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

Authors: Chi Nguyen Van
Journal: Journal of Control Science and Engineering
 (2019)

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## Abstract

The states of the suspension system including the road excitation depend on the road quality, the velocity of the car, and the sprung mass. Those states play a very important role in the control problem of stability, ride comfort, ride safety, and dynamic wheel load of the suspension systems. The velocities and deflections of the sprung mass and unsprung mass would not be measured fully in the practice. Therefore, 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. This 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. The mathematical model of the suspension system is rewritten for the state estimation problem, and the stochastic load profile is supposed the main noise input. The stochastic characteristic of the road excitation depending on the car’s velocity is taken into account in the model used for suspension system state estimation. The 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.


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## Body

## 1. 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 [1–4]. This work considers a typical active suspension system with the notations of the quantities depicted in Figure 1.Figure 1
Physical model of a typical suspension system.The mechanical system consists of sprung massms and unsprung mass mw, and the sprung mass and unsprung mass are connected by the suspension part including damping part cs and spring part ds in parallel. In case of active suspension, this part is added by an active actuator (electrical, electromagnetic, and hybrid devices driven by electrical signal ut) 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. The wheel is described by wheel damping part cw and wheel spring part dw. The states of the suspension system are deflections, velocities, and accelerations of sprung mass zs,z˙s,z¨s and unsprung mass zs,z˙s,z¨s, respectively. zr is the road excitation supported as the main input noise of the system; zr depends on the road profile and velocity of the car vt.The performance quality of the suspension system is reflected upon the comfort ride, and this quality is presented by 3 parameters: the accelerations of the car and the wheelz¨s,z¨c, respectively; the dynamic wheel load Fd and the suspension deflection zs−zw. The dynamic wheel load Fd is calculated as follows:(1)Fd=cwzr−zw+dwz˙r−z˙wN.The dynamic wheel loadFd depends on the mass of the car mc, velocity of the car v, and road profile as shown in Figures 2 and 3.Figure 2
The variation of the dynamic wheel load with the variation of the car velocity for the specific road profile.Figure 3
The variation of the dynamic wheel load with the variation of the car mass.The smaller values ofz¨s,z¨c,Fd are, the higher comfort ride of the passengers in the car is. To control actuator of active suspension system in order to get the desired values of z¨s,z¨c,Fd with the measured qualities z¨w, z¨s, zs−zw, all the states zr,zw,zs,z˙w,z˙s,Fd need to be estimated by using the state estimation. There are a lot of state estimation methods to implement this problem in some materials [5–8] 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. The state estimation quality is dependent of the road profile, velocity of the car vt, and the mass of sprung ms.The road excitation always is supposed as the main input noise, but we realize thatzr 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 zr will be changed.Figure4 shows the varying of the zr in the case velocity of the car changing from 5 km/h to 45 km/h for the same road profile. The road profile has stochastic characteristics as follows: min = −0.0232 cm; max = 0.0235 m; mean = 0.00131 m; variance = 0.033 m; μ=0.00131m; σ=0.058m.Figure 4
The variation ofzr with the velocity of the car for the same road profile.Figure4 shows that the velocity of the car is higher and the amplitude of the variation zr is lower. The maximum of zr is 0.06 m in the case vt=5km/h, and the maximums of zr are 0.023 m and 0.02 m in the cases vt=15km/h and 45 km/h, respectively.The changing of stochastic characteristicszr by the velocity of the car and road profile will affect strongly to the state estimation process.This paper presents the method using the SPKF to estimate the suspension system’s states. The 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. The 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. The remaining part of this paper is structured as follows: Section2 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.


