In discrete-time control architectures, the mathematical notation maps the ideal, unconstrained control law to the actual, physically limited control input delivered to the system at step
sat(vk)={umax,if vk>umaxvk,if −umax≤vk≤umax−umax,if vk<−umaxsat open paren v sub k close paren equals 3 cases; Case 1: u sub max of end-sub comma if v sub k is greater than u sub max of end-sub; Case 2: v sub k comma if minus u sub max of end-sub is less than or equal to v sub k is less than or equal to u sub max of end-sub; Case 3: negative u sub max of end-sub comma if v sub k is less than negative u sub max of end-sub end-cases; sat vk
Systems relying on dynamic controllers (such as PI or PID loops) continue accumulating tracking errors within their internal integrators even after the physical actuator has hit its maximum limit. This forces the control signal In discrete-time control architectures
is typically bounded within a linear sector or dead-zone nonlinearity the mathematical notation maps the ideal