Smith-Predictor type Structure for a Class of Infinite-Dimensional Systems: Optimal Control and Performance Limitation Formula

In this talk we investigate control problems for infinite-dimensional systems whose transfer matrices are expressible in terms of a rational transfer matrix and a scalar (possibly irrational) inner function. This class of systems is capable of describing many practical control problems, when weighting functions are rational and plants have at most a finite number of unstable modes or zeros. In the first half of this talk the concept of Smith-predictors, that was originally used for I/O delay systems, is extended to the aforementioned class of systems. This allows us to reduce the optimal control problems to easily checkable criteria that do not require the solution of operator-valued equations. Furthermore, the obtained (stabilizing or suboptimal) controllers are shown to have the structure of Smith-predictors, or their dual. In the second half of the talk we derive a new expression for the H performance limit, based on state-space representation. The resulting formula, given as a functional of the inner function, helps us to understand how achievable H performance deteriorates due to the plant’s nonminimum phase properties or unstable modes. The example of a linear quantum control system suffering from feedback delay is given to illustrate the result.