State space realizations robust to overloading for discrete-time LTI systems

Abstract

Abstract In digital signal processing, overloading of a value occurs if the amplitude of a signal exceeds a predetermined limit. The overloading should be avoided, since it may cause a catastrophic effect on the system. To mitigate the overloading, the l2 norm scaling has been utilized for many years. However, the l2 norm scaling does not always work well, since it is not directly related to the amplitude of the signal. This paper presents a design method of state space realizations for digital discrete-time LTI systems that are robust to overloading. Upper bounds of the amplitudes of state variables are evaluated by using the invariant set theory. When different numbers of bits are allocated to represent the values of state variables, our realization is given by minimizing the volume of an ellipsoidal invariant set. On the other hand, when an equal number of bits is assigned to each state variable, an upper bound on the magnitude of the state variables is minimized and the optimal realization is obtained by the spherical invariant set. Numerical examples are provided to show that our realizations are robust to overloading.