Unmeasured Outputs

Unmeasured Inputs

Noise free Measured Outputs

Estimated

Unmeasured

Outputs

In Eq. (26.4) yk is the vector of measured variables (outputs), uk is the vector of manipulated variables (inputs), and wk and vk are pure random vector signals, with an average of zero, that represent the process and measurement noise respectively. The above discrete model is written in state space form, where the state vector, xk, represents the minimum information that we require to predict the system evolution for a given sequence of future inputs. The matrices A, B, and C are of proper dimensions.

The Kalman filter gives the optimum estimation of the true state of the system, given the known covariances, Q and R, of the process and measurement noise vectors. Figure 26.4 summarizes the main algorithm. Here, the corrected state estimate is a compromise, defined by the Kalman gain, between the model estimates and the measured values. The larger the covariance of the process noise, Q, the larger the Kalman gain, therefore the model predictions are weighted less. On the contrary, the larger the measurement noise covariance matrix, R, the smaller the Kalman gain, so model predictions are corrected less.

Even though the Kalman filter is widely used in robotics and related fields to filter measurement noise, in bioprocesses it is more useful for filtering process noise or estimating unmeasured variables like biomass content or metabolite concentrations. Compared with electro-mechanical systems, bioprocesses are slow; hence a high time scan can be used to filter measurement noise with simpler low pass filters. Moreover, to estimate an unmeasured variable once the state estimates

(1) Estimate the new state using the model only xk = A ' xk-1 + B ' uk-1 (2) Estimate the new state covariance matrix

Corrective Equations

(1) Compute the correction gain (Kalman gain)

(2) Correct the state estimate using the new measurement xk = xk+ Kk Vk -C ■ xk )

(3) Correct the state covariance matrix _Pk ={l - Kk • C )• Pt-

Fig. 26.4. Kalman Filter algorithm are corrected is simple. This is so because the process model is linear and any unmeasured variable should be a linear combination of the states. If the process model is nonlinear, a linearized model should be derived and the matrices A, B, and C updated at each sample time. This is known as an Extended Kalman Filter and a detailed discussion is beyond the scope of this chapter. The interested reader is encouraged to visit the web page of Welch and Bishop (Welch and Bishop 2003), where it is possible to find a lot of reading material, Matlab programs to experiment with and also many interesting links.

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