Abstract
Currently, mechatronic systems are commonly used in automotive control systems to improve ride comfort and road safety characteristics. In these control systems, the sensors used to determine the input signals of controllers are very important to help the calculation process and give the correct control output signals for actuators. However, not all input signals can be easily measured directly by sensors or their cost is too high. This paper proposes one using the Kalman–Bucy estimator to estimate the state variables of a two-axle car to study the automotive suspension system. The disturbance input is the road profile at the four tires, while the car speed is considered from 40 km/h to 120 km/h. This research only uses four signals, namely, the vertical displacement of the four unsprung masses, through the estimator it can determine 14 variables of the vector state of the general vertical model of the car. The advantage of this research is that it is possible to reduce the number of the sensors but still ensure that the necessary input signals are obtained for studying the full car model. The signal estimation through some sensors available on the car is necessary for practice. Survey results show that the signal quantification through the Kalman–Bucy estimator unit has an accuracy of over 96% compared to the original car’s signals.
1. Introduction
The vibration of cars is generated during the movement affecting by the road surface. Evaluation and control for the car vibration are necessary and important because this factor affects directly the ride comfort and road holding characteristics. The car suspension system plays an important role in these characteristics. Therefore, when studying the car vibration, it is impossible to separate the structural elements of the suspension system.
Traditional car suspension systems using leaf or coil springs has linear properties designed on ensuring a compromise between the ride comfort and the road holding under operating conditions different movements of cars on the road. Currently, in modern cars, air suspension systems using pneumatic spring elastomers are increasingly widely used. The outstanding advantage of this suspension system compared to the traditional one is the ability to change the stiffness coefficient following the load and the movement modes, ensuring at the same time two criteria of the ride comfort and the road holding. It is possible to apply new control technology to change the system’s parameters, creating an active suspension system. Current automotive vibration research needs to be more general, focusing on three main purposes: optimizing the suspension system and controlling the suspension system; studying the effects of vibration on people, goods, vehicles, and roads; and studying vehicle dynamics and control.
To study the vibration of cars, as well as the automotive dynamics problems, theoretical studies are usually done through car models, which include: 1/4 car model built by Krishnasamy et al. [1], Marcu et al. [2]; 1/2 car model by Gao et al. [3]; and full car models by Darius and Yahaya [4]. These models are used to simulate and control the suspension system. Meanwhile, the full car model is used to apply the Skyhook control algorithm to evaluate the ride comfort criteria by Kim and Hong [5]. The evaluation of these models is built on the principles of dynamics and high accuracy which can be applied to study the real problems of dynamics and vibration of automobiles. The previous studies have mainly studied about automotive vibration by the road surface and solutions to improve the ride comfort and the road holding criteria.
The experimental research orientation on automobile vibration is also of interest to researchers. The difficulty of this research orientation is that many high-precision sensors are needed to measure and determine the input signals for the experimental model. In real world application, the accurate results and high-precision testing equipment are required; therefore, the new measuring methods are need to eliminate noises. In order to solve this difficulty for the experimental research on automotive vibration, many researchers have come up with a solution to use signal estimators (filters). These estimators are able to determine the desired signals through cheap, common, and highly reliable sensors [6, 7]. Thus, these estimation methods are of great interest in the research and industry community.
The first method for the optimal estimation from noisy datasets is the least-squares method. After that, other estimation methods continued to be developed and researched based on the Markov process. The proportional-integral (PI) estimator has also been widely implemented to estimate both system states and faults. In [8], H∞/PI estimator is proposed to handle the disturbances/Unknown Input (UI)/Unknown Input Observer Based Approach for Distributed Tube-Based Model Predictive Control. Then, Hamdi et al. [9] presented a more comprehensive design for LPV systems by combining UI and PI estimators. Unfortunately, its restrictive constraint on the UI-decoupling condition cannot be always satisfied. Hence, in order to overcome this issue, Hassanabadi et al. [10] have developed a H∞ proportional-integral (PI) estimator so that the UI effect on estimation error is also attenuated by H∞ synthesis. Meanwhile, Zhang et al. [11] have built a H∞ high-order PI filter to estimate the actuator faults and attenuate the perturbation impact. On the other hand, too few studies are mentioned for the impact of parametric uncertainty [12] on PI estimator, which causes estimator instability by generating state x in the dynamics of estimation error [13].
