INFLUENCE OF STATE SPACE MODELLING FOR MONITORING OF CAR TIRE PRESSURE

T.M.J.A. COORAY

Department of Mathematics

University of Moratuwa

Sri Lanka

ABSTRACT

Dynamical vehicle models and a time dependent state space model have successfully been used to detect tire pressure losses. When the pressure is decreased in a tire, the corresponding wheel radius is reduced and the angular velocity increases. The basic idea behind this paper, is to indirectly estimate the wheel radius via secondary existing sensors. Two dynamical vehicle models, in which the relative wheel radius is included as model parameters, are formed. With these models any combination of pressure loss in the tires can be detected.

Key Words: Wheel radius, Kalman filter, Longitudinal dynamics, Lateral dynamics, Change detection

INTRODUCTION

Tire pressure monitoring is emerging as a great business opportunity, which clearly has been discovered by a lot of companies. Many different systems have been and are under development, and they can be categorized into two groups, direct and indirect systems. In the direct systems the pressure in each tire is measured directly and this information is transmitted to a central computer. In indirect systems the tire pressure itself is not measured, but is estimated from other signals. One approach is to examine the resonance frequency of the tire speed signals as in Kalman (1960) or to examine the wheel speed itself. The simplest indirect systems only monitors the difference in wheel speed, but to improve the performance, model based signal processing can be used. This has been used successfully earlier to estimate the tire-road friction (see Gustafsson, 2000), and to improve the estimation of a vehicle's yaw rate (see Forssell et al., 2001). In this paper the idea is to combine these systems to detect a change in tire pressure.

METHODOLOGY

Signals of interest are often not measurable, and instead have to be estimated from other measurable physical quantities. In order to get an accurate estimation the relations between these signals are needed. In this paper, it is assumed that these relations can be described as linear discrete time state space models, in which the state vector xk, contains the signals of interest and the measurement vector yk the measurements. The subscripts used in these equations corresponds to time instants

Equation (1) determines the dynamical properties of the states that are strongly related to the Ak-matrix. The process is driven by the noise vector wk, that is related to the states by the Bk-matrix. Equation (2) models the relation between the measurements and the states, and ek describes the measurement noise. The noise sequences wk and ek are assumed to be zero mean, mutually uncorrelated and having the covariance matrices Qk and Rk.

Observability

When forming state space models, care has to be taken so that the connection between states and measurements are strong enough such that changes in the state values will be noticed in the measurements. If there exists a state vector that does not fulfill this condition, the model is not observable. For time invariant systems, a condition for observability is that the matrix defined in equation (3) has full rank.

In this system with a time invariant A-matrix but time variant C-matrix are of special interest. For these the discrete observability gain can be used instead, which for time invariant A-matrices takes the form

The observability of a system can be monitored by computing the condition number of the gains in different time windows, which can be used to examine conditions affecting the observability.

State-Space Kalman Filter

The Kalman filter produces an estimate of the state vector that minimizes the mean square error of the estimate. It produces the optimal linear state estimator, and it is defined as a set of recursive equations.

Different behavior can be obtained by adjusting the covariance matrices Qk and Rk from equation (1) and (2). This tuning can thus be related to different assumptions regarding the process and measurement noise sequences. Further if the noise is independent and Gaussian distributed, the Kalman filter produces the optimal state estimate of all filters, included nonlinear filters. A derivation of the Kalman filter can be found in Durbin and Koopman, 2001.

Cusum Detector

Cusum dector is a popular choice, when monitoring if the mean of a signal changes above a certain level. The difference between the signal and this level is then accumulated in a test statistic, g(t). An alarm is set if the test statistic exceeds a threshold. The threshold is used as a tuning parameter, which depends on the nature of the signal. An algorithm that performs this on the signal g(t) would look like

Here h and A are the threshold- and drift parameters. The test can also be designed as two sided tests, in which two test statistics g1(t) and g2(t) are formed

A good reference for a more in depth description of the detector can be found in Bryson and Ho, 1969.

Coordinate Axis and Wheel Numbering

Wheels are numbered as in Figure 1, the orientation of the coordinate system is also defined. Subscript of physical quantity such as speed, pressure, torque etc, refers to specific wheels. The angular velocities about the x, y and z-axis are called roll, pitch and yaw rate respectively, but in this paper only the yaw rate is of interest. The direction of the x-axis and y-axis are frequently referred to as the longitudinal and lateral direction.

