Optimal discrete/continuous control with LQR performance criterion and a general class of 3rd-order linear dynamic control variations with one zero eigenvalue

Discrete/continuous (D/C) control is a natural generalization of conventional discrete-time control in which the control is strategically varied with time, in a specified continuous manner, between each consecutive pair of sample-times. In series of previous papers by the authors (1996, 1999, 2000) the LQR optimization of discrete/continuous control variations that are restricted to be -linear-in-time” (LiT), “exponential-in-time”(EiT), and “sinusoidal-in-time” (SiT), respectively, were considered in detail. In this paper the results in those three previous papers are generalized to include the case of optimal intersample control variations that satisfy a general third-order linear time-invariant ordinary differential equation with at least one zero eigenvalue. The values of the other two eigenvalues determines qualitatively the control time-variations across the sampling intervals. An example is worked in detail to illustrate the results