## 2. Modelling of the Suspension System

The first-order model of the road excitationzr as the function of the road profile and velocity of the car depicted in Figure 5 can be written as [9, 10]:(2)dzrdt=−λvzr+wzr,where λandv are the optional feedback parameter and velocity of the car, respectively, and wzr is the input noise caused by the road profile.Figure 5
The first-order model of road excitationzr as the function of the road profile and velocity of the car.Most of estimators using in the materials [11, 12] consider zr as the input noise acting directly to the systems, and the model of zr related to the road profile and velocity of the car is not taken into account in the model of the system. So, to describe all the states of the system including the road profile and road excitation, the model of the active suspension system needs to be mentioned in the relationship in equation (2).Let us consider the state vector of the suspension system described in Figure1 as(3)z1=zs,z2=z˙s,z3=zw,z4=z˙w,z5=zr,z6=z˙rz¯=z1z2z3z4z5z6T.Supposeut as the active force, if ut=0, the system is called the passive suspension system, else it is called the active suspension system, when ut≠0. The model of the suspension system is written as follows:(4)dz¯dt=z2+wzs,csz3−z1ms−dcz2−z4ms+utmc+wdzs,z4+wzw,cwz5−z3mw−dwλvz5+z4mw−csz3−z1mw+dcz2−z4mw−utmw+wdzw,z6+wzr,−λvz5+wdzr.Equation (4) is written briefly as follows:(5)dz¯dt=fz¯t+gz¯tut+wt,in which the state functions and vectors are defined as follows:(6)fz¯t=z2csz3−z1ms−dcz2−z4msz4cwz5−z3mw−dwz4mw−csz3−z1mw+dcz2−z4mwz60,gz¯t=001mc000−1mw−dwλzrmw000−λz5,wt=wzswdzswzwwdzwwzrwdzr,ut=utvt.The vectorfz¯t reflects the natural dynamic of the suspension system, and the matrix gz¯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. The dynamic noise vector is wt, in which four first elements are the noises of the states zs,z˙s,zw,z˙w caused by the errors of the model parameters like the changing of the mass ms 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 wzr,wdzr. By the characteristics of the road and system, wt is Gauss white noise and satisfies the equation Ewtwt′=Qδt−t′, where Q is the positive defined matrix.Figure 6
The difference ofzs between two cases of ms=94kg and ms=144kg for same road profile and velocity of the car.Figure 7
The variation ofzs by mass of the car ms for same road profile and velocity of the car.The measured signals of the suspension are accelerations of the sprung mass, unsprung mass, and the deflection between the sprung mass and unsprung mass. The output of the model is formed as vector as follows:(7)y¯=z¨sz¨wzs−zwT.The model of the output vector is(8)y¯t=−cszs−zwms−dcz˙c−z˙wms+utms+vz¨c,cwzr−zwmw+dw−λvzr−z˙wmw+cszs−zwmw+dcz˙c−z˙wmw−ut+vz¨w,zs−zw+vzczw.The equation (8) is written briefly as(9)y¯t=hz¯t+ζz¯tut+nt,where(10)hz¯t=−cszs−zwms−dcz˙c−z˙wmscwzr−zwmw−dwmwz˙w+cszs−zwmw+dcz˙c−z˙wmwzs−zw,ζz¯t=1ms0−1mw−dwλzrmw00,nt=vz¨cvz¨wvzczw.The vectorhz¯t is the natural output of the system and the matrix ζz¯t describes the effects of velocity and active force to the accelerations and deflections of sprung and unsprung masses. Vector nt is the output noise caused by the changing of the sprung mass and the spring stiffness. nt is Gauss white noise and satisfies the equation Entnt′=Rδt−t′, where R is the positive defined matrix.In general, the dynamic model of the suspension system taking into account the road excitation model and used for state estimation is described by(11)dz¯dt=fz¯t+gz¯tut+wt,y¯t=hz¯t+ζz¯tut+nt,(12)Fd=cwz5−z3+dwz6−z4.In the static state,ut=0 and vt=0, and the system remains the natural characteristics fz¯t and hz¯t. When vt≠0, the suspension system is effected by gz¯t, ζz¯t, and road profile. The models (11) and (12) are used to design the state estimator, and this model is depicted in Figure 8.Figure 8
The model of the suspension systems taking into account the road profile and velocity of the car.


