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In this article, a promoted method of adaptive input shaping based on recursive least square with forgetting factor is proposed to achieve zero residual vibration of time-varying flexible systems. First, the zero residual vibration condition of the flexible system is reviewed. Then, the mathematical analysis of recursive least square–based adaptive input shaping is presented; it can be seen that the traditional recursive least square method could calculate the least square solution with all historical I/O data. That is to say, with the increase in time and larger amount of I/O data, the current data could hardly affect the results of the updated input shapers’ coefficients; thus, the problems of insufficient adaptability and noise accumulation occur. So, a forgetting factor is introduced in the recursive calculation to give less calculation weight on historical data and improve the sensitivity of the current data; thus, the above-mentioned problem could be significantly avoided. At last, the verification experiments of adaptive input shaping are implemented on a two-link flexible manipulator, which is a classical flexible system with severely time-varying dynamics; the results validate the effectiveness of the improved adaptive input-shaping method for the vibration control of these flexible systems

An improved recursive least square–based adaptive input shaping for zero residual vibration control of flexible system