Dual Kalman Filter

[Qiang1991]

I want to estimate a variable x, and the observable variable y can be obtained from sensors. These two variables has an approximately linear relationship,i.e., y=k*x + b, but k and b are difficult to get, so I use two kalman filters, one for state (x) estimation, another for parameter (k,b) estimation, which actually combines the dual kalman filter. The general scheme and code in Python is as attached, and state estimation kalman filter functions, parameter estimation kalman filter functions are also attached.

##Python code
#initialize state estimator
kf = KalmanFilter(dim_x=1,dim_z=1)
kf.x = numpy.array([x0])              # initial state
kf.F = numpy.array([1.])              # state transition matrix
kf.Q = 1000                           # state noise variance,
kf.R = 1                              # measurement noise variance
 
#initialize parameter estimator
dkf = KalmanFilter(dim_x=2,dim_z=1)
dkf.x = numpy.array([[-0.01,-0.1]]).T # initial state 
dkf.F = numpy.array([[1,0],
                    [0,1]])           # state transition matrix
dkf.Q = numpy.array([[1,0],
                    [0,1]])
dkf.R = 100                           # measurement noise variance
 
measurements = []
resultsDKF = []                                         #dual kalman filter state estimation result
ERRDKF = []                                               #estimation error
 
for i in xrange(N):   # N samples
    #### parameter estimator
    dkf.H = numpy.array([[kf.x,1]])   # measurement function
    y = Y[i]
    dkf.update(y,dkf.R,dkf.H)        
    dkf.predict()
 
    #### state estimator
    kf.H = numpy.array([dkf.x[0]])
    z = Y[i]-dkf.x[1] # Y[i]-b
    kf.update(z,kf.R,kf.H)    
    kf.predict()
 
    #### save data
    measurements.append(Y[i])
    resultsDKF.append (kf.x)
    ERRDKF.append(measurements[-1]-resultsDKF[-1])

It seems that the predicted state varies with the true value, but there still is a big gap between predicted and true value, and if I set the initial state of x (that is x0 in the code) near the true initial value, the results are much better,as attached.

I think the dual kalman filter doesn't really track true values of x, what's the problem? Anybody who offers advice will be much appreciated.

[Qiang1991]

Is there anybody can help me figure out this problem? I personally think there is something wrong with details.