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ABSTRACT

This paper presents a system-theoretic analysis of numerical methods used in approximating solutions of ordinary differential equations. By representing ordinary differential equations with system block diagrams (i.e., interconnections of static elements and integrators), a numerical method can be viewed as a process in which the integrators of a continuous system are replaced by discrete approximations to the integrators (i.e., by discrete subsystems made up of interconnections of static elements and delays). The main result of the paper establishes that if the system block diagram corresponding to the original differential equation has no static loops and if the discrete subsystems used to replace the integrators have no static loops and no static through paths, then the resulting discrete system can be characterized by explicit difference equations. A system-theoretic study is conducted of several of the more commonly used numerical methods (the Euler, trapezoidal, Runge-Kutta, and predictor-corrector methods) and the limitations of using these numerical methods in the real-time analysis of input-output systems are examined.

REFERENCES

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