Currently, the common estimation method in the study of automotive vibration can be mentioned as: the usual Kalman estimator (KE) [7, 14–17]; KE is understood as optimal regression data processing algorithm. There are many ways to determine the optimal depending on the criteria for choosing evaluation parameters. The KE processes all available values, except error. It estimates the current values of the parameters of interest using the understanding of the parameters and system dynamics, numerical description of statistical data. It also includes system noises, measurement noises, uncertainty in the dynamic models, and any information about the initial conditions of the parameters of interest and then collects the response in the form of a (noise) value. The value update equation is responsible for predicting the current (time) value in advance and correlating the estimated error to obtain a prior estimate for the next time point. The value update equation is responsible for the negative feedback, i.e., combining the new value and the predecessor estimate to get an improvement on the posterior estimate. In practice, in many cases where continuous state parameters are required to be estimated, the KE estimator cannot be performed. The extended Kalman estimator (EKE) [18] uses the first-order Taylor expansion to obtain a linear approximation of the polar coordinate measurements in the update. In this process, a Jacobian matrix is generated, representing a linear mapping from poles to Cartesian coordinates, which is applied at the update step. Thus, the transformation matrix becomes a Jacobian matrix, while using nonlinear measure and state transformation functions for both prediction and update, respectively. The rest is all the same as the Kalman filter, the Kalman–Bucy estimator [19] (KBE) which is the continuous-time filter form of the KE filter, which is further developed based on the KE estimator. The purpose of the KBE estimator is to determine the unmeasurable state parameters (assuming they are observable) and the actual output parameters of the process [20, 21].
Richard S. Bucy was the one who worked on nonlinear differential equations, similar to those of Jacopo Francesco Riccati (1676–1754), now known as the “Riccati equation.” One of the special features of Kalman and Bucy’s theory is that the cycle is proved through the Riccati equation. It is a reliable solution, even if the dynamic system is unstable. The researchers by Kalman and Bucy proposed and fully added Kalman estimator for nonlinear applications and called it extended Kalman estimator. It can be said that, up to now, there has been no research detailing the application of the Kalman–Bucy method in estimating the state variables of the full car model.
In this paper, the authors present the designing content of a Kalman–Bucy estimator for estimating the state variables of the full car model. By using this estimator, some signal measuring sensors can be replaced for surveying car vibrations and using the survey results to evaluate research models according to the criteria of automotive vibration assessment. Therefore, the authors only need to use velocity sensors for four unsprung masses such as 190501 Velomitor CT, 330500 Velomitor, Nevada 330505 Low-Frequency Velocity Sensor, 330525 Velomitor XA Piezo-Velocity Sensor. The important contribution of this study is practical application. When applying the Kalman–Bucy estimator to the survey of the automotive vibration model, it is possible to reduce the investment cost of sensor equipment, requiring only a minimum number of cheap-common sensors but still ensuring high accuracy for the required signals. The content of the article consists of 5 parts: Section 2 presents a full vertical car model. Section 3 presents the designing content for the Kalman–Bucy estimator. Section 4 presents the simulation results and evaluation. Section 5 presents the conclusions and proposes future research directions.
2. General Vibration Model of Automobiles
2.1. Full Vertical Car Model
A full car model with two axles is shown in Figure 1 with 7 Degree of Freedom (DOF) [22–24]. This model has the main elements: spring ci, damper ki, and actuator fi. This actuator is controlled to change the damping force with the aim of studying the active or semiactive suspension system. The tires are simplified by the elastic elements with the stiffness coefficient cti. The sprung mass is considered as a rigid block with mass m and has three movement directions: the vertical displacement z, the pitch angle θ and the roll angle φ. The wheel assembly is also considered as a solid block with mass mi, has only in the vertical direction zui. The contact between the wheel and the road surface is a point.
Applying 2nd Newton law and D’Alembert’s principle for the full vertical car model in Figure 1, the dynamic equations of the car are defined as follows.
