Approximate H∞ loop shaping in PID parameter adaptation

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10 Scopus citations


This paper discusses the use of H-infinity approximation for the online adaptation of PID controller parameters. For a frequency loop-shaping control objective, it is possible to adapt the PID parameters directly with linear model estimation algorithms. Standard least squares algorithms are common solutions for this problem, but their estimates exhibit a well-known strong dependence on the properties of the excitation. This drawback becomes more pronounced for systems where the modeling mismatch is large, as is frequently the case in PID control. In an alternative formulation of the estimation problem, we use a filter-bank to decompose the error signal to different components and minimize approximately the H-infinity norm of the sensitivity-weighted error operator. This approach results in a more consistent estimate of the optimal PID parameters, at the expense of higher excitation requirements. It also allows for the computation of a 'health indicator' to describe the confidence in the estimated parameters. The practical implication of this observation is that PIDs can be tuned more reliably, even in cases of large mismatch between the target and the feasible loop shapes. It also suggests a general theme where a min-max optimization of an operator error provides an advantage over signal error optimization. The key aspects of the algorithm are illustrated by numerical examples.

Original languageEnglish (US)
Pages (from-to)136-152
Number of pages17
JournalInternational Journal of Adaptive Control and Signal Processing
Issue number1-2
StatePublished - Jan 2013


  • Direct adaptation
  • Frequency loop shaping
  • H∞ estimation
  • PID tuning
  • Uncertainty bounds

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Signal Processing
  • Electrical and Electronic Engineering


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