For deterministic systems, Section 3.6 discussed the problem of state estimation. This section considers state estimation for stochastic, linear, discrete-time state space systems
where
: is a known signal. The standard notation and assumptions stated in Section 4.6.1 apply.
The state estimate is computed according to
where
is called the measurement residual and
is a designer specified gain vector. The notation
indicates the estimate of x at time
before correcting the estimate for the information in the measurement
and
denotes the estimate of x after correcting the estimate for the information in the measurement
Similar interpretations apply to
Defining the state estimation error vector at time
prior and posterior to the measurement correction, as
the state estimation error time propagation, output error, and measurement update equations are
To determine conditions for stability in the time invariant case, substituting eqn. (4.124) into eqn. (4.122), taking the expected value, and simplifying yields
where the notation
has been used for
Therefore, the stability of the system requires that the eigenvalues of the matrix
be strictly less that one in magnitude.
From eqn. (4.122), the covariance of the state estimation error prior to the measurement update is given by eqn. (4.99):
From eqn. (4.123), the covariance matrix for the predicted output error is
From eqn. (4.124), the covariance matrix for the state estimation error posterior to the measurement correction is
If the gain sequence Lk is known, then eqns. (4.125—4.127) can be iterated to quantify the expected accuracy of the state estimation design for the given Lk.
Eqn. (4.127) is true for any state feedback gain Lk; therefore, this equation allows the performance (as measured by state error covariance) of alternative gain sequences to be quantitatively compared. In fact, eqn. (4.127) expresses the covariance of the state estimate immediately after the measurement correction as a function of the estimation gain vector Lk ; therefore, a new gain vector could be selected at each time instant to minimize the state estimation error variance.