The force balance equation for the sprung mass in the vertical direction:
The moment balance equation for the sprung mass in the roll direction OY:
The moment balance equation for the sprung mass in the pitch direction Ox:
The force balance equation for the unsprung mass at each wheel in the vertical direction:
During normal movement, the pitch and roll angles both reach small values, so the displacement of the sprung mass at each wheel zi is determined as follows:
Equations from (1) to (8) are the dynamics equations of the full car model in the vertical direction, and they are converted to the state-space representation as follows [25]:where the state vector:
External input: ; Control vector of actuators: . In equation (9), the variables required to evaluate the car vibration are vertical displacement, roll angle, pitch angle of the sprung mass and their acceleration; and load transfer coefficient at the two axes; therefore, the output vector should be chosen as Y = X. Here, A, B1, B2, C, D1, and D2 are the matrices of the system defined in Appendix.
2.2. Criteria for Assessing Car Vibration
The ride comfort characteristic is evaluated by the sprung mass displacement , the sprung mass roll angle , the sprung mass pitch angle , and their variables acceleration , , .
The road safety characteristic is evaluated by unsprung masses vertical displacement , and their acceleration .
The relationship between the ride comfort and the road safety of a car has always the conflicting behavior, that is, increasing the ride comfort will decrease the road safety and it depends a lot on the suspension parameters. Figure 2 shows the influence of damping coefficient on the road safety and ride comfort characteristics of cars.
If the damping coefficient is low, the ride comfort increases but the road safety of the movement decreases and vice versa. Therefore, choosing the suitable suspension parameters as well as applying the controlled suspension system is necessary to get good performance for the car vibration.
3. Kalman–Bucy Estimator Design for General Vibration Model of Automobiles
Currently, there are many methods of designing estimator to estimate the value of the signal such as: Kalman, H2, H∞, PI… [25, 27]. The Kalman estimator is a discrete-time filter, using a predictive-correction estimation algorithm, used as a kinematic system model to predict state values and a measurement model for correction this prediction. However, in practice, many cases require estimation of continuously changing parameters including state parameters of the system. Kalman–Bucy estimator (KBE) [19, 20] is the continuous-time filter form of conventional KF filter [16] and used in this study.
A controller that optimizes the operation of the system is based on the complete measurement of the state, i.e., all the variables of the state vector have data. However, full-state measurement can be costly or technologically infeasible, especially for rapid state measurement that needs to be accurate.
Equation (9) generalizes the state-space form of full vertical car model. Instead of measuring the complete state of x, it may be possible to estimate the state from measurements from the output boundaries y. Full state estimation is mathematically possible as long as the pair of states (A, C) is observable, although the efficiency of the estimator depends on how well the observations are quantified by the ability to observe Gramian. The Kalman filter is the most commonly used full-state estimator, as it optimally balances the competing effects of measurement noise, and model uncertainty. It is possible to use the full-state measurement from a Kalman estimator in association with the full-state feedback rule of the optimal controller (LQR) [26].
With the disturbation , and sensor noise , the system can write in this form:
The Kalman filter observer assumes that:
Here, E is the expected value and δ is the delta function. The matrices and are semideterministic positive with the earnings of the covariances of the noise and noise limits. Extensions to the Kalman filter exist for the terms unknown noise.
It is capable of containing an estimate of the complete state x from the measurement of the input u and the output y, through the estimator dynamics system:
The matrices A, B, C, and D are already contained in the system models, and the filter gain Kf is determined through a common procedure such as LQR controller. The filter gain Kf is given by the following equation:
The Riccati differential equation:
With the general function:
In this general function, the effects of noise which are required to determine the optimal balance between the positive estimation and the noise attenuation. Therefore, the Kalman estimator is considered a Linear Quadratic Estimation (LQE), and there is a dual formula for LQR optimization. The generalized function in (15) is calculated as the population mean over multiple executions. The diagram applied to build the Kalman–Bucy estimator is shown in detail in Figure 3.
Substitute the output estimate , the system is written as
The estimated state variables are represented as with the inputs y and u. If the system is observable then it is possible to set arbitrary eigenvalues of A − KfC with the choice of Kf.
