State Estimation Based on Sigma Point Kalman Filter for Suspension System in Presence of Road Excitation Influenced by Velocity of the Car

*e states of the suspension system including the road excitation depend on the road quality, the velocity of the car, and the sprung mass. *ose states play a very important role in the control problem of stability, ride comfort, ride safety, and dynamic wheel load of the suspension systems. *e velocities and deflections of the sprung mass and unsprung mass would not be measured fully in the practice. *erefore, it must be estimated by other measured quantities from the system such as acceleration and deflection of sprung mass and unsprung mass. To control the active suspension system, its states need to be estimated accurately and guaranteed the response time. *is paper presents the method using the sigma point Kalman filter to estimate the suspension system’s states including the road excitation, the deflections, and the velocities of the sprung mass and unsprung mass. *e mathematical model of the suspension system is rewritten for the state estimation problem, and the stochastic load profile is supposed the main noise input. *e stochastic characteristic of the road excitation depending on the car’s velocity is taken into account in themodel used for suspension system state estimation.*e results calculated based on the practical experiment data for specific road profile with some particular velocities of the car show that the suspension system states are estimated quite accurately in comparison with the practice states.


Introduction
In recent years, the research in active suspension control has continued to advance with respect to its capability of driving flexibly the suspension damping and suspension spring stiffness to guarantee the ride comfort [1][2][3][4].
is work considers a typical active suspension system with the notations of the quantities depicted in Figure 1.
e mechanical system consists of sprung mass m s and unsprung mass m w , and the sprung mass and unsprung mass are connected by the suspension part including damping part c s and spring part d s in parallel. In case of active suspension, this part is added by an active actuator (electrical, electromagnetic, and hybrid devices driven by electrical signal u(t)) to change the natural characteristics of suspension part in order to get the desired ride comfort. Connection between the unsprung mass and the road surface is the wheel. e wheel is z s , F d need to be estimated by using the state estimation. ere are a lot of state estimation methods to implement this problem in some materials [5][6][7][8] in the literature; most of them now use the Kalman Filters (KF) and extended Kalman filter (EKF) for some modified models of the suspension systems. e state estimation quality is dependent of the road profile, velocity of the car v(t), and the mass of sprung m s . e road excitation always is supposed as the main input noise, but we realize that z r not only depends on the road profile but also on the velocity of the car. For the same road profile, as the velocity of the car changes, the stochastic characteristics of the z r will be changed. Figure 4 shows the varying of the z r in the case velocity of the car changing from 5 km/h to 45 km/h for the same road profile. e road profile has stochastic characteristics as follows: min � − 0.0232 cm; max � 0.0235 m; mean � 0.00131 m; variance � 0.033 m; μ � 0.00131 m; σ � 0.058 m. Figure 4 shows that the velocity of the car is higher and the amplitude of the variation z r is lower. e maximum of z r is 0.06 m in the case v(t) � 5 km/h, and the maximums of z r are 0.023 m and 0.02 m in the cases v(t) � 15 km/h and 45 km/h, respectively. e changing of stochastic characteristics z r by the velocity of the car and road profile will affect strongly to the state estimation process.
is paper presents the method using the SPKF to estimate the suspension system's states. e system model taking into account the road excitation model and velocity of the car is used in the SPKF algorithm. Statistical characteristics of the road excitation are updated in SPKF by using the correlation matrices depending on the velocity of the car. e results calculated by the using the practical data for particular road profile with some specific velocities of the car show that the suspension system's states are estimated quite accurately in comparison with the real states. e remaining part of this paper is structured as follows: Section 2 mentions the modelling of the suspension system. Section 3 introduces the method using SPKF to estimate the states of the suspension systems. Section 4 shows the estimated states calculated by using the practical data for particular road profile with some specific velocities of the car. Finally, Section 5 summarizes some conclusions.

