A Study on Singularly Perturbed Open-Loop Systems by Delta Operator Approach

In this paper, the open-loop state response of the two -time-scale systems by unified approach using the δ-operator is pre- sented with an example of the aircraft longitudinal dynamics. First, the δ-operator system unifies both the continuous system and the discrete system simultaneously, and the δ-operator approach improves the finite word -length characteristics. This saves more com- puting time than that of the discrete system. Second, the singular perturbation method by block diagonalization reduces the sizes and orders of the systems. This also reduces the floating-point operations (flops). The advantage of those two approaches is shown by comparing our results with the earlier ones in the illustrative example of the longitudinal motion of F-8 aircraft. I. Introduction The two topics are covered in this paper: One is the unified approach using the δ-operators to improve the finite word - length characteristics. The other is the matrix block diagonali- zation to reduce the sizes and orders of the two -time-scale systems. 1. The unified approach by using the δ-operators The discrete models are written in the form of the shift (q) operators . B ut, equations of the discrete systems by the q- operators are not simple like those of the continuous systems by the operator, d/dt. The δ-operator system unifies both the continuous system and the discrete system. In other words, the equation of the δ-operator system represents both the continu- ous and the discrete systems simultaneously. As the sampling time Δ in the δ-operator system approaches zero, the unified system becomes the continuous system. Therefore, the δ- operator system includes the whole characteristics of the dis- crete system and can be handled like the continuous system. The easiness of handling the δ-operator system means to r e- duce a number of equations in the discrete system. The normal q-operator systems have the problem of crowd- ing poles within the boundary of the stability circle at small sampling time and the difficulties of the truncation and round- off errors. If the discrete system is converted to the δ-operator system, the problems mentioned above are disappeared since the resolution of the stability circle is increased. Moreover it is the δ-operator system that has the finite word -length characteris- tics improved compared with the q-operator system (6) (12). Therefore, the δ-operator approach reduces the computing time of the discrete system; thus, improves the quality of the on-lined operating systems that requires higher accuracy. The analytical work of the unified approach using the δ-operators was fully founded by Middleton and Goodwin (13). Li and Gevers (7) showed some advantages of the δ-operator state- space realization of the transfer function over that of the q- operator on the minimization of the roundoff noise gain of the realization. They studied that the δ-operator implementation is