An explicit expression of the steady-state error in Smith Predictor applied to linear systems with integral action

Time delays in control systems are often present due to inherent delays of the instrumentation and signal processing computations. Among the proposed solutions for time-delay compensation in linear control systems, an important role is played by the Smith Predictor scheme, as it allows the design of the controller for reference tracking without considering time delay. However, the Smith Predictor presents poor disturbance rejection capabilities, in particular when the system to be controlled has an integral action, e.g. as typical of motion control systems. This paper illustrates the ineffectiveness of the Smith Predictor when applied on an linear time-delay system with integral action and provides an analytical expression of the steady-state error when a step disturbance acts on the control signal.