## 3. Estimation of the States of the Suspension System by Using the SPKF

The 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. Figure9 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 xk is calculated more accurately than by EKF [13–15].Figure 9
Explanation of the dynamic mapping to determine the distribution of thex at the sampling time k via the nonlinear function calculated by the distribution of x at the sampling time k − 1 for the (a) EKF and (b) SPKF.
(a)
(b)For the suspension system, the distribution of the road excitation depends on the velocity of the car for the same road profile, so the functiongz¯t in the dynamic model also depends on the variance zr; 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 zr.Consider the time sampling to be small enough, the system (11) can be converted to the discrete model at the sampling time k as(13)z¯k=fk−1z¯k−1+gk−1z¯k−1uk−1+wk−1,y¯k=hz¯k+ζz¯kuk+nk.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 zr. The measured output is y¯k, including the accelerations and deflections of two masses. The velocity of the car also is the input of the SPKF. The road excitation for the same road profile wzr has the covariance matrix depending on the velocity of the car.The process of the state estimationzs,z˙s,zw,z˙w,zr using SPKF at the sampling time k is conducted by two main steps:Step 1(i)
Forms the augmented state vector at the sampling timek−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(14)w¯vk−1=w¯zsw¯dzsw¯zww¯dzww¯zrvk−1w¯dzrvk−1T,z¯^k−1a,+=z¯^k−1a,+Tw¯vk−1n¯T.(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. The covariance matrix of system noise also depends on the velocity of the car:(15)Cz¯˜,k−1a,+=diagCz¯˜,k−1+Cw˜vk−1Cn˜.(iii)
The augmented sigma point matrix containingp+1 sigma points is established as(16)χk−1a,+=z¯^k−1a,+,z¯^k−1a,++λCz¯˜,k−1a,+,z¯^k−1a,+−λCz¯˜,k−1a,+≜χk−1z,+χk−1w,+χk−1n,+T.(iv)
Calculates priori sigma points at the sampling timek 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:(17)i=1:p+1χk,iz,−=fk−1χk−1a,++gk−1χk−1a,+uk−1+χk−1,iw,+.(v)
The priori state estimation is calculated by using the priori sigma points mapped via the system dynamicχk−1,ia,− as(18)z¯^k−≈∑i=0pαimfk−1χk−1,iz,++gk−1χk−1,iz,+uk−1+χk−1,iw,+=∑i=0pαimχk,iz,−.(vi)
The covariance matrix of the estimation errors is updated by the following equation:(19)Cz¯˜,k−=∑i=0pαicχk,iz,−−z¯^k−χk,iz,−−z¯^k−T.Note thatλ is the scaling parameter; αimandαic are the weighted mean and covariance of χ which are chosen by unscented Kalman filter [16].(vii)
The estimated output distribution is calculated by usingχk,iz,−:(20)i=1:p+1,ψk,i=hχk,iz,−+ζχk,iz,−uk+χk−1,in,+.(i)
The estimated output of the suspension system at the sampling timek is estimated by following equation:(21)ky¯^≈∑i=0pαimhχk,iz,−+ζχk,iz,−uk+χk−1,in,+=∑i=0pαimψk,i.Step 2(i)
Update the covariance matrices of the output errors at the sampling timek and the output errors and state estimation errors; these matrices depend on the sigma points and matrix ψk,i (calculated by equation (20)) and also z¯^k−, y¯^k:(22)Cy˜,k=∑i=0pαicψk,i−y¯^kψk,i−y¯^kT,Cz˜y˜,k−=∑i=0pαicχk,iz,−−z¯^k−ψk,i−y¯^kT.(ii)
Calculate the estimation gain matrix as(23)Gk=Cz˜y˜,k−Cy˜,k.(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 as(24)z¯^k+=z¯^k−+Gky¯k−y¯^k.(iv)
Estimate the dynamic wheel load:(25)F^dk+=cwz^5k+−z^3k++dwz^6k+−z^4k+.(v)
Update the covariance matrix of the estimation errors:(26)Cz¯˜,k+=Cz¯˜,k−−GkCy¯˜,kGkT.(vi)
Back to the step 1 for the next sampling timek+1.The 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.


