State Estimation of Steady Kalman Filter

Question

I am working with discrete Kalman Filter on a system.

x(k+1)=A_k x(k)+B_k u(k)

y(k)=C_k x(k)

I have estimated the state from the available noised y(k) , which one is generated from the same system state equations with Reference Trajectory of the state. Then I have tested it with wrong initial state x0 and a big initial co-variance (simulation 1). I have noticed that the KF works very well, after a few steps the gain k quickly converges to a very small value near zero. I think it is may be caused by the process noise Q . I have set it small because the Q stands for the accuracy of the model.

Now I want to modify it to a steady state Kalman Filter . I used the steady gain from simulation-1 as constant instead of the calculation in every iteration. And then, the five equations can be simplified to one equation:

x(k+1)^=(I-KC)A x(k)^+(I-KC)B u(k)+K y(k+1)

I want to test it with same initial state and co-variance matrix as the one in simulation-1. But the result is very different from reference trajectory and even the result of simulation-1. I have tested it with the co-variance matrix p_infi , which is solved from the Discrete Riccati Equation :

k_infi=p_infi C'/(C p_infi*C'+R)

This neither works.

I am wondering-

1. How should I apply steady state KF and how should I set the initial state for it?
2. Is steady state KF used for scalar system?
3. Should I use it with LQ-controller or some others?

Answer 1

Let me first simplify the discussion to a filter with a fixed transition matrix, A rather then A_k above. When the Kalman filter reaches steady-state in this case, one can extract the gains and make a fixed-gain filter that utilizes the steady-state Kalman gains. That filter is not a Kalman filter, it is a fixed-gain filter. It's start-up performance will generally be worse than the Kalman filter. That is the price one pays for replacing the Kalman gain computation with fixed gains.

A fixed-gain filter has no covariance ( P ) and no Q or R .

Given A , C and Q , the steady-state gains can be computed directly. Using the discrete Kalman filter model, and setting the a-posteriori covarance matrix equal to the propagated a-posteriori covariance matrix from the prior measurement cycle one has:

P = (I - KC) (APA^T + Q)

Solving that equation for K results in the steady-state Kalman gains for fixed A , Q and C .

Where is R ? Well it has no role in the steady-state gain computation because the measurement noise has been averaged down in steady-state. Steady-state means the state estimate is as good as it can get with the amount of process noise ( Q ) we have.

If A is time-varying, much of this discussion does not hold. There is no assurance that a Kalman filter will reach steady-state. There may be no Pinf .

Another implicit assumption in using the steady-state gains from a Kalman filter in a fixed gain filter is that all of the measurements will remain available at the same rates. The Kalman filter is robust to measurement loss, the fixed-gain filter is not.

Answer 2

Steady state KF requires the initial state matches the steady state covariance. Otherwise, the KF could diverge. You can start using the steady state KF when the filter enters the steady state.

The steady state Kalman filter can be used for systems with multiple dimension state.