Representation of state-space flows by circles in the complex plane

It is shown that the flow in the state plane associated with a linear constant-coefficient second-order system may be represented by a circle in the complex plane. The main properties of these circles are examined, and they are shown to possess an additive combination property which should be useful in the synthesis of flows. The representation of higher-order flows may be achieved by orthogonally projecting them on to a suitably chosen set of planes, and representing the projected flows by circles in the complex plane. The additive combination property still holds, so that the results should be useful in the design of multivariable feedback controllers. Forced sinusoidal response is considered in a similar way.