Digital linear control theory for automatic stepsize control

Numerical integration methods are used to find the numerical solution of the transient analysis of electrical circuits. Because the electrical circuits are modelled by stiff differential algebraic equations, the BDF-methods are very popular in circuit simulation. Error control is used to handle with the trade-off between efficiency and accuracy. For optimization purposes smooth behaviour of the errors and the stepsizes is wanted. Application of digital linear control theory Error control can also be considered from a control-theoretic approach. For onestep methods, the local error estimate n r satisfies the following asymptotical model, which only depends on the last stepsize . P n n n h r φ = In logarithmic form, we get the next linear model. . log log log n n n h P r φ + = (1) Here n r log is viewed as the output of this system, which depends on the input n h log and an unknown disturbance n φ log . The goal is to keep the output close to a reference level ε log means of the input. A simple controller with this task is ( ) log log 1 log log 1 1 1 1 1 − − − − − = −       = n n n n n n r P h h or h r h p