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Research Article | DOI: https://doi.org/10.31579/2637-8892/JMAE/001 Copyright
*Corresponding Author:
Citation:
Copyright: © 2019. Francis Assadian. This is an open-access article distributed under the termsof the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Received: 30 November -0001 | Accepted: 30 November -0001 | Published: 30 November -0001
Keywords: controller output observer; kalman filter; youla parameterization; yaw rate estimation; sideslip estimation
To reveal the benifits of a newly developed estimation technique Youla Controller Output Observer (YCOO) [1], this paper provides a comparative study of both SISO (Single Input Single Output) and MIMO (Multiple Input Multiple Output) YCOO and various types of Kalman Filters, such as (Linear) Kalman Filter (LKF), Extended Kalman Filter (EKF) and Unscented Kalman Filter (UKF), in estimating yaw rate and sideslip of a vehicle. We compare how each estimation technique performs and test the robustness of these estimators under plant parameters and environmental uncertainties, such as vehicle mass and tire-road coefficient of friction variations. We consider realistic situations such as sensor bias and provide analysis and potential solutions for YCOO when dealing with these practical problems. We will compare the yaw rate estimation result of these estimation approaches with a simple kinematic model methodology.
Over the past years, there have been tremendous amounts of work in the vast area of dynamic system state and parameter estimation. Virtual sensing has become very popular as the cost of various sensors are continuously on the rise. The strategies used to devise state and parameter estimations include well-developed estimation techniques such as Kalman filtering [2], and nonlinear observers which present a robust estimator by a BMIs (Bilinear Matrix Inequality) and finding a convex solution to this estimator by posing the problem as an LMI (Linear Matrix Inequality) [3].
In almost all these studies, practical implementations of these estimators such as computation and calibration overheads are ignored. Most importantly, there has not been any investigation to illustrate the benefits, such as performance and robustness, of one technique over the other.
V=Ffcosδ+Frm-Uω
ω=Ffcosδ⋅a+Fr⋅bJ
αf=δ-tan-1V+aωU
αf=tan-1bω-VU
Ff=Df⋅sinC⋅tan-1Bf1-Eαf+EBf⋅tan-1Bfαf
Fr=Dr⋅sinC⋅tan-1Br1-Eαr+EBr⋅tan-1Brαr
af=V+Uω-S2ω
ar=V+Uω+S1ω
V=Ffcosδf+Frcosδrm-Uω
ω=aFfcosδ-bFrcosδrJ
αf=δf-tan-1V+aωU
αf=δr-tan-1V-bωU
β=Ffcosδf+FrcosδrmU-ω
αf=δf-tan-1β+aωU
αf=δr-tan-1β-bωU
Then the lateral forces will be computed exactly the same as the SISO case.
The front and rear lateral accelerations can be related to the new state β
by modifying equations (8) and (9) as follows,
af=Uβ+Uω-S2ω
ar=Uβ+Uω+S1ω
ω=Ff⋅a+Fr⋅bJ
Vω=-Cf+CrmU-aCf-bCrmU+U-aCf-bCrJU-a2Cf+b2CrJUVω+CfmaCfJδ
ay=-Cf+CrmUaCf-bCrmUVω+Cfmδ
βω=-Cf+CrmU-aCf-bCrmU2+1-aCf-bCrJ-a2Cf+b2CrJUβω+CfmUCrmUaCfJ-brJδfδr
afar=-Cf+Crm-S1aCf-bCrJ-aCf-bCrmU-S1a2Cf+b2CrJU-Cf+Crm+S2aCf-bCrJ-aCf-bCrmU+S1a2Cf+b2CrJUβω+Cfm+S1aCfJCrm-S1bCrJCfm-S2aCfJCrm+S2bCrJδfδr
These models are utilized to estimate the vehicle yaw rate for the SISO case, and the vehicle yaw rate and sideslip angle for the MIMO case.
2.4 Uncertainties
The acceleration sensor measurements in the simulation environment are made from a nonlinear model, a representation of the actual vehicle. While the YCOO and LKF utilize a linear model to estimate the states of this nonlinear plant. Therefore, for these estimators, model uncertainties include the differences between linear and nonlinear models used for state estimation and acceleration measurements respectively. In addition to this inherent uncertainty, we will examine parameter and environmental uncertainties, such as mass and tire-road coefficient of friction variations, for all estimator approaches discussed below, in the robustness test section of this paper.
For simulating sensor noise, data sheet of a single axis accelerometer SCA830-D05 [7] is utilized. From the data sheet, we can assume that the sensor noise has a zero mean and a variance of R = 2.5×10-3
.
