Currently for my work I am thinking of using this (taking states derivatives as states) because, for my system, I need more measurements to extract data for filter, and the state derivative (acceleration) appears to be measured in practice. However, in literature I did not come-across with such an application of Kalman filters which one or more of the state derivatives are accepted as state.
A state derivative x' can be used as a new state if you know how to express x'' via x and x'. For a pendulum described by x'' + fx' + kx = 0 we obviously choose x1 = x and x2 = x' as states and write x1' = x2; x2' = - kx1 - fx2. Further, if the right-hand side of the equation for x'' does not contain x, you may completely omit x and use only x' as the state.
Yes, you can put anything to the state when it is described in the prediction model.
Dear Vasiliy,
I am talking about taking states as x1 = x and x2 = x' and x3=x'' (for example pendulum case). Can this be possible?
Altan,
In this particular example it is probably meaningless, since no physical law gives an explicit expression for the 3rd derivative, x'''. Nevertheless, you can imagine a system in which the you can control the rate of change of the applied force (e.g., by setting the hydraulic valve position in the aircraft rudder actuator). In this case it is the 3rd derivative which you can control directly.
Dear Vasiliy,
For my dynamical system, in practice only yaw rate and acceleration can be measured. That constitutes a problem for parameter estimation. The state vector for this system is x=[y, psi, ydot, psidot], x'=[ydot, psidot, ydoubledot, psidoubledot]. In practice only psidot, ydoubledot can be measured. The integration of acceleration and yaw rate is not a solution because of integration drift. Even I assume that all states can be measured the parameter estimation is unsuccessful and diverges after some time. As the parameter (maximum friction coefficient) directly effects acceleration. That is why I am asking such a question.
Altan,
Surely you can add an extra state a = y’’ with a fictitious model a’ = w, where w is the process noise, and a corresponding measurement z = a + v, where v is the measurement noise.
I am however afraid that this trick cannot help you since it does not solve the integration drift problem but only hides it inside the Kalman framework. As I guess, the only sensors you have are MEMS (?) accelerometers and gyros. If so, you have no chance to estimate the path, y, travelled by your vehicle, by integrating their measurements without an external reference such as an odometer or a GNSS receiver. If you actually have these external aids, you should also include y and/or y’ into your measurement vector.
On the other hand, why do you really need to estimate this path? Does any of the state model equations depend on it? If your purpose is to extract the friction coefficient by comparing two path estimates, one from inertial sensors and another from the odometer, this seems to be very difficult.
Dear Vasiliy,
Yes, even tough I am not considering currently the practice (only simulation), I want my solution to be compliant with practice. Also your are right, only sensors in practice that can be used are gyros and acceloremeters for this case. The "y" is in my dynamical system represents lateral shift and the reference systems would not work as this shift is in order of milimeters. The dynamical response of the wheelset is changing according to the friction and in the literature some authors revealed that friction (e.g. adhesion) conditions can be estimated from this data. However, their solution is also based on simulations and some authors also accept that they can measure this lateral shift somehow, but they are doing it in two stages. First they are using linear Kalman filters and then they are processing residuals or the states to estimate adhesion condition by fuzzy logic, black & gray box models etc. My aim is to do it with just Unscented Kalman filter. I achieved it with the assumption of lateral shift, yaw angle and lateral velocity somehow measured. However, with the assumption of just acceloremeter and yaw rate measurement it seems it is impossible in one step. I tried to embed acceleration to the model but I do not have any success.
Altan,
You definitely need a special sensor (e.g., like those used in optical mice) to measure such small a displacement. Consumer-grade accelerometers are not suitable for this purpose. I am afraid that it is a wrong belief that any fashionable estimation algorithms, from fuzzy logic or neural networks to H-infinity or unscented Kalman filters, can compensate for the lack of precise measurements.
Dear Dr. Velez,
You are right, that was my mistake I could not see that. I considered yaw rate and acceleration as inputs also.
Thank you all, for the answers. I understood that the problem is observability in my system. Just by using yaw rate and angular velocity (in this case they are named as modest cost sensors) it is impossible to realize parameter estimation. I must assume that somehow yaw angle, lateral position and lateral velocity can be measured as it is assumed also in literature (the literature for case of my system).
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