Here, we note that the value of Kf greatly affects the estimation result. So we choose the following value: Kf = diag ([10−6, 10−6, 10−6, 10−6, 10−6, 10−6]).
Because the use of the equation of state-space (9) has the output parameters as variables of the state vector, it is possible to use the LQG controller in combination with the Kalman estimator to evaluate through the variables themselves. This variable, including: for the ride comfort, the criteria are vertical displacement, roll angle, pitch angle of the sprung mass, and their derivation; for the road safety, it is the displacement of four unsprung masses.
Thus, given the input parameters, measured output parameters, and assumptions about process noise, the purpose of the KBE is to determine the unmeasured state parameters (with the assumption that they can be observable) and the actual output parameters of the process. The estimates of the state parameters (t) and the output parameters (t) of the KBE are shown in Figure 4.
Unlike the Kalman estimator, which uses a prediction and correction algorithms to update the state parameter estimates, the Kalman–Bucy estimator requires the Riccati differential equation to be integrated continuously over time. The system of estimated update equations of the Kalman–Bucy set is expressed in mathematical form as follows:
In the system of equation (18): -estimate the covariance of the measure satisfying the Riccati equation; K-efficiency matrix KBE; R-weight matrix (covariance matrix) of the measured noise; and Q-weight matrix (covariance matrix) of the process noise (state).
The authors note that the values of Q, and R greatly affect the estimation results. Usually, the value of Q must be less than R and is chosen as follows:
The measured signal through the y sensors will combine with the input excitation to become the input signals for the estimator. The output consists of the observed signals and . These signals will be compared with the original signal to evaluate the efficiency of the estimator. After many times of testing based on the above simulation model, the values of the Q, R matrices for the Kalman–Bucy estimator to estimate the parameters in the state vector x are selected in formula (19).
With the diagram as shown in Figure 5, using the built-in general model of the car whose input parameters are the displacement speed of four wheels, the remaining state parameters of the oscillating models can be determined by surveying models. The survey results for comparison and evaluation with the KBF signal estimator are the state vector, the measurement signals. The road excitation signal of the models is used as the system disturbance to build the KBE estimator. The estimator survey program used in Matlab/Simulink is shown in Figure 6. Results of evaluating the accuracy of the estimated signals through two blocks comparing the output signal of the car models and the KBE estimator.
4. Evaluation of the Designed KBE Estimator
Using the general model of the car as shown in Figure 1, conducting simulations and surveys built with Matlab/Simulink software, the working quality of the Kalman–Bucy estimator is evaluated when the car is used for running on a random road profile [23] with the excitation signal from the road surface at each wheel. The frequency is at 10 rad/s and the simulation time is 10 s. q1, q2, q3, and q4 are shown as Figure 7. The Kalman–Bucy estimator is evaluated through the exact comparison of the survey signals: -Displacement of the sprung mass, -Velocity of displacement of the sprung mass, -Roll angle of displacement of the sprung mass, and -Pitch angle of displacement of the sprung mass. Three cases are considered in this study as follows:(i)Case 1: the car is moving in a straight road with a speed of 40 km/h(ii)Case 2: the car is moving in a straight road with a speed of 80 km/h(iii)Case 3: the car is moving in a straight road with a speed of 120 km/h
To evaluate the accuracy of the signal from the KBE estimator, two criteria are used in this study, including:(i)The value of signals over time(ii)The Root Mean Square (RMS) of signals [25, 27]
The noise considered during the investigation is the weight matrix for the measured noise (e−3)Rn (4 signals are the velocity of the unsprung mass) and the weight matrix (e−5)Qn for process noise (14 variables of the system’s state vector).
The goal of building a signal estimator is that the estimated signals through the Kalman–Bucy estimator must be closest to the real signal of the car. The parameter values of the full vertical model of the surveyed car are listed in Table 1.
The objective of building the estimator is that the signals after using the Kalman–Bucy set to estimate must have the closest results to the real signal when measured with an error of less than 5%. When designing the estimator, it is important to find the estimated gain for optimal estimation in the presence of noise. The excitation input signal for the survey models is as shown in Figure 7.
