Time-Varying Internal Model Based Control


Student: Xingyong Song, Pradeep Gillella, Yu Wang


In the automotive propulsion system, the desired motion trajectories are often periodic with respect to an angular or linear displacement, but not periodic with respect to time in general, as the rotational or linear speed varies in real-time. For instance, the engine intake or exhaust valve profile has an invariant period (duration of valve opening) in the rotational angle domain, while the period changes in the time domain due to a varying engine speed. This time-varying signal dynamics can be formulated into an internal model based tracking control framework. If we model the signal dynamics in the rotational-angle domain, it becomes linear time-invariant. However the actuator (plant) model becomes time-varying. The transformation between time and rotational angle does not change the fundamental difficulty of the tracking problem: one either has a LTI plant with a LTV exo-system (signal generating dynamics) in the time domain, or a LTV plant with a LTI exo-system in the angle domain. The time-varying nature of either the plant or the exo-system makes the control design much harder than the LTI case.

Unlike the LTI case, the signal generating dynamics cannot be directly embedded as the internal model unit. This is due to the lack of Cayley-Hamilton theorem in the time-varying setting. So the challenge of internal model-based control for LTV systems is to develop a systematic way for the construction of a robust time-varying internal model. We have approached this problem from both the input-output (I/O) representation and the state space representation. A key innovation is to adopt a feedback structure with two time-varying subsystems as the internal model unit.This structure allows us to directly access signals inside the internal model and leverages unique system properties (uniform observability) that don’t exist at the output of the internal model. The outcome of this work provides a systematic design method of time-varying internal model unit for a large class of systems.


  1. Sun, Z., Zhang, Z. and Tsao, T.-C., “Trajectory Tracking and Disturbance Rejection for Linear Time-Varying Systems: Input/Output Representation”, Systems and Control Letters, 58, pp.452-460, 2009.
  2. Zhang Z. and Sun, Z., “A Novel Internal Model-Based Tracking Control for a Class of Linear Time-Varying Plants”, ASME Transactions on Journal of Dynamic Systems, Measurement and Control, Vol. 132, 011004, January, 2010.