Dynamic system for the study of bouncing vibrations of a car with active suspension model for an independent suspension

At the beginning of the paper, the quantities that characterize a dynamic system are presented. Based on them, the state equations of the dynamic system are obtained, a system that models an active suspension. The types of used controllers, their block diagrams and the influence of the adjustment parameters on the process are reviewed. Next, numerical applications are made for the study of vibrations on models with 2 degrees of freedom, for an active suspension equipped with PID and LQR controllers. These controllers, as well as the simulation of the two systems, passive and active, were modelled and simulated using MATLAB/Simulink software. The numerical results obtained after the simulation are presented in the form of diagrams. In the end of the paper, it is concluded that the designed active suspension system can improve the ride quality by minimizing the displacement of the suspended mass.


Introduction
The active suspension is developed from the passive suspension which, in addition to the components of the passive suspension, has a force actuator, respectively a device that controls the elastic and damping characteristics [1], [2], [8], [9].It, through a controlled supply of energy, quickly adjusts the elastic and damping characteristics to the specified suspension mode: normal, comfortable or sporty [2], [3].
To do this, the necessary information about the current driving/use conditions of the vehicle is first collected, using various sensors that collect data such as the type and condition of the road surface, driving parameters, driving style and other data [4].
The data provided by the sensors goes to the vehicle's electronic control unit, where the processed information -depending on the control algorithms -is transmitted to the active components.
The complexity of the system implies a wide range of control algorithms, and most of the control methods are based on the exact knowledge of the suspension system, thus, the requirement of sensors increases and also increases the cost of the suspension system [5].
The calculation of the command of an active suspension requires the knowledge of dynamic systems theory notions.Therefore, in the following, the theoretical notions are presented and applications are made on systems with two degrees of freedom.
The dynamic system consists of a set of elements that communicate with each other and the physical quantities that help to transmit information in technical systems are called signals.
The control force is one of the defining characteristics of semi-active and active suspensions.It is responsible for controlling the elastic and damping characteristics of the suspension, which means that it has a significant impact on the behavior of the vehicle while driving.The purpose of this force is to minimize oscillations and displacements of the sprung mass, which contributes to improve the comfort and stability of the vehicle.In the case of semi-active suspensions, the controlled force is generated by adjusting the damping characteristics of the dampers.These can be achieved by changing the viscosity of the fluid or by adjusting the size of the holes in the damper.Modern semi-active suspensions use technologies such as magnetorheological dampers, electrorheological dampers and hydraulic dampers to achieve rapid adjustment of damping characteristics.
Active suspensions use an actuator to provide external compensating force.These systems are able to adapt at road conditions changing and can satisfy the requirements of comfort and stability while driving.The controlled force in active suspensions is generated by actuators acting directly on or in parallel with the suspension components.These actuators can be of various types, such as pneumatic actuators, hydraulic actuators, and magnetic or electromagnetic actuators.The control force, managed through appropriate controllers, is essential to achieve optimal performance of the semi-active and active suspension system [10], [11].

The characteristics of a dynamic system
A dynamic system is characterized by the fact that the output () depends on the entire internal evolution of the system under the influence of the input () (figure 1).
A linear continuous system has the following form [2]: where, the first equation is the state equation and the second is the output vector equation.is the matrix of transmission parameters [2].
The structure of a system defined by the two equations, the state equation and the output equation, can be represented in the following form [6] (figure 2).Stability is an important characteristic of dynamic systems, being one of the conditions of existence and functionality.A dynamic system is unstable if it cannot achieve the purpose for which it was designed (that of being stable).
In the field of system stability, two notions are used: the notion of internal stability (referring to the state of the system) and the notion of external stability (referring to the output of the system).
A system is internally strictly stable if the state of the system evolves in a free regime towards the origin: IOP Publishing doi:10.1088/1757-899X/1303/1/012038 State controllability is the property of a system in which the defined input () transfers the state of the system () from the initial state   to the final state   .The controllability study is based on the controllability matrix: Observability is studied using the observability matrix   :