4.1. Survey Car Movement at Different Speeds
4.1.1. Case 1: The Car Moves along a Straight Road at a Speed of 40 km/H
The car is moving in a straight line with a speed of 40 km/h on a random road surface. The noise measured matrices , , the resulting signal is shown in Figure 8.
(a)
(b)
(c)
(d)
According to the simulation results in Figure 8 and Table 2, it can see that the signal through the KBE estimator closely matches the value of the actual signal of the car. This shows that the accuracy of the estimator is very reliable. Indeed, according to the value comparison table of RMS, it can see that the signal Z has an accuracy with an error of 0%, has an accuracy with an error of 0.28%, and and have a small error accuracy not more than 4%. The objective of estimating parameters has guaranteed accuracy, because all parameters do not exceed 4% of error accuracy compared with the actual signal.
4.1.2. Case 2: The Car Moves along a Straight Road at a Speed of 80 km/H
Survey is carried out with car moving in a straight line at a speed of 80 km/h on a random road surface. With the noise measured matrices , the results are obtained in Table 3.
According to the simulation results obtained in Table 3, it can see that when the car is traveling at 80 km/h, the signal through the KBE estimator closely matches the value of the actual signal of the car. This shows that the accuracy of the estimator is very reliable. Indeed, according to the value comparison table of RMS, it can see that signal Z has an error accuracy of less than 2%, has an error accuracy of less than 4%, and φ and θ have error accuracy less than 1%. The objective of estimating parameters has ensured accuracy in the case of 80 km/h, because all parameters are not more than 4% compared to the real signal.
4.1.3. Case 3: The Car Moves along a Straight Road at a Speed of 120 km/H
Survey is carried out with the car moving in a straight line with a speed of 120 km/h on a random road surface. The noise measured matrices are , the results are obtained in Table 4.
According to the simulation results obtained in Table 4, the car running at a high speed of 120 km/h, the signal through the estimator still closely follows the value of the actual signal of the car. The goal of estimating parameters has guaranteed accuracy, because at 120 km/h, all estimated parameters have an error accuracy less than 2% when compared with the actual signal.
Thus, with the survey results obtained when driving cars at different speeds, it shows that the accuracy of the built-in signal KBE estimator completely meets the set criteria, because the error accuracy compared to with the original signal is always less than 4%.
4.2. Survey and Evaluation of Signal Estimators with Different Weight Matrix of Noises
The objective of the survey is to evaluate the operation of the signal estimator in the case of different noises, to find out the operating limit of the signal estimator. The noise considered and investigated here is the weight matrix for the measured noise Rn (4 signals) and the weight matrix Qn for the process noise (14 states).
4.2.1. Effect of Weight Matrix Qn on Process Noise (14 States)
Survey with the assumption that the process noise weight matrix Rn of the system is constant (e−3) for the car running at different speeds from 50 to 150 km/h with different matrix Qn values. From e−1 to e−5. Figures 9–12 show the surveyed values: Series 1 for the case of e−1, Series 2 = for the case of e−2, Series 3 for the case of e−3, Series 4 for the case of e−4, Series 5 for the case of e−5. At a speed of 50 km/h Figure 9 shows the obtained values correspond to five changes in the Qn value. According to the survey results when the car is running at a speed of 50 km/h, it shows that the Zsd value has a big difference from the Qn noise value of 0.1 and 0.01, when the Qn value is reduced to less than 0.01, the accuracy of KBE estimator has a very small error. Meanwhile, other values do not have much effect. At a speed of 80 km/h According to the survey results when the car is running at 80 km/h, it shows that the Zsd value of the estimator has an effect when the Qn noise changes at the values of 0.1 and 0.01, if the Qn is further reduced, the accuracy is high. The other values are not affected much. At a speed of 100 km/h According to the survey results when the car is running at a speed of 100 km/h, it shows that the Zsd value of the signal estimator is effective when the Qn noise changes at the values of 0.1 and 0.01. The other values are not affected much. At a speed of 150 km/h According to the survey results when the car is running at 150 km/h, it shows that the Zsd value of the KBE estimator is influential when the Qn noise changes at the values of 0.1 and 0.01. The other values are not affected much.
