Note: There are certain cases where the discrete response does not match the continuous response generated by a hold circuit implemented in a digital control system. For further information, see the page Lagging Effect Associated with a Hold. There is a MATLAB function c2d that converts a given continuous system either in transfer function or state-space form to a discrete system using the zero-order hold operation explained above. Now, create a new m-file and enter the following commands. All constants are the same as before.

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The following m-file converts the above continuous-time state-space model to a discrete-time state-space model. For continuous systems, we know that certain behaviors result from different pole locations in the s-plane. For instance, a system is unstable when any pole is located to the right of the imaginary axis.

For discrete systems, we can analyze the system behaviors from different pole locations in the z-plane.

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The characteristics in the z-plane can be related to those in the s-plane by the following expression. The figure below shows the mapping of lines of constant damping ratio and natural frequency from the s-plane to the z-plane using the expression shown above.

A discrete system is stable when all poles are located inside the unit circle and unstable when any pole is located outside the circle. For analyzing the transient response from pole locations in the z-plane, the following three equations used in continuous system designs are still applicable. Create a new m-file and enter the following commands.

Running this m-file in the command window gives you the following plot with the lines of constant damping ratio and natural frequency. Let's obtain the step response and see if these are correct. Add the following commands to the above m-file and rerun it in the command window.

You should obtain the following step response. Course descriptions Factsheet.

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