Modelling of the Suspension System
e first-order model of the road excitation z r as the function of the road profile and velocity of the car depicted in Figure 5 can be written as [9,10]: where λ and v are the optional feedback parameter and velocity of the car, respectively, and w zr is the input noise caused by the road profile.      Suppose u(t) as the active force, if u(t) � 0, the system is called the passive suspension system, else it is called the active suspension system, when u(t) ≠ 0. e model of the suspension system is written as follows: Equation (4) is written briefly as follows: in which the state functions and vectors are defined as follows: e vector f(z(t)) reflects the natural dynamic of the suspension system, and the matrix g(z(t)) converts the road excitation; active force and velocity of the car to the vibrations of the sprung mass and unsprung mass describe the relationship between the road excitation with road profile and velocity of the car. e dynamic noise vector is w(t), in which four first elements are the noises of the states z s , _ z s , z w , _ z w caused by the errors of the model parameters like the changing of the mass m s of the car (Figures 6 and 7) and the last two elements are the noises caused by the road profile and velocity of the car w zr , w dzr . By the characteristics of the road and system, w(t) is Gauss white noise and e measured signals of the suspension are accelerations of the sprung mass, unsprung mass, and the deflection between the sprung mass and unsprung mass. e output of the model is formed as vector as follows: e model of the output vector is (8) is written briefly as where e vector h(z(t)) is the natural output of the system and the matrix ζ(z(t)) describes the effects of velocity and active force to the accelerations and deflections of sprung and unsprung masses. Vector n(t) is the output noise caused by the changing of the sprung mass and the spring stiffness. n(t) is Gauss white noise and satisfies the equation 4 Journal of Control Science and Engineering In general, the dynamic model of the suspension system taking into account the road excitation model and used for state estimation is described by In the static state, u(t) � 0 and v(t) � 0, and the system remains the natural characteristics f(z(t)) and h(z(t)).
When v(t) ≠ 0, the suspension system is effected by g(z(t)), ζ(z(t)), and road profile. e models (11) and (12) are used to design the state estimator, and this model is depicted in Figure 8.

Estimation of the States of the Suspension
System by Using the SPKF e SPKF known as one of the state estimation methods has some advantages better than EKF by application ranges, and this method estimates more accurately the states for the second-order system or higher; meanwhile, computational complexity is same as EKF. SPKF is easily implemented as there is no requirement of the Jacobi matrix calculation.
For the EKF, the system's nonlinear dynamic is linearized at the sampling time, the input distributions are mapped via the linearized dynamic function to calculate the output distributions, and the output distributions is less accurate if there is high nonlinear of the system dynamic.
For the SPKF, the whole of the input distributions (systems noise and load profile) are mapped thought the nonlinear dynamic functions to get the output distribution. Figure 9 explains the differences basically between the EKF and SPKF though the calculation of distribution of x at the time k based on the nonlinear function Φ(·) and the distribution of x at the time k − 1. We can see that the distribution of x(k) is calculated more accurately than by EKF [13][14][15].
For the suspension system, the distribution of the road excitation depends on the velocity of the car for the same road profile, so the function g(z(t)) in the dynamic model also depends on the variance z r ; it is nonlinear. So that, applying the SPKF coming with the covariance matrices update by the velocity of the car could improve the accuracy of the state estimation as well as the road excitation z r .
Consider the time sampling to be small enough, the system (11) can be converted to the discrete model at the sampling time k as Journal of Control Science and Engineering Model (13) is used to estimate the suspension system states which include the position, velocities of the sprung and unsprung mass, and also the road excitation z r . e measured output is y(k), including the accelerations and deflections of two masses. e velocity of the car also is the input of the SPKF. e road excitation for the same road profile w zr has the covariance matrix depending on the velocity of the car. e process of the state estimation z s , _ z s , z w , _ z w , z r using SPKF at the sampling time k is conducted by two main steps: Step 1 (i) Forms the augmented state vector at the sampling time k − 1 based on the estimated states at the previous time k − 1 and the vector of mean values of the system noise w and measured noise n. Note that vector w has the last element which is the mean of the road excitation acted by the velocity of the car, so those vectors are (ii) Forms the estimated covariance matrix of the estimation error in the previous step including the covariance matrices of the estimation error, the noise error, and the measured error. e covariance matrix