## 4. State Estimation Results of the Suspension System

The physical parameters of the suspension system are(27)λ=0.2,cc=8399N/m,cw=151176N/m,dc=665N sec/m,dw=50N sec/m,mc=94.2kg,mw=22.9kg.Figure10 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.Figure 10
Road profile with length 1000 m, sampled 10 cm per measured point. Road profile PDF: min = −2.319 cm; max = 2.348 cm; mean = 0.0131 cm; median = 0.013 cm; variance = 0.331678 cm.Figure 11
The distribution characteristics DPF ofzr by velocity of the car.Table 1
The variation of the road distribution parameterszr by the velocity of the car for the road profile shown in Figure 10.
v
=
5
km
/
h
v
=
10
km
/
h
v
=
20
km
/
h
v
=
40
km
/
h
v
=
60
km
/
h
mean
z
r
m
0.127
−0.0346366
−0.0254131
−0.00482362
4.77238e−05
variance
z
r
m
0.0418896
0.031236
0.00639788
0.0015011
0.000690728
μ
z
r
m
0.127002
−0.0346366
−0.0254131
−0.00482362
4.77238e−05
σ
z
r
m
0.20467
0.176737
0.0799867
0.038744
0.0262817The covariance matrices depending on the velocity of the car are valued as(28)Cw˜v=5km/h=1.850000000.1750000000.850000000.1250000003.010000000.001,Cw˜v=20km/h=0.70000000.20000000.70000000.20000000.70000000.012,Cw˜v=40km/h=0.850000000.1750000000.850000000.350000000.60000000.01,Cw˜v=60km/h=0.850000000.1750000000.850000000.350000000.350000000.002,Cw˜v=80km/h=Cw˜v=60km/h.The estimated statesz^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 5km/h, 20km/h, 40km/h, 60km/h, and 80km/h. The errors of the state estimation errors are shown in Figure 17. Distribution of the state estimation errors is shown in Figure 18.Figure 12
The real states (solid line) and estimated states using SPKF (dotted line) in the case of 5 km/h. (a)zs and zs hat. (b) dzs/dt and dzs/dt hat. (c) zw and zw hat. (d) dzw/dt and dzw/dt hat. (e) zr and zr hat. (f) wzr.
(a)
(b)
(c)
(d)
(e)
(f)Figure 13
The real states (solid line) and estimated states using SPKF (dotted line) in the case of 20 km/h. (a)zs and zs hat. (b) dzs/dt and dzs/dt hat. (c) zw and zw hat. (d) dzw/dt and dzw/dt hat. (e) zr and zr hat. (f) wzr.
(a)
(b)
(c)
(d)
(e)
(f)Figure 14
The real states (solid line) and estimated states using SPKF (dotted line) in the case of 40 km/h. (a)zs and zs hat. (b) dzs/dt and dzs/dt hat. (c) zw and zw hat. (d) dzw/dt and dzw/dt hat. (e) zr and zr hat. (f) wzr.
(a)
(b)
(c)
(d)
(e)
(f)Figure 15
The real states (solid line) and estimated states using SPKF (dotted line) in the case of 60 km/h. (a)zs and zs hat. (b) dzs/dt and dzs/dt hat. (c) zw and zw hat. (d) dzw/dt and dzw/dt hat. (e) zr and zr hat. (f) wzr.
(a)
(b)
(c)
(d)
(e)
(f)Figure 16
The real states (solid line) and estimated states using SPKF (dotted line) in the case of 80 km/h. (a)zs and zs hat. (b) dzs/dt and dzs/dt hat. (c) zw and zw hat. (d) dzw/dt and dzw/dt hat. (e) zr and zr hat. (f) wzr.
(a)
(b)
(c)
(d)
(e)
(f)Figure 17
The state estimation errors by the velocities of the car. (a)zs − zs hat. (b) dzs − dzs hat. (c) zw − zw hat. (d) dzw − dzw hat. (e) zr − zr hat.
(a)
(b)
(c)
(d)
(e)Figure 18
Distribution of the state estimation errors. (a)zs − zs hat. (b) dzs − dzs hat. (c) zw − zw hat. (d) dzw − dzw hat. (e) zr − zr hat.
(a)
(b)
(c)
(d)
(e)From the state estimation results and the distribution of the state estimation errors, we see the following.For the estimation ofzsandzw, the distributions zs−z^s and zw−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.The distribution ofzr−z^r is larger than the distributions of zs−z^s and zw−z^w, the distribution of the estimation error z˜r=zr−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 ofdzs/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.The distribution ofdzw/dt−dz^w/dt has the variance larger more than two times in comparison with the distribution of dzs/dt−dz^s/dt.Figure19 shows the estimated wheel dynamic load for the velocities of the car from 5 km/h to 80 km/h.Figure 19
TheFd estimation versus the velocity of the car. (a) 5 km/h. (b) 20 km/h. (c) 40 km/h. (d) 60 km/h. (e) 80 km/h.
(a)
(b)
(c)
(d)
(e)Figure20 presents the estimated states for the case of varying the mass of the sprung as ms=94.3kg,124.3kg,and154.3kg, respectively.Figure 20
The estimated states for the case of varying the mass of the sprung. Case 1:ms = 94.38 kg, blue solid line for suspension system states and red solid line for estimated states. Case 2: ms = 124.3 kg, orange solid line for suspension system states and purple solid line for estimated states. Case 3: ms = 154.3 kg, green solid line for suspension system states and light blue solid line for estimated states. (a) zs and zs hat. (b) dzs/dt and dzs/dt hat. (c) zw and zw hat. (d) dzw/dt and dzw/dt hat. (e) zr and zr hat.
(a)
(b)
(c)
(d)
(e)


## 5. Conclusions

This 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. The mathematical model of the suspension system is rewritten for the state estimation problem; the stochastic load profile is supposed the main noise input. The stochastic of the road excitation depending on the car’s velocity is considered in the model. The 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.



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*Source: 6898756-2019-11-03.xml*