4.1. Learning Goals#

4.1.1. Process Historians#

Process historians are widely used throughout the process industries. The core function is to record and provide access to time series data for process tags.

The tclab library includes the basic functions of a process historian implemented using SQLite, a widely used database for web applications and internet connected devices.

  • An instance of the tclab historian requires a list of data sources. There is no limit on the number of data sources.

  • The default data sources record T1, T2, Q1, and Q2

  • Each data source consists of (tag, function) pair, where the tag is a string that uniquely identifies a time series, and function, with no arguments, returns the current value of the tag.

  • The tclab historian always records time with the tag “Time”

  • Data from the historian can be stored in .csv files, converted to Pandas data frames, or accessed directly from the historian data log.

What you should be able to do:

  • Read a list of data sources, and describe what data is recorded.

  • Add a tag and a function to a list of data sources.

  • Save data from the historian to a .csv file, then load the saved data for analysis.

4.1.2. State-Space Models#

A linear state-space model is written in the form

\[\begin{split} \begin{align} \frac{dx}{dt} & = A x + B_u u + B_d d \\ y & = C x \end{align} \end{split}\]

where \(x\) is the state vector with \(n\) elements, \(u\) is a vector of manipulable inputs, \(d\) is a vector of disturbance inputs, and \(y\) is a vector of process measurements.

  • A state-space system is stable if all eigenvalues have negative real parts.

  • If any eigenvalue has a positive real part, then the system is unstable.

  • Eigenvalues can be computed with eigenvalues, _ = np.linalg.eig(A)

  • The negative inverse of an eigenvalue corresponds to a system time constant.

What you should be able to do:

  • Convert systems of linear differential equations into state-space form, identifying all relevant vectors and matrices.

  • For a given problem, determine if the vectors and matrices are of compatiable dimensions.

  • Compute eigenvalues and determine stability.

  • Simulate the response of a state-space model to inputs and disturbances.

4.1.3. State Estimation#

State estimation is a form of real-time process analytics employed in control systems. State estimation combines process models with real-time process measurements to provide estimates of process variables that may not be directly measureable.

At each time step \(t_k\) there are two calculations to perform:

  • Model Prediction: Use the model to update the state to the next time step, i.e., \(\hat{x}_{k-1} \rightarrow \hat{x}_{k}^{pred}\) with the equation

\[\begin{split} \begin{align} \hat{x}_k^{pred} & = \hat{x}_{k-1} + (t_k - t_{k-1}) ( A \hat{x}_{k-1} + B_u u_{k-1} + B_d \hat{d}_{k-1}) \\ \hat{y}_k^{pred} & = C \hat{x}_k^{pred} \end{align} \end{split}\]

  • Measurement Correction: Use measurement \(y_k\) to update \(\hat{x}_{k}^{pred} \rightarrow \hat{x}_{k}\) with the equation

\[\hat{x}_{k} = \hat{x}_{k}^{pred} - (t_k - t_{k-1})L (\hat{y}_{k}^{pred} - y_k)\]

\(L\) is a matrix of observer gains.

The estimation error is given by \(e = \hat{x} - {x}\) which satisfies the differential equations

\[ \begin{align} \frac{de}{dt} & = (A - LC) e + B_d (\hat{d}-d) \end{align} \]

where \(d\) is the unmeasured disturbance, and \(\hat{d}\) is an apriori estimate of \(d\). If the estimate is accurate, then \(d = \hat{d}\).