Dynamics

Longitudinal Dynamics

Relationships between front and rear wheel radii are used in discussing how tire losses can be noticed. These relations are also utilized in the road friction indication system in Gustafsson (2000). At the driven wheels of a car the circumferential velocities wi, ri are always greater than the absolute velocity vi. This difference is called wheel slip and is defined as

It has been shown that the slip is related to the coefficient of friction μ defined as the ratio between the friction force F and normal force N

To compute the slip, the absolute velocities of the driven wheels are needed. Since these are unknown, they are estimated from the circumferential velocities of the non driven wheels where the slip is assumed to be negligible. Two slip equations can be formed, one for each side of the car

Since the wheel radii are unknown, the slip equations are simplified further by only examining the relative difference in angular velocity. The measured slip, &scircumflexij, is hence a biased estimate of the slip in the linear model.

The bias, δij is introduced since the wheel radii not are equal

The model relating the measured slip with normalized traction force for a low slip values is therefore written as

State Space Equation for Longitudinal Dynamics

When forming state space models, only the slip equations for each side of the car are known. Since these can be separated from each other, two state space models with state vector can be formed. Since only measurement equations are known, the state vector evolution is modeled as random walk with state and measurement vectors

If the μ-measurements are constant, the model will not be observable. The observability will however increase when the normalized friction varies with time. This can easily be understood since the objective of the time dependent Kalman filter is to estimate the slope and offset of the slip curve. If the friction force is constant, only one point on the slip curve is measured. However, why infinitely many slip curves can be fitted to this point. With more points on the slip curve, the possibility to estimate the true slip curve improves.

Lateral Dynamics

In lateral dynamics relations between the wheel radii on each side of the car are studied in a similar way as in the longitudinal dynamics, but the difference in angular velocity due to cornering instead of slip is examined. The angular velocities of the wheels on each side of the car are different both when cornering and when the wheel radii are unequal. To separate between these a gyro is needed to measure the direction of travel, or yaw rate, &PSI. Gyro sensors are, as many sensors often are, subjected to errors. These errors are typically modeled as

Here ψ is the measured yaw rate from the gyro, δsc is a scale factor error and δgyro is an offset error. Both errors often vary with the temperature, but the scale factor error is assumed to be small and is neglected.

When a car is cornering, as in Figure 2, the wheels on opposite sides will have different curve radii and hence their velocities are different. This difference in velocity is proportional to the yaw rate, and consequently the yaw rate can be estimated from the wheel speed sensors. It is however also influenced by different wheel radii, which also needs to be taken into account. Equations for the yaw rate can be formed from both the front and rear wheels

Here rn is the nominal wheel radius, and Bf and Br are the distances between the front and rear wheels respectively.

These equations contains the relative differences between the left and right side tire radius, defined as

State Space Model for Lateral Dynamics

The state space model for this system is based on the yaw rate – and gyro sensor models and with sample time denoted Ts it takes the following form.

Detecting a Change in Wheel Radius

Parameters that relate the wheel radii are central in the longitudinal and lateral dynamics. With these, the state space models can handle changes in wheel radius. Models of the longitudinal dynamics can be used to detect changes in wheel radius between the front and the rear wheels on each side, but if the pressure is decreased equally it would not be detected. Similarly models of the lateral dynamics can detect changes between the wheels on the left and the right hand side. If the models are merged, any decrease except diffusion can be detected. A change in only one wheel, would also be noticed in both systems, hence the reliability can be improved by combining the systems. First the definition of the parameters that relate the different wheel radius, from now on referred to as the δij-parameters is repeated.

All wheel radius can be viewed as the sum of a nominal wheel radius and an offset
.
If the higher order terms are neglected, the equations can also be derived by expressing the δij-parameters in terms of nominal wheel radius and offset in wheel radius.

The other possibility results in a new parameterization in which the models are merged into one model. In this the wheel radius offsets are introduced as states replacing the δij-parameters.

State Space Model for Proposed Model

The Taylor expansion can be used to form new parameterization where the longitudinal and lateral dynamics is merged into one system. The measurement equations for the different systems can now be written as functions of &DELTAri instead δij.

With the new parameterization it is easy to obtain a new state space model from the previous

With this implementation instead, the &DELTAij-parameters are obtained directly from the Kalman filter and a Cusum detector can be used to detect changes. A third implementation have also been examined, in which correlation between the δij-parameters is introduced by,

The idea behind this implementation is to examine if the observability of the states would increase when more correlation between the states were introduced.

One should however be a bit careful when drawing conclusions from this analysis. It does help to determine that the extra measurement did not improve the observability and that implementation can be discarded. The extreme difference between the implementation with the systems separated and the new parameterization can be explained. It is known that diffusion cannot be detected since it does not generate any relative difference in wheel radius. In the new parameterization this is introduced as an unobservable state vector, which reduces the observability in this implementation. This can hence not be used as a performance criteria to determine if a new parameterization should be used or not. The choice is therefore empirically based, why the systems were kept separated. An advantage with this can also be found in a marketing aspect. If a gyro is not available longitudinal dynamics can be used alone, resulting in a simpler and maybe cheaper system. Then with a gyro, lateral dynamics can be incorporated to a more advanced and expensive system.