3 State Estimation, The SISO Case
In the state estimation for the SISO case, the linear model includes a front wheel steering vehicle with one accelerometer at the center of gravity of the vehicle for lateral acceleration measurement as shown in Figure 3. The input, in this case, is the front wheel steering angle (δ
), and the output is the lateral acceleration at c.g (ay
). This model is utilized to estimate the vehicle yaw rate.
3.1 Youla Parameterization Control Design
The transfer function of the plant is the ratio between lateral acceleration and the road wheel angle,
T(s)=ωn2(s2+2ζωns+ωn2)(τs+1)
Ys=1Gp⋅ωn2(s2+2ζωns+ωn2)(τs+1)
Gp,ω=ωδ(s)=aCfJs+a+bCfCrmJUs2+Cf+CrmU+a2Cf+b2CrJUs+a+b2CfCrmJU2+bCr-aC
Tω=GcGp,ω1+GcGp,ω
Gp=NrDr-1=N11N12N21N22DEN00DEN-1
GpSM=D1D000D2D11DEN001DEN=NSMDSM-1
GpSMs= Gp100Gp2= 1s+p1s+p200gs2
Y1=1Gp1ωn2s2+2ζωns+ωn2τs+1
Y2=Ks2+2ζωns+ωn2τ1s+12τ2s+12
K=τ2+2ζωnτ2+ωn2gτ22
The lateral tire forces are functions of the tire slip angles and their relationship has been studied for many decades for developing an appropriate mathematical tire model. There are many tire models which relate the tire slip angles to the lateral tire forces. One commonly used and popular model is the Pacejka tire model, also known as the “Magic Formula” tire model. From [6], the basic formulas for lateral forces are,
where B
, C
, D
and E
are various tire parameters, such as B is the stiffness factor, C is the shape factor, D is the peak factor which is related to the tire normal force (Fz
) and tire-road friction coefficient (μH
), and E is the curvature factor. These coefficients are extracted from various tire testing and measurement data. We have utilized typical tire data in this paper.
Given a steering angle input and system state initial conditions, tire slip angles are computed from equations (3) and (4). Knowing the tire slip angles, tire forces are computed from equations (5) and (6). Having the tire forces, the equations of motion, (1) and (2), are solved for the solutions of the new states. This cycle is then repeated at each time step.
The lateral acceleration at c.g. is related to the front and rear tire forces and is given by,
This SISO model then consists of one input, the front road wheel angle, δ, and one output, the vehicle lateral acceleration at the c.g., ay.
2.2 Kinematic Relationship
The lateral acceleration is kinematically related to the states V
and ω
as discussed below, and therefore, by measuring the lateral acceleration, an estimate of the yaw rate can be obtained.
Given the distances between the c.g. and the axles, by measuring the lateral accelerations at the front and rear axles, we can kinematically estimate the yaw rate. As shown inFigure 4, the lateral accelerations af
and ar
, with distances of S1
and S2
to the c.g., can be related to the states V and ω,
By rearranging equations (8) and (9), yaw rate can be derived from af and ar,
By adding another input, such as controlling the rear road wheel angle, and by measuring both front and rear axle lateral accelerations, we can transform the vehicle model from SISO to MIMO as shown inFigure 5. With one more input, the state equations (1) and (2) can be written as,
The equations (3) and (4) are re-written as,
Since we assume the longitudinal velocity, U
, to be constant, instead of using the lateral velocity, V, we can use the vehicle sideslip β as one of the states. In this way, our state estimation will consist of estimating the vehicle sideslip and the yaw rate. Then using equations (15) and (11), we can write,
Then by substituting β in place of V into equations (13) and (14), the front and rear tire slip angles can be expressed as,
This MIMO model then consists of two inputs, the front and rear road wheel angles, δf
and δr
, and two outputs, the vehicle lateral accelerations at the front and rear axles, af and ar.
2.3.1 Linearizing the Bicycle Model for SISO and MIMO Cases
For linearizing a tire model, it is assumed that the lateral tire forces are proportional to slip angles with front and rear cornering stiffness coefficients Cf
and Cr
,
Then the state-space representation of the equations of motion for the SISO case is written as,
The lateral acceleration at c.g ay can be expressed as,
Similarly, knowing that the states for the MIMO case are the vehicle sideslip β and yaw rate ω, the state-space representation for this case can be expressed as,
The model of the plant is stable with no unstable/non-minimum phase zeros, which makes the controller design by the Youla parmeterization a straightforward case. Based on the Youla parameterization technique in [1], the Youla parameter can be computed by,
where T
is the desired closed-loop transfer function, and we select this desired transfer function to be a second order well damped filter with two additional poles at s = 1/τ
for better shaping of the Youla parameter at high frequency,
Then the Youla parameter is simply written as
The sensitivity transfer-function S of the feedback loop can be derived using the algebraic constraint between this transfer function and the closed-loop transfer function, T,
and finally, the controller Gc
, to be used for the yaw rate estimation, can be calculated by,
With proper choice of ζ
and ωn
, the controller, Gc, is designed so that the closed loop transfer function, T, is at 0 dB at low frequency which indicates good target following, and the sensitivity transfer function, S
, is small in this frequency range which indicates good disturbance rejection. In selecting a bandwidth for the closed loop transfer function, T, a tradeoff between the transient response speed and the overshoot should be made. Figure 6 shows the bode plot of the plant model used in YCOO. The model has a high gain of 40 dB in the lower frequency region and this gain decreases starting around 1 rad/s and reaches the minimum at 20 dB around 6 rad/sec. If the bandwidth of the controller/estimator is designed to be greater than 1 rad/sec, then the controller gain will be higher in the frequency range where the model gain is lower (above 1 rad/sec). This may result in a poor transient behavior of the closed loop system due to a bigger overshoot.
