Asymptotic sampling errors in closed-loop hybrid systems

A vector differential equation for the error due to sampling in closed-loop hybrid systems is developed using several approximations derived with Taylor series expansions. The equation is valid for both linear and nonlinear hybrid systems under the condition that the sampling period is small. The error equation is linear. Its homogeneous part is independent of how the hybrid system is subdivided into subsystems, but its forcing function depends on the characteristics of the analog and digital subsystems. Both parts of the error equation depend on the system state variables. The form of the equation shows that asymptotic sampling error depends on the first power of the sampling period and that the execution time of the digital subsystem has a predominant effect similar to that of the sampling period but weighted twice as heavily. The equation can be applied in compensating as well as evaluating hybrid systems. An example is included to illustrate the use and accuracy of the equations.