State Estimation of Steady Kalman Filter
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.
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_infiC'/(Cp_infi*C'+R)

This neither works.
I am wondering-

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