Using the designed controller transfer function Gc, we can obtain the closed loop transfer function by,
Based on the YCOO MIMO control and estimation design in [1], with the resulting transfer function matrix Gp,e
, we derive the Smith-McMillan form of this TFM (transfer function matrix).
First, the plant TFM is written in the factorized form,
Nr
is the numerator polynomial matrix and Dr
is the decoupled denominator. Then, we utilize the standard approach for deriving the Smith-McMillan form of this TFM using the greatest common divisor (GCD) of all K×K
minors, where D0=1
, and DK = GCD
of all K×K minors. Using this approach, we can write,
where D0
, D1
, and D2
are the greatest common divisors of the polynomial matrix Nr, NSM=100D2
, and DSM = Dr
.
Given the following relationship between GpSM
and Gp
,
then, it can be shown that the unimodular matrices uL
and uR
can be computed by
where g
is a gain and the plant has two LHP poles and two zeros at the origin. The Youla transfer functions designed for the decoupled plant have the following structures,
The S-M form or the decoupled plant TFM has the following form,
where the gain K is selected to be,
The gain K
is selected so that the closed loop transfer function, T2
, associated with Y2
, is at 0db in the low frequency region for good target or reference following. The extra poles at s=1/τ1
in Y1
and Y2 transfer functions are included to reduce their gains at high frequency. The extra poles at s=1/τ2
in Y2 are for canceling the effect of zeros of GpSM at the origin so that the closed loop TFM target following at low frequency is not drastically compromised as discussed in detail below. The coupled Youla TFM can be computed from the decoupled Youla TFM using the previously calculated unimodular matrices,
Then we can derive the close-loop transfer function matrix Ty
and sensitivity transfer function matrix Sy
from the decoupled system using the following relationships,
Then the transfer function matrix of the resulting controller, Gc, can be computed by
The plot of singular values for Ty, Sy, and Y
are shown in Figure 23. Both maximum and minimum values of Ty are at 0db in the desired frequency region, which enforces a good target following for the closed loop system. Since the second decoupled plant transfer function, Gp2
, includes two zeros at the origin, we need to modify the closed-loop transfer function by adding two poles at s=1/τ2 so that the closed loop transfer function behaves as a bandpass filter for ensuring its target following capability at the low frequency is not significantly compromised, as shown in Figure 23. The singular values of Y are low at high frequency which results in good sensor noise attenuation at the plant input.
Besides the sensor noise, the sensor bias is another source of uncertainties as sensor measurements normally include an offset. This offset includes the calibration error and drift over lifetime due to temperature and supply voltage variations. The sensor bias of the accelerometer type proposed in this study is within the range of ±1 m/s2
[7]. If we assume both the front and rear accelerometers have the same value for sensor bias, then in the case of YCOO, the yaw rate estimation is not affected by this offset as illustrated in Figure 32.
Although we are using a dynamic model in YCOO, the observation and feedthrough matrices (Ce
and De
) are derived from the kinematic relationship between lateral accelerations and states. Therefore, as YCOO estimates the yaw rate, the sensor bias of the front and rear accelerometers are canceled out. But the story is different for the case of vehicle sideslip estimation, in this case, the bias has a large influence on the estimation result.
Since the YCOO tries to track the reference generator output signal (measured signal), the error due to the sensor bias will accumulate over time and estimation will have a significant offset with respect to the reference/measured signal as shown in Figure 33. In the case of Kalman filters, the LKF has relatively large steady state error due to the sensor bias. However, the EKF and especially the UKF, due to its statistical sampling technique for selecting appropriate sample points, are not relatively affected by the sensor bias.
We have designed a sensor bias estimation strategy to eliminate the negative impact of the sensor bias on the YCOO estimation strategy, as illustrated in Figure 34. The sensor bias estimation strategy is a subject of a current invention disclosure.
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