Controllers
The Linear Quadratic Regulator (LQR) is a variant of regulation in which the state equation of the model is linear, the performance index is quadratic and the initial conditions are known [2].LQR is an optimal control problem for linear systems.For a dynamic system of the form: a controller () must be found in order to minimize the performance index: The first term on the right side of relation ( 6) represents the error, and the second term represents the energy consumption.
The block diagram of an LQR controller is the one in figure 3, where the matrix K has the role of minimizing the performance index, which describes the cost function of the system.Thus, to find an optimal control law for the dynamic system, the reference signal will have to be () = 0.
The matrices  and  from relation (6) determine the relative importance of error and energy consumption.These are symmetric and positive definite or semi-definite and are known as weight matrices.The PID (Proportional Integral Derivative) control method, a method proposed since the 1940s, is still used today in the control of industrial processes in most industrial procesess.
PID control is combined with sequential functions, logic and functional blocks for the construction of automated systems [7], [2].
The optimal performance of a controller can only be achieved if its parameters are chosen correctly.
The PID controller has in its composition three elements P, I and D which are summed to calculate the control signal.The controller is characterized by the equation: 0 (7) where:   -is the tuning parameter, Proportional,   -is the tuning parameter, Integrated,   -is the tuning parameter, Derived and  -is the error (the process variable).The block diagram of a PID controller is shown in figure 4. To obtain optimal process performance, determining the PID control parameters is obviously a crucial part of closed-loop control systems.
Various methods have been developed to adjust these parameters.Regardless of the method used, it is necessary to monitor the key elements of the controller: load disturbances, sensor sensitivity and reference signals.Methods used to adjust a PID controller are: Ziegler-Nichols, Cohen-Coon and Software tools.The best known methods are those developed by Ziegler and Nichols.The two methods proposed by Ziegler and Nichols help to adjust P, PI and PID controllers [5].
These were presented in 1942, but are still used today as a starting point for adjusting controllers [6].Next, based on these notions, a numerical application on a model with 2 degrees of freedom will be created.

The mechanical model for the study of bounce vibrations
In figure 5, the active suspension example is presented for the study of vibrations on a model with two degrees of freedom, also called a quarter car.The following notations were used:  1 -the unsprung mass,  2 -the suspended mass,  1 -the elastic constant of the tire,  2 -the elastic constant of the spring,  1 -the damping constant of the tire,  2 -the damping constant of the shock absorber, ℎ -the unevenness of the running path,  1 -the displacement of the mass  1 ,  2 -the displacement of the mass  2 ,   -the force controlled by the controller.From the dynamic balance equations, the differential equations of motion are obtained: The numerical application will be made for a car for which the following data is adopted:  1 = 32,5kg,  2 = 345kg,  1 = 120000   ⁄ ,  2 = 27900   ⁄ ,  1 = 6000 Ns  ⁄ ,  2 = 1500 Ns  ⁄ .The purpose of the application is to compare the performances of an active system, by reducing the oscillations of the suspended mass, in which the damping force is controlled by two controllers: PID and LQR.
The main stages in determining the performance of an active system are the following: the representation of the state-space model (figure 2), the model based on equations (8), the determination of the properties of the linear system, the design of the two PID and LQR controllers.

Presenting the model space-state
To implement a controller, the state-space model for representing differential equations can be used.The general form of the model is given by equation (1).
Using the notations in figure 5, we can write the state vector , the output vector  and the input vector  in the following form: The components of the state vector  are: the suspension stroke, 2 the speed of the suspended mass,  1 − ℎ the deflection of the tire, 1 the speed of the unsprung mass.
The components of the input vector  are: ℎ ̇ the derivative of the road height,   the force of the actuator, and the components of the output vector  are: the stroke of the suspension, 2 the acceleration of the suspended mass.We assume that the tire and the shock absorber have linear characteristics and based on the differential equations of displacement we can calculate the matrices , ,  and  :

The properties of the linear system
The stability.A linear system is stable if all poles have the real part located in the negative complex half-plane.The poles represent the roots of the characteristic equation, which is of the form: where I is the unit matrix.
After substitution we determine the poles of the system: It is observed that all poles of the system have negative real part and we deduce that the system is stable.The controllability.The state of the system is controllable if the rank of the controllability matrix is equal to the size of the matrix A.
Matrix of the controllability of the state of the linear system is built based on the matrices A and B.
The general form of the controllability matrix is: With the help of functions in the MATLAB software, ctrb and rank, the controllability matrix   and its rank are determined:   = 1 ⋅ 10 9 [ 0 0 0 0 0 0 −0.0083 0 0 0 0 0 −0.002 0 0.0330 0 0 0 0 0 0 0 0.0082 0 0 0 0 0 0.0082 0 −1.7164 0.0003 ] (15) It is observed that the rank of the controllability matrix is equal to the size of the matrix A. The observability.A linear dynamical system is observable if the rank of the observability matrix is equal to the matrix size A.
The observability matrix of the state of the linear system is built based on the matrices A and C.
With the help of the MATLAB functions obsv and rank the observability matrix   is determined and it's rank.