Dynamic function Φ (·)
Distribution of x k

Dynamic function Φ (·)
Distribution of x k  Journal of Control Science and Engineering of system noise also depends on the velocity of the car: (iii) e augmented sigma point matrix containing p + 1 sigma points is established as (iv) Calculates priori sigma points at the sampling time k by using the suspension system dynamic (13) and the augmented sigma point matrix at the sampling time k − 1. For the sigma point i, estimation of the priori sigma points is calculated by following equation: (v) e priori state estimation is calculated by using the priori sigma points mapped via the system dynamic χ a,− k− 1,i aŝ (vi) e covariance matrix of the estimation errors is updated by the following equation: Note that λ is the scaling parameter; α (m) i and α (c) i are the weighted mean and covariance of χ which are chosen by unscented Kalman filter [16].
(vii) e estimated output distribution is calculated by using χ z,− k,i : (i) e estimated output of the suspension system at the sampling time k is estimated by following equation: Step 2 (i) Update the covariance matrices of the output errors at the sampling time k and the output errors and state estimation errors; these matrices depend on the sigma points and matrix ψ k,i (calculated by equation (20)) and alsoẑ − k ,ŷ(k): (ii) Calculate the estimation gain matrix as (iii) Update the priori estimation states calculated at the step 1 by taking into account the output errors by the following equation, we have the estimation state asẑ (iv) Estimate the dynamic wheel load: (v) Update the covariance matrix of the estimation errors: Journal of Control Science and Engineering 7 (vi) Back to the step 1 for the next sampling time k + 1.
e vector of process noise means and the covariance matrix of the process noise errors depend on the velocity of the car for the same road profile. At each sampling time, these matrices need to be updated by the velocity of the car and road profile. For the particular road profile, the practical examinations need to be done to determine the covariance matrix of the process noise errors with respect to the velocity of the car. (27) Figure 10 shows the road profile (length 1000 m, sampled 10 cm/point), and the distribution characteristics of the road profile are depicted in Figure 11 and Table 1.

State Estimation Results of the Suspension System
e covariance matrices depending on the velocity of the car are valued as Variation of road amplitude (cm)   Figure 11: e distribution characteristics DPF of z r by velocity of the car. Table 1: e variation of the road distribution parameters z r by the velocity of the car for the road profile shown in Figure 10. e estimated states z s , _ z s , z w , _ z w , z r by using the SPKF for the suspension system are depicted in Figure 12 to 16 for the velocities of the car 5 km/h, 20 km/h, 40 km/h, 60 km/h, and 80 km/h. e errors of the state estimation errors are shown in Figure 17. Distribution of the state estimation errors is shown in Figure 18.
From the state estimation results and the distribution of the state estimation errors, we see the following.
For the estimation of z s and z w , the distributions z s − z s and z w − z w are quite the same; the variance of the distributions is gradually reduced from 0.04 cm to 0.005 cm by the increasing of the velocity of the car from 5 km/h to 80 km/h, respectively. e distribution of z r − z r is larger than the distributions of z s − z s and z w − z w , the distribution of the estimation error z r � z r − z r is 0.25 cm when the velocity of the car reaches 5 km/h, and z r decreases to 0.015 cm when the velocity of the car reaches 80 km/h. For estimation of the velocity of two masses, the distribution of dz s /dt − dz s /dt has variance 1 cm/s for velocity of the car 5 km/h and 0.1 cm/s for the 80 km/h. e distribution of dz w /dt − dz w /dt has the variance larger more than two times in comparison with the distribution of dz s /dt − dz s /dt. Figure 19 shows the estimated wheel dynamic load for the velocities of the car from 5 km/h to 80 km/h.

Conclusions
is paper presents the method using SPKF to estimate the suspension system's states including the road excitation, the deflections, and the velocities of the sprung mass and unsprung mass. To estimate the states, we use the     covariance matrices of the estimation errors varying by the velocity of the car. By taking advantage of this, the states of the system are estimated better, especially the road excitation. e mathematical model of the suspension system is rewritten for the state estimation problem; the stochastic load profile is supposed the main noise input. e stochastic of the road excitation depending on the car's velocity is considered in the model. e estimated results calculated based on the practical road profile data with some particular velocities of the car show that the state estimation errors are getting smaller when the velocity of the car is higher; the suspension system's states are estimated quite accurately in comparison with the real states. State estimation of the suspension system taking into account the road excitation depending on the velocity of the car based on the SPKF is the main contribution of this paper.
Data Availability e practical data of the road profile used to support the findings of this study have been collected, measured, and processed by the research group under under grant no. B2016-TNA-06. e road profile data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.