If I have a standard state-space model where all parameters are unknown (coefficients and covariance matrices for both the state equation and observation equation) and I want to estimate it with the kalman filter. Is it possible to estimate all the parameters at once? Is there a minimum number of observed variables that I need?

For example, if I have a state vector with 20 variables, seems unreasonable to me that I could identify all these parameters with only 1 observed variable. What is the formality around the dimension of the system that can be estimated?