𝑟𝑎𝑛𝑔(𝜴 𝑂 ) = 4
It can be noted that the rank of the observability matrix is equal to the size of the matrix A.

The controller LQR
If the two matrices, system matrix A and input matrix B, are controllable we can use the LQR optimization method.A function must be found,   , so as to minimize the performance criterion.
The matrices Q and R are weight matrices.Since it is possible to intervene only on the force of the actuator   , not on the disturbance of the path h, the new matrices A and B are the following: IOP Publishing doi:10.1088/1757-899X/1303/1/012038 The solution to the problem is to find a vector K such that   = −.
So the new matrix of the system is of the form: The calculation of the Q matrix is done with the following relation: With the help of the calculation program made in MATLAB for a value of R = 1, the gain vector is obtained: = [0,1172 0,0282 0,0948 −0,0011].

The PID controller
The PID controller, like the LQR, continuously monitors the operation of the system and calculates the error.The general form of a PID controller is: Determining the coefficients and adjusting them is done with the PID Tuner function in the MATLAB/Simulink software.These coefficients have the following values:   = 1,05,   = 6,86,   = 0,04.

Numerical results
Based on the equations of motion and the previous numerical results, the following graphs will show the influence of the two controllers on the passive system with two degrees of freedom.The red color (the curve with the highest amplitude) represented the response of the passive system, with a solid green line the response of the system controlled by the LQR controller and with a broken blue line the response of the system controlled by the PID controller.In figure 6 a significant reduction, approximately 50%, of the displacement of the suspended mass is observed for the system controlled by one of the PID or LQR controllers.
In the case of the system controlled by the LQR controller, the oscillations of the suspended mass are eliminated after approximately 2 seconds.In figure 7 the speed of the suspended mass for the system controlled by the LQR controller has the lowest value among the three compared systems, and its value becomes zero after about 2 seconds.
To obtain the two graphs, a unity step type signal having the value of 0.1 m was used to observe the influence of the two controllers on the passive system.
The value of the amplitudes for the suspended mass decreased, but also the period of oscillation of the suspended mass was reduced.

Conclusions
The active suspension offers high comfort compared to a passive suspension, but this consistent improvement in comfort requires energy consumption.
Active suspensions are used especially in luxury class vehicles and sports cars.
In this work, an active suspension for a car was designed by designing a controller, which has the role of improving the performance of the system, in terms of the design objectives compared to the passive suspension system.
The mathematical modelling was carried out using an example with two degrees of freedom on the quarter car model for the passive and active suspension system.
The state of the road surface was considered to be a unit step type excitation.In order to increase passenger comfort and vehicle handling, two LQR and PID controllers were developed.These controllers and the simulation of the passive and two active systems were modelled and simulated using the MATLAB/Simulink software.
The paper presents a way of adjusting the parameters of the PID and LQR controllers with the help of specialized functions of the MATLAB software.The steps used in designing the controllers represent a model that can be used for an easy understanding of how the controllers work.Thus, it is shown that the presented active suspension system offers a reduction of the displacements of the sprung mass.

Figure 1 .
Figure 1.Representation of a dynamic system.

Figure 2 .
Figure 2. Block diagram of a dynamic system.

Figure 3 .
Figure 3. Block diagram of an LQR controller.

Figure 4 .
Figure 4. Block diagram of a PID controller.

Figure 5 .
Figure 5. Mechanical model for the study of bouncing vibrations.

Figure 6 .
Figure 6.Displacement of the suspended mass.Figure 7. The speed of the suspended mass.

Figure 7 .
Figure 6.Displacement of the suspended mass.Figure 7. The speed of the suspended mass.