4 shows the response of SDOF system and percentage reduction in response output for case 2.

Table 4.

Response of SDOF system and percentage reduction in response output.

Case 2
Type	Absolute maximum displacement (m)	Reduction (%)	Absolute maximum velocity (m/s)	Reduction (%)
Without controller	8.594 × 10⁻³	–	1.023 × 10⁻¹	–
With controller
FLC	1.499 × 10⁻²	42.668%	1.816 × 10⁻¹	43.667%
LPID	1.048 × 10⁻²	17.996%	1.059 × 10⁻¹	3.399%
NPID	9.369 × 10⁻³	8.271%	1.099 × 10⁻²	89.2570%

Case 3

Figures 14 and 15 show its ability in response of SDOF system and percentage reduction in response output of case 3.

Table 5 shows the response of SDOF system and percentage reduction in response output for case 3.

Table 5.

Response of SDOF system and percentage reduction in response output.

Case 3
Type	Absolute maximum displacement (m)	Reduction (%)	Absolute maximum velocity (m/s)	Reduction (%)
Without controller	8.465 × 10⁻³	–	1.206 × 10⁻¹	–
With controller
FLC	1.689 × 10⁻²	99.527%	1.99 × 10⁻¹	65.008%
LPID	1.070 × 10⁻²	26.402%	1.229 × 10⁻¹	1.907%
NPID	6.205 × 10⁻³	26.698%	5.640 × 10⁻³	95.323%

Generally, Figure 16 shows the control forces with using all these three different controllers (FLC, LPID, and NPID).

The performance of the proposed controllers is tested. The results prove the effectiveness of the FLC controller and its ability in response of SDOF system and percentage reduction in response output as clarified in chart of percentage reduction in response in Figure 17.

Chart of percentage reduction in response.

Although many studies have reported on the use of MSDS in various applications and its important role besides the growth of vibration technology and its take on a more interdisciplinary character, little research has been reported on the effectiveness of this use. Consequently, the below table shows the comparison between our system and the system in (Ahmed, 2018) as follows.

Conclusion

Particularly, the significant contributions of the MSDS are reflected in the fields of automation and mechatronics engineering. This paper extended to velocity and acceleration performance evaluation, moreover, the consideration of the nonlinearity. In addition, fuzzy membership functions are an important part of a fuzzy system usually defined by chartered experts. A comparative analysis of the proposed control algorithms is analytically examined. It is evident from the simulation result that the FLC is active and appropriate for refining the system’s response and suitable for hands-on controlling of the position and velocity of an MSDS. Simulation results showed that the FLC satisfied all control specifications except in some cases where the NPID showed superiority as clarified in Figure 3 that shows chart of percentage reduction in response. However, FLC response is more satisfactory in terms of robustness, response of system and percentage reduction in response output. Besides, the FLC displays more efficiency than the system without this controller after comparing the performance evaluation of the system analysis, i.e. a significant reduction occurs in the response output of displacement and velocity.

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Dynamical analysis and controllers performance evaluation for single degree-of-freedom system

Article Category: Research-Article

Pubblicato online: 12 ago 2020

Pagine: 1 - 12

Ricevuto: 02 lug 2020

DOI: https://doi.org/10.21307/ijssis-2020-018

Parole chiave
FLC, LPID, NPID, MSDS, SDOF, Mamdani system, Defuzzification

© 2020 Ivan Isho Gorial published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

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Dynamical analysis and controllers performance evaluation for single degree-of-freedom system

Article Category: Research-Article

Pubblicato online: 12 ago 2020

Pagine: 1 - 12

Ricevuto: 02 lug 2020

DOI: https://doi.org/10.21307/ijssis-2020-018

Parole chiaveFLC, LPID, NPID, MSDS, SDOF, Mamdani system, Defuzzification

© 2020 Ivan Isho Gorial published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

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Parole chiave
FLC, LPID, NPID, MSDS, SDOF, Mamdani system, Defuzzification