On-off-type control systems designed on the basis of step-input response sometimes exhibit poor frequency-response characteristics. When subjected to periodic inputs, such systems may oscillate at subharmonics of the input frequencies under certain conditions. An example of subharmonic response of an on-off-type system is shown in Fig. 1(a). Fig. 1(b) shows the corresponding fundamental response.

When an on-off system is designed with a proportional band, the frequency response of the system must be carefully investigated for the possibility of “jump resonance” and subharmonic oscillations. This is particularly necessary when the system is known to be subjected to occasional periodic inputs.

This paper presents an effective method for evaluating the frequency response, both open-loop and closed-loop, for on-off-type feedback control systems with or without proportional bands. Fig. 2 shows one type of the system under discussion.

The essential material presented follows the Laplace-transform method suggested by D. Kahn [1]. It is an extension of the work already reported by K. Ogata [2, 3].

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