Active suspension with accelerometers and displacement observer/estimator problem

Question

I faced a logical dilemma, while designing and simulating Active suspension system.

This is what I have planned:

1. My system will be controlling actuators of a vehicle, using PI.
2. It is 2nd order mechanical system with mass-spring-damper-actuator.
3. In Matlab and Simulink, I am using Transfer Functions for analysis (to manage overshoot, settling time, etc.) and State Space representation (to design observer/estimator).
4. Sensors used are ONLY accelerometers, that will produce acceleration of the suspension displacement. However, I require a displacement, which will be Observed/Estimated by designed estimator and fed back to compare with the reference point.
5. I cannot use double integration as drift occurs.
6. I have 2 states from 2nd order system. (x1 = displacement and x2 = velocity)
7. We are dealing with LTI SISO system.

My mechanical system matrices are as follows:

A = [0 1; (-ks/ms) (-bs/ms)]; 
B = [0; (1/ms)];
C = [1 0];
D = [0];

Problem:

How can I use an Observer/Estimator states to estimate the suspension displacement (in Simulink) from the acceleration that is provided from the accelerometer, if in my opinion, acceleration state does not exist?

Let me know, If you would like to see some of my code or derivations, but I think this is one of the Logical Understanding problems, so any clarification will help to get me back on track.

Answer 1

If you can apply a sinusoidal displacement to the suspension and measure the signal from the accelerometer, you can avoid some of the problems with drift in the accelerometer. You can filter the accelerometer signal to remove low frequencies, because the signal you want it the sinusoidal one. Then integrate twice to get displacement.

$$a=C\sin{\omega t}$$ $$v=-C \omega \cos{\omega t}$$ $$d=-C\omega^2 \sin{\omega t}$$

Then you have the magnitude of your displacement with the drift removed by filtering it out.