From these results, the car is running at different speeds, surveying the effect of noise measured Qn shows that the measured noise value Qn has a great influence on the accuracy of the signal estimator. The measurement noise value Qn must be less than 0.01 for the estimator to ensure the required accuracy.
4.2.2. Effect of Weight Matrix on Noise Measurement Rn (4 Signals)
Survey when the weight matrix of process noise Qn of the system is constant (e−1, e−5). The selected Qn value remains unchanged for the surveying process. Let the car run at different speeds from 50 to 150 km/h to change different Rn values (Rn: 1 ÷ e−4). Figures 13–16 show the result for surveying values: Series 1 for the case of 1, Series 2 for the case of e−1, Series 3 = for the case of e−2, Series 4 = for the case of e−3, and Series 5 for the case of e−4. At a speed of 50 km/h According to the survey results in Figure 13, the measured parameters when Rn changes when the car is running at a speed of 50 km/h, it shows that the Zsd value of the estimator is influential when the noise Rn changes at different values (less than 0.01 when the noise Qn is 0.1). The other values are not affected much. If Qn is selected at e−5, then Rn does not affect the estimator. At a speed of 80 km/h According to the survey results in Figure 13, the measured parameters when Rn change when the car is running at a speed of 50 km/h, it shows that the Zsd value of the estimator is influential when the noise Rn changes at different values (less than 0.01 when the noise Qn is 0.1). The other values are not affected much. If Qn is selected at e−5, then Rn also does not affect the estimator. At a speed of 100 km/h According to the survey results when the car is running at a speed of 100 km/h, it shows that the Zsd value of the estimator is influential when the noise Rn changes at values less than 0.01 in the case of the noise Qn is 0.1. The other values are not affected much. If Qn is selected at e − 5, then Rn does not affect the estimator. At a speed of 50 km/h According to the survey results, when the car is running at a speed of 150 km/h, it shows that the Zsd value of the estimator is influential when the noise Rn changes at values less than 0.01 when the noise Qn is 0.1. The other values are not affected much. If Qn is selected at e − 5, then Rn does not affect the estimator.
More detailed survey results are shown in Tables 5–7, when considering the effect of weight matrix on process noise Qn and weight matrix for measuring noise Rn, which vary with the forward velocity. Thus, surveying the influence of many measurements Rn and Qn, it shows that the Qn value has a great influence on the accuracy of the estimator. The weight matrix Rn of the measured noise does not affect the accuracy of the estimator much. The choice of the estimator’s parameterization affects the accuracy of the estimated signal. The selected value of Qn must be less than 0.1 and Rn greater than 0.1 will ensure the accuracy of the design estimated signal.
5. Conclusions
In this study, the authors applied the Kalman–Bucy estimator to the general full model of 2-axle passenger cars. To build an observer for 14 signals of the system’s state vector through the use of only 4 common sensors, which are the vertical displacement of the unsprung mass. The effectiveness of the Kalman–Bucy estimator is evaluated with the three movements of the sprung masses as well as their velocities, and the vertical displacements of the four unsuspended masses of the car at velocities between 40 and 120 km/h. The simulation results show that the difference in the root mean square error is less than 4%, that is, the observed signal has an accuracy of over 96% compared to the real signal. The research process also evaluates the influence of measured noises on the accuracy of the estimator. The guaranteed value for the estimator with an error of not more than 5% is when choosing the weight matrix Qn less than 0.01 and Rn greater than 0.1. Thus, the signal observer proposed above can be used in studying general vibration models of automobiles. Using experimental research, it is possible to reduce the number of sensors that need to be measured but still can ensure the necessary accuracy.
The next research direction is to conduct experiments on real cars to evaluate the effectiveness of the proposed method, as well as to accurately consider the effects of noises from the sensors.
Appendix
The equation of state (A.1) for the general oscillation of the car is determined as follows:
In which, the state vector:
Input excitation source: ;
control vector of actuators: ;
the state matrices of (A.1) are defined as follows:
Data Availability
The survey data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This study was carried out thanks to funding from the University of Transport and Communications through a university-level scientific research project with code T2022-CK-004, titled: research and design of signal estimator for the car vibration model.