Theory and design of high-order bang-bang control systems

In this paper the nonlinear equations which describe the switching hypersurface of an
N
th-order linear time-invariant system, with real negative distinct poles, are developed when the input to the system is restricted in amplitude, either intentionally through the use of a relay or due to saturation of the power element. Based on the equations of the switching hypersurfaces, a design procedure is offered which will result in the optimum transient system response for any initial output or error initial conditions in the absence of an input.