Quantization errors in digital control systems

An analytic study was conducted to develop a quantitative expression for the errors due to quantization (round-off) in digital control systems. An upper bound on the error in the output that was caused by quantization became available for two different sampling rates by describing a typical digital control system in terms of its state variables. The amount of the error introduced in the output by any one quantization operation was found to depend upon the sampling rate, system time constants, and the form of the controller chosen for system compensation. For a third-order plant, preceded by a zero-order hold circuit and compensated by a controller that provided a response dominated by a closed-loop pole pair with a damping ratio of 0.7, the error due to quantization was determined to be approximately ten times the value of the quantizing level. The error was reduced to six times the quantizing level by increasing the sampling rate by a factor of 2, to 20 per second. The results of the study furnished a convenient procedure for analyzing quantization in a digital control system and led to a specification on the precision required in the digital controller and the necessary encoding equipment.