A posteriori error bounds for discrete balanced truncation

Balanced truncation of discrete linear time-invariant systems is an automatic method once an error tolerance is specified and it yields an a priori error bound, which is why it is widely used in engineering for simulation and control. We derive a discrete version of Antoulas’s H2-norm error formula and show how to adapt it to some special cases. We present an a posteriori computable upper bound for the H2-norm of the error system defined as the system whose transfer function corresponds to the difference between the transfer function of the original system and the transfer function of the reduced system. We also present a generalization of the H2-norm error formula to any projection of dynamics method. The main advantage of our results is that we use the information already available in the model reduction algorithm in order to compute the H2-norm instead of computing a new Gramian of the corresponding error system, which is computationally expensive. The a posteriori bound gives insight into the quality of the reduced system and it can be used to solve many problems accompanying the order reduction operation. Moreover, it is often more accurate in floating point arithmetic.