This paper presents a mathematical study of the control of a second order plant having real distinct roots by use of pulse -width-modulation. An integrator connected between the pulse-width-modulator and the plant is a fixed element of the control loop. The integral-square error is used as the Index of performance. The ISE is minimized with the constraint that the pulse modulator is limited to a maximum of one output reversal for a step change in input. Derivation of the equation which will predict the exact switching time of the modulator output to minimize the performance index for any step amplitude is presented. An example problem is worked to illustrate the use of the switching equation in determining switching time. The example is concluded with graphs showing the optimal plant response for various step input amplitudes, variation of switching time as a function of step amplitude, and variation of integral-square error as a function of switching time
