### Abstract

The simplest form of learning control using integral control based learning in the repetition domain can exhibit poor transients. In response to this difficulty, a learning control law that alternates the sign of its learning gain every time step and every repetition was developed to prevent the build up of learning control action toward the end of the trajectory. It is shown here that the algorithm can be thought of as an error signal with a carrier frequency at the Nyquist frequency, which means that the algorithm has the intriguing property that high frequency signals are shifted to low frequency and vice versa, each repetition. The alternating sign algorithm is governed by difference equations with time varying coefficients. Nevertheless, here we reformulate the equations to make a time-invariant formulation. This conversion allows one to apply the vast body of control theory design methods based on time invariant systems. The characteristics equation governing stability of the learning process is obtained for the repetitive control problem. Application of root locus methods are discussed. A reformulated version of the Nyquist stability criterion is given which considerably simplifies the application of Nyquist to this class of discrete-time problems, and allows one to easily read off the range of gain for which the system is stable. Conditions to satisfy to obtain monotonic decay of the error transients are developed for the alternating sign algorithm, and shown to be sufficient conditions for asymptotic stability by arguments related to small gain theory. Examples, show improvement is error histories by using the alternating sign repetitive control algorithm, by comparison to the pure integral control based learning.

Original language | English |
---|---|

Pages (from-to) | 107-118 |

Number of pages | 12 |

Journal | Journal of the Chinese Society of Mechanical Engineers, Transactions of the Chinese Institute of Engineers, Series C/Chung-Kuo Chi Hsueh Kung Ch'eng Hsuebo Pao |

Volume | 21 |

Issue number | 1 |

Publication status | Published - 2000 Jan 1 |

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### All Science Journal Classification (ASJC) codes

- Engineering(all)

### Cite this

*Journal of the Chinese Society of Mechanical Engineers, Transactions of the Chinese Institute of Engineers, Series C/Chung-Kuo Chi Hsueh Kung Ch'eng Hsuebo Pao*,

*21*(1), 107-118.

}

*Journal of the Chinese Society of Mechanical Engineers, Transactions of the Chinese Institute of Engineers, Series C/Chung-Kuo Chi Hsueh Kung Ch'eng Hsuebo Pao*, vol. 21, no. 1, pp. 107-118.

**Analysis and frequency response of the alternating sign learning and repetitive control laws.** / W. Longman, Richard; Huang, Yi Cheng.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Analysis and frequency response of the alternating sign learning and repetitive control laws

AU - W. Longman, Richard

AU - Huang, Yi Cheng

PY - 2000/1/1

Y1 - 2000/1/1

N2 - The simplest form of learning control using integral control based learning in the repetition domain can exhibit poor transients. In response to this difficulty, a learning control law that alternates the sign of its learning gain every time step and every repetition was developed to prevent the build up of learning control action toward the end of the trajectory. It is shown here that the algorithm can be thought of as an error signal with a carrier frequency at the Nyquist frequency, which means that the algorithm has the intriguing property that high frequency signals are shifted to low frequency and vice versa, each repetition. The alternating sign algorithm is governed by difference equations with time varying coefficients. Nevertheless, here we reformulate the equations to make a time-invariant formulation. This conversion allows one to apply the vast body of control theory design methods based on time invariant systems. The characteristics equation governing stability of the learning process is obtained for the repetitive control problem. Application of root locus methods are discussed. A reformulated version of the Nyquist stability criterion is given which considerably simplifies the application of Nyquist to this class of discrete-time problems, and allows one to easily read off the range of gain for which the system is stable. Conditions to satisfy to obtain monotonic decay of the error transients are developed for the alternating sign algorithm, and shown to be sufficient conditions for asymptotic stability by arguments related to small gain theory. Examples, show improvement is error histories by using the alternating sign repetitive control algorithm, by comparison to the pure integral control based learning.

AB - The simplest form of learning control using integral control based learning in the repetition domain can exhibit poor transients. In response to this difficulty, a learning control law that alternates the sign of its learning gain every time step and every repetition was developed to prevent the build up of learning control action toward the end of the trajectory. It is shown here that the algorithm can be thought of as an error signal with a carrier frequency at the Nyquist frequency, which means that the algorithm has the intriguing property that high frequency signals are shifted to low frequency and vice versa, each repetition. The alternating sign algorithm is governed by difference equations with time varying coefficients. Nevertheless, here we reformulate the equations to make a time-invariant formulation. This conversion allows one to apply the vast body of control theory design methods based on time invariant systems. The characteristics equation governing stability of the learning process is obtained for the repetitive control problem. Application of root locus methods are discussed. A reformulated version of the Nyquist stability criterion is given which considerably simplifies the application of Nyquist to this class of discrete-time problems, and allows one to easily read off the range of gain for which the system is stable. Conditions to satisfy to obtain monotonic decay of the error transients are developed for the alternating sign algorithm, and shown to be sufficient conditions for asymptotic stability by arguments related to small gain theory. Examples, show improvement is error histories by using the alternating sign repetitive control algorithm, by comparison to the pure integral control based learning.

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UR - http://www.scopus.com/inward/citedby.url?scp=0033905948&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0033905948

VL - 21

SP - 107

EP - 118

JO - Journal of the Chinese Society of Mechanical Engineers, Transactions of the Chinese Institute of Engineers, Series C/Chung-Kuo Chi Hsueh Kung Ch'eng Hsuebo Pao

JF - Journal of the Chinese Society of Mechanical Engineers, Transactions of the Chinese Institute of Engineers, Series C/Chung-Kuo Chi Hsueh Kung Ch'eng Hsuebo Pao

SN - 0257-9731

IS - 1

ER -