What Is Batch Linearization?

Batch linearization refers to extracting multiple linearizations from a model for various combinations of I/Os, operating points, and parameter values. You can analyze the time-domain, frequency-domain, and stability characteristics of the linear models obtained by batch linearization. For information regarding the tools that you can use for such analysis, see Linear Analysis in the Control System Toolbox™ documentation.

Consider the magnetic ball levitation model, magball (for model details, see magball Simulink Model):

You can batch linearize this model by varying any combination of the following:

  • I/O sets — Linearize a model using different I/Os to obtain any closed-loop or open-loop transfer function.

    For the magball model, some of the transfer functions that you can extract include:

    • Magnetic ball plant model, controller model

    • Closed-loop transfer function, from the Reference Signal to the plant output, h

    • Open-loop transfer function for the controller and magnetic ball plant combined, that is, the transfer function from the Error Signal to h

    • Output disturbance rejection model or sensitivity transfer function, obtained at the outport of Magnetic Ball Plant block

  • Operating points — Because operating points can influence model dynamics, you linearize a model at different operating points and study their effects on the model. Consider a simple unforced hanging pendulum with angular position and velocity as states. This model has two steady-state points, one when the pendulum hangs downward, which is stable, and another when the pendulum points upward, which is unstable. Linearizing this model close to the stable operating point produces a stable model, whereas linearizing this model close to the unstable operating point produces an unstable model.

    For the magball model, which uses the ball height as a state, you can obtain multiple linearizations for varying initial ball heights.

  • Parameters — Parameters configure a Simulink® model in a variety of ways. For example, you can use parameters to specify various coefficients or controller sample times. You can also use a discrete parameter, like the control input to a Multiport Switch block, to control the data path within a model. Therefore, varying a parameter can serve a range of purposes, depending on how the parameter contributes to the model.

    For the magball model, you can vary the parameters of the PID Controller block, Controller/PID Controller. The linearizations obtained by varying these parameters show how the controller impacts the model. Alternatively, you can vary the magnetic ball plant parameter values to determine the controller robustness to variations in the plant model. You can also vary the parameters of the input block, Desired Height, and study the effects of varying input levels on the model response.

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