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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.APPROXIMATE H 1 LOOP SHAPING 137 ‡ Note that, although parsimony is not a strict requirement in adaptive control (e.g., over-parametrized controllers), the robustness of such schemes is questionable at best. § Situations of insufficient excitation are not of interest here, because we have shown that they are prone to large parameter misadjustment and adaptation bursting in the pre-sense of arbitrarily small disturbances [25].Notice that, because of the special PID structure, the estimation error has the familiar linear-in-theparameters formnew cost functional is quadratic in  kC1 and can be written as follows, regardless of the update method for  k142 K. S. TSAKALIS AND S. DASHwhere S and P are the gradient and Hessian of the functional J and can be computed recursivelyThe descent direction is now computed by solvingwhere I is the set of indices for which the partial costs are 'close' to the maximum, for example, within 10%. This is carried out to prevent the subsequent line search from producing zero step size caused by the non-smooth boundary of the constraints (other computations of a descent direction are also possible). It is the use of the expression (5) that this computation is fairly simple and can be performed by finding the unconstrained solution and then projecting on the constraint set. Finally, given the descent direction  kC1 , we compute the step size by solving the line-search problem opt D arg min

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