One of the most common and important tasks in control is to maintain a specified constant output value. While constant output is typically achieved for linear time-invariant systems by controlling a state equilibrium, it is also possible to achieve with nonequilibrium state trajectories. These include superpositions with the zero dynamics trajectories, as well as nonequilibrium non-zero-dynamics trajectories. We give a geometric description of the state subspace which allows a constant output. For a nondegenerate system with no invariant zero at s = 0, the subspace is the sum of the equilibrium subspace and the maximal output-nulling subspace. Furthermore, for minimal systems, a stronger statement in the form of a necessary and sufficient condition is given.

Averaging methods are widely used for the control-oriented modeling of systems where high-frequency actuation is used to steer the average of the control variables, such as switching power converters or mechanical systems with vibrational control. Higher-order averaging provides approximations for the trajectories with improved accuracy compared to the commonly used first-order averaging method. Furthermore, differential flatness is a property of dynamical systems which allows parametrizing state and input trajectories from a flat output without integration. In this contribution we investigate the relation between higher-order averaging and flatness via examples of electrical and mechanical systems.