real shift in the level of the process variables caused by nonstationary disturbances such as changes in the impurity level of the feedstock. Stochastic nonstationary disturbances force the process to drift away from deterministic model predictions. The presence of a disturbance state model in addition to white noise variables w^ and vt will provide the necessary information for tracking the trajectories of state variables. A common practice for eliminating the offset caused by nonstationary disturbances (instead of using the disturbance state model) is to increase the Kalman filter gain K/t either directly or indirectly by augmenting the magnitude of the state noise covariance matrix Qk i (Refer to Eqs. (4.80) and (4.83)). This will reduce the bias, but will also increase the sensitivity of the Kalman filter to measurement noise, just like the effect of increasing the proportional gain of a feedback controller with only proportional action. The addition of the non-stationary disturbance model will have an effect similar to integral action in feedback controllers to eliminate the offset.
Since most Kalman filter applications for processes with nonstationary disturbances are for processes with external inputs u and involve processes that are typically nonlinear, the incorporation of the nonstationary disturbance model will be illustrated using an EKF for processes with external inputs. The nonlinear process is described by d-xd
— = fd(x,u,i) xo = x(t = 0) y(i) = h(x, u, t) (4.105)
where x represents the complete vector of state variables, and is the modeled deterministic subset of x. The complete state vector x consists of xd and stochastic state variables xs which include model parameter and disturbance states that may vary with time in some stochastic manner and may be unknown initially [300, 356]. Performing local linearization and discretization of model equations xfc
where is the zero-mean Gaussian white noise vector with covariance Qd. The true dynamics of stochastic states Xs are usually unknown and often they are assumed to follow a simple nonstationary random walk behavior [300, 356]
where is a white noise vector with covariance Qs. Usually Qs is a diagonal matrix with elements representing the change of magnitude in stochastic states (disturbances) in one sampling interval. The elements of Qs are tuning parameters to give good tracking behavior. The optimal one-step-ahead prediction of the stochastic states is
If more information is available about the stochastic states, an ARIMA model can be constructed to represent their dynamics.
The combined linearized dynamic/stochastic model of the system is
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